U. S. GEOLOGICAL SURVEY PROFESSIONAL PAPER 86 PLATE I

APPARATUS ON CAMPUS OF UNIVERSITY OF CALIFORNIA.
DEPARTMENT OF THE INTERIOR 

UNITED STATES ^GEOLOGICAL SURVEY

GEORGE OTIS SMITH, DIRECTOR

PROFESSIONAL PAPER 86 .

THE

TRANSPORTATION OF DEBRIS BY RUNNING WATER

BY

GROVE KARL GILBERT

BASED ON EXPERIMENTS MADE WITH THE ASSISTANCE OF 

EDWARD CHARLES MURPHY

WASHINGTON

GOVERNMENT PRINTING OFFICE

1914

CONTENTS.

Preface..........................................

Abstract.........................................

Notation.........................................

CHAPTER I. The observations....................

Introduction.................................

General classification.....................

Stream traction..........................

Outline of course of experimentation ......

Scope of experiments.....................

Accessory studies........................

Flume traction..........................

Apparatus and material........!.............

Experiment troughs......................

Water supply.............................

The water circuit........................

Discharge................................

Sand feed...............................

Sand arrester............................

Settling tank............................

Gage for depth measurement.............

Level for slope measurement.............

Pitot-Darcy gage.........................

Sand and gravel..........................

Methods of experimentation...................

General procedure for a single experiment. 

Details of procedure.....................

Width of channel....................

Discharge............................

The feeding of sand...................

The collection of sand...............

Determination of load................

Determination of slope...............

Contractor...........................

Measurement of depth...............

Measurement of velocity..............

Modes of transportation......................

Movement of individual particles.........

Boiling..............................

Saltation...........................

Collective movement....................

Units......................................:.

Terms......................................

Load....................................

Capacity................................

Competence..............................

Discharge................................

Slope....................................

Size of debris...........................

Form ratio...............................

Duty and efficiency.....................

Symbols.................................

Table of observations on stream traction. .....

Page.

9 CHAPTER II. Adjustment of observations........... 55

10 Observations on capacity and slope ............ 55

13 The observational series................. 55

15 Errors................................... 56

 15 Intake influences. ................ .^..... 56

15 Outfall influences....................... 56

15 Changes in apparatus.................... 57

17 Rhythm................................. 58

17 Slopes of de'bris and water surface........ 59

18 The logarithmic plots.................... 59

18 Selection of an interpolation formula. .... 60

19 The constant a and competent sloped...... 65

19 Interpolation........................... 72

19 Precision............................... 73

19 Duty.................................... 74

20 Efficiency............................... 75

20 Observations on depth....................... 87

20 Mode of adjustment...................... 87

21 Precision............................... 94

21 Mean velocity........................... 94

21 Formratio............................... 94

21 Graphic computation.................... 94

21 CHAPTER III. Relation of capacity to slope........ 96

22 Introduction................................. 96

22 In channels of fixed width................... 96

22 The conditions..!..........,............. 96

22 The sigma formula...................... 96

22 The power function and the index of rela23 tive variation.......................... 97

24 The synthetic index..................... 99

24 Application to the sigma function......... 99

25 Variation of the index.................... 104

25 Formulation with constant coefficient..... 109

25 Effect of changing the unit of slope........ 112

26 Precision............................... 113

26 Evidence from experiments with mixed

26 debris................................. 113

26 Relation of index to mode of traction...... 115

26 In channels of similar section................. 116

30 The conditions........................... 116

34 Sigma and the index..................... 117

35 The synthetic index..................... 119

35 Summary............................... 120

35 Review.................................... 120

35 Duty and efficiency......................... 121

35 CHAPTER IV. Relation of capacity to form ratio..... 124

35 Introduction................................. ]24

35 Selection of a formula....................... 124

35 Maximum............................... 124

36 Capacity and width...................... 125

36 Capacity and depth...................... 127

36 Capacity and form ratio................. 128

	3
CONTENTS.

Paga. 

CHAPTER IV Continued.

Discussion of experimental data.............. 130

Scope and method of discussion.......... 130

Sensitiveness and the index of relative vari-

. ation.................................. 130

Control of constants by slope............. 131

Control of constants by discharge......... 132

;" Control of constants by fineness............ 133

Special group of observations.............. 134

Summary as to control by conditions..... 134

The optimum form ratio............:........ 135

Summary...:.'.'V...."............. 1.......... 136

CHAPTER V. Relation of capacity to discharge..... 137

Formulation and reduction................... 137

Measures of precision and their interpretation.. 142

Control of relative variation by conditions..... 143

Duty and efficiency.............. 1. ^:....... 144

Comparison of the controls of discharge and slope. 145

Controls of capacity..............'........ 145

Controls of duty..................:...... 147

Controls of efficiency..................... 148

Summary.................................... 149

CHAPTER VI. Relation of capacity to fineness of

debris.......................................... 150

Formulation................................ 150

Precision................................... 151

Variation of the constant <f>.................... 153

Index of relative variation................... 153

Duty and efficiency......................... 154

Summary................................... 154

CHAPTER VII. Relation of capacity to velocity.:.. 155

Preliminary considerations................... 155

The synthetic index when discharge is constant. 157

Mean velocity versus slope................... 158

The synthetic index when slope is constant.... 159

The synthetic index when depth is constant... 160

The three indexes............................. 160

Relative sensitiveness to controls.............. 162

Competent velocity........................... 162

CHAPTER VIII. Relation of capacity to depth...... 164

Introduction................................. 164

When discharge is constant. ................. 164

When slope is constant...................... 164

Depth versus discharge....................... 165

When velocity is constant.................... 165

The three conditions compared............... 166

Comparison of controls by slope, discharge, and

mean velocity............................. 167

CHAPTER IX. Experiments with mixed grades..... 169

Adjustment and notation..................... 169

Mixtures of two grades......................... 172

Control by slope and discharge................ 175

Mixtures of more than two grades.............. 176

A natural grade............................. 177

Causes of superior mobility of mixtures........ 178

Voids....................................... 179

Fineness................................... 180

Relation of capacity to fineness, for natural

grades.................................... 180

Definition and measurement of mean fineness.. 182

Summary................................... 184

Page.

CHAPTER X. Review of controls of capacity........ 186

Introduction................................. 186

Formulation based on competence............ 186

The form-ratio factor........................ 190

Duty and efficiency......................... 192

The formula of Lechalas..................... 193

The formula.........................'.... 193

Discussion.............................. 194

CHAPTER XI. Experiments with crooked channels .......................................... 196

Experiments................................ 196

Slope determinations......................... 196

Forms and slopes............................ 197

Features caused by curvature................. 198

CHAPTER XII. Flume traction.................... 199

The observations............................ 199

Scope................................... 199

Grades of debris......................... 199

Apparatus and methods................. 199

Processes of flume traction................ 200

Movement of individual particles.. . . 200

Collective movement................ 201

Table of observations.................... 202

Adjustment of observations................... 203

Formulation............................ 203

Precision............................... 206

Discussion.................................. 206

Capacity and channel bed................ 206

Capacity and slope....................... 208

Capacity and discharge................... 209

Capacity and fineness.................... 210

Mixtures:............................... 212

Capacity and form ratio................. 213

Trough of semicircular cross section. ..... 214

Summary................................ 215

Competence................................ 215

Work of Overstrom and Blue.................. 216

CHAPTER XIII. Application to natural streams. ... 219

Introduction......:.......................... 219

Features distinguishing natural streams........ 219

Kinds of streams........................ 219

Features connected with curvature of

channel.............................. 220

Features connected with diversity of discharge. ............................... 221

Sections of channel...................... 222

The suspended load..................... 223

The two loads........................... 230

Availability of laboratory results............. 233

.The slope factor.......................... 233

The discharge factor...................... 233

The fineness factor....................... 235

The form-ratio factor...................... 236

The four factors collectively.............. 236

The hypothesis of similar streams......... 236

Summary.-.............................. 240

Conclusion................................... 240

CHAPTER XIV. Problems associated with rhythm.. 241

Rhythm in stream transportation.............. 241

Rhythm in the flow of water.................. 242
CONTENTS.

Page.

CHAPTER XIV Continued.

The vertical velocity curve.................. 244

The moving field............................ 249

APPENDIX A. The Pitot-Darcy gage.............. 251

Scope of appendix............................ 251

Form of instrument........................... 251

Rating methods............................. 252

Rating formula............................... 253

Page.

APPENDIX B. The discharge-measuring gate and its 

rating......................................... 257

The gate..................................... 257

Plan of rating............................... 258

Calibration of measuring reservoir............. 258

The observations.......................... .. 259

INDEX. .......................................... 261

TABLES.

Page.

TABLE 1. Grades of de"bris...................... 21 TABLE 26.

2. Gate readings and corresponding discharges............................. 23 27.

3. Data connected with changes in mode

of transportation..................... 33

4. Observations on load, slope, and depth.. 38 28.

5. Values of capacity generalized from Table

4(G)............................... 61

6. Data for the construction of curves in 29. 

figures 20 and 21 .................... 62

7. Values of a in C=61 (S-<;) n ............ 66

8. Values of a, arranged to show variation 30. 

in relation to fineness of debris........ 68

9. Experimental data on competent slope.. 69

10. Experimental data on competent dis- 31. 

charge.............................. 70

11. Values of a, as adjusted for use in interpolation equations................... 72 32.

12. Adjusted values of capacity, duty, and

efficiency.......................... 75 33. b'

13. Values of nt in d= ................. 89

& u4.

14. Adjusted values of depth of current, with 35. 

values of mean velocity and form ratio. 89

15. Values of i, in relation to slope......... 100 35.

16. Values of;, in C=cS'i ................. 109 37.

17. Values of i, for mixtures of two or more

grades.............................. 114 33

18. Comparison of values of ir for mixtures 39. 

and their components................ 115

19. Values of it associated with the smooth 40. 

mode of traction .................... 115

20. Relations of ^ to discharge, debris grade,

and channel width.................. 116

21. Selected data showing the relation of capacity to slope when the form ratio is 43. 

constant............................ 117

22. Values of a corresponding to data in

Table 21............................ 118 44.

23. Values of synthetic index under condition that R is constant, and under con- ' 45. 

dition that w is constant............. 119

23a. Comparison of ( parameters in C=&! 46. 

(S-a)n and E=f^ (S-a) nu . ....... 123 47.

24. Relation of capacity to width when

slope and depth are constant......... 126

25. Values of capacity and depth for grade 48. 

(C), width 0.66 foot, and slope 1.0 per 

cent............................... 128

Values of m1} in Coc rf7"1 , when slope is 

constant............................

Quantities illustrating the influence of 

slope on the relation of capacity to form 

ratio................................

Quantities illustrating the influence of 

discharge on the relation of capacity 

to form ratio.........................

Quantities illustrating the influence of 

fineness on the relation of capacity to 

form ratio...........................

Quantities illustrating the influence of 

slope and discharge on the,relation of 

capacity to form ratio ...............

Estimated ratios of depth to width to 

enable a given discharge, on a given 

slope, to transport its maximum load.

Data on the relation of capacity to discharge ..............................

Values of «, computed from data of 

Table 32............................

Adjusted system of values of K. .........

Values of i3 , arranged to show variation 

in relation to fineness...............

Comparison of ^ with i3 ................

Variations of ij and i3 in relation to discharge ..............................

Variations of \ and i3 in relation to width.

Variations of i^ and i3 in relation to fineness ................................

Variations of ij and iz in relation to slope.

Variations of the ratio - ............

Page. 

128

132

133

133

Variations of the ratio i,-l

L-l

134

135

137

140

141

144

145

146

146

147

147

148

148

Yalues of capacity, arranged to illustrate 

the relation of capacity to grades of 

debris.............................. 150

Data connected with the plots in figure 50.............................. 151

Values of 74 , compared with slope, discharge, and width................... 154

Values of IVQ .......................... 157

Means based on Table 46, illustrating the 

control of IVQ by discharge, fineness, 

and width.......................... 158

Synthetic indexes, comparing the control of capacity by mean velocity with 

the control by slope................. 158
CONTENTS.

160

161

TABLE 49. Values of IVS - ........................ 159

50. Means based on Table 49, illustrating 

the control of TVS by slope, fineness, . 

and width .......................... 159

 51. Values of Ira ......................... 160

52. Means based on Table 51, illustrating the 

control of Iva by fineness and width..

53. Comparison of synthetic indexes for capacity and mean velocity, under the 

several conditions of constantdischarge, 

constant depth, and constant slope....

54. Values of Ids ......................... 164

55. Means based on Table 54, illustrating 

the control of las by slope, fineness, 

and width.......................... 165

56. Synthetic indexes comparing the control 

of capacity by depth with control by 

discharge........................... 165

57. Values of lav ......................... 165

58. Means based on Table 57, illustrating the 

control of lav by mean velocity, fineness, and width..................... 166

59. Comparison of synthetic indexes for capacity and depth under the several 

conditions of constant discharge, constant slope, and constant mean velocity 167

60. Adjusted values of capacity for mixtures

of two or more grades of d6bris....... 169

61. Capacities for traction, with varied mixtures .of two grades............... r .. 172

62. Percentages of voids in certain mixed

grades of debris ...................... 179

63. Fineness of mixed grades and their components............................. 180

63a. Data on subaqueous dunes of the Loire. 194

64. Comparison of slopes required for straight

and crooked channels................ 197

65. Grades of debris....................... 199

Page. Page. 

TABLE 66. Relative speeds of coarse and fine de"-

bris in flume traction .....:........... 200

67. Observations on flume traction, showing 

the relation of load to slope and other 

conditions.......................... , 202

68. Values of capacity .for flume traction, adjusted in relation to slope............. 204

69. Comparison of capacities associated with

different characters of channel bed.... 206

70. Values of /! for flume traction........... 208

71. Comparison of values of II for flume traction with corresponding values for 

stream traction....................... 209

72. Data illustrating the relation of capacity

for flume traction to discharge........ 209

73. Values of capacity for flume traction, 

illustrating the control of capacity by 

fineness.............................. 210

74. Capacities for mixed grades and their

components......................... 212

75. Depths and form ratios of unloaded

streams.............................. 213

76. Capacities for flume traction in troughs

of different widths .................. 213

77. Data on flume traction in a semicylindric

" trough............................ 1.. 214

78. Observations by Blackwell on velocity

competent for traction................ 216

79. Values of n in Coc Vmn based on Blue's

experiments.."...................... 218

80. Velocities of streams with and without

tractional loads....................... 230

81. Ratio of the suction at one opening of the 

Pitot-Darcy gage to the pressure at the 

other............................... 254

82. Values of K and u in H-H^KV" ..... 254

83. Values of A in V= A^H-I^........... 255

ILLUSTRATIONS.

Page. 

PLATE I. Apparatus on campus of the University

of California.................... Frontispiece.

II. Debris used in experiments............. 22

III. Rough surfaces used in experiments on

flume traction........................ 200

 FIGURE 1. Diagrammatic view of shorter experiment trough......................... 19

2. Diagram of water circuit.............. 20

3. The contractor........................ 25

4. Diagrammatic view of part of experiment trough with glass panels and 

sliding screen....................... 27

5. Appearance of the zone of saltation.... 27

6. The beginning of a leap, in saltation. . 28

7. Diagram of accelerations affecting a

saltatory grain..................... 28

8. Theoretic trajectory of a saltatory particle. ............................. 29

9. Ideal transverse section of zone of saltation at side of experiment trough.. 30

FIGUKE 10. Longitudinal section illustrating the 

dune mode of traction..............

11. Longitudinal section illustrating the 

antidune mode of traction...........

12. Profiles of water surface, automatically 

recorded, showing undulations associated with the antidune mode of 

traction............................

13. Plot of a single series of observations of 

capacity and slope.................

14. Logarithmic plot of a series of observations on capacity and slope.........

15. Diagrammatic longitudinal section of 

outfall end of experiment trough, 

illustrating influence of sand arrester 

on water slope.....................

16. Diagrammatic longitudinal section of 

debris bed and stream, in a long 

trough.............................

Page.

31

33

55

56

57

57
CONTENTS.

FIGURE 17. Diagrammatic longitudinal section of 

outfall end of trough, illustrating 

influence of contractor.............

18. Profiles of channel bed, illustrating 

tractional rhythms associated with 

dunes of greater magnitude.........

19. Logarithmic graph of C=f(S), for grade 

(G), w=0.66 foot, Q==0.734 ft. 3/sec...

20. Extrapolated curves of C=f(S) for various tentative equations of interpolation and for slopes greater than 2.4 

per cent............................

21. Extrapolated curves of C=f(S) for various tentative equations of .interpolation and for slopes less than 0.8 per 

cent...............................

22. The relation of a in C=f(S-a) to the 

curvature of the logarithmic graph..

23. Diagrammatic sections of laboratory 

troughs, illustrating relation of current depth to trough width.........

24. Ideal curve of competent slope in relation to width of trough ..............

25. Logarithmic plot of competent slope in 

relation to fineness of de'bris........

26: Illustration of the method used to adjust values of capacity in relation to 

slope by means of a logarithmic plot 

of observed values of capacity in relation to slope minus a .............

27. Observations of depth of current in relation to' slope.....................

28. Logarithmic computation sheet, combining relations of capacity, mean 

velocity, form ratio, and slope......

29. Logarithmic locus of the exponential 

equation..........................

30. Locus of log j/=/(log x), illustrating the 

nature of the index of relative variation. ...............................

31. Variations of i, in relation to slope.....

32. Variations of i, in relation to width of 

channel...........................

33. Variations of i, in relation to discharge.

34. Variations of i, in relation to fineness of 

de'bris..............................

35. Variations of exponents ilt jl; and k in 

relation to slope.....................

36. Variations of ij under conditions of uniform mode <5f traction and uniform 

slope...............................

37. Curves of C f(S) for three trough 

widths.............................

38. Illustration of the relation of capacity 

to width of channel and to form ratio, 

when slope and discharge are constant

39. Cross sections of stream channels.......

40. Capacity for traction in relation to 

width of channel, when depth and 

slope are constant..................

41. Plot of equation (55)..................

42. Logarithmic plots of

2 \ ~ p m-t-1

Page.

58

61

62

63

65

67

67

69

72

87

95

98

98

105

106

107

108

113

116

118

124

125

127

129

130

Page. 

FIGURE 43. Eelation of capacity, C, to form ratio","

R. The variation of the function 

C=bi(l+aR)Rm with slope......... 131

44. Relation of capacity, C, to form ratio, 

R. The variation of the function 

C=b2 (l-aR)Rm with discharge..... 132

45. Relation of capacity, C, to form ratio, 

R. The variation of the function 

C=6a (l-f*R)Rm with fineness of d6- 

bris................................ 133

46. Relation of capacity, C, to form ratio, 

i-'-::R. The variation of the function 

\ i'G=^(\ fxK)Rm with slope and discharge. ............................ 134

47. Logarithmic plots of the relation of

;.-capacity to discharge................ 139

48. Ideal cross section of a stream in the ex- 

< periment trough, illustrating the relation of competent discharge to width. 140

49. Logarithmic plot of *=/(!>)........... 141

50. Logarithmic plots of capacity,for traction in relation to fineness of debris.. 150

51. Average departures of original values 

of capacity from system of -values 

readjusted in relation to fineness of 

..debris.............................. 152

52. Vertical velocity curve, drawn to illustrate its theoretic character near the 

stream's bed......................... 155

53.. Ideal profile of a stream bed composed

of de'bris grains..................... 155

54. Ideal curves of velocity in relation to 

depth, illustrating their relation to 

the zone of saltation................ 161

55. Tractional capacity for mixed debris, 

.in relation to proportions of component grades, with associated curves of 

fineness and of percentages of voids.. 173

56. Tractional capacities of components of 

mixed grades in relation to the 

percentages of the components in the 

mixtures........................... 174

57. Curves of capacity in relation to slope 

for grade (A), grade (G), and mixtures 

of those grades..................... 175

58. Curves of capacity in relation to slope 

for grade (B), grade (F), and mixtures 

of those grades..................... 175

59. Curves of capacity in relation to slope 

for grade (C), grade (E), and mixtures 

of those grades..................... 176

60. Curves of capacity in relation to slope 

for a mixture of five grades, (C, D, E, 

F, G). Comparison of mixture curves 

for three discharges and of mixture 

curve with curves for component 

grades ............................ 176

61. Capacity-slope curves for related mixtures and capacity-discharge curves 

for mixture and component grades.. 177

62. Capacity-slope curves for a natural 

grade of d6bris, compared with curves 

for sieve-separated grades........... 178
CONTENTS.

Page.

FIGURE 63. Curves showing the relations of various 

quantities to the proportions of fine 

and coarse components in a mixture of 

two grades of debris, (C) and (G).... 179 

' 64. Logarithmic plots of capacity in relation 

to linear fineness, for related mixtures 

of debris........................... 181

65. Curve illustrating the range and distribution of finenesses in natural and 

artificial grades of debris............. 181

66. Typical curves illustrating the distribution of the sensitiveness of capacity 

for traction to various controlling conditions............................ 191

67. Plans of troughs used in experiments to 

show the influence of bends on traction................................ 196

68. Contoured plat of stream bed, as shaped

by a current....................... 198

69. Curves illustrating the relation of capacity for flume traction to fineness 

of debris........................... 211

70. Diagram of forces...................... 224

71. Interference by suspended particle

with freedom of shearing............ 226

72. Suggested apparatus for automatic feed

of debris........................... 241

73. Modification of vertical velocity curve

by approach to outfall............... 244

74. Modification of vertical velocity curve

by approach to contracted outfall.. 245

75. Plan of experiment trough with local

contraction........................ 245

76. Profile of water surface in trough shown

in figure 75......................... 245

77. Modification of vertical velocity .curve

by local contraction of channel...... 245

Page.

FIGURE 78. Modification of vertical velocity curve 

when mean velocity is increased by 

change of slope...................... 245

79. Modification of vertical velocity curve 

when mean velocity is increased by 

change of discharge.................. 246

80. Modification of vertical velocity curve 

by changes in the roughness of the 

channel bed........................ 246

81. Modification of vertical velocity curve 

by addition of load to stream, with 

corresponding increase of slope...... 246

82. Modification of vertical velocity curve 

by addition and progressive increase . 

of load............................. 246

83. Ideal longitudinal section of a stream, 

illustrating hypothesis to account 

for the subsurface position of the level 

of maximum velocity.-.............. 248

84. Diagrammatic plan of suggested moving-field apparatus................... 250

85. Longitudinal section of lower end of 

receiver of Pitot-Darcy gage No. 3, 

with transverse sections at three 

points.............................. 251

86. Cross section of prism of water in trough, 

showing positions given to gage opening in various ratings................ 255

87. Graphic table for interpolating values of 

A \nV=A -JH H1} for observations 

made with gage 3b in different parts 

of a stream.......................... 256

88. Arrangement of apparatus connected 

with the rating of the dischargemeasuring gate..................... 257

89. Elevation and sections of gate for the

measurement of discharge........... 258
PREFACE.

Thirty-five years ago the writer made a 

study of the work of streams in shaping the 

face of the land. The study included a 

qualitative and partly deductive investigation 

of the laws of transportation of d6bris by 

running water; and the limitations of such 

methods inspired a desire for quantitative 

data, such as could be obtained only by 

experimentation with determinate conditions. 

The gratification of this desire was long deferred, but opportunity for experimentation 

finally came in connection with an investigation 

of problems occasioned by the overloading of 

certain California rivers with waste from 

hydraulic mines. The physical factors of 

those problems involve the transporting capacity of streams as controlled by various conditions. The experiments described in this 

report were thus instigated by the common 

needs of physiographic geology and hydraulic 

engineering.

A laboratory was established at Berkeley, 

Cal., and the investigation became the guest 

of the University of California, to which it is 

indebted not only for space, within doors and 

without, but for facilities of many kinds most 

generously contributed.

Almost from the beginning Mr. E. C. Murphy 

has been associated with me in the investigation 

and has had direct charge of the experiments. 

Before the completion of the investigation I 

was compelled by ill health to withdraw from 

it, and Mr. Murphy not only made the remaining series of experiments, so far as had been 

definitely planned, but prepared a report. 

This report did not include a full discussion of 

the results but was of a preliminary nature, 

it being hoped that the work might be continued, with enlargement of scale, in the near 

future. When afterward I found myself able 

to resume the study, there seemed no immediate prospect of resuming experimentation, 

and it was thought best to give the material 

comparatively full treatment. It will readily 

be understood from this account that I am 

responsible for the planning of the experimental

work as well as for the discussion of results 

here contained, while Mr. Murphy is responsible 

for the experimental work. It must not be 

understood, however, that in assuming responsibility for the discussion I also claim sole credit 

for what is novel in the generalizations. Many 

conclusions were reached by us jointly during 

our association, and others were developed by 

Mr. Murphy in his report. These have been 

incorporated in the present report, so far as 

they appeared to be sustained by the more 

elaborate analysis, and specific credit is given 

only where I found it practicable to quote 

from Mr. Murphy's manuscript.

Mr. J. A. Burgess was for a short time a 

scientific assistant in the laboratory, and his 

work is described in another connection. 

Credit should be given to Mr. L. E. Eshleman, 

carpenter, and Mr. Waldemar Arntzen, mechanician, for excellent work in the construction of 

apparatus., I recall with sincere gratitude 

the cordial cooperation of several members of 

the university faculty, and the investigation is 

especially indebted to the good offices and 

technical knowledge of Prof. S. B. Christy 

and Prof. J. N. Le Conte.

Portions of my manuscript were read by 

Dr. R. S. Woodward and Dr. Lyman J. Briggs, 

and the entire manuscript was read by Mr. C. E. 

Van Orstrand and Mr. Willard D. Johnson. 

To these gentlemen and to members of the 

editorial staff of the Geological Survey I am 

indebted for criticisms and suggestions leading 

to the elimination of some of the crudities of 

the original draft.

While the aid which my work has received 

from many colleagues lias been so kindly and 

efficient that individual mention seems invidious, my gratitude must nevertheless be 

expressed for valuable assistance by Mr. Fran9013 E. Matthes in the examination of foreign 

literature, and for the unfailing encouragement 

and support of Mr. M. 0. Leighton, until 

recently in charge of the hydrographic work 

of the Survey.

G. K. G.
ABSTRACT.

Scope. The finer debris transported by a 

stream is borne in suspension. The coarser is 

swept along the channel bed. The suspended 

load is readily sampled and estimated, and 

much is known as to its quantity. The bed 

load is inaccessible and we are without definite 

information as to its amount. The primary 

purpose of the investigation was to learn the 

laws which control the movement of bed load, 

and especially to determine how the quantity 

of load is related to the stream's slope and discharge and to the degree of comminution of the 

debris.

Method. To this end a laboratory was 

equipped at Berkeley, Cal., and experiments 

were performed in which each of the three conditions mentioned was separately varied and 

the resulting variations of load were observed 

and measured. Sand and gravel were sorted 

by sieves into grades of uniform size. Determinate discharges were used. In each experiment a specific load was fed to a stream of 

specific width and discharge, and measurement 

was made of the slope to which the stream 

automatically adjusted its bed so as to enable 

the current to transport the load.

The slope factor. For each combination of 

discharge, width, and grade of debris there is a 

slope, called competent slope, which limits 

transportation. With lower slopes there is no 

load, or the stream has no capacity 1 for load. 

With higher slopes capacity exists; and 

increase of slope gives increase of capacity. 

The value of capacity is approximately proportional to a power of the excess of slope above 

competent slope. If 8 equal the stream's slope 

and a equal competent slope, then the stream's 

capacity .varies as (S o)n. This is not a deductive, but an empiric law. The exponent n 

has not a fixed'value, but an indefinite series 

of values depending on conditions. Its range 

of values in the experience of the laboratory

i Capacity is defined for the purposes of this paper as the maximum 

load of a given kind of ddbris which a given stream can transport. See 

page 35.

10.

is from 0.93 to 2.37, the values being greater 

as the discharges are smaller or the debris is 

coarser.

The discharge factor. For each combination 

of width, slope, and grade of debris there is a 

competent discharge, K. Calling the stream's 

discharge Q, the stream's capacity varies as 

(Q K)°. The observed range of values for o 

is from 0.81 to 1.24, the values being greater 

as the slopes are smaller or the debris is 

coarser. Under like conditions o is less than 

n; or, in other words, capacity is less sensitive 

to changes of discharge than to changes of 

slope.

The fineness factor. For each combination 

of width, slope, and discharge there is a limiting fineness of debris below which no transportation takes place. Calling fineness (or degree 

of comminution) F'and competent fineness <£, 

the stream's capacity varies with (F </>)?. 

The observed range of values for p is from 0.50 

to 0.62, the values being greater as slopes and 

discharges are smaller. Capacity is less sensitive to changes in fineness of debris than to 

changes in discharge or slope.

The form factor. Most of the experiments 

were with straight channels. A few with 

crooked channels yielded nearly the same estimates of capacity. The ratio of depth to width 

is a more important factor. For any combination of slope, discharge, and fineness it is 

possible to reduce capacity to zero by making 

the stream very wide and shallow or very narrow and deep. Between these extremes is a 

particular ratio of depth to width, p, corresponding to a maximum capacity. The values 

of p range, under laboratory conditions, from 

0.5 to 0.04, being greater as slope, discharge, 

and fineness are less.

Velocity. The velocity which determines 

capacity for bed load is that near the stream's 

bed, but attempts to measure bed velocity 

were not successful. Mean velocity was measured instead. To make a definite comparison 

between capacity and mean velocity it is neces-
ABSTKACT. 11

sary to postulate constancy in some accessory 

condition. If slope be the constant, in which 

case velocity changes with discharge, capacity 

varies on the average with the 3.2 power of 

velo'city. If discharge be the constant, in 

which case velocity changes with slope, capacity 

varies on the average with the 4.0 power of 

velocity. If depth be the constant, in which 

case velocity changes with simultaneous 

changes of slope and discharge, capacity varies 

on the average with the 3.7 power of velocity. 

The power expressing the sensitiveness of 

capacity to changes of mean velocity has in 

each case a wide range of values, being greater 

as slope, discharge, and fineness are less.

Mixtures. In general, de'bris composed of 

particles of a single size is moved less freely 

than de'bris containing particles of many sizes. 

If fine material be added to coarse, not only is 

the total load increased, but a greater quantity 

of the coarse material is carried.

Modes of transportation; movement of particles. Sonie particles of the bed load slide; 

many roll; the multitude make short skips or 

leaps, the process being called saltation. Saltation grades into suspension. When particles 

of many sizes are moved together the larger 

ones are rolled.

Modes of transportation; collective movement. When the conditions are such that the 

bed load is small, the bed is molded into hills, 

called dunes, which travel downstream. Their 

mode of advance is like that of eolian dunes, 

the current eroding their upstream faces and 

depositing the eroded material on the downstream faces. With any progressive change of 

conditions tending to increase the load, the 

dunes eventually disappear and the de'bris surface becomes smooth. The sm'ooth phase is in 

turn succeeded by a scjcoiid rhythmic phase, in 

which a system of hills travel upstream. These 

are called antidunes, and their movement is 

accomplished by erosion on the downstream 

face and deposition on the upstream face. 

Both rhythms of debris movement are initiated 

by rhythms of water movement.

Application of formulas. While the principles discovered in the laboratory are necessarily involved in the work of rivers, the .laboratory formulas are not immediately available 

for the discussion of river problems. Being 

both empiric and complex, they will not bear

extensive extrapolation. Under some circumstances they may be used to compare the work 

of one stream with that of another stream of 

the same type, but they do not permit an estimate of a river's capacity to be based on the 

determined capacities of laboratory streams. 

The investigation made an advance in the 

direction of its primary goal, but the goal was 

not reached.

Load versus energy. The energy of a stream 

is measured by the product of its discharge 

(mass per unit tune), its slope, and the acceleration of gravity. In a stream without load 

the energy is expended in flow resistances, 

which are greater as velocity and viscosity are 

greater. Load, including that carried in suspension and that dragged along the bed, affects 

the energy in three ways. (1) It adds its mass 

to the mass of the water and increases the 

stock of energy pro rata. (2) Its transportation involves mechanical work, and that work 

is at the expense of the stream's energy. (3) 

Its presence restricts the mobility of the water, 

in effect increasing its viscosity, and thus consumes energy. For the finest elements of load 

the third factor is more important than the 

second; for coarser elements the second is the 

more important. For each element the second 

and third together exceed the first, so that the 

net result is a tax on the stream's energy. 

Each element of load, by drawing on the supply 

of energy, reduces velocity and thus reduces 

capacity for all parts of the load. This principle affords a condition by which total capacity 

is limited. Subject to this condition a stream's 

load at any time is determined by the supply 

of de'bris and the fineness of the available 

kinds.

Flume transportation. -In the experiments 

described above experiments illustrating 

stream transportation the load traversed a 

plastic bed composed of its own material. 

Other experiments were arranged in which the 

load traversed a rigid bed, the bottom of a 

flume. Capacities are notably larger for flume 

transportation than for stream transportation, 

and their laws of variation are different. 

Rolling is an important mode of progression. 

For rolled particles the capacity increases with 

coarseness, for leaping particles with fineness. 

Capacity increases with slope and usually with 

discharge also, but the rates of increase are less
12 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

than in stream transportation. Capacity is 

reduced by roughness of bed.

Vertical velocity curve. The vertical distribution of velocities in a current is controlled 

by conditions. The level of maximum velocity 

may have any position in the upper threefourths of the current. In loaded streams its 

position is higher as the load is greater. In 

unloaded streams its position is higher as the

slope is steeper, as the discharge is greater, 

and as the bed is rougher.

Pilot tube. The constant of the Pitot velocity gage the ratio between the head realized 

and the theoretic velocity head is not the 

same in all parts of a conduit, being less near 

the water surface and greater near the bottom 

or side of the conduit.
NOTATION.

Certain letters are used continuously in the 

volume as symbols for quantities, a definition 

accompanying the first use. These are ar^ 

ranged alphabetically in the following list, 

with brief characterization and page reference. 

The list does not include letters used temporarily as symbols and denned in immediate 

connection with their use; and if the same 

letter has both temporary and continuous uses 

only the continuous use is here given.

Page. 

A............... A constant of the Pitot-Darcy gage. 254

(A), (B), etc.....Grades of debris, separated by

sieves.......................... 21

(A! G4), etc..:. .Mixtures of debris grades, the letters and figures indicating components and proportions........ 169

o............... A pressure constant (Lechalas)... 193

<T ..............Numerical constant in equation

C=62 (l-.TrjR)JRm.............. 125

Accents (/x A w ).An accent over the symbol of a 

variable indicates the nature of 

its influence on a function. 

Thus A=f(&, £, t>, E} states 

that A is an increasing function 

of B, a decreasing function of C, 

a maximum, function of D, and a 

minimum function of E. ........ 96

b. ..............A constant in equation (109)...... 191

b' .............. A constant depth in equation (21). 88

&!............... A constant capacity in equation

(10)............................ 64

62 ...............A constant capacity in equation

(54).......................... 129

63 ............... A constant in equation (64)...... 139

64 .............. -A constant in equation (75)...... 151

b6 ...............A constant in equation (91)...... 186

C............... Capacity of a stream for traction of

de"bris (in gm./sec.)............ 35

Cr............... Readjusted value of capacity... 141,151

c............... Constant coefficient in y cxj ..... 99

ct ............... Constant coefficient in C=c15'^... 109

D.............. .Mean diameter (in feet) of particles

of debris...................... 21

d. .............. Depth of current (in feet)........ 33

c?...............Differential .................... 97

E............. .Efficiency of stream for traction of

debris; capacity per unit discharge per unit slope; CIQS.... 36

e ............... Base of Naperian logarithms..... 61

Page.

F. .............Linear fineness of debris; the reciprocal of D................. 21,183

F2 . ............. Bulk fineness of debris; the number of particles in a cubic foot. 21,183 

ft./sec.......... Feet per second; unit of velocity.. 34

ft. 3/sec..........Cubic feet per second; unit of discharge. ....................... 34

g.............. .Acceleration of gravity.......... 225

gm./sec.......... Grams per second; unit of load and

of capacity for load............ 34

H. ............. Range of fineness............... 96

H, Hl .......... Readings of comparator, PitotDarcy gage..................... 254

7............... Synthetic index of relative variation........................... 99

II ............. ./for capacity and slope......... 122

J3 ............... I for capacity and discharge ...... 147

74 ............... Jfor capacity and fineness ........ 153

las .............. / for capacity and depth, slope

being constant.................. 164

JrtQ ............. J for capacity and depth, discharge

being constant.................. 164

lav ............. J for capacity and depth, mean

velocity being constant........ 164

Ie ............. .Ifor efficiency and slope ........ 122

IB ............... Jfor capacity and slope, form ratio

being constant................ 119

Iv.............. I for capacity and mean velocity. 157

TVS .............. 7 for capacity and mean velocity,

slope being constant............ 157

Ivq ............. 7 for capacity and mean velocity,

discharge being constant........ 157

Iva .............. 7 for capacity and mean velocity,

depth being constant........... 157

Iw..............I for capacity and slope, width

being constant ............... 119

i................ Index of relative variation;1 exponent in y=vxi ................ 99

i,............... .i for capacity and slope.......... 99

i2 . .............. .i for capacity and form ratio.-.....' 130

i3 . .............. .i for capacity and discharge...... 141

t'4 . ...............i for capacity and fineness......... 153

j............... .Exponent in y=cx*'. .............. 99

j,................Exponent in 0=0^^ ............ 109

K, k. ........... .Constants of the Pitot-Darcy gage.. 254

K................ .A constant discharge, corresponding to competent discharge..... 139

L ..............Length, as a dimension of units.. 139

L. ..............Load; mass of debris transported

through a cross section per unit 

time (in gm./sec.).............. 35

M. .............Mass,, as a dimension of units.... 139

13
14 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TO. .............. Exponent in C= b.2(l-aR)Rm .... 129

771................Exponent; =ra+l............. 128,129

n.............. .Exponent in C=6,(S-<j) n .... 61, 64,96

b' 

HI ............... Exponent in d= ............... 88

o.............. .Exponentin C=b3 (Q-K) ° ....... 139

p...............Exponent in C=bt (F-<j>) p ...... 151

p. e.'.............Probable error.................... 89

TT. ..............Ratio of circumference of circle to

diameter..................... 21

^>...............A constant linear fineness, corresponding to competent fineness. 151 

Q.............. .Discharge (in ft. 3/sec.)- -.......... 35

JR................Form ratio; djw. ................. 36

p.............. .Value of R corresponding to maximum capacity.................. 129

<S................ Slope, in per cent, of stream bed

or water surface................ 34

«.................Slope of stream bed or water

surface; = fall per unit distance. 34

a...............A constant slope, corresponding to

competent slope................ 64

I .............. Sum of......................... 228

T. ............ .Time, as a dimension of units... 139

Z7................Duty of water (in gin./sec.) for

traction of debris; capacity per

unit discharge; CIQ............ 36

F............... Velocity of current (in ft./sec.).... 163

F&..............Velocity of stream at contact with

bed (Lechalas)................ 193

Vm ............ .Mean velocity; discharge-j-area of

cross section; Qjdw......... 33, 94,155

V8 .............. Velocity of stream at surface

(Lechalas)..................... 194

v.............. .Variable coefficient in y=vx ..... 99

 j^i...............Variable coefficient in C=vlSil ... 99

v3 .............. .Variable coefficient in C=v3Q*3 ... 141

w.............. .Width of stream channel (in feet).. 67
THE TRANSPORTATION OF DEBRIS BY RUNNING WATER,

By GROVE KARL GILBERT.

CHAPTER I. THE OBSERVATIONS.

INTRODUCTION.

GENERAL CLASSIFICATION.

Streams of water cany forward debris in 

various ways. The simplest is that in which 

the particles are slidden or rolled. Sliding 

rarely takes place except where the bed of the 

channel is smooth. Pure rolling, in which the 

particle is continuously in contact with the 

bed, is also of small relative importance. If 

the bed is uneven, the particle usually does not 

retain continuous contact, but makes leaps, and 

the process is then called saltation, an expressive name introduced by McGee.1 With swifter 

current leaps are extended, and if a particle 

thus freed from the bed be caught by an 

ascending portion of a swirling current its 

excursion may be indefinitely prolonged. Thus 

borne it is said to be suspended, and the process 

by which it is transported is called suspension. 

There is no sharp line between saltation and 

suspension, but the distinction is nevertheless 

important, for it serves to delimit two methods 

of hydraulic transportation which follow different laws. In suspension the efficient factor is 

the upward component of motion in parts of a 

complex current. In other transportation, 

including saltation, rolling, and sliding, the 

efficient factor is the motion parallel with the 

bed and close to it. This second division of 

current transportation is called by certain 

French engineers entrainement but has received 

no name in English. Being in need of a succinct title, I translate the French designation, 

which indicates a sweeping or dragging along, 

by the word traction, thus classifying hydraulic 

transportation as (1) hydraulic suspension and 

(2) hydraulic traction.

i McQee, W J, Geol. Soc. America Bull., vol. 19, p. 199,1908.

The bed of a natural stream which carries a 

large load of debris is composed of loose grains 

identical in character with those transported. 

The material of the load is derived from and 

returned to the bed, and the surface of the bed 

is molded by the current. When debris is 

transported through artificial channels, such as 

flumes and pipes, the bed is usually rigid and 

unyielding. Trifling as this difference appears, 

it yet occasions a marked contrast in the quantitative laws of transportation, and in the laboratory the two kinds of transportation were the 

subjects of separate courses of experimentation. 

It is necessary, therefore, for present purposes, 

to base a second classification of hydraulic 

transportation on the nature of the bottom. 

As the bed is typically plastic in stream channels and typically rigid in flumes and other 

artificial channels, it is convenient to call the 

two classes stream transportation and jlume 

transportation.

The second classification traverses the first 

and their combination gives four divisions 

stream suspension, stream traction, flume suspension, and flume traction. This report 

treats of stream traction and flume traction. 

It contains the record and discussion of a 

series of experiments made in a specially 

equipped laboratory at the University of California, Berkeley, in the years 1907-1909.

STREAM TRACTION.

Previous to the Berkeley work little was 

known of the quantitative laws of stream 

traction. The quantity of material transported has sometimes been said to be proportional to the square of the slope, but I have 

failed to discover that the statement has a recorded basis in theory or observation. A stater

15
16 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

ment more frequently encountered is to the 

effect that the quantity varies with the sixth 

power of the velocity; and the origin of this 

assertion is not in doubt. It is an erroneous 

version of a deductive law commonly attributed -t6 Hopkins (1844) or Airy and Law (1885), 

although announced as early as 1823 by Leslie.1 The law, as formulated by Hopkins, is 

that "the moving force of a current, estimated 

by the volume or weight of the masses of any 

proposed form which it is capable of moving, 

varies as the sixth power of the velocity"; 

and this law pertains not at all to the quantity 

of material moved, but to the maximum size 

of -the grain or pebble or bowlder a given current is competent to move.

The subject of the competence of currents, 

or the relation of velocities to the size of particles they can move, has also been treated 

experimentally by several investigators, and 

some account of their work will be given in 

later chapters.

About the year 1883 Deacon made, in Manchester, England, a notable series of experiments in the field of stream traction. As a 

result of definite measurements of quantities 

of sand transported and of the velocities of the 

transporting currents, he announced 2 that the 

amount transported, instead of varying with 

the sixth power of the velocity, as had been 

supposed, actually varies with the fifth power.

In the field of flume traction the work has 

been somewhat more extensive, having as its 

special incentive the needs arising in ore mills 

for the conveyance of crushed rock; and a 

resume1 of results will be found in the chapter 

on flume traction.

A still greater body of investigation has been 

conducted by German and French engineers 

with the use of laboratory models of river channels. The French work was done largely for 

the purpose of testing certain rules formulated 

by Fargue 3 for the improvement of navigable 

streams. The German experiments were and 

still are addressed to the broader subject of

i Sir John Leslie's analysis is to be found in his Elements of natural 

philosophy, and in the edition of 1829 occurs at pages 426-427 of volume 1. 

David Stevenson mentions 1823, which probably indicates the first 

edition. William Hopkins gives a different analysis with practically the 

same result, and does not mention Leslie. The passage quoted occurs in 

Cambridge Philcs. Sec. Trans., vol. 8, p. 233,1844, and is probably from 

his earliest discussion of the subject. Wilfred Airy's later but evidently 

independent analysis appears in Inst. Civil Eng. Proc., vol. 82, pp. 25-26, 

1885, with expansion by Henry Law on pp. 29-30.

» Deacon, G. F., Inst. Civil Eng. Proc., vol. 98, pp. 93-96,1894. 

3 Fargue, L., Annales des ponts et chausse"es, 1894.

river engineering in general and include within 

their scope the scientific study of the ways in 

which rivers shape and reshape their channels. 

The quantitative laws of stream traction, 

which constitute the chief theme of the Berkeley work, thus fall within the province of the 

German investigators, but their study has not 

been taken up. There are three German laboratories, all well equipped, located severally 

at Dresden, Karlsruhe, and Berlin.4

The flow of a stream is a complex process, 

involving interactions which have thus far 

baffled mechanical analysis. Stream traction 

is not only a function of stream flow but itself 

adds a complication. Some realization of the 

complexity may be achieved by considering 

briefly certain of the conditions which modify 

the capacity of a stream toj transport debris 

along its bed. Width is a factor; a broad 

channel carries more than a narrow one. 

Velocity is a factor; the quantity of debris 

carried varies greatly for small changes in the 

velocity along the bed. Bed velocity is affected 

by slope and also by depth, increasing with each 

factor; and depth is affected by discharge and 

also by slope. If there is diversity of velocity 

from place to place over the bed, more debris is 

carried than if the average velocity everywhere 

prevails, and the greater the diversity the 

greater the carrying power of the stream. Size 

of transported particles is a factor, a greater 

weight of fine debris being carried than of 

coarse. The density of debris is a factor, a 

low specific gravity being favorable. The 

shapes of particles affect traction, but the 

nature of this influence is not well understood. 

An important factor is found in form of channel, efficiency being affected by turns and curvature and also by the relation of depth to 

width. The friction between current and 

banks is a factor and therefore likewise the 

nature of the banks. So, too, is the viscosity 

of the water, a property varying with temperature and also with impurities, whether dissolved or suspended.

The enumeration might be extended, complexity might be further illustrated by pointing

< The equipment and work of the laboratory of river engineering of the 

Technical High School of Dresden are discussed in the Zeitschrift fur 

Bauwesen, vol. 50, pp. 343-360,1900, and vol. 55, pp. 664-680, 1905; the 

equipment of the laboratory of the Technical High School "Fredericiana" of Karlsruhe in the same journal, vol. 53, pp. 103-136,1903, and vol. 

60, pp. 313-328,1910; and 'the equipment and work of the Laboratory for 

Kiver Improvement and Naval Architecture, Berlin, in vol. 56, pp. 123- 

151, 323-324,1906.
THE OBSERVATIONS. 17

out the influence of conditions on one another, 

and the difficulty of measuring the detrital 

loads of streams might be dwelt upon, but 

enough has been said to warrant the statement 

that an adequate analysis with quantitative 

relations can not be achieved by the mere observation of streams in their natural condition. 

It is necessary to supplement such observation 

by experiments in which the conditions are 

definitely controlled.

OUTLINE OF COURSE OF EXPERIMENTATION.

In the work of the Berkeley laboratory 

capacity for hydraulic traction was compared 

with discharge, with slope, depth, and width 

of current, and with fineness of debris; and 

minor attention was given to velocity and to 

curvature of channel. For the principal experiments a straight trough was used, the sides 

being vertical and parallel, the ends open, the 

bottom plane and horizontal. Through this a 

stream of water was run, the discharge being 

controlled and measured. Near the head of 

the trough sand was dropped into the water at 

a uniform rate, the sand grains being of approximately uniform size. At the beginning of an 

experiment the sand accumulated in the trough, 

being shaped by the current into a deposit with 

a gentle forward slope. The deposit gradually 

extended to the outfall end of the trough, and 

eventually accumulation ceased, the rate at 

which sand escaped at the outfall having be- 

-come equal to the rate at which it was fed 

above. The slope was thus automatically 

adjusted and became just sufficient to enable 

the particular discharge to transport the particular quantity of the particular kind of sand. 

The slope was then measured. Measurement 

was made also of the depth of the current; and 

the mean velocity was computed from the discharge, width, and depth.

In a second experiment, with the same discharge, the sand was fed to the current at a 

different rate, and the resulting slope and 

depth were different. By a series of such experiments was developed a law of relation 

between the quantity of sand carried, or the 

load, and the slope necessary to carry it, this 

law pertaining to the particular discharge and 

the particular grade of sand. The same experiments showed also the relations of the 

velocity of the current to slope and load.

20921° No. 86 14 2

Another series of experiments, employing a 

greater or a less discharge, gave a parallel set 

of relations between slope, load, and velocity. 

By multiplying such series the relations between discharge and slope, discharge and 

load, and discharge and velocity were developed.

Then a third condition was varied, the 

width of channel; and finally the remaining 

condition under control, the size of the sand 

grains. Thus data were obtained for studying 

the quantitative relations between load, slope, 

discharge, width, and fineness, as well as the 

relations of depth and mean velocity to all 

others. In all, the range of conditions included six discharges, six widths of channel, 

and eight grades of sand and gravel, but not 

all the possible combinations of these were 

made. The actual number of combinations 

was 130, and under each of these were a series 

of measurements of load, slope, and depth. 

There were also limited series of experiments 

involving a greater number of discharges and 

a greater number of widths. The separate 

determinations of load and slope numbered 

nearly 1 ; 200, and those of depth about 900.

SCOPE OF EXPERIMENTS.

Before proceeding to a fuller description of 

apparatus and experiments, let us consider to 

what extent the conditions of the laboratory 

were representative of the conditions which 

exist in the natural stream.

The sand used came from the beds of American and Sacramento rivers and was assumed to 

be -representative of river sand in general. 

No attention was paid to the influence on 

traction of the form and density of grains. 

Each sample used was separated from the 

natural mixture by means of two sieves and 

was composed of grains which passed through 

a certain mesh and were arrested by a mesh 

slightly smaller. In the sand carried by a 

river near its bed the range of size is much 

wider. The limit of coarseness is found in 

those particles which the current is barely 

able to roll, the limit in fineness in those particles which the swirls of the current are not 

quite able to lift into suspension; and the 

limits vary from point to point of the channel 

bed. This difference in^ condition was not 

wholly ignored, but a short supplementary
18 TRANSPORTATION OF DEBRIS BY RUNNING WATER,

series of experiments was made with definite 

mixtures of sand of various sizes, as well as 

with a natural mixture.

The straight channel of the laboratory differs 

materially from the curved channels of nature. 

It gives comparatively uniform depths and 

velocities from side to side and from point to 

point in the direction of the flow, while in a 

curved channel the depths and velocities vary 

greatly both across and along the channel. 

This difference in condition also received some 

attention, a short series of experiments being 

made with crooked and curved channels.

The vertical sides of the troughs did not well 

represent the sloping banks of rivers, and no 

attempt was made to measure the qualification 

due to this difference. The cross section of 

the laboratory current was essentially a rectangle and the ability of the current to transport was found to be definitely related to the 

ratio between depth and width; but satisfactory 

connection was not made between this relation 

and the forms of cross section in rivers.

The thalweg of a river channel traverses an 

alternation of deeps and shoals, the deeps being 

characterized by a different system of velocities 

and by a different line of separation between 

the grades of debris carried severally by suspension and traction; but these contrasts were 

touched only in a qualitative way in the work 

of the laboratory.

Each experiment dealt with a slope in adjustment with a particular discharge and a particular grade of sand. In a natural stream the 

discharge is subject to variation, and its 

changes cause changes in the fineness of. the 

material carried along the bottom. Load and 

the local slopes are ever in process of adjustment to the temporary conditions of discharge 

and fineness, but the adjustment is never complete. The general or average slope is adjusted 

to an indeterminate discharge which is neither 

the smallest nor the greatest. For this phase 

of disparity allowance is not easily made.

One of the conditions affecting velocities is 

friction on bed and sides of channel. Friction 

on the bed depends partly on the roughness of 

the bed and partly on the consumption of 

energy by traction. Its laws are the same in 

laboratory and in river. Friction on the sides 

depends on the character of the channel wall 

and must be materially greater on river banks 

than on the smooth sides of the experiment

trough. The magnitude of the difference was 

not determined. Velocities are affected by the 

viscosity of the water, variations in this factor 

being caused by differences in temperature and 

by impurities in solution and in suspension. 

The transportation of small particles is affected 

by adhesion, a property varying with the mineral character of the particles and with the 

impurities of the water. These factors were 

ignored but are probably negligible in comparison with the factors tested. It may be 

mentioned, however, that the water of the 

laboratory was practically free from suspended material, whereas that of rivers is usually highly charged at the tune of most active 

traction.

These comparisons serve to show that the 

investigation treats of a group of important 

factors of the general problem of stream 

traction but by no means comprehends all. 

Its results constitute only a contribution to 

the subject.

ACCESSORY STUDIES.

Incidental and accessory to the main inquiry were a number of minor inquiries. One 

pertained to the Pitot tube, a second to other 

methods of measuring velocity near the bottom, a third to the relation between the mean 

velocity of a loadless stream and the roughness of its channel bed, and a fourth to the 

mechanical process of hydraulic traction.

FLUME TRACTION.

In the experiments on flume traction the bed 

of the channel was not composed of loose 

debris but was the unyielding bottom of the 

trough. The same apparatus was used, with 

appropriate modifications. In each experiment slope of channel was predetermined, the 

trough being placed with definite inclination. 

The bed of the channel was given a definite 

quality of roughness or smoothness, and the 

material of the load was of a particular fineness or of a definite mixture of sizes. With a 

definite discharge flowing through the trough, 

de*bris was fed to the current at a definite rate, 

and the rate was gradually increased until 

clogging occurred. The rate of feed just before clogging was then recorded as the maximum load under the particular conditions. 

The series of experiments used two widths of
THE OBSERVATIONS. 19

channel, five textures of channel bed, six discharges, and seven grades of sand and gravel, 

besides mixtures. There were nearly 300 determinations of load.

APPARATUS AND MATERIAL.

EXPERIMENT TROUGHS.

The trough in which most of the experiments 

were made was of wood, 31.5 feet long, with an 

inside width of 1.96 feet. The height of the 

sides was 1.8 feet at the head and 1 foot at the 

end, the change being made by a series of steps. 

Its proportions and general relations are illustrated by figure 1. The surfaces were planed 

and painted. At the head, where water 

entered, the trough was connected with a 

tank by a flexible joint, a groove of the under 

side of the trough bottom resting on a semicylindric member of the tank, so as to constitute a hinge, and the walls of the trough being 

connected with the sides of the tank by a sheet 

of flexible rubber. Here also was a gate by 

which the flow from the tank could be stopped. 

Close to the opposite end of the trough was a 

cross trough 11 feet long, 2.5 feet wide, and 3 

feet deep, rigidly attached to the experiment 

trough and extending below it. A rectangular 

opening in the bottom of the experiment 

trough, an opening having the width of that 

trough, permitted sand in transportation by 

the current to sink into the cross trough, which 

contained boxes to receive it. The width of 

the experiment trough was varied by means 

of a longitudinal partition which was given 

various positions. Its width at the end was 

also varied by means of two oblique partitions, 

the "outfall contractor," which merged with 

the sides a few feet from the end and could be 

adjusted as desired. For certain experiments

FIGURE 1. Diagrammatic view of shorter experiment trough, showing relations to stilling tank (A), cross tank (J5), and settling tank (C).

false bottoms were added, with surfaces specially prepared as to roughness. These will be 

specifically described in connection with the 

corresponding observations. A second trough, 

having the same function as the one just described was 150 feet long but similar in width 

and style. Its sides were higher and it was 

not hinged at the head but remained horizontal. By temporary arrangements of partitions curved and crooked channels 1 foot 

wide were constructed within this trough. 

The shorter trough was installed in the basement of the Mining Building of the University 

of California; the longer one on the campus 

near by. (See PI. I, frontispiece.) The longer 

trough was remodeled for the experiments on 

flume traction.

A third trough 14 feet long and 0.67 foot 

wide had its wooden sides replaced for a space 

of 3.5 feet, at midlength, by plate glass, so 

that observation could be made from the side. 

It was provided with a sliding diaphragm, 

to be described in another place.

A fourth trough, of iron, was used only in the 

experiments on flume traction and will be 

described in connection with those experiments.

A few experiments were made also in a 

trough carrying the waste water of the 150-foot 

trough. This had a width of 0.915 foot.

WATER SUPPLY.

The water was taken from the municipal 

mains of Berkeley. As it was not practicable 

to draw freely on this source, a moderate 

supply was made to serve for a long series of 

experiments, being stored in a sump and 

pumped up as required. By repeated use it 

acquired a certain amount of fine detritus in 

suspension, but the quantity was not sufficient 

to obstruct the view of the experiments or 

rather, when it was found obstructive, a fresh 

supply of clear water was substituted.

THE WATER CIRCUIT.

Starting from the storage tank or sump, 

the water was lifted by a power pump to a
20 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

high trough 13 feet above. In passing through 

this trough it was first quieted by baffles 

and then regulated as to surface level by 

means of a spillway about 13 feet wide, the 

overflow returning directly to the sump. At 

the end it sank slowly through a vertical 

shaft, or leg, whence it issued in a jet through 

an aperture regulated by a measuring gate. 

After spending its force against a water cushion 

it passed through a stilling tank and then 

through the experiment trough. From that 

it fell a short distance to a settling tank, and 

thence returned to the sump. 

In figure 2 the circuit is shown diagrammatically but without accuracy as to the 

arrangement and relative sizes of the parts of 

the apparatus.

Baffles Cn;

DISCHARGE.

For the control and determination of the 

discharges used in the experiments, a measuring gate was provided. Near the lower end 

of the vertical leg of the high trough the water 

issued through a rectangular opening in a 

brass plate. The gate, also of brass, sliding 

along the plate, controlled the size of the 

opening, its motion being given by rack and 

pinion and its position shown by a suitable 

scale. The head was about 6 feet and was 

determinate. The gate and its calibration 

are described in Appendix B (pp. 257-259).

The head was regulated by means of the 

spillway in the high trough, and the amount of 

overflow on the spillway was controlled by a

---------___. <_____

. inn 

High

\

0.

o

u 

1)

\l

 

>

trough

--

I,

Measuring 

gate

FIGURE 2. Diagram of water circuit.

gate valve in the supply pipe just above the 

pump. As a check on the control, the position of the water surface was shown in an 

inclined glass tube outside of the high trough. 

The index tube being nearly horizontal, its 

meniscus had a magnified motion and the 

condition of the head could be seen at a 

glance. (See fig. 88, p. 257.)

SAND FEED.

Above the experiment trough, and near its 

head, was a hopper-shaped box from which 

sand was delivered to the current in the trough. 

The box ended downward in an edge which 

stood transverse to the trough. Along this 

edge were a series of openings whose size was 

determined by a movable notched plate of 

brass. Water was supplied to the sand in the 

hopper, both at the top and near the bottom,

the amount being regulated by valves. This 

water came from a small reservoir that was 

kept full by diverting part of the jet issuing 

from the measuring gate, and its use therefore 

added nothing to the measured discharge.

For some of the experiments debris was fed 

by hand, the quantity being regulated by 

means of a measuring box and a watch.

SAND ARRESTER.

The cross trough attached to the experiment 

trough and extending below it (see figs. 1 and 2) 

had along its bottom a track on which moved a 

platform car. This car carried two iron boxes 

to receive the sand. The boxes were rectangular and a little broader than the experiment 

trough. Openings protected by wire gauze permitted water to drain from them when they 

were lifted out.
THE OBSEKVATIONS. 21

SETTLING TANK.

The function of the settling tank was to 

catch sand which was carried past the cross 

trough. It was fitted with a system of partitions providing two alternative courses through 

which the stream could be turned, and with a 

hinged partition the "deflector" by which 

the diversion was made.

GAGE FOR DEPTH MEASUREMENT.

A frame resting on the experiment trough 

bore in vertical position a slender brass rod. 

This was raised and lowered by rack and 

pinion, and its relative height could be read on 

a scale. Depth of water was measured by 

reading the scale first with the rod end at the 

water surface and again with it at the debris 

surface.

LEVEL FOR SLOPE MEASUREMENT.

A surveyors'.level stood a few yards from the 

trough, about equally distant from the ends, 

and was used, with a light rod, to measure 

relative heights for the determination of slopes 

of water surface and sand surface.

PITOT-DARCY GAGE.

A pressure-gaging apparatus of the PitotDarcy type, but of special pattern, was used 

to measure velocities of current. Its two apertures were directed severally upstream and 

downstream. Its external form was designed 

to give the least possible resistance to the current. The reading scale, with rubber-tube 

connection, had a fixed position, while the 

receiving member was moved from point to 

point. A fuller description is contained in 

Appendix A (pp. 251-256).

SAND AND GRAVEL.

The debris used in the experiments was 

obtained from three streams Sacramento 

River 7 miles below the mouth of the American, 

American River 8 miles above its mouth, and 

Strawberry Creek in Berkeley. The debris 

from the creek was relatively coarse and was 

used only in the experiments on flume traction. The mean density of the river material 

was 2.69; that of the creek gravel 2.53. The 

forms of the grains of sand are shown in Plate 

II. To prepare the de"bris for use it was sorted 

into grades by a system of sieves, and in the 

laboratory records each grade was designated 

bv the limiting sieve numbers. Thus the

grains of the 40-50 grade passed through a 

sieve with 40 meshes to the inch and were 

caught by a sieve with 50 meshes to the inch. 

For the sake of brevity the grades are commonly indicated in this report by letters in 

parentheses (A), (B), etc. and the same 

notation is extended to mixtures of two or more 

sizes. Neither of these notations, however, is 

suited for the mathematical discussion of the 

laboratory data, and three others were devised. 

These are, first, the mean diameter of particles, 

designated by D; second, the reciprocal of the 

mean diameter, or the number of particles, side 

by side, in a row 1 foot long, designated by F; 

third, the number of particles necessary to 

occupy, without voids, the space of 1 cubic 

foot, designated by F2 . In the sense that the 

notation of D distinguishes by magnitudes, the 

notations of F and F2 distinguish by minitudes. F is otherwise called linear fineness, 

and F2 bulk fineness.

To determine the several constants for a 

grade of debris, a sample was weighed and its 

particles were counted. Then, N being the 

number of particles in the sample, W their 

weight, 0 their density, and Wo the weight of 

a cubic foot of water,

p_NGWo 2 ~ W

Defining mean diameter as the diameter of a 

sphere - having the volume of the average 

particle 

F=y^ 

#=-p =

In the following table the grades of sand and 

gravel are characterized by the several notations.

TABLE 1. Grades of debris.

Grade 

name.

A). .

B). .

C).'. .

D). .

E). .

F). .

(Cr). .

(H). .

(D-. -

(J).- -

(K). .

Sieves

used in 

separation 

(meshes 

to linch).

50-60

40-50

30-40

20-30

10-20 6-8

4- 6

3- 4

1- 2

4- 1

i-1

D, mean diameter of 

particles 

(foot).

0. 00100

.00123

.00166

.00258

. 00561

.0104

.0162

.0230

.0547

.110

.200

F, number of 

particles 

to linear 

foot.

1,002

812

602

388

178

95.9

61.8

43.4

[18. 3]

[9.1]

5.0]

Ft, number of 

particles to 

cubic foot.

1,910,000,000

1,023,000,000

417,000,000

111,500,000

10, 770, 000

1, 685, 000

451,000

156, 000

11, 900

1,440

239

Range 

otD 

or F.

1.13

1.17

1.44

1.56

1.95

1.40

1.43

1.36

[2. 00]

[2. 00]

[1.50]

Range 

of J<\.

1.44

1.60

2.99

3.80

7.41

2.74

2.92

2.51

[8.00]

[8.00]

[3.37]
22 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

The sixth column of the table contains an 

index of the range in diameter within each 

grade, and this is also the range in linear fineness. The index is the ratio between the 

diameters of the apertures of the two sieves by 

means of which the grade was separated.. It 

would ' express accurately the ratio of the 

diameter of the largest particle to that of the 

smallest particle in the same grade if all the 

particles were spherical, or if all had precisely 

the same shape. But there are actual differences of form sufficient to modify materially the 

character of the separation. For the same 

mean diameter a prolate form will pass a sieve 

that will arrest a sphere, and an oblate form 

may be arrested where a sphere will pass. By 

reason of this qualifying condition, the actual 

range is somewhat greater than the tabulated 

estimate. The range for bulk fineness, given in 

the last column, is the cube of the range of 

linear fineness.

METHODS OF EXPERIMENTATION.

The methods here described are those used in 

the investigation of stream traction. Those 

used in studying flume traction are set forth in 

the chapter on that subject.

GENERAL PROCEDURE FOR A SINGLE EXPERIMENT.

The experiment trough stands horizontal. 

The width of channel has been fixed by the 

placing or the omission of the partition. The 

head gate is open. The openings in the hopper 

have been set to a particular size. The outfall 

contractor has been adjusted to a width previously found suitable for the conditions of the 

experiment. The two sand-catching boxes 

stand on the car, one of them being in position 

under the opening in the trough bottom.

The pump is started: The measuring gate is 

opened until its index reaches the point corresponding to the desired discharge. A valve 

associated with the pump is turned, if necessary, to adjust the water level in the high 

trough. When the flow in the experiment 

trough has become steady, or nearly steady, the 

sand feed is started by opening the valves 

which admit water to the hopper.

The "run" has now begun, and it is continued without change until the slope of the 

deposit constituting the channel bed has

become stable. The car in the cross trough is 

now moved so as to bring the reserve sandbox 

into position, a stop watch is started, and the 

deflector in the settling tank is shifted. These 

changes begin the period of load measurement. 

During this period measurements are made of 

the depth of the current and, under certain 

conditions, of the slope of the water surface, 

and the character of the sand bed is recorded. 

The measuring period is terminated by again 

shifting the sand boxes and the deflector, and 

the watch is stopped. The head gate is now 

closed, sand feed and pump are stopped, and 

the discharge-measuring gate is closed. Next 

the slope of the channel bed is measured, and 

the sand caught during the period recorded by 

the watch is weighed.

DETAILS OF PROCEDURE.

In the following paragraphs some details and 

variants will be described, with comments on 

methods and apparatus. As the investigation 

had no precedents to follow, its methods were 

necessarily developed by a process of trial and 

and failure. Many defects of method were 

remedied as the work went on. A few that were 

recognized after much work had been done were 

tolerated to the end, because their correction 

would have made the body of results less homogeneous.

WIDTH OF CHANNEL.

The full width of the 31-foot trough and the 

150-foot trough was 1.96 feet. By the use of 

partitions the channel was given widths approximately two-thirds, one-half, one-third, 

two-ninths, and one-ninth of a full width. The 

scale of width was, in feet, 0.23, 0.44, 0.66, 1.00, 

1.32, 1.96. A few experiments employed also 

the widths 1.2, 1.4, 1.6, and 1.8 feet.

DISCHARGE.

As the discharge was determined by flow 

through an aperture of adjustable size, under an 

adjustable head, the precision of its measurement involved (1) the precision of setting the 

gate at the aperture, (2) the precision of calibrating the aperture for different positions of 

the gate, and (3) the precision of adjusting the 

head.

The rating (described in Appendix B) was 

effected by a volumetric method, believed to be 

adequate, the average probable error of its 

determinations being about 1 in 500.
U. S. GEOLOGICAL SURVEY PROFESSIONAL PAPER 86 PLATE II

-^^_ ^Hp . _

DEBRIS USED IN EXPERIMENTS
THE OBSERVATIONS.

The gate was set by bringing an index mark 

opposite a graduation mark on a scale, the two 

marks being on brass plates in contact. The 

gate was controlled by rack and pinion, and 

considerable force was necessary to move it. 

The limit of error may have been 0.002 foot. 

The ordinary error is believed to have been less 

than 0.001 foot. An error of 0.001 foot in the 

setting would cause an error of 0.002 f t.3/sec., or 

f^ of the medium discharge.

The determination of head is subject to an 

accidental error and a systematic error. The 

accidental error pertains to the adjustment of 

water level in the high trough, by means of the 

valve at the pump, with observation of the 

tube index of water level. It was possible to 

give this adjustment a refinement comparable 

with that of the hook gage, but in practice that 

refinement was not attained, because a close 

watch was not kept on the index. It was 

found by experience that the fluctuations of 

level (occasioned by fluctuations of the electric 

current supplying power to the pump) were 

small, and they were usually neglected, a practical calibration of the valve at the pump being 

arranged so that it could receive the proper 

setting for each setting of the discharge measuring gate. The ordinary error of the adjustment of the head is estimated at 0.003 loot, 

which would occasion an error in the discharge 

of 1 in 4,000.

The remaining possibility of error is connected with the history of the apparatus. At 

the time of the calibration of the measuring 

gate the laboratory occupied temporary quarters. In its removal to permanent quarters 

there was a measurement and readjustment of 

the vertical distance constituting the head. 

Also, for the work with the long trough the 

measuring gate was transferred to a replica 

of the high trough, which may have differed 

in some particular affecting the constants. As 

the work of calibration was at no time repeated, there was no check on the errors which 

may have been thus introduced. In a general way, they are probably of the same order 

of magnitude as the errors of adjustment 

of water surface. It is believed that all other 

errors affecting discharge are small in comparison with that connected with the measuring 

gate.

The vertical width of the aperture by which 

discharge was regulated was 2 inches. The

head, measured from the middle of the aperture, was 6.0 feet. The horizontal dimensions 

of the aperture, during experimental work, 

ranged from 0.1 inch to 6.0 inches, and the corresponding discharges are given in the following table:

TABLE 2. Gate readings and corresponding discharges.

Gate 

opening 

(inches).

0.1

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.1

1.2

1.3

1.4

Discharge 

(fU/sec.).

0.019

.039

.058

.075

.093

.111

.128

.146

.1C4

.182

.200

.218

.237

.255

Gate 

opening 

(inches).

1.5

1.6

1.7

1.8

1.9

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

Discharge 

(ft.3/sec.)

0.272

.290

.308

.327

.345

.363

.454

.545

.639

.734

.828

.923

1.021

1.119

THE FEEDING OF SAND.

The fact that the hourglass has been used to 

measure time suggests that the flow of dry 

sand through an aperture may be uniform. 

Such a flow was not tested in the laboratory 

because the plan for experimentation required 

that sand should be used over and over, and 

it was not practicable to dry it. The hopper 

was a device intended to produce a uniform 

flow of wet sand. Moist sand will not flow 

through a small opening; but if enough water 

is present to more than fill the voids, adhesion 

is overcome and flow takes place, as in a quicksand. The freedom of the flow depends on the 

amount of -water. It was found difficult to 

maintain a uniform condition in the hopper. 

Another difficulty arose from clogging of the 

openings, and this was occasioned by shreds of 

wood fiber and similar impurities in the sand. 

The second difficulty was largely obviated, 

after a time, by making the openings larger 

and fewer; but the hopper feeding was at best 

not sufficiently uniform to be used in measuring the load carried by the experimental 

stream.

For all experiments in which a large quantity 

of d6bris was carried, the material was fed to 

the current by hand and was measured in the 

feeding. A small box of known capacity was 

filled with the material and emptied into the 

current at regular intervals timed by a watch 

or clock. If the interval was long, the measured unit was dumped on a sloping table above
24 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

the trough and gradually fed to the current by 

means of a scraper. Hand feeding had the 

defect of discontinuity, as well as irregularity 

in detail, but it had the advantage of measurement, and in certain experiments its measurement of load gave an important check on 

the measurement of debris delivered at the 

outfall end of the trough. With a perfect and 

stable adjustment of conditions the two should 

agree, and their disagreement served to show 

that the slope of the channel bed had not 

become perfectly adjusted, or else that its 

adjusted condition was subject to rhythmic 

oscillation.

. In some of the later work the rate of feed 

was measured from time to time by intercepting the stream of sand falling from the 

hopper during a definite number of seconds and 

weighing the sample thus caught.

THE COLLECTION OF SAND.

In the original construction of the apparatus 

for arresting the sand the opening in the 

bottom of the trough was covered by a coarse 

wire screen, which lay flush with the trough 

bottom. This was intended to separate the 

current above from the still water below and 

prevent the formation of eddies, which might 

keep the sand from settling to the collecting 

box and might also check the current. It 

fulfilled its purpose and was altogether satisfactory for currents of moderate velocity, but 

with high velocities it interfered with the 

arrest of the sand, letting a considerable 

fraction pass on to the settling tank. It was 

accordingly removed, apparently without bad 

results. Eddies were formed, but the anticipated difficulties were not realized.

On the whole the apparatus for arresting 

sand was successful. It was only with the 

finer debris and at the highest velocities that 

the fraction of load escaping to the settling 

trough was too large to be neglected in the 

weighing.

DETERMINATION OF LOAD.

The sand collected, in sand box and settling 

tank, during the period recorded by the stop 

watch was weighed without drying, and the 

gross weight was afterward corrected by an 

aUowance for the contained water. In order 

to determine the proper allowance a preliminary study had been made, and as a result of

that study a definite procedure was adopted 

for bringing the wet sand to a particular 

"standard" condition. After the sand-collecting box had been lifted from the trough all 

water which would drain from it by gravity 

alone was allowed to escape. It was then 

removed to smaller boxes for weighing. These 

boxes were jarred by tapping, which caused 

the sand grains to readjust their contacts and 

settle together, excluding a part of the interstitial water, which appeared at the surface 

and was poured off. The sand was then 

weighed. It is of interest to note that in the 

condition thus adopted as a convenient standard sand occupies less space than when dry, 

moist, or supersaturated; its voids are at a 

minimum.

The period recorded by the stop watch was 

ordinarily about 10 minutes but was made less 

when the current was most heavily loaded, 

because of the limited capacity of the sandcollecting box, and was extended for the 

lightest loads. Its beginning was sharply 

defined by the shifting of the sand boxes, 

which could be made to coincide within a 

second with the starting of the watch. Its 

end was somewhat less definite, but the error 

in time is believed to be small in comparison 

with the whole period.

The load per second was computed by 

dividing the total load, namely, the corrected 

weight of sand, by the number of seconds in 

the stop-watch reading. Its error included 

(1) the error of timing, (2) the error of standardizing the sand and correcting for contained 

water, and (3) the error in weighing. There 

are no definite data bearing on its amount, 

and nothing better can be recorded than a 

general impression that the results are reliable 

within 2 per cent, that the precision is lower 

than that of the discharge measurement, and 

that the error in determination of load is 

notably less than the error, presently to be 

considered, in correlating load with slope..

When the rate of feed was regulated by the 

periodic contribution of a measureful of de"- 

bris, the weighings of the unit, from tune to 

time, showed inequalities from which precision could be estimated. A computation indicated the average probable error, for a run, as 

about 1 per cent. This depended chiefly on 

the standardization, and as that was less perfect for the debris as fed than for the debris
THE OBSERVATIONS. 25

as collected, the ordinary measurement of collected load is presumably affected by a smaller 

probable error.

DETERMINATION OF SLOPE.

The observations of slope were made with 

surveyors' level and rod. The rod, made for 

the purpose, without unnecessary length or 

weight, was graduated to hundred ths of a foot 

and read by eye estimate to thousandths. It 

was held by an assistant while the observer 

and recorder stood at the telescope. The positions were determined by a graduation of the 

trough, which was marked at every foot. To 

measure the water slope, heights of the surface were taken at several points along the 

trough. To measure the sand slope, heights 

were taken at intervals of either 2 or 4 feet, the 

shorter interval being used with the shorter 

trough. The water slope could not be measured when the surface was rough. When the 

de'br.is surface was rough, it was usually graded 

before measurement by scraping from crests 

into adjacent hollows.

The observed heights were plotted on section 

paper, with relatively large vertical scale, and 

a straight line was drawn through or among 

them. The line served the purpose of a preliminary determination of slope, and the plots 

were inspected for the detection of systematic 

errors. As a result of this inspection a portion 

of the profile was selected for the determination of slope, and from the observations on this 

portion the slope was computed by leastsqua.res method.

CONTRACTOR.

As will be explained more fully in another 

connection, the slope measurements were affected (1) by systematic errors connected with 

the conditions under which the water entered 

and escaped from the trough, and (2) by accidental errors arising from rhythm. One of the 

measures used to diminish the systematic errors was the contraction of the current at the 

outfall end of the trough. The apparatus for 

this purpose consisted of two boards as wide 

as the depth of the trough and arranged as in 

figure 3. Their attachment to the sides was 

flexible, so that the degree of convergence and 

the width of aperture at the outfall could be 

modified at will. This apparatus will be called

the outfall contractor. The theory and efficiency of the contractor will be considered in 

the discussion of the slope errors.

FIGURE 3. The contractor. 

MEASUREMENT OF DEPTH.

The depth of the current was measured ao 

midwidth and near micllength of the trough. 

The determination was made by means of the 

gage already described (p. 21), during the 

period of time for which the load was measured. 

As the water surface was subject to rhythmic 

fluctuation, a series of observations of its position were made, and their mean was used. A 

series of observations of the position of the d6- 

bris surface were sometimes made also, but 

usually only a single observation, and the reading obtained was subtracted from the mean of 

readings on the water, surf ace. The observations of the debris surface were subject to an 

error which was regarded as more serious than 

that of the observations of water surface because, being essentially systematic, it could 

not be eliminated by repetition. The prssence 

of the gage rod in the water modified the distribution of velocities, and this modification included an increase of the current's velocity a 

little below the end of the rod. As the bottom was approached by the rod, the current 

scoured a hollow in the bed immediately under 

it; and if the rod were lowered to actual contact, the reading would give an excessive estimate of depth. What was attempted was to 

lower the rod to a position as nearly as possible 

at the level .of undisturbed parts of the bed 

surrounding the visible hollow. This was a 

matter of judgment, but not of confident judgment, because the actual bed was concealed by 

a cloud of saltatory debris particles. It is 

therefore recognized that the measurements of 

depth are uncertain.

Whenever the water profile as well as the 

debris profile was surveyed, an independent
26 TRANSPORTATION OF DEBRIS BY RUNNING WATER,

estimate of depth was obtained by subtracting 

one profile from the other. This mode of determination avoided the error incident to the 

gage work and was on the whole satisfactory, 

but unfortunately the number of experiments 

to which it could be applied was not large. 

During the greater part of the experimentation, 

the importance of the water profile was not 

recognized, and this particular use of it was 

essentially an afterthought.

The values from profiles being assumed to 

have relatively small errors, both systematic 

and accidental, it is possible to measure by 

their aid the precision of the values from gage 

readings. Of 118 depths which were measured by both methods, the gage gave the 

greater value for 36, the lesser for 78; and the 

average for gage values was 0.0045 ±0.0007 

foot less than the average for profile values. 

Independently of this apparent systematic error, the probable error of a single measurement with the gage was ±0.007 foot.

MEASUREMENT OF VELOCITY.

The mean velocity of the current is computed by dividing the discharge by the area of 

the cross section, or the product of width and 

depth. Its precision depends on those of the 

determinations of discharge, width, and depth; 

and as the precision for discharge and width 

is relatively very high, the precision of mean 

velocity may be regarded as identical with that 

of depth.

The attempts to measure velocity close to 

the channel bed were not successful. This is 

much regretted, because it is believed that bed 

velocity is a prime factor in traction and that 

slope and discharge exert their influence chiefly 

through bed velocity. The mode of measurement to which most attention was given was 

that by the Pitot-Darcy gage, and special 

forms of that instrument were constructed for 

the purpose. The difficulty which seemed insuperable was essentially the same as that encountered in the measurement of depth. As 

the instrument approached the current-molded 

bed of debris, the bed retreated, with the formation of a hollow. In the pressnce of the 

instrument the normal velocity at the bed did 

not exist. Inseparable from this difficulty is 

a property of the instrument. When it is held 

close to ths bottom or side of a channel its constant is not the same as in the free current.

The system of flow lines ,and velocities with 

which the stream passes the obstructing object 

determines the instrument's constant, and when 

that system is modified by a neighboring object the constant changes. The nature of these 

difficulties is such that it was not thought 

worth while to experiment with other gages 

and meters which limit the freedom of the 

current.

Other devices tried were of one type. Small 

objects, such as currants or beans, only sjightly 

denser than water, were placed in the current 

and watched. The lighter ones would not 

remain near the bottom. The heavier ones 

were visibly retarded when they touched the 

bed and were also retarded when close to the 

bottom by the cloud of saltatory sand, which 

has a slower average velocity than the water it 

suffuses.

MODES OF TRANSPORTATION.

MOVEMENT OF INDIVIDUAL PARTICLES. 

ROLLING.

In stream traction sliding is a negligible factor. The roughness of the bed causes particles 

that retain contact to roll. When, as in most 

of the experiments, the grains are of nearly 

uniform size, each moving grain has to surmount obstacles with diameter like its own, 

and when it reaches the summit of an obstacle 

it usually possesses a velocity which causes it 

to leap. So rolling is chiefly the mere prelude 

to saltation. With mixed debris the same is 

true for the finer grains, but the coarser may 

roll continuously over a surface composed of 

the finer, and the coarsest of all, those close to 

the limit of competence, move solely by rolling.

The large particle, as it rolls over the bed of 

smaller particles, indents the bed, and its contact involves friction. The energy thus expended comes from the motion of the water, 

and its communication depends on differential 

motion between water and particle. Except 

under special conditions, to be mentioned later, 

the load travels less rapidly than the carrier, 

and it is also true that in a load of mixed debris 

the finer parts outstrip the coarser.

SALTATION.

In stream traction the dominant mode of 

particle movement is saltation. Because saltation grades into suspension it has often been
THE OBSERVATIONS. 27

explained in the same manner, by appeal to upward movement of filaments of current, but 

the recent studies have led me-to entertain 

a different view. Before this view is presented 

an account will be given of certain observations 

which were made with the use of the trough 

having glass sides.

Through the trough was passed a current 

transporting sand of uniform grade, and the 

conditions were such that the sand bed and 

water surface were smooth. In the same water 

floated a few fine particles and thin flakes of 

mica, illustrating suspension, but there was no 

intergradation of the two processes. Viewed 

from the side, the saltation was seen to occupy 

at the bottom of the current a space with a 

definite upper limit, parallel to the sand bed. 

Within this space the zone of saltation the 

distribution of flying grains was systematic, 

the cloud being dense below and thin above, 

but not perceptibly varying from point to 

point along the bed. Viewed from above, the 

surface of the cloud seemed uniform and level, 

and it all appeared to be moving in the same 

direction. There was no suggestion of swirls 

in the current.

When, in looking from the side, attention 

was directed to the base of the zone, it was 

easy to watch grains that traveled ( half by 

rolling and half by skipping, and these moved 

quite slowly; but higher in the zone the 

motions were so rapid and diverse that all was 

a blur. To resolve this blur a sliding diaphragm was arranged. This consisted of a 

short board with a hole in it. The board hung

FIGURE 4. Diagrammatic view of part of experiment trough with glass 

panels (.4) and sliding screen (B). C, Hole in screen.

outside the wall of the trough, being supported 

by a cleat above in such manner that it could be 

slid along the trough. (See fig. 4.) The hole, 

about 2 inches square, gave a restricted view of 

the saltation zone. By sliding the board in the 

direction of the current and keeping the eyes 

opposite, a traveling field of view was obtained. 

Manifestly if the field traveled at the same rate 

as the current, any object moving with the 

current would appear at rest to the observer, 

because there would be no relative motion of

observer and object; and if objects in the water 

were moving (horizontally) at different rates, 

those coinciding in rate with the field would be 

seen as if at rest, while the others would be seen 

as moving.

When the field was moved slowly the rolling 

grains ceased to be distinct but were replaced 

in distinctness by grains that seemed to bob 

up and down. These vibrated through a space 

of two or three diameters, as if repeatedly 

striking the bed and rebounding. In interpreting this appearance, allowance must be

FIOUKE 5. Appearance of the zone of saltation, as viewed from the side 

with a moving field.

made for the fact that the grains were distinctly seen because they were moving horizontally about as fast as the diaphragm. Their 

paths were really low-arching curves, and only 

the vertical factor remained when the horizontal was abstracted. It is probable also that 

the appearance of rebounding was largely 

illusory, most of the grams either stopping at 

the end of the leap, or else leaping next time 

with a different velocity.

When the field was moved somewhat faster, 

the bobbing grains disappeared and there came 

into distinct vision a set of grains quite free 

from the bed and occupying a belt within the saltation zone. All the zone above and below them 

was blurred. In the middle of the belt vertical 

motion was to be discerned but was less conspicuous than in the lower zone. Where distinctness graded into blurring, lines of motion 

could be seen which were oblique and curved, 

the lines above the belt curving forward and 

those below backward, as shown in figure 5.

With progressively faster motion of the field 

the belt of distinct vision rose higher, until the 

top of the zone was reached, when all the lower 

part was blurred.

The systematic gradation of velocity and 

other features from the bed upward and the
28 TBANSPOBTATION OF DEBRIS BY RUNNING WATEB,

sensible uniformity of process over the whole 

width of. channel are not consistent with the 

idea that the saltation zone is invaded by 

eddies of large dimensions, such as would be 

competent to sustain the grains by the upward components of their motions. If there 

were large ascending and descending strands 

of current the visible surface of the zone 

would be locally raised and depressed by 

them. We must, indeed, assume that the flow 

is turbulent, in the technical sense, because 

parallel or laminar flow is impossible with

o&"^ o^S

A £

FIGUKE 6. The beginning of a leap, in saltation.

velocities competent for traction, but the eddies may be assumed to be of small dimensions in relation to the depth of the zone, and 

the lines of flow with which saltation is concerned may be assumed to be approximately 

parallel to the general direction of the current.

The explanation I would substitute for that 

of the uplift of grains by rising strands is that 

each grain is projected from the bed with an 

initial velocity which gives it a trajectory analogous to that of a cannon ball. The following fuller statement, though given with little 

qualification, should be understood as largely 

hypothetic.

In figure 6 the current is supposed to move 

from left to right above the grains of debris 

shown in outline. A grain which in A is at 

rest appears in B in an advanced position, 

having been rolled upward and forward about 

an undisturbed grain which lies in its way. 

(The moving grain is doubtless more likely to 

roll against two other grains than a single one, 

but the principle is the same.) In moving to 

its new position the center of gravity of the 

grain describes a curve convex upward. The 

grain continuously gains in velocity, and the 

acceleration also increases as the direction of 

motion comes to make a smaller angle with 

the direction of the current. At each instant 

the accelerative force due to the current and 

that of gravity are combined and have a resultant direction; and the combined or resultant accelerative force may be resolved into 

two parts, one of which coincides in direction

with the motion of the center of gravity and 

the other with the line joining the center of 

gravity and the point of contact. The lastmentioned component presses the moving 

grain against the stationary grain. Opposed 

to it is the centrifugal force arising from the 

curvature of the grain's path; and the point 

is finally reached where the centrifugal force 

dominates and the grain is free. Under the 

ordinary conditions of saltation this point is 

not the crest of the obstruction, but is on the 

upstream side, so that the grain's direction of 

motion at the instant of separation is obliquely 

upward. Thus the free grain is initially moving upward as well as forward, and it has almost literally made a leap from the bed.

If the grain were at that instant released 

from all influences but gravity, its path before 

returning to the bed would be the arc of a 

quadric parabola with vertical axis. The 

actual deviation of its trajectory from the 

parabolic form is analogous to that observed 

in gunnery, for it arises from the resistance of 

a fluid; but the laws of resistance are not the 

same for air and water, and the frictional acceleration in one case is negative while in the 

other it is mainly positive. The trajectory in 

gunnery is shorter than the ideal parabolic 

arc; in saltation it is longer.

Figure 8 gives diagrammatically the trajectory of a saltatory grain. In figure 7 AB 

is a portion of the same trajectory. Let the 

space A 0 represent the instantaneous velocity 

of the grain, and let the line AD represent in 

direction and length, the velocity of the water

FIGURE 7. Diagram of accelerations affecting a saltatory grain.

about the grain. Then, C and D being connected by a line, CD represents in direction 

and magnitude the relative velocity of water 

and grain, or the velocity of the water as referred to the grain. By reason of the mutual 

resistance of water and grain, this relative motion accelerates the grain, the acceleration being a function of the differential velocity, the 

size of the grain, and other conditions. On the 

line CD, showing the direction of the acceleration, let the space CE represent its amount. 

Then from E draw the vertical EF, represent-
THE OBSERVATIONS. 29

ing, to the same scale, the acceleration of the 

grain by gravity. Connect C and F; the line 

OF represents in magnitude and direction the 

resultant acceleration of the grain. These 

relations are independent of the particular 

directions of motion of the grain and the water. 

Let us now introduce the assumptions, believed 

to be practically true for the laboratory conditions, that the water in the region of saltation 

moves parallel to the bed and that its velocity 

increases notably with distance from the bed. 

In the ascending part of its path the grain encounters filaments of the current with higher 

and higher velocity. This tends to increase 

the relative velocity, but the grain is at the 

same time gaining in horizontal velocity and 

the gain tends to diminish the relative velocity. 

Unless the leap is short in relation to the size 

of the grain, the second of these tendencies is 

the greater, and at the highest point of its 

path the grain is moving nearly as fast as the

FIGURE 8. Theoretic trajectory of a saltatory particle, the initial point 

being at 7. Arrows indicate acceleration.

water. In the descending part of its path it encounters slower moving filaments of current, 

and at some point (//, fig. 8) its horizontal motion may equal that of the adjacent water. 

Then beyond H it passes through filaments 

moving still more slowly, and its acceleration 

from the reaction of the current becomes negative. The acceleration due to gravity is of 

course uniform and downward, and its combination with that due to the current yields a system 

of directions and magnitudes of the type indicated in figure 8 by short arrows. In the 

shorter and lower trajectories it is probable 

that the critical point Jlis not reached.

If the position of the grain before leaping 

(fig. 6) is such that only a relatively short roll 

suffices to free it, then its initial velocity is 

small and the angle of ascent at which it is 

freed is low. It has a short, flat trajectory, and 

its velocity at the highest point is moderate. 

If the original roll is longer there is time to 

acquire speed before the leap; the initial velocity is large and the angle of ascent is relatively high. It has a long and high trajectory 

and when at the crest has been accelerated to 

liigh velocity. If a grain at the end of a leap

touches the bed at a favorable point it may leap 

again without coming to rest, and the impetus 

of the first flight will thus enhance the initial 

velocity of the second.

In the observations with the moving field 

the grains seen most distinctly were those which 

moved horizontally with the field and at the 

same time had little vertical motion. So each 

belt of distinctness contained grains at the tops 

of their trajectories and was practically made 

up of such grains. The grains producing the 

curved lines in figure 3 were ascending or descending obliquely, and their horizontal components of motion coincided with the motion of 

the field for an instant only.

In general the observations seem to show 

that the summit velocities of the leaping grains 

increase systematically with the height of the 

leap, and this generalization is in perfect accord 

with the hypothesis that the paths of grains 

are determined primarily by initial impulse.

Under the hypothesis the series of velocities 

observed by aid of the moving field are not 

velocities of current, for the initial velocities of 

grains, being caused by the current, require that 

the water outspeed all the grains at the bottom 

of the zone of saltation. At the top of the 

zone there must be at least a slight advantage 

with the current, provided the water velocities 

increase upward. That the water velocities do 

increase upward can hardly be doubted, for in 

sweeping along the sand the stream expends 

energy, and as its energy subsists in velocity, 

the expenditure involves retardation. Moreover, the grains of sand are at the same time 

most numerous and slowest near the bottom 

of the zone, so that their effect is there greatest.

In this connection it is to be observed that 

the width of the belt of distinct vision in the 

moving field (fig. 5) is greater for the upper 

part of the zone of saltation than for the lower. 

As distinct vision is limited to a certain (undetermined) range in horizontal velocities, this 

fact implies that the increase in horizontal 

speed of sand grains with distance from the 

bed is less rapid in the upper part of the zone 

than in the lower.

The preceding discussion is subject to two 

qualifications, the first of which is connected 

with the retardation of the current at the side 

of the trough. By reason of that retardation 

the zone of saltation is shallower near the side 

and does not include the longer and higher
30 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

leaps. Figure 9 gives an ideal conception of 

the cross section of the zone and the distribution of flying grains within it. Observation 

from the side penetrates but a short distance 

into the cloud, the distance being least where 

the cloud is most dense. The practical limit 

of visual penetration may be assumed to take 

some such form as the line AB. Thus the

FIGURE 9. Ideal transverse section of zone of saltation at side of experiment trough.

tract actually studied in the work with the 

moving field was somewhat superficial and was 

not in strictness a vertical section of the zone. 

The second qualification is connected with 

turbulence. In steady flow the motion at each 

point of a stream is constant in velocity and 

direction. When the general velocity exceeds 

a certain minimum, which for the streams we 

have to consider is very small, the flow is not 

steady, but involves eddies or vortices, which 

as a rule move onward with the current. In 

consequence of these eddies the course of each 

particle of water is sinuous, and the sinuous 

courses interweave. The flow is then said to 

be turbulent. Usually there are both large 

and small eddies, the minute ones being multitudinous. As the axes of whirling movements 

have all attitudes, the directions of motion, as 

a rule, have upward or downward components, 

and the suspension of particles of debris is due 

to the upward components. Particles so small 

that they can not come to rest on the bottom 

are thereby lifted and relifted and kept in the 

body of the water. Under the conditions 

arranged for the study of saltation there 

appeared to be no large eddies, but the zone 

was unquestionably pervaded by small ones, 

excited by the roughness of the bed and by the 

differential motions of water and leaping 

grains. With increasing strength of current

the texture of turbulence would enlarge and 

saltation pass into suspension. With a diversified debris, instead of the uniform material 

actually used, there would be phases of action 

in which the paths of small grains were made 

sinuous by turbulence, while those of larger 

grains remained simple in form. The trajectory of saltation, as described, may therefore 

be regarded as a simple type of path which 

combines in all proportions with the sinuous 

type of path characterizing suspension.

Through the entire zone of saltation motion 

is being communicated to particles of the load 

by the water, and there is a corresponding loss 

of motion by the water. That loss reduces all 

the stream's velocities but makes the greatest 

reduction in the lower part of the zone of 

saltation. The loss of velocity in the lower 

strands reduces their power to cause particles 

to leap. The greater the load the greater this 

reduction, and thus the quantity of load is 

automatically regulated.

COLLECTIVE MOVEMENT.

In the experiment used to illustrate saltation 

the collective movement of the sand was uniform, the conditions of the experiment having 

been adjusted to that end. But it is equally 

possible so to adjust them as to make the 

collective movement rhythmic. Uniformity is 

in fact an intermediate phase between two 

rhythmic phases, which are of contrasted types. 

These phases will be described.

In another experiment a bed of sand was 

first prepared with the surface level and 

smooth. Over this a deep stream of water 

was run with a current so gentle that the bed 

was not disturbed. The strength of current 

was gradually increased until a few grains of 

sand began to move and then was kept steady. 

Soon it was seen that the feeble traction did 

not affect the whole bed, but only certain tracts, 

and after a time a regular pattern developed 

and the bed exhibited a system of waves and 

hollows. As the waves grew the amount of 

transportation increased, showing that, under 

the given conditions, the undulating surface 

was better adapted to traction than the plane. 

With such waves and hollows are associated a 

special mode of transportation, which is illustrated in figure 10. A current reaching the 

bed at A follows the rising slope and crest of 

the wave to C and then shoots free, to reach
THE OBSERVATIONS.

the bed again at D. The space overleaped 

between C and D is occupied by one or more 

slow-moving eddies. From A to C there is 

traction, the material being derived from the 

slope between A and B. At C the de*bris, 

being abandoned by the current, is dumped, 

and it slides by gravity down the slope CE. 

So the upstream face of the wave is eroded and 

the downstream face built out, with the result 

that the wave, as a surface form, travels 

downstream. As this is precisely what takes 

place when a sand hill travels under the 

influence of the wind, the name of the eolian 

hill has been borrowed, and the waves are 

called dunes.1 In one of the narrower troughs 

of the laboratory the dunes formed a single 

line. In a wider trough their arrangement 

sometimes suggested a double line, the crests

of one being opposite the hollows of the other, 

but their arrangement continually changed. 

On the bed of a broad, shallow stream they are 

apt to have a subregular imbricated pattern.2 

In a deep stream a single dune may be nearly 

as broad as the channel. In the laboratory 

the forms were inconstant, but the type was 

about as broad as long, with the front edge 

convex downstream. In natural streams the 

dunes show great variety in outline, some being 

described as longest in the direction of the 

current and others as greatly extended in the 

transverse direction. They vary in size with 

the size of the stream, but especially with the 

depth, and are transformed and remodeled 

with increase and reduction of discharge. The 

horizontal dimensions of most laboratory 

examples may be conveniently described in

FIGURE 10. Longitudinal section illustrating the dune mode of traction.

inches, but river examples may require scores 

or hundreds of feet.3 The maximum height in 

the laboratory was probably 2 inches; for 

those in Sacramento River 2 feet has been 

reported, and for those in the Mississippi 22 

feet.

In each series of laboratory experiments to 

determine the relation of load to slope the 

initial run was made with a small load, while 

for the succeeding runs the load was progressively increased. Enlargement of the load 

caused increase of slope and velocity, with 

decrease of depth, and these changes were 

accompanied by changes in the mode of transportation. In the earlier runs dunes were 

formed, and these marched slowly down the 

trough. Then, somewhat abruptly, the dunes 

ceased to appear, and for a number of runs the

i This is the name chiefly used by Swiss investigators (see De Candolle, 

Arch. sci. phys. et nat., vol. 9, p. 242,1883, and Forel, idem, vol. 10, p. 43, 

1884), and many observers compare the subaqueous feature to the eolian; 

but the specific title commonly used in the United States and Great 

Britain is sand wave, and some French engineers employ gr&oe. In the 

present paper, dune is preferred to sand wave because there is occasion 

to distinguish two species of de'bria waves.

channel bed was without waves and approximately plane, although somewhat ruffled in the 

run immediately following the disappearance of 

dunes. Finally a third stage was reached in. 

which the bed was characterized by waves of 

another type. These are called antidunes, 

-because they are contrasted with dunes in 

their direction of movement; they travel 

against the current instead of with it. Their 

downstream slopes are eroded and their upstream slopes receive deposit. They travel 

much faster than the dunes, and their profiles 

are more symmetric. The water surface, 

which shows only slight undulation in connection with dunes, follows the profiles of anti-

! The imbricated pattern is frequently seen beneath tidal waters, 

where ripple marks due to the reaction of wind waves are transformed 

into dunes when the tidal current sweeps across them. It is then usually 

to be ascribed to a difference in direction of the two actions. An elaborate account of its development in rivers is given by H. Blasius, who has 

recently investigated the whole subject of the rhythmic features of river 

beds. See Zeitschr. Bauwesen, vol. 60, pp. 465-472,1910.

8 Arthur Hider, who studied dunes in the lower Mississippi, reported 

a maximum length, crest to crest, of 750 feet, a maximum height of 22 

feet, and a maximum progression of 81 feet in a day. See Mississippi 

River Comm. Kept., 1882, pp. 83-88 (= Chief Eng. U. S. A., Kept., 1883, 

pp. 2194-2199).
32 TRANSPORTATION" OF DEBBIS BY RUNNING WATER.

dunes closely and shares their transformations. 1 

(See fig. 11.) Usually each antidune occupied 

the full width of the experiment trough; and in 

natural streams, so far as I have observed, they 

either reach from side to side of the channel or 

else form well-defined rows in the direction of 

the current. Not only is a row of antidunes a 

rhythm in itself, but it goes through a rhythmic 

fluctuation in activity, either oscillating about 

a mean condition or else developing paroxysmally on a plane stream bed and then slowly 

declining. Paroxysmal increase starts at the 

downstream end of a row and travels upstream, 

gaining in force for a time, and the climax is 

accompanied by a combing of wave crests. 

Where the debris is very coarse, as on the outwash plains of glaciers, a din of clashing bowlders is added to the roar of the water. 2

Of the phases of process in the laboratory 

Mr. Murphy writes:

Their [the dunes'] form is continually changing as they 

move forward; they divide and again unite, the parts 

traveling at different rates, and new ones form on top of

the older ones. The grains roll up the gentle slope, fall 

over the crest, are covered by other grains, and rest until 

the dune again passes over them and they are again uncovered. Thus the time during which they are in motion is 

small compared to the time during which they rest. As 

the velocity of the current increases, the rate of feeding 

being correspondingly increased, the size of the dunes and 

their rate of movement increase. Thus we find that when 

the discharge is 0.363 ft.3/sec., load 11 gm./sec., and slope 

0.32 per cent, the dunes are 7 to 9 inches long and one-half 

inch high and move at the rate of 0.56 foot a minute, but 

when the discharge is 0.734 ft. 3/sec. and the load 30 

gm./sec., the dunes are 13 to 15 inches long, three-fourths 

of an inch high and move at the rate of 1.5 feet a minute. 

As the velocity of the current increases some of the grains 

leap as well as roll, and some, instead of dropping over 

the crest of a dune and resting, leap to the next dune. 

The dune grows less distinct in form and finally at a critical velocity it disappears, dune motion ceases, and the 

sand surface becomes comparatively even. This condition of even surface flow continues as the slope increases 

until at another critical velocity antidune movement 

begins. A profile of the sand surface for this kind of 

motion is shown in figure 11. For this experiment the 

trough width is 1.32 feet, discharge 0.734 ft.3/sec., load 

213 gm./sec. and the sand slope 1.23 per cent. These sand 

waves are from 2 to 3 feet in length from crest to crest, they 

extend the width of the trough, and some of them are 0.5

-1.0

Y

6 '9 \Z 15 

FIGURE 11. Longitudinal section illustrating the antidune mode of traction. The numbers show distance in feet from the head of the experiment

trough.

foot in height from crest to trough of the wave. They 

travel slowly upstream, some of the sand being scoured 

from the downstream face in the vicinity of Y (fig. 11) 

and deposited on the upstream face at X. Some of these 

waves remain for two minutes or longer, but most of them 

not longer than one minute. A whitecap forms, on the 

surface of the water when the larger waves disappear. 

Sometimes two or more will disappear at once and leave 

the surface without waves for a distance of 10 feet or more. 

Only a portion of the sand transported takes part in the 

formation of these sand waves. The velocity in a wave 

trough is greater than near the crest. The sand grains 

flow nearly parallel to the bed as they pass through the

1 Antidunes, though less common than dunes, are by no means rare 

under natural conditions. They are described by Vaughan Cornish 

(Geog. lour. (London), vol. 13, p. C24,1899; Scottish Geog. Mag., vol. 17, 

pp. 1-2,1901), and are mentioned by John S. Owens (Geog. Jour.,vol. 31, 

p. 424,1908).

2 The sequence of bed characters dune,. smooth, antidune was 

observed by John S. Owens in studies with natural currents hi 1907, and 

the characters were correlated with velocities. With depths of 3 to 6 

inches and a bed of sand, he noted sand ripples [dunes] when the velocities, measured by floats, were from 0.85 to 2.5 ft./sec., and the 

appearance of antidunes at a velocity of about 3 ft./sec. (Geog. Jour. 

(London), vol. 31, pp. 416, 424, 1908). Sainjon and Partiot, studying the movement of de"bris.in the Loire, had previously observed 

that whereas with low velocities the entire bottom load was transported 

through the progress of dunes, with higher velocities the debris was swept 

along from crest to crest and the dunes were reduced in height (Annales 

des ponts et chausse"es, 5th ser., vol. 1, pp. 270-272,1871).

trough, but at the crest they have an up and down motion 

as well as a forward motion. On the crest of the larger 

waves their forward motion is small compared with their 

vertical motion. * * *

There is a sand movement by rotation or whirls that 

aids transportation. These whirls have been observed 

during dune motion only for smaller sizes of sand. They 

are of short duration, lasting usually less than one minute, 

but in this time one of these may scour a hole 1 to 3 inches 

deep and 4 to 10 inches in length. They usually start 

near the side of the trough, the axis inclining downstream 

and toward the center, making an angle of 30° to 60° with 

the side and a small angle at the bottom. These whirls 

are 3 to 5 inches in diameter and the sand grains are thrown 

violently up as well as downstream by them. This movement aids transportation by its lifting action, some of the 

grains being carried in suspension for a short distance by it.

The change in the appearance of a loaded stream as the 

load is increased, the discharge remaining constant, is 

very striking. For no load the water surface is even and 

smooth. As fine sand is fed into the water at a slow rate, 

small sand dunes will form on the bottom and many little 

waves will form on the surface. As the rate of feeding is 

increased, the slope and velocity increasing, these waves 

become larger and fewer and have the shape of an inverted 

canoe. These canoes are side by side, the number 

depending on the trough width and size of waves. When 

the width was 1.0 foot two sets formed side by side; when
THE OBSERVATIONS. 33

the width was 1.32 feet three sets formed. As the critical 

velocity at which dune motion ceases is approached these 

waves begin todisappear, and when thisvelocity is reached 

the water surface is waveless. This waveless condition 

continues as the rate of feed increases until sand motion 

in antidunes begins, when large waves, the width of the 

trough and corresponding in length to the sand waves 

beneath them, are formed as illustrated in figure 11.

In order to show the magnitude of these surface waves, 

wave traces have been drawn. Some of these are given 

in figure 12. A sheet of galvanized iron 4 feet long and 1 

foot high was divided into inch squares by lines. This 

plate was moistened and covered with fine dust. It was 

held vertical at a given place over the experiment trough 

and on signal was dropped into the trough and taken out 

again as quickly as possible. The dust was removed from 

that part of the plate in the water, leaving a well-defined 

outline of the wave. This wave trace was quickly sketched 

on paper by means of the lines marking the squares. Trace 

A, figure 12, is for zero feed; B is for a very email feed; C is 

for a larger feed, the surface being covered with the canoeshaped waves; D shows one of the larger waves associated 

with antidunes.

The slopes at which the phases of traction 

change are lower for large streams than for 

small, and lower for fine debris than for coarse. 

The phase with smooth bed which may conveniently be called the smooth phase covers 

a greater range of conditions with mixed 

de'bris than with assorted.

The processes associated with dunes and 

antidunes were briefly studied in the glazed 

trough and with the moving field. Transportation by saltation follows the entire profile of the antidunes but traverses the dunes 

only from A to 0 of figure 10. The velocity 

of saltatory grains is greatest where erosion 

takes place, namely, along the upstream slopes 

of dunes and the downstream slopes of antidunes, and it may reasonably be inferred that 

the water velocities are greatest in those 

places. The eye detected no difference in water

21 22 23 24 

FIGURE 12. Profiles of water surface, automatically recorded, showing undulations associated with the antidune mode of traction. Numbers

show distance, in feet, from the head of experiment trough.

depth over the two slopes of the antidune, and 

if the depth is the same so also is the mean 

velocity; but the ratio of bed velocity to mean 

velocity is known to vary with conditions.

The cause of the changes in process has not 

been adequately investigated, but a few suggestions may be made. To assist in a search, 

for controlling conditions, the factors connected with the two critical points the change 

from dune phase to smooth and the change 

from smooth to antidune were tabulated 

from the experiments with sand of a single 

grade (C); the positions of the critical points

being estimated.by Mr. Murphy at a time when 

the details of the experiments were freshly in 

mind. In Table 3 w is the width of trough, in 

feet; Q the discharge, in cubic feet per second; 

S the per cent of slope; d the depth of water in 

feet; Vm the mean velocity, in feet per second; 

L the load, in grams per second; and L^ the 

load per unit width. The data in this table 

are taken from a preliminary reduction of the 

observations and are less accurate than the 

results of the final adjustment, which appear 

in Table 12 (p. 75). They suffice, however, 

for the present purpose.

TABLE 3. Data connected with changes in mode of transportation.

w

0. 66

.06

.66

.66

1.00

1.00

1.00

1.00

1.32

1.32

1.32

1.32

1.96

1.96

1.96

Q

0.093

.182

.363

.545

.182

.363

.545

.734

.182

.363

.545

.734

.363

.734

1.119

First critical point.

S

1.00

.93

.82

.74

1.15

.89

.81

.73

1.20

1.00

.87

.75

1.04

.84

.74

d

0.076

.115

.190

.272

.082

.134

.180

.227

.068

.107

.145

.178

.081

.133

.182

vm

1.86

2.41

2.91

3.05

1.76

2.71

3.03

3.24

2.03

2.57

2.84

3.13

2.29

2.82

3.14

L

13

28

48

70

41

70

95

113

33

76

106

140

67

138

191

Li

20

46

73

106

41

70

95

113

25

57

80

106

34

71

98

Second critical point.

S

1.86

1.72

1.51

1.34

2.02

1.70

1.51

1.34

2.15

1.76

1.53

1.33

1.91

1.48

1.22

d

0.058

.092

.165

.209

.056

.109

.142

.194

.051

.099

.111

.149

.062

.108

.150

vm

2.44

3.01

3.37

3.97

3.25

3.33

3.75

3.78

2.70

2.77

3.71

3.73

2.99

3.47

3.81

L

44

88

145

184

111

201

244

301

121

210

251

293

219

354

418

Li

67

134

221

281

111

201

244

301

92

159

190

222

112

181

213

20921° No. 86-
34 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

The factors were then plotted in various 

combinations on logarithmic section paper, and 

certain approximate numerical relations were 

thus discovered. The first critical point is 

reached when d = 0.016 Vm2 ' 3 , or when d = 

0.0045 Li0' 65 . The second critical point is 

reached when d = 0.004 Fm3 ' 3 , or when d = 

0.0003 JV' 15 . The coefficients and exponents 

are not well defined by the data, but the 

general indications are (1) that each change in 

phase occurs when the depth of water bears 

a certain numerical relation to a power of the 

mean velocity near the cube, and (2) that the 

changes occur when the depth bears a certain 

numerical relation to the amount of load 

carried in each unit of width of current.

At the bottom the stream is limited and 

restricted by the bed of de'bris; at the top by 

the water surface. To the space between these 

bounds, a space measured by the depth, the 

eddies or convolutions of the current are confined. Within the range of conditions covered 

by the experiments the normal mode of flow 

involves sinuosity of the filaments of current, 

and the tendency toward diversity of internal 

movement is strong in proportion as the velocity is high. A particular relation between 

depth and velocity corresponds to a sort of 

equilibrium between the factors of turbulence 

and restraint, in accordance with which the 

sinuosity of the lines of flow is reduced to a 

minimum and the water surface and channel 

bed are approximately plane. This gives the 

smooth phase of traction. When the depth is 

increased without increase of velocity, the 

reduction of restraint permits the development of internal diversity, and this carries with 

it diversity of the plastic bed, giving the 

dune phase of traction. When the velocity 

is increased without increase of depth, the

restraint is overpowered, and a diversified but 

systematic arrangement of flow lines develops, 

which carries with it-systematic diversity of 

both water surface and channel bed and gives 

the antidune phase.

There will be occasion to speak of these 

relations in another connection in Chapter 

XIV.

It may be noted, as a possible contribution 

toward an explanatory analysis, that gravity 

opposes the current on the upstream side of 

the antidune (fig. 11) and assists the current on

the downstream side. It is where gravity accelerates that load is increased by erosion, and 

where gravity retards that load is reduced by 

deposition. In the case of the dune, however, 

erosion occurs where gravity is a retarding 

force. But here the descending current, which 

is accelerated by gravity and which is observed 

to have gained speed, is free from the bed and 

bears no load (fig. 10). In order to transport 

when it resumes contact with the bed it must 

take debris from the bed, and by so taking it

erodes.

UNITS.

The system of units to which the laboratory 

data have been reduced and which will be employed in their discussion is hybrid in that it 

includes the foot and the gram. The foot is 

made the fundamental unit for length, area, 

and volume because it is the unit employed generally by English-speaking engineers. The 

gram is made the unit of mass, primarily because it is of convenient magnitude, but also 

because of the manifest advantage of introducing the metric system wherever no practical 

difficulties interfere. It happens that the two 

measures which are given in grams are of categories unfamiliar alike to the engineer and the 

general reader, so that the gram unit encounters 

no conflicting habit of thought. One measure 

is the mass of a grain of sand or a pebble, the 

other the mass of the de'bris carried by a stream 

in a second.

The unit of time, for the indication of rates, 

is one second. Velocity is given in distance per 

second, ft./sec.; discharge in volume per second, 

ft.3/sec.; and load in mass per second, gm./sec.

In hydraulic and hydrodynamic treatises 

slope of streams is measured by the quotient of 

fall by distance, or the tangent of its angle, and 

the unit slope is taken as 45°. For practical 

purposes this unit is inconvenient because it 

transcends experience, and engineers commonly 

avoid it by speaking of slope in percentage or 

in fractions of 1 per cent, thus making 1 per 

cent the actual unit. For most purposes I 

find the smaller unit most convenient, but I 

have occasional use for the larger unit and shall 

accordingly use both. To avoid confusion the 

symbol S will be used with the smaller unit, 

and where discrimination is important it will 

be called per cent slope, while the symbol s will 

be used with the larger unit. 8 = 100 s.
THE OBSERVATIONS.

TERMS.

Load. The quantity of debris transported 

by a stream through any cross section in a unit 

of time is its load at that section. A part is 

carried in suspension and a part by traction, 

but as we are here concerned with traction 

only, the fractional load is to be understood 

when the word is used without specification. 

Load is measured in grams per second, gm./sec. 

For certain engineering purposes it is desirable 

to consider load as volume, not as the sum of 

the volumes of individual grains, but as the 

gross space occupied by the d6bris as a natural 

deposit. For the debris used in the experiments in stream traction, with the mixture 

of sizes ordinarily found in a river deposit, the 

weight of 1 cubic foot is about 50,000 grams, or 

110 pounds. The symbol for load is L.

Capacity. The maximum load a stream 

can carry is its capacity. It is measured in 

grams per second, gm./sec. As the work of 

the laboratory was largely to determine 

capacity by measuring maximum load, the 

two terms are to a large extent interchangeable 

in the discussion of laboratory data, but the 

distinction is nevertheless important. The 

symbol for capacity is G.

Capacity is a function of various conditions, 

such as slope and discharge, and the chief 

purpose of the laboratory investigation was to 

discover the relations of capacity to conditions. 

When a fully loaded stream undergoes some 

change of condition affecting its capacity, 

it becomes thereby overloaded or underloaded. 

If overloaded, it drops part of its load, making 

a deposit. If underloaded, it takes on more 

load, thereby eroding its bed. Through these 

reactions the profiles of stream beds are 

adj Listed, so far as stream beds are composed of 

debris. If the bed is of rock in place, the underloaded stream can not obtain its complement 

of cle'bris, but nevertheless it attacks the bed. 

By dragging debris over the rock it files or 

corrades the bed of its channel. It is a general 

fact that the loads of streams flowing on bedrock are less than their capacities.

Competence. Under certain combinations of 

controlling factors capacity is zero or negative. 

If then some one factor be changed just enough 

to render capacity positive, that factor in its 

new condition is said to be competent, or else

to be a measure of the stream's competence. 

For example, a stream at its low stage can not 

move the debris on its bed; with increase of 

discharge a velocity is acquired such that 

traction begins; and the discharge is then said 

to be competent. A stream flowing over 

a too gentle slope has no capacity, but coming 

to a steeper slope it is just able to move debris; 

the steeper slope is said to be competent. A 

current flowing over debris of various sizes 

transports the finer but can not move the 

coarser; the fineness of the d6bris it can. barely 

move is the measure of its competence.

Discharge. The quantity of water passing 

through any cross section of a stream in a unit 

of time is the discharge of the stream at that 

point. It is measured in cubic feet per second, 

ft.3/sec. The symbol is Q.

Slope. The inclination of the water surface 

in the direction of flow is known as the slope of 

the stream. It is the ratio which fall, or loss 

of head, bears to distance in the direction of 

flow. Per cent slope, as explained on page 34, 

is numerically 100 times as great, being the 

fall in a distance of 100 units. The terms 

slope and per cent slope are also applied in 

this report to the inclination of the bed of the 

channel.

Size of debris. The relative magnitude of 

the d6bris particles making up the load may be 

considered from two opposed viewpoints. 

Thus we may say that the load varies inversely 

with the coarseness of the debris, or that it 

varies directly with the fineness. The second 

viewpoint is here preferred because it conduces 

to symmetry in formulation. Two very different measures of fineness have been considered, and on page 21 the material of the 

laboratory is listed under both. One defines 

fineness by the number of particles which, 

placed side by side, occupy the linear space of 

1 foot. The other defines it by the number of 

particles required to fill the space of 1 cubic 

foot. Linear fineness is the more readily 

conceived, because it appeals to vision. Bulk 

fineness is the more easily determined. The 

symbol of linear fineness is F, of bulk fineness F2.

Form ratio. Of the variable factors which 

in combination produce the multifarious channel forms of natural streams, the laboratory 

dealt extensively with but a single one, the
36 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

relation of depth to width. The relation is a 

simple ratio, and either of the two terms 

might be made the divisor. The width has 

been chosen because, as in the case of fineness 

its selection conduces to symmetry in formulatier, The ratio U be

form ratio. Its symbol is R.

Duty and efficiency. Capacity varies with 

discharge, but is not proportional to it. The 

load which may be borne by a unit of discharge varies with the discharge and also with 

the other conditions. It is the capacity per 

unit discharge, or the quotient of capacity by 

discharge, and will be called the duty. The 

symbol is U.

Capacity varies also with slope but is not 

proportional to it. As the product of discharge 

by slope (by the acceleration of gravity) 

measures the stream's potential energy per 

unit time per unit distance, capacity also varies 

with the energy but is not proportional to it. 

The load which may be borne by a unit of discharge on a unit slope varies with all the conditions of transportation. It is the capacity 

per unit discharge and unit slope, or the quotient of capacity by the product of discharge 

and slope, and will be called the efficiency. 

. It is a measure of the stream's potential work 

of transportation in relation to its potential 

energy. Its symbol is E.

The primary definition of efficiency in mechanics is the ratio of work done to energy expended; it implies that the work may be expressed in the same unit as energy. The ratio 

is -always less than unity. But there is an 

important secondary use of .the term, applied 

to cases in which the result accomplished can 

not be expressed in terms of energy. In such 

cases the ratio may have any magnitude, as it 

arises from the comparison of incongruous 

quantities. It does not measure economy of 

energy but relative accomplishment in respect 

to any condition selected for comparison. As 

capacity for transportation is not statable in 

units of energy, the use of the term efficiency 

in this connection falls under the second definition.

Symbols. An index of symbols, with brief 

definitions and references to pages for fuller 

definition, may be found on page 13.

TABLE OF OBSERVATIONS ON STREAM 

TRACTION.

The observations on stream traction are 

presented in Table 4. As the original notes 

are voluminous, certain combinations and reductions were made before tabulation. The 

reduction of the slope observations involved 

discrimination, and the mode of reduction is 

therefore described below. Each horizontal 

line of the table contains the record of a single 

experiment.

The observations are arranged according to 

(1) fineness of debris, (2) width of channel, 

(3) discharge, (4) load and slope. The categories of fineness begin with single sizes, 

taken in order from fine to coarse, and follow 

with mixtures; they constitute subtables, each 

designated by a letter in parentheses. The 

arrangement by load and slope is approximate 

only. In a general way the sequences of load 

and slope are parallel, each increasing as the 

other increases, but the data are not perfectly harmonious, and where the two sequences differ the arrangement is somewhat 

irregular.

The first column, for all divisions of the 

table except (J), gives width of trough; the 

second, discharge. The third, fourth, and 

fifth pertain to load and give, respectively, 

the load as measured by debris fed at the 

head of the trough, the load as measured by 

d6bris caught at the outfall end, and the 

period, in minutes, during which de'bris was 

collected at the outfall end. The precision of 

measurement is probably somewhat higher 

where the period is relatively long.

-The next three columns pertain to slope. 

The sixth contains the slope, in per cent, of 

the water surface; the seventh, the slope of 

the bed of de'bris as shaped by the current; 

and the eighth, the extreme distance between 

points at which were made observations used 

in computing the slope or slopes. Nearly all 

the experiments for which the recorded distance is 16 feet or less were made in the 

shorter trough, of which the gross length was 

31.5 feet, the distance between the de"brisfceding apparatus and the debris-arresting apparatus being 24.5 feet. The experiments for 

which the recorded distance exceeds 16 feet 

were made in the longer trough, the distance
THE OBSERVATIONS. 37

from the feeding station to the collecting station being at least 16 feet longer than the 

space covered by the observations used in the 

computation.

The ninth column shows the depth at a 

single point as measured by the gage (see 

p. 21); the tenth gives the mean depth as 

estimated from the records of water profile and 

bed profile (see p. 25).

In the eleventh column, headed "Character 

of bed," a generalization is given from notes 

on the mode of transportation, the condition 

of the water surface during the run, and the 

condition of the bed of debris after the withdrawal of the water. The original notes are 

somewhat varied in scope and nomenclature, 

and it seemed best to make the tabulated 

record more simple. The words dunes, smooth, 

and antidunes denote the three modes of traction described on pages 30-33. The word 

transition is used where the mode of traction 

was intermediate between one of the rhythmic 

modes and the smooth mode, and also where 

different modes of traction obtained in different parts of the trough.

In the final column the word free indicates 

that the contractor (p. 25) was not used, but 

the experiment trough retained its full width 

to the end.

The reduction of the slope observations was 

preceded by a careful study of them with reference to their systematic and accidental errors. 

As a result of that study certain criteria of 

exclusion were adopted, by means of which, it 

is thought, the influence of systematic errors 

was materially reduced. The criteria were 

applied through an inspection of the plotted 

profiles of water surface and debris bed. The 

observations excluded were those believed to 

be much affected either by the peculiar conditions near the head of the trough or by the

peculiar conditions near the outfall end, so 

that the retained observations constituted a 

continuous series covering the middle.

The observations to be reduced were ascribed equal weights. They consisted of a 

series of level readings (h0 , \, Ti2 , . . hnr.1} 

Jin), each giving the vertical distance of a point 

of the de*bris surface, or of the water surface, 

below a horizontal plane of reference, together 

with the horizontal distances of the same 

points from an initial point in the axis of 

the trough. The observed points being at 

equal intervals along the trough, and the zero 

of distances being made coincident with the 

first of the observed'points, the horizontal distances may be represented by 0, I, 21, . . . 

(n l)l. The following formula1 was used to 

compute the slope, the number of terms in

7?

numerator or denominator being K when n 

was an even number, and 9 when n was odd.

I w2 +(n-2)2+(n-4)2 + ... v '

The numerical operations were simple.

In division (J) of the table the arrangement 

is somewhat different, and a column is added 

at the left. The division records experiments 

made with de"bris prepared by mixing, in definite proportions, two or more of the grades of 

debris to which the preceding divisions pertain. 

The notation adopted to designate these mixtures is analogous to that for chemical compounds, subscript figures being used to indicate 

approximate proportions. This notation, together with the more precise indication of the 

proportions, is given in the left-hand column.

1 Essentially a least-squares formula, although developed from apparently independent considerations.
38 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 4 (A). Observations on load, slope, and depth, with debris having 25,200 particles to the gram, or grade (A).

Width.

Feet. 

0.66

1.00

1.32

charge.

Ft.3lsec.

O flQ3

.182

.545

.182

.363

.734

.182

.363

.734

.363

.734

1.119

Feed.

Gm.lsec. 

27

67

19

93

59

147

10.5

151

20

240

52

317

12

17

18

23

37

50

75

73

129

120

7.3

42

45

44

65

69

125

121

214

38

82

86

127

151 '191

176

8.1

20

19

33

58

81

128

143

128

192

215

227

7.7

32

74

70

105

148

214

221.

94

99

186

169

180

198

221

Load.

Collection.

Gm.lsec. 

24

74

18

- 96 .

58

157

9.4

155

19

245

51

317

12

18

17

21

38

51

68

73

120

135

8.9

38

41

43

74

76

133

140

177

202

240

38

84

89

156

154

192

215

240

327

365

8.7

20

21

32

60

92

127

128

179

193

218

264

283

8.3

33

83

80

105

163

167

179

231

279

91

102

130

188

221

229

258

291

341

346

Period.

Minutes. 

5

5

10

4

5

3

10

3

8

3

4

3

10

10

10

8

6

5

5

5

5

3

8

6

5

5

4

5

4

5

4

4

4

5

4

4

4

3

3

3

3

3

3

6

6

6

5

5

5

5

5

4

4

4

4

3

3

10

6

4

5

5

 4

4

3

3

3

4

4

4

3

3

3

4

3

3

3

Water 

surface.

Per cent. 

1.12

.56

.58

.65

.83

.80

1.11

.37

.60

.77

.35

.53

.50

.69

.67

.78

.97

.18

.58

.79

.30

.33

Slope.--

Bed.

Per cent. 

 1.22

2.14

.69

1.59

.55

1.07

.53

2.04

.42

1.64

.44

1.19

.63

.75

.73

.79

.99

1.10

1.37

1.41

1.79

1.86

.27

.51

.58

.67

.84

.90

1.18

1.14

1.43

1.55

1.65

.35

 .48

.51

.73

.83

.81

.97

.95

1.13

1.24

.36

.41

.55

.60

.78

.94

1.10

1.17

1.18

1.38

1.50-

1.59

1.73

1.77

.18

' .36

.45

.49

.56

.79

.75

.94

.98

1.01

.45

.39

.57

.59

.63

.60

.81

79

.95

1.02

Distance.

Feet. 

16

16

16

14

16

16

16

16

14

16

14

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

10

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

12

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

; Dei

By gage.

Feet. 

0.065

.126

.095

.272

.118

.175

.264

.091

.084

.075

.080

.063

.058

.061

.060

.191

.124

.115

.112

.112

.130

.115

.106

.089

.097

.205

.198

.190

.179

.118

.118

.103

.090

.082

.073

.077

.075

.072

.076

.067

.069

.070

.283

.180

.156

.156

.137

.142

.135

.129

.135

.130

.220

.207

.194

.196

.182

.187

.165

.179

.176

)th.

By profiles.

Feet. 

0.083

.284

.136

 

.106

.091

.083

.087

.079

.195.

.131

.267

.207

.187

.129

.107

.097

.084

.293

.152

.146

.214

.201

Character of bed.

....do......................

....do......................

....do......................

Antidunes .................

[AntidunesJ............. ..

....do......................

....do.....................

....do.....................

Dunes......................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

Smooth...................

.....do....................

.....do....................

.....do....................

.....do....................

Outfal,!.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Contracted.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.
THE OBSEBVATIONS. 39

TABLE 4 (B). Observations on load, slope, and depth, with debris having IS,400 particles to the gram, or grade-(B).

Width

Feet. 

0.23

.44

.66

Discharge.

Ft.»/sec 

0.093

.182

.093

.182

.093

.182

.363

' .545 .

Load .

Feed.

Om. /sec

4.8 

3.5 

14 

15 

23 

27 

24 

42 

47 

41 

51

55

81

166

29 

44 

52 

58 

73

16 

34 

64 

67 

79 

108 

105

Collection.

Gm./sec 

4.C

<j 1

on

33

4.0

1 R

31

09

5

0

n

18

34 

53 

64

8.5 

16 

27 

73 

76

5.1 

9.2 

17 

15 

22 

28 

34 

41 

53 

57 

93

4.2 

6.0 

12 

16 

20 

21 

19 

42 

39 

44 

47 

55 

55 

55 

67 

72 

88 

105 

112 

119 

126 

139 

159 

169 

213 

236 

254 

268 

377

29 

40 

39 

49 

55 

77 

91 

104 

148 

152 

164 

167 

186 

226

5.8 

16 

35 

54 

68 

81 

104 

111 

132 

169 

242 

271 

304

Period

Mmule.

1

i

\ 

8 

4

8 

6 

7 

9 

5 

5 

5 

5 

5 

5 

7

8 

8 

7 

6 

8 

4 

8 

7 

6 

9 

8 

5 

8 

5 

7 

7 

4 

5 

6 

6 

4 

5 

5 

5 

4 

3 

3 

. 4 

3

5 

8 

6 

6 

5 

4 

5 

5

4 .

5 .

4 .

5 .

3 .

4 .

6 

9 .

5

8 .

6 .

6 .

5 .

5 .

5 .

5 . 

3 '. 

3 . 

3 .

Slope.

Water 

surface

Per cent

&......

0.32

.37

........

.18

.47

Bed.

. Per cent 

0.&

.94

1.01

1 49

2.12

.73

1.12

1 W

1.55

.73

on

1.18

1.62

2.31 

2.38

.50 

.75 

.98 

1.66 

1.73'

.75 

.84 

1.23 

1.32 

1.41 

1.47 

1.63 

2.01 

2.09 

2.17 

2.96

.35 

.36 

.69 

.68 

.66 

.72 

.81 

.98 

1.06 

1.18 

1.17 

1.32 

1.35 

1.36 

1.51 

1.54 

1.51 

2.00 

1.88 

2.05 

2.24 

2.19 

2.46 

2.79 

3.14 

3.06 

2.91 

2.93 

4.03

.51 

.68 

.69 

.75 

. .80 

.98 

.99 

1.14 

1.54 

1.43 

1.63 

1.66 

1.71 

1.81

.21 

.23 

.46 

.63 

.74 

.72 

.87 

. .89 

1.00 

1.32 

 1.81 

1.81 

1.79

Distance

. Feet. 

1C 

1C 

16 

16 

16

16 

16 

16 

16

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16

16 

16 

16 

12 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

14 

16 

16

16 

14 

14 

. 16 

16 

14 

16 

16 

16 

16 

16 .

16 .

16 .

Depth.

By gage

Fed. 

0.192 

.184 

.174 

.146

.364 

.284 

.230 

.232

.136 

.110 

.093 

.070 

.072 

.068

.194 

.177 

.146

.139

.089 

.080 

.060 

.059 

.058 

.050 

.056 

.054 

.049 

.058 

.037

.175 

.175

.137 

.113 

.110 

.107 

.113 

.101

.112 

.107

.103

.091 

.082 

.083 

.077

.115 

.096 

.082

.216 

.212 

.205 

.202 

.188 

.178 

.189 

.191 

.177 

.155

.158

.471 

.367 .

.291 

.292 .

.258 .

.260 . 

.220 .

.226 .

.230 .

.231 .

 BLPsr°

Feet.

0.183 

.183

.221

.472

.300

Character of becl.

. .....do....................

. Smooth...................

. ....do.....................

. Antidunes ................

. Dunes ....................

. .....do....................

Transition................

.....do....................

.....do....................

Smooth ...................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

Antidunes .................

[Antidunes] ................

....do.....................

....do.....................

....do.....................

....do.....................

....do.....................

....do.....................

....do.....................

....do.....................

....do.....................

....do.....................

....do.....................

....do.....................

....do.....................

....do.....................

...'.do.....................

....do.....................

....do.....................

Smooth . .... 1 ..............

....do.....................

....do.......:.............

....do.....................

Outfall.

. Free.

Do.

Do.

Do.

Do.

. Free.

Do.

Do.

Do.

Free. 

Do. 

Do. 

Do. 

Do/ 

Do.

Free. 

Do. 

Do. 

Do. 

.Do.

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.
40 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 4 (B). Observations on load, slope, and depth, with debris having 13,400 particles to the gram, or grade (B) Con.

1

Width.

Feet. 

1.00

1.32

1.

charge.

Ft.3/sec. 

0.182

.363

. .545

.734

.182

.362

1 

Load.

Feed.

Om.lsec. 

9.9 

9.9

0

2.1 

1.9 

6.4 

6.8 

10.5 

9.6 

8.7 

9.3 

18 

25 

28 

37

=

0 

3.9 

4.7 

18 

37 

56

56

84

7.3 

5.8

8.3 

8.7 

17 

19

Collection.

3m. [sec. . 

11 

13 

28 , 

39 

51 

98 

116 

157 

182

0

1.3 

1.6 

6.2 

6.8 

7.2 

9.4 

9.7 

11 

21 

25 

28 

49 

46 

55 

79 

153 

206 

218 

268 

305 

347

107 

117 

167 

199 

219 

274 

306 

315 

396 

504

0 

4.7 

4.9 

17 

38 

52 

54 

59 

60 

62 

80 

102 

188 

244 

305 

370 

416 

433

5.2 

5.3 

14 

17 

34 

27 

43

56

68

81

80

93

115

127

142

1 59

5.2 

7.S 

17 

16 

22

99

oc

38

so

104

136

131

134

Period.

Minutes, j

5 .

5

6

5

5

4

4

3

4

390 

360 

60

44 

5 

71 

6 

62 

30 

15 

4 

8 

4 

9 

4 

4 

4 

3 

3 

3 

3

5 

5 

4 

3 

3 

3 

3 

3

9

2

60 

90 

2 

1

1

1

(

3 

2 

2 

2

8 

8 

6 

6 

6 

9 

6 

6 

6 

5 

5 

7 

4 

4 

4 

4

i 

« 

<

£ 

( 

(

Slope.

Water 

surface.

Per cent.

0-030 

.18 

.17

.27

.42 

.43

::::::::::

.1 

.1 

.2 

.36 

.41

. O-l 31

.40

.21 

.42 

) .5[ 

.64

3 ........

1 .. ....

5 ........

7

5 . ..

5

5 ........

Bed. 1

°er cent. 

0.61 

.61 

.97 

1.15 

1.29 

1.87 

2.14 

2.43 

2.63

.044 

.16 

.16 

.28 

.31 

.23 

.33 

.29 

.31 

.43 

.49 

.54 

. .63 

.78 

.65 

1.00 

1.51 

1.66 

1.78 

2.00 

2.14 

2.28

.79 

.83 

1.16 

1.31 

1.29 

1.46 

1.63 

1.65 

1.84 

2.10

.012 

.15 

.18 

.26 

.38 

.48 

.45 

.50 

.53 

.44 

.54 

.68 

.92 

1.16 

1.30 

1.46 

1.72 

1.56

.40 

.45 

.80 

.98 

1.12 

1.18

1.22

1.41

1.66

1.71

1.77

1.88

9 IT

2.33

2.32

.25 

.26 

.43

I .43 

.47

.66

.69

.82

1.07

1.20

1.28

1.33

1. 37

Dis- T 

,ance.

Feet. 

16 

16 

16 

16 

16 

16 

16 

16 

16

60 

60 

60 

48 .

44 

16 

40 

16 

44 

40 

36 

16 

32 

14 

12 

16 

16 

. 16 

16 

16 

16 

16

12 

16 

16 

16 

16 

16 

16 

16 

16 

16

40 

40 

36 

28 

24 

28 

Id 

28 

16 

16 

28 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

12 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

16 

1«ie

1C14 If 1<

15 

1( 

1 

1 

1 

1

Depth.

}y gage.

Feet. 

0.106 .

.102 .

.073 .

.081 .

.073 .

.060 .

.067 .

.066 .

.070 .

.475 .

.344 

.321

.278 

.214 

.224 

.208 

.254 

.188 

.195 

.164 "".'isi'

.150 

.120 

.107 

.129 

.112

.194 

.167 

.157 

.169

.144

.735 

.497 

.496 

.332 

.282 

. .257 

.257 

.257 

.262 

.257 

.250 

.237 

.207

.129

.081 

.061 

.063 

.057 

.056 

.053 

.041 

.051 

.038 

.045 

.058

.037 

.039

.227 

.203 

.153 

.147 

I .131 

.12* 

I .12' 

2 . 10<- 

5 .09! 

5 .09' 

5 .08 

3 .07 

5 .09

By proflies.

Feet.

.323 

.321

.271 

.244

.193 

' .176

..........

=:

.740 

.500 

.497 

.334 

.267 

.256

.241

.126

.22C 

.197 

.IK 

.16C

! ........

t

) ........

J ........

) ........

) ........

2 ........

3 ........

1 

Character of bed.

....do.....................

. .do.....................

....do.....................

....do.....................

....do.....................

. .do.....................

. .do.....................

. .do.....................

....do.....................

Transition. ................

....do.....................

.....do............ . .

.....do............ . ......

.....do.....................

....do.....................

Antidunes. ................ 

.....do.....................

.....do.....................

.....do.....................

Antidunes. ................

.....do.....................

[Antidunes]. ............... 

.....do.....................

.....do....................

Dunes....................

.....do....................

.....do....................

.....do....................

. .....do....................

. .....do....................

.....do....................

.....do....................

. Dunes....................

. .....do....................

. .....do...................

. [Antidunes]..............

Outfall.

Contracted. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do. 

Free. 

Contracted. 

Free. 

Do. 

Contracted. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

. Contracted.

Do.

Do.

Do.

. Free.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.
THE OBSERVATIONS. 41 

TABLE 4 (B) Observations on load, slope, and depth, with debris having 13,400 particles to the gram, or grade (B} Con.

Width.

Feet. 

1.32

1.96

Discharge.

Ft.yscc. 

0.363

.545

.734

.363

.545

.734

Load.

Feed.

Qm.lsec.

30 

40 

50

82 

82

26

8.1

47

80

86 

75

109

7.7 

38 

35 

101 

89

245

Collection.

Gm./sec. 

162 

173 

257 

261

29 

55 

54 

89 

92 

105 

104 

147 

218 

264 

278 

361

 30 

39 

55 

56 

63 

90 

129 

136 

134 

144 

167 

178 

182 

204 . 

255 

268 

280 

341 

342 

383 

485 

516

7.5 

18 

25 

36 

40 

44 

59 

66 

68 

96 

137 

147 

177 

186 

214 

249

53 

59 

70 

72 

90 

89 

94 

106 

144 

217 

264 

299 

307 

303 

356

7.6 

40 

40 

66 

88 

98 

103 

186 

185 

198 

239 

332 

341 

352 

348 

469 

520

Period.

Minutes. 

7 

5 

3 

3

5 

4 

4 

4 

5 

5 

4 

4 

3 

4 

3 

3

5 

7 

6 

6 

5 

5 

6 

6 

5 

4 

4 

5 

' 5 

4 

3 

3 

3 

3 

3 

2 

2 

3

8 

8 

12 

23 

12 

8 

4 

8 

6 

5 

3 

3 

3 

3 

3 

3

23 

6 

5 

18 

3 

5 

4 

4 

4 

4 

3 

3 

3 

3 

3

6 

5 

8 

8 

4 

9 

6 

3 

3 

3 

4 

3 

3 

3 

3 

2 

2

Slope.

Water 

surface.

Per cent.

O.C9 

.57 

.68

.33

.47

.31 

.46

.44

Bed.

Per cent. 

1.54 

1.63 

1.90 

2.12

.47 

.54 

.61 

.76 

.76 

.75 

.84 

.98 

1.28 

1.49 

1.49 

1.85

.31 

.41 

.50 

.50 

.53 

.58 

.87 

.67 

.78 

.85 

.90 

.96 

.92 

1.01 

1.10 

1.14 

1.17 

1.35 

1.45 

1.51 

1.65 

1.66

.31 

.65 

.62 

.59 

.77 

.92 

1.01 

1.03 

1.07 

1.21 

1.45 

1.48 

1.62 

1.64 

1.81 

2.00

.54 

.70 

.69 

.71 

.78 

.80 

.98 

.97 

1.12 

1.39 

1.60 

1.58 

1.54 

1.61 

1.73

.23 

.27 

.44 

.53 

.59 

.64 

. .56 

.80 

1.05 

1.03 

1.05 

1.40 

1.39 

1.38 

1.45 

1.64 

1.87

Distance.

Feet. 

16 

16 

16 

16

12 

14 

14 

16 

16 

14 

16 

16 

16 

16 

16 

16

12 

16 

16 

16 

16 

16 

16 

16 

 16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

14 

14 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

10 

16 

12 

14 

16 

14 

14 

16 

16 

16 

16 

16 

16 

16 

16 

16 

If.

Depth.

By gage.

Feet. 

0.092 

.082

.205 

.162 

.154 

.139 

.141 

.148 

.158 

.141 

.106

.289 

.226 

.219 

.195 

.206 

.199 

.174 

.170 

.172 

.165 

.173 

.169 

.166

.162 

.104 

.110 

.070 

.087 

.080 

.059 

.068 

.078 

.062 

.063 

.044 

.050

.060

.132 

.127 

.130 

.126 

.091 

.111 

.115

.099

.095

.250 

.188 

.171 

.161

.139 

.148

.150

.146 

...... ._

By profiles.

Feet.

0.207 

.165 

.152

.267

.257 

.196

Character of bed.

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

Transition ................. 

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

Transition ................. 

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

[Antidunes] ................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

Antidunes ................. 

[Antidunes] ................

.....do.....................

[Antidunes] ................

[Antidunes] ................ 

Antidunes .................

Outfall.

Free. 

Do. 

Do. 

Do.

Contracted. 

Do, 

Do. 

Do. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.
42 TRANSPORTATION OF DEBRIS BY RUNNING WATER,

TABLE 4 (B). Observations on load, slope, and depth, with debris having 13,400 particles to the gram, or grade (B) Con.

Width.

Feet. 

1.96

Discharge.

Ft.ijsec. 

1.119

Load.

Feed.

Gm.lsec. 

0.5 

14 

29 

53 

112

126

Collection.

Gm.lsec. 

6.4 

13 

32 

59 

109 

110 

129 

191 

230 

317 

310 

356 

355 

406 

418

Period.

Minutes. 

10 

10 

11 

4 

8 

6 

5 

4 

4 

3 

2 

2 

2 

3 

3

Slope.

Water 

surface.

Per cent. 

0.21

.28 

.37

Bed.

Per cent. 

0.19 

.18 

.32 

.28 

.59 

.61 

.65 

.72 

.73 

.91 

1.00 

.93 

.98 

1.13 

1.31

Distance.

Feet. 

16 

16 

16 

12 

16 

16 

12 

16 

16 

16 

16 

16 

16 

16 

16

Depth.

By gage.

Feet. 

0.411 

.353 

.345 

.229 

.315 

.211 

.188 

.188 

.184 

.187 

.176

.145

By profiles.

Feet. 

0.406 

.375 

.337

Character of bed.

[Dunes] ....................

Outfall.

Contracted. 

Do. 

Do. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

TABLE 4 (C). Observations on load, slope, and depth, with debris having 5,460 particles to the gram, or grade (C).

Width.

Feet. 

0.44

.66

Discharge.

Ft.*jsec. 

0.093

.182

.093

.182

.363

Load.

Feed.

Gm.lsec.

2.0

38

Collection.

Gm.lsec. 

3.2 

5.3 

9.4 

12 

19 

26 

35 

34 

46

9.7 

14 

38 

43 

47 

51 

97

3.5 

8.6 

10 

15 

15 

17 

16 

22 

35 

48 

61 

74 

100 

109 

142 

146 

156

2.3 

9.7 

16 

18 

18 

28 

' 39 

45 

72 

78 

105 

113 

123 

133 

146 

155 

163 

190 

205 

220 

217 

246

22 

32

, 34 

37

Period.

Minutes. 

7 

5 

5 

5 

7 

4 

5 

4 

4

6 

6 

6 

5 

4 

5 

3

10 

13 

7 

6 

6 

6 

6 

6 

5 

4 

4 

3 

3 

5 

3 

3 

3

420 

11

8

7 

7 

6 

5 

7 

5 

8 

4 

4 

4 

4 

3 

4 

3 

4 

3 

3 

3 

3

5 

5 

 6 

4

Slope.

Water 

surface.

Per cent.

0.29

1.20

1.69

Bed.

Per cent. 

0.64 

.85 

.94 

1.01 

1.26 

1.45 

1.59 

1.71 

2.24

.61 

.70 

1.33 

1.32 

1.38 

1.52 

2.16

.54 

.79 

.98 

.97 

1.05 

1.05 

1.11 

1.36 

1.56 

1.98 

2.38 

2.52 

3.02 

3.41 

3.73 

3.79 

4.04

.24 

.54 

.64 

.76 

.72 

.94 

1.16 

1.17 

1.54 

1.57 

1.95 

2.24 

2.32 

2.33 

2.58 

2.54 

2.63 

2.82 

2.99 

3.05 

3.20 

3.45

.44 

.64 

.70 

.61

Distance.

Feet. 

16 

16 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

64 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16

Depth.

By gage..

Feet. 

0. 161 

.141 

.131 

.122 

.117 

.111 

.111 

.100

.235 

.233 

.168 

.176 

.172 

.153 

.138

.108 

.084 

.077 

.078 

.075 

.077 

.073 

.066 

.063 

.055 

.050 

.052

.242 

.136 

.124 

.132 

.127 

.117 

.096 

.099

.089

.062

.241 

.222 

.211 

.213

Bv profiles.

Feet.

0.238

.110

.093

Character of bed.

....do.....................

....do.....................

....do.....................

....do.....................

....do.....................

....do.....................

....do.....................

....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

..:.. do. ....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

[Smooth]...................

.....do....................

.....do....................

.....do....................

.....do....................

.....do....................

.....do....................

.....do....................

.....do....................

.....do....................

.....do....................

(Transition! ...............

Outfall.

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Free. 

Do. 

Do. 

Contracted.
THE OBSERVATIONS. 43 

TABLE 4 (C). Observations on load, slope, and depth, with debris having 5,460 particles to the gram, or grade (C) Con.

Width.

Feet. 

0.60

1.00

Discharge.

Ft.ysec. 

0.363

.545

.734

.182

.363

Load.

Food.

Om.lsec. 

35 

37

82 

89 

92

2.8

44 

46 

46 

45

111 

119 

117

.67 

67 

67 

152 

164

28 

39 

53 

53 

79

105

1.9 

6.5 

19

42

46 

43

85 

75 

85 

75

140 

158

Collection.

Gm.jsec. 

35 

40 

39 

54 

78 

83 

98 

99 

97 

121 

121 

134 

161 

182 183 ' 

228 

241 

245 

240 

281 

292

2.8 

36 

45 

48 

49 

47 

41 

47 

53 

76 

88 

117 

115 

121 

130 

151 

152 

162 

196 

239 

241 

252 

275 

346

61 

64 

67 

168 

163

7.6 

11 

16 

17 

18 

26 

41 

54 

56 

70 

84 

97 

106 

114 

128 

131 

145 

166

1.9 

5.1

18 

24 

30 

39 

46 

48 

48 

58 

61 

83 

81 

86 

84 

86 

104 

104 

124 

131 

147

Period.

Minutes. 

4 

4 

5 

5 

6 

5 

3 

3 

3 

5 

4 

4 

3 

3 

3 

3 

3 

3 

3 

3 

3

410 

7 

4 

4 

4 

4 

7 

8 

8 

4 

3 

3 

3 

3 

3 

5 

4 

5 

4 

4 

3 

3 

3 

3

4 

4 

4 

3 

3

9 

12 

9 

5 

5 

10 

8 

6 

11 

3 

6 

4 

4 

4 

4 

4 

3 

3

94 

41 

10 

8 

6 

7 

6 

4 

4 

6 

5 

8 

3 

9 

3 

5 

5 

5 

3 

3 

5

Slope.

Water 

surface.

Per cent.

0.15

.17

.26 

.42

.61

Bed.

Per cent. 

0.63 

.66 

.67 

.87 

.99 

1.04 

1.08 

1.07 

1.08 

1.19 

1-.23 

1.41 

1.71 

1.87 

1.91 

1.97 

2.04 

2.08 

2.11 

2.29 

2.57

.20 

.54 

.51 

.56 

.57 

.58 

.60 

.60 

.57 

.78 

.82 

.96 

.98 

.99 

1.01 

1.12 

1.12 

1.21 

1.53 

1.56 

1.64 

1.69 

1.73 

2.04

.62 

.62 

.58 

.99 

1.04

.54 

.57 

.73 

.75 

.74 

.91 

1.10 

1.31 

1.32 

1.52 

1.68 

1.85 

1.96 

2.07 

2.24 

2.32 

2.48 

2.66

.19 

.24 

.42 

.44 

.52 

.65 

.73 

.70 

.73 

.82 

.83 

.95 

.97 

.93 

.96 

.97 

1.09 

1.09 

1.25 

1.26 

1.38

Distance.

Feet. 

16 

16 

16 

16 

14 

16 

14 

16 

16 

14 

14 

16 

16 

16 

16 

14 

16 

16 

16 

16 

16

56 

16 

16 

14 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

14 

14 

16

14 

14 

16 

14 

16

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

48 

48 

48 

16 

16 

40 

16 

16 

16 

16 

16 

32 

16 

32 

16 

16 

16 

 16 

16 

16 

16

Depth.

By gage.

Feet. 

0.216 

.212 

.205 

.186 

.174 

.167 

.168 

.173 

.164 

.174 

.168 

.161

.165

.579 

.312 

.288 

.284 

.278 

.288 

.303 

.308 

.300 

.255 

.260 

.223 

.223 

.228 

.242 

.226 

.239 

.222 

.234

.327 

.318 

.320 

.263 

.277

.117 

.116 

.110 

.107 

.108

.065 

.062

.051 

.065

.322 

.266 

.209 

.187 

.164 

.168 

.142 

.143 

.141 

.144 

.133 

.135

.123

.133 

.121 

.133 

.114 

.114 

.115

By profiles.

Feet.

0.577

.326 

.275 

.221

.170

Character of bed.

.....do.....................

.....do.....................

Transition. ................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

B

hmes]. ...................

[Transition] ................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

[Transition]................

.....do.....................

.....do.....................

[Smooth]. ..................

Smooth....................

[Antidunes] ................

Antidunes .................

[ Antidunes] ................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

[Antidunes] ................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

Smooth....................

Outfall.

Contracted. 

Do. 

Free. 

Do. 

Do. 

Do. 

Contracted. 

Do. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Free. 

Contracted. 

Do. 

Do. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Contracted. 

Do. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do.

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Free. 

Do. 

Contracted. 

Free. 

Contracted. 

Do. 

Free. 

Do. 

Contracted. 

Free. 

Contracted. 

Free. 

Do. 

Do. 

Do. 

Contracted. 

Do. 

Free.
44 TRANSPORTATION OF DEBRIS BY RUNNING WATER, 

TABLE 4 (C). Observations on load, slope, and depth, with debris having 5,460 particles to the gram, or grade (C) Con.

Width.

Feet. 

1.00

1.32

Discharge.

FtJ/sec. 

0.363

.545

.734 

1.119

.182

.363

Load.

Feed.

Om.lsec.

60 

61

146

iss

156

1.9 

2.6 

2.6

13

42

74 

76 

103

160

111 

114 

325 

325

11

7.8 

26

Collection.

Gm.lsec. 

166 

233 

257 

256 

255 

349 

362

48 

57 

68 

64 

71 

107 

107 

150 

153 

165 

169 

173 

181 

205 

266 

315 

292 

296 

281 

355 

391 

398 

431

1.6 

2.1 

3.0 

13 

15 

14 

39 

84 

85 

117 

77 

95 

132 

194 

181 

230 

267 

308 

309 

329 

444 

501 

118 

129 

348 

370

9.6 

7.8 

13 

21 

30 

43 

58 

99 

120 

126 

130 

141

8.8 

10 

26 

 27 

38 

51 

60 

85 

111 

118 

130 

129 

154 

158 

190 

237 

264 

266 

300 

308

Period.

Minutes. 

5 

4 

3 

3 

4 

3 

3

5 

5 

4 

4 

5 

5 

5 

3 

4 

4 

5 

' 3 

3 

4 

4 

3 

4 

4 

3 

3 

3 

3 

3

60 

170 

180 

12 

20 

12 

8 

4 

4 

5 

4 

6 

4 

3 

6 

4 

4 

4 

3 

3 

3 

3 

3 

3 

2 

2

10 

9 

6 

5 

6 

5 

5 

5 

5 

4 

4 

3

6 

6 

6 

6 

5 

5 

5 

5 

6 

5 

4 

4 

4 

4 

3 

3 

3 

3 

3 

3

Slope.

Water 

surface.

Per cent.

0.11 

.11 

.16 

.21 

.17 

.26

.66

.33 

.31

.55

Bed.

Per cent. 

1.50 

1.99 

1.98 

2.10 

2.11 

2.48 

2.65

.51 

.53 

.61 

.62 

.62 

.83 

.88 

1.02 

1.05 

1.09 

1.00 

1.12 

1.12 

1.28 

1.50 

1.60 

1.70 

1.75 

1.87 

1.95 

1.94 

1.97 

2.17

.13 

.16 

.14 

.21 

.27 

.33 

.34 

.58 

.58 

.57 

.61 

.59 

.81 

.99 

1.01 

1.15 

1.29 

1.43 

1.50 

1.51 

1.78 

2.07 

.57 

.66 

1.17 

1.22

.66 

.70 

.84 

.98 

1.14 

1.24 

1.64 

2.08 

2.23 

2.10 

2.29 

2.34

.32 

.35 

.56 

.56 

.72 

.79 

.89 

1.07 

1.23 

1.27 

1.32 

1.33 

1.46 

1.44 

1.80 

1.92 

2.14 

2.18 

2.30 

2.46

Distance.

Feet. 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

14 

16 

16 

16 

14 

14 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

48 

32 

40 

56 

40 

76 

36 

16 

14 

36 

14 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

14 

14 

16 

16

16 

12 

14 

16 

16 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

16 

14 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

Depth.

By gage.

Feet.

0.211 

.206 

.193 

.189 

.193 

.176 

.175 

.167 

.173 

.163 

.160 

.166 

.160 

.148 

.136

.581

.459 

.420 

.322 

.232 

.241 

.261 

.249 

.233 

.221 

.194 

.203 

.187 

.199

.312 

.305 

.256 

.253

.093 

.100 

.090 

.076 

.067 

.067 

.055 

.040 

.049 

.056 

.051 

.045

.163 

.171 

.140 

.127 

.123 

.122 

.120 

.106 

.104 

.100 

.097 

.085 

.090 

.089

By profiles.

Feet.

0.579 

.570

.537

.467 

.413

.099

.174 

.170 

.178

Character of bed.

.....do.....................

.....do......... ...........

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do......................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

Outfall.

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Free. 

Do. 

Contracted. 

Do. 

Free. 

Do. 

Do. 

Contracted. 

Free. 

Do. 

Do. 

Contracted. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Free. 

Contracted. 

Free. 

Do. 

Contracted. 

Do. 

Do. 

Free. 

Do. 

Do. 

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do. 

Free. 

Do. 

Do. 

Contracted. 

Do. 

Do. 

Do.

Contracted. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.
THE OBSERVATIONS. 45 

TABLE 4 (C). Observations on load, slope, and depth, with debris having 5.460 particles to the gram, or grade (C) Con.

Width.

Feet. 

1.32

1.96

Discharge.

Ft.ysec. 

0.545

.734

.363

.545

.734

1.119

Load.

Feed.

Gm.jscc.

4.3

11

25

36 

36

70 

68

121

117 

130 

133

129 

132

Collection.

Om. /sec. 

36 

55 

79 

96 

118 

159 

210 

' 248 

256 

340 

362 

407 

394

4.7 

14 

29 

53 

58 

74 

105 

113 

120 

159 

227 

252 

309 

368 

405 

432 

456

9.5 

16 

26 

40 

60 

77 

90 

124 

163 

175 

203 

227 

227

30 

55 

79 

101 

138 

165 

210 

241 

266 

321 

331 

355 

386

32 

41 

49 

55 

63 

73 

72 

87 

101 

96 

166 

241 

259 

302 

334 

350 

362 

368

90 

112 

131 

138 

140 

149 

158 

215 

253 

331 

360 

397

Period.

Minutes. 

6 

6 

5 

5 

5 

5 

5 

4 

. 3 

3 

3 

3 

3

120

6

5

4 

4 

4 

3 

3 

3 

3 

3

10 

6 

10

7 

4 

4 

4 

4 

4 

4 

4 

4 

4

8 

5 

5 

6 

5 

4 

4 

4 

4 

3 

3 

3 

3

8 

10 

5 

5 

6 

7 

5 

4 

6 

5 

3 

4 

4 

3 

3 

3 

3 

3

5 

6 

4 

6 

5 

5 

4 

5 

3 

3 

3 

3

Slope.

Water 

surface.

Per cent

0.18

.45

.73

.42 

.54

.53 

.46

.67

.68 

.60 

.65

.79 

.79

Bed.

Per cent 

0.56 

.62 

.69 

.83 

.85 

1.10 

1.37 

1.57 

1.58 

1.89 

1.95 

2.08 

2.20

.16 

o.20

o.35

o.Sl

.50 

o.55

.65 

a .75

o.73

.87 

1.05 

1.15 

1.37 

1.62 

1.72 

1.82 

1.90

.35 

.59 

.71 

.90 

.99 

1.09 

1.21 

1.41 

1.63 

1.67 

1.85 

1.93 

2.03

.56 

.70 

.79 

.92 

1.07 

1.10 

1.34 

1.38 

1.58 

1.85 

1.85 

1.96 

2.06

.45 

.41 

.37 

.49 

.60 

.58 

.61 

.60 

.70 

.75 

.92 

1.17 

1.22 

1.31 

1.53 

1.49 

1.45 

1.50

.53 

.61 

.64 

.56 

.65 

.69 

.71 

.78 

.78 

.97 

1.03 

1.12

Distance.

Feet. 

12 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

48

14

16

16 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

- 16 

16 

16 

16 

16 

16

16 

18 

16 

12 

16 

16 

16 

16 

16 

16 

14 

16 

16 

16 

16 

14 

16 

16

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

Depth.

By gage

Feet. 

0.186 

.172 

.152 

.152 

.141 

.132 

.118

.409 

.324

.292

.210

.214 

.189

.186 

.117

.183

.167 

.163 

.149 

.143

.128 

.106 

.105 

.100 

.089 

.080 

.078 

.067 

.063 

.064 

.067 

.061 

.060

.148 

.127 

.113 

.115 

.100 

.098 

.094 

.094 

.080 

.071

.205 

.194 

.177 

.171 

.149 

.165 

.155 

.155 

.120 

.147 

.139 

.117 

.113 

.106 

.105

.222 

.188 

.184 

.180 

.196 

.186 

.189 

.178 

.171 

.166 

.152

By profiles.

Feet.

0.416

.141

.113

.195 

.209

.161 

.175

.145

' . 205 

.202 

.180

.199 

.192

Character of bed.

.....do.....................

.....do.....................

Smooth...................

.....do.....................

.....do.....................

.....do.....................

[Dunes] ....................

.....do.....................

.....do.....................

.....do.....................

.....do......................

.....do............. ........

:....do............. ........

Transition . . ........

.....do......................

.....do............. ........

Smooth........... ........

.....do......................

.....do.......... ........

.....do............. ........

.....do............ ........

.....do......................

Dunes. ....... ........

.....do............. ........

.....do......................

Smooth......... ........

....do............. ........

....do......................

....do......................

Antidunes.. ........

....do............. ........

....do............. ........

[Dunes] ....... ........

Dunes............. ........

....do............. ........

....do............. ........

Dunes...... ......

....do......................

....do............. ........

[Antidunes] ................

....do............. ........

....do......................

Smooth....................

....do......................

Smooth ........... ........

....do......................

....do......................

....do......................

....do......................

....do......................

Outfall.

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Free. 

Contracted. 

Free. 

Contracted. 

Do. 

Free. 

Do.

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Free. 

Contracted. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Free. 

Do. 

Contracted. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Free. 

Contracted. 

Do. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

« Computed graphically from data hi a notebook afterward lost.
46 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 4 (D). Observations on load, slope, and depth, ivith debris having 1,460 particles to the gram, or grade (D).

Width.

Feet. 

0.66

1.00

Discharge.

Ft.3 l sec. 

0.093

.182

.545

.182

.363

.545

Load.

Feed.

Gm.jsec.

6.4

8.9 

8.9 

18 

32 

33

0 

1.8 

1.8 

5.9 

12 

12 

23

30 

32

Collection.

Gm.lsec. 

5.3 

8.1 

13 

20 

21 

29 

36 

45 

47 

62

5.8 

14 

17 

23 

30 

47 

53 

83 

108 

121 

127 

134

6.9 

9.5 

17 

27 

37 

64 

92 

145 

172 

203 

209 

218 

231 

331 

310

9.5 

16 

22 

27 

28 

48 

49 

56 

80 

98 

109 

126 

152

0 

2.3 

2.0 

5.4 

12 

12 

23 

28 

35 

86 

112 

130 

141 

170 

181 

229 

258

30 

31 

66 

143 

168 

229 

256 

281 

310 

340

Period .

Minutes 

6 

6 

5 

4 

5 

4 

6 

4 

4 

4

7 

6 

6 

6 

5 

4 

4 

8 

9 

4 

4 

4

6 

5 

5 

5 

4 

5 

4 

4 

4 

4 

4 

3 

3 

3 

3

7 

5 

5 

5 

4 

4 

4 

4 

4 

4 

4 

4 

3

81 

66 

20 

22 

26 

12 

6 

5 

4 

4 

4 

3 

3 

4 

4 

3

4 

4 

4 

4 

4 

4 

3 

3 

3 

3

Slope.

Water 

surface.

Per cent.

.30 

.28 

.61 

.61

.037

.16 

.19 

32 

.27 

.56 

.53

1.85 

2.00

.53

Bed.

Per cent. 

0.80 

.94 

1.19 

1.39 

1.68 

1.83 

1.88 

1.98 

2.25 

2.47

.39 

.77 

.82 

.95 

1.10 

1.26 

1.40 

1.95 

2.10 

2.26 

2.28 

2.39

.19 

.20 

.51 

.57 

.64 

.81 

.96 

1.25 

1.42 

1.55 

1.58 

1.57 

1.65 

1.98 

2.02

.69 

.84 

1.06 

1.08 

1.14 

1.34 

1.39 

1.57 

1.83 

2.09 

2.33 

2.55 

2.75

.18 

.18 

.25 

.35 

.43 

.58 

.65 

.80 

1.14 

1.32 

> 1.44 

1.49 

1.65 

1.75 

1.94 

2.11

.53 

.55 

.78 

1.11 

1.26 

1.58 

1.61 

1.62 

1.74 

1.91

Distance.

Feet. 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

16 

 16 

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

16 

16 

14 

16 

16 

12 

16 

16 

16 

12 

12 

12 

14 

12 

12

14 

16 

14 

16 

16 

16 

16 

16 

16 

16 

14 

14 

16

40 

28 

24 

24 

20 

20 

24 

16 

16 

16 

16 

14 

16 

14 

16 

14 

14

16 

16 

16 

16 

16 

16 

16 

14 

14 

14

Depth.

By gage.

Feet. 

0.077 

.077 

.075 

.076 

.063 

.058 

.061 

.063 

.056 

.056

.166 

.147 

.135 

.123 

.105 

.108 

.108 

.090 

.090 

090 

.093 

.090

.460 

.422 

.403 

.318 

.316 

.267 

.237 

217 

.211 

.202 

.209 

.191 

.177 

.196

.110 

.100 

.094 

.099 

.088 

.086 

.082 

.071 

.066 

.070 

.066 

.062 

.064

.418 

.292 

.283 

.245 

.224 

.235 

.209 

.170 

.164 

.136 

.120 

.116 

.117 

.115 

.103 

.104 

.101

.236 

.241 

.187 

.166 

.162 

.136 

.136 

.140 

.136 

.136

By proflies.

Feel.

 0. 464 

.425 

.382 

.322

Character of bed.

Dunes..................... 

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do......................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

..do ... . ..........

do .. .........

.289 

.286 

.252 

.224 

.229 

.199

.099 

.098

.240

C.

.....do......................

Transition ................. 

.....do.....................

.....do......................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do......................

.....do.....................

.....do.....................

Transition ................

.....do....................

.....do....................

.....do....................

.....do....................

.....do....................

.....do....................

.....do....................

.....do....................

.....do....................

.....do....................

.....do....................

.....do....................

.....do....................

Outfall.

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Free. 

Do. 

Do 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do 

Do 

Do 

Do 

Do 

Do 

Do

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.
THE OBSEKVATIONS. . 47 

TABLE 4 (D). Observations on load, slope, and depth, ivith debris having 1,460 particles to the gram, or grade (D) Con.

Vidth.

Feet. 

1.00

1.32

Discharge.

Ft.ysec. 

0.734

.363

.734

Load.

Feed.

Gm./sec. 

0 

5.8 

19 

20 

49

83

8.9 

10 

25 

59 

60 

100 

102 

154 

166 

169 

195 

202

17 

22 

48 

50 

53 

51 

65 

68 

86 

90 

99 

172 

202 

192 

192

Collection.

Gm./sec. 

0 

6.5 

12 

21 

56 

66 

82 

108 

170 

189 

193 

265 

293 

354 

377

11 

11 

23

53 

57 

89 

97 

135 

136 

161 

236 

247

15 

25 

43 

49 

47 

60 

72 

59 

72 

83 

84 

170 

174 

209 

200

Period.

Minutes.

29 

18 

15 

4 

4 

3 

4 

3 

4 

4 

3 

3 

3 

3

10 

10 

8 

5 

2 

4 

3 

3 

3 

3 

3 

3

10 

10 

10 

8 

8 

5 

5 

5 

5 

5 

4 

3 

3 

3 

3

Slope.

Water 

surface.

Per cent. 

0.085

.18 

.22 

.26

.50 

.40 

.46

.28 

.33 

.60 

.51 

.64

.75 

.66

1.28 

1.00 

.97

Bed.

Per cent.

0.21 

.32 

.37 

.61 

.61 

.75 

.83 

.98 

1.09 

1.19 

1.39 

1.46 

1.73 

1.76

.30 

.34 

/ .67 

1.00 

1.05 

1.25 

,1.25 

! 1. 54 

1.57 

1.67 

1.89 

1.91

.34 

.37 

.52 

.58 

.61 

.62 

.55 

.70 

.79 

.67 

1.15 

1.00 

.98 

1.15 

1,17

Distance.

Feet. 

40 

36 

24 

36 

16 

12 

16 

14 

14 

16 

16 

16 

16 

16 

14

16 

16 

16 

16 

16 

16 

16 

16 

14 

14 

16 

16

16 

16 

14 

16 

16 

14 

16 

16 

16 

16 

16 

12 

14 

12 

12

Depth.

By gage.

Feet.

0.491 

.423 

.413 

.285 

.250 

.241 

.220 

.206 

.198 

.195 

.189 

.185 

.175 

.162

.137 

.171 

.137 

..124 

'.129 

.107 

.111 

.090 

.094 

.089 

.088 

.087

.360 

. .272 

.243 

.224 

.233 

.202 

.201 

.206 

.184 

.181

.154 

.160 

.156 

.152

By profiles.

Feet.

0.482 

.411 

.410

.140 

.171 

.154

.354 

. .304 

.227 

.248 

.231

.228 

.192

.164 

.168

Character of bed.

Dunes forming. . ...........

.....do.....................

.....do.....................

.....do.....................

Smooth ....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

[Dunes]....................

.....do.....................

Dunes .....................

.....do.....................

(Smooth]...................

.....do.....................

.....do.....................

Dunes.....................

.....do.....................

.....do.....................

Dunes.....................

[Smooth] ...................

Smooth ....................

....do.....................

....do.....................

....do.....................

Outfall.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Free. 

Contracted. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Free. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.
48 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 4 (E). Observations on load, slope, and depth, with debris having 142 particles to the gram, or grade (E).

Width.

Feet. 

0.60

1 00

1.32

Discharge.

Ft?lsec. 

0.182

.363

.734

1.119

.182

.363

734

1.119

.363

.734

1.119

Feed.

Gm.lsec.

95

95

31

31

31

38

38

48

48

48

48

4R

96

96

50

91

91

101

101

101

101

193

193

193

6.4

19

19

47

50

50

78

2.1

2.1

6.7

6.6

24

50

95

142

2.1

6.9

6.9

21

21

24

41

41

47

87

95

142

142

203

47

47

189

189

20

20

86

33

86

86

50

50

172

172

Load.

Collection.

Gm.lsec.

91

21

2831

3n

31

31

42

46

47

4750

95

97

44

80

85

94

87

94

95

167

184

186

5.2

19

19

44

44

50

75

Tr.

2.9

2.2

. 7.7

7.9

22

53

104

168

Tr.

1.7

3.6

4.5

10

20

20

26

44

47

45

86

100

135

150

220

50

50

170

190

20

20

93

33

85

86

46

50

163

171

Period.

Minutes. 

15

19

15

13

9

9

9

10

10

6

6

4

4

6

3

3

3

3

3

3

3

3

3

21

10

16

8

8

9

5

61

CO

20

20

34

9

5

4

70

63si

25

8

18

16

10

6

8

6

4

3

3

2

6

8

3

3

15

10

5

9

4

3

6

4

3

3

Water 

surface.

Per cent.

0.065

.11

.23

.46

.39

.80

1.24

1.92

2.26

r\A

.07

.17

.31

.31

.43

.61

.82

1.17

1.50

1.47

1.80

Slope.

Bed.

Per cent. 

1.20

1.27

1.43

1 48

1.56

1 fU

1 04

1.27

1.26

1 9Q

1 03

1.09

1.43

1.46

.56

1 93

1 31

1.11

1.24

1.31

1 90

1.38

1.28

1.41

.59

1.11

1.12

1.80

1.80

1 <U

2.42

.22

95

44

.44

.87

1.28

1.83

2.29

.19

.18

.29

.26

.46

.48

.47

.73

.74

.83

1.19

1.23

1.46

1.50

1.61

.52

.59

1.15

1.18

.67

.73

1.78

.60

1.10

1.12

.58

.62

1.13

1.22

Distance.

Feet. 

16

16

16

16

16

16

16

16

. 16

16

16

16

16

16

14

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

32

32

48

48

48

32

16

16

16

16

32

32

48

48

32

40

28

16

16

36

36

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

Dep

By gage.

Feet. 

0.125

.117

.115

.115

.114

.200

.197

.200

.185

.372

.387

.562

.094

.087

.081

.077

.093

.357

.314

.241

.242

.218

.212

.161

.113

.110

.618

.562

.447

.442

.411

.403

.364

.303

.301

.324

.326

.273

.236

.223

.187

.189

.380

.406

.131

129

.097

.239

.198

.211

.324

.317

th.

By proflies.

Feet.

0.277

.215

.208

.167

.115

.459

.403

.407

.363

Character of bed.

smooth).

.....do......................

....do......................

....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do.....................

Outfall.

Contracted.

Do.

Do.

Do.

Do.

Contracted.

Do.

Do.

Do.

Contracted.

Do.

Do.

Do.

Contracted.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Contracted.(?)

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Free.

Contracted.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do.

 Do.

Do.

Do.

Do.

Do.
THE OBSERVATIONS. 49 

TABLE 4 (F). Observations on load, slope, and depth, with debris having 22.1 particles to the gram, or grade (F).

Width.

Feet. 

0.66

1.00

1.32

Discharge.

Ft?jsec. 

0.182

.363

.734

.182

.303

.734

1.119

.363

.734

1.119

Load.

Feed.

Gm.lsec.

11

58 

58

26 

26 

71 

71

35 

35 

106 

106

6.8 

41 

41

10 

10 

51 

51 

51 

120

26 

26 

104 

104

52 

52 

207 

207

21 

21

70 

. 70

26 

26 

26 

104 

104

58 

58 

58 

212 

212

Collection.

Gm.lsec. 

11 

12 

52 

53

26 

26 

80 

71

37 

38 

103 

108

6.8 

42 

44

9.5 

10 

47 

51 

56 

120

25 

25 

98 

114

53 

56 

197 

209

21 

21 

68 

72

26 

27 

27 

102 

106

50 

50 

59

188 

208

Period.

Minutes. 

17 

23 

11 

8

17 

12 

6 

10

9 

6 

3 

3

21

7 

7

18 

20 

14 

10 

8 

3

14

12 

5 

5

5 

5 

3 

3

7 

21 

9 

5

13 

12 

12 

3 

4

12 

9 

6 

2

2

Slope.

Water 

surface.

Per cent.

Bed.

Per cent. 

1.29 

1.31 

2.51 

2.50

1.12 

1.13 

1.89 

1.96

.97 

1.00 

1.68 

1.75

1.36 

2.49 

2.53

.85 

.91 

1.65 

1.70 

1.68 

2.47

.77 

.77 

1.52 

1.60

.80 

.85 

1.65 

1.67

1.16 

1.21 

2.05 

2.07

.83 

.85 

.86 

1.48 

1.58

.74 

.78 

.84 

1.55 

1.60

Distance.

Feet. 

16 

16 

16 

16

16 

16 

16 

16

16 

16 

16 

16

16 

16 

16

16 

16 

16 

16 

16 

16

16 

16 

16 

16

16 

16 

16 

16

16 

16 

16 

16

16 

16 

16 

16 

16

16 

16 

16 

16 

16

Depth.

By gage.

Feet. 

0.118 

.129 

.102 

.106

.207 

.204 

.176 

.170

.346 

.340

.090 

.078 

.080

.160 

.161 

.128 

.130 

.140

.268 

.265 

.217 

.211

.330 

.343

.116 

.118 

.114 

.108

.212 

.209 

.215 

.176 

.180

.275 

.288 

.284

By profiles.

Feet.

Character of bed.

.....do.....................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

.....do......................

....do.......................

Outfall.

Contracted. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do.

Free. (?) 

Do. 

Do.

Free. (?) 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do.

Free. (?) 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do.

Contracted.

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do.

TABLE 4 (G). Observations on load, slope, and depth, with debris having 5.9 particles to the gram, or grade (G).

Width.

Feet. 

0.66

Discharge.

Ftf/sec. 

0.363

.734

1.119

Load.

Feed.

Gm.lsec. 

10 

25 

25 

51 

50 

100 

100

10 

10 

25 

50 

50 

105 

203 

210

10 

50 

50 

100 

100 

203 

203 

304 

304

Collection.

Gm.lsec. 

11 

25 

28 

47 

50 

100 

105

8.5 

11 

26 

48 

49 

104 

210 

219

12 

49 

54 

94 

114 

212 

215 

311 

331

Period.

Minutes. 

15 

12 

15 

6 

8 

3 

3

14 

23 

11 

7 

11 

5 

3 

3

18 

6 

6 

4 

4 

3 

3 

3 

3

Slope.

Water 

surface.

Per cent.

Bed.

Per cent. 

1.11 

1.44 

1.48 

1.82 

1.90 

2.56 

2.70

.68 

.70 

.95 

1.19 

1.19 

1.71 

2.43 

2.35

.62 

.98 

1.02 

1.35 

1.32 

1.97 

1.95 

2.40 

2.36

Distance.

Feet. 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

14

16 

16 

16 

16 

16 

16 

16 

16 

16

Depth.

By gage.

Feet. 

0.198 

.186 

.192 

.175 

.175 

.160 

.158

.373 

.364 

.342 

.322 

.324 

.292 

.261 

.264

.558 

.460 

.451 

.414 

.411 

.374 

.378 

.354

By profiles.

Feet.

Character of bed.

-

Outfall.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

20921° No. 86 14-
50 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 4 (G). Observations on load, slope, and depth, with debris having 5.9 particles to the gram, or grade (G} Con.

Width.

Feet. 

1.00

1.32

Discharge.

Ft.ysec. 

0.363

.734

1.119

.363

.734

1.119

Load.

Feed.

Gm./sec. 

10 

20 

25 

25 

34 

51 

102

10 

20 

25 

25 

50 

101 

201 

201

10 

25 

25 

50 

50 

101 

102 

204 

306 

306

25 

25 

51 

51 

98 

98

10 

10 

25 

25 

51 

102 

203

10 

51 

51 

98 

98 

197 

295 

304

Collection.

Om./sec. 

10 

21 

25 

28 

34 

50 

97

11 

20 

25 

26 

49 

100 

189 

212

10 

24 

25 

53 

54 

100 

103 

214 

297 

310

24 

26 

49 

47 

95 

123

9 

12 

24 

27 

52 

102 

200

14 

46 

67 

95 

110 

200 

282 

302

Period.

Minutes. 

20 

15 

12 

18 

12 

9 

5

42 

15 

15 

11 

10 

5 

4 

3

20 

15 

17 

6 

7 

4 

4 

3 

3 

3

12 

9

 10

10 

3 

4

21 

18 

13 

13 

6 

4 

3

19 

4 

6 

3 

4 

3 

3 

3

Slope.

Water 

surface.

Per cent.

Bed.

Per cent. 

1.27 

1.48 

1.61 

1.62 

1.76 

2.09 

2.74

.78 

.86 

.95 

.97 

1.27 

1.69 

2.30 

2.37

.64 

.66 

.67 

.97 

.90 

1.22 

1.31

1.78 

2.04 

2.21

1.90 

1.97 

2.25 

2.34 

3.02 

3.10

.82 

.82 

1.08 

1.14 

1.41 

1.82 

2.44

.71 

.90

1.07 

1.31 

1.40 

1.83 

2.17 

2.26

Distance.

Feet. 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

16 

14 

14

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

16

Depth.

By gage.

Feet. 

0 143 

.139 

.136 

.141 

.132 

.129 

.114

.272 

.248 

.248 

.251 

.235 

.214 

.190 

.191

.389 

.357 

.359 

.324 

.326 

.308 

.307 

.275 

.252 

.253

.115 

.104 

.108 

.105 

.093 

.097

.210 

.210 

.194 

.200 

.193 

.171 

.163

.297 

.270 

.261 

.248 

.245 

. .230 

.201 

.202

By profiles.

Feet.

..........

Character of bed.

.....do.....................

.....do.....................

.....do.....................

[Smooth], ..................

Outfall.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

TABLE 4 (H). Observations on load, slope, and depth, with debris having 2.0 particles to the gram, or grade (H).

Width.

Feet. 

0.66

Discharge.

Ft.3lsec. 

0.363

.734

1.119

Load.

Feed.

Qm./sec. 

10 

10 

21 

21 

52 

52

10 

10 

21 

21 

52 

52 

105 

209 

209

10 

10 

10 

26 

52 

52 

52 

105 

105 

105 

209 

209

Collection.

Gm.lsec. 

9.2 

12 

17 

22 

51 

56

7.9 

11 

20 

22 

51 

52 

105 

209 

222

12 

10 

12 

26 

53 

54 

53 

98 

106 

113 

209 

209

Period.

Minutes. 

21 

16 

18 

14 

5 

6

18 

17 

17 

15 

8 

8 

4 

3 

3

16 

20 

21 

15 

13 

10 

9 

3 

6 

3 

3 

3

Slope.

Water 

surface.

Per cent.

Bed.

Per cent. 

1.49 

1.58 

1.80 

1.84 

2.43 

2.47_

.90 

.95 

1.10 

1.19 

1.50 

1.51 

2.02 

2.69 

2.92

.74 

.81 

.89 

1.03 

1.26 

1.28 

1.33 

1.65 

1.63 

1.62 

2.31 

2.38

Distance.

Feet. 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

14 

16 

16 

16 

16

Depth.

By gage.

Feet. 

0.184 

.184 

.183 

.173 

.167 

.173

.345 

.348 

.333. 

.344 

.320 

.310 

.289 

.250 

.253

.502 

.510 

.503 

.470 

.442 

.437 

.447 

.391 

.398 

.392 

.340 

.334

By proflies.

Feet.

Character of bed. Outfall.

Contracted. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.
THE OBSERVATIONS. 51

TABLE 4 (I). Special group of observations on load, slope, and depth, with ddbris of grade (C); for discussion of form ratio.

Width.

Feet. 

1.00

1.20

1.40

1.60

1.80

Discharge.

lWlsec. 

0.734

.923

1.119

.734

.923

1.021

1.119

.734

.923

1.021

1.119

.734

.923

1.021

1.119

.734

.923

1. 021

1.119

Load.

Feed.

Gm./sec. 

74 

76 

160

96 

94 

254

111

114

66 

67 

193 

199

99 

249 

246

105 

110 

304

108 

111 

322 

339

76 

76 

220 

216

72 

82 

271 

283

82 

292 

289

105 

123 

292 

284 

292

69 

265 

269

67 

69 

67 

298 

289

67 

63 

66 

310

82 

96 

85 

336

69 

70 

211 

205

70 

67. 

260 

260

70 

69 

70 

292 

296

64 

75 

386 

363

Collection.

Gm.lsec. 

84 

45 

194

103 

103 

280

118 

129

74 

74 

218 

223

104 

266 

274

107 

109 

325

114 

123 

292 

326

74 

75 

215 

226

79 

74 

279 

297

97 

319 

323

104 

130 

325 

337 

312

70 

236 

268

82 

69 

66 

299 

280

66 

73 

80 

311

83 

91 

98 

332

69 

72 

204 

203

75 

74 

258 

273

69 

76 

76 

323 

308

66 

74 

375 

3S5.

Period.

Minutes. 

4 

4 

3

3 

3

2

3 

3

3 

3 

3 

3

3 

3 

3

3 

4 

3

4 

4 

3 

3

4 

4 

3 

3

4 

4 

3 

3

. 4 

3 

3

4 

4 

3 

3 

3

4 

3 

3

4 

5 

4 

3 

3

4 

4 

4 

3

4 

4 

4 

3

4 

4 

3 

3

4 

4 

3 

3

4 

4 

4 

3 

3

4 

4 

3 

3

Slope.

Water 

surface.

Per cent.

0.56

Bed.

Per cent. 

0.58 

.58 

.99

.58 

.62 

1.09

.57 

.66

.51 

.60 

1.05 

1.07

.65 

1.06 

1.01

.57 

.54 

1.20

.62 

.61 

1.00 

1.03.

.63 

.59 

1.04 

1.07

.51 

.57 

1.03 

1.04

.58 

1.03 

1.08

.57 

.61 

.97 

1.01 

1.07

.60 

1.14 

1.12

.45 

.46 

.50 

1.05 

1.07

.39 

.47 

.51 

1.06

.49 

.47 

.55 

.99

.61 

.60 

1.04 

1.06

.53 

.58 

1.04 

1.04

.51

.52 

.57 

1.00 

1.01

.42 

.47 

1.05 

1.05

Distance.

Feet. 

16 

14 

16

16 

16 

16

14 

14

16 

16 

16 

16

14 

16 

16

16 

16 

16

12 

10 

16 

14

16 

14 

16 

1C

16 

16 

16 

14

16 

14 

16

16 

16 

14 

14 

16

12 

16 

16

12 

12 

12 

16 

16

12 

12 

12 

16

16 

12 

12 

12

16 

16 

16 

16

14

14 

16 

16

16 

14 

16 

14 

16

14 

14

16 

12

Depth.

By gage.

Feet. 

0.232 

.241 

.194

.270 

.282 

.233

.312 

.305

.222 

.215 

.173 

.175

.232 

.210 

.201

.250 

.262 

.197

.269 

.269 

.224 

.231

.199 

.196 

.158 

.151

.235 

.237 

.176 

.175

.248 

.181 

.180

.254 

.231 

.213 

.215 

.203

.191 

.121 

.122

.233 

.202 

.224 

.161 

.161

.243 

.244 

.232 

.169

.227 

.234 

.233

.188

.174 

.174 

.137 

.130

.206 

.204 

.157 

' .152

.224 

.227 

.225 

.155 

.158

.240 

.243 

.160 

.167

By profiles.

Feet.

.277

Character of bed.

.....do.....................

.....do.....................

[Transition] ................

Transition .................

[Transition] ................

Transition...........:.....

.....do.....................

.....do.....................

[Transition]. . ..............

Smooth....................

.....do.....................

.....do.....................

Transition .................

[Smooth]. ..................

[Smooth]. ..................

Smooth ....................

.....do.....................

.....do.....................

.....do.....................

[Transition]................

Dunes ..................... 

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

.....do.....................

Outfall.

Contracted. 

Do. 

Do.

Contracted. 

Do. 

Do.

Contracted. 

Do.

Contracted. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do.

Contracted. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do.
52 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 4 (I). Special group of observations on load, slope, and depth, with debris of grade (C) Continued.

Width.

Feet. 

1.96

Discharge.

FtJ/sec. 

0.734

.923

1.021

1.119

Load.

Feed.

Gm.lsec. 

85 

84 

234 

222 

234

96 

246

84 

85 

94 

328 

316

82 

85 

82 

345 

328

Collection.

Gm.lsec. 

83 

88 

236 

231 

250

97 

279

85 

87 

104 

343 

352

82 

84 

94 

341 

363

Period.

Minutes. 

4 

4 

3 

3 

3

3 

3

4 

4 

3 

3 

3

4 

4 

4 

3 

3

Slope.

Water 

surface.

Per cent.

Bed.

Per cent. 

0.61 

.69 

1.05 

1.09 

1.12

.62 

1.02

.50 

.57 

.54 

1.09 

1.09

.45 

.57 

.52 

1.00 

1.00

Distance.

Feet. 

16 

16 

16 

16 

16

16 

16

16 

14 

16 

16 

16

12 

16 

16 

16 

16

Depth.

By gage.

Feet. 

0.147 

.146 

.117 

.123 

.122

.176 

.143

.194 

.200 

.193 

.141 

.144

.214 

.222 

.219 

.150 

.147

By profiles.

Feet.

Character of bed.

.....do.....................

.....do.....................

Outfall.

Contracted. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do.

TABLE 4 (J). Observations on load, slope, and depth, with debris of two or more grades mixed.

Designation of mixture, component grades, 

and percentages by weight.

(AA)-(A)50:(C)50.........................

(A3Gi)-(A)75:(G)25........................

(A2Gi)-(A)67:(G)33. .......................

(AiGi)-(A)50:(G)50... .....................

(AjG2)-(A)33:(G)67........................

(AiG4)-(A)22:(G)78........................

(B4Fi)-(B)78:(F^22........................

(B2Fi)-(B)64 : (F)36. .......................

(B1Fi)-(B)47:(F)53. .......................

(B1F2)-(B)31:(F)69. .......................

(BiF«)?=(B)18:(F)82........................

(C4E1 )-(C)79:(E)21.. .......................

Width of 

trough.

Feet. 

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

Discharge.

Ft.3jsec. 

0.363

.363

.363

.363

.363

.363

.363

.363

.363

.363

.363

.363

Load.

Feed.

Gm-jsec. 

42 

90 

128 

171

20 

90 

80 

175 

175

20 

42 

42 

87 

110

22 

46 

91

42

22 

43

42 

84 

113 

113 

169

39 

78 

105 

157

45 

90 

120 

181

28 

41 

81

16

27 

40

43 

85 

115 

171

Collection.

Gm.lsec. 

43 

91 

140 

169

18 

74 

89 

164 

176

20 

42 

42 

88 

119

22 

43 

93

40

21 

42

43 

92 

105 

125 

159

42 

80 

109 

157

49 

95 

130 

162

35 

40 

82

17 

29 

43

49 

84 

118 

171

Period.

Minutes. 

8 

4 

4 

3

12 

5 

4 

3 

3

12 

6 

6 

3 

3

8 

6 

4

6

20

7

6 

3 

3 

3 

2

7 

5 

4 

2

7 

4 

3 

2

11 

6 

4

15 

10 

6

8 

4 

3 

2

Slope of bed.

Per cent.

0.58 

.92 

1.23 

1.47

.51 

.93 

.95 

1.38 

1.36

.53

.62 

.64 

1.00 

1.18

.68 

1.01 

1.42

1.36

1.30 

1.79

.59 

.86 

.99 

1.05 

1.38

.57 

.85 

1.11 

1.49

.73 

1.14 

1.41 

1.61

.86 

1.06 

1.56

.82 

1.16 

1.46

.70 

.93 

1.15 

1.'52

Distance.

Feet. 

16 

16 

1C 

16

16 

16 

16 

16 

16

16 

16 

16 

16 

16

16 

16 

16

16

16 

16

16 

16 

16 

16 

16

16 

16 

16 

16

16 

16 

16 

16

16 

16 

16

16 

14 

16

16 

16 

16 

16

Depth by 

gage.

Feet.

Character of 

bed.

Transition. 

Do. 

Antidunes. 

Do.

Smooth. 

Transition. 

Do. 

Antidunes. 

Do.

Smooth. 

Do. 

Do. 

Antidunes. 

Do.

Smooth. 

Transition. 

Do.

Smooth.

Smooth. 

Do. 

Do. 

Do. 

Transition.

Smooth. 

Do. 

Do. 

Do.

Transition. 

Do. 

Smooth. 

Do.

Dunes. 

Do. 

Do.

Dunes. 

Do. 

Do.

Smooth. 

Do. 

Do. 

Do.
THE OBSERVATIONS. 53

TABLE 4 (J). Observations on load, slope, and depth, with debris of two or more grades mixed Continued.

Designation of mixture, component grades, 

and percentages by weight.

(C!E1 )=(C)65:(E)35... ......................

(CjEi)=(C)48:(E)52.........................

(CiE2)=(C)31:(E)69.........................

(CiE<)=(C)19:(E)81.. .......................

(C«Gi)=(C)80 : (0)20.........................

(C2Gi)<=(C)67: (0)33.. .......................

(CiG])=(C)50:(G)50.. .......................

(CiG2)=(C)33 : (0)67.........................

(E4Gi)-(E)80: (0)20.. ..........;...........

(E2Gi)=(E)67: (0)33. .......................

(EiGi)= (E)50 : (0)50. .......................

Width of 

trough.

Feet. 

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

Discharge.

Ft*lsec. 

0.182

.363

.182

.363

.182

.363

.182

.363

.363

.363

.363

.363

.363

.303

.363

Load.

Feed.

Qm.lsec. 

78 

104

43 

85 

114 

155 

171 

155 

171

26 

40 

40 

53 

51 

53 

51 

81 

77 

81

39 

39 

78 

104 

104 

104 

157

26 

39 

52

78

39 

39 

78 

108 

157

20 

26 

39 

39 

52 

52

40 

53 

78 

109

31 

93 

140 

187

187

31 

31 

31 

61 

92 

92 

122 

183

17 

50 

100 

133 

133

16 

32 

32 

64

15 

31 

61

16 

32 

63

16 

31

62

Collection.

Gm.lsec. 

81 

105

42 

85 

117 

170 

169 

170 

173

28 

40 

41 

50 

60 

52 

61 

67 

78 

87

40 

41 

78 

93 

115 

119 

158

26 

39 

57 

76

39 

39 

81 

112 

158

20 

28 

40 

42 

51 

54

40 

56 

76 

109

32 

92 

140 

194 

195

29 

29 

31 

66 

90 

95 

120 

180

18 

50 

105 

128 

145

15 

34 

36 

69

17 

31

58

15 

30 

59

18 

34 

60

Period.

Minutes. 

3 

4

7 

4 

3 

3 

3 

4 

3

12 

10 

10 

6

6

4

5

6 

7 

5 

3 

3 

3 

3

10 

8 

6 

3

8 

9 

3 

3 

3

14 

10 

6 

6 

6 

12

8 

7 

4 

3

11 

5 

4 

3 

3

12 

10 

7 

4 

5 

3 

2

15 

10 

3 

3 

3

16 

10 

10

7

15 

10

8

18 

11 

6

18 

10

8

Slope of bed.

Per cent.

1.77 

2.06

.69 

1.03 

1.27 

1.57 

1.65 

1.60 

1.74

.99 

1.17 

1.27 

1.43 

1.50 

1.51 

1.55 

1.82 

1.95 

1.89

.74 

.76 

1.07 

1.37 

1.40 

1.43 

1.85

1.10 

1.37 

1.64 

2.22

.77 

.84 

1.25 

1.54 

2.10

1.02 

1.29 

-1.61 

1.65 

1.88 

1.93

1.00 

1.14 

1.45 

1.75

.62 

1.07 

1.30 

1.48 

1.62

.59 

.63 

.62 

.90 

1.03 

1.08 

1.20 

1.65

.71 

.99 

1.34 

1.62 

1.66

.95 

1.17 

1.27 

1.59

.72 

1.00 

1.39

.76 

1.03 

1.45

.87 

1.12 

1.60

Distance.

Feet. 

16 

16

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16

16 

16 

16 

16

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16

16 

16 

16 

16

16 

16 

16

16 

16 

16

16 

16 

16

Depth by 

gage.

Feet.

0.096 

.090 

.091 

 .081

.082

.150 

.151

.093 

.088 

.082

.085 

.079 

.084 

.078 

.082

Character of 

bed.

Smooth. 

Do.

Transition. 

Do. 

Smooth. 

Do. 

Do. 

Do. 

Do.

Transition. 

Do. 

Do. 

Smooth. . 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Transition. 

Do. 

Smooth. 

Do. 

Do. 

Do. 

Do.

Transition. 

Do. 

Smooth. 

Do.

Dunes. 

Do. 

Do. 

Transition. 

Do.

Dunes. 

Do. 

Do. 

Do. 

Transition. 

Do.

Dunes. 

Do. 

Transition. 

Do.

"Dunes. 

Transition. 

Do. 

Smooth. 

Do.

Smooth. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do. 

Do.

Smooth. 

Do. 

Do. 

Do. 

Do.

Transition. 

Smooth. 

Do. 

Transition.

Transition. 

Smooth. 

Transition.

Smooth. 

Do. 

Transition.

Smooth. 

Do. 

Transition.
54 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 4 (J). Observations on load, slope, and depth, with debris of two or more grades mixed Continued.

Designation of mixture, component grades, 

and percentages by weight.

(EiGj)-fE)33 : (G)67. .......................

(AiCiG2)=(A)25 : (C)25 : (G)50. ..............

(CDEFG)=(C)45 : (D)35 : (E)12 : (F)6 : (G)2.

Width of 

trough.

Feet. 

1.00

1.00

1.00

Discharge.

Ft?lsec. 

0.363

.363

.182

.363

.545

Load.

Feed.

Om.lsec. 

16 

23 

30 

30 

45 

45

16 

16 

16 

32

126 

126

21 

42 

84

67 

67 

84 

84 

84 

113 

113 

169 

169

84 

84 

169 

169 

 253 

253

CoIIect.onOm.lsec. 

15 

23 

29 

32 

49 

54

13 

17 

14 

33 

88 

131 

151

25 

45 

84

69 

69 

80 

83 

83 

120 

118 

172 

173

90 

92 

151 

177 

228 

242

Period.

Minutes. 

14 

12 

12 

11 

7 

7

15 

10 

15 

10 

5 

4 

4

12 

10 

5

8 

6 

5 

5 

5 

5 

5 

4 

4

6 

5 

3 

3 

3 

3

Slope of bed.

Per cent.

0.95 

1.18 

1.35 

1.33 

1.73 

1.74

.77 

.79 

.82 

.88 

1.32 

1.59 

1.67

.82 

1.23 

1.72

.81 

.85 

.91 

.93 

.95 

1.18 

1.19 

1.53 

1.54

.65 

.67 

1.07 

1.13 

1.40 

1.37

Distance.

Feet. 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16 

16

14 

16 

16

16 

16 

16 

16 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16

Depth by 

gage.

Feet.

Character of 

bed.

Transition. 

Smooth.

Transition. 

Do.

Transition. 

Do. - 

Smooth. 

Do. 

Do. 

Transition. 

Do.

Dunes. 

Transition.

Smooth. 

Do. 

Do. 

Do. 

Do.

Smooth. 

Do. 

Do. 

Do. 

Do.

TABLE 4 (K). Observations on load, slope, and depth, with unassorted debris. a

Width.

Feet. 

1.00

Discharge.

Ft. 3/sec. 

0.182

.363

Load.

Feed.

Gm./sec. 

19 

19 

38 

75 

145

22 

22 

38 

38 

77 

154

Collection.

Gm.lsec. 

18 

18 

38 

76 

147

20 

27 

34 

42 

74 

150

Period.

Minutes. 

15 

15 

8 

3 

2

10 

12 

6 

6 

5 

3

Slope of bed.

Per cent.

0.74 

.79 

1.00 

1.55 

2.28

.43 

.50 

.59 

.67 

.89 

1.31

Distance.

Feet. 

16 

16 

16 

16 

16

16 

16 

16 

16 

16 

16

Depth 

by gage.

Feet.

0.169 

.176 

.157 

.158

Character of bed.

Dunes. ........................................

.....do.........................................

.....do.........................................

.....do.........................................

.....do.........................................

Smooth. ...........:..........................

.....do.........................................

.....do.........................................

.....do.........................................

Outfall.

Contracted. 

Do. 

Do. 

Do. 

Do.

Contracted. 

Do. 

Do. 

Do. 

Do. 

Do.

a The very coarsest particles were removed by passing the sample through a &-mesh sieve, and the very finest by passing it over a 60-mesh sieve. 

It retained the equivalents of grades (A), (B), (C), (D), (E), and (F).
CHAPTER II ADJUSTMENT OF OBSERVATIONS.

OBSERVATIONS ON CAPACITY AND SLOPE.

THE OBSERVATIONAL SERIES.

Each of the experiments in stream traction 

involved six quantities (1) a fineness, or 

grade of debris, (2) a width oi trough, (3) a 

discharge, (4) a slope, (5) a load, or capacity, 

and (6) a depth of current. The experiments 

were arranged in series, for each of which grade, 

width, and discharge were constant, while 

within each the magnitudes of slope, capacity, 

and depth were varied. There will be frequent 

occasion to mention these secondary units of

the experimental work, and whenever the title 

series seems not sufficiently specific they will be 

called observational series. The number of 

such series recorded in Table 4 is 153.

The factors of grade, width, and discharge, 

which are related to an individual series as 

fixed conditions, or constants, do in fact assume 

the character of variables when series is compared with series; but their modes of determination and combination are not of such character that their numerical values can be 

checked and adjusted by means of recorded 

relations.

300

j200

100

 Slope

FIGURE 13. Plot of a single series of observations on capacity and slope. Capacity in grams of debris persecond.

traction are indicated.

Slope in per cent. The modes of

In each experiment the values of slope, load, 

and depth are mutually dependent; within each 

series they form a triple progression, the depth 

decreasing while slope and load increase; 

but the laws of these interdependent variations are partly masked by irregularities in the 

sequences. As a preliminary to the general 

discussion, the observational values were subjected to a process of adjustment, whereby the 

sequences were freed from irregularities. The 

irregularities are made manifest by the comparison of the sequences of two variables, and 

first consideration will be given to those of 

capacity and slope.

Figure 13 exhibits the relations of capacity 

to slope as observed in a single series of experiments (that for grade (C), with w=1.32 feet 

and Q = 0.363 ft.3/sec.). The ordinates indicate 

capacity, as measured by debris delivered 

at the lower end of the trough; the abscissas 

represent slope, as measured on the bed of 

the channel. The arrangement of the observational dots suggests that if the observations 

were harmonious the dots would fall in a 

line of simple curvature. Such a line would 

express the law connecting capacity and slope. 

The departures of the dots from such linear 

arrangement represent irregularities, or errors, 

in the experimental data. The adjustment 

proposed is the replacement of the imperfectly 

alined dots by a generalized or representative 

line, or the replacement of the inharmonious

 55
56 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

values of capacity and slope by a system of 

harmonious or adjusted values.

ERRORS.

As a first step in the treatment of the errors 

of the data they were studied with a view to 

the discrimination of the systematic and the 

accidental.

The three modes of traction the dune, the 

smooth, the antidune although intergrading, 

are mechanically different. It was surmised 

that they might differ in efficiency, so that the 

capacity-slope curve might show a step in

400

200

100

3*80 'o

60

.40

20

10 I

7

Slope

FIGURE 14. Logarithmic plot of a series of observations on capacity 

and slope. Compare figure 13.

passing from one to another; and it was also 

surmised that the law connecting capacity 

with slope might not be the same for the 

several modes. A suggestive observation had 

shown that on very low slopes slopes so low 

that capacity is minute the current changes 

an artificially smoothed bed of debris to a 

system of dunes, and that with the development of dunes the load is notably increased 

without any change in general slope. To test 

the surmises all the series were plotted on logarithmic section paper. Figure 14 shows a 

logarithmic plot of the same data which appear 

in figure 13; and it will be observed that the 

line suggested by the points has much less

curvature in the logarithmic plot. Its approximation to a straight line makes the study 

of its local peculiarities comparatively easy. 

The examination of the plots, while not disproving the surmises, showed that whatever 

diverse influences may be exerted by the modes 

of traction, they are too small to be discriminated from the irregularities due to other 

causes.

Other sources of systematic error are connected with the methods of experimentation.

INTAKE INFLUENCES.

As the water entered the experiment trough 

from the stilling tank it was accelerated, the 

gain in mean velocity being associated' with a 

quick descent in the surface profile. Beyond 

this descent the profile usually rose somewhat, 

and there was commonly a moderate development of fixed waves. This development was 

modified and the waves were on the whole 

reduced by the addition of the debris. As it 

fell into the water the debris had no forward 

momentum, and it therefore tended to retard 

the current. But the debris also accumulated on the bottom, reducing the depth of the 

water at that point, and this reduction necessitated an increase in mean velocity. In the 

immediate neighborhood of tne place where 

de"bris was fed the slope of the water was 

affected by an abnormality distinguishable 

from the intake abnormality proper, and the 

joint abnormality faded gradually downstream. 

The nature of these features varied with the 

discharge and load, with the gradual development of the adjusted slope, and also with the 

mode of feeding. During the greater part of 

the experimental work the feeding was either 

automatic and continuous or else manipulated 

by hand in such way as to make it nearly continuous, but for a minor part the feeding was 

intermittent, a measureful of debris being 

dumped into the water at regular intervals.

OUTFALL INFLUENCES.

In all the earlier work the trough had the 

same cross section at the lower end as elsewhere, and the water fell freely from its open 

end to the settling tank. As the resistance to 

its forward motion was less at the outfall than 

within the trough, the water flowed faster 

there. Its faster flow diminished the resistance just above, and thus the influence of
ADJUSTMENT OF OBSEBVAT10NS. 57

outfall conditions extended indefinitely upstream. An expression of this influence was 

found in the water profile, which was usually 

convex in the lower part of the trough, the 

degree of convexity diminishing upstream. 

Its effect on the profile of the bed is not readily 

analyzed, because that profile is adjusted 

through the velocity of water at the bottom of 

the current, and the bed velocity is not simply 

related either to mean velocity or to depth.

FIGURE 15. Diagrammatic longitudinal section of outfall end of experiment trough, illustrating influence of sand arrester on water slope.

A second factor at the outfall end was the 

arrangement for separating the debris from the 

current. This included a well, ABCD, figure 

15, which was sunk below the trough bed and 

into which the debris sank, while the current 

passed on to the outfall at E. In part of the 

work the space AD was entirely open; in 

another part a coarse screen was stretched 

across it. In either case the resistance of this 

part of the channel bed differed from the

resistance along the debris slope and may have 

been greater or less. From the well to the 

outfall, DE, the frictional resistance was less 

than elsewhere. As the fixed part of the 

channel bed, DE, was horizontal and the 

d6bris portion, OA, was inclined, the profile 

of the bed changed at A. Projected-forward, 

the slope GA passed below E to H, and when 

the debris slope, down which the transporting 

current flowed, was steep, the part DE was 

related to it somewhat as a dam. The tendency of the quasi-dam was to retard the current near the outfall and make the water 

profile concave, and in some of the experiments 

the profile actually became concave. Other 

outfall factors were recognized, but they are 

not here mentioned because they are believed 

to be of relatively small importance.

In the reduction of observations on slope an 

attempt was made to lessen the effect of intake 

and outfall influences by omitting from the 

calculations the profile data obtained near the 

ends of the trough. The data from a considerable number of experiments were finally 

discarded altogether and do not appear in the 

tables. To replace the discarded data experiments were afterward made with a modified 

apparatus.

FIGURE 1C. Diagrammatic longitudinal section of debris bed and stream, in a long trough.

CHANGES IN APPARATUS.

As the terminal influences of all kinds 

diminish with distance from the trough ends, a 

manifest mode of avoidance is to employ a 

very long experimental trough and determine 

slopes from observations in the medial portion 

exclusively. A trough length of 150 feet was 

tried and proved moderately successful for 

very low slopes. For steeper slopes there were 

practical difficulties the need either of a 

trough with adjustable slope or of a very deep 

trough and a correspondingly large stock of 

assorted d&bris and these led to the consideration of other plans. The one adopted was 

to contract the channel at the outfall and thus 

increase the resistance to flow at that point, 

and with, the contracted outfall to use a trough 

of moderate length. The theory of this plan 

may be illustrated by a diagram. In figure 

16, AB represents in profile the bed of a long

trough, CB the profile of debris, and DE' the 

water profile. In the tract EF the water and 

debris profiles are nearly parallel and depth and 

velocities are therefore practically uniform. 

From F to B' the water profile is notably convex because the resistance to flow afforded by 

the water itself steadily diminishes toward B''.

FIGURE 17. Diagrammatic longitudinal section of outfall end of 

trough, illustrating influence of contractor.

The plan undertook to introduce at F, by contraction, a resistance equivalent to that 

afforded by the water beyond" F and then dispense with the portion of the trough between 

F and B. In the longitudinal section, figure 

17, ADB is the bed of the trough, with the well 

for catching debris, as already shown in figure
58 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

15; EGB is the profile of one side wall. The 

walls converge from F to G (see fig. 3), producing the contraction at the outfall. It was 

found that the bed of debris, instead of running to a feather edge at D (compare fig. 16), 

held its thickness to C' and ended in a steep 

incline.

The device of contraction accomplished its 

purpose of avoiding terminal difficulties, but it 

was found to aggravate certain other difficulties, next to be described.

EHYTHM.

Whenever the profile of a current was determined by a series of measurements applied to 

the water surface, that surface was found to be 

in a state of unrest. Its position in any vertical fluctuated upward and downward rhythmically. The amplitude of oscillation, which 

might be great or small, was not constant;

that is, the rhythm was not simple, but compound. It consisted apparently of many 

rhythmic elements differing one from another 

in period and amplitude.

The rhythmic quality, thus easily appreciated by watching the play of the surface in 

relation to a fixed point, permeated every 

function of the current the slope of its profile, 

both local and general, the slope of its bed, the 

quantity of debris transported, the mode of its 

transportation. The rhythm of the dune has 

already been described, but associated with the 

dunes were greater debris waves, also traveling 

downstream and each involving the volume of 

many dunes. In the bed of the long trough a 

series of them could be seen; in the shorter 

trough one or two might be made out, or the 

effect might be only an alternate temporary 

steepening and flattening of the general slope. 

The rhythm of the antidune was accompanied

FIGURE 18. Profiles of channel bed, illustrating tractional rhythms associated with dunes of greater magnitude. Scales: Each horizontal 

space =20 feet; each vertical space =0.4 foot. For Nos. 1 to -5 the average slope is 0.2per cent, and the average load (of grade (C)) per foot 

of channel width is 7 gm./sec. For Nos. 6 to 9 the average slope is O.C per cent and the average load 37 gm./sec.

by rhythmic paroxysms and doubtless by 

other rhythms which escaped recognition 

because the steep slopes with which the antidune was associated were not studied in the 

long trough. Figure 18 shows a few channelbed profiles in which rhythmic features appeal 

to the- eye, but the greater number of such profiles merely show an irregularity in which 

periodicity is not conspicuous. It is .probable 

that the currents were affected by numerous 

coexistent rhythms, which served to confuse 

one another and thus masked periodicity except 

when some one rhythm was stronger than the 

rest.

The condition of relatively smooth channel 

bed which intervened between the conditions 

characterized severally by dunes and antidunes was also a condition of relative uniformity in all the activities of the current, and 

when it prevailed the rhythmic variations were 

at a minimum.

The rhythms of the transporting stream 

manifestly constitute a group of phenomena 

worthy of systematic study, but the Berkeley 

laboratory, having a definite and different 

theme, treated them only as difficulties interfering with its work. It sought the capacity 

for load inhering in the average of all the 

diverse slopes presented by the rhythms, and 

it necessarily treated the deviations of slope 

measurements from that average as accidental 

errors.

The rhythms affected the determinations of 

loads as well as slopes. The variations of 

profile were effected by erosion and deposition, and a current which was eroding or 

depositing carried more load at one point 

than at another. As the loads were largely 

determined by weighing the debris delivered 

at the trough end in a limited time, the amount 

obtained would depend in part on the phase of 

slope variation near the point of delivery.
ADJUSTMENT OF OBSERVATIONS. 59

In the earlier experiments with contraction 

at outfall as in other experiments it was 

necessary to continue a run, with uniform 

discharge and uniform feed of debris, until 

the slope of the sand bed had been automatically adjusted to the conditions. The criteria 

adopted for recognition of a state of adjustment were two that the water slope equal 

the bed slope and that the rate of delivery 

of debris equal the rate of feed. It was found 

impracticable to satisfy these tests, because 

both slopes and the rate of delivery fluctuated 

through a wide range, and an approximate 

adjustment, if attained, could not be made to 

continue. The state of affairs may be likened 

to the waving of a flag in the wind; at the 

outer margin the amplitude of the undulation 

is much greater than close to the staff. In 

the long trough the outfall end corresponded 

to the staff, giving a fixed position and element of uniformity to which the profiles conformed, and the rhythmic departures were 

greater with distance from the outfall. The 

shorter trough when combined with end contraction represented a segment of the long 

trough at a distance from the outfall and was 

correspondingly subject to great fluctuations. 

Despite these difficulties, the nature of which 

was not well understood at the time, a large 

number of experiments were made in this way.

The work with free outfall was affected 

chiefly by terminal influences, and as these 

produced systematic errors there was danger 

of false conclusions. The work with contracted outfall was affected by accidental 

errors of such magnitude as largely to mask 

the nature of the laws sought. Between 

these perils of Scylla and Charybdis a middle 

course was finally steered by using a moderate amount of contraction, whereby the recognized systematic errors were practically avoided 

without the introduction of insuperable rhythmic irregularities.

SLOPES OF DEBRIS AND WATER SURFACE.

The slope of the bed of debris to which 

measurement was applied had been established by the stream as that appropriate to 

the stream's load of debris. It was caused 

by the load, in conjunction with the discharge 

and other conditions, and it accurately sufficed 

to give the stream capacity for that load. 

This was my point of view in arranging the

experimental methods, and accordingly one of 

the principal measurements undertaken was 

that of the debris profile. But the slope more 

generally considered in hydraulic studies is that 

of the water surface. Head, the hydraulician's 

ordinary measure for the determination of 

power, is the vertical interval between two 

points of "the water surface, and slope is the 

loss of head in a unit of distance. Under 

conditions of uniform flow the two profiles 

are parallel, but for various reasons our experimental currents ordinarily lacked so much 

of uniformity that the two slopes were appreciably different. I do not find it easy to 

decide which slope should be regarded as the 

true correlative of capacity for traction, but 

as all our laboratory data include the de"bris 

slope, while the determinations of water slope 

were relatively infrequent, the discussion of 

results has adhered almost exclusively to the 

former. If the water slope is the true correlative, then the use of the debris slope 

involves a systematic error.

THE LOGARITHMIC PLOTS.

When the data of an observational series 

are plotted on ordinary section paper, as in 

figure 13, and a representative line is drawn 

through or among them, that line is the graphic 

equivalent of

(2)

When they are plotted on logarithmic section 

paper, as in figure 14, and a representative 

line is drawn, that line is the graphic equivalent of

(3)

The second equation, or line, is the logarithmic form of the first.

As already mentioned, a logarithmic plot 

was made of each observational series. The 

plot included primarily the slopes of the de"- 

bris bed and the determinations of load from 

the delivery of debris at the end of the trough, 

but it included also, with distinctive notation, such determinations as were available 

of water-surface slopes and of load based on 

the rate of feed at the head of the trough. 

The notation also classified observations with 

reference to the three modes of traction and
60 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

distinguished experiments made with use of 

the contractor from those with free outfall.

A critical study of these plots led to several 

conclusions. (1) Judged by the internal evidence of regularity and irregularity, the observations of debris slope and water slope 

have about the same quality. (2) Similarly, 

the observations of load fed and load delivered have about the same quality. (3) As 

already stated, the law of sequence is not 

discovered to change in passing from one 

.process of traction to another; the assumption of a continuous law is the best practicable. (4) Except for very low slopes, the 

results obtained with the use of the contractor do not differ widely from those without it. (5) There are differences between 

bodies of experimental data obtained at different stages of the work differences of uncertain source but presumably connected with 

modes of manipulation which make it desirable to treat such bodies separately whenever 

practicable. (6) The best representative line 

is not straight but curved, and its curvature is 

always in one direction.

If the representative line were straight, function (3) would have the form

log <7=log A + nlogS (4)

in which A is a constant capacity and n a ratio ; 

and function (2) would have the form

.(5)

On some of the plots the observational points 

are too irregular to afford trustworthy evidence 

of curvature. On most of them the indicated 

curvature is slight. From inspection of the 

data during the progress of the experiments it 

was thought that the true representative line 

would prove to be straight, in which case the 

accurate determination of two points on the 

line would suffice; and some of the experimental 

work was adjusted to that theory. The series 

giving data for but two points on the logarithmic plot of course furnished no evidence as to 

curvature of the representative line.

For all those cases in which the position of 

the best representative line could be inferred, 

with close approximation, from the arrangement of the observational points, a satisfactory 

adjustment could be made by simply drawing

the line and then converting its series of positions into figures; but this procedure would 

afford no control for the curvature of representative lines in cases where the observational points were few or inharmonious. In 

order to make the stronger series support the 

weaker, the plan was adopted of (1) connecting 

the lines with a formula of interpolation, (2) 

correlating the constants of the formula with 

conditions of experimentation, and thus (3) 

giving deductive control to the lines of the 

weaker series.

SELECTION OF AN INTERPOLATION FORMULA.

The best interpolation formula is one which 

embodies the true theory of the relation to 

which the observations pertain. In the present 

case the true theory is not known, but there 

are certain conditions which a theory must 

satisfy, and these may be used as criteria in 

the selection of a form for empiric formulas 

of interpolation. Subject to these criteria, 

the form selected should serve to minimize the 

discrepancies between observed and adjusted 

values.

For the study of the character of the curve 

to represent best the logarithmic plot of observations, the data for debris of grade (G) 

were selected. The experiments furnishing 

those data were all performed by one method, 

the method using moderate contraction of the 

trough at outfall; and for that grade the apparent curvature of the logarithmic graph is 

greater than for most others. The data were 

first plotted (on logarithmic paper) in groups, 

each group containing the data for three graphs 

which pertain to the same width of channel 

but to different discharges. It was assumed 

that the three graphs, if correctly drawn, would 

constitute a system, the one for the medium 

discharge being intermediate in form and position between the other two; and in drawing 

them on this assumption the forms were mutually adjusted. Then a rearrangement was 

made which grouped together data agreeing as 

to discharge but differing as to width of channel, and further adjustment was made. Selection was finally made of the graph for w = 0.66 

foot and $ = 0.734 ft. 3/sec., and this was drawn, 

through the selected positions, by the aid of a 

flexible ruler. Thus the curve in figure 19 is 

a graphic generalization not only from its
ADJUSTMENT OF OBSERVATIONS. 61

particular series of observations but from 

several related series. It was assumed to be a 

typical or representative curve for the function log C=fiQog $); and the corresponding

.200

100 

80

 if eo

CD 

OL. 

CD

u

40

20

/

/

/

/

/

//

/

/

/. . /

/

/

*

/

/

:6 .8 1.0 2.0 

Slope

FIGURE 19. Logarithmic graph of C=f(S), for grade (G), w= 0.66 foot, 

Q=0.734 ftjysec.

plot on ordinary section paper (the line BD in 

fig. 20) was assumed to represent typically 

the function C=f(S). The coordinates of the 

line BD are given in Table 5.

TABLE 5. Values of capacity for traction, graphically generalized from data of Table 4 (G), for w=0.66 foot and 

Q=0.734ft. 3lsec.; corresponding to the curve of log C= 

fi (log 8) in figure 19, and the curve BD in figure 20.

Slope.

Per cent. 

0.8 

1.0 

1.2 

' 1.4 

1.6

Capacity.

Om./sec. 

16.3 

31.2 

49.9 

72.0 

97.0

Slope.

Per cent. 

1.8 

2.0 

2.2 

2.4

Capacity.

Om./sec. 

124 

153 

1*5 

220

A number of tentative formulas were now 

compared with this empiric line, their parameters being computed so that they would fit, as 

nearly as practicable, the values of C'in Table 

5. Certain functions, including the simpler 

functions of the circular arc and the exponenlowing functions yielded curves closely resembling that of figure 20:

.(6) 

.(7) 

_ (8) 

. (9) 

(10) 

l}

Functions (6), (7), and (8) are special cases 

of the general formula of interpolation with 

integral exponents :

x = a + av y + a2y2

No. (11) is a somewhat involved power 

function suggested by results of a preliminary 

discussion of the laboratory data. Nos. (9) 

and (10) are special cases of the general parabolic function

(13)

and have the virtue of facilitating the graphic 

treatment of the material. Their logarithmic 

equivalents are, respectively,

(14)

(15)

log l> + n log S 

"log (7= log & + TI log (S-a)

and, as each of these is the equation of a 

straight line, the graphic derivation of the 

exponent, by means of logarithmic section 

paper, becomes a simple matter after the 

value of A or a has been determined.

The adjustment of equations (6) to (11) to 

the specific data in Table 5 gives them the 

following forms, (6a) being derived from 

(6), etc.:

C=-29.5S + Q7.59S*-7.194;S3 ...... (6a)

<7= -12.86S-I-44.08S2 .......... ___.(7a)

(7= - 19.25 + 16.94S + 34.4SS2 ....... (8aj

(7=70.5(5r -0.39) 1'63 ............... (lOa)

2.68

C-B1.28 ** .--...................(Ha)

When the curves corresponding to these 

equations are plotted for the region - covered

tial function tf=e «(«+«), could not be fitted, by the empiric line BD, they coincide very 

even approximately, to the data; but the fol- closely with that line. The greatest departure
62 TRANSPORTATION OF DEBEIS BY RUNNING WATER.

is in the curve for (7a), but its divergence is not 

sufficient to throw it out of apparent harmony 

with the series of points representing the 

original observations.

In order to exhibit further the properties of 

the formulas, their curves were extrapolated in

both directions from the locus BD. Table 6 

contains the numerical data used in plotting 

the extensions. Figure 20 gives the extensions of the curves for slopes greater than 

those of the experiments, and figure 21 the 

extensions for smaller slopes.

TABLE 6. Numerical data computed for the construction of curves in figures 20 and 21.

Values of C corresponding to 

s-o..................................................... s-o.i... ................................................

S-0.2.... ...............................................

S-0.3..... ..............................................

S-0.4..... ..............................................

S-0.5...... .............................................

S-0.6 ...................................................

S-0.7 ................................................... s-0.8..........:.........................................

S- 3.0..... ..............................................

S-4^)...... .................................. ...........

S-5.0. ..................................................

S-6.0...................................................

S-7.0. .................................................. -

5-8.0...................................................

S-9.0...................................................

s- 10.0...................... ............................

(to)

Gm.lsec. 

0

16.3

327

504

645

.705

645

409

-31

Per cent.

\ °

.45

I 8.94 

6.04

.22

3.33

(7a)

Gm./scc. 

0

-.845

-.810

.108

1.908

4.59

8.15

12.6

17.9

358

654

1,039

1,510

2,070

2,718

Per cent. 

0 

.29

(8a)

Gm.lsec. 

-19.25

-11.07

- 6.97

- 1.96

3.33

9.50

16.36

332

589

920

1,315

1,741

. 2,313

2,916

Per cent. 

-1.03 

.54

(9a)

Gm.lsec. 

-10.0

- 9.57

- 8.30

- 6.20

- 3.28

.45

4.98

10.33

16.48

353

631

990

1,422

2,519

3,926

Per cent. 

-0.49

(lOa)

Gm.lsec.

...do ........

...do........

...do........

.039

1.93

5.54

10.46

16.49

' 337

571

849

1,172

1,531

1,923

2,355

2,818

Per cent. 

0.39

(lla)

Gm.lsec. 

0

.0014

.073

.490

1.59

3.64

6.81

11.14

16.67

323

616

676

84.

998

1,140

1,276

1,400

Per cent. 

0

4.37

2000

1,500

1,000

D

0123456789

Slope

FIGURE 20. Extrapolated curves of C=/(S) for tentative equations of interpolation, and for slopes greater than 2.4 per cent.

The approximate range of this series of 

experiments is from a slope of 0.8 per cent to 

one of 2.4 per cent. The extrapolated curves 

pertain to slopes from 2.4 to 10 per cent and 

from 0.8 to 0 per cent. The prompt divergence 

of the lines as they leave the locus to which 

they were adjusted shows that they have 

widely different values for purposes of extrapolation, and therefore presumably for purposes of interpolation.

Attention being given first to the curves for 

higher slopes (fig. 20), it will be observed that 

four of them ascend with progressively increasing rate. The curve of formula (lla) ascends 

continuously, but its rate of ascent changes 

at the slope of 4.37 per cent from an increasing 

rate to a decreasing rate. The curve of formula 

(6a) exchanges its increasing rate of ascent for 

a decreasing rate at the slope of 3.33 per cent, 

attains a maximum at a slope of about 6 per
ADJUSTMENT OF OBSERVATIONS. 63

cent, and crosses the line of zero capacity 

before reaching the slope of 9 per cent.

The general characteristics of stream traction 

do not admit of a maximum in the relation 

of capacity to slope. Capacity for traction is 

clearly an increasing function of the stream's 

velocity, and the velocity is clearly an increasing function of the slope. There is reason also 

to believe that capacity increases at an increasing rate up to the slope corresponding to 

infinite capacity. There are three forces concerned in traction first, the force of the current, of which the direction is parallel to the 

slope; second, a component of gravity, when 

gravity is resolved in directions parallel and

normal to the slope; third, the resistance of 

the bed, which is a function not only of the 

others, but inversely of the slope. Within the 

range of experimental slopes the component of 

gravity is negligible in comparison with the 

force of the current, and the influence of slope 

on the resistance is relatively unimportant; 

but as the angle of stability for loose material 

is approached the resistance diminishes rapidly, 

and at the slope of instability (65 to 70 per 

cent for river sand) gravity is competent to 

transport without the aid of current, and the 

stream'.s capacity is infinite. All these factors 

depend on slope, and as the increment to 

capacity verges on infinity in approaching the

+ 15

+ 5

-5

-10

<7aJ

/(So)

.1 .2 .3 .4 .5 .6 .7 .8 .9

Slope

FIGURE 21. Extrapolated curves of C=/(S) for tentative equations of interpolation and for slopes less than 0.8 per cent.

slope which limits variation, it is highly probable that capacity grows continuously with 

slope.

This criterion suffices for the rejection not 

only of the specific formulas (6a) and (lla), 

but also of their types, (6) and (11). In formula (6a) the occurrence of the maximum 

value of 0 is determined by the negative coefficient of $3 ; and it is true as a general fact that 

equations of the class indicated by (12) yield 

maxima whenever the coefficient of the highest 

power of the independent variable is negative. 

It is possible, or perhaps probable, that if each 

series of laboratory values were to be formulated under (7) or (8) the conditions for maxima 

would be found to occur. On the whole, the

extrapolations for higher slopes tend to restrict 

choice to forms (9) and (10), with some reservation as to forms (7) and (8).

Figure 21 gives extrapolated curves for 

slopes less than 0.8 per cent and represents the 

same equations as figure 20, except that the 

curve for (6a) is omitted. It will be observed 

that it magnifies greatly the space between 0 

and B in figure 20, the scale of slopes being 10 

times and the scale of capacities 100 times as 

large. The implications of the functions for 

low slopes are specially important because 

extrapolation from laboratory conditions to 

those of natural streams will nearly always 

involve the passage from higher to lower 

slopes.
64 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

Curves (7a) and (11 a) reach the origin of 

coordinates that is, their equations indicate 

that at the zero of slope there is no capacity 

for traction. Formula (lla) gives small but 

finite capacities for very low slopes; but under 

formula (7a) finite capacities cease when the 

slope falls to 0.29 per cent, and for lower slopes 

there is indication of negative capacities. If 

the conditions of traction permitted, negative 

capacity might be interpreted as capacity for 

traction upstream; but as this is inadmissible, 

the negative values may be classed as surd 

results arising from the imperfection of the 

correlation between an abstract formula and a 

concrete problem. The curves of (Sa) and (9a) 

also intersect the axis of slope at some distance 

from the origin, and their extensions indicate 

negative capacity. The curve of (lOa) becomes 

tangent to the axis of slope at the point corresponding to a slope of 0.39 per cent and there 

ends, having no continuation below the axis. 

It is the real limb of a parabola of which all 

other parts are imaginary. It expresses to the 

eye the implication of formula (lOa) that traction ceases when the slope is reduced to 0.39 

per cent, and that its cessation is not abrupt 

but gradual; and also the implication of the 

general formula (10) that traction ceases when 

the slope is reduced to the value a.

It is a matter of observation that when slope 

is gradually reduced, the current becoming 

feebler and the capacity gradually less, the 

zero of capacity is reached before the zero of 

slope. For each group of conditions (fineness, 

width, discharge) there is a particular slope 

corresponding to the zero of capacity. It is 

also a matter of observation that the change 

in capacity near the zero is gradual. Formulas 

(7a), (8a), (9a), and (lOa) therefore accord 

with the data of observation in the fact that 

they connect the zero of capacity with a finite 

slope; formula (lla), which connects zero 

capacity with zero slope, is discordant. Also, 

formulas (lOa) and (lla) accord with the data 

of observation in that they make the approach 

of capacity to its zero gradual; while formulas 

(7a), (8a), and (9a), which make the arrival of 

capacity at its zero abrupt, are in that respect 

discordant.

But one of the formulas (lOa), shows qualitative agreement with both of the criteria 

applied through extrapolation to low slopes; 

and that formula is one of the two which

respond best to the criterion applied through 

extrapolation to high slopes. That type of 

formula, or

0=^(8-0)*.... ...... .(10)

was therefore selected for the reduction of the 

more or less irregular series of observational 

values of capacity to ' orderly series better 

suited for comparative study.

In rewriting the formula the "coefficient is 

changed from & to 11} because corresponding 

coefficients &2 , &3 , etc., are to be used in a series 

of formulas expressing the relations of capacity 

to various conditions. As slope is a ratio 

between lengths, (S a~)n is of zero dimensions 

and &! is of the unit C; it is the value of capacity 

when S o=l.

The slope which is barely sufficient to 

initiate traction has been defined (p. 35) as 

the competent slope. To whatever extent a 

represents the competent slope the formula 

has a rational basis. The local potential 

energy of a stream, or the energy available at 

any cross section in a unit of time, is simply 

proportional to the product of discharge by 

slope or, if the discharge be constant, is proportional to the slope. So long as the slope 

is less than that of competence the energy is 

expended on resistances at contact with wetted 

perimeter and air and on internal work 

occasioned by thpse resistances. When the 

slope exceeds the competent slope, part of the 

energy is used as before and part is used in 

traction. The change from competent slope 

to a steeper slope increases the available 

energy by an amount proportional to the 

increase of slope, and the increase of energy is 

associated with the added work of traction. 

Capacity for traction, beginning at competent 

slope, increases pari passu with the increase 

of the excess of slope above the competent, 

slope, and there is manifest propriety in treating it as a function of the excess of slope 

rather than of the total slope. It is of course 

also a function of the total slope; but an 

adequate formula for its relation to the excess 

of slope may reasonably be supposed to be 

simpler than a formula for its relation to total 

slope. If a represents competent slope, then 

the relation of capacity to S a should be 

simpler than its relation to S.

Instructive information as to the relative 

simplicity of the two functions is obtained by
ADJUSTMENT OF OBSERVATIONS. 65

comparing their logarithmic graphs. In figure 

22 the curved line A3 has been, copied from the 

curve in figure 19.. It is the graph of log 0= 

f Qog £) for grade (G), width 0.66 foot, and discharge 0.734 ft.3/sec. If for 8 we substitute 

5 0.3, we modify the graph by moving each 

point of it to the left by an amount equal to 

logs' -log (8 0.3); and we produce the lino 

CD, which is the graph of log C=fu (\og 

($ 0.3)). If in similar manner we derive 

the graph of log C=fm(\og (S-0.6)), the result is the line EF. CD curves in the same 

direction as AB but less strongly; EF curves 

in the opposite direction. It is evident that

the three curves belong to a continuous series, 

and that somewhere between CD and EF a 

member of the series is straight or approximately straight. That straight line, GH, is 

the graph of log ^/^(log (£-0.39)); but as 

it is straight, its equation may be written

log (7= log &! + » log (£-0.39),

in which log^ is the ordinate of the intersection of the line with the axis of log C, and n 

is the trigonometric tangent measuring the 

inclination of the line to the axis of log 8. 

This is identical with equation (15) except 

that '0.39 appears in place of a; and in fact

200

100

80

o 60

20

/

£G

77 /

LL_±

.1 .Z .4 £ .8 .UO 2.0 

Slope

FIGURE 22. The relation of a in C=f(S d) to the curvature of the logarithmic graph.

the value of a in equation (lOa) was computed 

graphically by means of the logarithmic plot.

The line AB, being the graph of log C= 

/i (log /S), is also the logarithmic graph of 

C=f(S). The line GH, being the graph of 

log C=frr(\og (S a)) is also the logarithmic 

graph of #=/v (S <r). Their relation in respect to simplicity is that of the curve to the 

straight line.

In view of these suggestions of harmony it is 

peculiarly pertinent to inquire whether a is 

actually representative of competent slope; 

and it will be convenient to make that inquiry 

in connection with the determination of values

20921° No. 86 14 5

of a for the several series of observations on 

capacity and slope.

THE CONSTANT a AND COMPETENT SLOPE.

In the experimental data for graded debris 

Table 4, (A) to (H) are 117 series of values 

of capacity and slope. After these had been 

plotted and inspected in a comparative way, 

it was decided to restrict the main discussions 

to 92 series only, the discarded series being all 

short as well as somewhat discrepant among, 

themselves. Of the 92 series retained, only 

30 afford information as to the corresponding values of a; that is, only 30 of the loga-
66 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

rithmic plots exhibit curvature so definitely 

that the approximate magnitudes of the constants necessary to eliminate it can be inferred. 

In some of the remaining series the observations are not so distributed as to bring out the 

curvature. For others the observational positions on the plots are too widely dispersed to 

give good indication of the character of the 

best representative line.

On the 30 logarithmic plots the curves approximately representing the observations were 

drawn, and the values of a necessary to replace 

the curves by straight lines were computed 

graphically in the manner just indicated. 

These values are given, to the nearest tenth of 

1 per cent of slope, in Table 7, where they are 

arranged with reference to the conditions of 

the experiments.

TABLE 7. Values of a in C^b^S a) , estimated from logarithmic plots of observations.

Grade

(A) 

(B)

(C)

(D) 

(E) 

(F) 

(0) 

(H)

F

25,200 

13,400

5,460

1,460 

142 

22.1 

5.9 

2.0

Range 

of F

1.77 

2.35

2.85

3.58 

12.65 

2.15 

2.92 

2.51

W

(feet).

0.66 

1.00 

1.32 

1.96

.23 

.44 

.66 

1.00 

1.32 

1.96

.44 

.66 

1.00 

1.32 

1.96

.66 

1.00 

1.32

.66 

1.00 

1.32

.66 

1.00 

1.32

.66 

1.00 

1.32

.66

Values of » for discharge (ft. '/sec.) of 

0.093

-

0.7 

.5

.4 

.2

-

0.182

-

0.2

.4 

.0

.1 

.3

-

.0

0.363

0.1

.1

.1 

.1 

.3

.0

' 

.6

.6

. 0.545

-

-

0.1

-

0.734

-

0.1

.1

.1

.0

.4

.3

.4

.fi

0.923 1.021 1.119

-

0.3 

.3

.4

NOTE. The horizontal dashes indicate series of observations to which values of a are to be assigned.

The same table shows the distribution of the 

experimental series which fail to give values of 

a but to which it is proposed to assign values. 

In order to assign these values properly it is 

necessary to know the laws of variation of a 

with reference to the conditions of experimentation. These are suggested in part by the 

roughly determined values of the table and 

are. otherwise indicated by general considerations. The variations are connected with at 

least three conditions discharge, width, and 

fineness and are less surely connected with 

range of fineness.

Considering first the variation of a with discharge apd giving attention in the table to 

values of a falling in the same horizontal line, 

we find by inspection that invariably the value

for a larger discharge is either less than or 

equal to the value for the corresponding smaller 

discharge. The indication is that a is a decreasing function of discharge. This relation 

might have been inferred from general considerations, on the theory that o represents 

competent slope. Competent slope is the slope 

giving competent velocity along the bed, but 

bed velocity also varies directly with discharge. 

With large discharge less slope is necessary to 

induce competent velocity; with small discharge more slope. In other words, competent slope varies inversely with discharge. In 

a preliminary discussion of the traction data 

for debris of grades (B) and (C) it was found 

that capacity is more sensitive to changes in 

slope than to changes in discharge. The rela-
ADJUSTMENT OF OBSERVATIONS. 67

tion of capacity to slope and discharge jointly 

being tentatively represented by

<7cc (16)

the values of n and m were computed for many 

different conditions, and it was found that on

nrv\

the average = 0.34. There is, however, considerable variation in the ratio, and it is relatively large when C is small. On plotting the 

values of the ratio in relation to C and graphically extrapolating, it was found that when 

(7=0 the ratio is about 0.5. The condition of 

zero capacity is that of competent bed velocity 

and competent slope.

For any particular value of C in (16) the 

product of Sn by Qm is constant, or

1

whence

But, as we have just seen, when <7has the parfff\j ticular value C=Q, = 0.5. Therefore competent slope varies inversely as Q°-5 . Its assumed 

representative, a, is assumed to vary inversely 

with the square root of discharge.1

Turning now to the relation of a to width, 

w, and examining Table 7, we see that a is not 

exclusively either an increasing or a decreasing function of w. Where the smallest widths 

are concerned, as with $ = 0.093 ft.3/sec. arid 

# = 0.182 ft.3/sec., for grades (B) and (C), the 

function is decreasing. Where the greater 

widths are concerned it is for the most part inr 

creasing. There is a rational explanation for 

such double relationship in the case of competent slope.

Figure 23 represents two troughs in cross 

section. Each has a bed of debris, and they 

are assumed to be carrying the same total 

discharge. In the wider there is less discharge 

for each unit of width, and the tendency of the 

smaller discharge is to reduce velocity. Therefore to maintain a particular velocity namely,

i The preliminary discussion on which is based a oc -f^ did not have 

the advantage of the formula using a. A rediscussion, to be found in 

Chapter V, yields a oc ^^; but it was not practicable to give the adjustment the benefit of this later work without repeating the greater part 

of the computations of the paper. It is not believed that the advantage 

to the results would be commensurate with the labor involved.

the competent velocity the tendency is to 

produce a relatively steep slope. So far as 

this factor is concerned, competent slope is 

relatively steep for a wider trough. But there 

is another factor; the velocity is influenced by 

the resistance of the sides of the trough. This 

resistance is greater where the water is deeper, 

because the surface of contact is broader; and 

the water is deeper in the narrower trough. 

The tendency of the resistance is to reduce 

velocities and therefore to make the slope for 

competent velocity steeper in the narrower 

trough.

Water'

FIGURE 23. Diagrammatic sections of laboratory troughs, illustrating 

relation of current depth to trough width.

In very wide troughs the influence of the 

sides is of minor importance, the influence of 

discharge per unit width dominates, and the 

competent slope varies directly with the width. 

In very narrow troughs the influence of the 

sides dominates, and the competent slope 

varies inversely with the width. For some 

intermediate width the two tendencies are 

balanced, and the competent slope has its 

minimum value. In figure 24 abscissas 

measured from 0 represent width of trough, 

and ordinates represent competent slope. The 

curve sketched, while not quantitative, has 

adequate basis for its broader features and

c. s.

FIGURE 24. Ideal curve of competent slope ( C. S.) in relation to width 

of trough (W).

shows the general character of the relation of 

competent slope to width. From its minimum 

it ascends gradually on the side of greater 

width, and on the side of lesser width rises with 

relative rapidity toward a vertical asymptote 

near, the axis of competent slope. These 

various characters are approximately paralleled 

by the variations of a as shown in the table.

The relation of a to fineness does not come 

out very clearly in Table 7, and a different
68, TRANSPORTATION OF DEBRIS BY RUNNING WATER.

arrangements is therefore, given in Table 8, 

w]ti<^|omits the. values of a not affording comparison1 , anol adds three interpolated values. 

The general fact thus shown is that a increases, 

as fineness diminishes, but to this there are two 

exceptions,,,,.. The more important exception 

is, in,, grade .-(!£), of which all values of a are less 

than the (corresponding values for. .either grade 

(C) or.grade (G). The .values for grades (B) 

and (C) appear to be about the same, although 

the two grades differ notably in fineness. The 

exceptions are associated with peculiarities of 

the grades with respect to range of fineness. 

In grade (E) the range for bulk fineness is about 

twice as great as in any of the other grades; 

and if this character indicates the cause of its 

abnormally small values of a, then the abnormally high values in grade (B) are explained 

by its small range, of fineness as compared to 

the range of grade (C).

TABLE 8. Values of a, from Table 7, arranged to show variation in relation to fineness of debris.

Fineness (F2). ....

Range of fineness..

Width.

0.44 

.44 

.66 

.66 

1.00 

1.00 

1.00 

1.32

Discharge.

0.093 

.182 

.363 

.734 

.182 

.363 

.734 

.734

(B).

13,400

2.35

(C):

5,460

2.85

(E).

142

12.65

(G).

5.9

2.92

(H).

2.0

2.51

Values of a.

0.5 

.0

.1

.1

0.4 

.2 

.1

.1

.3 

.1

.1 

.1

0.0 

.0 

.0

0.6

.4

.6 

.3 

.4

0.6

The values of o inferred from the logarithmic 

plots are not sufficiently precise to yield 

quantitative laws of the relations of a to 

fineness and the range of fineness, and it is 

again necessary to seek instead the laws 

governing competent slope. Does competent 

slope vary inversely with fineness, and what 

is its law? Does competent slope vary inversely with range of fineness, and what is its 

law ?

Inasmuch as fine debris is moved by a relatively slow current and as the force of the current is a direct function of slope, the competent 

slope for fine debris is less than for coarse. If 

we accept the thesis of Leslie and Hopkins (see 

p. 16) that competent bed velocity (F^) varies 

with the sixth root of the volume or mass of

the debris particle, then, since bulk fineness is 

the reciprocal of volume,

Vrh OC -f7

If we assume, from the Chezy formula, that 

mean velocity of current is proportional to the 

square root of slope and apply it to component 

mean velocity (Vcm) and competent slope 

($c ), we have

If we further assume that bed velocity is proportional to mean velocity, then, by combining 

the three proportions and reducing, we obtain

S'oc

T71 0.33" 

J 2

(17)'

As each of the three assumed laws is subject to 

important qualifications, the product of their 

combination must be regarded as but a rough 

approximation to the law connecting competent slope with fineness.

Further light on the law is afforded by some 

experiments made for the specific purpose of 

determining competent slope. In these experiments the discharge remained constant while the 

velocity was modified by changing the width of 

the outfall. The slope of the bed had been prepared in advance, and the slope of the water 

surface was measured for each width of outfall. At each stage of the experiment the 

movement of grains of debris along the bottom 

was noted by such phrases as "many," "several," "few," "very few," "none," the words 

being used in that order as a sort of scale. 

Competent slope was inferred from a comparison of these notes with the recorded slopes of 

the water surface. The results are given in 

Table 9.

In a closely related series of experiments, 

Table 10, a slope of debris was prepared in 

advance, a small discharge was passed over it, 

and the discharge was progressively increased, 

with notes on the movement of debris grains. 

These experiments gave competent discharge.

In each series the depths were measured, and 

from these the mean velocities (Vm) were 

computed.

The experimental determinations were indefinite for several reasons. In the first place,
ADJUSTMENT OF OBSERVATIONS. 69

a stream with the critical bottom velocity does 

not transport debris and therefore does not 

establish its own slope of bed. The slopes 

artificially prepared were of necessity imperfectly adjusted. In the next place, the prepared slopes did not imitate the natural 

diversity of detail but were plane. When a 

current of competent velocity passed over one 

of them it immediately began to shape the bed 

into dunes, and as the modeling proceeded the 

activity of transportation increased. After the 

dunes were formed, a smaller general velocity, 

or a less discharge, or a lower general slope 

was competent. The experiments being made

in sets, the first of a set gave a result from the 

plane bed and the others from a more or less 

diversified bed.

In the third place, the particles composing 

one of the experimental grades of debris were 

not of uniform mobility. Not only were they 

of diverse size, as indicated by the "range of 

fineness," but they were different in shape and 

in specific gravity, so that some were able to 

resist a considerably stronger current than 

others. The competent slope for the least 

mobile particles was materially steeper than 

that for the most mobile, and no mode of 

gaging average mobility was discovered.

TABLE 9. Experimental data on competent slope.

Grade 

of 

debris.

(B).....

m o;.....

(D).....

(E).....

(G).....

Width 

of 

trough.

Feet.

1.00

1.00

1.00

1.00

1.00

Width of contractor.

Feet.

0.30 

.40

.30

.30

.40 

.30

.30

.40

.30 

.40 

.40

.40

.45

.50

.55

.40

.40

.50

.50 

.60 

.60

.70

.87

.87 

.87 

.87 

.87

.87

Discharge.

Ft.ysec.

0.363 

.363

.363

.363

.363 

.734

.363

.363

.734 

.734 

.363

.363

.363

.363

.363

.363

.734

.363

.734 

.363 

.363

.363

.363

.363 

.734 

.363 

.363

.363

Slope 

deTjris.

Per ct.

0.048 

.03

.028

.10

.10 

.10

.02

.02

.02 

.02 

.06

.10

.10

.10

.10

.05

.035

.035

.035 

.05 

.10

.40

.40

.50 

.50 

.65 

.80

1.00

Slope 

of 

water.

Per ct.

0.01 

.06

.03

.01

.042 

.045

.21

.34

.33 

.57 

.03

.055

.054

.056

 .043

.025

.057

.069 

.086 

.096

.22

.29

.46 

.46 

.70 

.80

.93

Depth.

Feet.

0.490 

.425

.475

.451

.391

.465

.418

.743 

.658 

.392

.373

.357

.387

.618

.357

.562 

.330 

.314

.256

.238

.225 

.360 

.201 

.184

.173

Mean 

velocity.

Ft./sec.

0.74

.85

.76

.80

.93

.78

.87

.99 

1.12 

.93

.97

1.02

.94

1.19

1.02

1.31 

1.10 

1.16

1.42

1.52

1.61 

1.01 

1.80 

1.98

2.10

Notes on movement of debris.

A very few grains moving. Small dunes forming. 

Many grains moving. Dunes forming.

A very few grains moving. Some dunes forming.

A very few grains moving in a few places.

Many grains moving. Dunes forming. 

Dunes forming rapidly.

No grains moving.

An occasional grain moving.

Very few, if any, grains moving. 

Many grains moving. Dunes forming. 

A very few grains moving. No dunes forming.

Very few grains moving.

Some small grams moving.

Several small grains moving. No dunes forming.

Many grains moving.

No transportation.

Do.

Do.

Some transportation in lower half of slope. 

Very few grains moving in lower half of slope. 

A few grains moving.

No grains moving.

Do.

A few grains moving at one place. 

A few grains moving in places. 

Very few; if any, grains moving. 

An occasional grain moving.

Several grains moving in center half of slope.

When an experiment was begun with a velocity 

well below competence, and the velocity was 

gradually increased, the first movement detected would be the saltation of some small or 

light particle, and then the number of particles 

moving would gradually grow with the quickening of current.

An attempt to correlate the " notes on movement of de"bris" in Table 10 for a discharge of 

0.363 ft.3 /sec. gave the following values of 

water slope for equivalent phases of movement:

Grade.............. (B) (C) (D) (E) (G)

Slope of water (per 

cent)............. 0.03 0.03 0.06 0.10 0.93

By assuming the power function Sc = aFzn, or 

log $c = log a + n log F2, and plotting log Sc in

relation to log F2 , the value found for n is about 

 0.5, but it has a large uncertainty. The

-E

($.

2 

Log F

FIGURE 25. Logarithmic plot of competent) slope in relation to fineness 

of debris.

plot, figure 25, shows the competent slope for 

grade (E) smaller than would be indicated by
70 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

its neighbors, but not so much smaller as is the 

value of a for grade (E) in Table 7. It also 

makes the competent slope for grade (B) 

aberrant in the same sense as the corresponding

value of a. The data are too vague to give 

much value to such correspondences; but the 

broader resemblances aid in connecting a with 

competent slope.

TABLE 10. Experimental data on competent discharge.

Grade 

of 

debris.

fT?\

OF).....

(G)....

(H)...

Width 

of 

trough.

Feet.

1. 32

1.00

.00

.66

1.00

.66

1.00

1.32

.66

Width 

of contractor.

Feet.

0.90

.72

' 02

.43

.40

(?)

.45

.70

.92

.46

.45

Discharge.

Ft. 3/sec.

0.039

.019

.019

.039

.039

.019

.010

.019

.019

.029

.039

. 058

.075

.093

.010 

.019

.010

.019

.039

.039

.058

.075 

.093

.111

.140

.104

.140

.104 

.182

.200

.218

.164 

.182 

.200 

.140

.111

.093

.075 

.058

.075

.093

.182

.218

. 2n5

.290

.182

.218

. 255 

.182

.140

.111

.093

.111

.128

.MO

.272

.327

. 34ft

.3R3

.218 

.237 

.255

.272

.290

.218

.237

.255

.146

.164

.182

.363 

.454 

.545

.639

.272 

.308

.345

.182

.218 

.255

Slope 

of 

de"bris.

Perct.

2.00

1.95

1.00

1.00

1.00

1.00

2.00

1.94

1.00

1.00

.50

.50

.50

.50

2.00 

2.00

1.13

1.13

1.13

2.51

1.05

1.05 

1.05

1.05

1.00

1.00

1.10

1.10 

1.10

1.10

1.10

1.68 

1.58 

1.58 

2.05

2.05

2.05

2.05 

2.50

2.50

2.50

1.00

1.00

1.00

1.00

1.45

1.45

1.45 

1.90

1.90

1.90

2.52

2.52

2.52

2.52

1.05

1.05

1.05

l.Oo

1.40 

1.40 

1.40

1.40

1.10

2.00

2.00

2.00

2.50

2.50

2.50

1.10 

1.10 

1.10

1.10

1.30 

1.30

1.30

2.05

2.05 

2.05

Depth.

Feet.

0.015

.014

.026

.024

.018

.015

.025

.053

.060

.065

.070

.022 

.028

.042

.050

.043

.071 

.092

.086

.093

.119

.126 

.139

.143

.153

.112 

.104 

.104 

.106

.088

.088

.070 

.052

.054

.072

.124

.141

.153

.159

.089

.098

.105 

.090

.080

.059

.061

.067

.074 ""."iis

.125

.128

.130

.075 

.079 

.084

.090

.099

.079

.090

.097

.055

.061

.069

.212 

.260 

.399

.329

.163 

.171

.184

.132

.137 

.141

i 

Mean 

velocity.

Ft. /sec.

0.96

1.03

1.14

1.23

.80

.67

.76

.74

.96

1.15

'1.33

.09 

1.03

.09

1.19

' 1.36

1.59 

1.52

1.70

1.77

1.84

1.0.5 

1.98

2.10

2.13

2.20 

2.03 

2.88 

2.07

1.89

1.58

1.01 

1. 01 '

2.08

1.95

1.47

1.55

1.07

1.84

2.04

2.23

2.43 

2.02

1.83

1.88

1.52

1.66

1.73

""i.79

1.99

2.04

2.11

2.21 

2.27 

2.27

2.28

2.22

2.09

1.99

1.98

2.01

2.04

1.99

2.57 

2. r>2 

2.74

2.91

2.49 

2.70

2.82

2.05

2.39 

2.72

Notes on movement of debris.

Many grains moving. Dunes quickly formed.

Some grains moving; many moving as small channels are formed or surface becomes 

rough.

No grains moving.

Many grains moving.

Several grains moving. Dunes forming.

No grains moving.

A few grains moving in a few places. Dunes forming.

Many .grains moving. Dunes forming.

A few grains moving in a few places.

Several grains moving in several places.

No grains moving.

A few grains moving in a few places.

Severn! grains moving in several places.

Transportation.

A few grains moving in a few places. 

Several grains moving in nearly all parts of trough.

No motion.

A few grains moving in a few places.

Many grains moving. Surface becoming rough.

An occasional grain moving. Several grains moving after a time.

No grains moving.

A few grains moving in a few places. 

Several grains moving in some places.

Many grains moving.

Few grains moving in some places.

Several grains moving.

No grains moving.

Very few grains moving. 

A few grains moving near middle of trough.

A few grains moving.

Several grains moving in lower half of trough.

Some grains moving in lower half of trough. 

Several grains moving in lower half of trough, but very few in upper half. 

Many grains moving in lower half, but few in upper half. 

Several grains moving in middle of trough; many in lower part. Cutting of grade.

A few grains moving.

Very few grains moving.

No grains moving.

A few grains moving.

Many grains moving.

No grains moving.

No (or very few) "grains moving.

Very few moving.

Many grains moving.

Very few small grains moving.

Several grains moving. *" Many grains moving. 

Several grains moving.

Very few grains moving.

No grains moving.

1)0.

A very few grains moving in part of trough.

A few' grains moving, except near head of trough.

Many grains moving. Grade cutting in places.

No grains moving.

Very few grains moving.

A few grains moving.

Several grains moving.

Very few grains moving near center of trough. 

None moving except at middle of trough. 

Very few moving except at middle of trough.

A few grains moving.

Several moving in middle and lower parts; very few in upper.

A few grains moving.

Several grains moving in middle part of trough.

Several grains moving.

Very few grains moving.

Several grains moving.

Do.

Occasionally a grain moving. 

A few grains moving in lower half of trough. 

Several grains moving in lower two-thkds of trough, and occasionally a grain in upper

third.

Several grains moving in middle third of trough; a few in upper and lower parts.

Occasionally one grain moving. 

Some grains moving below middle; 1 to 3 grains in a cross section.

2 to 5 grains moving in a cross section.

No grains moving.

A few grains moving in lower three-fourths of trough. 

Many grains moving in lower three-fourths ol trough.
ADJUSTMENT OF OBSERVATIONS. 71

In treating the data of Table 9, the assumption was made that the note "several grains 

moving" corresponds to competent slope; and 

averages were found of the values of mean 

velocity and depth. 1 These are:

Grade.................. (E) (F) (G) (H)

Mean velocity (ft/sec.).- 1.10 1.52 2.14 2.83

Depth (foot)............ 0.020 0. 092 0.108 0. 218

When the logarithms of these numbers are 

plotted the positions fall well in line, and the 

representative line.gives (Vcm indicating the 

mean velocity corresponding to competent bed 

velocity)

o n

7 

F CTO 

= _.

Ty- 0.22 .........(18)

 L >

Assuming again that bed velocity is proportional to mean velocity, and again assuming 

the validity of the Chezy formula, we obtain 

from (18)

1 .........(19) 0.44

The two values of the exponent of F2 derived 

from the experiments, namely, 0.44 and 

 0.50, are both larger than the deductive 

value, -0.33, of equation (17), but the disparity is quite natural in view of the indefiniteness 

of the data and the uncertainties of the assumptions. Collectively the values indicate an order 

of magnitude..

The influence of range of fineness on competent slope appears to be of the same nature as 

its influence on a, though much less pronounced, 

but the determinations of competent slope are 

too indefinite to give the greatest value to the 

comparison. It is significant, however, that 

while the logarithmic plots for grade (E:) and 

width 1.00 and for three different discharges 

(Table 7) all yield values of a less than 0.05 per 

cent, the experiments on competent slope (Table 9) record for one of the discharges "no 

grains moving" with a slope of 0.21 per cent, 

and for another "very few, if any, grains moving" with a slope of 0.33 per cent. The values 

of a in this case fall far below those for the most 

mobile components of the debris grade which 

has the largest range of fineness. The general 

facts appear to be that a varies decreasingly

1 A few series of observations on competent velocity have been made 

by others. They pertain chiefly to flume traction and are cited in 

Chapter XII. Login, whose results are given in Chapter VII, used the 

methods of stream traction but omitted to measure the sizes of materials 

transported.

with range of fineness, and that competent 

slope is subject to a variation of the same kind, 

which may or may not be of the same magnitude.

The cause of this variation is not surely 

known, but a plausible suggestion in regard to 

it may be made. In the experiments with 

mixtures of two or more grades it was found that 

before the slope had been established, especially when low velocities were used, the current tended to sort the debris, building deposits 

with the coarser part and delivering the finer 

material at the end of the trough. In experiments with a single grade the same tendency 

doubtless existed. It was in fact observed in 

connection with dunes and antidunes, which 

sometimes showed a shading in color due to 

partial sorting with respect to density, the 

heavier particles being dark, the lighter pale. 

As the differences of size within a grade were 

not such as to appeal strongly to the eye, considerable sorting with respect to size might 

take place without attracting attention. With 

the ordinary routine of the experiments, which 

began in each series with low slopes and velocities and gradually increased them, the influences of such sorting may have been systematic, 

and thus may have modified that relation of 

values which finds expression in the constant a. 

The result of such influence would be more pronounced for grade (E) than for grades with 

smaller range of fineness. It is easy to see also 

that an allied influence may have affected to 

some extent the interpretation of the experiments on competent slope.

The variations of a are paralleled in so many 

ways by the variations of competent slope as 

to leave little doubt that the one is in some 

way representative of the other. It can not 

be said that the constant a, arbitrarily introduced to rectify curves and thereby facilitate 

interpolation, is the equivalent of the slope of 

competence if for no other reason than that 

the competent slope for a grade of debris made 

of unequal grains eludes precise definition 

but it may well be a complex function of the 

competent slopes of all the different sorts of 

grains contained in one of the laboratory 

grades.

For the practical purpose of obtaining values 

of a for use in formulas of interpolation, the 

preceding discussion yields a large body of 

pertinent information. Sigma varies inversely
72 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

with a power of discharge, approximately the 

0.5 power. It varies inversely with a somewhat smaller power of the bulk fineness of 

de"bris. It varies inversely with the range of 

fineness of the grades of debris, this variation 

serving to qualify the preceding. It varies 

with width of channel, the variation including 

a minimum. Without attempting to give 

definite symbolic expression to these laws of 

variation, they were applied to the practical 

problem, and by a series of adjustments the 

skeleton of values of a in Table 8 was developed 

into a system covering the whole range of 

experimental conditions. That system is presented in Table 11.

TABLE 11. Values of a, in per ce^t of slope, as adjusted for 

use in interpolation equations of the form C=b1 (S a)n.

1 

Grade.

(A).....

i

/Ty\

(C)..... 

(D).....

/T7\

/T?\

(G).....

(H).....

Width 

(feet).

0.66 

1.00 

1.32 

1.96

.23

.44 

.66 

1.00 

1.32 

1.96

.44 

.66 

1.00 

1.32 

1.96'

.66

1.00 

1.32

.66

1.00 

1.32

.66

1.00 

1.32

.66

1.00 

1.32

.66

Values of a for discharge ( ftflsec. ) of 

0.093

0.40

.70

.50 

.30

.40 

.16

.19

......

......

0.182

0.20

.12 

.17

.fiO

.10 

.10 

.12 

.17

.20 .11' 

.15 

.22

.14

.17

.08

.33

.44

0.363

"aw"

.10

.17

.08 

.08 

.12 

.18

'".08 

.11 

.16 

.24

.12 

.16

.09

.06 

.04

.21

.31 

.39

.50

.58

.71

.80

0.545

0.04

.07 

.06 

.10 

.14

".'oe"

.09 

.13 

.20

.08

.12

......

......

0.734

"o'os"

.07 

.10

.00 

.08 

.12

"."64" 

.07 

.11

.17

10 

.11

.06

.04 

.03

.17

.22 

.28

.36

.41

.50

.56

0.923

..'.'."

.'...'.'.

......

0.08

 >

......

....!!

1.021

......

......

......

 - - 

1.119

 --"

"aio

.06 "".u

.. .03 

.02

.17 

.23

.27 

.33 

.40

.48

INTERPOLATION.

The values of a having been assigned, the 

data of Table 4 were once more plotted on 

logarithmic paper, the ordinates again representing load or capacity, and the abscissas representing S a, or observed slope less the constant slope a. As in the preliminary plotting 

for inspection, the primary data were (1) the 

estimates of load from the quantity of de*bris 

delivered at the end of the trough and (2) the 

associated slopes of the bed of debris, and 

accessory data were added with distinctive

notation. The secondary data were used 

chiefly to indicate the relative precision of the 

primary data, the primary having greater 

weight when agreeing more closely with the 

secondary. The illustrative plot, figure 26, 

shows only the primary data.

The next step was to draw through and 

among the observational points the best representative straight line. It is a property of 

the logarithmic plot that its distances represent ratios, and the scale .of ratios is everywhere the same. Similar errors of observational positions are shown by similar distances 

in all parts of che plot, provided the errors are 

considered as fractional parts of the plotted 

quantities. Each observational point should 

be given the same influence in determining the

77

FIGURE 26. Illustration of the method used to adjust values of capacity, 

in relation to slope, by means of a logarithmic plot of observed values 

of capacity in relation to slope minus a.

representative line, provided the (fractional) 

probable errors of the observations are the 

same. In this case, however, the fractional 

probable errors for low capacities and slopes 

are much greater than for high capacities, and 

the best representative lines can not be drawn 

without consideration of weights. Adjustment by the least-squares method was considered and experiments were tried, but the 

labor entailed by the necessity of using weights 

was not thought to be warranted by the 

quality of the data. The following simpler 

and less rigorous method was employed:

A group of observational points corresponding to the highest slopes was selected by inspection of the plot, and its center of gravity 

was computed by a graphic method. In the
ADJUSTMENT OF OBSERVATIONS. 73

case shown by figure 26, the group includes 

six points and the center of gravity is the 

point indicated by an arrow. The representative line was made to pass through this 

point and was otherwise adjusted to position 

by eye estimate, with consideration of all the observational points and their supposed weights. 

By this method the observations of greatest 

weight were enabled to fix one point on the 

line and were also consulted, along with observations of less weight, as to its direction or 

attitude. The method obviously left much to 

personal judgment, but had a rigorous method 

been attempted it would have been difficult or 

impossible to avoid the use of nonrigorous 

judgment in the assignment of weights.

The line when drawn is a generalized expression, for a single series of observations, 

of the relation of capacity to slope less a. It 

is the graphic equivalent, or graph, of a specific 

equation of the form

log C=log l^ + n log (S-o).... . (20) 

wliich is the logarithmic equivalent of

0=^(8-0)*-........... .(10)

The inclination of the .line (tan 6} was next 

measured, or computed, giving the numerical 

value of n; and its point of intersection with 

the axis of log C was determined, giving the 

value of lr The values of n, b1} and a for the 

92 series are shown in Table 15.

The graph was used, instead of the equivalent equation, in computing values of 0 corresponding to systems of values of 8. The 

results are given in Table 12 and constitute the 

data for further generalizations. To compute 

values of C for a particular observational series, the value of a for the series was first subtracted from each value of S in the adopted 

system, then each remainder (8 a) was applied as an argument to the graph and the 

corresponding value of C read off. The range 

of values thus computed and tabulated was 

either limited by the range of observational 

values, or else included a moderate extrapolation, which never exceeded 10 per cent of the 

observational range.

PRECISION.

II' the position of the straight line in each of 

the logarithmic plots (fig. 26) had been determined by rigorous methods, it would be possible to compute by rigorous methods the probable errors of the quantities implied by its 

position. As only approximate methods were 

employed in placing them, only approximate 

measures of precision are attainable, and an 

elaborate treatment would be unprofitable. 

The precision of the attitude of the line, corresponding to the quantity n, has not been estimated; but computation has been made of the 

precision of its position in the direction of the 

axis of log C; and also of the precision of the 

observations, on the assumption that the distances of the observational points from the 

straight line represent errors of observation. 

The precision of the position of the line involves the precision of the adjusted values of 

capacity, and also the precision of the coefficient of the equation of adjustment. The 

method of computation, given below, is also 

the method employed in various other computations of precision, the results of which appear 

in later chapters.

Each logarithmic plot, corresponding to an 

observational series, was treated separately. 

The distance of each plotted observational 

point from the representative straight line, in 

a direction parallel to the axis of log C, was 

measured. This distance, interpreted by the 

scale of the section paper, gives the logarithm 

of the ratio between an observed capacity and 

the corresponding adjusted capacity. By using 

a strip of the section paper as measuring scale, 

it was possible to read the ratio directly. It 

was also possible, without computation, and as 

a simple matter of reading, to subtract unity 

from the ratio and multiply the remainder by 

100, thus recording directly the residual, or 

observational error, as a per cent of the quantity measured. The facility of this operation 

determined the estimation of all errors in percentage.

From the residuals thus obtained, probable 

errors were computed by "the following approximate formulas, in which m is the number of the 

residuals and [v] is the sum of the residuals, 

irrespective of sign:

Probable error of an observation =

0.845 -

Probable error of adjusted capacities = 

0.345 ^£L,
74 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

Table 12 contains the computed probable 

errors for all those series, 82 in number, in 

which the number of observations is not less 

than 4. For a smaller number of observations 

it was possible so to frame adjusting equations 

as to leave no residuals; and although this was 

not done, the propriety of applying the computation to such cases was not evident.

The arithmetical mean of the 82 values of 

probable error for adjusted capacities is ±2.50 

per cent. The corresponding mean for observed 

capacities is ±8.80 per cent. Another measure 

of the precision of the observations, collectively, 

is the arithmetical mean, irrespective of sign, 

of all the residuals (966 in number) involved 

in the 82 series. That mean is 11.5 per cent.

While these estimates of precision are computed specifically for capacities or loads, the 

sources of the errors are not restricted to the 

observations of loads but include also the observations of slope and discharge. With use of 

the same diagrams it would be possible to consider slope as a function of capacity and compute the probable errors of slope determinations. The relations are such that, within any 

series, the probable error of slope, considered 

as a percentage, is less than the probable error 

of capacity.

The residuals within a series are, as a rule, 

relatively great for the lower slopes; but this 

is true only when the residuals are considered 

as fractional parts of the capacity values to 

which they pertain. If the residuals be 

measured in the unit of capacity, then they are 

relatively great for the higher slopes. The 

gradation of precision in relation to slope was 

discussed with some care, and an elaborate 

system of weights was prepared for the adjusted values of capacity. While these weights 

were of service in connection with various 

combinations afterward made, it has not 

seemed necessary to include them in the 

printed tables. It is to be understood, however, that the probable errors associated with 

each series in Table 12 apply to the values of 

capacity as a group, the probable errors (in 

per cent) of the smaller values being relatively 

great and those of the larger values relatively 

small.

The residuals represent chiefly the errors of 

observation, but they include also errors introduced in the process of adjustment. To whatever extent the formula of adjustment misrepresents the actual relation between the 

variables, to whatever extent the assigned 

values of a are inaccurate, and to whatever 

extent the representative lines of the plot are 

misplaced, factors of error are introduced which 

tend to increase the residuals. The estimates 

of probable error of observations are therefore 

larger than they would be if the methods of 

adjustment were perfect.

The observations giving large percentage 

residuals were, as a class, treated in the graphic 

adjustments as of relatively low weight; but in 

the computations of probable error they were 

treated as of equal weight. The estimates of 

probable error, and their average values, are 

larger than they would be if computed with 

regard to weights. The influence of this factor 

can not be definitely evaluated, but ratios 

brought out in the discussion of residuals for 

the assignment of weights indicate that it has 

some importance. It is thought that the 

average probable error of the adjusted values 

may be as low as ±2.0 per cent, and that of the 

observations as low as ±7.0 per cent.

On the other hand, the computations. take 

account only of the discrepancies revealed by 

comparing observations of the same series and 

do not cover such discrepancies as exist between one series and another. The estimates 

of probable error are smaller than they would 

be if both classes of discrepancies were included. 

This matter receives further consideration in 

Chapters V and VI.

DUTY.

The duty of water traction, as defined by the 

units adopted for this paper, is the capacity in 

grams per second for each cubic foot per second

0 of discharge, and its formula is U=Q- The

duty corresponding to each adjusted capacity 

has been computed, and the values appear in 

Table 12.

It is some tunes desirable to treat duty as the 

ratio which the mass of the load, or capacity, 

bears to the mass of the carrier. To obtain the 

value of duty as a ratio of masses the corresponding value of U should be divided by 

28,350, the number of grams in a cubic foot of 

water.
ADJUSTMENT OF OBSERVATIONS. 75

It is sometimes desirable to treat duty as a 

volume of d6bris, including voids that is, as 

the space occupied by debris as a deposit, 

either before or after transportation. In such 

cases the American engineer commonly uses 

the cubic yard as a unit of volume and 24 hours 

as a unit of time. To obtain the approximate 

duty in cubic yards of de"bris per 24 hours, 

divide the duty in gm./sec. by 14. The exact 

value of the appropriate divisor depends on the

specific gravity of the debris particles and on 

the percentage of voids in the debris aggregate.

EFFICIENCY.

The measure of efficiency, as defined on page 

36, is E"=xc . The computed efficiencies are

tabulated in Table 12, along with the adjusted 

capacities and duties.

TABLE 12. Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency.

[Or., grade of ddbris; w, width of experiment trough in foot; Q, discharge in cubic feet per second; S, slope of channel bed, in per cent; C, capacity 

for stream traction, in grams per second; t/,duty of water for traction,=-; .^efficiency of current, = C-J

Conditions.

Gr.

(A)

(A)

(A)

(A)

(A)

(A)

(A)

w

0.66

.66

.66

1.00

1.00

1.00

1.32

Q

0.093

.182

.545

.182

.363

.734

.182

S

1.2

1.4

1.6

1.8

2.0

2.2

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

.5

.6

.7

.8

.9

1.0

1.2

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

.4

.5 '.6

.7

.8

.9

1.0

1.2

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

Adjusted data.

C

23.0

32.3

42.6

53.5

65.5

78.0

18.7

. 25.0

32.0

39.5

56.3

75.0

96.0

119

49.8

66.0

83.0

101

120

140

184

8.2

12.5

17.6

23.5

30.0

37.5

54.5

74.0

97.0

122

149

179

17.0

26.9

38.2

51.8

66

' 83

100

140

185

235

290

42.5

65

91

121

154

191

231

320

7.0

11.2

16.3

22.3

29.1

36.8

54.3

74.8

98

V

247

347

458

576

704-

840

103

137

176

217

309

412

528

654

91.4

121

152

185

220

257

358

45.1

68.6

95.7

129

165

206

300

407

534

670

819

984

46.8

74.1

105

143

182

229

276

386

510

648

799

58.0

88.6

124

165

210

260

315

436

38.5

61.6

89.6

122

160

202

298

411

538

E .

206

248

286

320

352

3S2

147

171

196

217

257

294

330

363

183

202

217

231

244

257

282

90.2

114

137

161.

183

206

250

291

334

372

410

447

117

148

175

204

228

254

276

322

364

405

443

145

177

207

236

262

289

315

363

77

103

127

152

178

202

248

294

336

. Probable error 

(per cent) of 

Adjusted 

data.

1.6

Observation.

5.2
76 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 12. Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency Con.

Conditions.

Or.

(A)

(A)

(A)

(A)

(A)

(A)

(B)

(B)

(B)

w

1.32

1.32

1.32

1.96

1.96

1.96

.23

.23

.44

Q

0.182

.363

-

.734

.363

.734

1.119

.093

.182

.093

S

1.8

2.0

2.2

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

.3

.4

.5

.6

.7

 .8

.9

1.0

1.2

1.4

.4

.5

.6

.7

.8

.9

1.9

1.2

1.4

. 1.6

1.8

2.0

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

.4

.5

.6

.7

.8

.9

1.0

.8

.9 i.o

1.2

1.4

. 1.6

1.8

2.0

2.2

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

Adjusted data.

C

124

152

183 .

3.6

10.5

19.5

30.2

42.6

56.0

71

87

104

142

183

227

276

328

27.3

48.2

73.5

103

135

170

219

250

344

445

12.6

22.4

34.0

47:3

72.2

79

96

136

178

227

278

335

11.2

29.5

52

78

105

136

169

204

240

320

100

138

180

222

266

311

359

2.8

5.3

7.8

12.5

17.1

21.6

26.1

30.5

39.4

3.3

6.7

10.0

13.3

20.0

26.6

33.2

39.9

4.6

7.4

10.2

13.2

19.5

26.2

33.0

40.0

47.2

54.7

62.0

70

U

681

835

1,010

9.9

28.9

53.7

83.2

117

154

196

240

287

391

504

626

760

904

37.2

65.6

100

140

184

232

285

341

469

606

34.7

61.7

93.7

130

171

218

264

375

490

625

766

923

15.3

40.2

70.8

106

143

185

230

278

327

436

89.4

123

161

198

238

278

321

30.1

57.0

83.9

134

184

232

281

328

375

18.1

36.8

55.0

73.1

110

146

182

219

49.5

79.6

110

142

210

282 '

355

430

508

588

667

753

E

378

 418

459

49.6

96.3

134

166

195

220

245

267

287

326

360

391

422

452

124

164

200

233

263

290

317

341

391

433

86.8

123

156

186

214

242

265

312

350

390

425

462

76.5

134

177

212

238

264

288

309

327

363

224

246

268

283

298

309

321

37.5

63.3

83.9

112

131

145

156

164

170

25.9

46.0

61.1

73.1

91.5

104

114

122

Probable error 

(per cent) of 

Adjusted 

data.

1.6

2.3

2.3

2.6

3.4

3.9

1.2

2.0

Observation.

5.2

7.6

7.2

'9.4

10.9

12.3

2.6

4.9
ADJUSTMENT OF OBSERVATIONS. 77 

TABLE 12. Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency Con.

Conditions.

Or.

(B)

(B)

(B)

(B)

(B)

(B)

to

0.44

.66

.66

.66

.66

1.00

Q

0.182

.093

.182

.363

.545

.182

5

0.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

4.2

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Adjusted data.

C

7.7

11.0

14.9

19.0

23.7

28.7

40.0

52.7

66

81

4.3

6.2

8.4

10.8

16.3

22.9

30.0

38.0

46.5

56.0

66.0

76. 5

88.0

100

2.8

5.5

8.9

12.8

17.3

22.2

27.6

33.5

46.7

61.4

78

96

113

134

156

179

202

228

255

281

310

340

371

402

26.0

35.2

45.6

56.9

68.3

81

107

137

168

200

234

7.9

17.4

28.7

41.3

55.0

70

86

102

120

158

198

240

284

330

7.3

10.8

15.0

19.5

24.6

30.1

42.8

57

73

90

108

129

150

U

42.3

. 60.5

81.9

104

130

158 "220

290

363

445

46.2

66.7

90.3

' 116

175

247

323

409

500

602

710

823

946

1,080

15.4

30.2

48.9

70.4

95.1

122

152

184

257

338

429

528

621

736

858

984

1,110

1,250

1,400

1,540

1,700

1,870

2,040

2,210

71.6

97

126

157

188

223

295

378

463

551

644

14.5

31.9

52.6

75.8

101

128

158

187

220

290

363

440

521

606

40.1

59.4

82.4

107

135

165

235

313

401

495

594

709

824

E

84.6

101

117

130

144

158

193

207

229

247

66.0

83.4

100

116

146

176

202

227

250

274

296

316

338

360

51.3

75.5

97.9

117

136

152

169

184

214

241

268

293

310

335

357

378

397

413

438

453

472

492

510

526

143

162

180

196

209

223

246

270

289

306

322

72.5

106

132

152

168

183

198

208

220

242

259

275

289

303

80.2

99.0

118

134

150

165

179

223

259

275

297

322

343

Probable error 

(per cent) of 

Adjusted 

data.

0.9

2.9

1.8

1.4

3.6

1.1

Observation.

2.1

9.7

9.7

5.4

13.0

3.4
78 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 12. Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency Con.

Conditions.

Gr.

(B)

(B)

(B)

(B)

(B)

(B)

(B)

(B)

w

1.00

1.00

1.00

1.00

1.32

1.32

1.32

1.32

Q

0.182

.363

.545

.734

.182

.363

.545

.734

S

2.6

2.8

3.0

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

.3

.4

.5

.6

.7

Adjusted data.

C

173

197

224

3.7

9.4

16.6

25.4

35.3

46.2

58.5

71

85

116

149

186

225

266

310

356

99

120

143

193

247

305

368

435

505

11.0

25.0

42.8

63

86

110

138

168

199

268

344

422

507

3.7

6.6

10.1

14.1

18.7

23.8

29.3

42.0

55.8

71

88

105

124

144

166

6.9

13.6

21.8

31.2

41.9

53.1

65.5

79

109

141

177

215

255

297

39.8

56

75

95

116

140

191

248

310

375

21.7

39.0

59.5

83

110

U

954

1.080

1,230

10.2

25.9

45.8

70

97.2

127

161

196

234

320

411

512

620

732

854

981

182

220

262

354

453

560

675

798

927

15.0

34.1

58.3

86

117

150

188

229

271

365

469

575

691

20.3

36.3

55.5

77.5

103

131

161

231

307

390

484

577

682

791

912

37.5

60.1

86.0

115

146

180

218

300

389

488

592

702

818

73

103

138

174

213

257

351

455

569

688

29.6

53.1

81.0

113

150

E

366

386

410

51.0

86.3

114

140

162

181

201

218

234

267

293

320

344

366

388

419

228

244

262

295

. 323

350

375

399

421

75

114

146

172

195

214

235

254

271

304

335

359

384

50.8

72.6

92.5

111

129

146

161

176

219

244

269

288

310

330

351

93.8

120

143

164

182

200

218

250

278

305

329

351

372

146

172

197

218

237

257

292

325

355

382

98.7

133

1G2

188

214

Probable error 

(per cent) of 

Adjusted 

data.

1.1

3.6

1.9

2.8

;

1.6

2.4

4.1

2.1

Observation.

3.4

16.5

6.0

11.4

5.4

10.9

18.3

9.7
ADJUSTMENT OF OBSERVATIONS. 79 

TABLE 12. Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency Con.

Conditions.

Gr.

(B)

(B)

(B)

(B)

(B)

(C)

(C)

(C)

w

1.32

1.96

1.96

1.96

1.96

.44

.44

.66

Q

0.734

.363

.545

.734

1.119

.093

.182

.093

5

0.8

.9

1.0

1.2

1.4

1.6

1.8

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

Adjusted data.

C

 139

170

204

279

361

450

545

7.7

14.3

22.4

32.0

43.0

54.8

67.6

97

130

167

208

253

300

29.8

44.2

60.5

79

98

120

168

221

282

345

18.6

35.8

56.0

78.5

103

130

158

190

257

330

407

488

575

13.4

36.1

65.0

98

135

175

218

263

313

417

2.3

4.1

6.1

8.4

10.9

16.4

22.6

29.4

36.6

44.2

52.3

8.7

12.1

15.9

20.1

24.7

34.3

45.3

57.2

70

84

98

3.3

5.0

6.9

9.0

11.3

13.8

19.4

25.7

32.4

39.7

U

189

232

278

380

492

613

742

21.2

39.4

61.7

88.2

' 118

151

186

267

358

460

573

697

826

54.7

81.1

111

145

180

220

308

406

517

633

25.3

48.8

76.3

107

140

177

215

259

350

450

554

665

783

12.0

32.3

58.1

87.5

121

156

195

235

280

373

24.7

44.1

65.6

90.4

117

176

243

316

394

475

562

47.8

66.5

87.4

110

136

188

249

314

385

462

538

35.5

53.8

74.2

96.8

121

148

209

276

348

427

E

236

258

278

317

351

383

412

50.3

76.8

103

126

148

168

186

222

256

287

318

348

375

109

135

159

181

200

220

257

290

323

352

84.3

122

155

178

200

221

239

259

292

322

346

369

392

60

108

145

175 202-

223

241

261

280

311

41.2

63.0

82.0

100

117

147

174

197

219

238

255

79.7

95

109

122

136

157

178

196

214

231

244

71.0

89.7

106

121

134

 148

174

197

217

237

Probable error 

(per cent) of 

Adjusted 

data.

2.1

3.9

2.9

4.1

5.4

3.3

2.3

1.7

Observation.

9.7

15.5

11.3

17.1

24.1

10.1

6.1

.6.8
80 .TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 12. Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency Con.

Conditions.

Or.

(C)

(C)

(C)

(C)

(C)

(C)

to

0.66

.66

.66

-66

.66

1.00

Q

0.093

.182

.363

.545

.734

.182

S

2.0

2.2

2.4

2.6

2.8

S

3.2

3.4

3.6

3.8

4

4.2

.3

.4

.5

.6

.7

.8

.9

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

.4

.5

.6

.7

.8

.9

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

.2

.3

.4

.5

.6

.7

.8

.9

1

1.2

1.4

1.6

1.8

2

2.2

.5

.6

.7

.8

.9

1

.4

.5

.6

.7

.8

.9

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Adjusted data.

C

47.5

56

65

74

84

95

105

117

128

. 140

153

166

3

5.7

9

12.9

17.1

21.8

. 26.7

32.1

44.1

57

71

87

102

120

138

157

177

198

219

241

266

289

16.5

23.3

31.9

41.2

51.1

62

73

98

124

152

183

215

248

283

320

6.6

14.8

24.8

36.5

49.2

63.3

78.5

95

112

150

190

233

280

328

380

51

68

87

107

129

152

4.5

7.6

11.4

15.6

20.4

25.6

31.2

44

57.7

72

90

106

125

145

166

189

U

511

602

699

796

904

1,020

1, 130

1,260

1,380

1,510

1,650

1,780

16.5

31.3

49.5

70.9

94

120

147

176

242

313

390

478

560

659

758

863

892

1,090

1,200

1,320

1,460

1,590

45.5

64.2

87.9

113

141

171

201

270

342

419

514

592

683

780

882

12.1

27.2

45.5

67.0

90.2

116

144

174

205

275

349

427

514

602

697

69.5

93

118

146

176

207

24.7

41.8

62.6

85.8

112

141

171

242

317

396

495

582

686

797

912

1,040

E

256

274

291

306

323

340

353

371

384

398

412

424

55

75.8

99

118

134

150

163

176

202

223

244

265

280

299

316

332

347

363

375

388

406

419

114

128

146

161

176

 190

201

225

244

262

280

296

310

325

339

60.5

90.7

114

134

150

166

180

193

205

229

249

267

285

301

317

139

154

169

182

196

207

61.8

83.6

104

123

140

157

171

202

226

247

275

291

312

332

351

372

Probable error 

(per cent) of 

Adjusted 

data.

1.7

1.2

1.8

2.1

3.8

1

Observation.

6.8

5.8

9.0

10.1

8.5

4.4
ADJUSTMENT OF OBSERVATIONS. 81

TABLE 12. Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency Con.

Conditions.

Gr.

(C)

(C)

(C)

(O

(C)

(C)

(C)

w

1.00

1.00

1.00

1.00

1.32

1.32

1.32

Q

0.363

.545

.734

1.119

.182

.363

.545

S

0.2

.3

.4

.5

.6

.7

.8

.9

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

.5

.6

.7

.8

.9

1.0

1.2

1 4

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

.6

.7

.8

.9

1.0

1.2

Adjusted data.

C

3.3

9.6

17.5

26.6

36.6

47.5

59

72

85

113

143

175

210

245

281

320

3fiO

401

48.2

64.9

82

100

119

140

183

228

275

326

377

430

12.5

27.3

44.2

63.2

84

106

130

155

ISO

235

294

352

416

483

550

96

128

162

198

236

276

361

451

5.6

8.7

12.3

16.6

21.4

32.7

46.5

62

80

99

120

144

6.4

13.6

22.1

31.9

42.2

53.8

66

79

105

135

166

199

233

270

307

56.3

73.3

91

109

129

171

U

9.1

26.4

48.2

73.3

101

131

163

198

234

311

394

482

579

675 '

774

882

992

1,100

88.4

119

150

183

213

257

336

418

504

598

692

789

17.0

37.2

60.2

86.2

114

144

177

211

245

320

401

480

567

658

750

85.8

114

145

177

211

247

323

403

30.8

47.8

67.6

91.2

118

180

256

341

440

544

659

791

17.6

37.5

60.9

87.9

116

148

182

218

289

372

458

548

642

744

846

103

134

.167

200

237

314

E

45.5

88

120

147

168

187

204

220

234

259

281

301

322

338

352

367

381

393 .

177

198

214

229

242

257

280

299

315

332

346

359

85

124

150

172

190

206

221

234

245

267

286

300

315

329

341

172

190

207

221

234

247

269

288

51.3

68.3

84.5

101

118

150

183

213

244

272

299

329

58.7

93.8

122

146

166

185

202

218

241

266

286

304

321

338

352

172

191

209

222

237

262

Probable error 

(per cent) of 

Adjusted 

data.

1.2

1.1

7.8

2.3

2.4

1.2

1.8

Observation.

0.2

5.5

37.7

5.1

8.4

5.5

8.5

20921° No. 86 14-
82 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 12. Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency Con.

Conditions.

Or.

(C)

(C)

(C)

(C)

(C)

(C)

(D)

(D)

w

1.32

1.32

1.96

1.%

1.96

1.96

.66

.66

Q

0.345

.734

.363

.545

.734

1.119

.093

.182

S

1.4

1.6

1.8

2.0

2.2

2.4

.2

.3

.4

 5,

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

.5

.fi

.7

.8

.9

1.0

1.2

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

.4

.5

.6

.7

.8

. .9

1.0

1.2

1.4

Adjusted data.

. C

215

263

312

362

415

470

9.3

25.0

43.2

63.8

86

109

133

160

187,

243

304

367

432

500

570

10.5

17.8

26.5

36.6

47.9

60.0

88

119

154

193

235

280

25.7

39.3

55.0

72

91

111

156

205

259

316

375

440

34.0

55

78

103

131

100

190

255

323

394

88

130

177

228

283

343

480

2.7

3.9

5.4

7.1

9.1

13.5

18.6

24.7

31.4

38.5

46.3

55.2

64.3

4.7

7.7

11.3

15.3

19.7

24.6

29.8

41.0

53.4

U

395

483

573

664

762

862

12.7

34.1

58.9

86.9

117

148 ,

181

218

255

331

414

500

589

681

777

28.9

49.0

73.0

101

132

165

242

328

424

532

648

771

47.2

72.1

101

132

167

204

286

376

475

580

688

808

46.3

74.9

106

140

178

218

259

347

440

537

78.7

116

158

204

253

307

429

29.0

41.9

58.1

76.4

97.9

145

200

266

338

414

498

594

692

25.8

42.3

62.1

84.1

108

135

164

 225

293

E

282

302

318

332

346

359

63.5

114

147

174

195

211

226

242

255

276

296

312

327

340

353

57.8

81.7

104

126

147

165

202

234

265

295

324

350

94.4

120

144

165

186

204

238

268

297

322

344

367

116

150

177

200

222

242

259

289

314

336

157

193

226

255

281

307

358

48.4

59.9

72.5

84.9

97.9

121

143

166

188

207

226

248

266

64.5

84.6

104

120 '

135

150

164

188

209

Probable error 

(per cent) of 

Adjusted 

data.

1.8

2.2

3.5

2.0

2.8'

2.2

2.6

4.2

Observation.

8.3

9.0

15.5

7.0

13.2

8.9

8.1

15.2
ADJUSTMENT OP OBSERVATIONS. 83 

TABLE 12. Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency Con.

Conditions.

Gr.

(D)

(D)

(D)

(»)

<D)

(D)

(D)

i

w

0.66

.CO

1.00

1.00

1.00

1.00

1.32

Q

0.182

.54.5

.182

.363

.545

.734

.363

S

1.6

1.8

2.0

2.2

2.4

2.6

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

Adjusted data.

C

67

82

98

114

132

150

4.7

11.7

20.7

31.2

43.1

56.2

70.2

86

101

137

175

217

261

308

357

11.3

15.1

19.2

24.0

34.5

46.4

59.8

74

90

106

124

144

104

6.9

13.2

20.9

29.6

39.0

49.6

61

73

99

127

158.

190

225

261

. 299

27.6

40.2

55.0

71

89

108

152

201

257

317

382

6.9

17.5

31.0

47.0

65.0

85

106

129

153

208

268

332

388

11.9

18.8

27.0

36.6

47.2

59.2

87

118

154

194

237

U

368

450

538

626

.726

824

8.5

21.5

38.0

57.2

79.1

103

129

158

185

251

321

398

479

568

655

62.1

83

106

132

190

255

329

407

495

582

682

792

902

19.0

3C.4

57.6

81.6

107

137

168

201

273

350

435

524

620

819

824

50.6

73.8

101

130

163

198

279

369

472

582

700

9.4

23.8

42.2

64.0

88.6

116

144

176

208

282

365

452

542

32.8

50.9

74.4

101

130

163

240

325

424

534

653

E

230

250

269

284

302

317

 42.5

71.6

95

114

132

147

161

176

185

209

229

249

266

284

298

88.7

104

118

132

158

182

206

226

248

264

284

304

324

63

91.0

115

136

153

171

187

201

228

250

272

291

310

327

343

101

123

144

162

181

198

232

264

295

323

350

47.0

79.3

106

128

148

166

180

196

208

236

361

283

302

65.6

84.8

106

139

144

163

200

232

265

297

326

Probable error 

(per cent) of 

Adjusted 

data.

4.2

3.4

2.5

3.3

1.2

2.5

7.0

Observation.

15.2

15.3

9.1

13.2

3.9

11.5

24.3
84 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 12. Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency Con.

Conditions.

Gr.

(t»

(E)

(E)

(E)

(E)

(E)

(E)

(E)

(E)

w

1.32

.66

.66

1.00

1.00

1.00

1.00

1.32

1.32

Q

0.734

. .363

.734

.182

.363

.734

1.119

.363

.734

S

0.3

.4

.5

.6

.7

.8

.9

1.0

1.2

1.0

1.2

1.4

1.0

1.2

1.4

1.6

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

.2

.3

.4

.5

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

.5

.6

.7

.8

.9

1.0

1.2

1.4

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

.5

.6

.7

.8

.9

1.0

1.2

Adjusted data.

C

10.2

20.9

34.0

49.2

67

87

108

131

185

24.8

36.2

49.4

40.0

59.0

82

108

2.4

3.8

5.5

7.4

9.6

12.0

14.8

20.7

27.6

35.3

43.5

52.5

62.4

73

84

96

1.3

3.3

6.0

9.2

13.1

17.5

22.5

28.0

33.8

47.2

62.0

78.5

96

115

137

159

4.2

9.1

15.2

22.4

30.8

40.0

49.8

60.5

72.0

98

126

. 156

188

222

44

CO

77

97

117

138

188

242

15.3

20.3

25.4

30.7

36.3

49.3

63.5

78.8

95

112

24.7

33.0

42.0

51.7

C2

73

97

U

13.9

28.5

46.3

67.0

91.3

118

147

178

252

68.3

99.7

136

54.5

80.4

112

147

13.2

20.9

30.4

40.6

52.7

66.0

81.4

114

152

194

239

288

343

401

461

527

3.6

9.1

16.5

25.3

36.1

48.2

62.0

77.2

93.1

130

171

216 '

264

317

378

438

5.7

12.4

20.7

30.5

42.0

54.5

67.8

72.4

98.1

134

172

213

256 . 302

39.3

54

69

87

105

123

168

216

42.2

55.9

70.0

84.6

100

136

175

217

262

309

33.7

45.0

57.2

70.4

84.5

99.5

132

E

46.3

71.2

92.6

112

130

148

163

178

210

68.3

83.1

97.2

54.5

67.0

80

92

33.0

41.8

50.7

58.0

65.0

73.3

81.4

95

108

121

133

144

156

167

177

188

18.0

30.3

41.2

50.6

60.1

68.9

77.5

85.8

. 93.1

108

122 .

135

147

158

172

182

28.5

41.3

51.8

61.0

70.0

77.9

84.8

91.6

98.1

112

123

133

142

151

76.6

89

98

108

117

123

140

154

70.3

79.9

87.5

94.0

100

113

125

136

145

. ISA

67.4

75.0

81.7

88.0

93.9

99.5

110

Probable error 

(per cent) of 

Adjusted 

data.

5.3

0.9

1.9

5.1

5.0

3.2

Observation.

21.4

2.1

5.0

14.6

19.5

6.4
ADJUSTMENT OF OBSERVATIONS. 85 

TABLE 12. Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency Con.

Conditions.

Or.

(E)

(F)

(F)

(F)

(F)

(F)

(F)

(F)

(F)

(F)

(F)

w

1.32

.66

.66

.66

1.00

1.00

1.00

1.00

1.32

1.32

1.32

Q

1.119

.182

.363

.734

.182

.363

.734

. 1.119

.363

.734

1.119

S

0.6

.7

.8

.9

1.0

1.2

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

1.0

1.2

1.4

1.6

1.8

2.0

.8

1.0

1.2

1.4

1.6

1.8

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.0

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

.7

.8

.9

1.0

1.2

. 1.4

1.6

1.8

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

1.2

1.4

1.6

1.8

2.0

2.2

.8

.9

1.0

1.2

1.4

1.6

.8

.9

1.0

1.2

1.4

1.6

1.8

Adjusted data.

C

49.4

65.0

82.5

102

123

171

9.3

13.7

18.9

24.7

31.4

38.7

46.8

55.7

20.5

29.9

41.0

53.2

67.0

81.5

31.6

39.0

55.3

73.0

93.5

115

4.2

7.3

11.3

16.4

22.5

29.8

38.2

47.7

7.4

10.5

14.0

22.8

33.4

40.0

60

76

94

112

133

20.0

27.3

35.1

42.8

63

85

109

136

37.5

51.0

64.8

79.5

112

150

191

236

21.4

31.0

41.5

53.0

66.0

80

22.9

31.0

40.2

61.6

87.0

115

43.3

56.0

70.0

101

138

178

221

U

44.2

58.1

73.7

91.1

110

153

51.0

75.3

104

136

172

213

257

306

56.5

82.4

113

147

185

225

43.0

53.2

75.3

99.5

127

157

23.1

40.1

62.1

90.1

124

164

210

202

20.4

28.9

38.6

62.8

92.0

127

165

209

259

309

367

27.3

37.2

47.8

58. 3

85.8

116

148

185

33.5

45.6

57.9

71.0

100

134

171

211

59.0

85.4

114

146

182

220

29.6

40.1

52.0

79.7

119

149

38.7

50.0

62.6

90.2

123

159

197

E

73.7

83.0

92.1

101

110

128

42.5

53.8

65.0

75.6

86.0

96.8

107

118

56.5

68.7

80.8

91.9

103

112

47.8

53.2

62.8

71.0

79.4

87.2

17.6

28.6

38.8

50.0

62.0

74.6

87.5

101

25.5

32.1

38.6

52.3

65.7

79.4

91.7

104

118

129

141

39.0

46.5

53.1

58.3

71.5

82.9

92.6

103

47.9

57.0

64.3

71.0

 83.3

95.8

107

117

48.2

 61.0

71.3

81.1

91.0

100

?7.0

44.6

52.0

66.4

85.0

93.2

48.4

91.1

62.6

75.2

87.9

99.4

109

Probable error 

(per cent) of 

Adjusted 

data.

1.8

.8

2.3

.4

2.0

.5

.9

JL2

3.2

3.2

Observation.

3.6

1.7

4.6

.8

4.8

1.0

1.8

2.4

7.2

7.2
86 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 12. Adjusted values of capacity, based on data of Table 4, with corresponding 'values of duty and efficiency Con.

Conditions.

Or.

(G)

(G)

(G)

(G)

(G)

(G)

(G)

(G)

w

0.66

.66

.66

1.00

1.00

1.00

1.32

1.32

Q

0.363

.734

1.119

.363

.734

1.119

.363

.734

S .

1.0

1.2

1.4

1.6

1.8

2.0

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

.6

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

Adjusted data.

C

. 8.5

14.8

23.0

32.7

43.7

56.0

10.3

16.0

22.8

30.4

48.3

69

93

120

149

181

216

253

11.9

19.0

27.7

37.6

49.0

75.0

107

143

182

227

276

325

382

9.7

16.3

24.4

34.1

45.2

57

71

86

102

120

8.0

13.3

19.5

26.9

44

65

89

115

145

177

212

249

13.5

22.7

33.6

46.0

61.0

92

128

171

217

268

321

380

18.1

28.3

39.9

53.8

69.8

88.0

109

134

7.9

12.8

18.9

33.8

52.0

74.0

98.0

125

155

187

222

260

U

23.4

40.8

63.4

90.1

120

154

13.3

20.7

29.5

39.3

62.5

89.2

127

155

193

234

280

327

10.6

17.0

24.7

33.6

43.8

67.0

95.6

128

163

203

247

290

341

26.7

44.9

67.2

94.0

125

157

196

237

281

331

10.9

18.1

26.6

36.6

60.0

88.6

121

157

198

241

289

339

12.1

20.3

30.0

41.1

54.4

82.2

114

153

194

239

287

340

49.5

78.0

101

148

192

242

300

389

10.8

17.4

25.8

46.1

70.8

101

134

170

211

255

303

354

E

23.4

34.0

45.3

56.3

66.7

77.0

19.0

25.9

32.8

39.3

52.1

63.7

79.4

86.2

96.5

106

117

126

17.7

24.3

30.9

37.3

43.8

55.8

68.3

80.0

90.6

102

112

121

131

22.2

35.0

42.0

52.2

62.5

71.4

81.7

91.2

100

110

15.6

22.6.

29.6

36.6

50.0

63.3

75.6

87.8

99.0

110

120

130

20.1

29.0

37.5

45.6

54.4

68.5

81.5

95.6

108

120

130

142

27.7

39.0

45.9

61.7

73.9

86.5

100

115

13.5

19.3

25.8

38.4

50.6

63.2

75.4

85.0

95.9

106

117

126

Probable error 

(per cent) of 

Adjusted 

data.

2.2

1.5

1.2

1.9

1.9

4.3

2.4

2.2

Observation.

. 5.8

4.3

3.6

5.0

5.5

13.6

4.8

5.8
ADJUSTMENT OF OBSERVATIONS. 87 

TABLE 12. Adjusted values of capacity, based on data of Table 4, with corresponding values of duty and efficiency Con.

Conditions.

Or.

(0)

(H)

(H)

(H)

w

1.32

.66

.66

.66

Q

1.119

.363

.734

1.119

S

0.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

.7

.8

.9

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

Adjusted data.

C

17.9

28.0

39.5

52.2

81

115

153

194

238

285

335

387

3.1

6.9

12.4

19.4

28.0

38.1

49.8

63

78

4.5

8.2

12.8

24.6

39.2

57.0

77.0

100

125

151

181

213

246

282

6.0

11.1

17.6

25.2

43.5

65.3

91.0

120

152

186

223

263

U

16.0

25.0

35.3

46.6

72.4

103

137

173

213

255

299

346

8.5

19.0

34.2

53.5

77.1

105

137

174

215

6.8

12.4

19.4

37.3

59.4

86.4

117 

152

189

229

274

323

373

427

5.4

 9.9

15.7

22.5

38.8

58.3

81.3

107

136

166

199

235

E

22.9

31.2

38.1

40.6

60.3

73.6

85.6

96.2

106

116

125

133

7.1

13.6

21.4

29.7

38. C

47.8

57.1

66.9

76.8

8.5

13.8

19.4

31.1

42.4

54.0

65.0

76.0

85.9

95. C

105

115

126

133

7.7

12.4

17.4

22.5

32.3

41.6

50.8

59.5

68.0

75.5

83.0

90.4

Probable error 

(per cent) of 

Adjusted 

data.

3.1

1.0

1.6

3.2

Observation.

8.7

3.5

4.9

11.2

OBSERVATIONS ON DEPTH.

MODE OF ADJUSTMENT.

To adjust the observations of depth it is 

necessary to deal with their relations to another 

variable, and either slope or capacity might be 

used. The selection of the particular variable 

for comparison was a matter of convenience 

only, because adjustment to either one would 

bring the depth values into orderly relation to 

the adjusted values of the other also; but the 

question of convenience was not unimportant, 

A fairly thorough preliminary study was therefore made, in which the depth measurements 

for many series were plotted in relation, severally, to measurements of capacity and measurements of slope.

In figure 27 the horizontal scale is that of 

slope, the vertical of depth. The round dots 

represent observations made with grade (C),

width 1.32 feet, and discharge 0.734 ft.3/sec. 

Despite irregularities, the grouping suggests as

+

">>..

\ 

\

\

\

>^ *

' . _

i-' 

V.

0.5 1.0 1.5 

Slope

FIGURE 27. Observations of depth of current, in relation to slope, 

plotted as dots. The crosses show logarithms of the same depths 

and slopes.

the representative line some such curve as that 

drawn. On taking account of the physical conr
88 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

ditions, it is evident that as the slope is flattened, the current is slowed and the depth increased, and that zero slope gives infinite depth. 

The theoretic curve, therefore, has an asymptote in the vertical line corresponding to zero 

slope. Similarly the depth is reduced by increase in slope but remains finite for very high 

slopes. The theoretic curve has as asymptote 

a horizontal line corresponding, exactly or 

appioximately, with the (horizontal) line of 

zero depth. These asymptotes relate the curve 

d =/($) to the hyperbola. In the same figure, 

but with use of a different scale, are a series of 

crosses which show the same observations as 

they appear when plotted on logarithmic section paper. They are the plot of observations 

on log d=fi (log $); and their arrangement suggests that the representative line may be 

straight.

Plots were made to show the relations of 

depth observations to associated capacity, and 

these also suggest the hyperbola and the 

straight line. If, however, the locus of d =fa (C) 

is a hyperbola it differs materially from that of 

d=f(S), for as depth increases and current 

slackens, capacity becomes zero when current 

reaches the value of competence, and depth 

is not then infinite. So the line of zero capacity is not an asymptote to the curve.

It is to be observed also that the representative lines for log d=fi (log S) and log d=frr 

(log CT) can not both be straight, for if they were 

there could be derived from them a straight 

line representing log C=fv (logS), and it has 

already been found that that line is curved. It 

is, indeed, probable that neither of the logarithmic plots involving depth is straight; yet 

there are cogent practical reasons for assuming 

one or the other to be so. One reason is the 

very .great convenience of the straight-line function, and another that the relatively small range 

of the depth data renders impracticable such a 

discussion of the curvature of logarithmic loci 

as was made in the case of capacity versus 

slope. Accordingly, the most orderly plots of 

log d =/i (log 8) and log d =/ (log (7) were compared with special reference to curvature. For 

the function of log C the plots were found to 

indicate curvature in one direction only, while 

for the function of log 8 they indicated slight 

curvatures in both directions, with the straight 

line as an approximate mean. The function

log d =fi (log S) was accordingly selected for the 

adjustment of the depth observations; and the 

representative line on the logarithmic plot was 

assumed to be straight. In accordance with 

that assumption, the adopted formula of interpolation was

(22)

with its logarithmic equivalent,

log d = \og I'-nJog 8...

The coefficient b' is a depth, the depth corresponding to a slope of 1 per cent.

The data were all plotted on logarithmic 

section paper. The notation was made to 

distinguish depth measurements made at a 

single point by means of the gage (see p. 25) 

from those based on full profiles of water surface and bed of debris. The former were used 

exclusively in the drawing of the representative 

lines, but not because they were regarded as 

of higher authority. It was thought best not 

to combine data which in certain cases were 

known to be incongruous; and the gage observations covered the whole range of the work, 

while the profiles did not. The measurements 

by profile were used in criticising the measurements by gage, and they determined the acceptance or rejection of certain gage measurements. 

It was noted that in some series of observations the depth measurements by the two 

methods were in close accord, while in others 

there was a large systematic difference; and 

certain series were rejected because of such 

large differences.

The plots were made to distinguish also the 

observations associated with different modes 

of traction the dune, smooth, and antidune 

modes. The observations with the smooth 

mode were assumed to be best, as a class; and 

these, together with the observations connected

with the transitional phases of traction, were 

used to fix an initial point of each representative line. The direction of the line was then 

adjusted by eye estimate to make it representative of all the points of the particular 

series. In this adjustment consideration was 

given to the conditions affecting the measure-' 

ments of both depths and slopes.

The lines were first drawn for those observational series which appeared from the plots 

to be most harmonious, and for these there was
ADJUSTMENT OF OBSERVATIONS. 89

discovered a tendency toward parallelism, 

within each group, of lines associated with a 

particular sand grade and channel width but 

differing as to discharge. For numerous other 

groups, involving very irregular observational 

positions, such parallelism was assumed; and 

under that assumption the better series of a 

group were made to control the directions of 

the lines for the poorer. For three groups the 

directions were interpolated by use of the 

directions found in affiliated groups.

For each observational series the direction 

of the representative line gave the value of % 

in. equations (21) and (22), and the intersection 

of the line with the axis of log d gave the value 

of I'.

v

TABLE 13. Values o/Wj in d~ jr.

Grade 

of 

ddbris.

f*

s

(E 

(F 

(0 

(H

Value of Wi for trough having width (in feet) of 

0.23

0.49

0.44

0.44

M4] 

.44

O.GG

0.40 

.56 

.40 

[-35] 

.30

.28 

.34

1.00

0.34 

.53 """si"

.26 

.30

1.32

0.50 

.70 

.48-. 58 

.40-. 59 i \y

1.96

0.60

.55 

.48

In Table 13 the determinations of nx are 

assembled, being arranged vertically with respect to grade of d6bris and horizontally with 

respect to width of trough. Collectively they 

indicate that, for constant discharge, the depth 

varies inversely with the slope and less rapidly 

than the slope. Comparatively they indicate 

considerable range in the rate of variation.

There is some suggestion of system in the distribution of values of nv and it is especially 

probable that to the coarser debris belong the 

smaller values, but irregularity is, on the whole, 

more in evidence than system. The irregularity is to be ascribed largely to the imperfection of the data, and imperfection is believed 

to inhere especially in the measurements of 

depths by means of the gage. (See p. 25.)

Table 14 contains series of adjusted or interpolated values of depth. These were computed for slopes with the constant interval of 

0.2 per cent, the range of slope within each series being approximately the same as the range 

through which the depth observations were 

made. The range is usually less than the 

range of slopes for which capacities were computed, the difference being occasioned by the 

fact that satisfactory measurements of depth 

could not be made during vigorous antidune 

traction. The computations were graphic, the 

values of adjusted depth being read directly 

from the logarithmic plot.

The number of series represented in the table 

is 73, being one-fourth less than the number for 

which capacities and slopes were tabulated. 

For 18 series the depth measurements were not 

reduced, the chief reasons for rejections being 

either (1) that the number of observations was 

very small, (2) that the observations included 

none made in connection with the smooth 

phase of traction, and (3) that the measurements with gage differed systematically by a 

large amount from corresponding measurements by the method of profiles.

TABLE 14. Adjusted values of depth of current (d), with values of mean velocity ( Vm] and form ratio (R= J. 

[ni, Exponent in adjusting formula (21); p. e., probable error, in per cent, of adjusted values.]

Srade.

,'.....

).....

i.....

. e...

S

0.2

.4

.6

.8

1.0

1.2

1.4

d

0.218

.154

.126

.109

.098

.090

(A)

1.32

.363

.50

1.5

vm

1.25

1.78

2.09

2.53

2.83

3.10

R

0.165

.117

.096

.083

.068

d

0.227

.185

.160

(A)

1.32

.734

.50

1.8

Vm

2.45

3.00

3.45

R

0.172

.141

.122

d

0.136

.107

090

079

.071

064

(A)

1.96

.363

.60

0.7

Vm

1.36

1.74

2.05

2.35

2.G2

2.87

R

0.170

.155

.046

.040

.036

.033

d

0.255

.167

.131

.110

(A)

1.96

.734

.60

1.7

Vm

1.47

2.24

2.88

3.41

R

0.130

.085

.067

.056

d

0.232

.182

.153

(A)

1.96

1.119

.60

3.4

Vm

3.35

4.25

5.05

R

0.118

.093

.078
90 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 14. Adjusted values of depth of current (d), with values of mean velocity (Vm) and form ratio ( R=~ J Con.

Grade. .

Q......

p.e....

S

0.4 

.6 

.8 

1.0 

1.2 

1.4 

1.6 

1.8 

2.0 

2.2 

2.4 

2.6

Grade. .

Q......

p.e....

S

0.2 

.4 

.6 

.8 

1.0 

1.2 

1.4 

1.6 

1.8 

.2.0 

2.2 

2.4 

2.6

Grade. .

Q......

ni......

p.e....

S

0.2 

.4 

.6 

.8 

1.0 

1.2 

1.4 

1.6 

1.8 

2.0 

2.2 

2.4

Grade..

Q......

Til......

p.e....

S

0.2 

.4 

.6 

.8 

1.0 

1.2 

1.4 

1.6 

1.8

(B) 

.23 

.093 

.49 

3.3

d

0.200 

.179 

.163 

.150 

.141 

.133

vm

2.02 

2.17 

2.49 

.2.68 

2.86 

3.04

R

0.880 

.780 

.715 

.660 

.620 

.583

(B) 

.66 

.182 

.40 

2.1

d

0.208 

.176 

.133 

.118 

.108 

.100 

.095 

.090 

.085 

.082 

.079

vm

1 

1 

2 

2 

2 

2 

2 

3 

3 

3 

3

.33

.76 

.08 

.34 

.56 

.75 

.94 

.10 

.25 

.38 

.50

R

0.317 

.240 

.203 

.181 

.165 

.153 

.144 

.137 

.130 

.125 

.120

(B) 

1.00 

.734 

.34 

1.4

d

0.347 

.275 

.239 

.217 

.200

vm

2.10 

2.68 

3.10 

3.41 

3.67

(B) 

1.96 

.545 

.55 

2.8

d Vm

0.168 1.63 

.135 2.04 

.115 2.38 

.102 2.70 

.092 2.30 

.085 2.26

R

0.347 

.275 

.239 

.217 

.200

(B) 

.23 

.182 

.49

d

0.327 

.292 

.266 

.247 

.231 

.218

Vm

2.43 

2.70 

2.96 

3.02 

3.42 

3.61

R

1.43 

1.28 

1.17 

1.08 

1.01 

.96

(B) 

.44 

.093 

[.44] 

5.5

d

0.128 

.113 

.102 

.094 

.088 

.083 

.079 

.075 

.072 

.069 

.067

(B) 

.66 

.363 

.40 

2.1

d

0.246 

.210 

.187 

.171 

.158 

.148

Vm

1.73 

1.96 

2.08 

2.36 

2.53 

2.67 

2.82 

2.95 

3.08 

3.20 

3.32

R

0.294 

.259 

.235 

.216 

.202 

.190 

.182 

.173 

.167 

.160 

.154

(B) 

.66 

.545 

.40 

2.0

Vm R d

................. 0.421 2.22 0.373 .32 

2.63 .318 .27 

2.95 .283 .24 

3.23 .259 .22 

3.49 .240 .20 

3.71 .226 .19

7 

3 

i 

5 

i 

7 

1

(B) 

1.32 

.182 

.70 

2.5

d

0.200 

.124 

.094 

.077 

.066 

.058 

.052 

.047 

.043 

.040 

.038 

.036

Vm

0.68 

1.12 

1.48 

1.82 

2.13 

2.42 

2.70 

2.96 

3.20 

3.45 

3.70 

3.92

(B) 

1.96 

.734 

.55 

5.4

R d Vm

n 2r

0. 086 .18 

069 . 14 

059 .12 

052 .11 

047

8 1.4C 

3 2.0, 

7 2.5, 

6 2.9J 

2 3.3,

043 ........ ......

R

) 0.136 

> .094 

) .075 

.064 

) .057

R

0.153 

.094 

.071 

.057 

.050 

.046 

.039 

.036 

.033 

.031 

.029 

.027

Vm

1.97 

2.60 

3.04 

3.40 

3.70 

3.97 

4.24

R

0.650 

.490 

.418 

.370 

.340 

.315 

.296

(B) 

.44 

.182 

[.44] 

2.6

d

0.220 

.184 

.163 

.146 

.135

Vm

1.90 

2.27 

2.56 

2.81 

3.04

R

0.502 

.421 

.371 

.336 

.310

(B) 

1.00 

.182 

.34 

3.3

d

0.101 

.091 

.084 

.079 

.075 

.072 

.069 

.066 

.064 

.062 

.061

(B) 

1.32 

.545. 

.70 

2.4

d

0.233 

.175 

.143 

.123 

.108 

.097

Vm

1.76 

2.35 

2.88 

3.38 

3.82 

4.27

(B) 

1.96 

'1.119 

.55 

3.8

d

0.40C 

.271 

.217 

.18J 

.162 

.147

vm

1.45 

2.13 

2.66 

3.11 

3.51 

3.89

R

0.203 

.138 

.110 

.094 

.083 

.075

JfJ

0.176 

.133 

.108 

.093 

.082 

.073

Vm

1.81 

2.00 

2.15 

2.39 

2.40 

2.51 

2.62 

2.71 

2.80 

2.92 

3.00

R

0.101 

.091 

.084 

.079 

.075 

.072 

.069 

.066 

.064 

.062 

.061

(B) 

1.32 

.734 

.70 

2.6

d Vm

-T

(B) 

.66 

.093 

.55 

2.3

d Vm

0.079 1.80 

.069 2.03 

.063 2.25 

.057 2.45 

.053 2.65 

.050 2.82 

.047 3.00 

.045 3.15

R

0.118 

.105 

.095 

.087 

.081 

.076 

.072 

.068

(B) 

1.00 

.363 

.34 

2.5

d

0.239 

.188 

.164 

.148 

.137 

.128 

.122 

.117 

.112 

.108 

.105

Vm

1.53 

1.94 

2.24 

2.46 

2.66 

2.83 

2.98 

3.12 

3.25 

3.36 

3.48

R

0.239 

.188 

.164 

.148 

.137 

.128 

.122 

.117 

.112 

.108 

.105

(B) 

1.96 

.363 

.55 

3.3

R d

0. 256 2. 10 0 

. 193 2. 83 

. 158 3. 48

.194 0.117 

.146 .094 

.120 .080 

..... .071

..... .064

..... .059

054

..... .051

..... .048

(C) 

.44 

.093 

.44 

0.6

d

0.163 

.143 

.130 

.120 

.112 

.106 

.101

Vm R

1.28 0.370 

1.46 .328 

1.62 .298 

1.75 .275 

1.87 .256 

2. 00 . 242 

2. 10 . 230

Vm

1.58 

1.98 

2.33 

2.65 

2.93 

3.20 

3.43 

3.67 

3.88

R

0.059 

.047 

.041 

.036 

.033 

.030 

.028 

.026 

.025

(C) 

.44 

.182 

.44 

0.7

d Vm

0.240 1.73 

.213 1.96 

.192 2.16 

.177 2.34 

. 165 2. 50 

. 157 2. 66

R

0.546 

.480 

.436 

.400 

.375 

.355
ADJUSTMENT OF OBSERVATIONS.

TABLE 14. Adjusted values of depth of current (d), with values of mean velocity (Vm) and form ratio ( R=~ ) '

91

Con.

Grade. .

Q. ......

Tit ......

p. e....

,S

0.2

.6 

.8 

1.0 

1.2 

1 4

1.6

1.8

!.0

!.2

!.4

1.6

Grade 

w.... 

Q....

»!....

p.e..

S

0.2

.4

.6

.8 

1.0 

1.2 

1.4

1.6

1 4

2.0

2.2

2.4

2.15

Grade. .

mi......

p.e....

S

0.2

.4

.6 

.8 

1.0 

1.2 

1.4 

1.6 

1.8 '2.0

2.2

Grade .

10.:.. .

Q.... .

Til.... .

p.e.. .

S

0.2

.4

.6

.8 

1.0 

1.2 

1.4

1.6

1.8

2.0

2.2

2.4

2.6

d

.105 

.090 

.079 

.071 

nfi<;

.061

.057

.054

.051

ftilft

.046

d

0.139

.113

.098 

.086 

.078 

.072

.067

.063

.059

.056

.054

d

0.168

.138 

.120 

.108 

.099 

.092 

.086

d

0.194

.159

.138 

.124 

.113 

.105

.099

(C) 

.66

.093

.56 

0.8

Vm

1.08

1.35 

1.58 

1.79 

1.97 

2.14

2

QA

2 4^

2 GO

2.72

2 op.

3.00

(C) 

1.00 

.182 

.53 

1.9

Vm

1.30

1.62

1.88 

2.13 

2.34 

2.55

2.74

2.90

3.08

3.24

3.38

(C) 

1.32

.363

.48

2.3

vm

1.66

2.00 

2.28 

2.54 

2.76 

2.96 

3.15

(C) 

1.96 

.734 

.48 

1.0

Vm

1.96

2.38

2.73 

3.04 

3.31 

3.56

3.80

R

0.199

.160 

.135 

.119 

.107 

099

AQO

.086

.082

.077

.074

.071

R

0.139

.113

.098 

.086 

.078 

.072

ftfi?

.063

.059

.056

ft ^4

 R

0.128

.105 

.092 

.082 

.075 

.070 

.065

R

0.099

.081

.070 

.063 

.058 

.053

.050

d

0.270

.183

.146 

.125 

.110 

.100 

new

AOC

.080

H7C

.071

d

0.307

.212

.171

.147 

.130 

.118 

ins

.102

d

0.174 

.152 

.136 

.125 

.116 

.108

d

0.240

.197

.172 

.153 

.140

(C) 

.66

.182

.56 

1.8

Vm

1.03

1.52

1.89 

2.21 

2.50 

2.77 

3.00

3.23

3.44

3.64

3.83

(C) 

1.00 

.363 

.53 

0.9

Vm

1.18

1.72

2.14

2.49 

2.80 

3.08 

3.34

3.60

(C) 

1.32

.545

.48

.8

vm

2.36 

2.71 

3.02 

3.30 

3.54 

3.78

(C) 

1.96 

1.119 

.48 

1.0

Vm

2.40

2.92

3.36 

3.73 

4.08

R

0.410

.277

.222 

.188 

.167 

.150 

.138

.128

.120

.113

.107

R

0.307

.212

.171

.147 

.130 

.118 

.108

.102

R

0.132 

.115 

.104 

.095 

.088 

.083

R

0.122

.100

.088 

.078 

.072

d

0.288

.230 

.196 

.173 

.157

d

0.190 

.170 

.153 

.141

.132

.123

d

0.344

.246

.203 

.177 

.158 

.145 

.135 

.124

d

0.087 

.080 

.074 

.069

.066

.063

.060 riEo

.056

.054

(C) 

.66

.363

.56 

1.4

Vm

1.88

2.38 

2.80 

3.20 

3.53

(C) 

1.00 

.545 

.53 

1.5

Vm

2.86 

3.22 

3.54 

3.84

4.14

4.52

(C) 

1.32

.734

.48

1.1

Vm

1.63

2.27

2.76 

3.17 

3.50 

3.84 

4.14 

4.40

(D) 

0.66 

.093 

.40 

1.3.

Vm

1.63 

1.78 

1.91 

2.02

2.13

2.23

2.33

2.42

2.50

2.58

R

0.438

.350 

.298 

.264 

.238

R

0.190 

.170 

.153 

.141

.132

.123

R

0.261

.187

.153 

.134 

.120 

.110 

.102 

.096

R

0.132 

.132 

.112 

.105

.100

.095

.091

noo

.085

OR9

d

0.562

.381

.304 

.260 

.228 

.206

d

0.450

.310

.250

.214 

.190 

.172

d

0.132

.108 

.095 

.085 

.078 

.072 

.068 

064

.061

.058

d

0.172

.143

.127 

.115 

.107 

.100

.094

.090

.086

.082

.079

.077

(C) 

.66

.545

.56 

1.3

vm

1.43

2.13

2.68 

3.16 

3.60 

4.00

(C) 

1.00 

.734 

.53 

2.2

vm

1.66

2.39

2.96

3.45 

3.88 

4.26

(C) 

1.96

.363

.48

1.2

vn

1.18

1.44 

1.66 

1.85 

2.02 

2.18 

2.32 

2.46

2.59

2.71

(D) 

0.66 

.182 

.40 

1.1

Vm

1.61

1.92

2.17 

2.40 

2.59 

2.77

2.92

3.07

3.21

^ ifi

3.48

3.60

R

0.850

.580

.461 

.392 

.346 

.312

R

0.450

.310

.250

.214 

.190 

.172

R

0.068

.056 

.048 

.043 

.040 

.037 

.035

(VM

.031

.030

R

0.258

.217

.192 

.174 

.161 

.151

.143

.135

.130

ioe

.120

.116

d

0.340 

.290 

.255 

.230

d

0.100

.084 

.074 

.067 

.061

.056

.053

.050

.047

.045

.043

d

0.166

.136 

.118 

.106 

.097 

.090 

.084

070

.075

d

0.461

.349

.296

.265 

.242 

. .224 

.212

.200

.191

.183

(C) 

.66

.734

.56 

1.9

Vm

3.26 

3.82 

4.34 

4.80

(C) 

1.32 

.182 

.58 

1.7

Vm

.1.38

1.65 

1.88 

2.09 

2.29

2.47

2.73

2 Q1

2.96

3.12

3.27

(C) 

1.96

.545

4ft

1.1

Vn

1.66

2.02 

2.34 

2.61 

2.86 

3.10 

3.30

3

CA

3.69

(») 

0.66 

.545 

.40 

1.5

Vm

1 Q1

2.38

9 an

3.14 

3.42 

3.69 

3.91

4.13

4.33

4.50

R

0.515 

.440 

.386 

.349

R

0.075

.064 

.056 

.050 

.046

(U1

.040

OT7

/we

.034

.032

R

0.085

.069 

.060 

.054 

.049 

.046 

.043 

040

ftQQ

R

.531

.450

.403 

.369 

.342 

.322

.305

.291

.280
92

TABLE 14.

TRANSPORTATION OP DEBRIS BY RUNNING WATER.

Adjusted values of depth of current (d), with values of mean velocity ( Fra) and form ratio ( R=~ J Con.

Grade. .

Q......

p.e....

S

0.2

.4

.6 

.8 

1.0 

1.2 

1.4 

1.6 

1.8 

2.0 

2.2

2.4

2.6

2.8

3 n

Grade. .

Q......

«!..... .

p.e....

8

O n

.6

1.0 

1.2 

1.4

1.6

1 8

9 n

2.2

2.4

Grade . 

w.... . 

Q.... .

Til.... .

p.e.. .

S

0.2

.4

.6

1.0 

1.2 

1.4 

1.6

1.8

2.0

2 9

2 A

2.6

d

0.116 

.103 

.094 

.087 

.081 

.077 

.073 

.070 

.067

.065

.063

.061

nio

d

0.423

.281

.222

.187

.164 

.147

d

0.395

.312

.272

.228 

.214 

.203 

.194

(D) 

1.00

.182

.42

.8

Vm

1.57 

1.76 

1.93 

2.09 

2.22 

2.34 

2.47 

2.57 

2.67

2.77

2.87

2.95

3.04

(D) 

1.32

.734

.59

1.7

Vm

1 30

1.97

2.50

2.97

3.38 

3.78

(E) 

1.00 

.734 

' .34 

2.6

Vm

1.85

2.34

2.70

2.98

3.23 

3.44 

3.63 

3.80

R

0.116 

.103 

.094. 

.087 

.081 

.077 

.073 

.070 

.067

.065

.063

.061

.059

R

0.323

.214

.169

.143

.125 

.112

R

0.395

.312

.272

.246

.228 

.214 

.203 

.194

d

0.278

.207

.174 

.154 

.140 

.130 

.122 

.115 

.108 

.104 

.101

.097

d

0.129 

.122 

.115

.109

d

0.134 

.127 

.122 

.117 

.113

.108

.106

1fM

.101

(D) 

1.00

.363

.42

1.8

Vm

1.32

1.77

2.10 

2.36 

2.60 

2.81 

3.00 

3.18 

3.33 

3.48 

3.63

3.77

(E) 

.66

.363

[.35]

1.6

Vm

2.14 

2.28 

2.40

2.52

(F) 

.66 

.182 

.30 

1.5

Vm

2.05 

2.16 

2.28 

2.36 

2.46

2.52

2.60

0 R7

2.74

R

0.278

.207

.174 

.154 

.140 

.130 

.122 

.115 

.108 

.104 

.101

.097

R

0.195 

.183 

.173

.166

R

0.205 

.194 

.185 

.178 

.172

.167

.162

.157

113

d

0.252

.213 

.188 

.172 

.158 

.149 

.141 

.134 

.128 

.123

d

0.206 

.194 

.183

.175

d

0.227

.213 

.201 

.192 

.185 

.178

.173

.167

(D) 

1.00

.545

.42

.9

vn

2.16

2.56 

2.89 

3.19 

3.43 

3.67 

3.88 

4.08 

4.27 

4.44

(E) .66-

.734

[.35]

Vm

2.66 

2.83 

3.00

3.14

(F) 

.66 

.363 

.30 

0.2

Vm

2 41

2.59 

2.74 

2.86 

2.98 

3.08

3.18

3.28

R

0.252

.213 

.188 

.172 

.158 

.149 

.141 

.134 

.128 

.123

R

0.313 

.293 

.285

.265

R

0.345

.321 

.303 

.290 

.280 

.270

.262

.254

d

0.260 

.231 

.210 

.195 

.183 

.173 

.164 

.157

d

0.097 

.091 

.087

.083

.080

.077

d

0.099 

.095 

.091 

.088 

.085

.083

.081

.079

.078

(D) 

1.00

.734

.42

1.5

vm

2.87 

3.21 

3.51 

3.80 

4.04 

4.28 

4.49 

4.69

(E) 

1.00

.182

.34

2.6

vm

1.88 

2.02 

2.13

2.22

2.31

2.39

(F) 

1.00 

.182 

.26

vm

1.83 

1.92 

2.00 

2.07 

2.14

2.20

2.26

2.32

2.37

R

0.260 

.231 

.210 

.195 

.183 

.173 

.164 

.157

R

0.097 

.091 

.087

.083

.080

.077

R

0.099 

.095 

.091 

.088 

.085

.083

.081

.079

.078

d

0.163

.138 

.124 

.113 

.105 

.099 

.094 

.090 

.086

d

0.252

.198

.173

.157

.146 

.137 

.129

.123

.118

.114

.112

.108

d

.153 

.145 

.139 

.135 

.131

.127

CD) 

1.32

.363

.40

2.1

vm

1.68

1.98 

2.22 

2.43 

2.62 

2.78 

2,93 

3.07 

3.20

(E) 

1.00

.363

.34

1.6

Vm

1 44

1 st

2.32

2.50 

2.66 

2.80

2.94

3.05

3.16

3.27

3.37

(F) 

1.00 

.363 

.26 

1.2

Vm

2.24

2.38 

2.50 

2.60 

2.69 

2.78

2.85

R

O i<w

.106 

.094 

.086 

.080 

.075 

.071 

.068 

.065

R

O w

1QQ

.157

.146 

.137

19O

.123

.118

.114

.112

.108

R

0.162

.153 

.145 

.139 

.135 

.131

, .127

r,
ADJUSTMENT OF OBSERVATIONS. 93

TABLE 14. Adjusted values of depth of current (d), with values of mean velocity ( Vm), and form ratio ( R=T::) Con.

Grade. .

Q......

p.e....

S

0.6

.8

1.0 

1.2 

1.4 

1.6 

1.8 

2.0 

2.2 

2.4

2.6

2.8

i n

Grade..

Q.......

Til ......

s

0.

1. 1. 1. 1. 1.

2. 

2. 

2. 

2. 

2.

3.

3.

Grade..

Q.......

TOl......

p.e.....

S

0,

1. 1. 1. 1. 1.

'!.

;) 

'j

3.

fi

8

0

?,

4

6

8 

0 

2

1 

6

K

(I

6

R

0 

2 

4 

6

8 

0 

2

4 

6

R

D

(F) 

1.32

.36,

[.26

2.1

d Vm

0. 127 2. 1 

.121 2.2 

.,116 2.3 

.112 2.4 

.108 2.5 

.106 2.5 

. 103 2. 6

d

0.287

.363

.246

.232

.222

.213 

.206 

.199 

.194 

.188 

.184 

.180

d

0.303

.281

.264 

.251 

.241 

.232 

.226 

.219 

.214 

.208 

.203

5

1

* 

R

1 0.096 

7 .092 

6 .088 

4 .085 

2 .083 

9 .081 

5 .079

(G) 

1.00

.734

.30

0.6

vm

2.57

2.80

2.99

3.16

3.30

3.43 

3.56 

3.68 

3.78 

3.87 

3.96 

4.05

(0) 

1.32

1.119

.26

0.7

vm

2.80

3.03

3.22 

3.38 

3.52 

3.66 

3.78 

3.88 

3.98 

4.07 

4.17

(G

.60

.36

[.20

0.2

d Vm

0.222 2.<

.218 2.( 

. 198 2. S 

.189 2. £ 

.183 3.( 

.176 3. 

.172 3.5 

.167 3.C 

.163 3.C

.158 3.4

R

0.287

.263

.246

.232

.222

.213 

.206 

.199 

.194 

.188 

.184 

.180

R

0.230

.213

.200 

.190 

.183 

.176 

.170 

.165 

.162 

.157 

.154

3

]

R

19 0.336

55 .315 

SO .300 

)1 . 287 

)2 .277 

3 .268 

!3 .260 

!0 .253 

!9 . 247

16 . 242

d

0.376

.344

.322

.302

.290 .279' 

.269 

.260 

.253 

.247 

.241

d

0.220 

.207 

.197 

.188 

.181 

.174 

.168 

.164 

.159

d

0.389

.358

.338 

.320 

.306 

.294 

.285 

.276 

.270 

.263

.257

.252

(G) 

1.00

1.11

.30

0.5

V*

(H) 

.66

.36,

.34

1.7

Vn

(

.

\<s.

vn

7:

3

3. 

3. 

3. 

3. 

3. 

4. 

4. 

4

4

4

}

n

3 00

3 27

3 50

3 70

3 86

4.04 

4.18 

4.31 

4.43 

4.55 

4.61

t

»

2.52 

2.68 

2.82 

2.96 

3.07 

3.18 

3.29 

3.39 

3.48

5) 

66

734

26]

2

, R

86 0.5

10 .5

30 .a 

48 .4 

63 .4 

78 .4 

90 .4 

02 .4 

13 .4 

22 .3

32 .3

41 .3

R

0.376

.344

.322

.302

.290

.279 

.269 

.260 

.253 

.247 

.241

R

0.336 

.316 

.300 

.286 

.276 

.267 

.258 

.250 

.243

d

88 0.4

40 .4

08 .4 

81 .3 

60 .3 

43 .3 

30 .3 

18 .3 

05 .3 

95 .3

87 .3

78 .3

d

0.114 

.112 

.108 

.105 

.103 

.101 

099

noft

nofi

d

O S89

.354 

.333 

.316 

.302 

.290 

.280 

.271 

.263 

.256

OCA

.244

(G) 

.66

1.119

[.26]

0.4

Vn

65 3.

36 3.

13 4. 

96 4. 

82 4. 

71 4. 

61 4. 

52 4. 

46 4. 

38 5.

32 5.

27 5.

(G) 

1.32

.363

.26

1.1

vm

2.41

2.48 

2.55 

2.62 

2.67 

2.74 

2.78

 2.83

2.87

(H) 

.66

.734

id

0 9

Vm

2.93

3.16 

3.36 

3.53 

3.70 

3.84 

4.00 

4.13 

4.25 

4.36 

4.46

*1 ^7

R

68 0. 705

92 .660

12 .626 

30 .600 

44 . 580 

58 . 561 

70 . 548 

80 . 532 

90 . 522 

00 .512

10 .502

19 .494

R

0.087 

.084 

.082 

.080 

.078 

.077 

.075

.074

H71

R

0.580

.538 

.504 

.479 

.458 

.440 

.425 

.411 

.399 

.388

07Q

OTfi

d

0.158 

.149 

.143 

.137 

.132 

.128 

.124 

.121

.118

.116

d

0.216

.203

194

.186

.180 

.174 

.169 

.165 

.162 

.158

d

O

e/m

.507

.470 

.441 

.420 

.400 

.385 

.372 

.360 

.349 

.341

(G) 

1.00

.363

.30

0.4

Vm

2.30 

2.42 

2.54 

2.65 

2.74 

2.83 

2.93 

3.00

3.07

3.14

3.22

(G) 

1.32

734

.26

1.2

Vm

2.60

2.75

2.89

3.00

3.12 

3.22 

3.30 

3.39 

3.47 

3.54 

3.01

(H) 

.66

1.119

34

0.5

 vn 

2

QO

3.12

3.36 

3.59 

3.78 

3.94 

4.10 

4.25 

4.40 

4.50 

4.64

R

0.158 

.149 

.143 

.137 

.132 

.128 

.124 

.121

.118

.116

E

0 164

.154

.141

.136 

.132 

,128 

.125 

.122 

.120 

.118

R

O

ocn

.770

.713 

.670 

.636 

.608 

.584 

.564 

.548 

.530 

.518

,*-.,>'

*'

;->-
94 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

PRECISION.

Probable errors were computed in the manner 

described on page 73. The largest error 

found for a series of adjusted values of depth 

is ±5.5 per cent; and the arithmetical mean 

of the 66 determinations is ±1.70 per cent. 

The mean of 66 determinations of the probable 

error of an observation of depth is ±5.47 per 

cent; and the arithmetical mean of the (710) 

residuals, or differences (irrespective of sign) 

between observed and adjusted depths, is 

6.65 per cent.

The residuals of any series, computed as 

fractional parts of the quantity measured, are 

greater for small depths than for large. Computed in fractions of a foot, they do not vary 

notably with depth.

The average value of the depths to which 

these measures of precision pertain is 0.176 

foot. Using this factor to convert the preceding average percentage errors into (approximately) equivalent linear errors, we have, 

for the average of probable errors of adjusted 

depths ±0.003 foot; for the average of probable 

errors of observed depths ±0.010 foot, and 

for the average residual 0.011 foot.

The errors thus estimated include (1) the 

strictly accidental errors of observation, (2) the 

more .or less systematic influences exerted on 

the measurements by the diverse modes of 

traction, and (3) errors occasioned by the assumptions underlying the method of reduction; but they do not include such systematic 

errors as are shown by comparing gage measurements with profile measurements. (See p. 

26.) In connection with the descriptions of 

methods of measurement it is stated that the 

probable error of a single measurement by gage 

was computed from the comparison of values 

by two methods (in all cases where both are 

used), and on the assumption that the measures 

by the profile method are relatively accurate. 

This computation gave a value smaller than 

± 0.010 foot, namely, ± 0.007 foot. The groups 

of measurements to which the two estimates of 

precision apply are not the same, although they 

overlap. One group includes all gage measurements of 66 series, their number being 710; the 

other group includes all those gage measurements of the 92 series which were checked by 

profile measurements, their number being 118.

By comparing these results with those reported on page 74, it is seen that the computed 

probable errors for depth are smaller than those 

for capacity. It is nevertheless believed that 

the measurements of load were more precise 

than those of depth. The discrepancy is accounted for by considering, first, that the estimates of precision, instead of applying simply to 

the measurements of load and depth, apply to 

capacity for load as a function of slope and to 

depth as a function of slope; and, second, that 

the relation of load to slope is subject to continual rhythmic variation, while the relation of 

depth to slope is little influenced by that variation.

MEAN VELOCITY.

As the discharge, Q, equals the product of the 

sectional area of the current, wd, by the mean 

velocity, Vm, we have

_£ (23)

For each observational series, Q and w are constant and

Vm *------------ (24)

Substituting for d its value in the interpolation 

equation (21), and remembering that &' is a 

constant for each series, we obtain

.............(25)

By means of (23) a value of mean velocity was 

computed for each adjusted value of d, and 

these values are given in Table 14. They involve all the assumptions of the formula for the 

reduction of the depth observations and have 

the same fractional measures of precision.

FORM RATIO.

The adjusted values of d were used also for 

the computation of the form ratio, R, which is 

the quotient of the depth of the current by its 

width; and a value of R is tabulated with each 

value of d. Within each observational series 

the form ratios are proportional to the depths, 

and they have the same measures of precision.

GRAPHIC COMPUTATION.

All these computations were made by graphic 

methods. For each observational series a plot

r
ADJUSTMENT OF OBSERVATIONS. 95

was made on logarithmic section paper. By 

way of illustration the plot for grade (B), 

width 1.96 feet and discharge 1.119 ft.3/sec. is 

reproduced in figure 28, the finer lines of the 

logarithmic net being omitted. The vertical 

lines represent values of slope; the horizontal 

lines, capacity, depth, mean velocity, and form

100 Vm ,1.0

7

z

-.2 .3 .4 .6 

SJope

1 R-0.1

.8 1.0 2.0

FIGURE 28. Logarithmic computation sheet, combining relations of 

capacity, mean velocity, form ratio, and slope.

ratio. For the convenience of having the 

graphs close together then* scales are made to 

differ, the ratio of one to another being 10, 100, 

or 1,000. In the particular instance shown in 

figure 28 the same line represents C= 100, 

<Z = 0,1, 7TO = 1.0, and# = 0.1. With use of this 

notation were plotted the equations for C, d,

Fm , and R, as functions of S. In their logarithmic forms these are

log d = log &' n: log 8 

log 7M -log §-log <Z=log §-log I'+n, log 8

w w

V log JK = log d-log w=log --MI log 8

log C'=log li + n log (S-o)

Their loci are the straight lines marked d, Vm, 

and R and the curve marked C.

After the preparation of these constructions 

the values of d, Vm, and R, corresponding to the 

selected series of values of S, were read from the 

sheets, affording the data of Table 14. The 

sheets had also many other uses, for in recording the relations of four interdependent variables to slope they also recorded then* relations 

to one another. The points of the four loci 

which lie in the same vertical represent corresponding values of the several variables, and 

this property made it possible to read from the 

plot the value of a variable corresponding to a 

particular value of one of the others. For 

example, if it is desired to learn the capacity 

corresponding to a mean velocity of 2 ft./sec., 

the intersection of the velocity line with the 

line representing 2 on the scale of velocities is 

first found. From that intersection a vertical 

is followed or drawn to the capacity line and 

the position of the second intersection is read on 

the capacity scale.

A large part of the numerical data cited in 

the following discussions were either taken 

directly from the computation sheets or based 

upon them.

The accuracy of the computations by logarithmic graph may be characterized by saying 

chat it is slightly below that by slide rule. The 

theoretic accuracy is the same, but tests of ihe 

logarithmic paper employed showed it to be a 

less perfect instrument than the slide rule, 

which was used for a large body of routine 

computations.
CHAPTER III. RELATION OF CAPACITY TO SLOPE.

INTRODUCTION.

A series of chapters, beginning with this one, 

are given to the discussion of the observational 

data. The discussions make use of the adjusted values of capacity for stream traction, 

slope of stream bed, and depth of current, 

with their derivatives, contained in Tables 12 

and 14. Associated with those adjusted values 

are certain grades of transported material, or 

degrees of fineness, certain widths of channel, 

and certain discharges of the transporting 

stream. The leading subjects of discussion are 

the relations of capacity to slope, discharge, 

fineness, and form ratio, but consideration is 

also given to the relations of capacity to depth 

and velocity, and to the relations which duty, 

efficiency, and depth bear to various conditions. The discussion is essentially empiric, its 

course being guided in small degree only by 

theoretic considerations.

The treatment of the relation of capacity to 

slope first views it as conditioned by channels 

of fixed width, and then as subject to the relatively ideal condition of fixed form ratio.

IN CHANNELS OF FIXED WIDTH.

THE CONDITIONS.

In each observational series the width of 

channel was constant, and so also were the 

discharge and the grade of debris constituting 

the load. As the load was changed, the slope 

responded; velocity responded to change of 

slope; and with variation of velocity went variation of depth. The ratio of depth to width, or 

the form ratio, was therefore a variable; so 

that the stream which dragged a large load 

down a steep slope differed in form, and to that 

extent in type, from the stream which moved 

a small load along a gentle slope. In a few 

cases it is possible so to combine data from 

different series as to discover the relation of 

capacity to slope for streams which have similar cross sections; and these will be examined 

in another place; but the principal discussion 

relates to streams with constant width and 

variable form ratio.

96

THE SIGMA FORMULA. 1

Those properties of the formula

(10)

which determined its adoption for the reduction 

of the observational data to a more orderly 

system led also to the consideration of its 

availability as an empiric formula for the general relation of the stream's capacity for traction to the slope of its bed. With a view to 

this second use, the specific values of 61; o, and 

n derived for the purpose of the reduction 

values which are recorded in Table 15 were 

arranged and combined in various ways, in 

order to discover, if possible, definite relations 

to the several conditions in accordance with 

which the experiments were varied. It has 

already been noted (p. 71) that the critical 

slope, o, varies inversely with fineness of 

de'bris, with discharge, and (probably) with 

range of fineness within a grade, and that it 

varies inversely with width of trough when 

that width is relatively small, but directly 

with width when width is relatively large.

=ftP, (26)

written to express these relations in symbols, 

introduces H to designate range of fineness and 

distinguishes trend of function by means of 

accents. As the notation by accents will be 

frequently used, its definition may be made 

explicit. Where the function is direct, or 

increasing, its value increasing with the increase of the independent variable, the symbol 

of the variable is given the acute accent (') 

Where the function is inverse, or decreasing, its 

value decreasing with increase of the variable, 

the grave accent 0) is used. For a maximum 

function, first increasing and then decreasing 

( A ) is used, and for a minimum function ( v ). 

The discussion of the values of &t showed (1) 

that they vary directly and in marked degree, 

but irregularly, with F2 , (2) that they vary

1 Since these lines were penned I have discovered that this title 

duplicates one in the field of higher mathematics. Nevertheless I 

retain it because of its mnemonic convenience. The "two fields of 

application are so distinct that serious confusion will not be occasioned.

T'
RELATION OF CAPACITY TO SLOPE. 97

directly and approximately in simple ratio, 

w.ith Q, and (3) that they vary increasingly, if 

at all, but very slightly with w, while (4) a relation to H could not be disentangled from the 

relation to F2 . The discussion of the values of 

n showed (1 ) that they vary inversely and irregularly with F2 , (2) that they vary inversely 

and more regularly with Q, and (3) that the 

variation in respect to w is direct for small 

widths and inverse for large widths, while (4) 

the relation to H is not separable.

The data on the three parameters are summarized in equation (27), which is an expansion of equation (10). In all probability ft and 

/n are as complex as/, but no factors are introduced of which the influence was not definitely 

shown by the discussion.

<?=/r (4 4) [8-f A

This equation is subject to a qualification 

connected with the assignment of values to a. 

It will be recalled that that assignment was 

somewhat arbitrary, and also that the values 

of a entered into the computation of the values 

of lt and n. A systematic error in the values 

of a would therefore cause systematic errors 

in the other parameters and might vitiate 

conclusions as to the laws of their variation. 

A search was made for evidence of such errors, 

the search making use of the principle (easily 

demonstrated) that a positive error in a would 

cause a positive error in \ and a negative error 

in n. While the result of the search wa,«i negative, it is not to be supposed that the values of 

a have high precision. To their errors, in 

combination with the obscure influences of the 

varying range of fineness and with the errors of 

observation, are to be ascribed the irregularities of the constants of the adjusting equations.

While the algebraic relations are such that 

minor errors in the values of a might have important influence on values of &x and n, their 

influence on the interpolated values of 0 would 

be small.

The uncertainties affecting the several elements of equation (27) are so great that no 

attempt will be made to develop from it a 

definite and quantitative expression for the 

relation of capacity to slope. For this reason 

the preceding paragraphs have attempted to 

present only the general tenor of the discussion

20921° No. 86 14 7

and have purposely omitted details. The 

results, despite important qualifications, show 

clearly that any general expression of the law 

connecting capacity and slope which might be 

based on formula (10) would be highly complex. 

With reference to the main subject of this 

chapter, the following section is'of.the nature 

of a digression, its purpose being to define a 

method and terminology used in several of the 

succeeding chapters.

THE POWER FUNCTION AND THE INDEX OF 

RELATIVE VARIATION.

One of the algebraic forms to which the title 

power function is applied is

y=axn ............ -..(28)

If the coefficient be suppressed, leaving 

y oc xn

this is the exact equivalent of the familar 

"y varies as the nih power of a;." This mode of 

comparing the rate of variation of one thing 

with the rate of variation of another is extensively employed, and it so commends itself by 

its simplicity that its use is constantly extended 

into fields where its applicability is approximate only. Having occasion to make much use 

of certain variants of this function, I find it 

important to obtain a clear conception of its 

properties and shall therefore give the matter 

somewhat elementary attention with due 

apology to the mathematical reader.

If we consider x and y merely as numbers, 

the rate of variation with y with respect to x 

is the ratio of the differential increment of y 

to that of x. That ratio is

dy

= (29)

If we consider x and y (and also the constant 

a) as powers of a common base, equation (28) 

becomes

log y = log a + n log x (30)

The rate of variation of log y with respect to 

log x is, differentiating,

'n ............(31)
98 TRANSPORTATION OF DEBRIS BY RUNNING WATER,

Now, in saying that one quantity varies as 

a certain power of another, or in using such a 

function as (28), the index of variation, or that 

by which is indicated the comparative rate of 

variation, is the exponent n; and the value of 

this exponent, in terms of the variables, is found 

not in (29) but in (31). The expression "y 

varies as the nth power of x" is equivalent to 

"the rate of variation of y, considered as a 

power, is n times the rate of variation of x, 

considered as a power."

FIGURE 29. Logarithmic locus of the power function.

An equivalent result would be attained if 

attention were given to the quality of the 

growth of x and y. Considering their growth 

as occurring by additive increments, equation 

(29) gives the ratio between their rates of increase. Considering their growth as a matter 

of multiplication by ratios, the additive increments are increments to logarithms, and equation (31) gives the ratio between rates of increase.

Equation (30), the logarithmic equivalent of 

(28), is the equation of a straight line. Representing it by AB in figure 29, its inclination

CO ftfi equals n, and its intersection with the axis

of log y gives CO = log a. These familiar properties enhance the utility of function (28) by 

enabling the investigator to discuss its constants on logarithmic section paper.

In many, probably a large majority, of the 

physical problems to which the power function 

is applied, it is found that the exponent, n, 

does not have a constant value through the 

entire range of observed values of x and y. 

The locus of log y=/(log x} is then not a 

straight line but a curve, which we may represent by AB in figure 30. At any point of the 

curve C, its minute element, not distinguishable

CE from a straight line, has an inclination,

and which we may call n2. The value of n2 

varies from point to point of the curve, so that 

if we try to express the relation of y to x in the 

form of equation (28) we must regard the exponent as a variable.

The element at C may also be thought of as 

part of the tangent line CD, of which the equation is

log y = FO + n2\og x 

whence

On comparing this "with (28), it is seen that 

\og~lFO corresponds to a. Let us replace it 

by oj,-. giving

y^aX*'------ ------ -(32)

It is evident that for any other point of the 

curve, as C' ', the tangent intersects the axis of 

log y at a point different from F, and this corresponds to a different value of at. In other 

words, if we would express in an equation of 

the type of (28) the same relation between two 

variables that is expressed by the logarithmic 

locus in figure 30, we must make the coefficient 

as well as the exponent variable.

which is homologous with n in (31) and (28)

FIGURE 30. Locus of log y=f(log x), illustrating the nature of the index 

of relative variation.

The values of ax and n2 are evidently functions of the independent variable, x.

It is possible to give to the relation shown 

by the logarithmic locus an algebraic expression 

which is identical in form with (28) and in 

which the coefficient is a constant. That is to 

say, it is possible to segregate the variability of 

parameters in the exponent; but when that is 

done the exponent no longer corresponds to the 

expression "varies as the nth power of," and
RELATION OF CAPACITY TO SLOPE. 99

the expression thus lacks the essential quality 

of (28). To illustrate, let us assume that the 

position of the point F is fixed, so as to give a 

constant value to ar The values of n2 then 

correspond to the inclinations of lines FC, FC', 

etc., drawn to points on the curve; but these 

lines are not tangents and their inclinations are 

not those of the corresponding elements of the 

curve.

In order to satisfy the condition, from equation (28), that y = a when x l arid logz = 0, 

the fixed point must be at the intersection, G, 

of the curve with the axis of log y. The lines 

connecting it with other points of the curve are 

therefore chords.

In the discussions of our laboratory data use 

will be made of both these variants of the power 

function; and they will be distinguished, from 

one another as well as from (28), by the following notation. In

y=vx' ..........(33)

the coefficient and exponent are both variable 

and are functions of x. The symbol v is chosen 

for the coefficient to signalize its variability. 

The exponent, i, denotes the instantaneous ratio 

of the variation of y to the variation of x, when 

those variations are viewed as ratios. It is the 

first differential coefficient of log y with respect 

to log x, and it will be spoken of as the index of 

relative variation. 

In

y = cxJ.. ............ (34)

the coefficient is constant and the exponent 

variable.

Whenever, in the investigation of the natural 

law connecting two variables, pairs of simultaneous values of the variables are known by observation, it is possible to plot a curve representing empirically log y =/ (log x) such a curve as 

ACGB in figure 30. The directions of the elements of that curve, or the values of i, are essentially facts of observation. They depend 

exclusively on the phenomena and are independent of the units in which observational 

data are expressed. It is different with the 

values of j, for those depend on the position of 

the point, G, in which the curve intersects the 

axis of log y, and therefore on the position of 

the axis. The position of the axis corresponds

to log x = 0 or x = 1 and is thus dependent on 

the magnitude of the unit by which the independent variable is measured.

In other chapters of this report much attention is given to the values of i, and the discussion of the variations of such values is used as 

a mode of treating empirically the relations 

between the various factors of the general 

problem of traction.

THE SYNTHETIC INDEX.

Recurring to figure 30, let us give attention to 

a restricted portion of the curve, for example, 

the part between C and C'. The value of i corresponding to the point 0 is the inclination of 

the line OF; the value of i corresponding to C' 

is the inclination of C'F'. Between the two are 

a continuous series of other values. The inclination of the chord connecting C and Cf , conconsidered as .a ratio or exponent, is intermediate between the extreme values of i. If the 

sequence of values follows a definite law, the 

value given by the chord equals some sort of a 

mean derived from the others; and, in any case, 

it is in a sense representative of the group. It 

may be called a synthetic index of relative variation between the indicated limits.

If the coordinates of C be log xf and log y', 

and the coordinates of C' be log x" and log y", 

then, representing the synthetic index by /,

/=logy"-logy' 

log x" log x' (35)

As the direction of the chord depends on the 

positions of C and C' upon the curve, so the 

value of / depends on the limits between which 

it is computed. As the direction of the chord 

gives no information concerning the direction 

of any part of the curve, so the value of / can 

not be used to determine any particular value 

of i. It is used in the following pages for the 

comparison of different'functions for which the 

data span approximately the same range of 

conditions.

APPLICATION TO THE SIGMA FUNCTION.

Let us now represent the relation of capacity 

to slope by an equation of form (33),

/"/ Oft - /o /* \

6'=^ £''-__..___._._. (36)
100 TRANSPORTATION OE DEBRIS BY RUNNING WATER.

and develop the value of \ from equation (10). 

The logarithmic equivalent of (10) (p. 64) is

log Cr =log-61 + nlog (S-<j)____(15)

Differentiating, and dividing both members by 

dlog-S;

dlog C_ndlog(S-a) 

dlogS~ dlogS. .

Introducing these values'in (37) and reducing, 

we have

nS

Sr-C (38)

The first member of this equation is the value 

of il} and \ may be substituted for it. Also

dS

 a

and

dS

The values of n and a being known for each 

observational series, it is possible, by means of 

equation (38), to compute \ for any value of 8. 

The values of \ have been computed for series 

of values of 8 having the uniform interval of 

0.2 per cent (and below 1.0 per cent the interval of 0.1 per cent) and having such range in 

each observational series as to correspond with 

the range of the observations. These values 

are recorded in Table 15. It is proposed to 

discuss their relations to various conditions, 

beginning with slope.

TABLE 15. Values ofi^, the index of relative variation for capacity in relation to slope.

Conditions of experimentation .....

Parameters of adjusting equation. .

Grade. .

w .....

Q......

flT.......

6....... 

[n.......

5

0.2................................

.3................................

.5................................

.6................................

.7................................

8

.9 . .... ..... . .........

1.0................................

1.2.............................!..

1.4..............................:.

1.6................................

1.8................................

2.0................................

'2.2................................

(A)

0. 60

0.093

0.40 

32.2 

1.50

,0.182

0.20 

56.2 

1.60

0.545

0.04 

150.3 

1.42

1.00

0.182

0.12 

47.4 

1.81,

0.363

0.07 

113.3 

1.70

0.734

0.05 

252.3 

1.70

1.32

0.182

0.17 

50.1 

. 1.80

0.363

0.10 

122.8 

1.53

.0.734

0.07 

282.4 

1.59

1.96

0.363

0.17 

128.9 

1.58

0.734

0.10 

280.1 

1.40

1.119

0.08 

397.0 

1.21

Values of »i. .

2.25 

2.10 

2.00 

1.93

1.88

1.84

2.24 

2.12 

2.05 

1.99

1.92 

1.86 

1.82 

1.79

1.54

1.52 

1.50 

1.49 

1.48 

1.48

1.47

2.39

2.27 

2.19 

2.13 

2.09 

2.06

2.02 

1.99 

1.96 

1.95 

1.93

1.92

2.07 

1.98

1.92 

1.89 

1.87 

1.85 

1.83

1.81 

1.79 

1.78 

1.77

1.94 

1.89

1.85 

1.83 

1.81 

1.80 

1.7.9

1.78

2.72

2.51 

2.38 

2.28 

2.22 

2.17

2.10 

2.05 

1.02

1.99 

1.97

1.95

3.05

2.29 

2.04 

1.91

1.83 

1.78 

1.75 

1.72 

1.70

1.67 

1.65 

1.63

1.62 

1.61

2.08 

1.93 

1.85

1.80 

1.77 

1.74 

1.72 

1.71

1.69 

1.68

2.74 

2.39

2.20 

2.08 

2.00 

1.94 

1.90

1.84 

1.80 

1.76

1.74

1.72

2.79

2.09

1.86 

1.74

1.68 

1.63 

1.60 

1.57 

1.55

1.52

1.63

1.57 

1.53 

1.50 

1.48 

1.47

1.44 

1.43 

1.42 

1.41 

1.40

1.39
RELATION OF CAPACITY TO SLOPE. 

TABLE 15. Values ofilt the index of relative variation for capacity in relation to slope Continued.

101

Conditions of experimentation .....

Parameters of adjusting equation . .

Grade..

w .....

Q......

L......

(n.......

S

0 9

.3................................

4

.5................................

.6................................

.7................................

.8................................

.9................................ i.o................................

1.2................................

1.4. ...............................

1.6. ...............................

1 8

'<>.. 0. ...............................

'.\.i. ...............................

 > 4

 > g

,f 0

3.0................................

3.2................................

3.4................................

3.8................................

4.0................................

4.2................................

Conditions of experimentation .....

Parameters of adjusting equation . .

Grade..

w...--.

Q......

S

0.2..................................

.3..................................

. 6..... .........'.... ...............

.7..................................

.8..................................

.9.................................. i.o..................................

1.2.................................

1.4.................................

1.6.................................

1.8..................................

2.0..................................

9 9

2 4

O Q

3.0..................................

3.2..................................

0 A

3.6..................................

 J C

4.0.................................

4.2..................................

(B)

0.23

0.093

0.70 

23.8 

0.93

0.182

0.60 

33.2 

0.99

0.44

0.093

0.50 

29.7 

1.15

0.182

0.10 

34 3 

1.63

0.66 -

.0.093

0.30 

19.4 

1.64

0.182

0.10 

39.8 

1.64

0.363

0.08 

91.4 

1.45

0.545

0.07 

132.8 

1.38

1.00

0.182

0.12 

37.6 

1.69

0.393

0.08 

97.5 

1.54

0.545

0.06 

157.2 

1.53

0.734

0.06 

219.3 

1.52

Values of i'i.

»

7.41 

4.17 

3.10

2.22 

1.85 

1.65 

1.52 

1.43

1.36

6.% 

3.98 

2.98 

2.49

1.99 

1.74 

1.59 

1.49

4.04 

3.08 

2.60 

2.31

1.98 

1.80 

1.68 

1.60 

1.54

1.50

1.46

1.43

2.08

1.96

1.90 

1.86 

1.83 

1.81

1.78 

1.76 

1.74 

1.73

2.87 

2.63 

2.46 

2.34

2.19 

2.09 

2.02 

1.97 

1.93

1.90 

1.88 

1.86 

1.84

1.82

2.46

2.19

2.05

1.97 

1.92 

1.88 

1.85 

1.82

1.79 

1.77 

1.75 

1.74 

1.73

1.72 

1.71 

1.71 

1.70

1.70 

1.69

1.69

1.69

1.69

1.68

1.68

1.73

1.67 

1.64 

1.61 

1.59 

1.58

1.55 

1.54 

' 1.53 

1.52 

1.51

2.13

1.80

1.68

1.61

1.57 

1.54 

1.51 

1.50 

1.49

1.47 

1.46 

1.45 

1.44 

1.-43

(B)

1.32

0.182

0.17 

39.5 

1.61

0.363

0.12 

96.5 

1.54

0.545

0.10 

165.4 

1.55

0.734

0.08 

233.5 

1.57

1.96

0.363

0.18 

93.9 

1.64

0.545

0.14 

153.5 

1.60

0.734

0.12 

228.2 

1.46

1.119

0.10 

363.3 

. 1.43

2.22

2.11 

2.04 

1.98 

1.95 

1.92

1.88 

1.85 

1.83 

1.81 

1.80

1.79

1.78 

1.77

1.76

1.76

2.57

2.10

1.93

1.84

1.78 

1.74 

1.71 

1.69 

1.68

1.65 

1.63 

1.62 

1.61 

1.60

1.60 

1.59

1.65 

1.63 

1.62

1.61 

1.59 

1.58 

1.58 

1.57

1.57

2.17 

1.89 

1.78 

1.72

1.69 

1.66 

1.64 

1.62 

1.61

1.60 

1.58 

1.58 

1.57

. (0

0.44

0.093

0.40 

22.6 

1.43

0.182

0.20 

34.4 

1.50

0.66

0.093

0.16 

18.2 

1.58

0.182

0.11 

38.6 

1.54

Values of ii.

2.80 

2.44

2.25 

2.13 

2.05 

1.99 

1.94

1.88 

1.84 

1.80 

1.78 

1.76

1.75 

1.74

1.73

2.56

2.20 

2.02

1.92 

1.85 

1.81 

1.77 

1.75

1.71 

1.68 

1.66 

1.64 

1.63

1.62

1.94

1.86 

1.81 

1.78 

1.75 

1.73

1.70 

1.67 

1.66 

1.65

2.14

1.96.

1.87

1.81 

1.78 

1.75 

1.72 

1.71

1.68 

1.67 

1.65 

1.64

2.99

2.57

2.35 

2.21 

2.12 

2.06 

2.01

1.94 

1.89 

1.85 

1.83 

1.81

1.79

2.23

2.09 

2.01 

1.94 

1.90 

1.86

1.82 

1.78 

1.76 

1.74

2.44 

2.09 

1.93

1.83 

1.77 

1.72 

1.69 

1.66

1.63 

1.60 

1.58 

1.57 

1.56

2.86

2.15 

1.91 

1.79

1.73 

1.68 

1.64 

1.62 

1.60

1.57

4.29 

3.39 

2.86 

2.57 

2.39

2.15 

2.00 

1.91 

1.84 

1.79

1.75

2.22 

2.10 

2.00 

1.93 

1.88

1.81 

1.76 

1.72 

1.69 

1.67

1.65

2.33

2.16 

2.05 

1.98 

1.92

1.87

1.82 

1.79 

1.76 

1.74 

1.72

1.70 

1.69 

1.69 

1.68 

1.67

1.66 

1.66 

1.65 

1.65 

1.65

1.64

2.43 

2.12 

1.97

1.88 

1.83 

1.79 

1.75 

1.73

1.70 

1.67 

1.65 

1.64 

1.63

1.62 

1.61 

1.61 

1.60 

1.60

1.59 

1.59 

1.59 

1.59
102 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 15. Values ofi^, the index of relative variation for capacity in relation to slope Continued.

Grade. .

Conditions of experimentation ..... w ......

Q......

It.......

Parameters of adjusting equation . .{61 ...... 

In.......

S

0.2..........................................

.3..........................................

.4..........................................

.5..........................................

.6..........................................

.7..........................................

.8..........................................

.9.............:............................

1.0....:.....................................

1.2..........................................

1.4..........................................

1.6..........................................

1.8..........................................

2.0..................................:.......

2.2......'....................................

2.4..........................................

2.6..........................................

2.8..........................................

3.0..........................................

3.2..........................................

3.4..........................................

3.6..........................................

3.8..........................................

Parameters of adjusting equation. . ....... .-i

(C)

0.66

0.363

0.08 

82.9 

1.46

0.545

0.06 

122.8 

1.48

0.734

0.04 

162.1 

1.50

1.00

0.182

0.15 

40.4 

1.59

0.363

0.11 

100.1 

1.41

0.545

0.09 

159.2 

1.33

0.734

0.07 

199.2 

 1.35

1.119

0.06 

301.6 

'1.39

1.32

0.182

0.22 

33.7 

1.85

0.363

0.16 

99.9 

1.40

0.545

0.13 

156.1 

1.35

0.734

0.11 

218.5 

1.30

Values oft'i.

1.83 

1.74

1.68 

1.65 

1.62 

1.60 

1.59

1.56 

1.55 

1.54 

1.53 

1.52

1.52 

1.51

1.51

Grade. .

Q......

\«. ......

4i

........

S '

0. 2. .................................................

.3..................................................

.5..................................................

.6..................................................

.7..................................................

.8..................................................

.9.................................................. i.o..................................................

1.2..................................................

1.6..................................................

1.8..................................................

2

n

2. 2. .................................................

2.4..................................................

2.6..................................................

2.8..................................................

2.12

1.86

1.75 

1.69

1.65 

1.62 

1.60 

1.59 

1.58

1.56 

1.55 

1.54 

1.54 

1.53

1.53

1.63

1.61 

1.59 

1.58 

1.57 

1.57

2.54 

2.27

2.12 

2.02 

1.95 

1.90 

1.87

1.81 

1.78 

1.75 

1.73 

1.71

1.70 

1.69 

1.68 

1.68

3.14

2.23

1.95 

1.81

1.73

1:68

1.64 

1.61 

1.59

1.55 

1.53 

1.51 

1.50 

1.49

1.49 

1.48 

1.47 

1.47

(C)

1.90

0.363

0.24 

93.4 

1.62

0.545

0.20 

155.7 

1.50

0.734

0.17 

245.1 

1.34

1.119

0.14 

438.3 

1.59

1.63

1.57 

1.53 

1.50 

1.48 

1.47

1.44 

1.43 

1.42 

1.41 

1.40

1.39

2.18

1.76

1.64

1.57

1.53 

1.50 

1.48 

1.46 

1.45

1.43 

1.42 

1.41 

1.41 

1.40

1.39

1.58

1.55 

1.52 

1.50 

1.49 

1.48

1.46 

1.45

2.92 

2.70 

2.55 

2.45 

2.37

2.27 

2.20 

2.14 

2.11 

2.08

2.06 

2.04

i

3.00

2.33

2.06

1.91 

1.82 

1.75 

1.70 

1.67

1.61 

1.58 

1.55 

1.54 

1.52

1.51 

1.50

1.72 

1.65 

1.61 

1.57 

1.55

1.51 

1.49 

1.47 

1.45 

1.44

1.43 

1.42

2.90 

2.12 

1.80 

1.67

1.60 

1.55 

1.51 

1.49 

1.47

1.44 

1.42 

1.40 

1.39 

1.38

1.37

(D)

0.66

0.093

0.19 

13.2 

1.80

0.182

0.14 

37.2 

1.55

0.545

0.08 

115.2 

1.51

1.00

0.182

0.17 

32.8 

1.67

0.363

0.12 

87.9 

1.49

0.545

0.10 

129.4 

1.68

0.734

0.08 

174.6 

1.52

Values of /i.

3.12

2.71 

2.47 

2.32 

2.21 

2.14

2.03 

1.96 

1.91 

1.87 

1.84

1.82

2.49

2.24 

2.09 

1.99 

1.92

1.87

1.79 

1.74 

1.71 

1.68 

1.66

1.65

2.33

2.03

1.87 

1.77 

1.70 

1.65 

1.62

1.56 

1.53 

1.50

2.21

2.07 

1.99 

1.93

1.88. 

1.85

1.80

2.64 

2.47 

2.36 

2.28 

2.22

2.14 

2.08 

2.04 

2.01 

1.99

1.97 

1.96 

1.94

2.38 

2.15

2.02 

1.93

1.88 

1.83 

1.80

1.75 

1.72 

1.70 

1.68 

1.66

1.65 

1.64 

1.63

2.52

2.06

1.89 

1.80

1.74 

1.71 

1.68 

1.66 

1.64

1.62 

1.60 

1.59 

1.58 

1.57

1.57

2.20 

2.12 

2.06 

2.01

1.94 

1.90 

1.86 

.84 

.82

.81 

.80 

.79

1.78

2.48

2.12

1.95

1.86 

1.80 

1.75 

1.71 

1.69

1.65 

1.63 

1.61 

1.59 

1.58

1.57 

1.56

2.11

2.04 

1.97 

1.93 

1.90 

1.87

1.84 

1.81 

1.80 

1.79 

1.78

2.54 

2.12 

1.90 

1.81

1.75 

1.72 

1.69 

1.67 

1.65

1.63 

1.61 

1.60 

1.59
RELATION OF CAPACITY TO SLOPE.

TABLE 15. Values ofi1; the index of relative variation for capacity in relation to slope Continued.

103

Grade. .

Q......

(*.......

fc......

In.......

s

0.2.........................................

.3..

.4.........................................

.5.........................................

.6.........................................

.7

.8.........................................

.9.........................................

1.0.........................................

1.2.........................................

1.4.........................................

1.6.........................................

1.8.........................................

2.0.........................................

2.2.........................................

2.4.........................................

2.6.........................................

2.8.........................................

Conditions of experimentation.... . ..........

Grade..

Q.:....

........ &,......

w.......

S

0.2..........................................

.3..........................................

.4..........................................

.5..........................................

.6..........................................

.8..........................................

.9..........................................

1.0..........................................

1.2..........................................

1.4..........................................

1.6...................................:......

1.8.........................................;

2.0..........................................

2.2..........................................

2.4.......................................:..

2.6..........................................

2.8..........................................

3.0..........................................

(D)

1.32

0.363

0.16 

80.1 

1.77

0.734

0.11 

160.0 

1.65

(E)

0.66

0.363

0.16 

33.8 

1.93

0.734

0.11 

50.2 

1.73

1.00

0.182

0.08 

17.0 

1.72

0.363

0.06 

37.3 

1.70

0.734

0.04 

76.5 

1.58

1.119

0.03 

145.9 

1.58

1.32

0.363

0.04 

38.8 

1.57

0.734

0.03 

76.1 

1.50

1.119

0.02 

126.1 

1.77

Values of it.

2.60

2.41 

2.29 

2.21 

2.15 

2.11

2.04 2 fin

1.97

1.94

1 Q7

2.60

2.27

2.11

2.02 

1.95 

1.91

1.88 

1.85

1.81

2.09

2.03 

1.99.

2.17

2.13 

2.10 

2.08

2.16 

2.06

1.99 

1.95 

1.92 

1.90 

1.88

1.85 

1.83 

1.82 

1.81 

1.80

1.79 

1.79

1.78

1.78

2.43 

2.13 

2.00 

1.93

1.89 

1.86 

1.84 

1.82 

1.81

1.79 

1.78 

1.77 

1.76 

1.76

1.75 

1.75

1.98 

1.83 

1.76 

1.72

1.69 

1.68 

1.67 

1.66 

1.65

1.64 

1.63 

1.62 

1.62 

1.61

1.69

1.67 

1.65 

1.64 

1.64 

1.63

1.62 

1.62 

1.61 

1.61 

1.61

1.61

1.60

1.68 

1.67 

1.66 

1.65 

1.64

1.63 

1.62 

1.61 

1.61 

1.60

1.59

1.57 

1.56 

1.55 

1.55 

1.54

1.54

(F)

0.66

0.182

0.33 

12.02 

1.86

0.363

0.21 

30.5 

1.69

0.734

o.ir

52.2 

1.61

1.00

0.182

0.44 

7.96 

2.31

0.363

0.31 

28.2 

1.88

0.734

0.22 

65.0 

1.61

1.119

0.17 

106.9 

1.60

1.32

0.363

0.39 

30.3 

1.63

0.734

0.28 

71.1 

1.73

1.119

 0.23 

126.4 

1.49

1.83 

1.82 

1.82 

1.81 

1.81

1.80

(0)

0.66

0.363

0.50 

27.5 

1.74

Values of /i.

2.57 

2.44 

2.35 

2.28 

2.23

2.19

2.16

2.14

2.14

2.04 

1.99 

1.94 

1.91 

1.89

1.98

1.94

1.88 

1.83 

1.80 

1.78

3.65 

3.38 

3.20 

3.07 

2.97

2.90 

1.84 

1.79

3.07 

2.87 

2.72

2.54 

2.42 

2.33 

2.27 

2.23

2.19 

2.16 

2.14

2.35 

2.22 

2.13 

2.06

1.98 

1.91 

1.87 

1.84

2.08 

2.00 

1.94 

1.90

1.83 

1.79 

1.76 

1.74

t "." ,

i

2.42 

2.27 

2.16 

2.08 

2cn

1 00

2.65 

.2.50 

2.40

2.25 

2.15 

2.09

2.09 

2.00 

1.94

1.85 

1.78 

1.74 

1.71

3.48

2.98 

2.71 

2.53 

2.41 

2.32

2. 25 

2.20 

2.16 

2.12

2.08
104 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 15. Values ofil} the index of relative variation for capacity in relation to slope Continued.

Parameters of adjusted equation ...........

Grade. .

Q......

£.......

n1:.:::::

S

0. 2. .........................................

.3..........................................

.4..........................................

.5..........................................

.6..........................................

.7..........................................

g

.9... .............. ... . .........

1.0..........................................

1.2..........................................

1.4..........................................

1.6..........................................

1.8..........................................

2.0..........................................

2.2.....................................:....

2.4..........................................

2.6..........................................

9 S

3 n

3.2..........................................

(G)

0.66

0.734

0.36 

64.5 

1.69

1.119

0.28 

85.0 

1.78

1.00

0.363

0.58, 

23.6 

1.85

0.734

0/41 

65.7 

1.70

1.119

0.33 

115.5. 

1.63

1.32

0.363

0.71 

15.39 

2.37

0.734

0.50 

62.3 

1.71

1.119

0.40 

114.5 

1.54

(H)

0.66

0.363

0.80 

19.38 

2.01

0.734

0.56 

.52.9. 

1.72

1.119

0.48 

75.1 

1.67

Values of ti.

3.49 

3.08 

2.82 

2.65

2.42 

2.28 

2.19 

2.12 

2.07

2.03 

1.99 

1.96

3.23

2.89 

2.68 

2.54 

2.43

2.29 

2.20 

2.14 

2.09 

2.05

2.02 

2.00 

1.98

3.57 

3.15 

2.90 

2.71 

2.60

2.51 

2.43 

2.38 

2.33

2.29

4.11 

3.49 

3.12

2.88

2.58 

2.40 

2.29 

2.20 

2.14

2.09 

2.05 

2.02

3.63

3.09 

2.78 

2.58 

2.44

2.25 

2.14 

2.06 

2.00 

1.96

1.92 

1.89

3.91 

3.67

3.50 

3.36 

3.26 

3.18 

3.11

3.05

3.00

4.58 

3.86 

3.43

2.94 

2.64 

2.49 

2.38 

2.29

2.22 

2.17 

2.12 

2.08

3.61

3.09 

2.78 

2.57

2.32 

2.16 

2.06 

1.99 

1.93

1.89 

1.85 

1.83

6.03 

4.69 

4.02 

3.62 

3.35

3.16 

3.01 

2.90 

2.81

5.75 

4.56 

3.92

3.23 

2.88 

2.65 

2.50 

2.40

2.32 

2.25 

2.20 

2.16 

2.12

2.09

5.31 

4.17 

3.58 

3.21

2.78 

2.54 

2.38 

2.27 

2.20

2.14 

2.09 

2.05

VARIATION OF THE INDEX.

Each column of the table contains a set of 

values of \ which pertain to the same grade, 

fineness, width, and discharge and of which the 

changes are related to slope only. In figure 31 

a number of these sets are plotted in relation to 

slope. The curves have a strong family likeness, arising from the fact that the data were 

all adjusted by the sigma formula; but the 

likeness would not altogether disappear if the 

assumptions of that formula were abandoned. 

The general relations of the index to slope are 

as follows:

(1) It varies decreasingly with slope.

(2) Its rate of change is greater for low 

slopes than for high.

The upper group of curves all pertain to 

grade (C) and width 1.00 foot, but represent 

different discharges. They show (3) that the 

rate of change for similar slopes is greater for 

small discharges than for large.

The second group .of curves all pertain to 

grade (C) and discharge 0.363 ft. 3/sec., but 

represent different widths. They show (4) 

that the rate of change for similar slopes is 

greater for broad channels than for narrow, or,

as the depth varies inversely with the width, 

that the rate of change is greater for shallow 

streams than for deep.

! The third group of curyes all pertain to 

width 1.00 foot and discharge 0.363 ft.3/sec., 

but represent different grades of de"bris. They 

show (5) that the rate of change is greater for 

coarse debris than, for fine.

In the third group the curves for grades (A), 

(B), (C), (D), and (E) lie close together, while 

those for the coarser grades (F) and (G) are 

well separated. This is probably connected 

with the fact that the range of fineness gradually increases from (A) to (E) and then drops 

abruptly from (E) to (F). The influence of 

increasing range approximately neutralizes 

that of decreasing fineness, and the inference 

is (6) that the rate of change in the index is 

greater for small range than for large.

Consider now the variations of the index in 

relation to width. In figure 32 (p. 106) the 

ordinates, as before, represent values of ^ and 

the abscissas represent width of channel. The 

points fixed by the data are shown by the dots.

(7) The upper group of curves all pertain to 

grade (C) and discharge 0.182 ft.3/sec., but represent different slopes. Then- common charA.
RELATION OF CAPACITY TO SLOPE. 105

acter is .a distinct .minimum. From the neighborhood of width 0.66 foot there is increase of 

\ in the direction, of greater width, and also in 

the direction.of less width.

(8) The position of the minimum is apparently the. same for low slopes as for high.

(.9) The minimum is most strongly marked 

in case of the gentler slopes.

The second group of curves all pertain to 

grade (C) and a slope of .1.0 per cent but differ 

in respect to discharge. Each of them shows 

a minimum, except the curve for discharge

 f

4-

(F)

Grade (C)

(G)

W-lft 

O =.363 ftMec.

Slope 

FIGURE 31. Variations of ii in relation to slope.

0.093 ft.3/sec., which has but two fixed points. 

They show also that 

(10) The position of the minimum is related 

to discharge. For large discharges it is associated with relatively large widths, for smaller 

discharges with smaller widths.

(11) The minimum is more pronounced, or 

the associated rates of change in the index .are 

higher in case of small discharge than of largeVarious analogies, .which appear in another 

part of this paper, render it probable that all 

the preceding inferences .are of a general character; but those in regard to width are not sustained by all the data.

The curves of the third group are based on 

observations with grade (B) and are drawn, 

like those of the first group, to contrast the 

relations of the index to width for different

->- 

>
106 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

slopes. With the gentler slopes they give 

indications of a minimum, but not with the 

steeper. The character of the discrepancy is 

such as to suggest that the values of the index 

computed for width 0.23 foot vary too rapidly 

with slope; and this result might be brought 

about by assigning too large a value to a. A 

critical review of the data, however, failed to 

find warrant for any material change in that 

constant.

It is believed that a group of discrepancies 

which this instance illustrates are connected

with the history of the experimental work. 

The first grade to be investigated was (B), and 

the methods of manipulation were subject to 

various minor changes, which were not always 

recorded; but the range of conditions was large. 

Grade (C) was next taken up, and again the 

range of conditions was large. Other grades 

followed, with less elaborate range of conditions; but the work on grade (G) was somewhat 

expanded, in order to learn the influence of 

coarser debris on various factors. The work 

on (G) also had the advantage of the fullest

.£3 .44 .66 1.00 1.32 

Width

FIGURE 32. Variations of i\ in relation to width of ehannel.

1.96

development of experimental method as well as 

that of uniformity of method. Because of this 

history it is believed that the results for grades 

(C) and (G) are of higher authority than those 

of other grades; and the belief is strengthened 

by the general symmetry and internal consistency of the '(C) and (G) results. The inferences, given in preceding paragraphs, from 

data of grades (C) and (G) are therefore accepted, and the discordance of data for grade 

(B), while not specifically explained, is ascribed 

in a general way to unrecorded differences in 

laboratory methods.

The curves of the fourth group of figure 32 

all pertain to grade (G) and discharge 0.734 

ft.3/sec. but differ in respect to slope. Compared with the first and second groups they are 

seen to be consistent with the inference as to .a 

minimum value of \, but the minimum falls 

outside the range of widths for which data were 

obtained. With grade (C) and discharge 0.734 

ft.3/sec. the minimum falls between widths 1 

foot and 1.32 feet, but nearer to the former. 

With grade (G) and the same discharge it apparently falls with some width less than 0.66 

foot. This indicates that 
RELATION OF CAPACITY TO SLOPE. 107

(12) The position of the minimum is related 

to fineness. For the finer debris it is associated 

with relatively great width; for the coarser, 

with smaller width.

The curves of the fourth group support the 

ninth inference, that the minimum is most 

strongly marked for the gentlest slope.

In the study of the data many other comparisons of the influence of width were made, 

but they are not here illustrated. Their chief 

service was in indicating the comparative value 

of different divisions of the body of data. The 

general fact brought out and one emphasized 

in various other ways is that the measures of

.093 .182 .363 .545 .734

Discharge

FIGURE 33. Variations of f i in relation to discharge.

1.119

precision derived from discrepancies of observations within a single series by no means cover 

the whole field. The discrepancies discovered 

when properties of different series are compared 

are quite as important and must be given consideration in connection with the broader generalizations.

 Let us now consider the relations of the varying value of \ to discharge. These are illustrated by figure 33. The curves of the upper 

group all pertain to grade (C) and slope 1.0 

per cent but differ in respect to width of channel. Those of the second group pertain to 

grade (C) and slope 1.8 per cent; those of the 

third group to grade (G) and slope 1.8 per cent. 

The general fact is that 

(13) As discharge increases the value of it diminishes. There are three exceptions, of which
108 TRANSPORTATION OF DEBEIS BY RUNNING WATER.

two do not exceed the computed probable 

errors of the data, and the third is connected 

with a value of \ to which the lowest weight 

is ascribed.

(14) The rate of change in the index is 

greater for small discharges than for large.

(15) For the same discharges the rate of 

change in the index is greater for wide channels than for narrow, and is therefore greater 

for shallow streams than for deep.

. FIGURE 34. Variations of «i in relation to fineness of debris.

The curves of the lowest group all pertain to 

width 0.66 foot and slope 1.8 per cent, but 

differ as to grade of debris. They: indicate 

that 

(16) The rate of change in the index is 

greater for coarse debris than for fine. The 

peculiarities of spacing, as in a previous instance, may show the influence of the factor of 

range in fineness within the several grades.

To consider now the relations of the values 

of \ to the fineness of debris, the comparison is 

made with linear fineness instead of bulk fineness, as in discussing a, and it is found convenient to plot the logarithms of the quantities 

instead of the quantities themselves. In figure

34 the curves of the upper group are derived 

from experiments conducted with a trough 

width of 0.66 foot, and each one pertains to a 

particular combination of slope and discharge. 

Those of the second group are derived from experiments with a trough width of 1 foot.

(17) In the main they show decrease of \ 

with increase of fineness, but the finer grades 

give the opposite indication. The data are not 

sufficiently harmonious to determine whether 

the law of change is continuous or involves a 

reversal. If it is continuous, it is an inverse 

function of F.

In view of the fact that the double variation 

of \ in relation to width is a complicating factor 

and of the further fact that that variation is 

less pronounced with high slopes than with low, 

two curves (the lowest group of fig. 34) were 

constructed from data pertaining to the highest 

practicable slope, 2.4 per cent. Each curve 

belongs to a particular width of trough, and 

each is a composite with respect to discharge. 

Their indication is practically the same as that 

of the other groups.1

The character of the material has not seemed 

to warrant a quantitative discussion of the 

variations of the index of variation, and a summary of the qualitative discussion is necessarily limited to generalities. The index of 

relative variation or the sensitiveness of capacity for traction to change of slope is a decreasing function of the slope, the discharge, the 

fineness of debris, and the range of fineness 

and is a minimum function of width of channel.

In symbols,

If we assume tentatively that the function replacing i{ in the exponent is the product of 

functions of the individual conditions that is, 

if we write

then we must also recognize that in/iOS), /jis 

itself a function of Q, F, H, and w, that /a is a 

function of F &nd w, and that/v is a function of

1 In the data on flume traction the relation of capacity to fineness 

exhibits peculiarities quite analogous to those here found in the relation 

of ii to fineness. The capacity is larger for very fine and very coarse 

ddbris than for intermediate grades. A tentative explanation (see 

Chapter XII) connects the larger capacity for fine debris with a transition in process from traction to suspension.

4-
RELATION OF CAPACITY TO SLOPE. 109

S, Q, and F. Parallel complexities would also 

arise if attempt were made to formulate the 

relations by means of such an expression as

From equation (1.0) (p. 96),

(41)

The sensitiveness of capacity to slope appears 

to be a- function of the conditions jointly 

rather than severally.

The development of complexity within complexity suggests that the actual nature of the 

relation is too involved for disentanglement by 

empiric methods, but that conclusion does not 

necessarily follow. Just as a highly complex 

mathematical expression may be the exact 

equivalent of a fairly simple expression of a 

different type, so a physical law may defy formulation when approached in a certain way 

yet yield readily when the best method of 

approach has been discovered.

FORMULATION WITH CONSTANT COEFFICIENT.

For the relation of capacity to slope the 

formula equivalent to (34) is

(4.0)

Also as (10) and (40) give equivalent expressions for 0, We may equate their second members:

Substituting the value of ct from (41), 

whence

(42)

in which the constant coefficient cx is the value 

of capacity when S = 1.

By means of this formula the values of j\ in 

Table 16 were computed. These values have 

been subjected to systematic comparison with 

the associated conditions (S, Q, w, etc.) in the 

same manner as were the values of the exponent it } ' but the results of the comparison 

need not be given in detail because they are 

parallel to those for ir They are in fact identical so far as verbal statement is concerned 

and differ only in quantitative ways. It is 

true in the main, and almost without exception, that the variations of ^ are less rapid 

than those of ^; and their average range is 

about half as great.

TABLE 16. Values ofjl , in C c^Sh, the coefficient cl being a constant.

Grade. .

Q......

S

0.2........... ....................

.4................................

.6................................

.8................................

1.0................................

1.2................................

1.4................................

1.6................................

1.8................................

2.0................................

2.2...'.............................

(A)

0.66

0.093

15

0.182

 39.4

0.545

142

1.00

0.182

37.6

0.363

100

0.734

231

1.32

0. 182

36.5

0.363

104

0.734

252

1.96

0.'363

96.1

0.734

242

0. 119

359

Values ofji.

2.36 

2.27 

2.20 

2.16

9 19

2.09

2.06 

1.99

1.95 

1.93 

1.91 

1.88

1.50 

1.49 

1.48

1.47

2.15 

2.10 

2.06

2.04 

2.02 

2.01 

2.00

1 QO

1.99

1.93 

1.88 

1.85 

1.83

1.82 

1.81 

1.80 

1.80

1.85 

1.82 

1.80 

1.79

1.78

2.32 

2.23 

2.17

2.13 

2.10 

2.08 

2.06 

2.05

2.04

2.09

1.86 

1.76 

1.72 

1.70

1.68 

1.67 

1.66 

1.65 

1.65

1.80 

1.75 

1.73 

1.71

1.70 

1.68

2.2i 

2.03 

1.96 

1.90

1.87 

1.84 

1.82

1.81

1.80

i

1.91

1.68 

1.61 

1.57 

1.55

1.54

i.'si)

1.35 

1.33 

1.31
110 TRANSPORTATION OF DEBRIS BY RUNNING WATER. 

TABLE 16. Values of jlt in C=clSii, the coefficient cl being a constant Continued.

Cl

Grade. .

w......

Q......

s

0.2................................

.6................................

.8................................ i.o................................

1.2................................

1.4................................

1.6................................

1.8................................

2.0................................

2.4................................

2.6..... ........................

2 ft *

 ? n

3.2................................

3.4................................

3.6................................

3.8................................

4.0................................

4.2............................:...

Grade. .

Q......

S

0.2................................

.4................................

.6................................

.8................................

1.2..............................'..

1.4..;...............,..............

1.6................................

1.8................................

2.0................................

2.2.................................

2.4.................................

2.6.................................

2.8.................................

3.0.................................

3.2................................

3.4................................

3.6................................

3.8................................

4.0................................

4.2.....:..........................

(B) .

0.23

0.093

7.8

0.182

13.4

0.44

0.093

13.3

0.182

28.9

0.66

0.093 0.182 0.363 0.545

1.00

0.182 0.363 0.645 0.734

Values of j\.

4.57 

3.10

2.60 

2.34 

2.18 

2.05 

1.96

1.89

3.09 

2.49

2.21 

2.05 

1.94 

1.86

2.63 

2.31

2.16 

2.02 

1.94 

1.88 

1.83

1 79

1.76

1.73

1.88

1.84 

1.81

1.79 

1.78 

1.77 

1.77

2.47 

2.34

2.26 

2.20 

2.16 

2.13 

2.10

2.08 

2.06 

2.04 

2.03 

2.02

1.96

1.89 

1.85 

1.82

1.81 

1.80 

1.79 

1.78 

 1.77

1.77 

1.76 

1.76 

1.75 

1.75

1.75

1.75

1.74

1.74

1.74

1.74

1.63 

1.60 

1.58

1.56 

. 1.55 

1.54 

1.54 

1.53

1.69

1.57

1.53 

1.50 

1.49

1.48 

1.47 

1.46 

1.46 

1.45

(B)

1.32

0. 182

29.2

0.363

79.3

0.545

140

0.734

205

1.96

0.363

67.8

0.545

120

0.734

189

1.119

312

2.00 

1.95 

. 1.92

1.90 

1.88 

1.87 

1.86 

1.85

1.84 

1.84

1 SQ

1 QQ

1 fit

1.95

1.78

1.73 

1.70 

1.68

1.66 

1.65 

1.64 

1.64 

1.64

1.63 

1.63

1.64 

1.62

1.61 

1.61 

1.60 

1.60 

1.59

 1.59

1.79 

1.68 

1.64 

1.62 

1.61

1.61 

1.60 

1.59 

1.59

(0)

0.44

0.093

10.9

0.182

24.7

0.66

0.093

13.8

0.182

32.3

Values of Ji.

2.26 

2.08 

2.00 

1.94

1.92 

1.89 

1.87 

1.85 

1.84

1.83

1.82

1.82

1.92 

1.82 

1.78 

1.75

1.72 

1.71 

1.70 

1.69 

1.68

1.68

1.79 

1.75 

1.73

1.71 

1.70 

1.69 

1.68

1.81 

1.76 

1.73 

1.71

1.69 

1.68 

1.67 

1.67

2.36 

2.16 

2.06 

2.01

1.97 

1.94 

1.92 

1.90 

1.89

1.88

""i.'gs"

1.91 

1.87

1.84 

1.82 

1.81 

1.80

i

1

1.82 

1.74 

1.70 

1.66

1.64 

1.63 

1.62 

1.61 

1.61

1.96

1.69 

1.65 

1.62 

1.60

 1.58

3.08 

2.60 

2.39

2.26 

2.17 

2.11 

2.06 

2.03

1.99

2.05 

1.95 

1.88

1.83 

1.80 

1.77 

1.76 

1.75

1.74

2.00 

1.93 

1.88

1.85 

1.83 

1.81 

1.80 

1.79

1.78 

1.77 

1.76 

1.76 

1.75

1.75 

1.75 

1.74 

1.74 

1.74

1.73

1.88 

1.80 

1.76 

1.73

1.71 

1.70 1.69- 

1.68 

1.67

1.67 

1.66 

1.66 

1.66 

1.65

1.65 

1.65 

1.64 

1.64
BELATION OF CAPACITY TO SLOPE. 

TABLE 16. Values of ji, in 0=0^8^, the coefficient c, being a constant Continued.

Ill

Grade. .

Conditions of experimentation ..... w ......

Q......

5

0.2.........................................

.4.........................................

.6.........................................

.8.........................................

1.0.........................................

1.2.........................................

1.4.........................................

1.6......................:..................

1.8.........................................

2.0.........................................

2.2.........................................

2.4.........................................

2.6.........................................

2.8.........................................

Grade..

Q......

S

0.2..........................................

.6..........................................

.8..........................................

1.0..........................................

1.2..........................................

1.4..................;.......................

1.6..........................................

1.8..........................................

2.0..........................................

2.2..........................................

2.4..........................................

2.6..........................................

2.8..........................................

Grade..

Q......

S

0.2... .......................................

.4..........................................

.6..........................................

.8.......................................... i.o..........................................

1.2..........................................

1.6..........................................

1.8..........................................

2.0..........................................

2.2..........................................

2.4..........................................

2.6..........................................

2.8..........................................

(C)

0.66

0.363

73.4

0.545

112

0.734 0.182 0.363

153 31.2 84.9

1.00

0.545

140

0.734 1.119

181 277

1.32

0.182

21.3

0.363

78.4

0.545

130

0. 734

184

Values of ./i.

1.68 

1.63 

1.61 

1.59

1.58

1.57 

1.56 

1.55

1.55

1.54 

1.54

1.54

1.76 .

1.65 .

1.61 

1.59 

1.58

1.57 .

1.56 .

1.56 .

1.55 .

1.55 .

1.55 .

............... 2.03

2. 12 1. 73

1.58 

1.58 

1.57

1.98 1.66 

1.91 1.62 

1.87 1.59

1. 84 1. 57 

1. 82 1. 56 

1. 80 1. 55 

1.79 1.54 

1.78 1.53

1. 77 1. 53 

1. 76 1. 52

1. 76 1. 52 

1. 75 1. 51

(C)

1.96

0.363

59.8

0.545

111

0.734

191

1.119

345

1.51 

1.49 

1.47

1.46 

1.45 

1.44 

1.43 

1.42

1.42

1.65 .......

1.5 

1.4 

1.4 

1.4

1.4 

1.4 

1.4 

1.4 

1.4

1.4

3 1.55 

9 1.51 

7 1.49 

5 1.48

4 1.47 

3 1.47 

3 .......

2 .......

2 .......

2

2.61 

2.46 

2.37

2.32 

2.28 

2.25 

2.23 

2.21

2.19 

2.17

1.92

1.78 

1.71 

1.67

1.64 

1.62 

1.60 

1.59 

1.58

1.57 

1.56

1.62 

1.58 

1.55

1.53 

1.51 

1.50 

1.49 

1.49

1.48 

1.47

1.86 

1.60 

1.54 

1.49 

1.47

1.45 

1.44 

1.43 

1.42 

1.42

1.41

(D)

0.66

0.093

9.1

0.182

29.5

0.545

102

1.00

0.182

24.0

0. 363

72.5

0.545

108

0.734

154

Values of ji.

2.38 

2.22 

2.14

2.08 

2.04 

2.01 

1.99 

1.97

1.95

2.03 

1.94 

1.87

1.83 

1.80 

1.78 

1.76 

1.75

1.74

. (D).

1.32

0.373

58.9

0.734

132

1.88

1.73 

1.66 

1.62

1.60 

1.58 

1.56

1.95 

1.89 

1.85

1.82

2.41 

2.29 

2.22

2.18 

2.15 

2.12 

2.10 

2.09

2.08 

2.07 

? 03

2.02 

1.90 

1.84 

1.80

1.78 

1.76 

1.74 

1.73 

1.72

1.71 

1.71 

1.70

1.91

1.74 

1.69 

1.66 

1.64

1.63 

1.62 

1.61 

1.61 

1.60

1.60

2.06 

2.01

1.98 

1.95 

1.93 

1.91 

1.90

1.89 

1.88 

1.87

1.87

1.84

1.77 

1.72 

1.69

1.67 

1.65 

1.64 

1.63 

1.63

1.62 

1.61

1.94 

1.90 

1.87

1.85 

1.84 

1.84 

- 1.83 

1.82

1.82

1.92 

1.75 

1.70 

1.67 

1.65

1.64 

1.63 

1.63 

1.62

(E)

0.66

0.363

24.8

0.734

40.1

1.00

0.182

14 7

0.363

33.6

0.734

71.5

1.119

139

1.32

0.363

36.4

0.734

72.6

1.119

122

Values of ji.

2.25 

2.16 

2.11

2.07 

2.05

2.03

2.02

2.01

1.93 

1.88 

1.85

1.83

2.09

2.06 

2.03

2.17

2.15 

2.13 

2.12

1.99 

1.94 

1.98 

1.88

1.86 

1.85 

1.85 

1.84 

1.84

1.83 

1.83 

1.82

1.82

2.03

1.89 

1.85 

1.82 

1.81

1.80 

1.79 

1.79

1.78 

1.78

1.77 

1.77

1.76 

1.70 

1.68 

1.66 

1.65

1.64 

1.64 

1.63 

1.63 

1.63

1.65 

1.64 

1.63

1.63 

1.62 

1.62 

1.62 

1.62

1.62

1.62

1.66 

1.65 

1.64

1.63 

1.62 

1.62 

1.62 

1.62

1.56 

1.55 

1.54

1.54

1.82 

1.81 

1.81

1.80
112 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 16. Values of j1; in-0=0$ h, the coefficient Cj, being a constant Continued.

Grade. .

v> ......

Q......

s

0.2.................................

.4.................................

.6..........................................

.8

1.0.................................

1.2.................................

1.4.................................

1.6.................................

1.8.................................

2.0.................................

2.2.................................

2.4.................................

2.6.................................

2.8.................................

3.0.................................

Ci ..................................

Grade..

w . . . . .

Q......

S

0.2............................. ..

.4........... ......................

.6.................................

.8. .................................

1.0.................................

1.4.................................

1.6.............'....................

1.8.................................

2.0.................................

2.2.................................

.2.4.................................

2.6.................................

2.8.................................

3.0.................................

3.2.................................

3.4.................................

(F)

0.66

0.182

' 5.7

0.363

20.4

0.734

38.7

1.00

0.182

2.1

0.363

13.9

0.734

43.6

1.119

79.6

 1.32

0.363

13.5

0.734

40.3

1.119

86.5

Values of j\.

2.67 

2.60 

1.53 

2.47 

2.43

9 41

2.39

2.38

2.14

2.08 

2.05 

2.03 

2.01 

1.99

1.94

1.91 

1.88 

' 1.86 

1.85

3.87 

3.71 

3.59 

3.50 

3.43

3.36 

3.31 

3.27

2.85 

2.73

2.65 

2.58 

2.52 

2.48 

2.45

2.42 

2.39 

2.37

2.14 

2.06

2.02 

1.98 

1.95 

1.94

1.95 

1.90

1.87 

1.84 

1.82 

1.81

.........

2.54 

2.45 

2.38 

2.33 

2.29

2^26

(G)

0.66

0.734

30.3

1.119

48.6

1.00

0.363

4.8

0. 734

26.8

1.119

60.1

1.32

0.363

0.82

0.734

19.0

1.119

52.0

2.52 

2.40

2.32 

2.27 

2.22

2.01 

1.94

1.88 

1.85 

1.83 

1.80

(G)

0.66

0.363

8.3

""S." 48

3.21 

3.04 

2.91 

2.82 

2.76

2.70 

2.65 

2.61 

2.58 

2.55

(H)

0.66

0.363

0.76

0.734

12.8

1.119

25.2

Values of ji.

2.85 

2.65

2.53 

2.45 

2.38 

2.34 

2.30

2.27 

2.24 

2.22

2.77

2.55 

2.43

2.36 

2.31 

2.27 

2.24 

2.21

2.19 

2.17 

2.16

3.94 

3.67 

3.48 

3.35 

3.25

3.16 

3.10 

3.04 

2.99

2.94

3.15

2.88

2.72 

2.61 

2.63 

2.48 

2.43

2.39 

2.36 

2.34

2.91

2.60 

2.44

2.34 

2.27 

2.22 

2.18 

2.15

2.12 

2.10

5.34 

5.10

4.92 

4.77 

4.65 

4.55 

4.45

4.38

4.32

3.93 

3.43

3.22 

3.00 

2.88 

2.79 

2.72

2.66 

2.61 

2.57 

2.54

2.81 

2.57

2.44 

2.35 

2.27 

2.23 

2.19

2.15 

2.12 

2.10

7.65 

6.57 

5.93 

5.50 

5.20

4.96 

4.77 

4.63 

4.49

4.68 

3.92

3.54 

3.31 

3.15 

3.04 

2.95

2.88 

2.82 

2.76 

2.72 

2.69

2.66

3.63 

3.21

2.98 

2.83 

2.75 

2.64 

2.58

2.53 

2.49 

2.45

Y

The greatest contrast between the rates of 

variation of the two exponents is found when 

relations to slope are considered, the least 

when relations to fineness are considered.

The variations of the two exponents with 

respect to slope are illustrated by figure 35. In 

the example from which the curves in this 

figure were computed the range of variation, 

within the experimental limits, is considerably 

above the average, but the example is otherwise 

typical. The rate of change is everywhere 

smaller for j1 than for ilt and the total change, 

or range of variation, is therefore smaller. This 

relation prevails throughout the range of slopes 

covered by the experiments but would not be

found to hold for slopes far outside of that 

range. The average range of jlt for the investigated cases, falls between one-half and onethird of the average range of ir

It thus appears that a modification of the 

plan of formulation which dispenses with variation in the coefficient and thereby concentrates 

all expression of variation in the other parameter, the exponent, tends also to diminish diversity in that parameter.

EFFECT OF CHANGING THE UNIT OF SLOPE.

The values of ^ are functions not only of the 

conditions of experimentation and of the constant slope a but also of the unit used for the

 -v
RELATION OF CAPACITY TO SLOPE. 113

measurement of slope. (See p. 99.) In the 

term log (I a), which enters into the value of 

7\ (equation 42), 1 is the unit of slope; and the 

relative magnitude of 1 and a changes as the 

unit changes. Other terms of the formula are 

also (implicitly) functions of the unit, bub the 

various influences are not compensatory, and 

the resultant is of such nature that the values 

of jl vary inversely with the magnitude of the 

unit. It will be recalled that in the notation of 

this paper the symbol S pertains to the unit 

height ...-. . , . , , 

distance ' symbol s to the unit

height IT../. distance' curve marked ^ in figure 35 represents values of the exponent computed with 

use of the smaller unit, and the curve marked 

Tc represents a coordinate system of values cornSlope

FIGURE 35. Variations of exponents t'i, ;'i, and It, in relation to slope. 

The scale of slope is in per cent.

puted with use of the larger unit. The latter 

curve, produced, would intersect the curve of \ 

at a point corresponding to s = 1 or 8 = 100.

It would be possible, by employing the larger 

unit in the notation for slope, to construct a 

table equivalent to Table 16 in which the values 

of 7\ would all be smaller and would have in each 

series less range. For many of the series they 

would approach closely the associated values of 

n. This reduction of exponents would be accompanied by an enormous increase in the 

values of coefficients, each value of ct in the 

table being magnified by the" factor 100n. The 

increase would result from the fact that ct is the 

capacity corresponding to unit slope, while the 

unit slope in that case would be 45°. Considered as capacities, the values of c^ would be 

in a sense fictitious, because the laws of traction wifh which we are dealing do not apply 

(see p. 63) to so high a slope as 45°.

20921° No. 86 14 8

To recur for a moment to the general account 

of the index of relative variation, it will be 

recalled that the index was shown to be independent of the units of the observational quantities. In this particular instance the value of

cr

the index, nw^~, has two factors, of which the

first is an abstract number and the second is a 

ratio between slopes and is independent of the 

unit of slope. The ordinates of the curve 

marked \ in figure 35 are therefore independent 

of the slope unit, and the curve is a fact of observation, plus the assumptions of the formula 

of adjustment.

PRECISION.

Because the values of \ were computed from 

values of n and o, their precisions are involved 

with those of n and a, but the relation is not 

simple. The precision of n depends partly on 

the harmony of the observations of load and 

slope and partly on the precision of o. It is 

not feasible to measure the precision of a for 

individual series of observations, and the precisions of individual values of n and ^ are 

therefore indeterminate. All that has been attempted is to derive a rough estimate of average 

precision from average values of the quantities 

involved. The estimated average probable 

error of values of n is 3.9 per cent, and the corresponding estimate for values of \ is 4.6 per 

cent.

The precision of it varies with slope within 

each series represented by a column in Table 

15, being relatively high for the steeper slopes. 

The probable error, if measured in the same 

unit with il} is much larger for gentle slopes 

than for steep; if measured in percentage, it is 

somewhat larger for the gentler slopes. Measured in percentage, its value for the steeper 

slopes approximates the corresponding probable error of n.

EVIDENCE FROM EXPERIMENTS WITH MIXED 

DEBRIS.

The observations on capacity and slope when 

the debris transported consisted of a mixture 

of two or'more grades were reduced in the same 

general manner as those for single grades. It 

was not thought advisable to make any adjustment of the values of a, but each logarithmic 

plot was treated independently. For about 

one-third of the mixtures the best value
114 TRANSPORTATION OF DEBBIS BY RUNNING WATER.

appeared to be zero. For another it fell between 

the values which had been assigned to the finer 

and coarser components of the mixture. For 

the remaining third it exceeded the values associated with the components. When the load 

was a mixture of many grades combined in approximately natural proportions, and also when 

it consisted of unsorted debris from a river bed,

the values of a were small and for the most part 

zero.

These features were scrutinized with special 

interest because the properties of grade (E) 

had suggested that great range of fineness 

might determine very low values of a. On the 

whole, the data from mixtures favor that view, 

but their support is by no means unanimous.

TABLE 17. Values ofi^for mixtures of two or more grades of debris.

(Grade.. 

Conditions of experimentation .... .<w ...... IQ......

Parameters of interpolation equation. . .{*' '

S

0.4.........................................

.6.........................................

.8.........................................

1.0.........................................

1.2.........................................

1.4.........................................

1.6.........................................

1.8.........................................

9 n

(Grade.. 

Conditions of experimentation . .... <w ...... 

IQ......

Parameters of interpolation equation -{^"

5

0.4..........................................

.6..........................................

i.o..........................................

1.2..........................................

1.-6............................ -.........- ...

1.8.................'.-.......................

2 9

1 Grade.. 

w Q::::::

Parameters of interpolati'jn equation ---j^"

*

0.4..........................................

.6..........................................

.8..........................................

1.0..........................................

1.2..........................................

1.4..........................................

1.6..........................................

1 fl

2.0..........................................

2.4..........................................

2.6..........................................

2

O

(AiCi) 

1.00 

0.363

0.25 

1.06

(A3Gi) (A2Gi) 

1.00 1.00 

0.363 0.363

0.30 0.20 

1.38 1.53

l.W LOO l.W l.W 1*00 

0.363 0.363 0.363 0.363 0.363

000 0.25 0.25 

1.98 2.09 2.21 1.08 0.96

(BiF,) 

1.00 

0.363

0.50 

1.51

(BiFj) 

1.00 

0.363

0.50 

1.59

(B,F4 ) 

1.00 

0.363

0.50 

1.60

(C4Ei) 

1.00 

0.363

0.30 

1.10

Values of ii.

1.82 

1.55 

1.42

1.34 

1.30 

1.26

(CiEi) 

1.00 

0.182

0

1.78

2.76 2.30 

2.21 2.05 

1.97 1.92

1.84 1.84 

1.76 1.79 

1.70 ........

(C2E,) 

1.00 

0.363

0.30 

1.13

1.98 ........ ........ 1.8

1.98 . ......... ... 1-5

1.98 2.09 2.

1.98 2.09 2. 

1.98 2.09 2. 

1.98 2.09 2. 

............ 2.

21 1.4

21 1.3 

21 1.3 

21

5 1.64 

7 1.39 

4 1.28

6 1.21 

1 1.17 

1.14

21 ...............

........ ........ 2.21 ........ .......

(CiE,) 

1.00 

0.182

0 

1.60

L 00 

0.363

0.40 

1.12

(CiE2 ) 

1.00 

0.182

0.20 

1.40

(CiE,) 

1.00 

0.363

0.30 

1.25

(C,E4 ) 

1.00 

0.182

0 

1.55

4.02 

3.02

2.59 

2.35 

2.19 

2.09

(CiE4 ) 

1.00 

0.363

0.30 

1.33

4.23 

3.17

2.72 

2.47 

2.31

(C4GO 

1.00 

0.363

0 

1.97

4.27 

3.20

2.75 

2.49 

2.33

(C*G,) 

1.00 

0.363

0.30 

1.24

2.20 

1.76 

1.57

1.47 

1.40 

1.36 

1.32

1.00 

0.363

0.50 

1.13

Values of »i.

1.78 

1.78 

1.78

1.78

(CiGf)

1.00 

0.363

0.60 

1.47

2.26

1.81

1.62

1.51 

1.44 

1.39 

1.36

(E4G,)

1.00 

0.363

0 

1.87

1.60

1.60 

1.60 

1.60 

1.60 

1.60

1.60

1.00 

0.363

0.30 

1.90

1.86

1.68 

1.56 

1.49 

1.44 

1.40

1.00 

0.363

0.40 

1.28

1.75

1.68 

1.63 

1.60 

1.57 

1.55

1.54

(EiGi)

1.00 

0.363

0 

2.03

2.02 

1.79

1.67 

1.60 

1.54 

1.51 

1.48

1.45

1.00 

0.363

0.48 

1.67

1.55 

1.55

1.55 

1.55 

1.55 

1.55 

1.55

1.55

1.90

1.77 

1.69 

1.63 

1.59 

1.56

1.97 

1.97 

1.97

1.97 

1.97 

1.97 

1.97

(CDEFG)

1.00 

1.82

0 

1.63

1.00 

0.363

0 

1.48

1.00 

0.545

0 

1.22

2.49 

1.99 

1.78

1.66 

1.58 

1.53 

1.49

3.01 

. 2.26

1.93 

1.76 

1.64 

1.56

Natural.

1.00 

0.182

0.30 

1.43

1.00 

0.363

0 

1.84

Values of ii.

3.68

2.94 

2.58 

2.36 

2.21

1.87

1.87 

1.87

1.87 

^ 1.87 

1.87 

1.87

3.04 

2.71

2.53 

2.42 

2.34

2.56 

2.13

1.92 

1.79 

1.70 

1.64

2.03

2.03 

2.03 

2.03 

2.03 

2.03

4.19 

3.22

2.79 

2.55 

2.39 

2.28

1.63 

1.63

1.63 

1.63 

1.63 

1.63

1.48 

1.48

1.48 

1.48 

1.48 

1.48

i.22

1.22 

1.22

1.22 

1.22 

1.22 

1.22

2.29 

2.04

1.90 1.82- 

1.76 

1.72 

1.68

1.65

1.63

1. 62

1.84 

1.84 

1.84 

1.84

1.84 

1.84 

1.84

J.
BELATION OF CAPACITY TO SLOPE. 115

The adjusted values of capacity may be 

found in Table 12, together with a variety of 

other data. The values of \ are contained in 

Table 17, having been computed for the range 

of slopes covered by the observations. In 

order to compare these with the indexes of 

relative variation for the single grades from 

which the mixtures were made, Table 18 has 

been prepared, containing indexes corresponding to the uniform slope of 1.2 per cent. An

analysis of this table shows that 3 mixtures 

are more sensitive to variation of slope than 

are components, 13 are less sensitive, and 18 

show sensitiveness of intermediate rank. A 

general average shows the mixtures 11 per cent 

less sensitive than the (means of) components, 

but the contrast is much more pronounced for 

natural river debris and for the most complex 

artificial combination than for the simpler 

mixtures.

TABLE 18. Comparison of values of \ for mixtures and their components.

Component grades.

Finer.

(A).. ........................

!

A S

TJ\

ir\

fC\

crA

/w\

(A),(C)......................

Mixture of (C), (D), (E), (F),s 

Natural combination ranging i

Coarser.

r\\ Sv::::::::::::::::::::::::::

F). ..........................

E). ..........................

E). ..........................

G). ..........................

G). ..........................

(G). ..........................

md (G)........................

Q

0.363 

.363 

.363 

.182 

.363 

.363 

.363 

.363 

[ .182 

.363 

I .545 

/ .182 

\ .363

Value of ii when ratio of finer to coarser is 

1:0

1.81 

1.81 

1.65 

1.81 

1.55 

1.55 

1.79 

1.34 

1.81 

1.55 

1.44 

2.02 

1.81

4:1

1.84 

1.36

1.47 

ol.97 

ol.87

2: 1

1.84 

1.21 

ol.78 

1.51 

1.66 

2.53

1:1

1.34

ol.98 

2.59 

01.60 

1.68 

1.93 

1.92

9 7Q

ol.63

ol.22

1.90

ol.84

1:2

o2.09 

2.72 

1.68 

1.67 

2.94 

"2.03

i : 4

o2.21 

2.75 

ol.55 

1.77

0: 1

1.55 

3.57 

2.54 

1.85 

1.79 

3.57 

3.57 

3.57

3.57

3.65 

2.54

o For these, <r=0 and »i=n.

The general tenor of the evidence from mixtures is to show that in passing from the laboratory conditions (with graded debris) to natural 

conditions it will be proper to reduce the estimates of the sensitiveness of capacity to changes 

of slope.

RELATION OF INDEX TO MODE OF TRACTION.

One of the possible ways of bridging the 

chasm between laboratory conditions and river 

conditions is through the consideration of 

modes of traction. River discharges are enormously greater than the experimental, river 

slopes are relatively minute, and river channel 

sections are very dissimilar in form and proportions, but the three modes of traction have 

the same sequence in rivers as in the laboratory troughs. This consideration has led to the 

separation of certain data connected with a single mode of traction the one characterized by 

smooth surfaces of water and debris. Table 19 

contains values of the index of relative variation, ilt associated with smooth traction and 

grouped with reference to various conditions. 

In the upper division of the table all the values 

pertain to the same grade of d6bris (C) and

are arranged according to discharges and channel widths; in the lower division all pertain to 

the same width of channel and are arranged according to discharges and grades of debris.

TABLE 19. Values of\ associated ivith the smooth mode of 

traction.

I. Values pertaining to the same grade of dfibrls, (C).

Width 

(feet).

0.44 

.66 

1.00 

1.32 

1.96

Value of »i for discharge (ft. '/sec. ) of 

0.093

2.15 

1.79

0.182

1.88 

1.70 

1.81 

2.20

0.363

1.59 

1.59 

1.61 

1.96

0.545

1.59 

1.48 

1.55 

1.79

0.734

1.58 

1.48 

1.51 

1.62

1.119

1.93

II. Values pertaining to the same width of channel, 0.66 feet.

Grade.

(B, <;§ 8

(H

Value of »'i for discharge (ft. 8/sec.) of 

0.093

2.09 

1.79 

2.01

0.182

1.82 

1.70 

1.72 

1.99

0.363

1.61 

1.59

2.13 

2.41

3.35

0.545

1.54 

1.59

1 fi4

0.734

1.58

2.19 

2.65

1. 119

2.20 

2.54
116 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

Inspection of the table shows that under 

this special condition the index (1) varies inversely with discharge, (2) varies inversely 

with fineness, and (3) varies both inversely 

and directly with width. In all these respects 

the variations are similar to those noted when 

the constant condition is slope, but the rates of 

variation are not the same. To illustrate the 

differences in rate I introduce Table 20, in 

which data of Table 19 are compared, in parallel columns, with similar data (from Table 

15) conditioned by constant slope. The same 

data are also presented graphically and with 

some generalization in figure 36. On giving 

attention to the first division of the table and 

to the upper diagram, it will be seen that the 

variation of the index with discharge is somewhat less when the constant factor is mode of 

traction than when it is slope. The second 

division and second diagram show that the 

variation with fineness is also less for constant 

mode of traction. The third division and 

diagram indicate that the variation with width 

of channel is of the same order of magnitude in 

the two cases, but that for constant mode of 

traction the minimum value of the index is 

associated with greater width.

The comparison does not indicate that the 

condition of uniform mode of traction is

FIGURE 36. Variations of ii under condition of uniform mode of traction ( T, T, T), and under condition of uniform slope (S, S, S). Vertical distances represent ii, with a common scale but different zeros. 

Horizontabdistances represent, for the upper pair of curves, discharge; 

for the second pair, logarithms of fineness; for the lower pair, width of 

channel; the values increasing from left to right.

greatly to be preferred to that of uniform slope 

as a basis for the extension of laboratory generalizations to large natural streams.

TABLE 20. Relations q/\ to discharge, debris grade, and channel width, when conditioned (1) by a constant mode of traction

and (I?) by a constant slope.

Grade, (C) 

w, 0.66 foot.

Q

0.093 

.182 

.363 

.545 

.734

Values of ii with 

Smooth 

traction.

1.79 

1.70 

1.59 

1.59 

1.58

Constant 

slope 1.0 

per cent.

1.87 

1.73 

1.59 

1.58 

1.57

w, 0.66 foot. 

Q, 0.303 ft.s/sec.

Grade.

(B) 

(C) 

(E) 

(G) 

(H)

Values of ii with 

Smooth 

traction.

1.61 

1.59 

2.13 

2.41 

3.35

Constant 

slope 1.4 

per cent.

1.54 

1.55 

1.99 

2.71 

4.69

Grade, (C) 

Q, 0.182 ft.s/sec.

w

0.44 

.66 

1.00 

1.32

Values of »i with 

Smooth 

traction.

1.88 

1.70 

1.81 

2.20

Constant 

slope 1.2 

per cent.

1.81 

1.70 

1.81 

2.27

IN CHANNELS OF SIMILAR SECTION.

THE CONDITIONS.

It will be shown in the following chapter that 

one of the important conditions affecting capacity is the relation of stream depth to stream 

width, or the form ratio R. The matter has, in 

fact, already received some attention in connection with the variation of o. Now, in each 

observational series the width is constant while 

the depth varies, so that the form ratio is a 

variable. Its variations accompany and are

inseparable from those of slope; and the variation of capacity (within an observational 

series), which up to this point has been treated 

as if it were purely a function of slope, is in 

reality a function of slope and form ratio 

jointly. To separate the two factors and thereby discover the relation of capacity to slope for 

streams of similar section, it is necessary to 

bring together data obtained by use of troughs 

with different widths, selecting points of two 

or more adjusted series which are characterized 

by the same ratio of depth to width. In every
RELATION OF CAPACITY TO SLOPE. 117

such comparison the capacity and slope associated with the narrower trough are relatively 

large, while for the wider trough they are relatively small. The opportunities for comparison 

are not abundant, because in the main the 

observational series which are of like conditions 

except as to width do not overlap in respect 

to form ratio.

SIGMA AND THE INDEX.

In the records of the main body of experiments 24 cases of overlap are found, all associated with the finer grades of de"bris, from (A) 

to (D). In each of these the comparison includes two widths only, no instance occurring 

in which it can be extended to three. There 

is however, a special group of experimental

series, planned in part for this particular purpose, in which the trough-width interval is so 

small that triple overlaps occur. The special 

experiments were made with de'bris of grade 

(C); the trough widths were 1.0, 1.2, 1.4, 1.6, 

1.8, and 1.96 feet; and the experiments yield 

nine triple overlaps. In six of these the extent 

of overlap is such that numerical comparisons 

have been made for more than one value of E. 

With the aid of the computation sheets 

described on page 95, a table was compiled in 

which adjusted capacities and slopes were 

arranged with respect to form ratio, and in 

this table, which has not .been printed, , the 

matter of overlaps was canvassed. Table 21 

contains the data involved in the triple 

overlaps.

TABLE 21. Selected data, for grade (C), showing the relation of capacity to slope when the form ratio is constant.

R

0.07.................

.08................

09

.10.................

.11................

.12................

.13................

.14................

.IS................

.10................

w

1.60

l.SO

1 Qfi

1 GO

1 SO

1.96

1.40

1.00

1.80

1.40

1.60

1.80

1.20

.1.40

1.60

Q= 0.734

5

1.38

1.17

7^

1 f&

on

.49

1.33

.72

.55

1.06

.60

.45

1.43

.72

.44

C

354

262

118

ooc

151

48

346

102

57

224

71

35

382

103

37

w

1.60

1.80

1.96

1.40

1.60

1.80

1.40

1.60

1.80

1.40

1.60

1.80

Q= 0.923

S

1.29

.94

.58

1.44

.83

.60

1.17

.69

.50

.96

.58

.42

C

413

210

88

522

185

83

357

130

56

243

94

36

w

1.60

1.80

1.96

1.40

1.60

1.80

Q=1.021

S

1.19

.82

.50

1.16

.65

.47

C ' 

405

195

75

380

135

58

'IV

1.60

1.80

1.96

1.60

1.80

1.96

1.40

1.60

1.80

1.40

1.60

1.80

1.40

1.80

Q=l. 119

S

1.07

.74

.50

.87

.62

.41

1.20

.60

.45

99

.51

.39

.83

.44

.34

C

380

1<W

88-

261

134

470

60

331

QC

47

71

OQ

To illustrate the use made of such data, the 

case of # = 1.119 ft. 3/sec. and # = 0.15 is 

selected. In that example the values of 

slope and capacity are given for the trough 

widths 1.4, 1.6, and 1.8 feet. These values 

come from three adjusting equations, which are, 

for the widths severaUy, 0=386 (tf-0.08) 1 -72, 

(7=400 (tf-0.10) 1 -60, and (7=430 (5-0.12) 1 -67 . 

The graphs of the equations are shown in

figure 37. On each graph is a dot indicating 

the point which corresponds to the tabulated 

values of C and 8, and for each of these points 

the ratio R is 0.15. The curves represent the 

relations of 0 to S under the condition of 

uniform width. The three dots are points on 

an undrawn curve to express the relation of 

0 to S under the condition of constant form 

ratio.
118 TRANSPORTATION OF DEBRIS BY RUNNING WATER,

It is convenient, in discussing this undrawn 

curve, to assume that its equation involves a 

and is otherwise of the same type as the equations used in discussing the data for constant 

width. In this case, moreover, the assumption is countenanced by the fact that traction 

is limited by the competent slope. By making 

the assumption, it is possible to compute all 

the parameters of the curve from the coordinates of the three known points and write its 

equation :

have been computed for each of the cases of 

Table 21. They are assembled in Table 22.

TABLE 22. Valttes of a corresponding to data in Table 21.

In the present connection the most significant 

of the parameters is a; and the values of a

R

0.07... 

.08... 

.09...

.10... 

.11... 

. 12. . . 

. 13. . . 

. 14. . .

.15...

.16...

Value of a when Q is 

0.734

/ °

i o

/ +.20 

\ +.20 

.... ..

0.923

-.44

'[""+."35" 

{ +.33 

I +.31

1.021

+.04

.... .. 

1.119

r+.'os"

\ +.11

| +.21 

\ +.24 

I +.24

400

200

Slope

FIGURE 37. Curves of c=/(S) for trough widths of 1.4,1.6, and 1.8 feet, showing points which agree as to form ratio.

Grade (C). Q=1.119 fU/sec. E=0.15.

One value of o has a negative sign. As this 

yields positive capacity for negative slope, 

thus transcending the physical conditions of 

traction, and as the value is numerically large, 

an error of importance is betrayed. The 

error appears not to be one of computation 

and must be ascribed either to the observational data or to assumptions made in the 

work of reduction and adjustment. In either 

case its existence serves to qualify the values 

of a as of low precision.

In view of that qualification, the study of the 

values of a in detail appears unprofitable and 

only their mean will be considered. In deriving the general mean a subsidiary mean was 

first found for each case of overlap and these 

were afterward combined, this course being 

followed because when several values of a are 

derived from the same overlap of series they 

are not independent. The general mean thus 

derived is +0.08, and this may be compared 

with a mean similarly obtained from the asso-
RELATION OF CAPACITY TO SLOPE. 119

elated equations under condition of constant 

width, namely, +0.12. So far as this moderate difference in magnitude has significance, it 

indicates that the index of relative variation, it , 

has the smaller range under the condition of 

constant form ratio; for the index's range in 

magnitude is a direct function of a.

THE SYNTHETIC INDEX.

Another mode of viewing the data of Table 

21 ignores the middle term of the triple overlap and considers only the first and last. From

the values of capacity and slope in these two 

terms a value of the synthetic index can be 

computed by equation (35). It is possible in 

this way to broaden the range of data by including the cases of simple overlap; and it is 

possible to make many pertinent comparisons 

by computing the corresponding values of the 

synthetic index for observational series in 

which the constant condition is width instead 

of form ratio. The indexes with form ratio 

constant and with width constant may be 

symbolized severally by Is and Iw.

TABLE 23. Values of synthetic index under condition that R is constant, (IB); with coordinate 'values, under condition that

W is constant, (Iw).

Grade.

(A) 

(B)

(C)

(":>)

Adj

Q.

0.363 

.734 

.182

.363 

.545

.734 

.734 

.182 

.182

.182 

.363

.363 

.545

.734 

.734

.734 

.734 

.734

.734 

.923

.923 

1.021 

1.021 

1.119

1.119

.363 

.734

ustcd mea

Width.

1.32 

1.32 

1.00

.66 

1.32

1.00 

1.32 

.44 

.66

1.00

.66 

1.00

1.32 

1.00

.66 

1.00

1.20 

1.32 

1.40

1.60 

1.40

1.60 

1.40 

1.60 

1.40

1.60

1.00 

1.00

US

1.%

1.96 

1.32

1.00 

1.96

1.32 

1.96 

.66 

1.00

1.32

1.00 

1.32

1.96 

1.32

1.00 

1.32

1.60 

1.96 

1.80

1.96 

1.80

1.96 

1.80 

1.96 

1.80

1.96

1.32 

1.32

R.

.06 

.07 

.10 

.12 

.06 

.07 

.08 

.09 

.20 

. .07 

.08 

.20 

.12 

.30 

.10 

.12 

.14 

.05 

.06 

.07 

.20 

.30 

.10 

.12 

.06 

.12 

.14 

.40 

.16 

.18 .20" 

.13 

.09 

.10 

.11 

.07 

.08 

.11 

.12 

.13 

.09 

.13 

.10 

.14 

.15 

.16 

.11 

.12 

.10 

.16 

.18 

.20

'* 

2.01 

2.23 

1.77 

1.90 

1.95 

2.09 

2.22 

2.49 

1.69 

2.00 

2.11 

1.80 

2.05 

1.71 

1.74 

1.88 

2.02 

2.11 

2.36 

2.67 

1.53 

1.96 

1.72 

1.87 

2.22 

1.56 

1.64 

1.46 

1.47 

1.54 

1.59 

2.01 

1.60 

2.06 

2.17 

1.81 

1.96 

2.10 

2.18 

2.30 

1.93 

2.08 

1.95 

2.00 

.2.10 

2.20 

1.93 

2.07 

2.02 

1.86 

1.94 . 

2.11

1.88

Iw.

Channel 

narrow.

1.74 

1.80 

1.82

1.87 

2.00 

2.06

1.75

1.69

1.69 

1.75 

1.82 

1.70 

1.77 

1.87 

1.62

1.58 

1.65 

1.72 

1.44 

1.48 

1.20 

1.48 

1.52 

1.73 

1.91

2.00

1.92 

1.98

1.91

1.86 

1.99 

1.86

1.49

1.87 

1.94 

1.67

1.66 

1.72

1.72

Channel 

wide.

1.99 

2.17 

1.75

1.92 

2.02 

2.16

1.75 

1.96

1.82

1.88 

1.79 

1.92 

2.04

2.26 

2.43 

1.64 

2.07 

1.66 

1.88 

2.24 

1.52

1.60

2.01

1.95 

2.07,

2.08

1.98

1.99

2.13

2.07 

2.13

1.97

1.89

Table 23 contains values of 1R computed for 

the 9 cases of overlap shown in Table 21 and 

for 24 cases of simple overlap previously mentioned. Where the overlap is extensive, values 

of IB were computed for two or more values

of R. Each of these values is based on data of 

two width-constant series, pertaining to different trough widths, and whenever the extent 

of such an associated width-constant series permitted, a value of Iw was computed from its
7 120 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

data for the same range of slope as that covered 

by the value of IR. The table thus comprises 

data for the coordinate values of the indexes 

under conditions of constant form ratio and 

constant width.

The most important generalization from the 

data in Table 23 is contained in the adjusted 

means which appear at the bottom. In the 

derivation of these means due account was 

taken of the fact that the several individual 

values from the same overlap are not independent arid also of the fact that the coordinate 

data for constant width are incomplete. The 

mean for constant form ratio, 1.88, is practically the same as that one for constant width 

which is associated with the wider channels, 

1.89, and is notably greater than the mean associated with the narrower channels, 1.72. On 

the whole it is indicated that the sensitiveness 

of capacity to slope is the greater for traction 

conditioned by constant form ratio, in the proportion of 1.88 to 1.805, or as 1.04 to 1.

When the values of IB and Iw are arranged 

according to the associated values of R, the following relations are brought out:

B

0.05 to 0.08. ........

.14 to .40.........

Mean 

IR2.14 

1.93 

1.80

Mean Ia.

Narrower 

trough.

1.86 

1.81 

1.63

Wider 

trough.

2.10 

1.94 

1.86

/*

Iw

1.08 

1.03 

1.03

/ SUMMARY.

Only a small fraction of the observational 

data are available for the discussion of the 

capacity-slope relation under the condition of 

constant form ratio, and the discussion is therefore limited to a comparison of its features under that condition with corresponding features 

under the condition of constant channel width. 

The results of such comparison may be summarized as follows: The sensitiveness of tractional capacity to variation of slope is in general greater under the condition of constant 

form ratio, but the difference is of moderate 

amount. The difference is somewhat less (at 

least within the limits of available data) for 

broad and shallow streams than for streams 

that are narrow and deep. The range of sensitiveness, or its variation with variation of slope, 

appears to be somewhat less under the condition of constant form ratio. The generalizations in regard to traction by currents of variable depth but invariable width may be extended, with only moderate qualification, to the 

case of currents which retain geometric similarity of section while slope is varied.

/ ' REVIEW.

With increase of the slope of descent goes increase of a stream's energy (per unit time, per 

unit distance). With the increase of energy 

goes increase of capacity for the transportation 

of debris along the channel bed. The increase 

of energy is strictly proportional to the increase 

of slope, but the increase of capacity follows a 

different law. The law is not simple, but one 

feature persists through all its manifestations: 

The capacity for traction increases more rapidly than the slope. The difference in rapidity, 

or the magnitude of the difference between the 

rates of change for capacity and slope, is itself 

a variable, depending on a variety of conditions. The study of the relation of capacity to 

slope is here treated as a study of the influence 

of conditions on the magnitude of the difference between the two rates of change.

The magnitude of that difference is indicated 

by a quantity, of the nature of an exponent, 

called the index of relative variation (of capacity, as compared to slope), and designated by 

it . The index may be defined as the first 

differential coefficient of log C with respect to 

log S. It is illustrated by saying that the 

capacity varies, instantaneously, as the \ 

power of the slope.

For the greater part of the field covered by 

the experiments the index falls between 1.4 

and 3.0, but under some conditions it is considerably higher. It varies with slope, being 

higher for low slopes and lower for high. It 

varies with discharge, being relatively high for 

small discharges. It varies with fineness, being 

relatively high for coarse debris. Briefly, it 

varies inversely with slope, discharge, and 

fineness. It varies also with width of channel, 

decreasingly for relatively narrow channels and 

increasingly for relatively broad channels; so 

that, for any particular combination of slope, 

discharge, and fineness there is a width characterized by a minimum value of the index.

It is furthermore true that no0 one of these 

variations is itself constant in rate, the rate of 

each having its own law of variation. Thus 

the complexity of the relation of capacity to
RELATION OF CAPACITY TO SLOPE. 121

slope is such as to be characterized by variations of the rates of variation of rates of variation; and there are even vistas of higher orders 

of variability. The law connecting capacity 

with slope may be susceptible of much more 

compact expression, but such formulation must 

probably await the development of a mechanical theory of stream traction.

Formulation founded on the index of relative variation, while bringing out clearly certain general features of the law, is not able to 

afford a complete quantitative statement. It 

may be likened to a map in definite hachures 

as contrasted with one in definite contours. 

As the hachure tells the direction and rate of 

slope but omits the absolute altitude, so the 

index tells the relative change under given conditions but omits the absolute capacity. To 

remedy this defect, the experiment was tried 

of substituting for the equation C=v1 Sii, in 

which vl is variable, the equation 6l =c1 5l/1 , in 

which cx is constant for each series of observations. Formulation by means of cx and ^ is 

more nearly analogous to the contour map, 

but the variability of the exponent ft is no less 

formidable than that of \ while the definition 

and derivation of /t are less simple and its 

significance is less clear.

Further utilizing the analogy of the map, we 

may think of the capacity-slope relation as an 

undulating topography, in which the vertical

C element is -~- or

o

and the horizontal elements are qualifying conditions. Formulation 

is a mode of representing this topography, the 

hills and valleys of which do not depend on the 

mode but are real. Two modes have been 

tried, each with limitations, but the ideal mode 

is not known. The contour map or the relief 

model would serve admirably if the qualifying 

conditions were two only, but as they number 

at least four, a graphic or plastic expression is 

possible only in space of n dimensions.

By reason of the complexity of the relation 

of capacity to slope and because of the lack of 

a mechanical theory of flow arid traction, the 

laboratory data do not warrant inferences as to 

the quantitative relations of 'capacity to slope 

for rivers.

Of various attempts to evade the complexity, 

two are thought worthy of record. In each 

series of experiments the mode of traction 

changes with increase of slope, first from dune

to smooth, then from smooth to antidune, but 

the critical slopes are not the same for different 

discharges or widths or degrees of fineness. It 

appeared possible to gain in simplicity by 

treating separately the data associated with a 

single mode of traction, and data for the 

smooth mode were accordingly segregated and 

discussed. Greater simplicity was not found, 

but the range of variation is somewhat smaller 

for the single mode of traction.

The second attempt was connected with the 

form of cross section of the current. Within a 

single series of experiments the width was constant and the depth varied, so that the capacity 

was conditioned not only by slope but by form

ratio, R = ~. By comparing one observational

series with another it is possible to obtain data 

conditioned by difference in slope, but without 

difference in form ratio. The discussion of 

such data developed only moderate modification of the results previously obtained and no 

reduction in complexity.

DUTY AND EFFICIENCY.

For the purposes of this paper duty has been

defined as the ratio of capacity to discharge:

C 'v 

U= Q. Combining this with 0=f(S) , the most

general expression for the relation of capacity

to slope, we have TJ= Under no form discovered for/($) is this expression reducible to 

simpler terms. For each value of discharge, 

duty is simply proportional to capacity; and 

the entire discussion of this chapter applies to 

duty as well as capacity. The parameters, n, 

*i> ju Iw> and o may be transferred, without 

modification, to formulas for duty.

Efficiency has been defined as the ratio of 

capacity to the product of discharge by slope:

" = 08-"

The combination of this with

yields

-(44)

-(36)

-(45)

The transformation of the exponent is important. While capacity varies as the ii power
122 TRANSPORTATION OF DEBRIS BY RUNNTJSTG WATER.

of slope, efficiency varies as the it l power. 

The index of relative variation is 1 less forefficiency than for capacity. Table 15 could 

therefore be adapted to efficiency by diminishing all its values by unity. As the lowest 

known values of \ are greater than unity the 

lowest in Table 15 is 1.31 it follows that efficiency is an increasing function of slope under 

all tested conditions. 

The combination of (44) with

yields

C = .(40) 

-(46)

showing that in passing from the field of capacity to that of efficiency the exponent associated with a constant coefficient also is reduced 

by unity.

Following the form of equation (35), we have

log (7"-log (7 

2l ~logS"-logS''

where C' and C" are specific values of capacity 

corresponding severally to the slopes S' and S". 

Designating by Ie the synthetic index of efficiency in relation to slope, and by E' and 

E" the efficiencies corresponding to C' and 

C" we have

/,= log £"-log E' 

log £"-log S' .(35b)

As C= ESQ, log <7'=log E' +logS' + log Q, and 

log <7"=log E" +log S" + log Q. Substituting 

these values in (35a) and reducing, we have

7 _logff"-logff' 1

1 log S" log $'

Subtracting the members of this expression 

from those of (35b) and transposing, we have

L = L-l. ............ (47)

That is, the synthetic index of relative variation of efficiency with reference to slope is less 

by unity than the corresponding index for 

capacity.

It is evident that at competent slope, when 

capacity is zero, efficiency also is zero. Like 

capacity, it increases with increase of slope. 

Under the assumption that its law of increase 

is of the same type, its value varying with a

power of (S a), two expressions have been 

derived, but neither has been found reducible 

to simple form. The algebraic work being 

omitted, they are

S-o

log S-2 

log (S-<r)-2

In the first of these expressions the coefficient is variable, being a function of 'S; so that 

the exponent may be regarded as the index of 

relative variation for efficiency in relation to

(S o). As falls to zero when 8 falls to

the limiting value a, and as it approximates 

unity when S is indefinitely large, the values 

of the exponent He between n and n l for all 

practical cases. In the second expression the 

coefficient is constant, with respect to slope, 

but the exponent is transcendental and intractable.

Thus it appears that the derived expression 

for efficiency as a function of S a is not 

simply related to the coordinate expression 

for capacity and is not available for practical 

purposes; but it does not necessarily follow 

that the actual relation of efficiency to slope 

can not be formulated for practical purposes by 

an equation of the sigma type. All that is 

really shown is that if capacity and efficiency 

are both formulated in that way, the results 

are not consistent. Formula ( 10) was adopted 

for the capacity-slope relation, not because it 

expresses a demonstrated law of relation, but 

because it so far simulates the real law of relation as to be available for the marshaling of the 

observational data. It seems quite possible 

that had the data been first translated from 

terms of capacity into terms of efficiency, the 

type of formula would have been found equally 

available.

By way of testing the matter a few comparative computations were made, observational series being selected for the purpose 

from those which in the adjustment gave small 

probable errors. From the original data in 

Table 4 values of efficiency were computed, and 

these were plotted on logarithmic paper in relation to S a, the values of a being those employed in the adjusting equations. In four of 

the nine cases treated the locus indicated was a
BELATION OF CAPACITY TO SLOPE. 123

straight line; and the drawing of the line gave 

values of J5 and na in E=B (S-Oi) n». In 

each of the remaining five cases the indicated 

locus was a curve, and the curvature was such 

as to indicate a larger constant in place of a.

This larger constant, au was determined graphically, and the other parameters were computed as before. The results are given in 

Table 23a, together with comparative data 

from Table 15.

TABLE 23a. Comparison of parameters in the associated functions of capacity and efficiency, C=b1 (S a)n and

E=B S oJnu.

Grade.

(B

(B

(C

<c

(C

(0

(G

(G

(H

Meaj

w

0.66

1.00

.66

.66

1.00

1.32

.66

1.00

.66

Q

0.182

.363

.182

.545

.363

.363

.734

.734

.734

61

39.8

97.5

38.6

123

100

100

64.5

65.7

52.9

B

196

268

187

193

280

240

78

63

58.6

a

0.10

.08

.11

.06

.11

.16

.36

.41

.56

<n

0.10

.18

.11

.06

.31

.22

.56

.41

.76

<ri a

0

.10

0

0

.20

.06

.20

0

.20

.09

n

1.64

1.54

1.54

1.48

1.48

1.40

1.69

1.70

1.72

«H

0.67

.56

.63

.56

.37

.54

.74

.96

.80

n-n,,

0.97

.98

.91

.92

1.11

.86

.91

.74

.92

.92

The proximate inferences from these plots 

and comparisons are, first, that efficiency may 

be formulated, with sufficient accuracy for 

practical purposes, as proportional to a power 

of S o: ; second, that, when it is thus formulated, the approximate values of ai are in general larger than the values of a obtained in the 

formulation of capacity; and, third, that the 

values of the exponent are smaller than the 

equivalent values for capacity, the differences 

usually being somewhat less than unity.

The field of these inferences was also traversed by a mathematical inquiry, of which 

the results are more definite. If the relation 

of efficiency to slope be formulated by

the exponent Un is always less than n, but never

so small as n 1. For the range of conditions 

covered by the experiments, it is little greater 

than n 1. The value of ax is always greater 

than the corresponding value of a, the difference being usually small. The difference is 

greater when the value of the exponent is 

relatively small. Equation (47a) is incompatible with the corresponding equation for 

capacity, (10). If the-locus of E=f(S) be 

separately plotted by means of the two equations, the resulting curves are not coincident, 

but they intersect at three points and He close 

together elsewhere (in the practical field) 

unless the difference between a and at is large. 

.On the whole, it appears entirely feasible 

to formulate efficiency by means of equation 

(47a).
CHAPTER IV. RELATION OF CAPACITY TO FORM RATIO.

INTRODUCTION.

Details of channel form in a natural stream 

are highly diversified. In connection with the 

bendings to right and left the current is thrown 

to one side and the other, with the result that 

the cross section is not, for the most part, symmetric about a medial axis but shows greater 

depth on the side of the swifter flow. In the 

straight channels of the laboratory there was 

little departure from bilateral symmetry and 

the cross section was approximately rectangular. For this reason those relations of traction 

to form of cross section which are found to exist in the laboratory can not, in general, be inferred of natural streams. Nevertheless there 

is probably an approximate correspondence between the two types when the tractional properties of a broad, shallow channel are compared 

with those of a narrow, deep channel; and to

that extent the discussion of form ratio (R = )

v w'

is pertinent to the problems of natural streams.

In connection with the study of the laboratory data the form ratio is a factor of great importance, for not only is capacity for traction 

directly conditioned by it, but it affects every 

law of relation between capacity and another 

condition.

In the discussion of capacity in relation to 

slope the effects which might have been referred 

to form ratio were treated instead as due to 

width, while small account was taken of the coordinate influence of depth. For many purposes the choice of viewpoint is indifferent, but 

when large and small channels are to be compared there is decided advantage in taking account of form ratio. The form ratios of laboratory channels and river channels, for example, are of the same order of magnitude, but the 

widths are not.

SELECTION OF A FORMULA.

MAXIMUM.

When identical discharges are passed through 

troughs of different width and are loaded with 

debris of the same grade, and the loads are 

124

adjusted so as to establish the same slope, it is 

usually found, not only that the capacity varies 

with the width, but that some intermediate 

width determines a greater capacity than do 

the extreme widths. That is, the curve of 

capacity in relation to width exhibits a maximum. The form ratio varies inversely with the 

width; and the same maximum appears when 

the capacity is compared with form ratio. The 

curves in figure 38, introduced to illustrate this 

fact, show data from Tables 12 and 14 for grade 

(C), with Q = 0.363 ft.3/sec. and £=1.0 percent. 

In the upper curve capacities are compared 

with widths; in the lower one the same capacities are compared with form ratios.

Width

O.I

Form ratio

FIGURE 38. Illustration of the relation of capacity to width of channel 

and to form ratio, when slope and discharge are constant.

The formula for the discussion of such relations must be one affording a maximum. It 

must also satisfy various physical conditions, 

as will presently appear.

The explanation of the maximum, so far as 

its main elements are concerned, is not difficult. 

The phenomenon was in fact anticipated in the 

planning of the experiments, and certain 

courses of experimentation were arranged with 

special regard to the discovery of the form 

ratio of highest efficiency.

Conceive a stream of constant discharge and 

flowing down a constant slope but of variable 

width. The field of traction is determined by 

the width, and the evident tendency of this 

factor is to make the capacity increase as the 

width increases. The rate of traction for each 

unit of width is determined by the bed velocity 

in that unit, and the bed velocity is intimately
BELATION OF CAPACITY TO FORM KATIO. 125

associated with the mean velocity. Velocity 

varies directly with depth, and, inasmuch as 

increase of width causes (in a stream of constant discharge) decrease of depth, the tendency of this factor is to make capacity 

decrease as width increases. Velocity is also 

affected by lateral resistance, the retarding 

influence of the side walls of the channel. The 

retardation is greater as the wall surface is 

greater, therefore as the depth is greater, and 

therefore as the width is less. As capacity 

varies inversely with the retardation, and as the 

retardation varies inversely with width, it follows that the tendency of this factor is to make 

capacity increase as width increases. Thus the 

influence of width on capacity is threefold: Its 

.increase (1) enlarges capacity by broadening 

(the field of traction, (2) reduces capacity by 

reducing depth, and (3) enlarges capacity by 

,' reducing the field of side-wall resistance. Now, 

without inquiring as to, the laws which affect 

the several factors, it is evident that when the 

width is greatly increased a condition is inevitably reached in which the depth is so small 

that the velocity is no longer competent and 

capacity is nil. It is equally evident that when

the width is gradually and greatly reduced 

the field of traction must become so narrow 

that the capacity is very small, and eventually 

the current must be so retarded by side-wall 

friction that its bed velocity is no longer 

competent and capacity is nil. For all widths 

tjetween these limits capacity exists, and somewhere between them it attains a maximum.

The forms of algebraic function which afford 

a maximum are many; but no general examination of them is necessary, because the physical 

conditions of the problem serve to indicate the 

appropriate type. As just observed, the variation of form ratio (when discharge and slope are 

constant) involves simultaneous variations of 

width and depth. To develop an expression 

for the relation of capacity to form ratio, it is 

convenient first to determine separately the 

relations of capacity to width and to depth, 

and then to combine the two functions.

CAPACITY AND WIDTH.

To consider separately the response of 

capacity for traction to variation of width it is 

necessary to relinquish, for the time being, the 

assumption of constant discharge and variable 

depth, and substitute for it the assumption of

constant depth and slope, with variable discharge. That is, we are to conceive a stream 

of constant slope, of which the width is progressively increased or fHTrnnishp.fi and of 

which the discharge is varied in such way as to 

maintain a constant depth. Figure 39 represents the cross section of such a stream, whether 

natural or of the laboratory type.

Near the sides the current is retarded by side 

friction. Also, the freedom of its internal 

movements is restricted by the sides, just as it 

is everywhere restricted by the upper surface 

and the bed. These lateral influences diminish 

with distance from the sides and finally cease to 

be perceptible. We may thus recognize, in a 

broad stream, two lateral portions, AB, in 

which capacity is affected by the sides, and a 

medial portion, AA, in which capacity is not 

thus affected. In the medial portion total 

capacity is strictly proportional to the distance 

AA; or, in other words, the capacity per unit

f

D

FIGURE 39. Cross sections of stream channels; to illustrate the relation of capacity to width.

distance, C1} is uniform. In a lateral portion 

the capacity per unit distance diminishes as 

the side is approached. Whatever the law of 

diminution, the total capacity of a lateral portion is equivalent to the capacity per unit distance in the medial portion, multiplied by some 

distance AD, less than AB. Therefore the 

total capacity for the whole stream is

It is evident that for a shallow stream the 

distances AB and DB are less than for a deep 

stream; and while the assumption may not be 

strictly accurate, it must be approximately 

true that DB is proportional to the depth. 

Making that assumption and introducing the 

numerical constant «, I replace 2DB by ad, 

and write

G=Cl (w-ad)

As we are here concerned only with the law of 

variation of C, we may conveniently replace 

this by the proportion
126 TRANSPORTATION OF DEBRIS BY RUNNING WATER,.

Substituting for d its equivalent Rw,

As d is by postulate constant, and as w = -^f,

itf ocrpr. We may therefore substitute -^ for w in R J R

the proportion above, obtaining

<7oc

1-aR 

R

-(49)

This expression gives the relation of capacity 

to form ratio, so far as that relation depends 

on variation of width. Eventually it is to be 

complemented by an expression similarly 

dependent on variation of depth.

The preceding analysis involves the assumption that the stream is so broad, in relation to 

its depth, that its medial portion is unaffected 

by lateral influences. The resulting proportion, 

(49), is not necessarily applicable to narrower 

streams. It is quite conceivable that when 

the channel is so narrow that the reaction of 

the sides affects all parts of the current the 

"variation of capacity follows a different law. 

The analytic consideration of the case of 

narrower channels has not been attempted, 

but some information has been obtained from 

the experiments. The following examination 

of experimental data is directed toward this 

question and also toward that of the magnitude 

of the constant a.

TABLE 24. 'Relation of capacity for traction to width of channel, when slope and depth are constant.

Grade.

(B)

(C)

(B) and 

(C).

Slope 

(per

cent).

 0.8

1.2

.8

1.2

0.8 and 

1.2

Depth

(feet).

0.08 

.10 

.12 

.15 

.20 

.08 

.10 

.12 

.10 

.12 

.14 

.20 

.10 

.12 

.14

.08 

.10 

.12 

.14 

.15

Value of C when width (feet) is 

0.23

2.8

0.44

8.6 

15.2 

32.0

22.1 

32.0

[5.3]

20.0

0.66

8.4 

15 

21.0 

36.7 

62.5 

29 

46 

65 

12.5 

19.2 

27 

52 

45 

[64] 

81

1.00

[17.4] 

26.6 

38.0 

60.0 

113 

50 

74 

103 

22.0 

34 

51 

111 

80 

115 

156

1.32

20.4 

35.4 

57.5 

112

88 

156

25.0

78

109 

151 

221

1.96

43 

66 

95 

145

142 

211 

290 

42- 

76 

135

17i 

[300] 

480

Geometric means.

}......... 11.7

15.6 

29.6 

36.1

43.1

29.4 

50.7 

62.5

78.. 2

42.4 

75.2

124

78.0 

115 

158

211

The assumptions of the present section 

include constant slope and constant depth, 

with discharge and capacity adjusted to variation of width. The experiments involve constant width and constant discharge, with 

automatic adjustment of slope and depth to 

variation of load. In order to check the 

analysis by means of the laboratory data it is 

necessary to employ some method of interpolation. Two methods were tried, but only 

one need be described.

Attention being first restricted to a particular grade of debris and a particular slope 

of channel, the computation sheets (p. 95) 

for the different discharges were entered with 

a particular depth as argument, and the 

associated values of capacity and slope were

taken out. These values were plotted on 

logarithmic section paper as a series of points. 

Through these points was drawn a curve the 

locus of log C=f(\ogS), under the condition

that d is constant. By means of this curve 

values of (7 were interpolated, corresponding to 

selected values of S. The process was then 

repeated with other depths, other widths, and 

other grades; and in this way were obtained 

sets of values of capacity in relation to width, 

under the condition of constant depth and 

slope. Such interpolated values of capacity 

are presented in Table 24. It was found that 

the data for grades (B) and (C) only are full 

enough to serve the present purpose.

The tabulated capacities are also plotted, in 

relation to width, in the upper and second divi-
RELATION OF CAPACITY TO FORM RATIO. 127

sions of figure 40. If the data were precise, 

and if equation (48) were strictly accurate, the

300

200

100

4-00

300

200

100

200

100

Grade

Grade

Means

'£)

C)

 2

FIGURE 40. Capacity for traction in relation to width of channel, when 

depth and slope are constant. Scale of capacities, vertical; widths, 

horizontal.

oblique lines of the figure would all be straight 

and would all intersect the line of zero capacity 

somewhere to the right of the origin. The

irregularities of the lines are of such distribution 

as to indicate that they are occasioned chiefly 

by the imperfection of the data, and so far as 

may be judged by their inspection the formula 

is substantially correct.

As the individual lines do not well indicate 

the points of intersection with the horizontal 

axis, a set of composites were prepared, each 

combining the data for a particular depth, 

without distinction as to grade of debris or 

slope of channel. In the computations for 

these a few interpolations were first made, and 

then the capacities were combined by taking 

their geometric means. The numerical results 

appear at the bottom of Table 24, and these 

are represented by dots in the lower division of 

figure 40. The indications of the dots were 

then generalized by drawing straight lines 

among them, and the intersections of these 

lines with the line of zero capacity gave points 

corresponding to D in figure 39. More strictly, 

the distance of each intersection from the 

origin gave an estimate of the quantity 2 DB 

in equation (48). As each estimate is asso2 757? ciated with a particular depth, and as tx ~ -, ,

the intersections give also values of the constant a. in (49).

d

0.08

.10

.12

.143

2DB

0.30

.24

.325

.36

<T

3.7

2.4

2.7

2.5

As the plotted dots are so irregular as to 

admit of much latitude in the drawing of the 

lines, these values of the constant are far from 

precise. No inference may be drawn from their 

differences, and collectively they serve only to 

indicate an order of magnitude. For fine sand, 

with slopes of about 1 per cent, the constant a 

has a probable magnitude of 2 or 3.

CAPACITY AND DEPTH.

To consider separately the response of capacity to variation of depth, the assumption of 

constant discharge and variable width must 

again be laid aside, and there must be substituted for it the assumption of constant width 

and slope, with variable discharge. That is, 

we are to conceive a stream of constant width, 

of which the discharge and load are simultane-
128

ously varied in such way as to maintain the 

slope unchanged, and the changes in depth are 

to be compared with the changes in load or 

capacity. To make the results strictly coordinate with those for the control of capacity by 

width, we should deal only with channels so 

broad and shallow that the lateral portions 

(fig. 39) are small in comparison with the medial 

portions, but this is not practicable. It is possible, however, to minunize the influence of 

lateral retardation by selecting groups of data 

in which the form ratio is small.

Table 25 contains data pertaining to debris 

of grade (C), a channel width of 0.66 foot, and 

a slope of 1 per cent. The values of capacity 

are taken from Table 12 and the values of 

depth from Table 14. It appears by inspection 

that the capacity increases with the depth. 

On plotting the pairs of values on logarithmic 

section paper, it is found that they may be 

represented approximately by a straight line. 

The examination of many such plots showed 

that the most accurate representative line has 

a gentle curvature, but for the present purposes 

it suffices to assume that the line is straight. 

That is to say, it is found to be approximately 

true, and the assumption is made, that C" varies 

with some power of d, or

Coed11 (50)

In the particular case m^l.90, and other 

values of mt are shown in Table 26.

TABLE 25. Values of capacity and depth for currents transporting debris of grade (C), when the width is 0.66 foot and 

the slope. 1.0 per cent.

0.093

13.8

.079

0.182

32.1

.110

0.363

73

.173

0.545

112

.228

0.734

152

.255

TABLE 26. Valuesofm1 in C<X dm', when slope is constant.

Grade.

(C) 

(G)

Slope, 

(per 

cent).

0.6 

1.0

1.0 

1.6 

2.4

Value of T»I when width (feet) is 

0.66

2.20 

1.90

2.67 

2.05 

1.80

1.00

2.50 

2.10

2.96 

3.43 

2.26

1.32

3.10 

2.32

3.95 

2.83 

2.48

1.96

3.41 

2.97

Comparison of the tabulated values of m^ 

shows that they have considerable range and

that then- variations in magnitude are definitely related to several conditions. The sensitiveness of capacity to variation of depth (when 

width and slope are constant) is greater as the 

slope is less, is greater as the fineness is less, 

and, with one exception, is greater as the width 

is greater:

P,w) ..........(51)

As w is by postulate constant, and as d = 

it follows that docR. R may therefore be substituted for d in proportion (50), giving

tfoctfV............ .(52)

This is an expression for the relation of 

capacity to form ratio, so far as that relation 

depends on variation of depth. It is the complement of proportion (49) on page 126.

CAPACITY AND FORM RATIO.

We how return to the assumption of constant discharge. Having obtained an approximate expression for the law of capacity's variation in response to change of width and a 

coordinate expression for variation on response 

to change of depth, we next inquire how these 

may be combined into an expression for the 

response of capacity to simultaneous changes 

of width and depth, when those changes are of 

such character as to leave discharge constant.

If the functions (49) and (52) be assumed 

to be independent, their simple and direct 

combination gives

tfcc R

(53)

This formula, as may readily be shown, satisfies an important condition by providing for a 

maximum value of 0, but the assumption involved in its construction is not strictly warranted. The law embodied in

 aR

 JK is based

on the constancy of d and may not be valid 

when d is made variable, while the law embodied in Rmi may not be valid when w is made 

variable.

Viewing the matter from another side, we 

may say that the difficulty would not exist if 

the values of a and m1 depended only on fineness, slope, and discharge and were independent 

of R; but they are in fact functions of R. (We 

have already seen that ml is an increasing 

function of w, and the variation of a with d
RELATION OF CAPACITY TO FORM RATIO. 129

will appear in another connection.) An inquiry as to the nature of the functions an 

inquiry here unrecorded except as to its 

result served to determine that the variations of a and m1; as functions of R, affect 

capacity in opposite senses, so that their influences are at least partly compensatory. It 

appeared also that each influence is relatively 

great when R is large and relatively small 

when R is small, so that their laws of distribution include a compensatory factor.

Despite the existence of this difficulty, 

wlu'ch is palliated rather than cured by features 

of compensation, the function in (53) has been 

adopted as the best practicable formula for 

the relation of capacity to form ratio. The 

logical defect in the combination of its two 

factors may mean that its accuracy is inferior 

to that of the factors; but as the factors are 

confessedly only approximate, the defect is not 

inconsistent with the possession of even superior 

accuracy by the combination. In the treatment of so intricate a subject by methods 

which are dominantly empiric altogether adequate formulation is not to be hoped for; but 

the modicum of physical foundation afforded 

to formula (53) is believed to give it advantage over a purely mathematical expedient.

A slight transformation gives it more convenient form. Moving R from the denominator to the numerator and making mj 1 = m,

we have

C cc (1 - a R) Rm

It is convenient also to change from a proportion to an equation; introducing a coefficient, 62 ,

0=1, (l-aR}Rm . -(54)

As R is i a ratio between lengths, and a (p. 125) 

is also a numerical quantity without dimensions, 62 ig °f *he um't °f @- -ft ig the value of

capacity when (1 a R} Rm = l.

Let us now consider the properties of the 

formula. For the sake of giving a visible illustration, it has been applied to the example 

already used in figure 38. As the equation has 

three parameters, its constants require for their 

determination three pairs of values of R and (7. 

The example furnishes four pairs, and these, by 

approximate adjustment, give

(7=346 (1-2.185) #°-52 (55)

The corresponding curve is shown in figure 41. 

20921° No. 86 14 9

This conforms to the physical conditions by 

indicating two values of R for which capacity 

is nil, and an intermediate value for which 

capacity is at maximum. On referring to (54), 

it is evident that (7=0 when R = 0, and also

when l (xR = 0, or R = -.

a

In ascribing finite capacity to all small 

values of R the formula is inaccurate, for when 

the form ratio is gradually reduced the velocity 

must always fall below competence before the 

ratio reaches zero. The true function might be 

represented by some such curve as the broken 

line D in the figure. It has not seemed advisable to complicate the formula by a modification which might remedy this defect.

In the region of the larger values of R the 

formula is subject to a qualification already 

mentioned on page 125. The larger values are 

associated with narrow channels, in which the

0 .1 .Z, .3 A

It- i v.,S

FIGURE 41. Plot of equation (55). Capacity, vertical; form ratio, horizontal.

influences of the side walls or banks are dominant. So far as the laboratory data give indication, the formula is applicable to these values; 

but the adjustment might be less satisfactory 

with channel walls of a different character.

The region of the maximum value of capacity, which constitutes the chief field for the 

application of the formula, is little affected by 

the qualifications which have been mentioned.

Differentiating equation (54) and equating 

the first differential coefficient with zero, we 

have

1 777/ whence R = -v, which is the condition giv- ' b

ing C its maximum value. Designating this 

value of R by p, we have

1 ___m

a m + 1

1 m

p m + 1

-(56)

-(57)
130

Substituting from (57) into (54), we have

Equations (54) and (58) are alternative expressions of the same relation, the one involving 

a without p, the other p without a. Each has 

its field of superior convenience, alike for computations and discussion.

To obtain the maximum capacity, p is substituted for R in (58). The equation reduces to

-(59) .-A-,-

m+r

DISCUSSION OF EXPERIMENTAL DATA.

SCOPE AND METHOD OF DISCUSSION.

The equations adopted for the formulation of 

C=f(R), namely, equations (54) and (58), involve four constants. &2 is a quantity of the 

unit of capacity; p is a ratio, the form ratio corresponding to maximum capacity; or is a ratio 

connected with side-wall resistance; and m is 

an exponent. There is a mutual dependence 

between a and p; but Z>2 , a, and m, grouped 

together in (54), are independent; and so are 

&2 , p, and m, grouped in (58). In the following 

discussion of the relation of capacity to form 

ratio, equations of the form of (54) and (58) are 

derived from groups of experimental data; and 

these are compared in such way as to show the 

control of their constants by the conditions of 

slope, discharge, and fineness.

A "group" of experimental data, for this purpose, includes values of capacity and form ratio 

from at least three observational series, all pertaining to the same fineness, slope, and discharge, but to different widths. Three pairs of 

observational values suffice to determine the 

three independent parameters; with a greater 

number the problem is usually one of adjustment, to determine the most probable values of 

the parameters.

The computation of the constants by algebraic methods is tedious, and a graphic method 

was substituted. When an equation of the 

form of .(58) is plotted on logarithmic section 

paper the shape and size of the curve are determined wholly by the exponent m: its position 

measured in the direction of the axis of log R 

is determined by the value of p; and its position

with respect to the axis of log C by the value 

of &2 . A graphic process based on these properties gave solutions of sufficient approximation for the purposes of the discussion.

SENSITIVENESS AND THE INDEX OF RELATIVE 

VARIATION.

The sensitiveness of capacity to variation of 

form ratio is indicated graphically by the inclination of the logarithmic locus C=f(R). As 

that locus is a curve, the sensitiveness varies 

with R. The form of the curve, as mentioned 

in the last paragraph, is determined by the exponent m, and the steepness (see fig. 42) of its 

legs varies directly with m. The exponent is 

thus a general index of sensitiveness.

FIGURE 42. Logarithmic plots of C=62 (l - j .Rm, corresponding 

to the same value of 68 and p, but different values of m.

The logarithmic equivalent of (54) is

log C=log &2 + log (1 aR) +mlogR 

Differentiating, we have

d log C=d log (1 - aR) +md log R 

Dividing by d log R, we have

dlog C_dlog (1-aR) 

d log R ~ d log R

Making substitutions from 

d log C .

and

d log R = -

and reducing, we have

1-aR (60)
RELATION OF CAPACITY TO FORM RATIO. 131

It is evident that aR increases with 1-aR

increase of R; therefore i2 decreases with increase of R. Also i2 is positive when m > T p

j A - -u aR T . and negative when <<-r^ # ^n Posing

from positive to negative, i2 passes through

, aR , n 1 m

zero when m = , 5 or when H = r-r = P

1 «r/( or m +1 r

(cf. equation 56). When R = Q, the Hmiting

or5 condition for a broad, shallow stream,.. _ = 0

and i2 = m. As this condition may only be

approached but not reached, the positive 

values of the index are ah1 less than m. When

! = 1, or R == -, the condition limiting traction

aR

a'

for a narrow, deep stream, 

and i

;= + oo 1-aR'

= oo. The negative values of the 

index therefore range to infinity.

The progressive changes of the index are 

conveniently illustrated by the characters of 

the curves in figure 42. Consider, for example, 

the curve corresponding to m=1.0. As this 

curve is the logarithmic locus of C=f(R), its

200

100

Gracte(G) 

Q-,734

\

2.0

0.3

S'l.8-%

0.6

FIGURE 43. Relation of capacity, C, to form ratio, R. The variation of the function c-=bi(l <tR)Rm with slope. Scale of C vertical;

scale of JB horizontal.

inclination to the horizontal represents iv At 

the extreme left, corresponding to a very small 

value of R (a shallow, wide stream), its inclination is a little less than 45°; or the value of i2 is 

a little less than 1.0, the value of m. As the 

curve ascends i2 is positive. The inclination 

then diminishes gradually and becomes zero at 

the apex of the curve, which corresponds to 

R = p. As the curve begins its descent the 

forward inclination corresponds to negative 

values of i2) and these values increase numerically, becoming very large as the curve 

approaches verticality.

CONTROL OF CONSTANTS BY SLOPE.

To illustrate the way in which the index of 

sensitiveness and other constants defining the 

relation of capacity to form ratio are jLnfluenced

by the condition slope, two sets of computations were made. The first set used data pertaining to debris of grade (C) and to a discharge 

of 0.182 ft.3/sec.; the second, data for grade (G) 

and 0.734 ft.3/sec. The observational data and 

the computed constants appear numerically in 

Table 27 and graphically in figure 43. The 

dots in the figure represent points fixed by the 

observations, and through these have been 

drawn curves conforming to formulas (54) and 

(58). Short vertical lines note the positions of 

the maxima, where R=p.

On referring to the table, it is seen that as the 

slope increases the values of m, p, arid a decrease, while the value of &2 increases. With 

steeper slopes of channel, capacity is relatively 

less sensitive to changes in the form of cross 

section; with steeper slopes, the form of sec-
132 TBANSPOBTATION OF DEBRIS BY BUNKING WATEB.

tion giving highest capacity for traction is relatively shallow and broad; with steeper slopes, 

the reduction of capacity through retardation 

of the current by the channel sides is relatively 

small; and the general effect of steepened slopes 

is increased capacity.

Each set of values of a constant exhibits an 

orderly progression, marred only by minor irregularities due to the inaccuracies of graphic

TABLE 27. Observed and computed quantities illustrating the

computation. The results recorded in Tables 

28 and 29 are much less orderly. It is to be 

noted that the remarkable orderliness in the 

present case is- not due to high precision of 

the data but to the fact that the values of C 

and R assembled in Table 27 had previously 

been adjusted, as described in Chapter II, in a 

manner which made them orderly with respect 

to slope.

influence of slope on the relation of capacity to form ratio.

Grade

and discharge 

(ft.3/sec.).

(C) 

0.182

(G) 

0.734

Observational data.

Slope 

(per 

cent).

0.6

1.0

1.4

1.8

.8

1.4

2.0

2.6

Width 

(feet).

0.44 

.66

1.00

.44 

.66 

1.00

.44 

.66 

1.00

.66 

1.00

.66 

1.00

1.32

.66 

1.00 

1.32

.66

1.00

1.32

.66

1.00

1.32

Form 

ratio.

0.546 

.222

.113

.436 

.167 

.086

.375 

.138 

.072

.120 

.063

.540 

263

.164

.460 

.222 

.141

.418

.199

.128

.387

.184

.120

Capacity 

(gm./sec.)

8.7 

12.9

11.4

24.7 

32.1 

31.2

45.3 

57.0 

57.7

87 

90

16.0 

13.3

7.9

69 

65 

52

149

145

125

253

249

222

Constants of equations.

m

\ 0.44

1

| .20

I .08

} [-05

1 1.55

)

1 .80

1 f . 55UU

1

i

[ .45

'

P

0.25

.158

' .079

.05

.44

".36

.32

.29

a

1.22

1.05

.94

.95

1.38

1.23

1.11

1.07

6s

34.4

55.5

75.2

110]

170

296

451

662

'it

i -1.55 

\ + .07

I + .28

f- .65 

] - .01 1 + .10

{

- .46 

- .07 

+ .01

/ - .08 

\- .01

{

-1.42 

+1.07

+1.26

f - .50 

\ + .42 

I + .59

f - .32

\ + .27

I + .41

f - .26

\ + .20

I + -30

CONTEOL OF CONSTANTS BY DISCHARGE.

Table 28 and figure 44 exhibit quantities and 

curves illustrating the influence of discharge on 

the relation of capacity to form ratio. The 

data employed are those of grade (C) with a 

channel slope of 1 per cent, and of grade (G)with 

a channel slope of 2 per cent. Of the seven 

curves in the figure, four were determined by 

three observational points. As three is the 

minimum number of given points for the determination of a curve of this type, each curve 

passes through all the points. In each of the 

other cases four points are given by the observations, and it was not found possible to pass 

one of the curves through all the points. If 

the formulas are correct, the incompatibility 

indicates errors of the data. Errors of similar 

magnitude inferably affect the three-point 

groups of data, and this inference qualifies the 

(determination of the curves and their equations. 

Of the same general tenor is the fact that the 

four curves under grade (C), pertaining to discharges which increase in arithmetic progression, do not exhibit an orderly progression as 

to shape.

Grade fC)

--£00

I____.____I

Grade (G)

.734

j_______I

0 .2 ..4- 0 .2 .4- 

FIGURE 44. Relation of capacity, C, to form ratio, R. The variation 

of the function C=6a (1 a R )Rm with discharge. Scale of C vertical; 

scale of R horizontal.
BELATION OF CAPACITY TO FORM EATIO. 133 

TABLE 28. Observed and computed quantities illustrating the influence of discharge on the relation of capacity to form ratio.

Grade 

and 

slope 

(per 

cent).

(C)

1.0

(G)

2.0

Observational data.

Discharge 

(ft.a/sec.)

0.182

.363

.545

.734

.363

.'734

1.119

Width 

(feet).

0.44

.66

1.00

.66

1.00 

1.32

1.96

.66

1.00 

1.32

1.96

.66

1.00 

1.32

1.96

.66

1.00

1.32

.66

1.00

1.32

.66

1.00

1.32

Form 

ratio.

0.436

.167

.086

.264

.130 

.082

.043

.346

.170 

.104

.054

.386

.190 

.120

.063

.260

.128

.082

.418

.199

.128

.532

.260

.165

Capacity 

(gm./sec.).

24.7

32.1

31.2

73

85 

79

60

112

140 

129

111

152

180 

187

190

56

45.2

28.3

149

145

125

227

268

238

Constants of equations.

m

0.20

.52

.40

.05

1.52

.55

.58

p

0.158

.157

.174

.061

.215

.320

.322

a

1.05

2.18

1.64

.78

2.80

1.11

1.14

6 2

55.5

340

392

229

1,611

451

.830

h

-0.65

- .01

+ .10

- .84

+ .13 

+ .30

+ .42

- .91

+ .01 

+ .19

+ .30

- .38

- .12 

- .05

- .00

-1.15

+ .96

+1.22

- .32

+ .27

+ .41

- .96

+ .16

+ .35

The same incongruities of course characterize the tabulated numerical results. Increase 

of discharge is accompanied, in the main, by 

diminution of m and a, but the indications are 

inconclusive as to p and &2 .

CONTROL OF CONSTANTS BY FINENESS.

With reference to the influence of fineness of 

debris, the comparative data are restricted to 

three grades. Table 29 and figure 45 record 

the data and results for grades (B), (C), and 

(G), with use of a single slope and a single discharge. The curves for grades (B) and (C), 

which fall close together, are so different in 

form as to intersect; and the difference shown 

by the forms is otherwise expressed by incongruities among the values of constants. With 

the exception of &2 > the constants for grade (B) 

are intermediate between those of the other 

grades, whereas the intermediate grade is (C). 

If we ignore this incongruity and compare the

constants for grades (B) and (C) with those for 

the coarse grade (G), we find that m, p, and a 

are all relatively small for the fine grades, or

ISO

100

50

.z .4- :e

FIGURE 45. Relation of capacity, C, to form ratio, R. The variation 

of the function C=bt (l aR) Rm withflneness of ddbris. Scale of C 

vertical; scale of R horizontal.

vary inversely with fineness, while &2 varies 

directly with fineness. The variations of m 

and p are more pronounced than that of a.

TABLE 29. Observed and computed quantities illustrating the influence of the fineness of debris on the relation of capacity

to form ratio.

Slope

(per cent) 

and discharge 

(ft.»/sec.).

0.8

0.734

Observational data.

Grade

and 

fineness

(Ft).

(B)

13,400

(C)

5,460

(G)

5.9

Width 

(feet).

1.00

1.32

1.96

.66

1.00 

1.32

1.96

.66

1.00

1.32

Form 

ratio.

0.217

.120

.064

.440

.214 

.134

.070

.540

.263

.164

Capacity 

(gm./sec.).

138

139

130

107

130 

133

131

16

13.3

7.9

Constants of equations.

m

0.20

.12

1.55

p

0.10

.126

.44

a

1.04

.85

1.38

62

242

191

170

it

-0.09

+ .06

+ .13

- .48

' - .10 

- .01

+ .06

-1.42

+1.07

+1.26
134 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

SPECIAL GROUP OF OBSERVATIONS.

The special group of observations recorded 

in Table 4 (I), page 51, were arranged largely 

for the purpose of defining p, the optimum form 

ratio. They differed from the main body of 

observations in that the interval between the 

discharges employed and the interval between 

the widths employed were both smaller; and 

they were restricted to a single grade of debris. 

They had the advantage of an experimental 

method believed to be the best developed in the 

laboratory; and in view of this advantage their 

series were constituted of fewer individual observations than those of the main body of experiments. The results have not been satisfactory, and attempts at formulation in the 

present connection have developed marked incongruities. In figure 46 three curves are 

given, and the corresponding numerical data 

appear in Table 30. Each curve is based on 

five observational points, but they are so irregularly placed that their control is feeble.

On comparing the two cases having the 

same discharge, it is seen that the greater 

slope is associated with the smaller values of 

m, p, and a and with the greater value of &2 ,

the difference being most strongly marked for 

m and p. On comparing the two cases having 

the same slope, it is seen that the greater dis300

200

q-0.734

.1 .2 .3

FIGURE 46. Relation of capacity, C, to form ratio, R. Variation of 

the function C=bi(\ a.R)Rm with slope and discharge. Data from 

special group of experiments with de"bris of grade (C). Scale of C 

vertical; scale of R horizontal.

charge is associated with the smaller values of 

m and p and with the larger value of Z> 2 , while 

the values of a are nearly equal.

TABLE 30. Observed and computed quantities illustrating the influence of slope and discharge on the relation of capacity

to form ratio.

Grade.

(C)

Discharge 

(ft. «/sec.)-

0.734

1.110

1.119

Observational data.

Slope 

(per 

cent).

1.1

.6

1.1

Width 

(feet).

1.0

1.2

1.4

1.6

1.8

1.2

1.4

1.6

1.8

1.96

1.2

1.4

1.6

1.8

1.96

Form 

ratio.

0.190

.143

.108

.079

.072

.216

.180

.140

.122

.102

.185

.145

.108

.091

.078

C

230

240

240

231

230

114

125

131

125

130

350

400

400

418

420

Constants of equations. ,

771

0.3

.5

.1

p

0. 123

.129

.048

a

1.88

2.58

1.90

62

586

546

643

r,

-0.25

- .07

+ .05

+ .12

+ .14

- .72

- .35

- .06

+ .05

+ .15

- .34

- .28

-.16

- .11

1 - .07

SUMMARY AS TO CONTROL BY CONDITIONS.

The treatment of the observational data by 

means of a formula specially designed to show 

the relation of capacity for traction to the 

proportions of the cross section develops 

incongruities. These are of such distribution 

as to indicate that they are due in chief part 

to the observations and their methods of 

adjustment. Discrepancies which manifestly

pertain to the data are so large that it is not 

practicable to determine whether the imperfections of the formula are important.

The exponent m varies inversely with slope, 

with discharge, and with fineness. Thus all 

the conditions which tend to increase capacity 

tend also to make capacity less sensitive to 

changes in form ratio.

The optimum form ratio, p, varies inversely 

with slope and with fineness. As to its varia-
BELATION OF CAPACITY TO FORM EATIO. 135

tion with discharge the evidence is not unanimous; either it varies inversely under some 

circumstances and directly under others, or 

else its proper variation is inverse and data 

of contrary import are erroneous. The latter 

view is thought more probable, because in 

many other connections the controls of slope 

and discharge follow parallel lines.

The constant a, which represents the resistance of side walls or banks to the flow of the 

stream, is also a decreasing function of slope, 

discharge, and fineness.

The constant &2 > which is of the unit of 

capacity, varies directly with slope and inversely with fineness, and the evidence as to 

its variation with discharge is conflicting. As 

&2 is the value of capacity when

>=1

it corresponds to a rather complicated relation 

between R, m, and a, or E, ra, and p; and this 

relation makes the interpretation of the lack of 

order among its tabulated values a difficult 

matter.

If & 2 is left out of the account, it is possible 

to generalize by saying that the constants of 

equations (54) and (58) vary decreasingly with 

the conditions which affect capacity increasingly.

m=/(£, 4 / )-------- -(61)

<* = /,(£, 4 *')-'---------1 --(62)

#) ------------ (63)

THE OPTIMUM FORM RATIO.

The ratio of depth to width which gives to 

a stream its greatest capacity for traction is of 

importance to the engineer whenever he has 

occasion to control the movement of de"bris. 

The title optimum ratio is especially appropriate 

when his desire is to promote that movement.

The range of values for the ratio, under laboratory conditions, is from 1:2 to 1 : 20. One 

effect of this wide range, when taken in connection with the variety of conditions by which 

the ratio is controlled, is to complicate the 

formulation of practical rules; but this difficulty is not insuperable. It is qualified to an 

important degree by the consideration that 

capacity, in the region of its maximum, changes 

very slowly with change of form ratio, so that

an approximate determination of the ratio has 

practical value.

The values of the ratio given in Table 31 are 

appropriate to the conditions of the Berkeley 

laboratory that is, they pertain to troughs a 

few inches or a few feet wide, with smooth 

vertical sides. It is important to note also that 

they apply only to transportation of de"bris 

over a bed of debris, and not to flume traction, 

which has a different law. (See p. 213.)

TABLE 31. Estimated ratio* of depth of current to width of 

trough, to enable a given discharge, on a given slope, to 

transport its maximum load.

Material transported.

Slope 

(per 

cent).

| 0.5 

] 1.0 

I 2.0

f 1.0 

\ 2.0

I 3.0

Ratio for discharge (ft.8/sec.) of 

0.25

1:4 

1:6 

1:10

1:2

0.50

1:6 

1:9 

1:16

1:25 

1:3

0.75

1:8 

1:12 

1:20

1:25 

1:3 

1:4

1.00

1:9 

1:15

1:3 

1:5

1:7

No way has been found to extend the quantitative results to rivers. It can hardly be 

questioned that the optimum ratio for rivers 

varies inversely with slope, discharge, and 

fineness of debris, but its absolute amount can 

not be inferred from the experimental results. 

River slopes are relatively very small and river 

discharges are relatively very large, and the 

two differences affect the ratio in opposite 

ways. To compute the joint result we should 

have definite and precise information as to the 

laws of dependence, but our actual knowledge 

is qualitative and vague.

In this connection it is of interest to record a 

single observation on river efficiency. Where 

Yuba river passes from .the Sierra Nevada to 

the broad Sacramento Valley its habit is rather 

abruptly changed. In the Narrows it is narrow and deep; a few miles downstream it has 

become wide and shallow. Its bed is of 

gravel, with slopes regulated by the river 

itself when in flood, and the same material 

composes the load it carries.

In the Narrows the form ratio during high 

flood is 0.06 and the slope is 0.10 per cent. 

Two miles downstream the form ratio is 0.008 

and the slope is 0.34 per cent. Thus the 

energy necessary to transport the load where 

the form ratio is 0.008 is more than three times 

that which suffices where the form ratio is
136 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

0.06; and it is evident that the larger ratio is 

much more efficient than the smaller. The 

data do not serve to define the optimum form 

ratio, but merely show that it is much greater 

than 0.008. In this instance the slopes and 

fineness are of the same order of magnitude as 

those realized in the laboratory, but the discharges are of a higher order.

When water without detrital load is conveyed by an open rectangular conduit, the 

form ratio of highest efficiency is that which 

yields the highest mean velocity. It is approximately 1:2. This corresponds to the 

maximum value of the optimum ratio for traction, and the correspondence might have been 

expected on theoretic grounds. The two factors which, in ultimate analysis, determine 

capacity for traction are velocity of current 

along the bed and width of bed. When discharge and slope are such as barely to afford 

competence with the most favorable form 

ratio, that ratio is one giving the highest 

velocity, namely, 1:2. The other factor, 

width of bed, is evidently favored by lower 

values of R; and therefore, as the conditions 

recede from the limit of competence, the optimum form ratio becomes smaller. This line 

of reasoning might, in fact, have been used to 

show a priori what has actuaUy been shown 

by the experiments that the value of p varies 

inversely with slope, discharge, and fineness.

SUMMARY.

Capacity for traction varies with the depth 

of the current, being approximately (though 

not precisely) proportional to a power of the 

depth. Capacity varies also with the width 

of the current, being approximately proportional to the width less a constant width. 

This constant width is equivalent to the product of the depth by a numerical constant. 

When the discharge is constant, any change of 

width causes a change of depth and also a

change & of form ratio, R - . A formula for the j w

variation of capacity in relation to form ratio, 

when the discharge is constant, is based on the 

above-mentioned properties and takes the 

form

C=12 (l-aR}Rm ... -........(54)

in which or is a numerical constant; or 

R m

in which p is the optimum form ratio, or the 

form ratio giving the highest capacity.

That capacity should have a maximum value 

corresponding to some particular value of form 

ratio is made to appear from theoretic considerations, and the fact of a maximum is 

shown by the experimental data. The same 

data show that the optimum form ratio has 

different values under different conditions, its 

values becoming smaller as slope, or discharge, 

or fineness increases.

The sensitiveness of capacity to the control 

of form ratio is indicated in the formulas by 

the exponent of R, and that also varies with 

conditions. It becomes smaller as slope, or 

discharge, or fineness increases.

It is believed that all the generalizations 

from the laboratory results may be applied to 

natural streams, but only in a qualitative way; 

the disparity of conditions is so great that the 

numerical results can not be thus applied.
CHAPTER V. RELATION OF CAPACITY TO DISCHARGE.

C

FORMULATION AND REDUCTION.

As a condition controlling the capacity of a 

stream for the traction of debris, discharge is 

the coordinate of slope. Each of the two factors is proportional to the potential energy of 

the stream, on which traction depends; and the 

control of each is exercised through the control 

of velocity. Their fundamental difference in 

relation to traction is connected with depth of 

current. When velocity is augmented by increase of slope, the depth is reduced; when it is 

augmented by increase of discharge, the depth 

is increased. Notwithstanding this difference, 

the relations of capacity to discharge parallel 

those of capacity to slope to a remarkable 

extent. Thanks to this parallelism, the discussions of the present chapter may be based in 

considerable part on those which have preceded.

The data for the comparison of capacity with

discharge are contained in Table 12. In each 

division of that table assigned to a grade of 

d6bris are a number of subdivisions pertaining 

severally to particular widths of channel. Each 

column of such a subdivision pertains to a particular discharge and contains a 'series of adjusted capacities corresponding to an orderly 

series of slopes. The values of capacity connected with the same slope and comprised in 

the same subdivision constitute a group illustrating the relation of capacity to discharge. 

Table 32 contains in its upper part a number of 

such groups, selected and arranged for the present purpose. So far as practicable they pertain to the same slope, but it was not possible to 

secure absolute uniformity in this respect and 

at the same time make the representation include data of all the grades of debris. For 

grades (A) to (E) the slope is 1.0 per cent; for 

grades (F) to (H), 1.2 per cent.

TABLE 32. Data on the relation of capacity to discharge, with readjusted values of capacity, Cr, and values of the index of

relative variation, i3 .

(Grade..-.- --..--.

I Width (

Data..........................

Adjusting equations ...........

Index of relative variation. ....

Probable error (per cent),. ....

er cent) . . 

feet).....

Q

0.093 

.182 

.363 

.545 

.734 

1.119

1 ' 

1 63

Q

0.093 

.182 

.363 

.545 

.734 

1.119

Q

0.093 

.182 

.363 

.545 

.734 

1.119

/ c

\ Cr

(A) 

1.0 

.66

C

[14.8] 

39.5

140

.028 

1.08 

290

Cr

15.1 

38.5 

89 

142

is

1.54 

1.28 

1.17 

1.14

(A) 

1.0 

1.00

c

37.5 

100

231

.048 

1.05 

370

Cr

37.5 

99 

167 

231

»a

1.43 

1.21 

1.15 

1.12

(A) 

1.0 

1.32

C

36.8 

104

250

.067 

1.09 

389

Cr

37.0 

103 

174 

250

*V

1.72 

1.33 

1.24 

1.20

(A) 

1.0 

1.96

C

96

240 

359

.105 

1.00 

367

Cr

96 

159 

234 

370

«3

i.4i

1.24 

1.17 

1.10

(B) 

1.0 

.23

C

7.8 

13.3

.002 

.79 

49.5

Cr

7.8 

13.3

t's

0.81

.80

(B) 

1.0 

.44

C

13.2

28.7

.017 

1.00 

170

Cr

13.2

28.7

it

1.21 

1.10

(B) 

1.0 

.66

C

10.8 

33.5 

81 

120

.033 

1.13 

270

Cr

11.1 

31.6 

77 

127

»8

1.75 

1.38 

1.24 

1.20

4.8 

2.4

(B) 

1.0 

1.00

c

30.1 

85 

143 

199 .

.057 

1.12 

313

Cr

30.3 

83 

140 

201

ia

1.63 

1.33 

1.25 

1.21

1.4 

0.7

(B) 

1.0 

1.32

C

29.3 

79 

140 

204

.080 

1.06 

313

Cr

28.5 

83 

140 

201

i*

 1.89 

1.36 

1.24 

1.19

2.4

1.2

(B) 

1.0 

1.96

C

67.6 

120 

190 

313

.125 

1.08 

312

Cr

67.6 

123 

185 

311

13

1.64 

1.40 

1.30 

1.21

1.4 

0.7

137
138

TABLE 32. Data on the relation of capacity to discharge, with readjusted values of capacity, Cr. and values of the index of

relative variation, i3 Continued.

Conditions ............ .-{Slope (per cent) . . 

iWidth(feet).....

Data..........................

Index of relative variation. ....

Probable error (per cent) ......

Q

0.093 

.182 

.363 

.545 

.734 

1.119

1 " \ ° I 63

Q

0.093 

.182 

.363 

.545 

.734 

1.119

Q

0.093 

.182 

.363 

.545 

.734 

1.119

/ C \ Cr

Conditions. . ......... . .-{Slope (per cent). . 

Iwidth(feet).....

Data..........................

Index of relative variation. ....

Probable error (per cent) ......

Q

0.093 

.182 

.363 

.545 

.734 

1.119

I «0

[ 63

Q

0.093 

.182 

.363 

.545 

.734 

1.119

Q

0.093 

.182 

.363 

.545 

.734 

1.119

( ° \ Cr

(C) 

1.0

.44

C

10.9 

24.7

.020 

1.02 

157

Cr

10.9 

24.7

is

1.30 

1.14

(E) 

1.0 

1.00

C

14.8 

33.8

72 

138

.188

Cr

is

(C) 

1.0 

.66

C

13.8 

32.1 

73 

112 

152

.041 

.94 

213

Cr

13.2 

33.9 

74 

112 

151

- »3

1.68 

1.21 

1.06 

1.02 

1.00

2.2 

1.0

(E) 

1.0 

1.32

C

36.3

73 

123

.263

Cr

is

(C) 

1.0 

1.00

C

31.2

85 

140 

180 

276

.071 

1.00 

285

Cr

33.0

84 

135 

187 

299

' it

1.64 

1.24 

1.15 

1.11 

1.07

3.6 

1.6

(F) 

1.2 

.66

C

9.3 

29.9

55.3

.139

Cr

is

(C) 

1.0 

1.32

C

21.4 

79 

129 

187

.099 

1.05 

305

Cr

22.0

75 

130 

190

it

2.30 

1.44 

1.28 

1.21

2.7 

1.3

(F) 

1.2 

1.00

C

4.2 

22.8

63 

112

.242

Cr

is

(C) 

1.0 

1.96

C

60 

111 

190 

343

.156 

1.14 

354

Cr

59 

120 

190 

340

*i

2.00 

1.60 

1.45 

1.32

2.9 

1.5

(F) 

1.2 

1.32

C

21.4

61.6 

101

.339

Cr

is

(D) 

1.0 

.66

C

9.1

29.8

101

.059 

.90 

195

Cr

9.2 

29.6 

67 

102

it

2.46 

1.33 

1.07 

1.01

(G) 

1.2 

.66

C

14.8

48.3 

75

.199 

94

.83

Cr

15.1

46 

76

is

2.04

1.29 

1.22

(D) 

1.0 

1.00

C

24.0 

73 

108 

153

.102

.88 

224

Cr

24.3 

69 

110 

151

*3

2.00 

1.22 

1.08 

1.02

2.4 

1.2

(CO 

1.2 

1.00

C

9.7

44.0 

92

.345

Cr

is

(D) 

1.0 

1.32

C

59.2

131

.143 

.81 

201

Cr

59.2 

97 

131

it

1.33 

1.10 

1.01

(G) 

1.2 

1.32

C

33.8 

81

.433

Cr

is

(E) 

1.0 

.66

C

24.8

40

.108

Cr

is

(H) 

1.2 

.66

C .

3.1

24.6 

43.5

.233 

1.24 

.54

Cr

4.2

22.5 

46

is

3.46

1.82 

1.57
RELATION OF CAPACITY TO DISCHARGE. 139

The same data are plotted in figure 47, where 

horizontal distances represent logarithms of discharge and vertical distances logarithms of capacity. For each of the above-mentioned 

"groups" the plotted points are connected by a 

series of straight lines, and each of the broken

FIGURE 47. Logarithmic plots of the relation of capacity to discharge. 

The horizontal scale is that of log Q, the vertical of log C. The zeros 

of log C for the different plots are not the same.

lines thus produced is the rough logarithmic 

graph of an equation C=f(Q). The graphs 

are arranged according to grades of debris, and 

secondarily according to widths of channel, 

their order from left to right corresponding to 

the sequence from narrower to broader channels. 

In effecting this arrangement graphs, were

moved bodily up or down but not to the right or 

left.

It appears by inspection that the graphs 

bend toward the right as they ascend. To this 

rule there are a few exceptions, but the only 

strongly marked exceptions are connected with 

grade (E), the data for which have previously 

been recognized as anomalous. As the inclination of each line indicates the sensitiveness of 

capacity to the control of discharge, the general 

bending to the right, or the reduction of inclination in passing from lower to higher discharges 

shows that sensitiveness diminishes as discharge increases. This feature is similar to one 

observed in studying the relation of capacity to 

slope, and as that feature was found to be connected with competence, the resemblance leads 

at once to' the suggestion that here also is a connection with competence.

If a very small discharge be made to flow 

over a sloping bed of debris and the discharge 

be gradually increased, transportation of debris 

will commence when the competent discharge 

is reached and will increase with further increase of discKarge. It is a plausible hypothesis that the capacity is more simply related 

to the excess of discharge above the competent 

quantity than to the total discharge. Following the procedure in the case of capacity and 

slope we may assume that capacity is proportional to a power of the excess of discharge 

above a constant discharge, the constant discharge being closely related to competent discharge. In the following formula, constructed 

on the plan of equation (10),

<7=63 (Q-*)0 --....... ..(64)

K is a constant discharge and &3 is a constant 

numerically equal to the value of capacity 

when the discharge equals K + l, although it is 

not strictly a capacity. The dimensions of 

capacity are M+1 T~l, of discharge L+s T~\ and 

of (Q-K~)° L+3° T~°; and these values give to 

&3 the dimensions L~*° M+1 T°~\

As a preliminary to the adjustment of the 

observational data of Table 32 by this formula, 

values of K were graphically computed by the 

method previously employed in connection 

with a. The computations were applied to all 

groups of values of capacity in the table, 

except such as comprise less than three capacities and except also the aberrant data of grade 

(E). In Table 33 the results are arranged
140 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

with reference to grade of de'bris and channel 

width.

TABLE 33. Values of the discharge constant, K, computed 

from the discharges and capacities of Table 32.

Grade.

SB 

&(F 

(G 

H I

Value of * when width of channel 

(in feet) is 

0.66

0.056 

.025 

.060 

.149 

.280 

.326

1.00

0.070 

.095 

.100 

.130 

.130 

.160

1.32

0.075 

.035 

.030

.180

1.96

0.215 

.000 

.060

The tabulated values of K show great irregularity. It is not difficult, however, to recognize a tendency to grow larger in passing from 

finer de'bris to coarser; and there is also, 

though it is ill defined, a tendency to increase 

with increasing width. The cause of the firstmentioned tendency is readily understood, 

because relatively coarse de'bris requires a 

relatively swift current to move it; and the 

reality of the second also finds .support when 

the conditions affecting competent discharge 

are considered.

Postulating, initially, a channel of some 

particular width, containing a stream of which 

the discharge is barely competent, let us assume 

that the width is increased. In spreading to 

the new width the stream loses depth and 

velocity, and its velocity is no longer competent. That the velocity may again become 

competent the discharge must be increased. 

Thus it is in general true that competent discharge increases with increase of width. In 

the cross section of a broad laboratory current, 

figure 48, a medial portion, AA, is unaffected

FIGURE 48. Ideal cross section of a stream in the experiment trough, 

illustrating the relation of competent discharge to width.

by side-wall resistance and has competent 

velocity. Two lateral portions, AB, have less 

than competent velocity. Let us imagine 

these lateral portions replaced by narrower 

divisions, AD, in which velocities are the same 

as in A A. The effective width for the maintenance of competent velocity is then DD =

w 2 ED. If now width and discharge be 

increased or diminished, with maintenance of 

competent velocity, the quantity 2 BD is 

unaffected, so that it may be regarded as a 

constant. Velocities being, by hypothesis, 

uniform through the whole space DD, the discharge is proportional to the width of that 

space.

Qc oc w & constant ..----.. (65)

This result assumes a channel so wide that a 

medial portion is unaffected by side-wall resistance. It may not be true for narrower channels. If it were to be refined, the constant 

would be found to be a function of depth (compare p. 129); but in applying its principle to 

the values of K no allowance was made for 

that factor.

On the theory that K is closely related to 

competent discharge and has similar properties, 

and with the assumption that (65) may be 

applied to narrow channels as well as broad, the 

principle of (65) was used in adjusting the 

tabulated values of K in relation to width of 

channel. The formula assumed was

K <x 0.2

The value of the constant was arbitrarily 

fixed, after several trials, no criterion for selection being discovered except the harmony of 

results.

With the aid of this formula it was possible 

to combine the data of Table 33 in such way as 

to afford a better view of the relation of K to 

grade, or fineness ; and this was done. By multiplication or division, each value was reduced 

to its equivalent for a channel width of 1 foot, 

and means were taken. These means appear 

in the second line of Table 34. They were compared with the mean diameters of particles for 

the several grades, the comparison being made 

on logarithmic section paper. In figure 49, 

showing this plot, the numbers indicate 

weights. While the plotted points do not fall 

well into line, they leave no question that the 

value of K rises as the debris becomes coarser. 

On the assumption that the function is a power 

function, a straight line was drawn to represent 

it; and by this line the mean values of K were 

adjusted. Values for the other trough widths 

were then computed. These appear in Table 32.
RELATION OP CAPACITY TO DISCHARGE. 141 

TABLE 34. Adjustment of system of values of K for a channel width of 1 foot and slopes of 1.0 and 1.2 per cent.

Mean diameter of particles (foot). ............................

Mean « computed from Table 33 ..............................

(A)

0-00100

.074

.048

(B)

0.00123

.078

.057

(C)

0.00166

.047

.071

(D)

0.00258

.117

.102

(E)

0.00561

.188

(F)

0. 01040

.188

.242

(G)

0. 01620

.367

.345

(H)

0.02300

.925

.406

In making these adjustments due account 

was taken of the fact that not all the data used 

pertain to the same slope. Allowance for slope 

difference was made in plotting the values of 

K to adjust for fineness, and again in applying 

the results of that adjustment. The adjustments for slope are based on the following 

consideration.

By use of the data in Table 12, values of « 

pertaining to the same grade and width but 

differing as to slope were compared. In one 

case K appeared to increase with increase of 

slope, but in most cases it appeared to decrease.

FIGURE 49. Logarithmic plot of «=/(!>).

Little weight, however, was given to such 

determinations, because it was evident that 

they were controlled by the values of a, which 

had been somewhat arbitrarily assigned in the 

first adjustment of the observations. A more 

satisfactory indication was obtained by reasoning based on the relative sensitiveness of 

capacity to the several controls of slope and 

discharge. The index of relative variation of 

capacity with respect to slope being il} we may 

designate the corresponding index with respect 

to discharge by i3 . As capacity varies simultaneously with S*1 and Q*3 it is evident that for 

any particular value of capacity Sil and Q*3

vary inversely, 

vields

The proportion Q'3 oc

««A

8'

From a discussion of the relations of the two 

indexes, to be found on a later page of the 

present chapter, it appears that the average 

ratio of it to i3 is 1.36 but that the ratio varies 

somewhat with conditions. For the condition

(7=0 it is assumed to be 1.3; or « oc ^-^

While considerable uncertainty attaches to 

the numerical relations, there can be no question that K varies directly with width and 

inversely with fineness and slope. As any 

change in width affects the form ratio in the 

opposite sense, we may write

F, £)_.__.......(66)

A system of values of AC having been thus 

arranged, logarithmic plots were made of the 

capacities of Table 32, in relation to Q K, and 

straight lines were drawn representing equations of the type of (64). In cases where the 

plotted points did not fall well in line, recourse 

was had to weights based on the probable 

errors recorded in Table 12. The lines thus 

drawn gave, by their inclinations and intersections, the values of the other parameters, o and 

&3, as well as the adjusted values of capacity, 

Cr, all of which are presented in Table 32. The 

plots for data of grades (E) and (F) and for 

two of the three groups under grade (G) 

exhibited so great irregularities that it was 

decided not to use them.

The similarity of equation (64) to equation 

(10) makes it unnecessary to repeat the reasoning by which (38) was educed, and the equation 

for the index of relative variation for capacity 

in relation to discharge may be written directly:

oQ (67)

By its use the values of ia in Table 32 were 

computed.
142. TRANSPORTATION OF DEBRIS BY RUNNING WATER.

MEASURES OF PRECISION AND THEIR 

INTERPRETATION.

In 16 of the columns of Table 32 the values 

of capacity before and after adjustment constitute data suitable for the computation of 

probable errors. As the number of residuals 

is in each case small, the particular values of 

probable errors (recorded at the bottom of the 

table) are not themselves of high precision; but 

the averages have greater claim to attention. 

It should be added that the residuals were 

treated as of equal weight, despite the fact 

that in the adjustment which determined them 

the relative weights of the observational data 

were recognized.

The probable errors of the upper line, marked 

" 0," pertain to the adjusted values of capacity 

taken from Table 12, which are here treated as 

observations. The probable errors of the lower 

line, marked " CT," pertain to the groups of 

readjusted values of capacity.

The average probable error (or, strictly, the 

mean of the nine tabulated errors) of the readjusted capacities is ±1.3 per cent. The corresponding average for the capacities before 

readjustment is ±2.64 per cent.

It. is of interest to compare the last figure 

with the previously computed average probable 

error of the adjusted values of capacity in 

Table 12, namely, ±2.50 per cent. The earlier 

estimate applies to 66 series of values, the later 

to 36 values taken from 36 of the series. The 

36 values may be regarded as properly representative of the series from which they come. 

Of the unrepresented series, a portion escaped 

because the values they furnished to Table 32 

fell in groups of less than four, and such a 

group did not afford suitable data for computation of probable error. The others were omitted 

because the groups they constituted involved 

incongruities so great that they were rejected in 

the readjustment. The omission of incongruous groups evidently had the effect of lowering 

the estimate of average probable error, and to 

give validity to the comparison due allowance 

should be made for that effect. The discarded 

values were accordingly treated so far as necessary to compute their probable errors, and it was 

found that by the inclusion of these the estimate 

of average probable error was raised from 2.64 to 

3.2 per cent. The revised .estimate is believed

to be properly comparable with the earlier estimate of 2.50 per cent. It will be recalled that 

the original observations were characterized by 

accidental errors, ascribed chiefly to rhythm, and 

by systematic errors, ascribed chiefly to methods 

of observation. The nature of the first adjustment was such that its computed probable 

errors were little affected by the errors of the 

second class. The adjustment was conditioned by a formula involving the assumption 

that capacity varies as a power of (8 a'), and 

also by various assumptions involved in the 

arrangement of a system of values for a. Whatever errors were introduced in connection with 

these assumptions tended to increase the estimates of probable error; but they may also be 

supposed to have aggravated somewhat the 

errors of the class not covered by the computations of probable error.

The errors falling outside the estimates of 

probable error were largely of such nature as to 

affect an observational series in its entirety, 

and it was expected that they would be revealed in the failure of groups of quantities 

taken from different series to exhibit an orderly 

sequence. Abundant evidence of their existence has been encountered in various discussions, including the control by conditions of the 

sensitiveness of capacity to slope, of the sensitiveness of capacity to form ratio, of the 

value of the optimum form ratio, and of the 

value of the constant K; but the present discussion is the only one affording an estimate of 

their magnitude.

Assuming that the estimate of 2.5 per cent 

represents the influence of a restricted class of 

errors, and that the estimate of 3.2 per cent 

represents the joint influence of that class and a 

second class, the independent influence of the 

second class is represented by

V3.22 -2.52 = ± 2,0 per cent

The indication is that the two classes of errors 

are of nearly equal importance. 1

These results are qualified by the fact that 

the estimate of 3.2 per cent includes not only 

the errors of the adjusted values of capacity in

1 The considerations making it probable that ±2.5 per cent is an overestimate for the average error of the first class (p. 74) do not apply to the 

estimate of ±3.2 for the combined"error. If the average error for the first 

class is as low as ±2.0 per cent, the computed average for the second 

class becomes ±2.5 per cent.
RELATION OF CAPACITY TO DISCHARGE. 143

Table 12, but also whatever errors affect the 

method of adjustment in connection with the 

discussion of the control of capacity by discharge. That method includes the assumption that capacity varies as a power of (Q K) 

and also certain assumptions as to the control 

of K by various conditions. If the errors involved in those assumptions could be eliminated or discriminated, the general estimate 

for the values in Table 12 would be somewhat 

reduced.

As the chief result of this discussion, the 

general precision of the main body of material 

contributed by the Berkeley experiments is 

characterized by an average probable error 

slightly in excess of 3 per cent.

This estimate applies specifically to the adjusted capacities of Table 12 as those capacities 

are related to slope and discharge, and it can 

not be extended to derivatives of those capacities without qualification. It is believed that 

the precision of the readjusted capacities of 

Table 32 is higher, and also that of the values 

of /j in Table 23, but that the values of i1} jlf 

and i,, in Tables 15, 16, and 32, rank lower.

In the adjustment of the observations on 

capacity in relation to slope, and also in the 

adjustment of observations on depth and slope, 

many cases were treated in which the observations were either not sufficiently numerous or 

not sufficiently harmonious to afford good control of the parameters of the adjusting equations. In such cases the parameters were estimated in groups, with orderly sequences of 

values. A similar method was employed also 

in readjustments in relation to discharge. 

While this procedure appeared, and still appears, to be the best practicable, it can hardly 

fail to introduce a certain amount of error. 

The terms of the adjusted sequences are inevitably associated with different form ratios; 

and the laws connecting form ratio with 

capacity are so different from the other laws 

of the system as to determine sequences less 

simple than those actually used. The ideal 

adjustment would take simultaneous cognizance of the complicated interrelations of 

capacity, slope, discharge, and form ratio; 

but such comprehensive treatment can not 

be attempted -with profit until we have a 

better theoretic knowledge of the physical 

reactions.

CONTROL OF RELATIVE VARIATION BY 

CONDITIONS.

The sensitiveness of capacity to changes of 

discharge is indicated by i,, the index of relative variation. Inspection of the values of that 

index recorded in Table 32 shows that they 

vary inversely with discharge and directly 

with width, and both these tendencies are also 

to be inferred from the lines of figure 47.

The table further indicates that the rate of 

change in the index in response to change of 

discharge is greater for small discharges than 

for large. This feature might also have been 

inferred from the plotted lines, but its systematic expression in the table is a product of the

formula. i. = /-> , bv which the values were ; 3 Q K' J

computed.

The relation of the index to width of channel 

is not similarly dominated by the formula, 

although somewhat influenced by the assignment of values of K. The variation of the 

index with width is in general more pronounced 

for low discharges than for high, but in the 

data for high discharges occur two exceptions 

to the general law. These exceptions are 

ascribed to irregularities in the data. As the 

width for any particular discharge varies inversely with the form ratio, it follows that the 

index is a decreasing function of form ratio.

The relation of the index to slope is not shown 

by the table. It was the subject of a special 

inquiry, including 32 comparisons. In each 

comparison a value of the index for a particular 

slope was contrasted with the value for a slope 

twice as great, the other conditions being the 

same. In 25 instances the greater index was 

associated with the smaller slope; in 7 instances 

with the larger. The mean of the indexes 

computed for smaller slopes was 1.51; the mean 

for larger slopes 1.17. The general law appears 

to be that the index varies inversely with 

slope. The seven instances of opposite tenor 

are all associated with large discharges; and 

their occurrence is ascribed to a systematic 

error connected with the assignment of values 

to K.

Table 35 compares values of the index with 

fineness. Despite irregularities, it is evident 

that the values tend to increase in passing from 

finer -to coarser grades that is, their variation 

in respect to fineness is inverse.
144

TABLE 35. Values of the exponent i3 arranged to show variation in relation to fineness of debris.

Grade.

(A) 

(B) 

(C) 

(D) 

(G

(H)

w=0.66

1.17 

1.24 

1.06 

1.05 

2.04

3.46

i

Q=0.363

W=1.00

1.21 

1.33

1.24 

1.25

Values of is

w=1.32

1.33 

1.36 

1.44 

1.33

Q=C

w=0. 66

1.28 

1.38 

1.21 

1.30

.182

tt>=1.00

1.43 

1.63 

1.64 

2.02

In summary, the sensitiveness of capacity to 

variation of discharge is greater as slope, discharge, form ratio, and fineness are less.

(68)

DUTY AND EFFICIENCY.

The variation of capacity with discharge is 

indicated in general terms by an equation of 

the type of (33):

<7=^_ ____________ (69)

Duty being the quotient of capacity by discharge, this gives

and, as efficiency is the quotient of capacity by 

discharge and slope,

E=% Q^............(71)

That is, the index of relative variation for both 

duty and efficiency, in relation to discharge, is 

less by unity than the corresponding index for 

capacity. Therefore the values of the index 

in Table 32 need only to be reduced by unity 

to apply to duty and efficiency.

Under ordinary conditions the index for duty 

and efficiency falls between unity and zero; or, 

in other words, duty and efficiency increase 

with increase of discharge, but their increase is 

less rapid than that of discharge. Exceptionally the increase is much more rapid, the exceptions being associated with discharges little 

above the limit of competence. On the other 

hand, there appear to be conditions under which

the index falls below zero, so that duty and 

efficiency diminish with increase of discharge. 

The diminution indicated by the figures in the 

column (of Table 32) for grade (D) and width 

0.66 foot is only of the order of magnitude of 

the probable error; but a pronounced diminution would be inferred from the values of the 

index for grade (B) and width 0.23 foot. As 

the results from the last-mentioned group of 

observations stand by themselves in various 

respects, some reservation is felt in regard to 

them, and there is at least room for doubt 

whether the diminution is actually demonstrated.

With respect to all conditions the variations 

of the index for duty and efficiency follow the 

same laws as the index for capacity; but, as a 

consequence of the uniform reduction by unity, 

the rates of variation are higher. If, for example, in passing from a smaller to a larger discharge, the index for capacity falls from 1.40 to 

1.20, a reduction of one-seventh, the index for 

efficiency falls from 0.40 to 0.20, a reduction 

of one-half.

Lines of reasoning strictly parallel to those 

employed in the last section of Chapter III 

yield the following conclusions:

The synthetic index of relative variation for 

the duty of water in relation to discharge is 

less by unity than the corresponding synthetic 

index for capacity in relation to discharge.

The synthetic index of relative variation for 

efficiency in relation to discharge is less by 

unity than the corresponding synthetic index 

for capacity in relation to discharge.

If the duty of water, or if efficiency, be assumed to vary as some power of Q AC, the exponent of that power (expressing the instanta0 K

neous rate of variation) equals o -~ As discharge increases from K toward infinity, the 

exponent diminishes from o toward o 1 .

If the relation of efficiency to discharge ( and 

similarly for the relation of duty to discharge) 

be expressed by

the value of on is always less than the corresponding value of o, the difference approaching 

but not exceeding unity. The value of /q is
EELATION OF CAPACITY TO DISCHARGE. 145

always greater than the corresponding value 

of K, usually much greater. It was found by 

trial that, within the range of conditions realized in the laboratory, the difference between 

values of efficiency computed directly by means 

of (71a) and values computed indirectly by 

means of (64) is not large, its order of magnitude 

being that of the probable errors.

The control of duty and efficiency by discharge is further considered in the following 

section.

COMPARISON OF THE CONTROLS OF DISCHARGE AND SLOPE.

CONTROLS OF CAPACITY.

We are now in position to compare the influences exerted by slope and discharge, severally, on capacity. The general fact brought out 

by the comparison is that capacity is more sensitive to changes of slope than to changes of 

discharge, but the difference in sensitiveness is 

not the same for all conditions.

TABLE 36. Comparison of the index of relative variation,^, for capacity and slope, with the index, i3 , for capacity and discharge. 

[For grades (A) to (D) the data are for S-1.0; for grades (G) and (H) for 5=1.2.]

(A)

(B)

/p\

(D)

(0)

(H)

0.182

.363

.545

.734

1.119

.093

.182

.363

.545

.734

1.119

.093

.182

.363

.545

.734

1.119

.093

.182

.363

.545

.734

.363

.734

1.119

.363 

.734

1.119

w=-

n

2.31

1.81

2.39

1.88

0.44

»3

1.21

1.10

1.30

1.14

w=

ii

1.99

1.48

2.34

1.82

1.58

1.49

1.88

1.73

1.59

1.58

1.57

2.22

1.80

1.64

2.98

2.42

2.29

6.03

3.23

2.78

0.66

*3

1.28

1.14

1.75

1.38

1.24

1.20

1.68

1.21

1.06

1.02

1.00

2.46

2.33

1.01

2.04

1.29

1.22

3.46

1.82

1.57

w=

ii

2.06

1.83

1.79

1.92

1.67

1.62

1.61

1.87

1.59

1.47

1.48

2.61

1.69

1.87

1.65

1.00

h

1.43

1.21

1.12

1.63

1.33

1.25

1.21

1.64

1.24

1.15

1.11

1.07

2.00

1.22

1.08

1.02

W=

ii

2.17

1.70

1.71

1.94

1.75

1.73

1.71

2.37

1.67

1.55

1.47

2.11

1.85

1.32

h

1.72

1.33

1.20

1.89

1.36

1.24

1.19

2.30

1.44

1.28

1.21

1.33

1.01

w=

ii

1.90

1.55

1.31

2.01

1.87

1.66

1.60

2.14

1.87

1.62

1.85

1.96

»3

1.41

1.17

1.10

1.64

1.40

1.30

1 91

2.00

1.60

1 A<5

1.32

In Table 36 values of the index of relative 

variation are brought together from Tables 15 

and 32. The selection includes all such as correspond in respect to d6bris, trough width, 

slope, and discharge, with the exception of 

those of trough width 0.23 foot, which appear 

to be anomalous. There are 64 pairs of values.

Of the 64 comparisons, 62 show capacity as 

more sensitive to slope, 2 as more sensitive to 

discharge. The two exceptional cases are from 

experiments with debris of grade (D) and with 

channel width 0.66 foot; and the data from 

those experiments were reexamined in search 

for an explanation of what seems an anomaly. 

No explanation was found, and, as the observations are supported by the estimates of pre20021° No. 86 14 10

cision, it remains probable that there are real 

exceptions to the general rule.

The means of the 64 values of ^ and i3 are, 

severally, 1.93 and 1.42; and the ratio of the 

first to the second is 1.36. On the average, 

the sensitiveness of capacity to slope is onethird greater than the sensitiveness of capacity 

to discharge.

n

To ascertain the variation of the ratio -^

^

with discharge, the values in Table 36 were 

specially grouped for the taking of partial 

averages. The first group gave comparative 

ratios for discharges of 0.093 and 0.182 ft. 3/sec., 

by means j*£Jour sets of index values, each set 

agreeing as to all conditions other than dis-
146 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

charge. The second group gave comparative 

ratios for discharges of 0.182, 0.363, and 0.734 

ft. 3/sec., by means of five sets of index values; 

and two other groups made other comparisons, 

as shown in Table 37. The upper division of 

the table gives mean values of it and is ; the 

lower division, their ratios. It will be observed

that within each group the values of the 

indexes decrease as discharge increases, while 

the values of the ratio, as a rule, increase with 

the increase of discharge. To the first rule 

there are no exceptions; the exceptions to the 

second are not so important as to leave the 

principle in doubt.

TABLE 37. Variations of the indexes it and i3 , and their ratio, in relation to discharge.

Group.

1 

2 

3 

4

1 

2 

3

4

Sets.

4 

5 

8 

7

Q= 0.093

ii

2.23

ia

1.48

ti/is

1.50

Q= 0.182

«i

1.81 

2.02

13

1.21 

1.74

Jl/«3

1.50 

1.16

Q=0.363

ii

1.73 

1.76 

2.67

«3

1.30 

1.41 

1.97

ti/is

1.33 

1.25 

1.35

Q= 0.545

h

1.69

is

1.26

»1/J3

1.35

Q=0. 734

ii

1.65 

1.59 

1.97

is

1.16 

1.19 

1.32

ii/»3

1.42 

1.34 

1.49

Q= 1.119

ti

1.85

is

1.22

ii/ts

1.52

The variations of the indexes with discharge as a factor controlling capacity, is more prohave already been illustrated in another way nounced for large discharges than for small, 

(pp. 107, 143). The new fact brought out is under like conditions of slope, width, and 

that the superiority of slope over discharge, fineness.

TABLE 38. Variations of the indexes it one? i3 , and their ratio, in relation to width of channel.

Group.

1 

2 

3

1 

2 

3

Sets.

4 

8 

8

w=0.44

ii

2.10

is

1.19

»l/«3

1.77

w=0.66

ii

1.94 

1.67

is

1.50 

1.17

ii/is

1.29 

1.40

«J=1.00

ii

1.71 

1.63

is

1.35 

1.20

ii/is

1.27 

1.36

w=1.32

ii

1.83 

1.66

is

1.55 

1.28

ii/t's

1.18 

1.30

w=1.96

fi

1.83

fa

1.50

ii/fs

1.22

A different grouping of the index values, 

but similar in principle, gave the means and 

ratios of Table 38, which is related to channel 

width just as Table 37 is related to discharge. 

With a single exception, the mean values of 

indexes increase with width; thus illustrating 

general facts previously noted on pages 104 

and 143. Without exception, the ratios of 

ij_ to i, decrease with increase of width. The 

new fact brought out is that the superiority of 

slope over discharge, as a factor controlling 

capacity, is. more pronounced for narrow channels than for wide, under like conditions of 

slope, discharge, and fineness.

As form ratio varies inversely with width, it 

follows. that the superiority of slope is more

pronounced, under like conditions, when form 

ratio is large than when it is small.

A third grouping of the index values, making 

a similar comparison of their averages and 

ratios with fineness of de"bris, is reported in 

Table 39 (p. 147). The mean values of indexes 

increase on the whole in passing from finer to 

coarse grades, but there is much irregularity. 

The same irregularity was encountered when 

these relations were examined in other connections. (See pp. 108 and 143.) The ratios 

decrease on the whole from finer to coarser, 

and there is but one discordance among the 

sequences.

To compare the variations of the indexes 

with changes in slope, the 32 pairs of values of
RELATION OF CAPACITY TO DISCHARGE. 147

i, mentioned on page 143 were contrasted with of the three groups indicates that the ratio

corresponding values of iv The results are of the indexes increases with increase of

summarized in Table 40, below, in which each slope.

TABLE 39. Variations of the indexes i{ and i3 , and their ratio, in relation to fineness of debris.

1

2

3

1

2

3

11

8

3

(^

t'i

1.77

h

1.

0"

is

1.28

h

38 '

(1

!'l

1.74

1.79

ii

1.

1.

3)

13

1.39

1.38

k

25

30

(<

t'i

1.76

1.65

ill

1.

1.

2)

>3

1.45

1.32

«3

21

26

(I

' ii

1.87

ill

1.

5)

ii

1.54

?3

21

«

ii

2.70

hi

1.

3)

«3 '

1.C6

»3

09

(I

ii

4.63

til

1.

I)

»3

2.64

it

76

TABLE 40. Variations of the indexes ^ and %, and their ratio, in relation to slope.

Group.

1 

2 

3

1 

2 

3

Sets.

18 

10

4

5=0.5

ii

1.90

»3 

1.56

ii/is

1.22

S=1.0

ii

1.68 

1.87

is

1.23 

1.45

ii/is

1.35 

1.29

S=1.2

ii

2 CO 

 O-u

«3

1.47

ii/is

1.91

S=2.0

)'i

1.66

h

1.12

tl/<3

1.49

S=2.4

ii

2.13

»'3

1.08

iilk

1.98

To sum the results of the preceding paragraphs: The sensitiveness of capacity for traction to changes of slope, as measured by the 

exponent il} is in general greater than its sensitiveness to changes of discharge, measured 

by v The superiority of the control by slope 

persists through nearly (or perhaps quite) the 

entire range of conditions realized in the 

laboratory. If the superiority is measured by

ft

the ratio ?*, its average value (based on 64 com- l? 

parisons) is 1.36, and it increases with increase

of slope, discharge, form ratio, and fineness of 

d6bris.

(72)

Another mode of comparing the controls by 

slope and discharge is by means of the synthetic index of relative variation (p. 99), and 

in some respects this mode is more satisfactory 

than the one given above. The synthetic index, 

73 , of the relative variation of capacity and discharge was computed for 21 cases, in each of 

which the greater discharge was approximately

double the lesser. The synthetic index Ilf of 

the relative variation of capacity and slope, 

was computed for 21 pairs of cases, the greater 

slope being double the lesser. Each value of 73 

was joined to two discharges and a slope and 

associated with a pair of values of Iv Each of 

the two values of 1^ was joined to one discharge 

and two slopes, the slopes being so chosen, 

whenever possible, that their mean coincided 

with the slope of the associated 73 . The mean 

of the 42 values of 7X is 1.92; that of 21 values 

of 73 is 1.42; and the ratio of these means is 1.35. 

The result is practically identical with that 

obtained by the discussion of values of it and i3 .

CONTROLS OF DUTY.

The index of relative variation of the duty 

of water in relation to slope (p. 121) is i1} the 

same as the index for capacity and slope. 

The corresponding index for the duty of water 

in relation to discharge (p. 144) is ia 1. The

n

ratio of these indexes, - ^-r, is evidently greater *s~ 

n

than the ratio -r, which has just been consid-'
148

ered that is, the superiority of the control by 

slopes, as compared with the control by discharge, is more strongly marked in the case of 

duty than in the case of capacity.

For general averages, -^ = ^=1.36, arid 

L 1.93

OA2 = 4.3.

In Table 41 the values of . * correspond to

those of -^ in Tables 37 to 40 that is, they

are based on the partial means of those tables 

and are arranged under the same groups, the 

purpose being to show how the dominance of 

control by slope, as expressed by a ratio, varies 

with certain conditions.

o

TABLE 41. Variations of the ratio . ' in relation to discharge, width of channel, fineness of debris, and slope.

Q= 0. 093

.182 

.363 

.545 

.734 

1.119

w= 0. 44 

.66 

1.00 

1.32 

1.96

Grade (A) 

(B) 

(C) 

(D 

(G) 

(H)

S= 0.5 

1.0 

1.2 

2.0 

2.4

Values of T ^ for group 

1

4.6 

8.6

11.2 

3.9

6.3 

4.5 

3.9

3.4 

7.3

2

2.7 

5.8

10.3

9.8 

4.9 

3.3

8.0 

5.1 

3.5

4.2

13.8

3

 4.3 

6.5

8.4

8.1 

5.9 

3.7

4.9

3.1

6.0

26.6

4

2.8

6.1 

8.4

In comparing this table with the tables of

t\

-1 , the most conspicuous feature noted is that the

variation of with all conditions is much

more pronounced than the variation of A

^3

Associated with this is the fact that the exceptions or apparent reversals observed in Tables 

37 and 39 are not repeated in Table 41.

In verbal generalization of the tabulated 

results it is to be borne in mind (1) that the 

alphabetic order in which the grades are 

arranged is the order from fine to coarse, and 

(2) that variation with respect to form ratio

is the inverse of variation with respect to 

width of channel. The general fact is that 

the dominance of control by slope, as compared with control by discharge a dominance 

always pronounced is notably increased by 

increase of slope, discharge, fineness, or form 

ratio.

(73)

CONTROLS OF EFFICIENCY.

The index of relative variation of efficiency 

in relation to slope (p. 144) is ^ 1; and the 

corresponding index for efficiency in relation to 

discharge is i3 l. The ratio of these indexes

^i is greater than ^, the corresponding ratio

in the case of capacity, with exception of those 

doubtful cases in which i1 <i3 . That is, the 

superiority of the control by slope, as compared to the control by discharge, is more 

strongly marked in the case of efficiency than 

in the case of capacity. It is also true that

n __ 1 n

T1 -< . 1 or that the superiority of control

t/q JL b/Q JL

by slope is less strongly marked for efficiency 

than for duty. For the general averages 

it = 1.93 and i, = 1.42 the computed values of 

the ratios expressing superiority of control by 

slope are, as to capacity 1.36, as to efficiency 

2.21, as to duty 4.3.

i 1

TABLE 42. Variations of the ratio *__, in relation to discharge, width of channel, fineness of debris, and slope.

Q= 0.093

.182

.363 

.545 

.734 

1.119

w= 0. 44 

.66 

1.00 

1.32 

1.96

Grade (A) 

(B) 

(C) 

(D) 

(G) 

(H)

S= .05 

1.0 

1.2 

2.0 

2.4

Values of r = for group 

1

2.6 

3.9

5.5 

1.9

2.7 

1.9 

1.7

1.6 

2.9

2

1.4 

2.4

4.1

3.9 

2.0

1.5

3.6 

2.0 

1.6

1.9

5.5

3

1.8 

2.7

3.1

3.1 

2.4 

1.7

3.0

2.4

1 Q

14.1

4

1.7

3.0 

3.9
RELATION OF. CAPACITY TO DISCHARGE. 149

i 1 In Table 42 the values of 1 _ 1 correspond to

n

those of -^-~ in Table 41 and are similarly

derived from data of Tables 27 to 40. The purpose of the table is to show how the dominance 

of control by slope, as expressed by a ratio, 

varies with certain conditions.

A comparison of tabulated values for the 

several ratios shows that the ratios associated 

with efficiency vary with conditions more rapidly than those associated with capacity, but 

less rapidly than those associated with duty. 

In Table 42, just as in Table 41, there are no 

exceptions as to the direction of the trend of 

variation.

Bearing in mind that the alphabetic order in 

which the grades are arranged is the order from 

fine to coarse, and that variation with respect 

to form ratio is the inverse of variation with 

respect to width, we see that the general fact 

shown by the table is that the dominance of 

control by slope a dominance always pronounced is notably increased by increase of 

slope, discharge, fineness, or form ratio.

(74)

SUMMARY.

With d6bris of a particular size and a channel bed of a particular slope, there is a parties 

lar discharge which is barely competent to 

cause transportation. With increase of discharge above this barely competent discharge, 

there is a proportional addition to the stream's 

potential energy. The relation of capacity to 

discharge is formulated on the assumption that 

the capacity is proportional to some power of 

the added energy, and therefore to the same 

power of the added discharge. As each grade 

of debris is somewhat heterogeneous as to the 

size of its grains, this assumed principle can not 

be applied strictly; the practical assumption is 

that capacity varies with a power of the difference between the discharge and a constant

discharge, the constant being so chosen as best 

to harmonize the data.

By means of such formulation the data were 

readjusted and the rate of variation of capacity 

with discharge, or the index of relative variation, i,, has been computed for a variety of 

conditions. It is found to be greater as the 

slope of channel, the discharge, the fineness of 

debris, and the form ratio are less. The average of the values computed ior laboratory conditions is 1.42 and the ordinary range is from 

1.00 to 2.00.

The rate at which the efficiency of the stream 

and the duty of the stream's water vary withdischarge is denoted by an index which is less 

by unity than that for capacity. Its average 

is 0.42 and its ordinary range is from 0 to 1.00.

It has previously been shown that the corresponding indexes showing the relation. of 

capacity to slope are larger. In other words, 

capacity is more sensitive to changes of slope 

than to changes of discharge. If relative sensitiveness to the two controls be expressed by a 

ratio, the average value of that ratio is 1.36. 

The ratio varies with conditions, being relatively large when slope, discharge, fineness, 

and form ratio are relatively small.

The primary adjustment of observations of 

capacity, described in Chapter II, was an adjustment with respect to slope. The probable 

errors computed from differences between ad-r 

justed and unadjusted values were influenced 

by only a portion of the observational errors. 

In readjusting values of capacity with respect 

to discharge, another division of the observational errors was encountered and its importance was estimated. The probable errors 

computed from the results of the two adjustments are believed to represent with sufficient 

approximation the order of precision of the adjusted values of capacity, which constitute the 

main body of data for the discussions of the 

report. The order of precision is expressed by 

saying that the average probable error of the 

adjusted values is a little more than 3 per cent.
CHAPTER VI. RELATION OF CAPACITY TO FINENESS OF DfiBRIS.

FORMULATION.

To study the laws affecting the control of capacity for traction by fineness of debris, capacities should be compared which are subject to 

like conditions in all other respects. For this 

purpose data from Table 12 were arranged as 

in Table 43, where the capacities in each horizontal line are conditioned by the same slope, 

discharge, and width of channel. All the data 

of that table pertain to a slope of 1.0 per cent; 

but similar tables were constructed for slopes 

of 0.5, 0.7, 1.4, and 2.0 per cent.

The same data were also plotted on logarithmic paper; and, after a preliminary examination, five sets were selected for special investigation. The plots of these appear in figure 50, 

where ordinates are logarithms of capacity and 

abscissas are logarithms of linear fineness. It 

is to be noted that the zero of log C is not the 

same for the different graphs. The graphs were 

moved up or down, so as to avoid confusion 

through intersection.

The first law illustrated by the plots is that 

capacity increases as fineness increases; the 

second, that it increases more rapidly for small 

fineness than for great fineness. Despite irregularities of the data it is evident that the

locus of log C=f (log F) is a curve, and that the 

function has a general resemblance to those

fjfj (G) (F) (D) (O (S)fA)

FIGURE 50. Logarithmic plots of capacity for traction in relation to 

fineness of debris; corresponding to data in Table 44.

found in comparing capacity with slope and 

discharge.

TABLE 43. Values of capacity for traction, arranged to illustrate the relation of capacity to grades of debris.

Conditions.

S

1.0

w

0.66 

1.00 

1.32 

1.96

Q

0.093 

.182 

.363 

.545 

.734 

1.119

.093 

.182 

.363 

.545 

.734 

1.119

.093 

.182 

.363 

.545 

.734 

1.119

.093 

.182 

.363 

.545 

.734 

1. 119

Value of C for grade 

(A)

39.5

140

37.5 

100

231

36.8 '104

250

96

240 

359

(B)

10.8 

33.5 

81 

120

30.1 

85 

143 

199

29.3 

79 

140 

204

67.6 

120 

190 

313

(C)

13.8 

32.1 

73 

112

152

31.2 

85 

140 

180 

276.

21.4 

79 

129

187

60.6 

111 

190 

343

(D)

9.1

29.8

101

24.0 

73 

108 

153

59.2

131

(E)

24.8

40

14.8 

33.8

72 

138

36.3

73 

123

(F)

20.5

39

14.0

42.8 

79.5

40.2 

70

(C)

8.5

30.4 

49

26.9 

61

18.9 

52.2

(H)

12.8 

25.2

150
KELATION OF CAPACITY TO FINENESS OF DEBRIS. 151

For any velocity, as determined by slope, 

discharge, and width, there is a competent 

fineness, marking the limit between traction 

and no traction; and to this extent, at least, the 

relation of fineness to traction is analogous to 

the relations of slope and discharge. It is not 

easy to carry the analogy further, because slope 

and discharge are conditions of active force, 

and fineness is a condition of reactive force, or 

resistance; but an experiment in formulation 

reveals a parallelism quite as striking as that 

between the capacity-slope and capacity-discharge functions.

Assuming

(75)

in which 0 is a constant fineness and &4 a constant of the numerical value of capacity when 

F= <£ + !,* the five sets of data in. Table 44 

were treated graphically for the determination 

of <j> and p. The methods were such as have 

already been described (pp. 65 and 139). The 

formulas were then used to compute readjusted values of capacity, Cr, and values of the 

index of relative variation, i4 , and probable 

errors were also computed.

TABLE 44. Numerical data connected with the plots in figure 50, and illustrating the relation of capacity for traction to grades

of debris.

.

Conditions of experiments ......... J Q ...... Is.......

U....... 

Parameters of equations ............ <r> ...... 

[bt ......

fGrade(A)..

Grade(B). 

Grade(C).

Data and computed results . . . . QJ^JQ >p<" 

Grade (F)! 

Grade (G). 

Grade (H).

1

0.66 

0.363 

1.4

41 

0.62 

4.5

6.8 

2.8

C Cr »4

137 145 0.65 

124 123 . 67 

69

49. 4 59 .81 

41 37 1.08 

23 22 1.82 

6.9 7.1 13.50

U

1.54 

1.55

2.10 

1.99 

2.71 

4.69

w...... 

Q......

S......

(*...... 

.......

1&4......

rc......

(Cr.....

Gra< 

Gra 

Grac

Grac 

Grac 

Grac

Je(A) 

le?B) 

le(C) 

3e(D) 

le(E) 

le(F) . 

ie(G) . 

ie(H)..

2

1.00 

0.734 

0.7

55 

0.60 

2.4

7.6 

2.9

C

121 

110 

106 

85 

40 

20 

8

Cr

140 

123, 

102 

76 

42 

21.7 

7.4

14

0.63 

.64 

.66 

.70 

.87 

1.42 

5.45

Ji

1.83 

1.66 

1.50 

1.72 

1.68 

2.35 

4.11

4

1.32 

0.734 . 1.0

48 

0.61 

3.8

1.7 

0.6

C

250 

204 

187 

131 

73 

40.2 

18.9

Cr

250 

219 

180 

133 

74 

40.2 

19

<«

0.64 

.65 

.66 

.70 

.84 

1.22 

5.20

n

1.71 

1.71 

1.47 

1.85 

1.54 

2.40 

3.43

3

1.00 

0.363 

1.4

49 

0.58 

3.7

4.9 

1.9

C

185 

149 

143 

127 

62 

33.4 

16.3

Cr

196 

173 

143 

108 

62 

34.4 

16.3

»4

0.61 

.62 

.63 

.66 

.80 

1.18 

2.80

ii

1.79 

1.63 

1.53 

1.63 

1.78 

2.42 

3.15

5

1.32 

0.363 

2.0

55 

0.50 

11.4

6.2 

2.4

C

328 

255 

 233 

237 

112 

66 

28.3

Cr

319 

285 

242 

190 

115 

67 

27.4

«4

0.53 

.54 

.55 

.58 

.72 

1.19 

4.55

t

1.97 

1.63 

1.52 

1.92 

1.60 

2.03 

3.67

PRECISION.

The average probable error of the readjusted capacities was found to be ±2.1 per 

cent. This error is to be ascribed in part to 

discordance of the data among themselves, and

in part to discordance of the formula with the 

data; but the distribution of the residuals is 

not such as to imply important discordance of 

the formula.

1 The dimensions of capacity aro Jf+i r-», and of fineness I*-3. The 

constant 64, being equal to Cj (F <j>)v, has" dimensions L+WJtf+i r-i. ;
152

There are, however, important discordances 

among the data. Considered as errors, the 

discordances constitute a group which were not 

detected in the adjustments of capacities to 

slopes and to discharges, but which escaped 

those tests because related peculiarly to the 

grades of debris. From the residuals of the 

present readjustment the average probable 

error of capacities before readjustment is estimated at ±5.4 per cent, whereas the average 

probable error of the body of once-adjusted 

capacities from which these were selected was 

estimated at 2.5 per cent. On the assumption 

that the estimate for the whole body of values 

applies to the selected group, the share of error 

associated with the grades is estimated as

V5.42 -2.52 = ±4.8 per cent.

Inspection of the logarithmic plots suggested 

that part of the discordance of the data is systematic. To bring out the systematic element 

the original values of capacity in Table 44 were

divided by the readjusted values, and means 

were taken of the quotients. The means are 

listed below, and in figure 51 they are plotted 

logarithmically in relation to fineness. The 

plotted points conspicuously out of line are 

those for grades (B) and (D), the capacities 

determined for grade (B) being relatively too 

small and those for grade (D) too large. The 

same result was obtained from a canvass of a 

wide range of data.

Ratios oj original values of capacity to adjusted values.

Grade...................

Linear fineness. . ........

/ A ^

1,002

0.96

(B).

812

0.89

(C)

602

1.01

(D)

388

1.13

(E)

178

0.98

(F)

95.9

0.97

/ (~\\

43.4

1.02

It is surmised that these errors arise in part 

from variations of experimental method, and 

this suspicion attaches especially to grade (B), 

which was the first to be treated in the laboratory. It attaches much less to grade (D), for

_1_

(G)

* --. ' +

(F) (E)

+

 4 ___

----- ---^-T--

CD) (C) CB) (A)

FIGURE 51. Average departures of original values of capacity from the system of values readjusted in relation to fineness .of debris. The horizontal line represents the readjusted system. The broken lines, above and below, correspond to departures of 10 per cent.

which the experimental method was about the 

same as for grades (A), (E), and (F). So far as 

the quality of the experimental work is gaged 

by the probable error, that on grade (D) would 

appear to be considerably below the average.

It is surmised that systematic error may also 

be connected with properties of the grades 

other than that of fineness. Range of fineness 

has already been appealed to as an explanation 

of apparent anomalies as to competent slope. 

From the experiments with mixtures (Chapter 

IX) we know that great range within a grade 

would tend to increase capacity for traction. 

As the range is small for grade (A), regularly 

increases to grade (E), and is again small from 

(F) to (H), the influence of this factor tends to 

enhance the capacity of grade (E); but the 

effect of that influence is not apparent in the 

diagram.

The separation of grades, being effected by 

sieves, was a gaging of grains by certain dimensions. The grains may have differed also as to 

shapes and densities, and each of these properties would affect capacity. The de'bris first 

obtained for the laboratory had been washed 

from the bed of Sacramento River when a flood 

broke the levee below the mouth of American 

River. This furnished material for the finer 

grades. Other lots of de'bris were taken from 

the bed of American River, and these furnished 

material chiefly for the coarser grades. So far 

as the separations from the different lots coincided, they were used indiscriminately. It is 

possible that grade (D) was composed chiefly

of the coarser particles of a fine alluvium, while 

grade (E) was composed chiefly of the finer 

particles of a relatively coarse alluvium. As 

any natural body of fractional de'bris is the 

result of a sorting process in which tractionable 

particles are separated from the immovable on 

one side and from the suspendible on the other, 

'it is likely to include among its coarser grains 

many which are tractionable only because of 

low density or favorable shape, and among its 

finer grains many which escape suspension 

because of high density or unfavorable shape.
RELATION OF CAPACITY TO FINENESS OF DEBEIS. 153

Thus it appears possible that the superior 

mobility of grade (D) was determined by 

properties other than size. Unfortunately the 

record is not of such character that the value 

of this suggestion can now be tested.

A third suggestion pertains to the gaging of 

fineness. The method of gaging included a 

weighing and involved certain assumptions as 

to homogeneity in average density and in shape 

which may not have been fully warranted.

These various suggestions, while not susceptible of test at the present time, are sufficiently 

plausible to show the possibility of definite 

causes for the discordances discovered by the 

comparison of data from different grades. In 

m7 judgment it is proper to ascribe the greater 

discordances to such causes, and to view them 

as abnormalities with respect to the law connecting capacity with fineness.

In view of the magnitude of the abnormalties 

or discordances, it does not appear profitable 

to extend the readjustment of data to other 

and shorter sets. The five sets in Table 44 

were selected because they included great range 

in fineness, and because they were qualified to 

yield fairly definite values of the constant <j>.

VARIATIONS OF THE CONSTANT <£.

The laws which control the variation of <j> 

have not been developed from the observations, 

but their general character may be inferred 

deductively by considering the relations of competent fineness to various conditions it being 

assumed that </> is intimately related to competent fineness. Postulate a current of which 

the velocity is determined by a particular slope, 

a particular discharge, and a particular width. 

For this current a certain fineness is competent. 

Increase of slope or discharge increases the 

velocity and makes a lower fineness competent. 

Decrease of width, which corresponds to increase of form ratio, increases velocity and 

makes a lower fineness competent. Thus competent fineness, and therefore (f>, varies inversely 

with the slope, discharge, and form ratio.

£) .-.-......(76)

capacity and fineness. The formula for the 

index (cf. pp. 100 and 141) is

.(78)

INDEX! OF RELATIVE VARIATION.

Framing an equation of the type of (33) 

C=v^....... ...... (77)

in which \ is the index of relative variation for

With, this formula, values of \ were computed 

from the data in Table 44, and they are given 

in the lower part of that table.

By inspection it appears that the index increases as fineness diminishes, its growth being 

at first slow but becoming rapid as competent 

fineness is approached. Because of the discordances of the data it is not easy to derive a 

body of values of the index for discussion in 

relation to other conditions, but it is relatively 

easy to obtain comparative values of the synthetic -index, 74, and the variations of these 

values may be assumed to show the same trends 

as the variations of \. Values of 74 were computed between corresponding data of grades 

(C) and (G) by the formula

* log ft- log Fn

in which Cj. and Clt are specific capacities corresponding to the finenesses FI and Fn ; and the 

results are given in Table 45.

From these results it is inferred (1) that the 

index varies inversely with the slope, (2) that 

it varies inversely with discharge, and (3) that 

it varies directly with, width, and therefore 

inversely with form ratio. The response is in 

general of a very pronounced character, but to 

this there is exception in one of the comparisons with width. It is possible that the index 

is a maximum function of width and a minimum function of form ratio. With some reservation on this point, we may generalize:

/;£) ' ~~ (79)

If equation (79) be compared with, equations 

(39) and (68), it will be seen that the variation 

of the capacity-fineness index observes the same 

laws of trend as the variations of the capacityslope index and the capacity-discharge index. 

In view of this general parallelism of variation, 

it is thought that the relative magnitudes of 

average i4 , average il} and average ia may be 

adequately discussed by means of a moderate 

number of comparisons. Accordingly only 

those values of -i4 computed from the five equations of Table 44 are used. The corresponding 

values of i3 are not available, but those of it
154

are given in Table 15. They have been copied 

into Table 45, so as to be conveniently compared.

Inspection shows that \ is in general much 

smaller than \ but that it becomes greater as 

the limit of competence is approached. As to 

the first of these generalizations there can no 

question, but the second is not equally satisfactory. In the vicinity of competence the 

value of \ is highly sensitive to the influence 

of (/>; and in the same region \ is highly sensitive to <r. The features of the table might be 

produced by slight overestimates of <f> or by 

slight underestimates of o. In view of this 

consideration it is probably best to leave the 

higher values of the index out of the account 

and base a computation of averages wholly on 

the lower values. Including only the 28 values 

of each index associated with grades (A) to (F), 

the means are, for \ 0.77, for \ 1.79; and the 

ratio of the second to the first is 2.4; that is, 

the sensitiveness of capacity to control by slope 

is estimated to be 2.4 tunes as great as its sensitiveness to control by fineness. The ratio of

sensitiveness for slope and discharge iji3 having 

been estimated at 1.36, it follows that the ratio

for discharge and fineness is 24

= 1.

Mean \ : mean i3 : mean i4 : : 2.4. : 1.8 : 1.0.

It is to be understood that these estimates 

are of the most general character. The ratios 

doubtless vary notably with conditions.

The property with which capacity has been 

compared in this chapter is linear fineness, F, 

defined as the reciprocal of diameter, or as the 

number of grains to the linear foot. Bulk fineness, F2 , defined as the reciprocal of volume, is 

proportional to the cube of linear fineness. It 

follows that the index of relative variation 

when capacity is compared with bulk fineness 

is one-third the corresponding index, i4 , for 

capacity and linear fineness; and the same 

factor applies to synthetic indexes. If bulk 

fineness were substituted for linear fineness in 

equations of the form of (75), the values of <j> 

would be quite different and the values of p 

would be uniformly one-third as great.

TABLE 45. Values of J4 , the synthetic index of relative variation for capacity and fineness, compared with slope, discharge,

and width of channel.

Discharge (ft.3/sec.). .

Width (feet).........

.((

((

0., i.;

s

1.8

2.4

>)

3)

363

52

It

1.05

.76

((

((

0.'

i.:

s

1.0

2.0

>) -^i

r34

52

It

0.99

.61

«

f

1.

Q

0.363

.734

1)

3)

8

32

It

1.05

.65

((

((

2.

1.

Q,

0.363

.734

')i\

#

32

It

0.92

.61

(

(<

2.(

0..

0.66

1.00 

1.32

1 \

3

1 }

363

It

0.59

.74 

.92

((

((

0 f

O/

1.00 

1.32

1 \

'{

 ^4

It

0.57 

.61

DUTY AND EFFICIENCY.

The relations of duty and efficiency to capacity involve discharge and slope but are independent of fineness. Fineness, therefore, has 

exactly the same control of duty and efficiency 

that it has of capacity, and the conclusions of 

this chapter apply without qualification to duty 

and efficiency.

SUMMARY.

Capacity for traction is greater for fine debris 

than for coarse that is, capacity increases with 

fineness. The law of increase admits of formulation in a manner strictly analogous to that 

employed in comparing capacity with slope 

and discharge that is, it is found that capacity varies approximately with a power of 

the fineness less a constant fineness. The value

of the constant fineness varies with conditions, 

being greater as slope, discharge, and form 

ratio are greater. The rate at which capacity 

varies with change of fineness, or the index of 

relative variation, is not the same for all conditions, being greater as slope, discharge and 

form ratio are less. Under similar conditions 

the rate is less than the corresponding rate for 

capacity and slope, the average ratio between 

them being as 1 to 2.4.

The arrangement of capacities in accordance 

with the assumed law of increase develops discrepancies which are believed to be of the nature of systematic errors. The largest of these 

have a magnitude of about 10 per cent. They 

are tentatively ascribed to peculiarities of the 

debris used in experiments and to imperfectly 

developed laboratory methods.
CHAPTER VII. RELATION OF CAPACITY TO VELOCITY.

PRELIMINARY CONSIDERATIONS.

The work of stream traction is accomplished 

by the movement of water along the bed of the 

channel. For that reason the system of water 

movements and water velocities near the bed 

is intimately related to the load or capacity for 

load. In certain parts of this paper and in 

the writings of some other investigators use is 

made of the term "bed velocity," or its equivalent, but the term has no satisfactory definition. The difficulties which are encountered 

in this connection have to do also with the 

vertical velocity curve.

In all the streams with which we are here 

concerned the flow is eddying or turbulent. 

At any point the direction of motion and the

FIGURE 52. Vertical velocity curve, drawn to illustrate its theoretic 

character near the stream's bed. OD is the origin of velocities.

velocity are constantly changing. If a mean 

be taken of the instantaneous forward components of velocity the components parallel 

to the axis of the stream it gives for the 

point a mean velocity coordinate with the 

mean velocity for the cross section obtained 

by dividing the discharge by the sectional area. 

It will be observed that the mean at a point is 

a mean with respect to time, while the sectional mean is primarily a mean with respect to 

space. The mean at a point, as thus defined, 

being called Vp> the vertical velocity curve 

may be defined as the curve obtained by plotting the values of Vp for any vertical of the 

current in their relation to depth. As commonly 

drawn by hydraulic engineers, it terminates 

downward at some distance from the origin of 

velocities, OD say at B in figure 52 connoting 

a finite velocity for the water in actual contact with the bed. This implication contravenes a theorem of hydrodynamics that the 

velocity at contact with the wall of a conduit is either zero or indefinitely small. The 

theorem is believed to have been established 

experimentally by the work of J. L. M. 

Poiseuille 1 and is generally accepted. In the 

direct study of the velocities of streams instrumental observation is not carried from surface 

to bed, but ceases at some point, C, and the 

drawing of the curve below that point is a 

matter of inference. The inference accordant 

with the hydrodynamic principle is that the 

curve changes its course below C and reaches 

the origin at or near O. 2 This inference accords 

also with our observations in connection with 

the study of saltation (see p. 29); and those 

observations suggest likewise that the curve is 

materially modified by the resistances to the 

current involved in the work of saltation.

It thus appears that in the region with which 

traction studies are specially concerned the 

range of Vp is great. The work of traction 

depends on a system of velocities and nob on a 

single one, and there is no individual value of 

Vp with special claim to the title "bed velocity." It would be possible to define bed 

velocity as the value of Vp at some particular 

distance from the bed or at a distance constituting some particular fraction of the depth of 

current; but such a definition would be hard to 

apply.

However smooth a stream bed of debris may 

be in its general aspect, it is never smooth as 

.regards details. Figure 53 gives an ideal proFIGURE 53. Ideal profile of a stream bed composed of ddbris grains.

file, the intersection of a bed by a vertical plane. 

Not only are there salients and reentrants, but 

some of the reentrants communicate with the 

voids within the mass of debris. In many of

1 See Lamb's Hydrodynamics, 3d ed., p. 544,1906.

2 See Cunningham, Allan, Hydraulic experiments at Roorkee, p. 46, 

^75, and Inst. Civil Eng. Proc., vol. 71, p. 23,1882, where ho discusses 

t he horizontal velocity curve; and Von Wagner, idem, p. 90.

155
156

the reentrants are doubtless stationary eddies, 

with reversed currents where the value of Vp is 

negative. It appears equally difficult to give 

definition to the bed as a datum from which to 

measure upward, and to select and define a 

locus for bed velocity. There is reason to suspect also that the problem as thus stated is 

unduly simplified by the assumption that the 

bed is a stable entity, clearly separate from the 

zone of saltation above. It did indeed so appear when the process of saltation was studied 

through the glass wall of the observation 

trough, but what was witnessed was the phase 

of the process at the edge of the channel bed, 

where the current was retarded by the resistance of the channel wall. At a distance from 

that wall, in the region where the cloud of saltatory particles effectually precludes visual observation, the passage from stability to mobility may be less definite. I am led to this suggestion by the observations, quoted byMcMath,1 

of a civil engineer who descended in a diving 

bell to the bottom of the Mississippi at a point 

where the depth was 65 feet and the bottom of 

sand. Stepping to the bed, he sank into it 

about 3 feet, and then thrusting his arm into the 

yielding mass, .could feel its flowing motion to a 

depth of 2 feet, the velocity diminished downward. In interpreting these phenomena, allowance must be made for the fact that the presence of the diving bell created an abnormal 

condition and if it rested on the bed put a stop 

to saltation. The flow of the sand is then to be 

ascribed to the difference in water pressure on 

the two sides of the bell. But the fact of the 

flow seems to indicate an antecedent state of 

mobility, a layer of the bed being supersaturated so as to have the properties of quicksand. 

If such a layer exists, then the transition from 

the bed to the saltation zone is not abrupt but 

gradual.

The difficulties in attempting to define bed 

velocity are supplemented by those which affect 

the measurement of velocities near the bed 

while traction is in progress (p. 26), and together they have served to prevent the use of 

bed velocity as a factor for quantitative comparison with capacity. This result has been 

regretted because the forces which accomplish 

traction are applied directly through the velocities of water near the bed, and it was admitted

i McMath, B. E., Van Nostrand's Mag., vol. 20, p. 227,1879.

only after the failure of repeated attempts to 

obtain serviceable estimates of bed velocity.

In the present chapter observed or interpolated capacities are compared with mean velocities of the stream, mean velocity being computed as the quotient of measured discharge by 

measured sectional area. The measurements of 

discharge and width being relatively simple and 

accurate, the determinations of mean velocity 

have the same degree of precision as the measurements of depth. (See p, 26.)

In comparing capacity with mean velocity, 

it is convenient always to treat fineness of 

debris and width of channel as constants, but 

it is also advantageous to recognize three 

separate points of view as to the status of 

discharge, slope, and depth.

First, We may treat discharge as constant, 

in which case slope and depth vary, along with 

velocity and capacity. Each of the observational series (Tables 4, 12, and 14) conforms 

to this viewpoint. When discharge is constant, the increase of power necessary to 

increase velocity is given by increase of slope, 

and the increase of velocity causes the unchanged discharge to occupy less space. As 

velocity and capacity increase, slope increases 

and depth decreases.

Second, we may treat slope as constant. 

With slope constant, the increase of power 

necessary to increase velocity is given by 

increase of discharge, but the rate at which 

discharge is increased is greater than the rate 

of increase given to velocity, and the increased 

discharge therefore requires more space. As 

velocity and capacity increase, both discharge 

and depth also increase.

Third, we may treat depth as constant. 

To increase velocity by increasing slope will, 

as we have seen, reduce depth. To increase 

velocity by increasing discharge will, as we 

have seen, increase depth. To increase velocity 

without changing depth, it is necessary to 

enlarge both slope and discharge. No experiments Were conducted with fixed depths, but 

the data for this comparison are readily 

obtained by interpolation.

It is proposed to examine the relation of 

capacity to velocity from each of these viewpoints, developing the results so far as necessary to give a basis for a comparison of the 

viewpoints.
RELATION OF CAPACITY TO VELOCITY. 157

A preliminary remark applies to all. For 

each grade of debris and width of channel, 

and for each specific assumption of a constant 

discharge, slope, or depth, there is necessarily 

a competent mean velocity, below which no 

traction takes place. The conception of such 

a competent velocity has underlain all the 

discussions of competent slope, competent discharge, competent fineness, and competent 

form ratio. A broad analogy therefore points 

to the propriety of formulating the capacityvelocity relation as other relations of capacity 

have been formulated. And the inference 

from analogy finds support in logarithmic 

plots of C=f( Vm) under each of the three abovementioned conditions. It may fairly be assumed, therefore, that the index of relative 

variation for capacity and velocity itself varies 

with velocity, being relatively small for high 

velocities, being relatively large for low velocities, and becoming indefinitely large as competent velocity is approached.

For the purposes of this chapter, however, 

it has seemed best to employ a simpler method, 

using only the synthetic index of relative 

variation characterized by the symbol 7. 

Calling the synthetic index for the variation 

of capacity with respect to mean velocity Iv, 

we may conveniently distinguish by Ira, Ivs 

and Ivd the values associated severally with 

the special cases of constant discharge, constant slope, and constant depth.

The computations of the index are made 

chiefly by the formula

log (7-log I" 

: logFTO '-logFm'

(80)

in which C' and G" are specific capacities, and 

Vm' and Vm" are the corresponding mean 

velocities. Graphically, Iv is the inclination of 

a line connecting two points of which the coordinates are, for the first, log C' and log Vm', and 

for the second, log C" and log Vm". Where 

the available data serve to place more than two 

points on the logarithmic plot of G f( Vm), definite suggestion may thereby be made that the 

one connecting the extreme points does not 

constitute the most probable location of the 

chord theoretically corresponding to Iv ', and in 

such cases a line is drawn with regard to all 

the data, and its inclination is measured on the 

plot.

The subject of competent velocity, which is 

of interest independently of the formulation of 

capacity and velocity, will be considered at the 

end of the chapter.

THE SYNTHETIC INDEX WHEN DISCHARGE IS 

CONSTANT.

In Table 14 are 73 series of values of Vm , 

each value corresponding to a stated value of S. 

The coordinate series in Table 12 contain values 

of C corresponding to the same values of 8. 

From each pair of series were taken the highest 

and lowest values of Vm and the corresponding 

values of 0, and from these four quantities was 

computed a value of IVq. The 73 values of the 

index are shown in Table 46, where the arrangement is such as to exhibit the variation of the 

values with respect to discharge.

TABLE 46. Values of lyq, the synthetic index of relative 

variation for capacity in relation to mean velocity, when 

discharge is constant.

Grade.

(A) 

(B)

(C)

(D)

(E) 

(F)

(G) 

(H)

w

1.32 

1.96

.23 

.44 

.66 

1.00 

1.32 

1.96

.44 

.66 

1.00 

1.32 

1.96

.66 

1.00 

1.32

.66 

1.00

.66 

1.00 

1.32

.66 

1.00 

1.32

.66

Value of lyo when discharge (in ft.8/sec.) is 

0.093

5.45 

4.32 

5.76

6.31 

3.36

5.38

........

0.182

4.50 

4.45 

4.61 

5.68 

2.92

4.30 

3.29 

3.63 

2.54

4.30 

4.16

"s.'io"

7.50 

11.46

0.363

4.05

3.55

2.93 

5.37

4.59

2.82 

3.56 

3.90 

4.36

4.13 

5.34

5.98 

5.35

6.67 

9.63 

8.41

9.22

8.80

13.60

12.57

0.545

4.22

2.48 

3.43

3.04 

2.52 

3.27 

3.74

4.59

4.34

0.734

3.62

 3.22

5.15

2.51 

3.38

2.92 

3.11 

3.69 

3.69

3.94 

3.34

6.00 

3.43

8.30 

8.49 

10.67

8.97

1.119

2.38

3.49

3.89

10.60 

8.05 

8.26

8.49

The values which lie in any horizontal line 

agree as to all conditions except discharge. On 

comparing the columns for discharges 0.093 and 

0.182 ft.3/sec., it is seen there are six lines 

carrying values in both columns. The means 

of these values are 4.80 and 4.21, the greater 

mean belonging with the smaller discharge. In 

the column for discharge 0.182 are nine values 

coordinate with values in the column for discharge 0.363, and the means for the two groups
158 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

of nine are 5.31 and 5.04. Again the greater 

mean is associated with the smaller discharge, 

and the same relation is found by other combinations. The partial means resulting from 

these reductions are arranged in the upper division of Table 47. The general fact exhibited is 

that the index, other conditions being the same, 

varies inversely with discharge.

The same method was employed to discover 

the nature of the control of the index, first, by 

grade of debris and, second, by width of channel; and the partial means are grouped in the 

middle and lower divisions of Table 47. Of 

the eight pairs of means connected with grade, 

seven agree in testifying that the index varies 

inversely with fineness. The exceptional testimony comes from the comparison of grades (B) 

and (C); and although it is emphatic, it serves 

rather to illustrate the general discordance of 

data from the experiments with grade (B) 

than to qualify the general law as to the index.

TABLE 47. Partial means based on Table 46, illustrating 

the control of IVQ, by discharge, fineness, and width.

Number 

of 

values.

6 

9 

7 

. 16 

8

4 

C 

9 

5 

5 

4 

3 

3

2 

4 

15

12

7

Means of IY(i for discharge (ft.s/sec.) of 

0.093

4.80

0.182

4.21 

5.31

0.363

5.04 

3.90 

6.58

0.545

3.51

0.734

5.47

5.81

1.119

5.64

Means of IVQ for grade 

(A)

3.45

(B) (C)

3. 

4.

49 ......

01 3.79 

... 3.34 

3.21

...... 3.44

(D)

~4.39

CE) (F) .(G)

"5.17' ..'.'.'.'. '.'.'.'.'.'.

...... 8.73 :.....

5.19 ...... 8.71

...... 8.24 10.54

............ 9.39

(H)

10.01

Means of IVQ for width (feet) of 

0.23

4.97

0.44

4.38 

4.84

0.66

3.80 

5.59

1.00

5.84

5.56

1.32

5.87 

3.36

1.96

3.62

Of the five pairs of means connected with 

width, three show increase of the index with 

increase of width and two give the opposite 

indication. The contrasts are not strong in 

any case, and the nature of the law is not clear.

It may be that the normal variation with 

width is so slight as to be masked by accidental 

errors of the data; or it may be that the index, 

like the constant a, is a minimum function of 

form ratio.

By dividing the slope interval covered by a 

computed value of the index and computing 

separately the indexes for the two subintervals, 

it was found that the index associated with the 

higher slopes has a smaller value than that for 

the lower slopes that is, the index varies 

inversely with slopes.

To compare the control of capacity by mean 

velocity with its control by slope the 73 values 

of Iw (the synthetic index of relative variation 

of capacity and slope, under condition of constant width) corresponding to the tabulated 

values of Iva were computed. Each value of 

IVQ is greater than the corresponding value of 

Iw, the ratio ranging from 1.4 to 4.0. The 

mean of 73 values of IVQ is 5.33; the mean Iw 

is 2.05; and the ratio of the means is 2.60. It 

is in general true that the greater the indexes 

the greater the ratio between them.

In Table 48 means of the two indexes are 

shown for the several grades of debris. Each 

index varies with fineness and so, too, does the 

ratio of indexes.

TABLE 48. Synthetic indexes, comparing the control of capacity by mean velocity with the control by slope, and comparing both controls with grades of debris.

Grade.

(A 

(B 

C] 

(D 

(E

J

) 

)

Number 

of 

separate 

determinations.

5 

18 

19 

9 

5 

5 

9 

3

73

Mean IYq

3.62 

4.35 

3.57 

4.50 

5.17 

8.73 

9.56 

10.01

5.33

Mean Iw

1.87 

1.94 

1.85 

1.87 

1.83 

2.27 

2.60 

3.20

2.05

Mean 7K(j

Mean Iw

1.93 

2.24 

1.93 

2.41 

2.83 

3.84 

3.68 

3.12

2.60

Mean Iw

Mean/K<2

0.52 

.45 

.52 

.42 

.35 

.26 

.27 

.32

.38

MEAN VELOCITY VERSUS SLOPE.

The comparison of IVq with Iw affords, 

incidentally, an estimate of the relative variation of mean velocity and slope. The rate of 

variation of capacity with mean velocity being 

IVQ, the rate of variation of mean velocity with 

capacity is l//r«; and the rate of variation of 

capacity with slope being Iw, the rate of 

variation of mean velocity with slope is
BELATION OF-CAPACITY TO VELOCITY. 159

\IIVqXlw = IwIIv(i. The values of IJIva listed 

in Table 48 are therefore estimates of the relative variation of mean velocity in relation to 

slope, and they have the same quality as the 

corresponding values of Iw and TVo,. They are 

subject to the limiting conditions of constant 

discharge, fineness, and width, and they are averages of variability for practically the entire 

range of conditions realized in the laboratory, 

with the exception of those in the immediate 

neighborhood of competence. For this range 

of conditions mean velocity varies, on the 

average, as the 0.38 power of the slope.

This result is comparable with the generalization embodied in the Chezy formula, which 

makes mean velocity vary with the 0.5 power 

of the slope. The two are not inconsistent 

because they pertain severally to loaded and 

loadless streams. In a loaded stream increase 

of slope augments load and thus develops 

rapidly a factor of resistance from which the 

loadless stream is free. Velocity, being limited by resistances, develops less rapidly when 

the conditions are such that the resistances 

develop more rapidly.

THE SYNTHETIC INDEX WHEN SLOPE IS 

CONSTANT.

For a selected slope, values of Vm may be 

found in Table 14 which agree as to width of 

channel and grade of debris and differ only as 

to the discharges with which they are associated; and in Table 12 may be found the corresponding values of capacity. Such parrs of 

values, when occurring in series of two to five, 

constitute data for the computation of values 

of Ivs. To cover the entire range of tabulated 

data without needless repetition choice was 

made of slopes 0.6, 1.0, 1.4, and 2.0 per cent. 

The data associated with these slopes gave 66 

values of the index, the values being essentially independent except in a few instances. 

They are shown in Table 49. From them were 

derived the sets of partial means arranged in 

Table 50, the method of derivation being that 

described in connection with Table 47.

The means show that Ivs varies inversely 

with slope, the variation being of moderate 

amount. They leave little question that it 

varies inversely with fineness, though the evidence is somewhat conflicting. They indicate

also a direct variation with trough width, the 

opposite tendency being indicated only by the 

means for the greatest widths. The plots of 

the data, not here reproduced, show that the 

index varies inversely with discharge.

TABLE 49. Values of Ivs, the synthetic index of relative 

variation for capacity in relation to mean velocity, when 

slope is constant.

Grade.

(A) 

(B)

(C)

(D) 

(E)

(F)

5

.06

.6 

1.0 

1.4 

2.0

.6 

1.0 

1.4 

2.0

.6 

1.0 

2.0

.6 

1.0 

1.4 

2.0

1.0 

1.4 

2.0

1.0 

1.4 

2.0

1.4 

2.0

Value of Ivs for width (feet) of 

0.23

2.42 

3.30

0.44

2.60

4.38 

2.83 

2.89

0.66

3.82 

4.39 

3.80 

4.95

3.20 

2.98 

2.34 

1.59

3.55 

3.40 

2.72

3.68 

2.29

2.21 

4.86 

4.92

3.97 

3.63 

3.50

9 78

6.84

1.00

4.60 

4.10 

4.45 

4.17

3.31 

3.01 

3.35

2.55 

2.49 

2.85

3.38

3.06 

2.85 

2.79

2.94 

5.80 

4.72

4.62 

4.92 

3.10

1.32

2.18

 3.24 

3.40 

3.24

4.20 

3.48 

3.10

4.10

2.39

6.50 

2.06 

5.05

1.96

1.87

5.15 

4.62

2.80 

2.45 

2.03 

1.33

TABLE 50. Partial means based on Table 49, illustrating 

the control of Ivs by slope, fineness, and widtli.

Number 

of 

values.

12 

16 

11

2 

13 

6 

4 

5 

2

1

4 

18 

11 

C

Means of Isy for slope (per cent) of 

0.6

3.68

1.0

3.35 

3.63

1.4

3:4, 

4.2,

2.0

J 3.81

Means of Ivs for grade 

(A)

2.02

(B)

4.19 

3.81

(C)

2.87 

30.5

(D)

3. is '.

2.82

(E) (F)

3.23' ...... 

2.93 4.11

(0)

4.05

3.56

(H)

7.81

Means of IYS for width (feet) of 

0.23

2.42

0.44

2.60 

3.17

0.66

3.23 

3.57

1.00

3.47 

3.72 .

3.46

1.32

3.54

3.71 

3.27

1.96

2.97

3.15
160 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

THE SYNTHETIC INDEX WHEN DEPTH IS 

CONSTANT.

The capacities and mean velocities corresponding to particular depths were derived from 

the computation sheets described on page 95. 

The selected depths were 0.10, 0.14, 0.20, and 

0.28 foot, and the data available for these depths 

afforded 32 values of the synthetic index, IvdThese are recorded in Table 51, and partial 

means derived from them are arranged in 

Table 52 so as to show variation in relation to 

fineness of debris and width of channel.

TABLE 51. Values of Ivd, the synthetic index of relative 

variation for capacity in relation to mean velocity, when 

depth of current is constant.

Width 

(feet).

0.44 

.66

1.00 

1.32 

1.96

Depth 

(feet).

0.14

.10 

.14 

.20 

.28

.14 

.20 

.28

.10 

.14 

.20

.10 

.14 

.20

Value of Ivd for grade 

(A)

3.23 

3.65

2.36 

2.23

(B)

4.12 

3.51

4.02

3.05 

2.17 

2.03

3.92 

3.41 

4.50

(C)

3.73

2.59

3.16 

3.39 

2.62

3.59

2.80 

3.44

3.86 

3.10

3.02 

3.60 

4.27

(D)

3.17 

3.51

3.52

(E)

4.46

(G)

7.86

7.22

TABLE 52. Partial means based on Table 51, illustrating the 

control of IYH, by fineness and width.

Number 

of 

values.

4 

8 

3 

1 

2

1 

4 

3

7

Means of Ivd for grade 

(A)

2.87

........

(B)

3.03 

3.59

"~3."64"

(C)

3.40 

.3.16 

3.44

(D)

3.40

(E)

4.46

(G)

"7."54"

Means of Ira for width (feet) of 

0.44

3.73

0.66

3.17

1.00

3.28

3.46 

3.71

1.32

2.88 

3.00

1.96

3.09

With a single exception in each group of 

means, the indication is that the index varies inversely with fineness and directly with width. 

Under the condition of constant depth each 

change in mean velocity ij accompanied by 

changes of both slope and discharge, and the 

influences of the two can not be examined separately. The features of the logarithmic plots, 

not here reproduced, show that the index varies 

inversely with slope and discharge, considered 

together.

THE THREE INDEXES.

In bringing together the results outlined in 

the preceding paragraphs we may replace width 

of channel by form ratio, bearing in mind that 

the two factors are so related that their variations are in opposite senses. So far as qualitative statement is concerned, the three synthetic indexes are identical in properties. The 

sensitiveness of capacity for traction to the 

control of mean velocity of current varies inversely with slope of channel, discharge, fineness of debris, and form ratio.

As to the first three conditions the generalization is unqualified, but it is possible that the 

function as to form ratio is of the minimum 

class instead of inverse.

The three indexes are not of the same magnitude. In comparing them Ivd was made the 

standard, partly because the values computed 

for it are fewer. To compare with each of its 

32 values, that value of Ivs which most nearly 

represents the same group of conditions was 

selected, and also a pair of values of IVq which 

collectively represent nearly the same conditions. Means were then derived for each index 

for each of the six grades to which they pertain, 

and also general means all of which are shown 

in Table 53. In the same table are the ratios 

between general means and also between the 

corresponding grade means. For the general 

means Ivq is 7 per cent greater than IYd, and IVq 

is the greater for all the partial means but one. 

For all the partial means but one Ivs is smaller 

than Ivd, and for the general means it is 9 per 

cent smaller. Capacity is most sensitive to 

the mean velocity conditioned by constant discharge and least sensitive to the mean velocity 

conditioned by constant slope.

These results have a theoretic connection 

with the fact that capacity is more sensitive 

to changes of slope than to those of discharge. 

When discharge is constant the changes of 

velocity are caused by changes of slope, and 

the changes in capacity are "those due to the 

changes of slope. When slope is constant the 

changes of velocity are caused by changes of
RELATION OF CAPACITY TO VELOCITY. 161

discharge, and the changes in capacity are 

those due to changes of discharge. When 

depth is constant the changes of velocity are 

caused by concurrent changes of slope and 

discharge, and the changes in capacity are 

intermediate between those caused by slope 

alone and by discharge alone.

TABLE 53. Comparison of synthetic indexes of relative variation for capacity arid mean velocity, under the several conditions of constant discharge,, constant, depth, and constant slope.

Grade.

G

A>

C) 

D) G! G) 

eneral

Number of comparisons.

4 

9 

13 

3 

1 

2

means....

Means.

/ 

3.34 

3.55 

3.60 

4.14 

4.39 

8.84

3.98

Ivd

2.87 

3.41 

3.55 

3.40 

4.46 

7.54

3.68

i»

1.97 

4.19 

2.96 

2.63 

3.06 

3.86

3.21

Ratios.

£

1.16 

1.04 

1.01 

1.22 

.99 

1.17

1.07

^

O.C9 

1.23 

.83 

.77 

.69 

.51

.91

It is profitable to consider the same facts 

also in relation to depth of current. Postulate 

an initial status, with a particular discharge 

and slope, determining a certain velocity, 

depth, and capacity. First,, increase the slope 

until the velocity is doubled. The capacity is 

increased, let us say (borrowing mean ZK« 

from the table) to 15.8 times its initial amount. 

At the same time the depth is reduced onehalf. Second, after returning to the initial 

status, increase slope and discharge by such 

amounts as to double the velocity without 

changing the depth. The capacity grows 

(mean Ivd) to 12.8 times its original amount. 

Third, starting from the initial condition as 

before, increase the discharge until velocity is 

doubled. The capacity grows (mean IV3) to 

9.2 times its original amount; and the depth 

is at the same time increased, being more than 

doubled. Thus, for the same (doubled) mean 

velocity, the capacity is greater as the depth 

is smaller. Mean velocity is more efficient for 

traction as depth is less.

Now, the primary direct cause of stream 

traction is bed velocity. A concurrent cause 

theoretically exists in the component of gravity 

parallel to the slope, acting directly on the 

load, but for all ordinary stream slopes this 

factor is negligible. Slope and discharge are 

(essentially) indirect causes and are causes 

only in so far as they occasion bed velocity. 

They also determine mean velocity, and, from

20921° No. 86 14 11

one point of view, mean velocity may be said 

to control capacity by controlling bed velocity.

Let us assume, for the moment, that bed 

velocity determines capacity irrespectively of 

depth. Then the variations of capacity above 

described imply corresponding variations of bed 

velocity, and, as the mean velocity does not 

change, we may infer that the ratio of bed velocity to mean velocity is a function: a decreasing function of depth.

This proposition is, to say the least, worthy 

of consideration, but it fails of demonstration 

because the assumption which paved the way 

for it is not valid. It is not true that the relation of capacity to bed velocity is independent 

of depth. In the first place, change of depth, 

when not accompanied by change of mean velocity, causes change in the mode of traction. 

Within the range of the above hypothetic conditions may occur both the dune rhythm and 

the antidune rhythm; and at least one of these 

has an influence on capacity. Moreover, these 

rhythms involve diversity of velocity from 

point to point along the bed, so that " bed velocity" has not a simple definition.

In the second place, the load, or the work of 

traction, reacts on the vertical distribution of 

velocities. In figure 5-1 the line ABC is asFIGURE 54. Ideal curves of velocity in relation to depth, illustrating 

their relation to the zone of saltation. 0=zero of velocity (horizontal) 

and of distance from the bottom (vertical).

sumed as the vertical velocity curve of a stream 

flowing in a straight conduit and bearing no 

load. A'B'C, identical except as to vertical 

dimensions, is assumed as the curve corresponding to the same mean velocity in a current onehalf as deep. Introducing, now, the condition 

of traction, we may represent the upper limit 

of the zone of saltation by the line DE. The 

potential velocities within the zone are evidently quite different for the two depths of current, and they give advantage, for traction, to 

the shallower current. The work of saltation 

tends to retard the lower filaments of current 

and through these the higher filaments, reduc-
V

162 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

ing the mean velocity and increasing the depth. 

If mean velocity and depth be restored by suitable changes in slope and discharge, a new and 

different vertical velocity curve will be obtained 

for each depth. While I am not able to deduce the exact nature of the changes, it seems 

clear that those portions of the new curves 

within the zone of saltation will be contrasted 

in some such way as are the potential curves of 

the drawing, and that for purposes of traction 

the advantage will still be with the shallow 

current. Provided the portions of the velocity curves within the zone of saltation show 

steeper gradient for the current of less depth, 

tractional capacity will for that current bear 

a higher ratio to mean velocity.

RELATIVE SENSITIVENESS TO CONTROLS.

The synthetic index of relative variation for 

capacity in relation to mean velocity, when the 

limiting condition is constant discharge, namely, 

IVQ, has been estimated (p. 158) as 2.60 times 

the corresponding index, lw for capacity in 

relation to slope. The same index has been 

estimated (Table 53) as 1.07 times IVd and 1.19 

times Iva. Combination of ratios indicates 

that Ivd is 2.43 times* and IVs 2.18 times as 

great as Iw. While these figures have an 

appearance of. exactitude, their order of precision is really low. They are built on averages of individual values of indexes, which 

among themselves are highly diversified. At 

best they represent the average of values 

covered by the range of experiments in the 

laboratory, but in part they are based on 

values covering much narrower ranges. Moreover, it was not possible, except in the case of 

IVQ and Iw, to derive the compared series of 

values of the index from data representing 

exactly the same conditions. For these reasons the numerical results should .be accepted 

only as indicating an order of sequence and an 

order of magnitude. The quantitative response of capacity to the change of mean 

velocity is much larger than its response to 

change of slope, probably more than twice as 

large. Minor differences depend on the conditions under which mean velocity varies. 

The response is greatest when velocities are 

subject to the condition of constant discharge, 

less when the restrictive condition is constant 

depth, and least when it is constant slope.

COMPETENT VELOCITY.

The. demonstrations by Leslie, Hopkins, 

Airy, and Law of the proposition that the 

diameter of the largest particle a current can 

move is proportional to the square of the 

velocity involve the principle that the pressure 

of a current is proportional to the square of its 

velocity, and also the assumption that the 

forward pressures on different parts of the 

particle are the same, so that the total pressure 

is proportional to the sectional area of the 

particle. 1 Under that assumption the total 

pressure may be conceived as applied to the 

center of gravity of the particle, a consideration of importance when the motion given to 

the particle is of the nature of rolling or overturning. These assumptions are not strictly 

true, because in the immediate vicinity of the 

channel bed the velocity increases with distance from the bed. Moreover, as we have 

seen (p. 29), the rate of increase.is a diminishing rate. As a consequence of the inequality 

of velocity and its mode of distribution (1) 

the average pressure on the upstream face of a 

large particle is greater than the corresponding 

average pressure on a small particle, (2) the 

point of application of the total pressure (the 

point which determines the lever arm in overturning and rolling) is always above the center 

of gravity, and (3) the point of application may 

be differently related to the center of gravity 

in particles of different sizes. The general 

effect of these qualifying circumstances is to 

reduce the difficulty of moving large particles, 

and thus to make the rate at which competent 

diametsr of particle increases with velocity (at 

any particular level) somewhat greater than 

that of the square of the velocity.

On the other hand, it is to be observed that 

in stream traction the roughness of the channel 

bed is defined by the coarseness of the load, 

and the system of velocities near the bed is a 

function of several things, one of which is the 

roughness. It is by no means impossible that 

the vertical velocity curve of a stream flowing 

over a bed of coarse debris is an enlarged 

replica of the curve of a shallower stream 

flowing over a bed of finer debris, in which case 

the law of Leslie might hold despite the qualifications mentioned above. The problem is too

1 For references see footnote on page 16.
RELATION OF CAPACITY TO VELOCITY. 163

complicated, for the present at least, for full 

deductive treatment, and there are no adequate 

experimental data.

Most of the earlier experiments on competence pertained to flume traction, but it is 

probable that in those of T. Login (1857) the 

conditions were such as to give stream traction. 

The currents he employed were shallow, and he 

measured velocities by means of floats which 

occupied half the depth of the water.1 As 

dimensions of transported particles are not 

included in his report of observations, a numerical formula can not be based on it.

Experimental data on competent velocity for stream traction.

[By T. Login.]

Velocity 

in ft. /sec.

Brick clay, mixed with water and then allowed to 

settle. ........................................ 0. 25

Fresh-water sand. . ............................... .67

Sea sand ......................................... 1. 10

Rounded pebbles, size of peas .................... 2. 00

John S. Owens, experimenting on the transporting power of sea currents, made use of 

small streams flowing from one tide pool to 

another. Measuring velocities by means of 

floats, he tested the ability of currents to move 

pebbles, 0.5 inch to 6 inches in diameter, over 

a channel bed of sand.2 His results are formulated in

where Dl is diameter of pebble in inches, V 

velocity in ft./sec., and TPthe weight in pounds 

of a cubic foot of the material of the pebble. 

This is equivalent to

(82)

where D is the diameter in feet and G the density of the material. 3

Of the Berkeley experiments on competence, recorded in Tables 9 and 10, a single 

series bears on the point under consideration, 

but its bearing is less direct than could be

1 Royal Soc. Edinburgh Proc., vol. 3, p. 475, 1857.

2 Gcog. Jour., vol. 31, pp. 415-420, 1908.

3 See also experiments by T. E. Blackwell, cited in Chapter XII.

desired. By selecting from Table 9 data corresponding to a trough width of 1 foot and a 

discharge of 0.363 ft.3/sec. I was able to make a 

logarithmic plot of diameters of particles of 

grades (B), (C), (D), (E), (G) in relation to 

competent mean Velocities, and this plot gave 

the following equation:

> = 0.0025 __ (83)

The exponent 2.7 is connected with the fact 

that Vm in this case is mean velocity with discharge constant. It is easy to infer from the 

data assembled in Table 53 that if either depth 

or slope had J^een the constant condition in the 

experiments a smaller exponent would be indicated. So the possibility remains that the 

Leslie law is true of mean velocities provided 

the depths increase along with the velocities, 

and the experimental data manifestly do not 

apply to bed velocities.

In order to compare the Berkeley observations with Login's, the diameters of his materials have been computed by equation (83). 

The diameter found for brick clay, 0.00006 foot, 

is much too large; but this result is readily 

accounted for by the fact that adhesion is an 

important factor in the resistance of clay to 

the action of the current. If the computed 

diameters for his "fresh-water sand," 0.00085 

foot, and "sea sand," 0.0032 foot, are correct, 

those materials correspond severally to our 

grades (A) and (D), a very fine sand and a 

coarse sand. The diameter found for "pebbles, size of peas," is 0.016 foot, or one-fifth of 

an inch.

The coefficient obtained by Owens, 0.059 in 

equation (82), is 14 times as large as our coefficient, 0.0042 in equation (83), a contrast 

which accords in a general way with the difference between the classes of phenomena observed. The pebbles he tested were rolled 

over a bed of relatively fine material, which 

gave them a smooth pathway with little resistance, while the grains to be moved in our 

experiments rested among similar grains and 

were less readily dislodged.
CHAPTER VIII. RELATION OF CAPACITY TO DEPTH.

INTRODUCTION.

As a condition controlling capacity for traction, depth, has several distinct aspects; and the 

nature of its control depends altogether on the 

character of associated conditions. Three aspects will here be, considered. They all assume 

that size of. debris and width of channel are 

constant, and they are severally characterized 

by the limiting conditions of constant discharge, 

constant slope, and constant mean velocity. 

In examining the nature of the controls, and in 

comparing them with one another and with 

other controls, use will be made of the synthetic index; and the method of discussion will 

be similar to that of the preceding chapter. 

The symbols for the synthetic index, under the 

three limiting conditions, will be severally

Idd, Ids, and IdY.

WHEN DISCHARGE IS CONSTANT.

A stream of constant discharge, flowing in a 

channel of constant width, can change its depth 

only by changing its mean velocity, and depth 

and velocity vary in opposite senses. Whatever the ratio by which the mean velocity is increased or diminished, the depth diminishes or 

increases in the same ratio. It follows that 

the law of change for capacity in relation to 

mean velocity is the inverse of the law of change 

for capacity in relation to depth. So far as 

these laws are expressed by values of the synthetic indexes,

*dQ~ ^KQIndependent computations of IdQ are therefore unnecessary, as the values of IVQ in Tables 

46, 47, and 48 need only change of sign to 

become the corresponding values of IM . The 

following summary statement of the general 

features of the control of capacity by depth 

is but a condensation and adaptation of the 

statement on pages 157-158.

Under the condition of constant discharge

capacity varies inversely with depth- Its rate

of variation is more than twice the rate at

which it varies with slope of channel. The

164

rate responds to changes in discharge, slope of 

channel, fineness of debris, and form ratio, 

diminishing as those factors increase (with 

possible exception as to form ratio).

Depth, when subject to the condition of constant discharge, varies inversely with slope. 

Estimates of the average rate of variation are 

contained in the last column of Table 48.

WHEN SLOPE IS CONSTANT.

A stream flowing down a constant slope, in 

a channel of constant width, and transporting 

debris of a particular grade changes its depth 

when the discharge is changed. The depth is 

greater as the discharge is greater. The 

capacity for traction also is greater as the 

discharge is greater. Therefore the capacity 

varies in the same sense as the depth.

Table 54 contains 68 values of Ids, computed from data of Tables 12 and 14. The 

method of derivation was identical with that 

already described for Table 49, with the exception that values of depth were used instead of 

values of mean velocity.

TABLE 54. Values of Ids, the synthetic index of relative variation for capacity in relation to depth of current, ivlien slope 

is constant.

Grade.

(A) 

(B)

(C) 

(D) 

(E)

(F) 

(G)

(H)

S

0.6

.6 

1.0 

1.4 

2.0

.6 

1.0 

1.4 

2.0

.6 

1.0 

1.4 

2.0

.6 

1.0 

1.4 

2.0

1.4

2.0

1.0 

1.4 

2.0

1.0 

1.4 

2.0

Value of Ids for width (feet) of 

0.23

1.07 

.89

0.44

2.17

3.44 

2.12 

1.80

0.66

2.02 

2.05 

1.78 

1.59

2.27 

1.98 

2.34 

2.27

1.84 

2.18 

2.01 

1.87

1.02 

1.09

2.40 

2.03

2.74 

2.18 

1.96

2.37

2.95

2.22

1.00

2.40 

2.18 

1.97

1 QQ

2.51 

2.22 

2.04 

2.22

1.97 

2.30 

2.16 

2.75

1.90

1.70 

1.79 

2.00

3.59 

2.87

3.02 

2.91 

2.51

1.32

2.31

2.93 

2.52 

2.30

2.56 

2.85 

2.36

2.03 

2.12

3.88 

3.07 

3.01

1.96

2.28

2.14

1.85 

1.41

3.29 

2.13 

2.65
RELATION OP CAPACITY TO DEPTH. 165

The derivation of means in Table 55 followed 

the precedents of Tables 47 and 50. An examination of logarithmic plots shows that Ids 

varies inversely, but only slightly, with discharge. The means in Table 55 show that it 

varies inversely with slope and fineness, but 

they give no clear indication of its essential relation to width.

TABLE 55. Partial means based on Table 54, illustrating 

the control oflds, by slope, fineness, and width.

Number of 

values.

14 

18 

13

2 

15 

10 

6 

4 

8 

3

2 

4 

19 

11 

6

Means of Ids for slope (per cent) of 

0.6

2.66

1.0

2.13 

2.12

1.4

2.10 

2.40

2.0

2.24

Means of Ids for grade 

(A)

2.29

(B)

2.53 

2.08 

2.13

1.81

(C)

2.39 

2.33

2.22

2.22

2.28

2.20

(D)

2.12

2.23

2.20

2.02

(E)

1.55

(F)

2.72

(G)

2.78

2.29

(H)

2.51

Means of Its for width (feet) of 

0.23

0.98

0.44

2.38

0.66

1.91

2.16 

1.98

1.00

2.36 

2.37

1.32

2.70 

2.59

1.96

2.24

For each of the 68 values of lds, the corresponding value of the synthetic index of the 

relation of capacity to discharge, 73 , was computed, the relations being such that the paired 

values represent exactly the same conditions. 

It was found that, without exception, the values 

of Ids are greater than the companion values 

of 73 . Table 56 contains a series of comparative means and their ratios, the ratio of general 

means being 1.62.

TABLE 56. Synthetic indexes, comparing the control of capacity by depth with the control by discharge, and comparing 

both controls with grades of debris.

Grade.

£

9 f 

g

Number 

of 

separate 

determinations.

2 

17 

17 

10 

6 

4 

9 

3

68

Mean Id-s2.29 

1.95 

2.41 

2.12 

1.58 

2.72 

2.81 

2.51

2.25

Mean la.

1.37 

1.30 

1.36 

1.26 

0.99 

1.81 

1.76 

1.70

1.39

Mean Ids

Mean /a

1.67 

1.50 

1.77 

1.68 

1.59 

1.50 

1.59 

1.48

1.62

Mean /3

Mqan/ds

0.60 

.67 

.56 

.59. 

.63 

.67 

.63 

.68

.62

DEPTH VERSUS DISCHARGE.

The comparison of Ida with 73 affords, incidentally, an estimate of the relative variation 

of depth and discharge. The rate of variation 

of capacity with depth being Ids, the rate of 

variation of depth with capacity is I/Ids', and, 

the rate of variation of capacity with discharge 

being 73 , the rate of variation of depth with discharge is 11 Us + 73 = 73 /7,iS . The values of 73 /7ds 

in Table 56 are therefore estimates of the variation of depth in relation to discharge, under the 

limiting condition of constant slope. They are 

of the quality of the synthetic index and are 

based on the general range of conditions 

realized in the laboratory, except those in the 

neighborhood of competence. For this range 

of conditions depth varies, on the average, with 

the 0.62 power of discharge.

WHEN VELOCITY IS CONSTANT.

By interpolation from the data recorded in 

Tables 12 and 14, values of capacity and depth 

may be found corresponding to selected values 

of mean velocity. Such values were derived 

for mean velocities of 2, 3, and 4 ft./sec., and 

from them were computed the 42 values of hv 

in Table 57.

TABLE 57. Values of Idv, the synthetic index of relative 

variation for capacity in relation to depth of current, when 

mean velocity is constant.

Grade.

(A) 

(B)

(C) 

(D)

(E) 

(G) 

(II)

vm

3 

4

2 

3

4

2 

3

4

2 

3 

4

2 

3

3 

4

3

Value of lav for width (feet) of 

0.23

-0.26

0.44

-0.48

- .75

0.66

+0.09 

- .14

_ -)A

0

- .57

+ .09

- .34 

- .22

-1.08

-2.69

- .75

1.00

-0.26 

-1.02

+ .21 

- .36

- .25

- .27 

- .75 

-1.50

-1.03

- .93

-1.35

-1.19

1.32

-0.32

+ .25 

+ .18 

- .37

+ .22 

- .32

- .35

+ .08 

- .92

-3.17

1.06

-0.75 

- .69

+ .32 

+ .19

-1.50 

- .44

Most of these values are negative, but nine 

are positive. The mean of the 42 values, combined with regard to sign, is 0.54. A positive value, considered by itself, indicates that 

capacity varies directly with depth, and a negative value that the variation is inverse. At
166

least two interpretations may be considered 

(1) that the actual variation is usually inverse 

but under some conditions is direct; (2) that 

the law of inverse variation is general and that 

the apparent exceptions are due entirely to the 

imperfection of the data. The second interpretation accords the better with the analogies 

afforded by other branches of the subject and 

is provisionally accepted.

The, interpretation directs attention to the 

general discordance of the computed values of 

I<ir, and a suggestion may be made as to the 

cause of that discordance. First, the measurement of depth was the most difficult of the 

direct measurements performed in the laboratory, and it was peculiarly subject to possibilities of systematic error. Second, the range 

of slopes through which depth could be observed was much less than in the case of 

capacity, and this made the work of adjustment less satisfactory. Third, in the computation of this particular index the depth data 

enter twice, and they enter in such way that 

their errors cumulate instead of canceling.

In Table 58 the relations of !<& to various 

conditions are shown by the comparison of 

corresponding means. Idv is seen to vary 

directly with mean velocity, and it may fairly 

be inferred to vary inversely with fineness, but 

the data as to width of channel are contradictory and inconclusive.

The control of the index by mean velocity 

affords information as to its relation to slope 

and discharge. Mean velocity varies directly 

with both slope and discharge, and it does not 

change without corresponding change in at 

least one of these factors. It follows that the 

index, which varies directly with mean velocity, 

also varies directly with at least one of the 

factors slope and discharge. In the groups of 

data from which values of the index were computed, increase of velocity was associated 

either with increase of slope or with increase of 

both slope and discharge but in no case with 

decrease of slope or discharge. It seems, therefore, proper to infer that l#r varies in magnitude directly with both slope and discharge. 

In this respect it stands as an exception among 

the indexes of relative variation connected 

with traction. All other species of 7 and also i 

vary inversely with slope and discharge. As a 

check on this exceptional result, certain probable errors have been computed. In the upper 

division of Table-58 0.19, the mean of 12 

values of Idv with a velocity of. 2 ft./sec., is 

compared with 0.41, the corresponding mean 

for a velocity of 3 ft./sec. The difference 

between these means, 0.22, has a probable 

error of ±0.13. A similar difference, appearing in the comparison of indexes for velocities 

of 3 and 4 ft./sec., is -0.24±0.18. As the 

two differences give testimony of the same 

tenor, their joint evidence is stronger than 

that of either separately, so that the discussion 

of the residuals leaves the presumption in favor 

of the conclusion that this particular index 

constitutes a real exception in its relation to 

slope and discharge.

TABLE 58. Partial means based on Table 57, illustrating the 

control of I<iv, by mean velocity, fineness and width.

Number of

values.

12 

9

2 

10 

7 

2 

5 

1

1 

2 

9

8 

5

Mean of lav for mean velocity (ft./ses.) of 

2

-0.19

3

-0.41 

- .57

4

-0.81

Mean of lav for grade 

(A) (B) (C) (D) (E)

-0.54 +0.18 .......................

........ - .11 -0.32 ...............

................ - .15 -0.57 .......

(F) (0) . (H)

...... -l.CO .......

...... -1.08 -0.75

Mean of lav for width (feet) of 

0. 23 0. 44

-0. 26 -0. 48 

.......... - .61

0.66 1.00

-0.07 ..........

- .64 -0.58

.......... - .49

 1.32 1.96

-0.61 ..........

.......... -0.44

THE THREE CONDITIONS COMPARED.

When depth is increased without change of 

slope (or width or grade of debris), its increase 

is effected by increase of discharge, with the 

result that capacity is increased, so that capacity is an increasing' function of depth. 

When .depth is increased without change of 

discharge, its increase is effected by reducing 

slope, with the result that capacity is reduced, 

so. that capacity is a decreasing function of 

depth. When depth is increased without 

change of velocity, its increase requires increase
RELATION OF CAPACITY TO DEPTH. 167

of discharge accompanied by diminution of 

slope; and as these changes have opposite influences on capacity, it is not evident a priori 

whether capacity will be enlarged or reduced. 

The experimental data show that it is slightly 

reduced, so that capacity is a decreasing function of depth.

When depth is reduced without change of 

slope, and the reduction is continued progressively, a stage is eventually reached in which the 

velocity is no longer competent for traction. 

It is probable, therefore, that, under this condition, an approximate formula for C=f(d) 

might involve a depth constant and be similar 

to the formula (64) used for 0=f(Q).

Reduction of depth without change of discharge involves increase of velocity, and it is 

evident that competence does not lie in that 

direction. But increase of depth involves reduction of velocity and leads eventually to a 

competent velocity. The limiting depth corresponding to competence is therefore a great 

depth instead of a small one. As mean velocity now varies inversely with depth, a coordinate formula might take the form

B being used as a general constant and d as a 

constant depth.

When depth is reduced without change of 

mean velocity, the efficiency of the mean 

velocity is enhanced and competence is not 

approached. When depth is increased, the 

efficiency of the unchanged mean velocity is 

diminished and a (large) competent depth 

may, under some conditions, be realized.

To show the numerical relations of the indexes by which these three capacity-depth functions are severally characterized, certain means 

are assembled in Table 59. It was not found 

possible to procure values of the different indexes representing closely the same conditions, 

and what was done was to derive for each index 

the mean of all determinations made for each 

particular grade of debris.

The arrangement by grades points again to 

the fact that all the indexes vary inversely with 

fineness of d6bris, but the rate of variation is in 

fact somewhat greater than these series of 

values suggest. The observations on the 

coarser grades were made with steeper average

slopes and larger average discharges than those 

on the finer grades, and the effect of high slopes 

and large discharges (except in case of ldv) is 

to reduce the indexes of relative variation.

TABLE 59. Comparison of synthetic indexes of relative variation for capacity and depth, under the several conditions of 

constant discharge, constant slope, and constant mean 

velocity.

Grade.

(A}

$(E)

m

(G) 

(H)

Number of separate determinations.

IdQ

5 

18 

19 

9 

5 

5 

9 

3

73

Ids

'2 

17 

17 

10 

6 

4 

9 

3

68

IdV

3 

12 

12

7 

2

""5" 

1

42

Means.

/dQ

- 3.62 

- 4.24 

- 3.58 

- 4.39 

- 5.17 

- 8.73 

- 9.45 

-10. 01

- 5.28

Ids

2.29 

1.95 

2.41 

2.12 

1.58 

2.72 

2,81 

2.51

2.25

IdV

-0.59 

- .15 

- .33 

- .56 

- .98

-1.30 

- .75

- .56

IdQ 

Ids

-1.57 

-2.17 

-1.49 

-2.07 

-3.29 

-3.21 

-3.36 

-3.99

-2.34

From the general means at the bottom of the 

table it appears that, for the range of laboratory 

conditions, capacity is 2.34 times as sensitive 

to the control of depth when the limiting condition is constant discharge as when the limiting condition is constant slope, and about nine 

times as sensitive as when the limiting condition is constant mean velocity.

One of the results of the discussion is to 

emphasize the importance, when considering 

the relation of tractional work to . depth, of 

sharply discriminating the conditions under 

which depth is regarded as a variable.

So far as the variations of the capacity-depth 

indexes admit of formulation in the symbols 

used for other indexes,

(84) 

(8'4a)

COMPARISON OF CONTROLS BY SLOPE, DISCHARGE, MEAN VELOCITY, AND DEPTH.

In Chapter V the general sensitiveness of 

capacity to slope is compared with that of 

capacity to discharge by means of coordinate 

values of the exponent i. In Chapter VII the 

sensitiveness to slope is compared with sensitiveness to mean velocity (with discharge constant) by means of coordinate values of /. 

The two methods of comparison are so far
168 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

related that their results should be fairly har- measure last mentioned has been compared

monious. In the present chapter the same with a coordinate measure of the sensitiveness

measure of sensitiveness to mean velocity, con- of capacity to discharge, thus completing a

verted by change of sign to a measure of sensi- circle and affording opportunity for a checking

tiveness to depth (with discharge constant), of estimates in respect to consistency. The

has been compared with a similar measure ratios resulting from the four comparisons are

for depth with slope constant. Finally, the as follows:

index of control by slope 

index of control by discharge

index of control by velocity (discharge constant) _ 

index of control by slope ~~"

index of control by depth (slope constant) _, fi9 

index of control by discharge

index of control by velocity (discharge constant) _ _ 

index of control by depth (slope constant)

The combination, by multiplication, of the first and second equations gives

index of control by velocity (discharge constant) 

index of control by discharge

The combination of the third and fourth gives an equation with identical first member, but 

the second member is 3.79. The two results differ by 8 per cent.
CHAPTER IX. EXPERIMENTS WITH MIXED GRADES.

ADJUSTMENT AND NOTATION.

The fractional load of a natural stream includes particles with great range in size. The 

grades of debris used in the laboratory had narrowly limited ranges. Although the reasons 

for this limitation were believed to be adequate, 

the possibility was recognized that the laws 

discovered by the use of artificial grades might 

not apply without modification to natural 

grades; arid in view of this possibility a series 

of experiments were arranged to bridge over 

the interval between the artificial and the natural in this particular respect. The same apparatus and the same general methods being used, 

observations were made first on mixtures of two 

grades, then on mixtures of three or more, and 

finally on a natural combination of sizes.

When work on mixtures was begun, need was 

soon found for a modification of the experimental procedure. During the automatic process of adjusting the slope to the load the current acted unequally on the components of the 

mixture, carrying forward an undue share of 

the finer part and depositing an undue share of 

the coarser. To escape the difficulties introduced by this partial re-sorting, a run was interrupted after it had gone far enough to indicate the approximate slope, the debris was 

taken from arrester and experiment trough and 

was throughly remixed, and then the approximate slope was artificially constructed in the 

trough, after which the run was continued.

The observational data are contained in divisions (J) and (K) of Table 4. The values of 

capacity in relation to slope were adjusted in 

the same manner as with individual grades, 

except that each value of o for an adjusting 

equation was derived from the data of the particular observational series, without influence 

from related series. The adjusted capacities 

are recorded in Table 60, and with them are 

fractional capacities computed for the several 

grades composing the mixtures. Table 17 contains the constants of the adjusting equations 

and also series of values of the index of relative 

variation.

In each set of experiments with mixture of 

two grades the proportions of the components 

were varied. The usual series of proportions 

was approximately 4:1, 2:1, 1:1, 1:2, and 

1:4. To denote these mixtures, a notation 

has been adopted similar to one employed in 

chemistry; for example, a mixture of grades 

(B) and (F) in the proportion 4:1 is designated (B4Ft).

The proportions of some of the mixtures are 

told accurately by the subscript figures; those 

of others only approximately. The theoretic 

or standard mode of apportionment was by 

weight of dry material, but when this was not 

convenient the material was weighed or measured in moist or saturated condition, and the 

actual ratio for dry weight was afterward 

learned by computation. The actual proportions are given in percentages in Table 4 (J).

TABLE 60. Adjusted values of capacity in relation to slope, for mixtures of two or more grades of debris and for an unsorted

natural alluvium, based on data, of Table 4 (J) and (K).

Q.............. .............................

S

0.5 .........................................

.6 .........................................

.7 .........................................

.8 .........................................

.9 .........................................

1.0 .........................................

1.2.........................................

1.4... ............... .................o.....

1.6..........................................

1.8.........................................

2.0.........................................

Probable error (per cent) ........;..........

(A,C,)

0.363

(A). (C) Total.

(A8GO

0.363

(A) (G) Total.

(A2Gi)

0.363

(A) (G) Total.

(Aid)

0.363

(A) (G) TotaL

Value of C.

16 

23 

30 

37 

44 

51

65 

80 

95

16 

23 

30 

37 

44 

51

66 

81 

96

32 

46 

60 

74 

88 

102

131 

161 

191

13 

22 

32 

44 

57 

71

99 

131 

165

4 

7 

11 

15 

19 

23

33 

44 

55

17 

29 

43 

59 

76 

94

132 

175 

220

2.8

13 

21 

29 

39 

49 

60

85 

113

7 

10 

15 

19 

24 

30

42 

56

20 

31 

44 

58 

73 

90

127 

163

4.0

8 

11 

14 

18 

22

32 

44 

57

9 

12 

15 

19 

23

33 

45 

58

17 

23 

29 

37 

45

65 

89 

115

169
170 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 60. Adjusted values of capacity, in relation to slope, for mixtures of two or more grades of debris and for an unsorted 

natural alluvium, based on data of Table 4 (J) and (K) Continued.

Q... ........................................

5

0.5.........................................

.6.........................................

.7.........................................

.8.........................................

.9.........................................

1.0.........................................

1.2.........................................

1.4.........................................

1.6.........................................

1.8.........................................

2.0.........................................

Probable error (per cent) ...................

Q..... ......................................

5

0.5..........................................

.6..........................................

.7..........................................

.8..........................................

g

i.o..........................................

1.2.......:.!!................................

1.4........................:.............-...

1.6..........................................

1 fi

Q... ..........................:...........:.

(A,G2 )

0.363

(A) (G) Total.

(AiGO

0.363

(A) (G) Total.

(B4Fi)

0.363

(B) (F) Total.

(BiF,)

0.363

(B) (F) Total.

Value of C.

7

10 

14 

19

14

21 

28 

37

21

31 

42 

56

(BiF,)

0.363

(B) (F) Total.

3

4 

5 

7 

9 

12

9

14 

20 

26 

34 

42

1

12

18 

25 

33 

43 

54

(BiF2 )

0.363

(B) (F) Total.

26 

37 

48 

60 

72 

83

107 

133

7 

10 

14 

17 

20 

23

30 

37

33 

47 

62 

77 

92 

106

137 

170

2.1

(B,F<)

0.363

(B) (F) Total.

22 

29 

38 

47 

54 

62

78 

93 

109

12 

17 

21 

25 

30 

35

43 

52 

61

34 

46 

59

72 

84 

97

121 

145 

170

1.1

(C«E,)

0.363

(C) (E) Total.

Value of C.

22 

26 

31 

37

48 

61 

75 

89

24 

30 

. 35 

41

55 

.68 

83 

100

46 

56 

66

78

103 

129 

158 

189

0.1

(CsE.)

0.182

(C) (E) Total.

i

9 

11

12

17 

21 

26

19 

23

28

37 

48 

59

28 

34 

40

54 

69 

85

(C2E,)

0.363

(C) (E) Total.

2 

3 

4 

5

6

7 

9

ii

13 

15 

18

25 

32 

40

13 

16 

19 

23

31 

39 

49

(CiEO

0.182

(C) (E) Total.

29 

40 

51 

62

74

97 

122 

146 

171

8 

11 

14 

17 

20

26 

32 

39 

46

37 

51 

65 

79 

94

123

154 

185 

217

0.7

(C,E,)

0.363

(C) (E) Total.

0.5..........................................

.6... .. .....................................

. 7.. ....................................... -

.8..........................................

q 1.0..........................................

1.2................................... ......

1.4..........................................

1.6..........................................

1.8..........................................

2.0..........................................

2.2..........................................

44

55

66

77

24

29

35

42

68

84

101

119

20 88

36

44

53

70

88

106

124

11

15

19

24

28

37

47

57

67

31

43

55

68

81

107

135

163

191

0.7

12

14

19

24

29

36

42

53

12

15

20

26

32

38

46

58

24

29

39

50

61

74

88

111

1.8

14

19

25

30

42

53

65

78

90

15

21

26

32

44

57

71

84

97

on

40

51

62

86

110

136

162

187

2.3

V
BXPEBIMENTS WITH MIXED GRADES. 171

TABLE 60. Adjusted values of capacity, in relation to slope, for mixtures of two or more grades of debris and for an unsorted 

natural alluvium, based on data of Table 4 (J) and (K) Continued.

Q........ ..................................

Component.. . .............................

S

0.5.........................................

.6.........................................

.7.........................................

.8.........................................

.9.........................................

1.0.........................................

1.2.........................................

1.4.........................................

1.6.........................................

1.8.........................................

2.0.........................................

2.2.........................................

Grade. .....................................

Q.................... .......................

S

0.5.........................................

.6.........................................

.7.........................................

.8.........................................

9

1.0.........................................

1.2.........................................

1.4.........................................

1.6.........................................

1.8.........................................

2.0.........................................

Q..........................................

S

0.6.........................................

.7.........................................

.8.........................................

.9.........................................

1.0.........................................

1.2.........................................

1.4.........................................

1.6.........................................

1.8.........................................

2.0.........................................

Probable error (per cent). ............

(CiEs)

0.182

(C) (E) Total.

(CiE2)

0.363

(C) (E) Total.

(C!EO

0.182

(C) (E) Total.

(CiE<)

0.363

(C) (E) Total.

Values of C.

7

9 

12 

15 

18 

22 

25

15

21 

27 

34 

41 

48 

55

22

30 

39 

49 

59 

70 

80

2.2

(C^Gi)

0. 363

(C) (G) Total.

8 

11 

13 

16

22

28 

34 

42 

48 

56

17 

23 

29 

35

.48 

63 

77 

92 

108 

124

25 

34 

42 

51

70 

91 

111 

134 

156 

180

5.6

(C2G,)

0. 363

(C) (G) Total.

3 

4

5 

6 

8 

9 

11 

12

13 

15

20 

26 

32 

39 

45 

53

16 

19

25 

32 

40 

48 

56 

65

0.1

(CiG,)

0.363

(C)' (G) Total.

6 

8

11 

14 

18 

21 

25

28 

33

46 

61 

76 

92 

109

34 

41

57 

75 

94 

113 

134

1.6

(C,G 2 )

0. 363

(C) (G) Total.

Values of C.

17 

24 

32 

42 

54 

65

93 

126 

164 

' 208

4 

6 

8 

. 10 

13 

16

23 

32 

41 

52

21

30 

40 

52 

67 

81

116

158 

205 

260

1.9

(E«Gi)

0.363

(E) (G) Total.

11 

19 

27 

36 

45 

55

74 

96 

119 

. 141

6 

9 

14 

18 

23 

27

37 

48 

59 

71

17 

28 

41 

54 

68 

82

111 

144 

178 

 212

1.2

(E2Gi)

0.363

(E) (G) Total.

10 

15 

21 

27

40 

53 

67 

80

10 

18 

22 

28

40 

53

67 

81

20 

31 

43 

55

80 

106 

134 

161

4.4

(E,Gi)

0. 363

(E) (G) Total.

4 

6

11 

17 

23 

31

8 

12

22 

33 

47 

61

12 

18

33 

50 

70 

!)2

2.2

(E,G8)

0.363

(E) (G) Total.

Value of C.

10 

13 

17 

21 

25

35 

47 

60 

75

2 

3 

4 

5 

6

9 

12 

15 

19

12 

16 

21 

26 

31

44 

59 

75 

94

6 

9 

13 

18

29 

43 

59

3 

5 

7 

9

15 

22 

30

9 

14 

20 

27

44 

65

89

8 

10 

13

18 

24 

31 

37

8 

10 

13

19 

25 

31 

38

16 

20 

26

37 

4.9 

62 

75

4 

6

8 

11 

14 

18 

23

9 

11

16 

22 

29 

37 

45

13 

17

24 

33 

43 

55 

68

1.4
172 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

TABLE 60. Adjusted values of capacity, in relation to slope, for mixtures of two or more grades of debris and for an unsorted 

natural alluvium, based on data of Table 4 (J) and (K) Continued.

Q.... ....................

5

0.6.......................

.7.......................

.8.......................

g

i.o.......................

1.2.......................

1.4.......................

1.6.......................

1.8.......................

9 n

(AAG2)

0.363

(A) (P) (G) Total.

(C-D-E^CH)

0. 182

(C) (D) (E) (F) (G) Total.

0.363

(C) (D) (E) (F) (G) Total.

Value of C.

2 

4 

7 

9

16 

24 

34 

45

2 

4 

7 

9

16 

24 

34 

45

5 

9 

13 

20

33 

50 

69 

91

9 

17 

27 

38

65 

98 

137 

181

4.0

8.2 

10.6 

12.8 

15.2

20.3 

26.6 

32.8 

39.6 

47.3

Q.. ...................'......................................

*

0.4..........................................................

.5..........................................................

.6..........................................................

.7..........................................................

.8...........................................................

.9........................................................... i.o...........................................................

1 9

1. 4 ...........................................................

1.6..........................................................

1 O

9 n

2. 2.... .............. '.. ..................................... -

2.4..........................................................

26

Probable error (per c

6. 6 2. 3 

8.2 2.8 

10. 0 3. 4 

11. 8 4. 1

16.0 5.5 

20. 6 7. 1 

25. 6 8. 8 

30. 8 10. 6 

36. 8 12. 6

1.1 0.3 18.9 

1.4 .5 23.5 

1. 7 .6 28. 4 

2. 0 .7 33. 9

2.7 .9 45.8 

3. 5 1. 2 59 

4. 4 1. 5 73 

5.3 1.8 88 

6. 3 2. 1 105

24.2 

29.2 

35.0 

40.9

53.5 

67.5 

81.9 

95.4

18.9 6.5 3.2 1.1 

22.8 7.8 3.9 1.3 

27.3 9.4 4.7 1.6 

31.8 10.9 5.4 1.8

41.6 14.3 7.1 2.4 

52.5 18.0 9.0 3.0 

63.7 21.8 10.9 3.6 

74.2 25.4 12.7 4.2

(C^E^G.).

0.545

(C) (D) (E) (F) (G) Total.

54 

65

78 

91

119 

150 

182 

212

0.4

Natural.

0. 182 0.363

Values of C.

36.4 

44.1 

50.4 

59.4 

68.0

85.0 

103 

121.5

28.4 

34.3 

39.9 

46.2 

52.8

66.1 

80.2 

94.5

9.7 

11.8 

13.7 

15.8 

18.1

22.7 

27.5 

32.4

4.8 

5.9 

6.8 

7.9 

9.0

11.3 

13.7 

16.2

1.6 

2.0 

2.3 

2.6 

3.0

3.8 

4.6 

5.4

81 

98 

114 

132 

151

189 

229 

270

0.4

15 

20 

26 

33

47 

63 

80 

98 

117

138 

159 

181

2.6

17 

26 

36 

47 

60 

75 

91

126 

168 

213

0.8

MIXTURES OF TWO GRADES.

The relation borne by the traction of mixtures to the traction of separate grades is most 

clearly shown by the records from varied 

combinations of two grades only. In all 

experiments with mixtures the same width of 

channel, 1 foot, was used; and from the adjusted capacities for mixtures of two it is 

possible to select a full set associated with the 

same discharge, 0.363 ft.3/sec., and the same 

slope, 1.4 per cent. These capacities are 

arranged for comparative examination in 

Table 61, and to them are added, from Table 

12, the corresponding capacities for the separate grades.

TABLE 61. Capacities for traction, with varied mixtures of two grades.

(A) 

(A3G,) 

(A2Gi)

(A,Gi)

(AiGs)

(A,G4)

(G)

Capacities.

Total.

185 

175 

169

89

42

25

16

(A)

185 

131 

113

44

14

5

0

(G)

0 

44 

56

45

28

20

16

(C) 

(C4 G, 

(C2G,

(C,G!

(QGj

(G)

Capacities.

Total.

143 

158 

144

106

50

16

(C)

143 

126 

96

53

17

0

(G)

0 

32

48

<53

33

16

1

1

(B) 

(B4F,) 

(BzF,)

(B,F,)

(B!F2)

(B,F4 )

(F)

Capacities.

Total.

149 

170 

145

129

69

39

33

(B)

149 

133 

93

61

21

7

0

(F)

0 

37 

52

\68

48

32

33

(C) 

(C4EO 

(CsEO

(C,EO

(C,E2)

(CtE4)

(E)

Capacities.

Total.

143 

154 

135

110

91

75

62

(C)

143 

122

88

53

28

14

0

(E)

0 

32

47

57

63

61

62.

(E) 

(E4Gi) 

(E2G,)

(E!Gi)

(E,G2)

(G)

Capacitias.

Total.

62 

59 

65

49

33

16

(E)

62 

47 

43

24

11

0

!

(G)

0 

12 

22

25

22

16
EXPERIMENTS WITH MIXED GRADES. 173

Each vertical column of capacities shows the 

tractional power of the current, first for the 

finer component alone, then for mixtures with 

progressively increasing shares of the coarser 

component, and finally for the coarser alone. 

The order of the different groups is that of the 

contrast in fineness between the finer and 

coarser components. If the linear fineness of 

the finer be divided, in each case, by the fineness of the coarser, the ratios obtained are 

(AG) (CG) (BF) (CE) (EG) 

16.2 9.7 8.5 3.4 2.9

(A)'

FIGURE 55. 'rractional capacity for mixed debris, in relation to proportions of component grades.

The data of the table are plotted, with a 

different arrangement, in figures 55 and 56. 

In figure 55 the total capacities are plotted in 

relation to the proportions of fine and coarse 

in the mixtures. The vertical scale being of 

capacity, the horizontal, if read from right to 

left, is of percentage of the finer component, 

or if read from left to right, is of percentage of 

the coarser. Figure 56 (p. 174) shows, in parallel 

columns, the capacities for the component 

grades, those for the coarser components being 

at the left.

Attention may first be directed to the total 

capacity curves of figure 55. If the capacities

for mixtures were related in the simplest 

manner to the proportions of components, the 

curve of capacities would be a straight line 

joining the .points (at the extreme left and 

extreme right) given by the capacities for the 

components. The (EG) curve that placed 

lowest in the diagram lying wholly above such 

a hypothetic line, would suggest that all mixtures give an advantage in traction, but this 

suggestion is. not supported by the other 

curves. The upper curve, for example, would 

He as much below as above a straight line 

joining its extremities. The capacity for the 

mixture appears always to exceed what may 

be called the pro rata.estimate when the finer 

component has the higher proportion, but it 

may fall below that estimate when the coarser 

component predominates.

The curves of figure 56 show the capacities 

apportioned to the components of the mixtures. Each one represents that portion of 

the total load which consists of the material 

of one component. In each curve of the lefthand column the ordinate at the right represents the load of the coarser material when by 

itself, and the successive ordinates toward the 

left show how the load is modified by the 

admixture, in the material fed to the current, 

of - gradually increasing percentages of the 

finer de*bris. There are no observations with 

very small percentages of the finer material, 

but in each case the second fixed point of the 

curve shows an increase of load. The addition 

of the finer de"bris not only increases the total 

capacity but increases the capacity for the 

coarser debris. The amount of the latter 

increase appears to be greater as the contrast 

in fineness of components is greater, and in 

the extreme case the capacity for the coarser 

is multiplied by 3.5. The capacity of the 

current for debris of grade (G) is 16 gm./sec.; 

but when that debris is mixed with twice its 

weight of grade (A), which is 16.2 times as 

fine, the capacity for the mixture is so large 

that one-third of it, apportioned to grade (G), 

is 56 gm./sec. In three of the five suites of 

experiments the highest capacity recorded for 

the coarser debris corresponds to the mixture 

of 1:1. In the others it corresponds approximately to 2:1 and 1:2. If the position of the 

maximum is related to contrast in fineness, it 

is associated with a larger ratio of the finer 

component when the contrast is great.
174 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

The corresponding curves for the finer component (at the right in fig. 56) show that 

capacity for finer debris is influenced by the 

admixture of coarser, but not in the same 

way. Each addition of the coarser reduces 

capacity, the rate of reduction being at first 

gradual, then more rapid, and afterward 

gradual.

The general effect of the addition of fine 

material being to increase capacity, and. the 

effect of adding coarse material to reduce 

capacity, let us now inquire the effect of

diversifying the material of the load by exchanging part of it for a finer grade and part 

for a coarser. Probably no general answer to 

this question may be derived from our data, 

but a partial answer is possible if we assume 

that the initial grade is separated from each 

of the substituted components by the same 

contrast of fineness, expressed as a ratio. 

To take a concrete example, from Table 61, 

the capacity for grade (A) is 185 gm./sec., 

for grade (G) 16 gm./sec., and for (AjGj), 

their equal mixture, 89 gm./sec. When for

..(GJ

 (G.)

FIGURE 56. Tractional capacities of components of mixed grades, in relation to the percentages of the components in the mixtures. Curves 

at left show capacities for coarser component, at right for finer. Ratios at left show percentages of finer component, at right of coarser.

half of (A) there is substituted an equal 

amount of (G), which is 16.2 times coarser,

89 the capacity is changed in the ratio

When for half of (G) there is substituted an 

equal amount of (A), which is 16.2 times finer,

89 the capacity is changed in the ratio v^ = 5.56.

The geometric mean of these ratios, 1.64, may 

plausibly stand for the effect of substituting 

an equal mixture of (A) and (G) for a grade 

symmetrically intermediate between (A) and 

(G). The method is easily criticized, and its 

assumptions will certainly not bear close

scrutiny, but it nevertheless yields a sort of 

composite which is of use in showing that the 

general effect of diversifying a stream's tractional load is to enlarge capacity. Corresponding composite ratios have been obtained 

from the other examples of Table 61 and are 

given below.

Grade.

(AG

(CG

(BF

(CE

(EG^

Ratio of 

fineness.

16.2

9.7

8.5

3.4

2.9

Ratio of

capacity 

change 

attributed

to mixture.

1.64

2.22

1.72

1.17

1.56
EXPEEIMENTS WITH MIXED GBADES. 175

The composite ratios have been arranged in 

the order of the ratio of fineness, but the comparison shows no correspondence. They prove 

equally inharmonious when compared with the 

fineness of the finer component, the fineness of 

the coarser component,, or the fineness of the 

mixture. Their irregularities must be ascribed to observational errors and to causes not 

at present to be discriminated from observational errors. It is of interest, however, to note 

that the large measure of capacity change associated with the (CG) combination might be 

inferred also from a comparison which involves 

a different viewpoint and also some practically 

independent data. If we think of grades (A), 

(C), and (E) as modifiers of capacity for grade 

(G), we may compare their efficiencies by means 

of the following quantities, taken from Table 61:

Grade.

(AiGi) 

(CiO,) 

(EiGi)

Capacity.

89 

106 

49

Grade.

(A,G 2) 

(CiG,) 

(EiG,)

Capacity.

42 

50 

33

The superiority of grade (C) as a modifier for 

(G) is thus brought out without making use of 

the capacities for the uncombined grades (A), 

(C), (E), and (G); and the result from mixtures of 1:1 is supported by that from mixtures 

of 1:2.

CONTROL BY SLOPE AND DISCHARGE.

The preceding comparisons are conditioned 

by a discharge of 0.363 ft.3/ sec., a slope of 1.4 

per cent, and a channel width of 1 foot. With 

a different set of conditions a different set of 

quantitative relations would be found, and the 

qualitative also would doubtless be modified. 

The observations on mixtures included no other 

width, and there was but a single set of experiments using a different discharge, but the 

range in slope was coordinate with that for the 

separate grades.

Figure 57 shows the capacity-slope curves 

for the (AG) set of experiments, figure 58 for 

the (BF) set, and figure 59 for the (CE) set. 

In figure 57 the curves for mixtures form a 

graded series between those for the component 

grades, and there is almost perfect harmony of 

form and attitude. As the points representing 

capacities associated with a slope of 1.4 per 

cent all lie in the same vertical line, and as 

similar points for another slope lie in some

other vertical line, it is evident by inspection 

that inferences from data of any other available 

slope would be practically identical with those 

from the slope of 1.4 per cent. It is also evident

200

012

Slope -

FIGURE 57. Curves of capacity in relation to slope for grade (A), grade 

(G), and mixtures of those grades. The ratios of components in the 

mixtures are indicated.

that indexes of relative variation, \ or 71; for 

the mixtures constitute, with those for the 

components, an orderly system. The same 

remarks apply also to the (CE) groups of

200

J:2

0 I 2

Slope

FIGURE 58. Curves of capacity in relation to slope for grade (B), grade 

(F), and mixtures of those grades. The ratios of the components in 

the mixtures are indicated.

curves, but they do not apply to the (BF) 

group in figure 58. The attitudes of the curves 

for mixtures are there out of harmony with the 

attitudes of the * (B) and (F) curves. The 

curves for the mixtures seem to belong to a
176 TBANSPOBTATIOJST OF DEBBIS BY BURNING WATEB.

different system, intersecting or tending to 

intersect the curves for the components. Evidently the indexes of relative variation are 

inharmonious, and evidently the inferences 

drawn from data for the slope of 1.4 per cent 

would not be duplicated by a discussion of 

data from a slope of 1.0 per cent or 1.6 per cent.

Slope

FIGURE 59. Curves of capacity in relation to slope for grade (C), grade 

(E), and mixtures of those grades. The ratios of the components in 

the mixtures are indicated.

Graphic comparison has been extended to the 

remaining data of Table 60, so far as dual mixtures are concerned, but no marked discordance 

has been discovered outside of the (BF) group. 

Although the cause of the exceptional discordance has not been found, I believe that it 

should be ascribed to some exceptional though 

unknown circumstance and not be permitted 

to nullify the otherwise harmonious testimony. 

The tenor of that testimony is that the relation 

of capacity to slope is substantially the same 

for mixtures as for simpler grades of debris. 

There is, however, a noteworthy qualification 

to this statement, in that the values of a have 

a smaller average for mixtures than for component grades.

The single comparison possible between 

results obtained with different discharges indicates that with mixtures, just as with their 

components, capacity increases with discharge 

in more than simple ratio.

MIXTURES OP MORE THAN TWO GRADES.

Experiments were made with a mixture of 

three grades, and with a mixture of five. The

former was observed with discharges of 0.182 

and 0.363 ft.3/sec., the latter with discharges of 

0.182, 0.363, and 0.545 ft. 3/sec. (See Tables 4 

(J) and 60.)

Brief consideration only will be given to the 

data from the mixture of three, (A1C1G2), 

because the points on which they bear are more 

fully covered by the data from the mixture of 

five. Grades (A) and (C) differ from one 

another in fineness much less than either 

differs from grade (G). The triple mixture may 

therefore be thought of as half coarse and half 

fine, with the distinction that the fine half is 

made up of two grades. By comparing data 

from it with data from the closely related mixtures (A A) and (CA)> in which the fine portion 

is of a single grade, we may throw light on the 

question whether the advantage to traction 

which is obtained by substituting two grades 

for one may be augmented by further diversification. With discharge 0.363 ft. 3/sec. and 

slope 1.4 per cent the capacities for the three 

mixtures are

(AA) (AAG,) (CA) 

89 98 106

As the capacity for the triple mixture has a 

value midway between those for the two dual 

mixtures, no advantage is indicated for the 

greater diversification.

-300

(C),

Slppe

I Z 

Slope

FIGURE GO. Curves of capacity in relation to slope for a mixture of 

five grades, (CDEFG). Comparison of mixture curves for three 

discharges, and of mixture curve with curves for component g:ades.

In the mixture of five grades the proportions 

were so arranged as to approximate a natural 

combination. The components and their percentages are shown by writing (C^DggE^FA). 

The curves of capacity in relation to slope are 

shown, for the three discharges, in the left-
EXPERIMENTS WITH MIXED GKADES. 177

hand diagram of figure 60. In the right-hand 

diagram the curve for discharge 0.363 ft.3/sec. 

is repeated, and with it are placed the corresponding curves for four of the component 

grades that for grade (F) being omitted 

because it is nearly coincident with the one for 

grade (E).

The curve for the mixture is not only of the 

same type with the others, but runs nearly 

parallel with those nearest. For all slopes the 

capacities obtained for the mixture are greater 

than for any component. The fact that the 

mixture gives a greater capacity than does the 

fine grade (C) alone shows that the addition to 

(C) of 35 per cent of (D) and 20 per cent of

still coarser grades works an advantage instead 

of a detriment.

In figure 58 it is seen that the addition of 20 

per cent of coarser debris to grade (B) increases capacity, and in figure 59 that a 

similar increase accompanies the addition of 

20 per cent of coarser debris to grade (C). 

So far as the case of the more complex mixture 

is comparable to these, there is no indication 

that a mixture of great complexity has advantage for traction over a mixture of two 

components only. As to this point the inference from data of the five-part mixture is 

supported by that already drawn from a 

datum of the (A^G,,) mixture, and it is

300-

200-

100

f <

200

0 

TO 

Q. 

TO 

U

100

Slope

A .8 

Discharge

1.2

FIGUKE Cl. Capacity-slope curves for related mixtures and capacity-discharge curves for mixture and component grades.

further supported by the facts brought together in the left-hand diagram of figure 61, 

which shows the capacity-slope curves of mixtures (C4Ej) and (C4GJ along with that for the 

five-part mixture.

The right-hand diagram of figure 61 is a 

group of plots of capacity as a function of discharge. These plots pertain to the complex 

mixture and its five components, and all are 

conditioned by a slope of 1.2 per cent. They 

show that the tractional superiority of the 

mixture is not confined to the use of a particular discharge, and they indicate also that the 

capacity-discharge relation is essentially the 

same for the mixture as for separate grades. 

The locus of C =/($) for the mixture is approximately a straight line, and if produced it intersects the- axis of Q to the right of the origin. 

It might be expressed by an equation in the 

form of (64) with an exponent near unity.

20021° No. 80 14 12

A NATURAL GRADE.

Two series of experiments were made with an 

alluvium in its natural condition, except that 

the very finest constituents had been removed 

by passing it over a 60-mesh sieve. The observations are recorded in Table 4 (K), and the 

adjusted capacities in Table 60. In figure 62 

the capacity-slope curves are plotted, and each 

is accompanied, for comparison, by the corresponding curves for grades (A) and (C).

The approximate mechanical analysis of this 

material, stated in terms of the separated 

grades of the laboratory series, is as follows:

Per cent. 

(A).................... 6

(I?).................... 13

(C). 27

Per cent. 

(D)................... 42

(E).................... 10

Coarser than (E)....... 2

The fact that the capacities (fig. 62) are 

greater than those'for grade (C), notwithstanding the dominance of a component correspond-
TRANSPORTATION OF DEBRIS BY RUNNING WATER,

ing ;< to the coarser grade (D), testifies again to 

the advantage for traction of a mixture as 

compared to a grade of narrow range in fineness. 

It will be observed also that each of the 

curves from the natural grade resembles closely 

its neighbors from artificial grades. So far as 

their evidence goes the type is the same for 

both, and the tendency of their evidence is to 

show that the laws connecting capacity with 

slope, as developed by the study of sorted 

debris, apply also to unsorted stream alluvium.

300

£.200

0.

ro 

0

100

fA)

0 I Z

Slope

FIGURE 62. Capacity-slope curves for a natural grade of de'bris, compared with curves for sieve-separated grades.

Table 63 (p. 180) gives computed finenesses 

for various mixtures and for this natural grade 

of de'bris. The fineness of the natural grade is 

nearly identical with that of grade (C), and the 

two thus afford a direct comparison between the 

capacities of a natural grade and a narrowly 

limited grade. For the same discharge and 

slope their tabulated capacities are respectively 

168 and 143 gm./sec., the ratio of advantage to 

the natural grade being 1.17. The computed 

fineness of grade (CDEFG) does not correspond to that of any simple grade, but a comparison made by means of interpolation gives 

150 and 131 gm./sec. as corresponding capacities for the mixture and a simple grade of the 

same mean fineness, and the ratio of the first 

to the second is 1.15. These ratios are smaller 

than those estimated from data for binary 

mixtures (p. 174), but are coordinate in value. 

The question of relative authority will be considered later.

CAUSES OF SUPERIOR MOBILITY OF 

MIXTURES.

When a finer grade of de'bris is added to a 

coarser the finer grains occupy interspaces

among the coarser and thereby make the surface of the stream bed smoother. This quality 

of smoothness appealed to the eye during the 

progress of the experiments. One of the 

coarser grains, resting on a surface composed 

of its fellows, may sink so far into a hollow as 

not to be easily dislodged by the current, but 

when such hollows are partly filled by the 

smaller grains its position is higher and it can 

withstand less force of current. In other 

words, the larger particles are moved more 

readily on the smoother bed, and this fact also 

was a matter of direct visual observation. The 

promotion of mobility applies not only to the 

starting of the grain but to its continuance in 

motion. It encounters less resistance as it 

rolls or skips along the bed, and it is less apt to 

be arrested. When a single large particle 

travels along a bed composed wholly of grams 

much smaller it rarely leaps, but rolls instead, 

 and it must in general be true that the larger 

particles in mixtures roll more and skip less 

than their smaller companions.

The admixture of finer de'bris thus changes 

the mode of traction for the coarser, and it is 

believed that the enhanced capacity is due 

mainly to this change. Capacity for the 

coarser is increased because the new condition 

reduces its resistance to the force of the 

current.

The fact that under some conditions the 

capacity for fine material is slightly increased 

by the addition of coarser is not so easily explained. The coarser grains do not make the 

bed smoother but rougher. The rougher bed 

retards the current. Even while rolling the 

larger grains are holding back the water, and 

the larger grains reduce the area of bed on 

which the traction of the smaller takes place. 

In these ways the presence of the coarse material tends to reduce the capacity of the current 

for the fine, and these factors certainly seem 

adequate to explain the general fact that 

capacity for the finer debris is reduced by admixture of the coarser.

Two factors may be named with the opposite 

tendency. The first is the impact of the 

coarser particles. In rolling and leaping they 

disturb the finer, tending thus to dislodge them 

from their resting places and either start them 

forward or else give them new positions from 

which they may be more easily swept. The 

second is the production of diversity in the
' EXPERIMENTS WITH MIXED GRADES. 179

current. Every obstruction diversifies the 

current. The' deflection necessary to pass it 

both constitutes and causes diversity of direction, and diversity of direction necessitates 

diversity of velocity. If a pebble is placed on 

the sandy bed of a small stream, the transformation of the adjacent parts of the bed by 

the diversified current is obvious. The building up of the bed in the lee of the pebble testifies to lowered velocity, and the scouring at the 

side to heightened velocity. In the same way 

each coarser grain of a heterogeneous stream 

load diversifies the current about it and gives 

to such of the filaments as are accelerated 

greater power for the traction of finer grains. 

With reference to the transportation of the 

finer c!6bris, the coarser grains have the function of obstructions whether they are partly 

embedded or lie on the surface or are rolled 

along.

If the factors concerned in the traction of the 

finer components of mixtures have been correctly stated, it is not difficult to understand 

that under most conditions the net result of 

their influences will be a reduction of capacity, 

and also that special conditions may determine an increase.

VOIDS.

' The packing together of larger and smaller 

grams which tends toward a smooth stream 

bed tends also toward- the reduction of interstitial spaces within the bedded debris, thus 

reducing the percentage of voids. It was suggested in the laboratory that the percentage of 

voids, used inversely, might serve as a sort 

of index of mobility, and estimates of the voids 

were accordingly made for most of the materials employed in the experiments with mixed 

grades. Partly for this purpose, and partly to 

obtain the factors needed to correct the weighings of load for interstitial water, a series of 

special weighings were made.

A vessel holding 535 cubic centimeters was 

filled with saturated debris and weighed. Afterward the same debris was weighed in a dry 

condition. The computation was made by the

formula

W' W 

Percentage of voids =

ages are smaller for the mixtures than for the 

component grades; and when the percentages 

were plotted in relation to the proportions of 

component grades (after the manner of the 

capacities in figs. 55 and 56) each series was 

found to indicate a minimum. The positions 

of the minima correspond, to mixtures with 30 

to 40 per cent of the finer grade of d6bris. In 

comparing voids with capacities, the minima 

of the void curves are to be considered in relation to the maxima of the capacity curves.

Capacity

Percentage 

of voids

Linear 

fineness

Bulk 

fineness

FIGURE 03. Curves showing the relations of various quantities to the 

proportions of fine and coarse components in a mixture of two grades 

of debris, (C) and (G). The horizontal scale, when read from left to 

right, shows the percentage of the coarser component in the mixture.

Those maxima, however, are associated with 

mixtures having 60 to 90 per cent of the finer 

grade; and the attempt at correlation therefore 

fails. A single void curve is reproduced in 

figure 63, together with the corresponding 

capacity curve.

TABLE 62. Percentages of voids in certain mixed grades of 

debris, compared ivith the percentages in the component 

grades.

in which W is the weight of saturated d6bris 

in grams and Wthe weight of dry debris.

Table 62 contains the estimated voids for the 

binary mixtures. In each series the percentPercentage of 

finer 

grade in 

mixture.

0 

20 

33 

SO 

67 

80 

100

Percentage of voids in grade 

(AC)

44

40

44

(AG)

38 

28 

24 

29 

31 

37 

44

(BF)

37 

31 

29 

29 

34 

39 

46

(CE)

40 

33 

32 

33 

35 

37 

44

(CG)

38"

26 

30 

 33 

38 

44

(EG)

38

29 

28 

34 

36 

40
180 TRANSPOETATION OF DEBEIS BY RUNNING WATER.

FINENESS.

A similar attempt was made to correlate the 

capacity curves of figures 55 and 56 with fineness. In computing the fineness of a mixture, 

the finenesses of its components were used as 

data, and the combination was made with bulk 

finenesses (p. 21) as follows: Denoting the proportions of components by a, b, c, etc., their 

bulk finenesses by F2 f , F2 ", F2 '", etc., and the 

fineness of the mixture by F2m,

: ....(85)

Linear fineness, F, was then computed by 

formula (88).

The results are listed in Table 63, together 

with the corresponding capacities for traction 

when the- discharge is 0.363 ft.3/sec. and the 

slope is 1-.4 per cent.

TABLE 63. Finenesses of mixed grades and their components. 

[Computed from data in Tables 1, 4 (J), and 4 (K).]

Grade.

(A)

(AA)

(C)

(A)

(AgGi)

(A2Gi)

(AiGi)

(AiG2)

(AiCU)

(0)

(B)

(B«Fi)

(BjFi)

(BiFi)

(B,Fj)

(B,F4 )

(F)

(C)

(C4Ei)

C2Ei) C,EO

CiE5 )

C,E4)

(E)

F-tllVfl

1,910 1,163

417

1,910

1,432

1,273

955

637

420

0.451

1,023

798

656

482

318

186

1.685

417

332

275

206

137

88

10.77

F

1,002

848

602

1,002

908

873

794

678

604

61.8

812

747

700

632

538

459

95.9

602

557

524

476

406

358

178

C

185

161

143

185

175

169

89

42

25

16

149

170

145

129

69

39

33

143

154

135

110

91

75

62

Grade.

(C)

(C4G,)

(CiOi)

(CiGi)

(dO»)

(G)

(E)

(E4G,)

(E,Gi)

(E,Gi)

(E,GS )

(0)

(A)

(AiC,G2)

(C)

(G)

(C)

(CDEFG)

(D)

(E)

(F)

(G)

Natural.

Fs/10«

417

334

278

209

139.3

0.451

10.77

8.71

7.33

5.61

3.89

0.451

1,910

582

417

0.451

417

228

111.5

10.77

1.685

0.451

419

F

602

557

526

478

418

61.8

178

166

157

143

127

61.8

1,002

673

602

61.8

602

492

388

178

. 95.9

61.8

603

C

143

158

144

106

50

16

62

59

65

49

33

16

185

98

143

16

143

150

127

62

33

16

168

The comparisons of capacity with fineness 

for series of mixtures of two grades are illustrated by figure 63, where the horizontal scale 

is that of the proportions of fine and coarse in 

the mixture. The capacity curve is identical 

with the second in figure 55, and the other 

curves pertain to the same series of (C G) mixtures. The curves for capacity and linear fineness are strongly discordant, capacity changing 

most rapidly with mixtures approximately in 

the ratio of 1:1, and fineness changing most

rapidly when the proportion of the finer grade 

is minute. The graph of bulk fineness is a 

straight line and betrays no sympathy with the 

sigmoid curve of capacity, though somewhat 

less discordant than the curve of linear fineness. 

It is quite evident that the peculiar relation 

of capacity to the proportions of a binary mixture is not to be either accounted for or formulated as a relation of fineness, and we have just 

seen that it can not be formulated in terms of 

the percentage of voids. The elimination of 

those two associated factors leaves it so far 

as our recognized alternatives are concerned 

to be ascribed wholly to modifications of the 

texture of the channel bed and the consequent 

modifications of the mode of transportation, 

and to these factors it is not practicable to give 

numerical expi ession.

RELATION OF CAPACITY TO FINENESS, FOR 

NATURAL GRADES.

There is another way of comparing the capacities pertaining to mixed grades with the 

fineness of the grades, which largely avoids the 

influence of changing mode of traction and 

which throws a side light on the relation of 

capacity to fineness in the case of natural 

grades. Instead of comparing the data for 

different mixtures of the same two simple 

grades, it compares data from similar mixtures 

of different pairs of simple grades.

Figure 64 has been compiled from data in 

Table 63 < Its upper group of five dots represents the logarithms of capacity in relation to 

the logarithms of linear fineness, for combinations of fine and coarse in the ratio of 4:1 

(one ratio of 3:1 being included). The line 

drawn among them gives, by its inclination, 

an estimate of 74 , 'the synthetic index of 

capacity in relation to fineness, for a range in 

fineness from 166 to 908, the value being 0.70. 

The next group of dots corresponds to mixtures of two fine to one coarse and gives 0.57 

as a value of the index. The next group, distinguished by crosses, corresponds to mixtures 

of one part fine with one of coarse. It includes 

six points, but one of these stands far from the 

line suggested by the others. The line, as 

drawn, represents an index value of 0.62. In 

the fourth group, distinguished by X's and 

corresponding to mixtures of one fine to two 

coarse, the points are so irregularly placed 

that no line can be drawn; and a fifth group,
EXPERIMENTS WITH MIXED GRADES. 181

not reproduced, is equally irregular. Fineness seems to control capacity when the finer

Mixture 4'-l

1:2

(CDEFG) /' Natural,

FIGURE 64. Logarithmic plots of capacity in relation to linear fineness> 

for related mixtures of de*bris.

component of the mixture is the more important, but other factors mask its influence 

when the coarser component dominates.

In preparing the mixture (C^DggE^FoGJ the 

endeavor was made so to apportion the components as to approximate a natural grade. 

To whatever extent that effort succeeded, the 

data obtained with use of the mixture are comparable with those afterward obtained with a 

natural grade. The capacities and finenesses 

of the complex mixture and the natural grade 

are represented by two plotted points near the 

bottom of figure 64, and the line drawn 

through them gives 0.57 as a value of the 

index. Unfortunately the range in fineness 

covered by the two is small from 492 to 603 

so that their determination of 74 has little 

weight; its close agreement with other values 

may be largely accidental.

In Table 45 mean values of 74 based on work 

with the sieve-separated grades are so arranged 

as to show their control by slope, discharge, and 

trough width. On making the indicated allowances for differences in condition, and comparing those values with the values indicated by 

figure 64, it appears that the latter are somewhat smaller but that the differences are not 

great. The index values from data of the 

experiments with mixtures range from 0.57 to 

0.70. Coordinate values from the experiments 

with single, sieve-separated grades would range, 

from 0.70 to 0.90.

500

Linear fineness 

FIGURE 05. Curve illustrating the range and distribution of Qnenesses in natural and artificial grades of ddbris.

In considering the bearings of these estimates there is advantage in giving graphic 

expression to the conditions they severally 

represent. When the results of mechanical 

analyses of sands and similar materials are 

presented graphically, the usual practice is to 

plot integrated quantities, or proportions, of 

ingredients on a scale of fineness or coarseness; 

but for the present purpose it is convenient to 

use the quantities without integration. In 

figure 65 the horizontal scale is that of linear 

fineness, and the ordinates are relative quantities of components of different finenesses. 

The curve AB C represents the composition of 

the debris constituting the tractional load of a 

river and is based on a sample of alluvium 

from a river bed. The area ABCD, between 

the curve and the axis of fineness, represents 

the total load and corresponds to the 100 per 

cent of integrative diagrams. The range in 

fineness is limited at A by competent fineness 

for traction, coarser material on the river bed 

not being moved by the particular discharge 

to which the diagram corresponds. It is
182 TRANSPORTATION OP DEBRIS BY RUNNING WATER.

limited in the opposite direction by competent 

fineness for suspension, the area GED representing suspendible, material entangled with 

the bed load. The.mean fineness of the tractional load; is-marked at M.,\;- > = \ - . ..-.: > t . 

In. sorting;debris f or r the-laboratory experiments, the material of the river's ; load was 

divided -by sieves, and this partition, might be 

represented in 'the diagram by,a -series of 

vertical lines slightly flexed-.to.-take account 

of the influence on the separation of irregularity 

in shape of particles. The shaded areas X and 

Y may represent the constitution as,to fineness 

of two of the sieve-separated grades. ,.(> * o/

The experiments show that for a narrowly 

limited grade, such as X or Y, which has the 

same mean fineness as the river alluvium, 

ABCD, the capacity for traction is much less 

than for the alluvium. They indicate also, 

though less decisively, that for a nearly equal 

mixture of a fine grade with a coarse, both 

being narrowly limited, the capacity is nearly 

.the same as for the unsorted alluvium, provided the mean fineness is the same. The 

essential property appears to be abundance of 

both coarse and fine, and not multiplicity of 

grades of fineness.

. Passing now from capacity to capacity's 

rate of variation in respect to fineness or to 

the valuation of i4 , we find a certain parallelism. For the narrowly limited grades the 

sensitiveness of capacity to fineness, as measured by the index i4 , extends upward from 

about 0.60; while values of the synthetic 

index, 74 , range from 0.70 to 1.00 or more-. 

As to these values the data are not abundant. 

For mixtures of two narrow grades we have a 

few estimates of 74 , of which the largest is 0.70; 

and for grades similar to ABCD a single weak 

estimate of 0.57. If we conclude that the 

sensitiveness of capacity to fineness is less for 

natural grades than for the narrowly limited 

grades, we must base the inference almost 

wholly on the data from the mixtures of two 

grades, connecting the latter with natural 

grades by aid of the analogy outlined above. 

This I am willing to do, but at the same time I 

would record my recognition of the weakness 

of the evidence and reasoning. It is estimated 

that, on the average, the capacity of streams 

for natural grades of debris varies with the 

0.60 to 0.75 power of linear fineness. This is 

equivalent to saying that capacity varies with

the 0.20 to 0.25 power of bulk fineness<r<or 

with the fifth or fourth root of bulk.finenesst:n

While the curve uTfigure 654s based on the 

mechanical analysis of material which constituted the tractional load of a river current, 

there is no reason to believe that its form presents a dominant type. Inspection of other 

analyses, in fact, suggests that such curves exhibit much variety and may sometimes even 

present two maxima. The load which a natural current carries is determined not only by 

the two limits of competence, but by the character of the material within its reach. Neighboring affluents of a river may bring to it 

strongly contrasted grades of debris, or their 

tribute may at one time be much finer than at 

another. Moreover, a river is not a simple current, but a complex of currents, which vary in 

competence and in the character of their loads. 

It is true that, the channel being considered as 

a whole, its load at one point is essentially the 

same as just above or just below, but the mode 

of movement involves a continual remodeling 

of the bed and a sorting and re-sorting of the 

material. The load at any particular point 

and time is conditioned by many factors of the 

complex. For this reason a representative 

sample of a river's load is not easy to define or 

to collect.

In view of this complexity it is difficult to 

apply even a simple formula to problems in 

river engineering, and refinement in formulation would be of little avail. For the same 

reason it is not practicable to derive a formula 

'directly from river data, and the product of 

.the laboratory is the best available, despite the 

artificial simplicity of its conditions.

DEFINITION AND MEASUREMENT OF MEAN 

FINENESS.

The term "mean fineness," as here used, is 

not free from the possibility of misapprehension. As the fineness of debris is a property 

depending on the size of component particles, 

it is not unnatural to think of fineness as a 

property of the particles and there is, for that 

matter, a fineness of particles. To obtain the 

mean fineness of particles, one would first determine the finenesses of the individual particles, and then the mean of those finenesses. 

The basal unit would be the particle. In deriving the mean fineness of a body of debris 

the basal unit is some unit by which quantity
EXPERIMENTS WITH MIXED GRADES. 183.

of debris is measured. It may be a unit of 

weight or a unit of volume. In this report a 

body of d6bris is conceived to be composed of 

equal volume units, each of which has a determined or determinable fineness, and its mean 

fineness is the mean of the finenesses of the 

volume units. In dealing with bulk fineness 

the mean computed is the arithmetical mean 

of the finenesses of units. In. dealing with 

linear fineness the mean computed is the cube 

root of the arithmetical mean of the cubes of 

the finenesses of units.

The intricacy of the definition of mean linear 

fineness arises from the relation of linear fineness to bulk fineness. The fundamental concept is that of bulk fineness, and the definition 

of linear fineness rests upon it. Linear fineness 

is essentially a derivative of bulk fineness, and 

mean linear fineness is an exactly similar derivative of mean bulk fineness. To pass from 

an assemblage of linear finenesses to their 

mean, it is necessary to pass through bulk 

fineness, and that passage involves cubes and 

cube root.

Bulk fineness is defined as the number of 

particles in a unit volume (1 cubic foot), it 

being assumed there are no voids. It is the 

reciprocal of the volume of the particle which 

might be called bulk coarseness. There are 

two practical modes of measuring it. If the 

specific gravity of the debris be known (or 

assumed), measurement includes a weighing 

and a counting. Then, W being the weight, 

N the number of particles, Q the specific 

gravity, W0 the weight of a cubic foot of 

water, and F2 the bulk fineness,

(86)

If the specific gravity be not known, measurement includes two weighings and a counting. 

Then, W being the weight in air and Wl the 

weight in water,

F = * 2 (87)

This procedure determines bulk fineness 

when all particles have the same volume; when 

they are of different volumes it determines 

mean bulk fineness. As a matter of fact, all 

our measurements in the laboratory were of 

mean fineness. It is not possible by any

method of sorting with which I am acquainted 

to separate from a natural alluvium..a grade 

which,is really uniform in fineness.

When the mean fineness of a sample of 

debris is desired, there is no need to'separate 

it into grades, because the process for measuring the mean fineness of the whole is identical 

with that for measuring the fineness of a grade, 

When bodies of debris of known finenesses .'are 

mingled, the mean fineness of the mixture is 

computed by a formula (85), which sums the 

finenesses by unit volumes' (or weights) .<and 

then divides by the number of unit volumes. . Linear fineness is defined as the reciprocal 

of the mean diameter of the particles of the 

debris. Like bulk fineness, it is treated as a 

property of the body of debris and not as a 

property of the particle. Mean diameter is 

defined as the diameter of a sphere having the 

same volume as the particle. . Defined thus, 

linear fineness is a function of volume of particle, and as bulk fineness is also a function of 

that volume, the two have a fixed relation:

"(I)' (88)

Substituting in (88) from (86) and (87), we 

have

The computations of fineness for this report 

used (89) or (90); or, what is equivalent, they 

first determined bulk fineness by (86) or (87), 

and then derived linear fineness by (88). The 

computations of mean linear fineness applied' 

(88) to mean bulk fineness. 

. It would have been possible to formulate fineness in such away that the definition of linear. 

fineness would be direct and comparatively 

simple, but any such formulation would encounter complexity in some of its parts, pro-. 

vided it established a logical relation between 

linear fineness and bulk fineness. Its adoption 

would also involve the sacrifice of simplicity in 

the measurement of fineness. Any system requiring the direct measurement of diameters 

would be inferior for practical purposes to the 

one here used. . The subject of scales of fineness has been 

elaborated because nearly all the results as to
184 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

the control of capacity by fineness would be 

quite different if a different scale were used. 

If, for example, mean linear fineness had been 

defined as the arithmetical mean of linear finenesses, the curve for linear fineness in figure 63 

would be a straight line, while the line for bulk 

fineness would be a curve similar to that shown 

for linear fineness but turned through 180°.

When the fineness of the tractional load of a 

stream is to be determined by means of a sample of the debris constituting its bed, account 

must be taken of another factor. Omitting 

considerations affecting the selection of a sample, which belong to Chapter XIII, let us assume that the sample hi hand is representative 

of the stream's tractional load. In addition 

to the debris which was carried along the bed, 

it inevitably includes finer material which was 

carried in suspension. Suspended particles are 

arrested along with the coarser and form part 

of every stream deposit. Once lodged in the 

interstices of coarser particles, they are sheltered from the current and are not again disturbed so long as the coarser material remains. 

If the deposition of the coarser debris is very 

rapid the amount of entangled finer stuff may 

be small, but when deposition is slow the interstices act continuously as traps and catch suspendible debris until they are filled. The latter is the usual condition, and the tractional 

sample therefore ordinarily contains a considerable percentage of suspensional material. 

To separate the two it is necessary to draw an 

arbitrary line, for the graduation in fineness is 

complete. As regards interstitial space, the 

tractional part of the sample is comparable with 

the more complex mixtures of the laboratory, 

and its voids may be estimated as 25 per cent 

of the whole space. The suspensional de"bris 

packed in these voids may be assumed itself 

to include 25 per cent of voids, so that the net 

volume of its particles is three-fourths that of 

the containing voids, or 18.75 per cent of the 

whole space. The net volume of the tractional particles being 75 per cent of the whole 

space, the two divisions of the sample bear the 

relation, by net volume or by weight, of 75 to 

18.75, or of 4 to 1. This gives a practical rule 

for separation. The sample should be divided, 

with aid of sieves and scales, into a coarser fourfifths and a finer one-fifth, and only the coarser 

part should be used in estimating mean fineness. In figure 65 the entrapped suspensional 

material is represented by the triangular area 

ODE.

SUMMARY.

The purpose of the experiments with mixtures was to bring the results from work with 

separate grades into proper relation with 

phenomena of unsorted natural material. The 

indications given by these experiments are in 

part direct and in part conditioned by the 

principle adopted in framing a scale of fineness. 

The adopted principle makes the conception of 

bulk fineness fundamental and that of linear 

fineness derivative.

The capacities for traction observed in the 

experiments with narrowly limited grades are 

less than for equivalent grades with greater 

diversity in fineness. A study of data from 

mixtures of two narrow grades indicates that 

the ratio of advantage for diversified debris is 

from 1.17 to 2.22, the mean of five estimates, 

from different groups of data, being 1.66. Two 

comparisons of results from highly diversified 

grades, with results from nearly homogeneous 

grades of the same fineness, give as estimates 

of the ratio of advantage 1.15 and 1.17. The 

larger estimates were made by an indirect 

method but are independent of the scale of 

fineness. The smaller estimates were made 

by a direct method but involve the theory of 

the scale of fineness. In combining the two 

groups of estimates, greater weight is assigned 

to the smaller, not because they are of recognized higher authority, but because the same 

scale of fineness will almost necessarily be 

used in applying the results of the investigation to practical questions. The compromise 

value of 1.2 is adopted, as a correction to be 

applied to values of capacity in Table 12 in 

estimating capacities for diversified grades of 

like fineness.

The advantage of diversification appears to 

arise largely from the fact that the finer 

particles, by filling spaces between the coarser, 

make a smoother road for the travel of the 

coarser, and it is not proved that a highly 

diversified debris gives higher capacity than 

one containing only two sizes of particles.

It is especially notable that when fine material is added to a previously homogeneous 

coarse material not only is the total capacity 

increased, but the capacity for the coarser part
EXPERIMENTS WITH MIXED GRADES. 185

of the load is increased, and it may even be 

enlarged several fold. The general effect of 

adding coarse to fine is to reduce the stream's 

capacity for the fine, but under some conditions 

there is a slight increase.

The general relations of capacity to slope, 

discharge, and fineness (and presumably to 

form ratio also) are the same for natural and 

other complex grades of debris as for the sievesorted grades of the laboratory, but some of 

the constants are not quite the same.

The sensitiveness of capacity to slope is on 

the average the same for both classes of debris 

grades, but the variation of sensitiveness in 

relation to slope, as determined by the constant a, is somewhat less for natural grades.

As to the relation of capacity to discharge 

comparison was limited to a single example, 

and that suggested no modification of the 

constants derived from work with laboratory 

grades.

The sensitiveness of capacity to fineness is 

somewhat less for natural grades than for the 

the laboratory grades. No values of the 

constant <£ were obtained for complex grades, 

and comparisons of sensitiveness were made 

only by means of the synthetic index of relative 

variation. The average value of that index, 

for natural grades of debris transported under 

laboratory conditions, is estimated at 0.20 to 

0.25 for bulk fineness.
CHAPTER X. REVIEW OF CONTROLS OF CAPACITY.

INTRODUCTION.

In the preceding seven chapters the relations 

of capacity for stream traction to a variety of 

factors have been examined one at :a time. It 

is now proposed to bring together some of the 

discovered elements of control. The experimental data thus far considered pertain to 

straight channels, and the factors- of control 

connected with bending channels have not received attention. Those factors must be included when the attempt is made to bring 

laboratory results into relation with river phenomena, but as they constitute a category by 

themselves it is convenient to leave them out 

of the account in correlating the results from 

straight-channel work.

The immediate determinants of capacity are 

(1) the velocities of the current adjacent to the 

channel bed, (2) the widths of channel bed 

through which those velocities are effective in 

moving debris, and (3) the mobility of the 

debris constituting the bed and the load. It 

was not found practicable to measure bed velocity, but measurement was applied to its two 

chief determinants, slope and discharge, and 

also to its intimate associate, mean velocity, 

and these have been discussed separately. 

Width has entered into the discussion chiefly as 

an associate of depth in the determination of 

form ratio. By reason of these and other interrelations the six controls of capacity which have 

been discussed slope, discharge, fineness, 

depth, mean velocity, and form ratio are not 

independent, and not all should appear in a 

general equation. Slope, discharge, and fineness being accepted as of primary importance, 

it is feasible to add but one of the others, and 

choice has been made' of form ratio.

FORMULATION BASED ON COMPETENCE.

The functions used in discussing the relations 

of capacity to slope, discharge, and fineness are 

similar, and each involves a conception of competence. Competence enters also the theory of

186

the relation of capacity to form ratio, but it 

enters in a different way. It is convenient to 

omit at first the form-ratio function and consider together the three which are similar. 

They are:

' ' ' - ^6^-a)".... ...... ..(10)

c=i,(q- KY ---'.-.-----(64)

. . 0=1^-^... ...... ..(75)

Each of these equations expresses the law of 

variation of capacity with respect to one condition when the other two conditions are constant, and in that sense they are independent ; 

but there is a mutual dependence of parameters 

which is of so complete a character that they 

are essentially simultaneous. The dependence 

of parameters is more readily stated by means 

of a specific instance than in general terms. 

In equation (10) 11} o, and n are constant so 

long as Q and F hold the same values ; they do 

not vary with variation of S. But when the 

values of Q and F are changed those of & 1? a, 

and n are modified. Through this control of its 

parameters the equation involves the relation 

of capacity to discharge and fineness.

The coefficient Z> t is the value of capacity 

when (S a) = 1 ; Z> 3 when (Q K) = 1 ; Z> 4 when 

(F <£) = !. Replacing them by Z> 5 , as the 

numerical value of capacity when (S o) = l, 

(Q «) = !, and (F </)) = !, we may combine 

the three equations into

0=&B (£-*) (Q-*)° (F-&P-. ..(91)

The constant &5 is not of the same unit with 

either 1) 1 , l>3 , or &4 . Its dimensions, derived from 

those of the variables of (91), are !**-*>]& " T°~\

From the experimental data have been computed 92 values of n, 20 values of o, and 5 

values of p. (See Tables 15, 32, and 44.) All 

these are positive. The following statistical 

summary gives a general idea of their relative 

magnitudes. Its figures are not based on the 

same range of observational data; but the
KEVIEW OF CONTROLS OF CAPACITY.1 187.

ranges for o and y correspond approximately 

with the middle part of the range for n.

. . Exponent.

n

0 ,

P

 Number

of deter- 

 minations.

92

. 20

5

Mean 

value.

1.59

1.02

.58

Range of 

values.

0. 93-2. 37

. 81-1. 24

. 50- . 62

It will be recalled that while the forms of the 

equations involving a, K, and <£ were based on the 

conception of competence, it was not found 

possible to correlate those.parameters strictly 

with competent slope, discharge, and fineness. 

The correlations were obstructed by phenomena 

of dune rhythms and of diversified fineness and 

could not be completed, but the forms of equation were found to be well adapted to the combined expression of observational data above 

the region of competence. Their relation to 

competence is not absolute but intimate, and 

it is so intimate that certain properties of the 

parameters may properly be inferred from the 

physical theory of competence.

When the swiftest velocity on the bed is 

barely able to move debris, there is a threefold 

condition of competence. For the particular 

discharge and fineness, the slope is competent; 

for the particular slope and fineness, the discharge is competent; for the particular slope 

and discharge, the fineness is competent. The 

conditions of competence for the three factors 

controlling capacity are thus not only similar 

but simultaneous and coincident. Neither 

factor can sink alone to the limiting level of 

competence, but the three arrive together. 

This is an important principle and lies at the 

foundation of the systematic interdependence 

of parameters and variables.

In equation (91) the quantities 

(S-o), (Q-K), and (F- <£) become zero simultaneously. When 

S = a, then also Q = K, and F= $ ; 

and vice versa.

(92)

As capacity varies directly with (S CT), 

(Q K), and (F fy, it is also true that each of 

these varies directly with capacity. Any 

change of condition which affects capacity 

affects those three quantities in the same sense. 

For example, suppose discharge to be increased.

This not only increases (Q K) and thereby 

increases capacity, but it also increases (S a). 

and (F <f>). One mode of expressing this fact 

is to say that capacity measures the remoteness 

of each controlling factor from the initial status 

of competence, and all recede or approach 

together.

Let us now make a more definite assumption, 

that discharge is increased while slope and 

fineness remain the same. The resulting increase of (S o), as & is unchanged, implies a 

diminution,of o; and the increase of (F <p) 

implies a diminution of. <£.. That is, a and <£ 

vary inversely with discharge. Parallel reasoning shows that a and « vary inversely with 

fineness, and that K and <j> vary inversely with 

slope.

These relations are here developed deductively from the theory of competence. They 

have been developed inductively from the observational data; for equations (26), (66), and 

(77) include

(93)

No way has been found in which to study the 

exponents deductively. The only evidences of 

order discovered by comparison of observational data pertain to n, which has been found 

(equation 27) to vary inversely with discharge 

and fineness. The question whether o and p 

follow similar trends could not be answered by 

the adjusted data because of the cumulative 

effect of accidental errors. There is, however, 

considerable force in analogic reasoning, based 

not only on equations (93), but on other elements of symmetry in the relations of capacity 

to the several factors elements to be noted 

later. The state of the evidence may be expressed by

n =/, (& h 

[o =//v(£, F)]

[P =/K (3, $)]

It is convenient to have a name for the group 

of constants designated by Greek letters, and 

as they define the conditions of competence, 

they may be called competence constants.

The exponent n and the associated competence constant a, as they vary with Q and F

(94)
188 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

and do not vary with S, are controlled by Q 

and F. As they both vary inversely with Q 

and with F, it follows that they vary directly 

one with the other. It is evidently true in 

general that each exponent varies directly with 

the associated competence constant.

(95)

With a constant, the-variation of S a is 

determined by variation of S. Considered as 

additive, their variations are identical; but if 

we regard the changes as ratios, the changes in

(S a) are proportional to o_ and those in S to 

m The ratio between these fractions, which is

Of

t? . is a measure of the sensitiveness of (S a) b a

to changes in S. It is evident that as S increases the sensitiveness diminishes. As capacity varies with a power of (S o), the sensitiveness of capacity to slope becomes less as the 

slope increases.

Any change in S causes, according to (92) 

and (93), a change of opposite character in K 

and o. When S is increased, K and o are reduced. As the sensitiveness of (Q K) to

change in Q is measured by Q_ , it is evident

that the reduction of K lessens the sensitiveness. 

This has the effect also of lessening the sensitiveness of capacity to discharge; and that sensitiveness is further lessened by the reduction 

of o. Parity of reasoning shows that increase 

of slope lessens the sensitiveness of capacity to 

fineness, so that the effect of increasing slope 

is to reduce the sensitiveness of capacity to 

all three of its controlling factors. It is evident also that a similar result would be reached 

if the analysis began by assuming an increase 

of discharge or fineness.

It is a general principle that any change in 

one of the control factors, slope, discharge, and 

fineness, causing capacity to increase, has the 

effect also of making capacity less sensitive to 

changes in each and all of the control factors; 

and the inverse proposition is of course equally 

true. The statement being phrased to include 

both, the sensitiveness of capacity to the three 

controlling conditions varies inversely with 

capacity.

The term " sensitiveness," as used in the preceding paragraphs, is equivalent to the more 

specific "index to relative variation," for 

which the symbol i has been used; and by 

reference to various studies of the control of 

the index by conditions it may be seen that 

the entire scope of the general principle just 

stated has been covered by essentially inductive generalizations. From equations (39), 

(68), and (79),

(96)

This checking of deductive by inductive results 

helps to establish the second and third equations of (94), which were inferred from analogies.

Very little is known of the nature of the 

functions in (93) to (96), beyond the fact 

that those of (95) are increasing and the 

others decreasing. Deductive reasoning has 

not been successfully applied, and induction 

has escaped the entanglement of accidental 

errors in only a single instance and to a limited 

extent. The exceptional instance is that 

represented by the first equation of group (96). 

The symbols being translated into words, that 

equation reads: The index of relative variation for capacity in relation to slope varies 

inversely with slope, with discharge, and with 

fineness. There are in fact three distinct 

propositions, and each of these might be 

expressed by a separate equation. As to the 

first proposition, that the index varies inversely with slope, it was found, inductively, 

that the rate at which it varies with slope is 

itself a decreasing function of slope and also 

of discharge and fineness; and knowledge of 

similar character was gained as to the second 

and third propositions (pp. 104-108). Representing by dils, diig, and diiff the rates of variation of the index in relation to slope, discharge, 

and fineness, severally, we have

& ft

ft (97)

These fragmentary determinations are all of 

one tenor, and in view of the remarkable 

symmetries already discovered among the ele-
REVIEW OF CONTROLS OF CAPACITY. 189

ments of equation (91), they render probable 

the general proposition:

The rate at which capacity varies' 

inversely with each of the three 

controlling conditions, slope, discharge, and fineness, itself varies 

inversely with each of the conditions.

(98)

Returning to (92) and (93), we may indicate 

certain corollaries.

Starting from the status of competence, let 

us assume that slope is increased, with destruction of the status, and that the status is restored by reducing discharge. In the restored 

status a is greater than in the original, « is less, 

and <f> is unchanged. It is evident that the 

nature of the result does not depend on the 

particular assumptions, and that we may pass 

to the general proposition:

When capacity is zero, the competence constants are so related 

that a change in any one of them ...... (99)

involves a change of contrary 

sign in some other.

Starting from a status characterized by a 

particular value of capacity, we may first 

break it by increasing slope and then restore it 

by decreasing discharge (fineness remaining 

unchanged). The first change reduces K and 

<j>; the second increases a and <j>. It does not 

appear whether the net result for <j> involves 

change in its value, but if so the change is 

probably small in relation to the increase in a 

and the decrease in K. It is evident that the 

nature of the result does not depend on the 

particular assumption, and that we may pass 

to two general propositions, each of which 

includes (99) as a special case:

Under the condition that capacity is constant, the competence 

constants are so related that a 

change in any one of them involves a change of contrary sign 

in some other.

Under the condition that capacity 

is constant, the values of slope, 

discharge and fineness are so 

related that a change in any one 

of them involves a change of 

contrary sign in some other.

(100)

(101)

It follows also that

Under the condition that capacity 

is constant, the value of each 

controlling condition (S, Q, or 

F) is so related to the corresponding competence constant 

(a, K, or <f>) that the two vary in 

same sense.

(102)

Propositions (100) and (102) are deduced 

from equations (93). By parity of reasoning 

equations (94) yield (103) and (104), but these 

two propositions share whatever uncertainty 

attaches to (94).

Under the condition that capacityis constant, the exponents n, o, 

p are so related that a change in 

any one of them involves a 

change of contrary sign in some 

other.

Under the condition that capacity 

is constant, the value of each 

controlling condition (S, Q, or 

F) is so related to the corresponding exponent (n, o, or p) 

that the two vary in the same 

sense.

(103)

(104)

As capacity can not be increased under (91) 

without increasing S, Q, or F, and as the 

increase of one of these involves under (93) the 

decrease of two competence constants, without 

any change of the third, it follows that the 

competence constants, collectively, vary ininversely with capacity. The same reasoning, 

if applied to (91) and (94), yields a similar 

conclusion as to the exponents. To combine 

the two in a single statement:

The competence constants a, K, 

and <p, taken as a group, and the 

exponents n, o, and p, taken as 

a group, vary inversely with 

capacity.

(105)

I find it not easy to bring into combination 

the laws of internal relation between parameters of a group and the laws which connect 

the groups with capacity; but if these laws be 

regarded as conditions, it is possible to frame 

more comprehensive theorems of tentative 

character. Equations (106) are of this class
190 TRANSPORTATION 0$ DEBRIS BY RUNNING WATER.

and are thought worthy of examination, although the data at hand do not suffice for their 

testing.

= 0

The equation under discussion,

(91)

is an expression of relation between capacity 

and three of its controls, namely, slope ($), 

discharge (Q), and fineness (F). It involves 

seven parameters, of which six are functions 

of the independent variables, S, Q, and F. It 

is thus a bare framework, and the completion 

of the structure calls for the replacement of the 

six parameters by their values in terms of the 

variables. The laws contained in the equations and propositions numbered (92) to (98), 

with their corollaries, (99) to (105), are contributions toward the completion of the structure, but they are largely of the nature of restrictions. They impose conditions to be satisfied by the perfected equation.

Some of the conditions are already embodied 

in the form of (91), and with reference to such 

conditions it is important that the origin of that 

form be not overlooked. The form assumes 

that the three competence constants .are the 

values of the corresponding variables when capacity is zero, whereas their identification with 

those values is by no means complete. The 

definition and recognition of the status of competence are so obstructed by the complicating 

conditions of nonhomogeneous debris and dune 

rhythm that no more can be asserted than an 

indefinitely representative relation. For most 

purposes, however, we are little concerned with 

conditions in the immediate neighborhood of the 

competence limit, so that this qualification is of 

small practical moment. Outside of the neighborhood of competence the support of the form 

is empiric; it has served well as a scheme for the 

marshaling of the observations. The support 

is qualified, in turn, by the fact that the observations are not of such harmony and precision 

as to discriminate nicely among formulas of 

adjustment. In view of these qualifications, 

the possibility has been recognized that some 

of the laws above enumerated might emanate 

from the form of the equation and have no 

other basis; and in view of this possibility the

foundations of each conclusion have been scrutinized. I believe that all the inferred laws, 

from (92) to (105), are essentially inductive.

It is easy to understand that .any constructive effort which should hang all supplementary 

conditions on the framework of (91) would result in a formula so unwieldly as to be useless. 

It is a matter of faith with me that if our data 

were so precise as to substitute definite quantitative relations for the fascicle of trends and 

indefinite parallelisms they have actually furnished, some way would be found leading from 

complexity to simplicity. I am not without 

hope that the presentation here made may suggest to the mechanist, familiar with the aspects 

of solved problems of similar difficulty, a rational theory under which the data may advantageously be recombined.

In an effort to discover unities among the 

complexities of the capacity relations, equation 

(91) was given the following form:

The three factors making the second division 

of the second member, being independent of the 

units of measurement, seemed well adapted to 

the expression of comprehensive harmonies, if 

such exist.

The following negations were demonstrated :

9 O T? 

The quantities -, -, and -7 are not equal, nor

are the ratios between them constant.

^ O 7? The quantities - 1. 1. and -, 1 are not 1 a K ' (f>

equal, nor are the ratios between them constant.

/S \n /Q \° The quantities f - 1 j , ( - 1 ) , and

(

F V 

-r 1] are not equal, nor are the ratios between them constant.

It was also found that the symmetric factors 

in equation (91), namely, (S a)n, (Q K)°, and 

(F (j))v, are not equal, nor are the ratios between them constant.

THE FORM-RATIO FACTOR.

In its relation to form ratio capacity has 

two zeros, one corresponding to a high ratio, 

the other to a low. Each of these corresponds 

also to a competent bed velocity, so that into
REVIEW OF CONTEOLS OF CAPACITY. 191

a perfect formula competence would enter 

twice. The formula adopted, however, ignores 

the element of competence, chiefly because its 

recognition, which would add a complication, 

was not seen to be of advantage for the expression of the control of capacity in the more important regions outside the vicinity of competence. The accepted formula is

0=6/1 ^ -\RTO -------(5S) \ m+1 pj

The quantity p is that value of R which corresponds to the maximum value of 0 the maximum standing between the two zeros and &2 

is a capacity constant. The function as a 

whole qualifies capacity by means of a numerical factor and may be combined with (91) by 

multiplication of the factors:

m R

m+1 pj 

.........(109)

The coefficient &, replacing &5 and &2 , is a quantity of the same unit with 65 (see p. 186), but 

numerically independent.

The function now added is of distinct type 

from the others, for instead of advancing by a 

continuous law from zero to infinity it first 

rises to a finite maximum and then returns to 

zero. The first three factors are harmonious; 

the fourth discordant. At every stage in the 

investigation the discussion of the laboratory 

data has been hampered by this discordance. 

In order to treat adequately the relation of capacity to either slope, discharge, or fineness, it 

was necessary to isolate that relation by equalizing other conditions, and slope or discharge or 

fineness could readily be equalized; but the 

form-ratio factor was intractable. By means 

of interpolation it was possible to assemble 

varied data characterized by the same form 

ratio, but that did not meet the difficulty. It 

was necessary to take account of the relation 

of the particular ratio to the optimum ratio, p; 

and the value of <o« varies with all other conditions.

The sensitiveness of capacity to form ratio, 

as measured by the index of relative variation, 

is less than its sensitiveness to other conditions. 

The average of 48 values tabulated in Chapter 

IV is 0.24, while similar averages for fineness, 

discharge, and slope are three, five, and seven

times as great. The distribution of sensitiveness, in relation to values of the independent 

variables, is illustrated by figure 66, where four 

curves are plotted, each representing a particular instance, selected as typical. The vertical 

scale is the same for all; and the ordinates represent values of the index of relative variation. 

The horizontal scale is that of slope for the 

curve SS, of discharge for the curve QQ, of 

fineness for-the curve FF, and of form ratio 

for the curve RpE. The vertical cc represents 

the competence constants and is an asymptote 

to three of the curves. The horizontal line mm 

gives the value of the exponent m corresponding to the form-ratio index. For values of R 

greater than p the index is negative, but its 

curve is drawn above the zero line to represent 

sensitiveness, which is not affected by sign.

FIGURE 66. Typical curves illustrating the distribution of the sensitiveness of capacity for traction to various controlling conditions. 

Ordinates represent values of the index of relative variation: abscissas, 

to four different scales, represent values of slope, discharge, linear 

fineness, and form ratio.

The two parameters of the form-ratio factor 

have laws of variation similar to those of the 

other parameters; each varies inversely with 

values of all independent variables except its 

own,

P=f(&,Q, ^) ...... -..-(61)

m=f, (&,&&) -'.--.-,-. .62)

It follows that each varies directly with each 

of the other parameters (o, K, <j{>, n, o, and p).

The relations of the parameters to form ratio 

are less simple. In all cases the evidence from 

the observational data is conflicting, and as the 

several cases have come up for consideration 

the trend of evidence has seemed now in one 

direction and now in another. Impressions as 

to those trends are recorded in the preceding 

chapters, but when assembled they fail to indicate any general principle. Recourse is there-
192 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

fore had to theoretic considerations alone, and 

the conclusions reached can claim no direct 

support from the tabulated values of exponents 

and competence constants.

The conclusion (105) that the competence 

constants collectively vary inversely with capacity is based on the fact that Individually 

they vary inversely with things which promote 

capacity, namely, slope, discharge, and fineness. Let us now assume that those three conditions remain constant and consider the effect 

of varying form ratio. Initially let the form 

ratio be so small that the bed velocity is competent; o, K, and $ are severally equal to the 

competent values of S, Q, and F. Now change 

width and depth so as to increase the form 

ratio and capacity becomes finite. That capacity may be finite, a, K, and 0 must be less 

than 8, Q, and F, and as the latter have not 

changed the competence constants have been 

reduced by the increase of form ratio. By 

parity of reasoning it can be shown that if the 

initial form ratio be so large as to make the 

bed velocity competent a reduction of form 

ratio will cause a reduction of a, K, and <£. 

Somewhere between the two form ratios of 

competence lies p, the form ratio of maximum 

capacity, and between the same limits lie minimum values of the competence constants.

The greater the capacity induced by adjustment of form ratio, the greater the reduction 

of slope, for example, necessary to reduce 

capacity to zero, and as this reduction varies 

directly with the depression of a below the 

initial value of S, it follows that the minimum 

value of a (and similarly of « and 0) coincides 

with the maximum of capacity.

The conclusion that the competence constants vary inversely with capacity is therefore 

true for the case in which changes in capacity 

are caused by changes in form ratio. It can 

be shown also that the exponents, n, o, and p, 

follow the same law.

The extension of this principle to the domain 

of form ratio gives assurance that the conclusions embodied in equations and propositions 

(99) to (108), conclusions which were reached 

from phenomena of slope, discharge, and fineness, are not vitiated by the traversing phenomena of form ratio.

The function in the form-ratio factor of (109) 

being characterized by a maximum, the variations of parameters with respect to form ratio

are characterized by a minimum. This la\? 

may be so combined with those of (93), (94) 

(61), and (62) as to yield the following sys 

tern of equations for the trends of changes in 

parameters consequent on changes in the fout 

independent variables of equation (109):

P =

n =/B (& P, R) 

o =/6 ($, P, R) 

P =/,(£,& 5)

 =fs ($, 4 P)

--(HO)

In the development of the form-ratio factor 

of equations (58) and (109), detailed in Chapter IV, the factor first appeared as (1 aR}Rm} 

the quantity a being a numerical coefficient introduced to represent the resistance to the current occasioned by the sides of the channel. It

1YI 1

was afterward shown that a = - -. and that

m+ 1 p'

form of coefficient was substituted. These relations show that either a or ra varies in value 

with the character and amount of the resistance by the channel sides; and, in point of fact, 

both do. Nor is that control restricted to the 

parameters of form ratio. Lateral, resistance 

affects also, and in comparable degree, the parameters of slope, discharge, fineness, and capacity. The sides of the laboratory channels 

were vertical and were of wood, planed and 

painted. Had they been smoother or rougher, 

or had they been inclined, the whole system of 

values given.by the experiments would have 

been different. There is no reason, however, 

to question that they would have yielded the 

same qualitative results.

DUTY AND EFFICIENCY.

The discussions of duty and efficiency in 

Chapters III and V give reason for the belief 

that the variations of either quantity in relation to controlling conditions may advantageously be expressed by an equation identical 

in form with (109). Such an equation would 

not be interconvertible with (109), nor would 

an equation for duty be the exact equivalent 

of one for efficiency. By the aid of reasonable 

assumptions the parameters of either equation 

might be derived from the parameters of another, but the results of computations by the 

several equations would not be strictly compatible. These discordances may be demonstrated as properties of the algebraic forms.
EEVIEW OF CONTROLS OF CAPACITY. 193

They are also connected with the fact, that the 

algebraic forms imperfectly represent the physical phenomena. It is -possible that the physical facts are closely represented by some oiie of 

the equations, in which case the incompatibilities ,of the other two equations are wholly 

of the nature of errors; but if this be so, our 

data do not enable us to indicate the truer 

equation.

The exponent in the slope factor is greater 

for the capacity and duty equations than for 

the efficiency equation, the difference being a 

large fraction of unity. The exponent in the 

discharge factor is greater for the capacity 

equation than for those of duty and, efficiency, 

the difference being a large fraction of unity. 

The competence constant in the slope factor 

is smaller in the capacity and duty equations 

than in that for efficiency, the difference being 

a small fraction of the constant. The competence constant in the discharge factor is 

smaller for the capacity equation than for the 

others, the difference, being relatively large. 

The interrelations of parameters and the relations of parameters to independent. variables 

are qualitatively the same for the equations .of 

duty and efficiency as .for the equation of 

capacity. . .

THE FORMULA OF LECHALAS.

THE FORMULA.

The only earlier serious attempt to formulate 

the transportation of debris, so1 far as I am 

informed, is that of C. Lechalas, who wrote in 

1871, under the title "Note sur les rivie'res a 

fond de sable."1 His discussion makes use of 

Dubuat's experiments ori competent velocity 

(1786), Darcy and Bazin's formulas for velocities in conduits and rivers (1878), and observational data accumulated by the French engineering corps in connection with projects for 

the improvement of navigable rivers. In the 

following abstract of the more elementary 

part of his discussion the symbolic notation 

and tKe terminology of the present paper are 

to some extent substituted for those of the 

original.

Postulate a stream of fixed width and constant discharge, traversing a bed of uniform 

sand, unlimited in quantity but without

i Annales dcs ponts et cftiauss&s, Me*m. et doc., 5th ser., vol. L, pp. 381- 

431,1871.

accessions. So -long as the velocity along the 

bed exceeds a certain value the. current transports sand. Below that limit the sand is 

undisturbed.

The discharge, Q, and width, w, . being 

known, the mean depth, d, the slope, s, the 

mean velocity, Vm, and the bed velocity, Vb , 

are given by the following three equations, of 

which (1) and (3) are from Darcy and Bazin. 

The constants are in meters.

f .0.00028 +

'

(2)

(3)

The sand travels (1) by rolling, (2) by suspension; A particle of water impinging on 

the bottom gives motion to a sand grain, the 

motion having a direction which depends on 

the impact and on the positions of adjoining 

particles, solid aiid liquid. The grain is projected free from the bottom 'or is rolled along 

it, the particular result depending on the 

inclination and force of the impact and oh 

various conditions which affect the resistance. 

Suspension corresponds especially to impacts 

associated with high velocities. Suspension is 

rare below a certain critical velocity for each 

density and size of sand grain. Transportation is slow at low stages of a variable stream, 

rapid and by suspension at 'high stages. The 

grains describe trajectories analogous to those 

of the water particles, but shorter; and there 

are frequent returns to the bottom, as well as 

restings between excursions. Larger ' grains 

are lifted less high, or are rolled only, or remain 

at rest.' Small grains afford a better hold 

(prise) in relation to their weight. The 

smallest of all are carried in the body of ^he 

current.

The amount by which the pressure on the 

upstream face of a grain immersed in a current 

exceeds the pressure on the downstream face 

is proportional to the square of the velocity. 

Represent it by aVb2 , the coefficient a depending' oh size, form, and position. For the sand 

of the Loire, the resistance developed, equals 

a 0.252, as that sand is immobile when Vb < 0.25. 

"The difference is equal to the product of the 

mass of the grains by their velocity, projected 

on the same axis as Vb that is to say, on the 

axis of the stream. This product, being pro20921° No. 86 14 -13
194 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

portional to the discharge [load] of sand, C, 

may be represented by the expression JiC. We 

have then:

The value of the m will be sought from observation, which will correct in a measure for the 

introduction of F& into the equation without 

allowance for the speed of the grains, etc. 

Subtraction of 0.06 ceases when the sands are 

prevented by suspension from rubbing on the 

bottom; therefore the formula becomes 

C=mVb2 for velocities above a certain limit. 

(It is readily understood that m, like a, is only 

approximately constant.)"

The rate at which dunes advance has been 

measured, in the French rivers, in relation to 

the velocities of the associated currents. It 

rises with Vb until the critical velocity is 

reached, and then drops as the change is made 

from rolling to suspension. The advance of 

dunes, depending on the fall of grains into the 

eddy (fig. 10) when they have been rolled to 

the crest, is affected by the introduction of 

suspension because then only a part of the 

traveling grains are received by the eddy.

Observations made on the Loire give as the 

limiting bed velocity for transportation [competent velocity for transportation], Fbc = 0.25 

meters per second. According to an engineer 

who has discussed those observations [H. L. 

Partiot?], the corresponding surface velocity, 

V8 , is equal to -<J0.ll; and the formula for the 

rate of advance of the dunes,

Rate of advance = 0.00013 (F/-0.11) .(5)

is good for all values of Vs up to 1.016. One 

might base on this a formula for load in relation to surface velocity, but the formula would 

be incomplete unless developed so as to take 

account of the depth; and it is best for the 

present to adhere to equation (4), which connects load with bed velocity.

Lech alas, however, for a temporary purpose, 

uses formulas of Darcy and Bazin to connect 

Vb with V8 , under certain assumptions as to 

depth, and with their aid computes for the 

Loire the critical bed velocity, Vbcc, at which 

suspension of the sands begins. F6cc = 0.55

meters per second. This is the velocity corresponding to Fs = 1.016, the surface velocity 

which limits the applicability of formula (5). 

The following table contains the observational data on dunes of the Loire and compares 

the observed rates of dune advance with rates 

computed by formula (5).

TABLE 63a. Data on subaqueous dunes of the Loire.

Surface velocity.

Met./sec.

0.58

.64

.73

.75

.81

.81

.83

1.00

1.016

1.016

1.03

1.05

1.11

1.13

1.33

Height of

dunes.

Meters.

0.900

.300

.300

.782

.967

.967

.760

.953

.920

.580

.487

.612

1.198

.650

.950

Rate of dune advance.

Observed.

Met.lsec.XW-*

3.0

3.3

5.1

6.3

6.7

7.5

7.6

10.5

12.4

12.0

6.2

7.0

5.8

8.7

5.6

Computed.

Met.lsec.XW-*

3.0

3.9

5.5

5.9

7.1

7.1

7.5

11.6

12.0

12.0

12.35

12.9

14.6

15.2

21.6

In later passages Lechalas recognizes the 

variations of velocity in passing from one 

vertical to another of the same stream sectio*n 

and makes (4) the formula for a division, one 

unit wide, of the cross section. Thus modified, 

it is applied in a variety of ways to practical 

engineering problems of the Loire.

DISCUSSION.

Lechalas's classification of transportation 

processes differs from that adopted for our 

work in that he makes saltation, at least 

verbally, a part of suspension. I am led, however, by a study of the more detailed descriptions Of his colleague Partiot, to believe that 

the line practically drawn between rolling and 

suspension differs in small measure only from 

the line we have drawn between traction and 

suspension.

The lower critical velocity of Lechalas is the 

exact equivalent of our velocity competent for 

traction, and his upper critical velocity corresponds approximately to our velocity competent for suspension. The two attempts at 

formulation likewise agree in giving prominence to the factor of competence. They 

differ in the mode of using that factor, and 

they are actuated by different preconceptions.
REVIEW OF CONTROLS OF CAPACITY. 195

In the first sentence of the passage (pp. 193- 

194) which has been inclosed in quotation marks 

to indicate its literal translation, Lechalas appears to equate the velocity of a particle of the 

load with the difference between the forward 

pressure of the current and the resistance given 

by the particle. Hooker 1 suggests that the acceleration of the particle instead of its velocity is 

intended; but with or without such.emendation 

the author's reasoning is obscure to me, for I 

5ee no necessary physical relation between the 

number or mass of debris particles moved and 

the pressure of the current. The load may be 

defined as the product of the mass of particles 

by their average speed; and their speed, being 

produced by the pressure of the current, may 

be simply related to it, but any relation of the 

mass to the pressure is necessarily indirect and 

presumably involved.

Whatever the strength or weakness of the 

postulates on which the formula is based, the 

manner in which it incorporates the principle 

of competence gives it a rough resemblance to 

those we have developed, while the characterization of its constant m gives to it a 

large empiric factor; and it is. in order to 

inquire whether, as an empiric formula, it 

finds support in the Berkeley data. As the 

Berkeley observations do not include bed 

velocities, the most direct comparison is 

impracticable; but an indirect relation may 

readily be established.

The difficulty we have found in defining bed 

velocity may be avoided, for the purpose of

l Hooker, E. H., Am. Soc. CivQ Eng. Trans., vol. 36, p. 256,1896.

the present comparison, by accepting the 

definition used by Lechalas in

........ (3)

and by assuming depth to be constant. According to the Chezy formula this assumption 

makes Vm approximately proportional to 

 \fa so that IQ-Jds, in (3), is proportional to 

Vm. It follows that Vb is proportional to Vmf 

and this permits us to substitute Vm for Vb in 

equation (4) by changing the constants:

C=K(Vm2 -Jc) (6)

This expression implies that capacity for traction varies with mean velocity at a rate which 

diminishes as mean velocity increases but is 

never so low as that of the second power of 

mean velocity. The corresponding data from 

our experiments, namely, the data for capacity 

in relation to mean velocity under the condition of constant depth, are in accord with this, 

except that they indicate a limiting index of 

relative variation somewhat less than 2. In 

Table 51 the values of the synthetic index, J.Vd, 

range from 2.03 to 7.86; and a value of 2.03 

for the synthetic index implies smaller values 

of the instantaneous index. This discrepancy 

is not important, and the formula of Lechalas, 

regarded as empirical, is probably adequate 

for the discussion of a body of observations 

on capacity and velocity. It could not, however, be used in connection with the Berkeley 

data unless both K and fc (or m and 0.06 in 

equation (4)) were permitted to vary with 

conditions.
CHAPTER XI EXPERIMENTS WITH CROOKED CHANNELS.

EXPERIMENTS.

In order to study the influence which, bends 

in the channel exert on capacity for traction, a 

short series of experiments were made with 

channels having angular bends and others with 

channels having curved bends. Each of these 

channels had a width of 1 foot and was shaped 

by means of partitions within a trough 1.96 

feet wide. (See fig. 67.) Above and below the 

bends were straight reaches of the same width. 

AH the experiments were made with debris of

grade (C) and with a discharge of 0.363 ft.3 /sec. 

The loads were measured. In some experiments the head lost in the region of the bends 

was measured by means of level readings on the 

water surface above and below.

After each experiment the profile of the bed 

was. determined by levelings at intervals of, 1 

foot, and in several cases the region of the 

bends was . covered by such levelings and 

sketches as to make it possible to construct a 

contour map of the bed.

I T

n

m

FIGURE 67. Plans of troughs used in experiments to show the influence of bends on traction.

SIOPE DETERMINATIONS.

The profiles of the channel beds as shaped 

by the current were plotted. Through that 

part of the profile corresponding to the straight 

channel above the bends was drawn a straight 

line representing the mean slope for that region, 

and a similar straight line was drawn below the 

region of the bends. From a point on the first 

line near the position of the first bend to a point 

on the second line corresponding to a position 

several feet below the last bend a straight line 

was drawn, and this was assumed to represent 

the mean slope of the channel in the region 

affected most by the bends. The profiles above 

the bends showed evidence of the rhythms commonly observed in the straight-channel experiments. The profiles below the bends showed 

steeper undulations, which were ascribed to 

196

the influence of the strong agitation of the 

water in passing the bends._ In estimating 

the slopes in the region of the bends the distance used was the length of the medial line of 

the channel.

The observations for head were made at 

points 2 feet and 4 feet above the first bend 

and 3 feet and 5 feet below the last bend, and 

the slopes were computed for the distances, on 

the medial line, between the points.

Check estimates of slope for straight channels were obtained by interpolation from Table 

12, the determinations of load being used as 

arguments.

The data are assembled in Table 64, where 

the stronger and weaker determinations of the 

slope of the debris surface are severally indicated by the letters a and 6. The measurements of water slope are thought to be coordi-
EXPERIMENTS WITH CROOKED CHANNELS. 197

nate in value with those of debris slope. The 

slopes computed from load measurements are 

probably of less weight.

TABLE 64. Comparison of slopes required for straight and 

crooked channels, respectively, under identical conditions 

of discharge, fineness, width, and load.

[Ddbris of grade (C); discharge, (U63 ft.s/sec.; width, 1 foot.]

Shape of 

crooked 

channel. 

(See flg. 

67.)

I'........

II...... 

III..... 

IV......

V.......

Load.

Om./sec. 

19 

75 

56

60 

61

60 

60

63 

63 

66 

64 

64

Slope in straight 

channel. 

Computed 

from load 

by Table 

12.

Percent. 

0.42 

.92 

.77

.80 

.62

.80 

.80

.83 

 . 83 

' .85 

.84 

.84.

Profile of 

ddbris 

above first 

bend.

Per cent. 

0.306 

.896 

.746

 .^..70 o 

.946

.76 a 

.85 a.

.726 

.626 

.876 

.87 a 

.86 a

Slope in crooked 

channel.

Average 

slope of 

debris.

Per cent. 

0.456 

.89ffl 

.80 a

.816 

1.15o

.80 a 

.856

.836 

.806 

.81 ft 

.77 a 

.80 a

Average 

slope of 

water 

surface.

Per cent. 

0.46 

.93

.78

.72

NOTE. Values marked a are given greater weight than those marked 6.

FORMS AND SLOPES.

The combinations of bends in the experiment 

channels are shown in figure 67: In three of 

the channels the bends were angular; in two 

curved. In channel I the angle of deflection 

was 10.5°; there was a single group of four 

bends, returning the course to its original direction; and the short reaches were approximately 

5 feet long. In channel II the arrangement 

was the same, 'with reaches of about 2.5 feet 

and deflection angles of 21.5°. In channel III 

were three groups of four deflections each, the 

angles being of 40.9° and the length of reach 

about 1.4 feet. Channel IV had the same proportions as No. II, with the substitution of 

curves for angles. The radius of curvature for 

the medial line was 6.55 feet. Channel V contained two groups of curves, each similar to the 

group in No. IV.

From the data (Table 64) connected with 

channels I, II, and III, it appears that with 

angular bends a greater slope is necessary to 

transport the load than when the channel is 

straight that is,, the capacity is reduced by 

angular bends. The reduction is greatest when 

the angle of deflection is greatest, and it is so

small for an angle of 10° as to leave doubt 

whether it might not disappear altogether with 

a somewhat smaller angle.

The single group of curves (IV) appears to 

reduce capacity slightly (increase of slope for 

'same load): but the double group (V) gives 

slopes indicating an increase of capacity.

As the bends of alluvial streams are curved, 

the curved experiment channels may be assumed to represent them better than do the 

angular channels, and it is possible that meandering channels have a greater capacity for 

traction than straight channels of the same 

length. There are, however, certain elements 

of incompleteness in the representation which 

make definite inference hazardous. The course 

of a stream which shapes its own channel 

through an alluvial plain is made up of bends 

and reaches. In passing from reach to bend 

there is a gradual increase of curvature until 

the radius of curvature, for- the medial line, is 

between twice and three times the width of the 

channel, and the change from bend to reach is 

also gradual. The forms are automatically 

adjusted to the system of accelerations and 

velocities within the current. The angular 

change of direction in the bend may be one of 

a few degrees only, but in meandering streams 

it is commonly from 90° to 180°. In the artificial channels all curves were circular arcs with 

a radius of 6.55 times the channel width; there 

was. no graduation in the radial acceleration 

due to deflection; the change from right-hand 

deflection to left-hand deflection was abrupt, 

without the intervention of a reach) and the 

changes of direction .were through angles of 

21.5° and 43°. That such differences are competent to affect transportation to a material 

extent is indicated by the relations of deeps 

and shoals (crossings) to bends. Fargue,1 from 

a discussion of an artificially adjusted portion 

of the Garonne, reached the conclusion that the 

distance downstream from the apex of a curve 

to the deepest point of the associated deep and 

the distance downstream from a point of inflection to the associated crossing are each normally one-fourth the length of a stream unit 

defined as the portion between two points of 

inflection. In our experiments (see fig. 68) 

each of these distances is one-half the length 

of the stream unit instead of one-fourth.

i Fargue, L., La forme du lit des rivteres a fond mobile, Paris, 1908.
198 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

Contour interval O.OZfoot 

Width, of trough*foot

FIGURE 68. Contoured plot of a stream bed, as shaped by a current.

In view of these facts, the apparent results 

of the experiments with crooked channels must 

be received with caution, and probably nothing 

more should be claimed for them than a general 

indication that the capacity of a moderately 

bent channel does not differ greatly from that 

of a straight channel.

FEATURES CAUSED BY CURVATURE.

Incidentally the experiments illustrated several consequences of curvature in addition to 

the influence on slope and capacity. At each 

turn the swiftest part of the current was thrown 

to the outer or concave side of the channel, and 

the slower parts moved toward the opposite 

side, the transfers giving to the current as a

whole a twisting motion. The action on the 

debris became exceptionally strong near the 

outer side and exceptionally weak near the 

inner. A result of the strong action was that 

part of the load was thrown upward, so as to 

be temporarily suspended, and a result of the 

diversity of velocity was the maintenance of 

deep places near the outer wall and of shoals 

near the inner. Associated with the twisting 

motion were many whirls or eddies; and the 

general obliquity of motion had the effect of 

reducing the mean velocity in the direction of 

the general flow. The reduction of mean 

velocity was recorded in an increase of mean 

depth, which amounted, in the average of all 

examples, to 7 per cent and ranged from 2 to 

14 per cent.
CHAPTER XII. FLUME TRACTION.

THE OBSERVATIONS.

SCOPE.

That which distinguishes flume traction from 

stream traction is the fixity of the channel bed. 

In stream traction the shapes of the bed are 

adjusted to the rhythms of the mode of transportation, and its texture is that of the de'bris 

in transit. In flume traction the bed is unresponsive, but its texture, being independently 

determined, has an important influence on the 

mode of transportation. The experiments 

were arranged to determine the influences of 

different textures of bed on mode of traction 

and capacity for traction and were otherwise 

varied in respect to slope, width, discharge, 

and the character of debris transported.

GRADES OF DEBRIS.

The material transported in the experiments 

included most of the grades already described 

(see Table 1 and Plate II) and also several 

mixtures not previously mentioned. In order 

conveniently to show the relations of the mixtures to their components, all the grades of 

de'bris used in the flume experiments are listed 

below in Table 65, the data of Table 1 being 

repeated so far as necessary. The elements of 

the table are defined at page 21. The material 

of the coarse grades (I) and (J) differs from 

that of the finer, being about 2 per cent less 

dense. Its particles also are somewhat less 

thoroughly rounded, their journey from the 

parent rock bed having been short.

TABLE 65. Grades of debris.

Grade name.

B). .......

C).........

E). .......

G). .......

H). .......

I).........

J).. .......

E,G,).....

EjH,!,)...

(EgHsIaJ,)..

Sieves 

used in 

separation 

(meshes 

to 1 inch).

40-50

30-40

10-20

4-6

3-4

1-2

*-l

D.

Mean diameter of 

particles 

(foot).

0.00123

.00166

.00561

.0162.

.0230

.0547

.110

.00698

.00706

.00836

F.

Number of particles to 

linear 

foot

812

602

178

61.8

48.4

18.3

9.1

143.2

141.7

119.6

Fa.

Number of 

particles to 

cubic foot.

1.023.000.000

417,000,000

10.770.000

451,000

156.000-..

11,900

1,440

5,610,000

5,430.000

3,266,000

APPARATUS AND METHODS.

The experiment trough, a modification of 

that represented in the frontispiece, was 60 

feet long and 1.91 feet wide, with vertical 

sides. The sides and bottom were of wood, 

planed and painted. For a portion of the 

experiments the bottom was covered by a 

false bottom, specially prepared to present a 

definite character of roughness, the sides remaining smooth. The trough was so arranged 

that it could be given various determinate 

slopes up to 3 per cent, and by means of an 

inclined false bottom a slope of 4.5 per cent 

was made. By means of a partition the width 

was reduced, for the greater part of the work, 

to 1.00 foot.

The width of trough at the outfall end was 

regulated by a contractor, as described on 

page 25. The de'bris was delivered at the 

outfall to a settling tank, which had two 

divisions; and a deflecting apparatus was so 

arranged that the delivery could be instantaneously diverted from one division to the other.

Above the trough near its head was a sloping 

platform on which measured units of de'bris 

were dumped at regular intervals, determined by 

a watch, and from which the de'bris was fed to 

the current by hand, with the aid of a scraper. 

The rate of feed was modified by changing the 

interval between dumpings, and by successive 

trials it was adjusted to the capacity of the 

current.

In accelerating the de'bris, as it fell into the 

water, the current was retarded, so that close 

to the feeding station it was slower than elsewhere. When the load was approximately 

adjusted to the general capacity of the current 

it constituted an overload in this particular 

tract, with the result that a portion was deposited. The load would then traverse an 

upper division of its course on a bed of de'bris, 

while in the lower and principal division it was 

in direct contact with the bottom of the trough. 

A tendency of the stream of de'bris to clog near 

the upper end of the trough, although moving 

freely beyond, was the ordinary criterion of the

199
200 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

proper adjustment of the feed, and when this 

condition existed the load delivered at the outfall was assumed to represent the capacity of 

the stream. The outfall was then directed for 

a measured interval of time to a reserved 

division of the settling tank, and the debris 

thus separately received was weighed.

Besides the deposit connected with the feeding of debris there were transitory deposits of 

a rhythmic-character, as described later.

Five characters of channel bed .were used, 

namely, a planed and painted wood surface; a 

rough-sawn, unplaned wood surface; a surface 

of wood blocks, with grain vertical-; a pavement 

of sand grains, set in cement; and a pavement 

of pebbles: (See PL III.) ' : '

PROCESSES OF FLUME TRACTION. 

MOVEMENT OF INDIVIDUAL .PARTICLES;

Flume traction differs 'f rbm stream traction 

in its extensive substitution of rolling for saltation and in the important place it gives to 

sliding. The relative importance ' of these 

modes of particle niovement is determined (1) 

by the texture of the bed surface in relation to 

the size of the particle, (2) by the velocity of 

the water in relation to the size of the particle, 

and (3) by the shape of the particle.

In stream traction the order of roughness of 

the bed is given by the fineness or coarseness of 

the material of the load, and this fact determines saltation as the dominant process. 'In 

flume traction the bed may be much smoother. 

On a smooth stream bed any particle with a 

broad facet is apt to slide, and a well-rounded 

particle to roll. Rolling is determined (rather 

than sliding) not only by the fact that the propulsive force of the current and the resistance 

given by the bed constitute a couple, but also 

by the fact that the current applies a greater 

force to the; upper part of the particle than to 

the lower. The less smooth the bed surface, 

the greater its resistance and the' more effective 

the couple in causing the particle to roll. With 

any particular texture of bed, the sizes of particles may determine their modes of progress 

the largest sliding, those of smaller size rolling, 

and the smallest leaping. Increase of velocity 

tends to increase saltation at the expense of 

rolling and to increase rolling at the expense 

of sliding. * '

A particle rolled slowly is in continuous contact with the bed. A round particle may roll

rapidly on a smooth bed without parting from 

it. Roughness of the bed causes changes of 

direction in the vertical plane, and such changes 

combined with high velocity cause leaps. If 

the particle is not round its rolling involves 

rise and fall of the center .of mass, and such 

changes combined with high velocity cause 

leaps. Shape of particle may thus be a determinant between saltation .and rolling; as well 

as between rolling and sliding.' 

. A flattish particle, which may either slide or 

roll, travels faster when rolling. This-is due 

partly.to the fact that when it rolls it rolls on 

edge and;thus.projects farther into the current, 

and partly to the fact that the resistance at 

contact with .the bed is greater for sliding than 

for .rolling. It is also true that rolling particles as .a class outstrip sliding particles as a 

class, the difference in speed being-marked.

For particles of similar size those which doininantly roll .outstrip those- which domiriantly 

leap. This is part of a -more general fact that 

the better-rounded particles travel faster than 

the more angular. In the traction of mixed 

debris, where rolling is characteristic.of larger 

particles and saltation of smaller, .the larger 

travel faster than the smaller. There .are thus 

two .important .ways in which. rolling gives 

greater speed than saltation.. It was not 

learned whether, a particle which alternately: 

rolls and leaps travels faster in one way than 

in the other. :

A suspended particle, having the same speed 

as the water, outstrips all others. It is therefore possible that as saltation approaches the 

borderland of suspension its speed exceeds that 

of rolling.

When samples of different grades are fed to 

the same current in succession, it is found that 

the coarser travel the faster, whatever the mode 

of progression (except suspension). In the following record of experiments the speeds of 

grade (J) constitute an apparent exception, but 

their slowness is ascribed to the fact that the 

particles of that grade were relatively angular.

TABLE 66. Relative speeds of coarse and fine debris in flume 

traction.

Depth of 

water.

Foot. 

'. 0. 127 

.182 

.110

" °. 182

Mean velocity of water.

Ft./sec. 

4.35 

3.04 

2.52 

1.52

Average speed of debris particles.

Grade (E).

Ft./sec. 

3.2 

2.3 

1.85 

.70

Grade (H).

Ft./sec. 

3.9 

2.4 

2.0 

.85

Grade (J).

Ft./sec. 

3.8

2.0
U. S. GEOLOGICAL SURVEY PROFESSIONAL PAPER 86 PLATE

ROUGH SURFACES USED IN EXPERIMENTS ON FLUME TRACTION.
FLUME TRACTION. 201

It may be .observed in passing that the recorded speed of the particles is on the average 

75 per cent of the mean velocity of the water, 

the ratio being gi eater as the velocity is greater 

and as the depth is less. A higher ratio would 

of course.be found if the speed of .particles 

were to be compared with that of the.lower part 

of the current. The percentage might be less 

than 75 if the stream were fully loaded.

In watching the traction of mixed debris it 

was observed, as already mentioned, that the 

larger particles traveled faster than the smaller. 

As it is difficult in such an observation to avoid 

giving attention largely to the more active particles, the observation applies especially to those 

which roll, but there is probably a similar contrast between the speeds of, less active particles also. In an experiment with glass balls of 

different sizes it was found that the larger were 

rolled by the current somewhat faster than the 

smaller. '

In attempting to understand the more rapid 

propulsion of the larger particles it is natural 

to compare the phenomenon with the more 

familiar fact that in the absence of water a 

large stone or ball will descend an incline with 

greater speed than a small one, but there is an 

important difference between the two cases. 

The object descending a dry incline is impelled 

by gravity, acting directly, and part of the 

resistance comes from the fluid in which it is 

immersed, whereas the rolling debris pebble is 

impelled chiefly by the moving fluid which 

surrounds it. Tb e advantage in speed accruing 

to the larger pebble in flume traction is probably connected with the fact that the velocities 

of the current increase from the bottom 

upward. If the velocities were all the same 

the pressures applied to similar pebbles of 

different diameter would be proportional to 

their sectional areas, or to the squares of their 

diameters; but because of the gradation of 

velocities the increase of pressure in consequence of increase of diameter is more rapid 

than the increase of the square of the diameter. 

The chief resistance to the forward movement 

of the pebble is engendered at its contact with 

the bed and is of the nature of rolling resistance. 

For uniform speed, the rolling resistance of a 

wheel is proportional to its downward pressure 

divided by its diameter; and since in the case 

of the pebble the pressure is proportional to 

the cube of the diameter, the rolling resistance 

is proportional to the square. Increase in size 

of the rolling pebble is thus accompanied by

increase of both propulsive force and rolling 

resistance, the increment to propulsive force 

being somewhat the larger; and the equality of 

force and resistance is restored by an increase 

of speed, which has the effect of reducing the 

propulsive force and increasing the resistance.1 

The reduction of propulsive force is connected 

with the fact, that that force is determined at 

each, instant by the velocities of the -water, not 

as referred to the fixed bed, but as referred to 

the moving pebble.

The analysis of forces might be further developed, but the foregoing brief outline serves 

to indicate a mechanical principle -underlying 

the observed fact that a current rolls large 

particles more swiftly than small. The principle is of fundamental importance in accounting for certain contrasts between the laws of 

flume traction and those of stream traction. 

The discrimination of traction and suspension, usually easy in the experiments with 

stream transportation, was difficult in the 

flume work. The zone of saltation grew 

deeper with progressive increase of slope until 

it occupied the whole depth of the stream. 

Further increase of slope, with increase of load, 

made the cloud of particles denser, but there 

seemed no way of telling when the condition 

became that of a flowing mixture of water 

and sand.

COLLECTIVE MOVEMENT.

To the general fact that in flume traction 

the particles of the load either roll or slide in 

continuous contact with the fixed bed or else 

skip from point to point along it, there are two 

noteworthy exceptions.

When the transported debris includes particles which are small compared to the projections constituting the roughness of the bed, 

some of the debris finds lodgment among the 

projections. The roughness of the bed is .thus 

diminished and the process of transportation 

is modified. The bed comes to be constituted 

in part of the fixed summits of projections and 

in part of mobile debris, and the process becomes a blending of flume traction proper and 

stream traction. When such conditions existed 

in the experiments it was found that the capacity 

was essentially that due to stream traction.

i The formula given by W. J. M. Rankine, in his "Applied mechanics" 

and elsewhere, for the rolling resistance of a wheel is R=--\a+b(v 3.28)| 

where R is resistance, Q gross load, r radius of wheel, v velocity in ft./sec., 

and a and 6 constants. The experimental values of a and 6 are such as to 

give velocity only a moderate influence on the resistance.
202 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

The second exception is a phenomenon of 

rhythm. It is probable that the rate of flume 

traction is always affected by rhythm, just as 

is that of stream traction, and the rhythm in 

load implies a rhythm in efficiency of current. 

Under certain conditions, of which the most 

important is small slope, the rhythm is manifested by the making of a local deposit from the 

load, a patch of debris appearing on the bed of 

the trough. Such a patch travels slowly downstream, being succeeded after an interval by 

another. With suitable variation of conditions 

the patches become more numerous, are regularly spaced, and occupy a greater share of the 

bed surface. They may even exceed the interspaces in area. When wide apart they 

are usually shaped in gentle slopes, but as their 

ranks close they assume the profiles of dunes, 

with steep frontal faces.

The advance of the debris patches is like that, 

of typical dunes, in that the upstream slopes 

are eroded while the downstream slopes are 

aggraded; and in this way their travel is a 

factor in debris transportation. It does not, 

however, become the dominant factor, as in

the dune phase of stream traction. There is 

always a large share of the load which passes 

over the deposits without being arrested and 

continues its journey across the intervening 

bare spaces.

In this rhythmic process there is a combination of elements belonging distinctively to 

flume traction and to stream traction, and it is 

possible that the process constitutes a transition 

from one system to the other; but, so far as developed by the experiments, it appears as a 

phase, a rhythmic phase, of flume traction. The 

slopes with which it was associated were much 

flatter than those necessary to carry the same 

load by the method of stream traction.

TABLE OF OBSERVATIONS.

In the following table the data are arranged 

primarily by character of bed surface and subordinately by width of channel, discharge, 

grade of debris, and slope of channel bed. 

The characters of bed surface are illustrated in 

Plate III. Load is given in grams per second, 

width in feet, discharge in cubic feet per second, 

and slope in percentage.

TABLE 67. Observations on flume traction, showing the relation of load to slope and other conditions. 

a. Over a surface of wood, planed and painted.

w

1.00 

1.91

Q

0.363 

.734

.363 

.736

S

0.32 

.50 

.75 

1.00 

1.28 

1.50 

1.93 

2.00 

2.50 

2.77 

3.00 

3.41 

3.93 

4.01 

4.03 

4.50 

.32 

.50 

.75 

1.00 

1.28 

1.50 

1.93 

2.00 

2.50 

2.77 

3.00 

3.41 

3.93 

4.01 

'4.03 

4.50 

.75 

1.28 

1.93 

2.77 

3.44 

.75 

1.28 

1.93 

2.77 

3.44

Value of L for grade 

(C)

9.1

58

154

294 

294

566

850

22.8

120

261

561

1,075

49 

110 

230

134

282 

432

(E)

6.1

50

107

183 

.202

318 

366 

444

660 

20.5

98

212

3'2 

381

590 

640 

771

1,081 

41.6 

103 

193 

324 

439 

114 

222 

368 

641 

857

(G)

6.1

55

126

199

360 

386 

479

685 

18.1

110

219

348

580 

726 

765

1,127 

58 

117 

222 

392 

471 

147 

248 

430 

725 

846

(H)

243

268 

453 

384 

438 

524

801

449 

474 

899 

721 

814 

976

1,468 

30 

137 

255 

463 

569 

115 

325 

501 

799 

1,028

(I)

248

334

636

660 

1,248.

1,170

(J)

625

1,095

778 

570

1,595

2,480

(Bid)

28

95

166

271 

342

484

622

68

179

297

427

790

1,278

(E2HiT2 )

138

. 397

783

1,025

315

909

1,557

1,980

(ESH2I3J2)

117

406

793

1,092

335

949

1,730

2,360
FLUME TRACTION. 203

TABLE 67. Observations on flume traction, showing the relation of load to slope and other conditions Continued, 

b. Over a surface of wood, rough-sawn, unplaned, and unpalnted.

w

1.00

Q

0.363 

.734

S

1.07 

1.88 

2.11 

2.20 

2.88 

3.09 

3.81 

4.15 

1.07 

1.88 

2.11 

2.20 

2.88 

3.09 

3.81 

4.15

Value of L for grade 

(E)

133

247

358

269

461

673

(G)

141

284

390

313

513

720

(H)

178

342

526

343

545

772

(I)

304

515

813

632

982

1, 605

(J)

930

i,55S

2,140

(EsHiI»)

186

355

609

881 

302

646

1,550

1,665

c. Over a surface of rectangular wood blocks, with grain vertical.

1.00

Q

0.734 2.00

3.00

4.00

Value of L for grade 

(E)

315

580

911

(H)

337

693

1,026

(I)

561

962

1,497

(J)

750

1,397

2,060

(EsH,I2)

620

1,089

1,660

(E3H2I3J2)

789

1,360

2,020

d. Over a pavement of sand grains, grade (G), set in cement, the d brls being also of grade (6).

w

1.00

Q

0.363

.734

5

1.00

2.00

3.00

4.00

1.00

2.60

3.00

4.00

L

12.1

71

175

317

61.2

214

413

656

e. Over a pavement of pebbles, grades (H) and (I), set lu cement.

w

1.00

Q

0.734

S

2.00 

3.00 

4.00

Value of L for grade 

(E)

225 

425 

630

(G)

'144 

316 

531

(H)

125 

273 

463

(I)

i96 

401

(J)

47l

(EiG,)

190 

483 

714

(E2HJ2)

228 

636 

992

(E3HSI3J2)

605 

1,059

ADJUSTMENT OF OBSERVATIONS.

FORMULATION.

In flume traction, as in stream traction, 

there is a finite slope competent slope corresponding to the zero capacity. An inspection of the observational data by plotting 

served to show that they could advantageously be adjusted by means of the formula 

based on the theory of competent slope:

0=^(8-0)*. (10)

Forty-two observational series were found to

give information as to the value of a. Of these, 

26 indicated positive values, three negative, 

and the remainder values so small as to be of 

uncertain sign. The mean of the 42 values is 

+ 0.29 per cent of slope. Eleven could be 

compared directly with values adopted in the 

adjustments of stream traction data, the mean 

of the eleven values being, for flume traction 

on a smooth surface +0.14 per cent, and for 

stream traction +0.28 per cent. This difference is consonant with observed differences in 

competent slope for the two modes of traction. 

With the aid of this information, and with use 

of considerations connected with modifications
204 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

of the mode of flume traction by rough surfaces, but here omitted, a scheme of values of a 

for the observational series of Table 67 was 

made out. The adjustments were then made, 

by the graphic methods described in Chapter 

II; and their results appear in Table 68. The 

same table records the parameters of the adjusting equations, and also the probable errors.

No adjusted values are given for grade (I) 

in the first division of the table, for the reason 

that the data, although apparently based on 

good observations, are strongly discordant. 

The observations were retained in the record 

because an aberrant fact, if established, may 

prove peculiarly valuable; but in this case the 

interpretation has not been discovered.

TABLE 68. Values of capacity for flume traction, adjusted in relation to slope of channel. 

a. Traction over a surface of wood, planed and painted.

w

1.00

Q

0.363

S

0.5 

1.0 

1.5 

2.0 

2.5 

3.0 

3.. 5 

4.0 

4.5

Probable error (per cent). . . .

Parameters of adjusting!^" ' " 

equations. |b"

w

1.00

Q

0.734

S

0.5 

1.0 

1.5 

2.0 

2.5 

3.0 

3.5 

4.0 

4.5

Probable error (per cent) ....

Parameters of adjusting fa - 

equations. j ^ "

w

1.91

Q

0.'363

Probable error (pe

Parame 

. equat

8

1.0 

. . 1.5 

2.0

: 2.5

3.0 

3.5

tersof adjusting]*'"

ons.............j6iValue of

(C)

28 

 93 

186 

300 

435 

590 

760 

945

2.9

0.04 

1.63 

100

(E)

25.5 

74 

134 

205 

283 

366 

453 

553 

654

3.7

0.06 

1.40 

80

(G)

25 

87 

158 

234 . 

315 398 ' 

486 

575 

665

0.8

0.25 

1.16 

122

(H)

254 

350 

451 

558 

666 

780

1.1

0.35 

1.22 

138

C for grade 

(I) (J)

625 

860 

1,085

1.80 

.96 '520

(E,Gi)

29 

93 

170 

255 

350 

450 

558 

666

1.3

0.15 

- 1.30 

114

(E,HiI»)

138 

266 

410 

568 

732 

910 

1,085

1.8

0.25 

1.29 

199

(E3H2I3J2)

117 

254 

408 

580 

763 

955 

1,155

1.2

0.40 

1.28 

223

Value of C for grade 

(C)

46 

177 

345 

548 

790 

1,050 

1,360 

1,700

2.1

0.03 

1.60 

185

(E)

.54 

145 

258 

382 

518 

665 

820 

985 

1,150

1.8

0.04 

1.35 

154

(G)

50 

151 

268 

393 

527 

668 

815 

965 

1,113

1.7

0.16 

1.20

187

(H)

338 

470 

645 

830 

1,027 

1,235 

1,450

0.7

0.22 

' 1.29 

221

Value of C for grade 

, .c°)

77 

145 

224

- --

0.07' 

1.44 

86

(E)

'69 

.130 

" 200 

281 . 

370 

464

0.5

0.10 

1.42 

80

(P). (H)

(I)

1 

1

653 

910 

,170 

,440

0.65- 

1.01 

470

85 '75 1. 91 

160 . 171 

241 275 

. 328 381 

415 491 

' 507 602

(J)

790 

1,220 

1,630 

2,030 

2,430

0.7

1.20 

.90 

965

Q J3

(EjOi)

65 

179 

310 

453 

610 

770 

940 

1,108

1.7

0.10 

1.24 

202

(EjjHiIj)

336 

592 

870 

1,160 

1,465 

1,790 

2,120

2.7

0.16 

1.20 

415

(E3H2I8J2)

335 

635 

970 

1,310 

1,675 

2,060 

2,450

0.7

0.28 

1.21 

500

Value of C for grade 

(C) (E) (G) (H)

0.734 1.0 197 158 

1. 5 324 281 

2.0 459 415 

2. S ........ 560

3 

3

3. 0 1.5 Probable error (per cent)

0-30 0-55 Paramet 

130 * 182" °9 *^

ers of adjusting!

0 -......- 720

5 ........ 880

.....'.:....: i-.4

185 204 

320 378 

460 550 

607 730 

755 915 ' 

910 1, 095

3.3 2.4

,... 0.05 0.08 0.20 0.35 

n... 1.16 1.30 1.12 1.06 

61.. 210 176 238 322
FLUME TRACTION. 205

TABLE 68. Values of capacity for flume traction, adjusted in relation to slope of channel Continued, 

b. Traction over a surface 'of wood, rough-sawn.

w

1.00

Q

0.363

Probable error (pe

S

1.5 

2.0 

2.5 

3.0 

3.5 

4.0

Parameters of adjusting K ' 

equations. K'"

. ' IV

:' 1.00

Q

0.734

" Probable error (pe

S

1.5 

2.0 

 2.5. 

3.0 

3.5 

..4.0

Parameters of adjusting I*"'" 

. . equations. k""

Value of C for grade 

(E)

97 

145 

200 

259 

331 

385

0.10 

1.34 

62

(G)

160 

220 

285 

355 

427

0.30

a. 12

82

(H)

120 

195 

279 

366 

460 

560

0.40 

1.28 

106

(I)

249 

374 

500 

630 

755

1.00 

1.01 

248

(J)

895

(E2HiI2 )

229 

341 

460 

582 

710 

840

4.6

0.30 

1.16 

184

Value of C for grade 

(E)

290 

390 

495 

600 

712

 0.08 

1.25 

128 '

(G)

331 

434 

540 

648 

758

0.20 

1.11 

173

(H)

354 

462 

570 

680

786

0.30 

1.02 

207

(I)

538 

752 

970 

1,100 

1,325

0.65 

1.06 

 390

(J) ;

1,490 

1,800 

2,100

1.20 

.76 

 950 .

(EsH,l!i)

 450 

660 

885 

1,115 

1,360 

1,615

2.3

0.20 

1.18 

325

c. Traction over wood blocks with grain vertical.

1.00 

Q

- 0.734. - 2.0

2.5

.3.0

3.5

4.0

Parameters of adjusting]* 

 equations. i -" \o\ - -

Value of C for grade 

(E) .

313

442

583

740

900

0.10 

1.46 

123

' (H)

354

507

673

850

1,026

0.45 

1.48 

201

(I)

540

770

1,008

1,245

1,500

0.75 

1.05 

430

' (J)

730

1,090

1,415

1,740

2,040

1.20

.81 

880

(E2H,12)

 613

860 Vi 10

1,375

1,660

0.50 

1.16 

385

(E3H2l3J2)

780

1,075 '

 .. 1,390

1,700

2,010

0.60 

1.06 

545

d. Traction over pavement of sand grains, grade (G).

1.00 1.0

1.5

2.0

2.5

3.0

3.5

4.0

Parameters of ad- [ff . . .

justing equa-<n...

tions. [bi . .

Probable error (per

Value of C.

Q=0. 363

11

36

72

119

175

240

311

0.50

1.70

36

1.1

Q= 0.734

61

131

215

310

413

525

650

0.30

1.72

101

0.1
206 TRANSPORTATION OF DEBRIS BY RUNNING WATER,

TABLE 68. Values of capacity for flume traction, adjusted in relation to slope of channel Continued, 

e. Traction over pavement of pebbles, grades (H) and (I).

w

1.00

Q

0.734

S

2.0 

2.5 

3.0 

3.5 

4.0

Parameters of adjusting K' ' ' 

equations. l^" "

Value of C for grade 

(E)

220 

313 

416 

529 

650

0 10 

1.50

84

(G)

143 

220 

310 

412 

522

0.40 

1.60 

67

(H)

122 

192 

272 

363 

460

0.55 

1.52

68

(I)

209 

290 

393

1.00 

1.65 

65

(EiGi)

200 

325 

460 

600 

740

1.00 

1.19 

200

(E2HiI2)

250 

420 

605 

800 

1,015

1.00 

1.26 

250

(E3H2l3J2)

605 

830 

1,060

1.20 

1.28 

280

PRECISION.

Probable errors were computed from the 

residuals of 25 series, the residuals being 

measured on the plots as percentages. The 

greatest probable error computed for a series 

of adjusted values of capacity is ±4.6 per cent, 

and their average is ±1.3 per cent. The average of the 25 determinations of the probable 

error of an observation is ±3.8 per cent. The 

residuals number 139, and then- average value, 

which also is a measure of the precision of the 

observations, is ±4.3 per cent.

A comparison of these measures with those 

obtained from the data for stream traction 

shows that the flume-traction data are decidedly the more harmonious. The average 

residual is more than twice as great for stream 

traction as for flume. Part of this difference 

may be due to the fact that the experiments 

with flume traction came last and had the 

benefit of previous experience, but it is to be 

ascribed chiefly to the fact that in flume traction the slope is constant, while in stream

traction it is subject to rhythmic fluctuations.

DISCUSSION.

CAPACITY AND CHANNEL BED'.

Data illustrating the influence of the character 

of the channel bed on the quantity of debris 

which a stream can transport have been assembled in Table 69. They are taken chiefly 

from the preceding table, but a few items arc 

from Table 72 and Table 12. A single item, 

marked as interpolated, is based on a combination of data from Tables 68 and 72. They 

pertain to all the simple grades and mixed 

grades with which experiments were made in 

flume traction; and stream traction is represented, so far as possible, by coordinate data. 

Comparisons are made for slopes of 2 and 3 per 

cent and discharges of 0.363 and 0.734 ft.3/sec.

The greatest capacity is in each case associated with the smoothest of the tested channel 

beds, the surface being that of a plank, planed 

and painted, with the grain running parallel 

to the current.

TABLE 69. Comparison of capacities for flume traction associated with different characters of channel bed.

[Width of trough, 1 foot.]

Q

0.363 

.363 

.734

.734

S

2.0 

3.0 

2.0

3.0

Character of channel bed.

Planed wood ..................

Debris...... ...................

De"bris.........................

Sand pavement. ...............

DSbris........ ..................

Gravel pavement. .............

Value of C.

Simple grades.

(B)

388

266

(C)

300

245

548

483

(E)

205 

145

115

366 

259

382 

290 

313

220 

222

665 

495 

583

416

(G)

234 

160

72 

45

398 

285 

175 

120

393 

331

215 

143 

145

668 

540

413 

310

(H)

254 

195

451 

366

470 

354 

354

122

830 

570 

673

272

(I)

364 

249

600 

500

711 

533 

540

[1, 060] 

970 

1,008

209

(J)

790

730

1,630 

1,490 

1,415

Mixtures.

(Eid)

453

200

770

460

(E8HiI2)

410 

341

732

582

870 

6'60 

613

250

1,465 

1,150 

1,110

605

(E3H2I3J2 )

970

780

1,675

1,390

605
FLUME TRACTION. 207

Next in order are two varieties of unplaned 

wooden surface; the first being that of boards 

or planks, paiallel to the current, retaining the 

roughness left by the saw; the second a pavement made by sawing planks of Oregon pine 

into short equal blocks and setting them on 

edge. Both these surfaces, as well as those 

described below, are illustrated in Plate III. 

These two varieties proved to have approximately the same properties in respect to traction, and the capacities associated with them 

are 23 per cent less than those for planed 

lumber. The range in ratio is not large for 

the different experiments, and the value given 

may be taken as a constant representing the 

difference in efficiency between new unplaned 

and planed wooden flumes. The difference 

tends, however, to diminish with wear, the 

unplaned lumber becoming smoother and the 

planed rougher.

The next grade of roughness was given by 

coarse sand debris of grade (G) set in 

cement, so as to constitute a pavement resembling sandpaper. The only material run 

over this was de*bris of the same grade, the 

special purpose being to compare flume traction with stream traction the condition of 

fixed bed with that of mobile bed when the 

degree of roughness is the same. The experiments gave the streams 50 per cent greater 

capacity when sweeping the debris over the 

fixed bed than when moving it at the same 

slope by the method of stream traction.

The sand pavement gives capacities half as 

great, on the average, as the surface of planed 

lumber, but the contrast is stronger for the 

smaller discharge and lower slope and less 

marked for the larger discharge and steeper 

slope.

The roughest surface used, ,a .pavement of 

pebbles prepared by setting in cement a mixture of grades (H) and (I), gave still lower 

capacities. These range from 20 to 62 per 

cent of the corresponding capacities given by 

planed lumber. The obstructing influence of 

the rough bottom is most strongly manifested 

when the material transported has a coarseness 

corresponding to the texture of the pavement. 

For finer material its roughness is mitigated by 

the lodgment of debris, which has the effect of 

establishing a pavement of the finer material. 

The word "debris" in the table indicates a 

channel bed composed of loose debris, the

debris in transit, and the associated process is 

that of stream traction. The available data 

afford comparison only for the four finer 

grades, (B), (C), (E), and (G), the grades 

which would be designated sand. Each comparison, with an apparent exception to be 

considered immediately, shows stream traction to be less efficient than flume traction. 

When stream traction is compared with flume 

traction over a smooth surface, the observed 

ratio of efficiency ranges from 19 to 88 per 

cent, the smaller ratios being associated with 

the coarser grades of debris.

The exception occurs when capacity over a 

bed of debris is compared with capacity over 

a pavement of pebbles, the two capacities 

being found to be the same. The cases which 

afford this comparison are for grades (E) and 

(G), and these fine materials, by filling the 

hollows of the pavement, create a condition of 

bed in which stream traction dominates. The 

comparison is really between normal stream 

traction and stream traction modified by the 

appearance of crests 'of fixed pebbles in the 

channel bed. In harmony with this interpretation is the fact that capacities for stream traction and for traction over the gravel pavement, 

when compared severally with capacity for traction over smooth wood, both show contrasts 

which increase with coarseness of the load.

The important general facts brought out by 

the comparisons are (1) that with a given discharge, channel width, and slope, the process 

of flume traction is able to transport more 

debris than that of stream traction, and (2) that 

a stream's capacity for flume traction varies 

inversely with the roughness of the flume bed.

The first of these principles serves to explain 

certain phenomena of clogging. When there 

is fed to a flume a load greater than its stream 

is able to transport, a portion is deposited. 

This changes the character of the bed in such 

a way as to substitute stream traction for 

flume traction. Stream traction, being less 

efficient, can carry still less load, and a larger 

fraction is deposited. If the conditions remain unchanged the bed is built up until its 

slope becomes that necessary to carry the entire load by stream traction. Unless the trough 

is deep or short, overflow results.

When clogging is initiated by a temporary 

overloading, the stream loses power to carry its 

normal tractional load, and deposition con-
208 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

tinues iipless the load is reduced considerably 

below the normal. In practical operations 

the first step toward the abatement of a clog 

is te stop all feeding of load above the deposit, 

that the stream may be able to take on load 

and thereby reduce the deposit.

CAPACITY AND SLOPE.

The rate at which capacity for flume traction 

is increased by increase of slope is contained 

implicitly in the values of n and o assembled in 

Table 68; for n gives the rate of variation of 

capacity with S a, and the instantaneous rate 

of variation of capacity with slope is given by

nS i = _ _,. All the tabulated values of n except

o o

two are greater than unity; and in each of the 

cases where n is less than unity all'values of ix 

computed for the range of slopes covered by the

experiments are greater than unity. The general fact is thus indicated that, within the practical range of conditions, capacity increases 

with slope in more than simple ratio. Efficiency also increases as slope increases.

The sensitiveness of capacity to changes in 

slope varies with changes of condition;'and this 

variation might be illustrated, as in treating of 

stream traction, by the tabulation and discussion of values of the index of relative variation, 

V It will suffice, however, in this case to 

make comparisons by means of the synthetic 

index, 'Ir Table 70 contains values of that index computed between the limits of 8 = 2.0 and 

$ = 3:5. They represent 42 of the 51 series of 

values given in Table 68, the other nine series 

not having sufficient range for the computation. The arithmetical mean of the 42 values 

of 7t is 1.46, and the range is from 1.08 to 2.08.

TABLE 1-0. Values of Jt for flume traction, computed between the limits S=2.0 and S=S.5.

w

1.00 

1. 00

1.91 

1.91

Q

, 0.363 

.,. -734

' .363 

- : 734

Character of channel bed.

......do.................:....

Value of /i.

Simple grades.

(C)

.1.66

1.66

 <E)

1.44 

1.43 

1.37 

1.30 

1.54

1.56 

1.50 

1.34

(G) -

1.30 

' 1.42 

1.30 

1.20

1.60

1.89 

1.32 

1.22

' (H)

1.41 

1.54 

1.40 

1.17 

1.56

1.95 1:40

1.23-

'(I)

1.66

1.27 

1.49

(J)

1.69

1.55

Mixtures.

(EjGO

1.40

1. 30

1.96,

(EiBiI,)

1.42 

1.32 

1.29 

1:08 

1.44

2.08

(E3H2I3J2)

1. 52

1.35

1.39

Inspection of these data shows, first, that 

the > values are always greater for $ = 0.363 

than for Q = 0.734. The experiments deal 

with no other discharges, but it is probably 

true in general (as in case of stream traction) 

that-increase of discharge is accompanied by 

decrease of the 'Sensitiveness . of capacity. to 

slope.' . -,'..- 

If the index varies iii a systematic-Way with 

fineness Of debris, its increase is connected 

with decrease of fineness, but the finest debris 

of the table, (C), carries large values of the 

index. The apparent conflict of'evidence has 

its parallel in the fuller" data for stream traction (see p. 108 and fig. 34), and it is possible 

that the sensitiveness increases' in two directions from a minimum value. Its variation 

might in that case be connected with the law 

relating capacity to fineness, as brought out in 

a later section.

The relation of the index to roughness of bed 

does not.follow a simple law. Its values .are 

in general least for the bed of rough lumber and 

progressively greater for planed lumber, wood 

blocks, and gravel pavement. . . . . The greater sensitiveness of capacity to slope 

when the channel bed is a coarse pavement 

may be connected with the fact that the mode 

of transportation over such a bed is approximately stream traction; and this suggests that 

in flume traction the sensitiveness may be less 

than in stream traction. Direct comparison 

can not be made wiih use of the values of /t 

in Table 70, because the slopes used in stream 

traction experiments have less range; but special computation's were made, sd far as the data 

were found to overlap. The results are contained in Table 71 and indicate that the sensi- 

. tiveness is greater for stream traction than for 

flume traction over a smooth bed, in case of 

grades (E) and (G), but less in case of grade (C).
FLUME TBACTION. 209

TABLE 71. Comparison of values of I^for flume traction 

over a bed of planed wood, with corresponding values for 

stream traction.

Width 

of trough 

(foot).

1.00

Grade of 

debris.

(C)

(E)

(G)

Discharge 

(ft.s/sec.).

0.363

.734

.363

.734

.363

.734

Limiting 

values of 

slope (per 

cent).

0.5-2.0

'.5-2.0

.5-2.JO

.5-2.0

2.0-3.0

1.0-2.0

/ifor 

flume 

traction.

1.71

1.79

1.50

i.41

2.27

1.38

7i for

stream 

traction.

l.CO

1.46

1.99

1.65

2.55

2.42

Various qualifications and doubts being 

omitted, the preceding paragraphs may be 

generalized by saying that the sensitiveness of 

capacity to slope is somewhat less in flume 

traction than in stream traction. It varies in 

both directions from a mean value expressed. 

by the exponent 1.5, being greater as the slope 

is less, as the discharge is less, as the fineness

is less, and as the channel bed is rougher. 

Efficiency for flume traction increases with 

slope. 

CAPACITY AND DISCHARGE.

Special series of experiments were made to 

determine the variation of capacity with discharge. In each series the conditions of slope, 

width, and grade of de*bris were kept constant 

and the discharge was varied. The observations are given in Table 72, and the same table 

contains the adjusted values of capacity, together with the parameters of the adjusting 

equations. Inspection of logarithmic plots 

showed the propriety of adjusting by means of 

the formula used with the data for stream 

traction,

C=lz (q-KY............ (64)

and the computations were graphic.

TABLE 72. Observations and ad justed data illustrating the relation of capacity for flume traction to discharge, for a rectangular

flume of planed wood 1 foot wide.

[L, observed load; C, adjusted value of capacity; Q, discharge.]

Parameters of adjusting equations . .

Q

0.039 

.093 

..182 

.363 

.545 

.734

1 ;

1 63

Q

0.039 

.093 

.182 

.363 

.545 

.734

2 per cent.

(B)

L

54.5 

151 

388 

619

0.002 

1.34 

1,430

0.9

C

58 

145 

371 

640

(E)

L

96 

202 

322 

446

0.007 

- 1.09 

630

0.2

C

95 

205 

322 

445

(H)

L

136 

268 

417 

546

0.040

.88 

760

0.4

C

134 

279 

415 

549

(I)

L

110 

385 

528 

738

0.140 

.68 

1,000

0.9

C

116 

364 

548 

711

(EiHiIi)

L

144 

397 

617 

907

0.060 

1.05 

1,345

0.5

C

146

384 

027 

890

3 per cent.

(E)

L

28.0 

79 

169 

366 

580 

785

0.004 

1.10 

1,125

0.3

C

28 

79 

169 

364 

572 

795

(H)

L

25.0 

128 

228 

446 

635 

914

0.025 

. ..... - -8g -

1,175 

1.0

C

27 

108 

230 

450 

670 

870

(I)

L

50.6 

272 

567 

922 

1,170

0.080 

.79 

1,650

0.7

C

53 

270 

600 

880 

1,170

(EjH,I2 )

L

87 

325 

718 

1,147 

1,500

O.«040 

.88 

2,220

1.4

C

123 

325 

730 

1,130 

1,550 .

Several values of capacity in Table 72 agree 

as to conditions with values in Table 68, and 

these values would be identical if the experiments were homogeneous. A comparison 

shows that the values given by the experiments 

comparing capacity and discharge are in general the greater, the average difference being 

6 per cent. This is evidently of the nature of 

systematic error and is probably connected 

with some change in apparatus or in detail of 

20921° No. 86 14 14

experimental method which occurred between 

the making of the two groups of experimental 

series.

The sensitiveness of capacity to changes of 

discharge varies with conditions. It is greater 

as the discharge is less, as the slope is less, and 

as the channel bed is rougher. It is relatively 

great for the coarsest and finest of the debris 

used and less for intermediate grades. Under similar conditions it is less for flume trac-
210 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

tion than for stream traction.. Expressed as 

an exponent, 73 , its average value for the range 

of the experiments recorded in Table 72 is 1.26. 

Values of 73 were also obtained from data in 

Table 68 by comparing the capacities for Q = 

0.363 ft.3/sec. with those for Q = 0.734 ft.3/sec.; 

and the'mean of such values is 0.97. These 

mean values are not necessarily inconsistent, 

for the synthetic index varies with the" range in 

discharge for which it is .computed and is lower 

as the discharges are higher; but "a study of individual Values shows that under identical conditions the data of Table 68 give the 16wer estimates of' sensitiveness.. The data as a whole 

indicate that, for the range of 'conditions realized in the experiments, the average value' 'of 

the exponent expressing sensitiveness to discharge is 1.2. There can be no question that 

for the larger discharges used it falls below unity. " ' "" 

-,. As the sensitiveness of the duty of water, and 

also the efficiency, to discharge is expressed by

an exponent which is less by unity than the 

corresponding exponent for the sensitiveness of 

capacity to discharge, it follows that duty and 

efficiency vary little with discharge. In general 

they gain slightly with increase of discharge, 

but they lose when the discharge or slope is 

relatively large. This accords with a result 

obtained by G. A. Overstrom, 1 who found from 

experiments with launders that duty rose with 

increase of depth to a limited extent only.

CAPACITY AND FINENESS.

In Tables 68 and 72 the values of capacity 

standing in any horizontal line constitute -a 

series illustrating the variations which are related to grades of debris, and if those in the 

columns for mixtures be excepted they illustrate the relations of capacity to fineness. 

Table 73 contains a selection of data from those 

tables, together with a single line taken from 

Table 69.

TABLE 73.: Values of capacity for flume traction, illustrating the control of capacity by fineness of debris.

Character of channel bed. w. -.-

1.91 

1.00 

1.00 

1.00 

1.00

1.00 1.00' 

1.00 

1.00

Q

0.734 

.363 

.363 

.734 

.734

.734 

.734 

.734 

' .363

S

2.0 

' 2.0 

3.0 

2.5 

3.0

3.0 

3.0

2.0

Valu»of (7 for grade 

(B)

388

266

(C)

459

590 

790 

1,050

245

(E)

415

202 

366 

518 

665

495 

583 

416 

115

(G)

460

398 

527 

668

540

310 

45

(H)

550 

268 

451 

645 

830

570 

673 

272

(I)

383

653 

910

970 

1,008 

200

(J)

625 

1,220 

1,630

1,490 

1,415

As the tables are examined, one of the features arresting attention is that in most of the 

series the smallest value of capacity does not 

appear at one end of the line but at some intermediate point. The occurrence of a minimum 

is in fact characteristic of all tested varieties 

of flume traction except that in which the bed 

is a pavement of pebbles. To give the feature 

graphic expression the data of the last five 

lines of Table 73 are plotted in figure 69, where 

the horizontal scale is that of linear fineness, F. 

The plotted points are far from, regular, but 

the general character of the representative 

curves is unmistakable, and freehand lines have 

been drawn. On another sheet, hot reproduced, 

the same data were plotted in relation to mean 

diameter of particles the reciprocal of F 

with similar result, except that the lower two 

curves became concave upward.

These curves illustrate the most important 

difference between the laws of flume traction 

and those of stream traction. In stream traction capacity increases continuously as fineness 

increases. In flume traction capacity increases 

with fineness when the grades of debris compared are relatively fine but increases with 

coarseness when the grades are relatively 

coarse. So far as these experiments show, the 

minimum of capacity corresponds to a coarse 

.sand, but its position on the scale of fineness 

may be assumed to vary with slope, discharge, 

and roughness of bed.

The curve for flume traction over a bed 

paved with gravel shows no minimum but is 

of the same type as the curve for stream traction. This fact is confirmatory of an inference

1 Quoted by R. H. Richards in Ore dressing.
FLUME TRACTION". 211

already drawn (p. 207), that the transportation 

of fine debris over a fixed bed of coarser debris 

particles is. essentially of the nature of stream 

traction. It may fairly be inferred that if we 

were able to extend this curve into the region 

of debris coarser than the gravel of the bed, a 

minimum would be developed.

If the curves were to be traced toward the 

right, by means of additional experiments with

1,400

200 400 OOU

Linear fineness

FIGURE 69. Curves illustrating the relation of capacity for flume traction to fineness of ddbris. Data from bed of planed wood are recorded 

by crosses; from wood-block pavement by circles; other data by dots.

finer debris, there can be little doubt that they 

would be found to continue their ascent; but 

eventually, as curves of traction, they would 

come to an end with the passage of the process 

of transportation from traction into suspension. 

In the opposite direction they may be conceived 

to attain a maximum and then drop suddenly 

to the base line; for despite the law of increase 

of capacity with coarseness, there must be a 

degree of coarseness for which the force of the 

current is not competent, and when that is 

reached the ordinate of capacity becomes zero. 

The position of this limit, which I have in earlier 

pages called competent fineness, evidently 

depends on slope and discharge, as determining 

the force of the current, and on the degree of 

Founding of the de"bris.

The double ascent of the curve of flume 

traction is susceptible of plausible explanation, 

by means of considerations connected with the

process of rolling. The process of rolling, involves a question of space. Each rolling pebble 

occupies .an area of the channel bed somewhat 

larger than its sectional area, even if the 

pebbles are .arranged in the closest .possible 

order. If we conceive the channel bed to be 

occupied by rolling pebbles of a particular size, 

separated by spaces which bear a definite ratio 

to the diameters of the pebbles; and .again 

conceive it to be. occupied by rolling pebbles 

of a larger size, with the same ratio between 

interspace and diameter; it is evident that the 

total volumes or masses of pebbles in the two 

cases will be proportional to the diameters. 

If the larger pebbles have twice the diameter 

of the smaller, then a given area of bed will 

contain twice as much rolling load of the larger 

pebbles as of the smaller. It is also true, as 

stated on an earlier page, that the rolling speed 

is somewhat greater for larger pebbles than for 

smaller. The tendency of these two factors is 

the same, to make the load greater for large 

particles than for small, when the process of 

transportation is rolling. The analysis is doubtless- too simple the degree of crowding on the 

bed, for example, may not be the same for 

different sizes, and the degree of crowding may 

affect the speed of rolling but qualifying 

factors can hardly impair the qualitative 

inference that the rolling load increases with 

coarseness.

It is a matter of observation that, under 

similar conditions determining force of current, 

the dominant' process in flume traction is for 

coarse debris rolling and for fine debris saltation. When the process is rolling, as just 

shown, capacity increases with coarseness of 

debris. When it is saltation, as illustrated by 

the body of experiments on stream traction, 

capacity increases with fineness. With the 

passage from saltation to suspension the effect 

is even heightened, and it is probable that in 

a number of the recorded experiments the 

process was largely that of suspension. Thus 

the double ascent of the capacity-fineness curve 

is determined by the distinctive properties of 

two (or three) modes of propulsion.

If the preceding explanation is well founded, 

the nature of the law connecting capacity with 

the degree of comminution of the debris in any 

particular case depends on those conditions 

which determine the dominant process of conveyance. If the channel bed is smooth, and if
TRANSPORTATION OF DEBRIS BY RUNNING WATER.

slope and discharge are so adjusted as to give 

a moderate velocity, the progression of sand 

may be by rolling, and in that case the capacity 

for different sands will vary inversely with their 

fineness. But if over the same smooth bed the 

current runs swiftly, sand will be made to 

travel by saltation, or by saltation and suspension, and then the capacity for different sands 

will vary directly with their fineness. The particular velocity with which the function reverses will depend on the quality of the bed,

being lower if the bed is somewhat rough, 

because roughness changes rolling to saltation. 

The critical velocity will be higher for gravel 

than for sand, because higher velocity is 

needed to make coarser debris leap.

The experiments which have been made were 

not sufficiently varied to afford test for these 

inferences, and as there is no present opportunity for continuance of laboratory work the 

inferences must be regarded as largely hypothetic.

TABLE 74. Capacities for flume traction of mixed grades and their component simple grades.

Character of bed.

Do...............................................

Do.... ................................ ....i......

Do...............................................

Do............................................... 

Wood block. . ........................................ 

Do...............................................

Do...............................................

Do...............................................

Q

0.363

.734 

.734 

.734 

.363 

.734 

.734 

.734 

.734 

.734 

.734

Mixed^rade.

(F,Gi)

 >--

(EsHiIa)...... 

(EsHiIs)...... 

(EsHzIsJs)..... 

(EiGi). .......

Capacity 

for 

mixture 

(gm./sec.).

450 

770 

1,465 

1,675 

582 

1,115 

1,110 

1,390 

460 

605 

605

Capacity for single grade (gm./sec.).

(E)

225 

366

385 

665

586 

665

502 

665

233 

259

446 

495

444 

583

417 

.583

230 

416

242 

416

181 

416

(G)

225 

398

385 

668

230 

310

. (H)

293 

830

335 

830

116 

366

223 

570

222

.. 673

278 

673

121

272

121 

272

(I)

586 

910

502 

910

233 

500

446 

970

444 

1,008

- 417 

. 1,008

242 

209

181 

209

(J)

-

335 

1,630

278 

1,415

121

MIXTURES.

Three mixtures of simple grades of debris 

were treated in the laboratory. The components of one, (E^), were a medium sand and 

a coarse sand. The others, (E-jHJ-j) and 

(E3H2I3J2 ), combined medium sand with fine 

gravel and coarser gravel. All three were 

tested in relation to slope on the smooth channel bed and on the gravel pavement one on 

the rough-sawn bed and two on the bed of 

wooden blocks. One entered into the experiments on capacity in relation to discharge. 

The results are contained in Tables 67, 68, and 

72. The data from those tables which pertain 

to a channel width of 1 foot and a channel slope 

of 3 per cent are assembled in Table 74. The

table contains also the capacity quota for each 

constituent, computed from the capacity for 

the mixture; and beneath each of these quotas 

is printed the capacity for the constituent when 

the entire load is composed of it.

A general fact brought out by this table is 

that the current can transport more of a 

mixture than it can of any one of the constituent grades. The table records a single exception, the capacity for the mixture being 1,390 

gm./sec., while that for its coarsest constituent 

grade is 1,415 grams.

Another general fact shown is that the capacity for each component as part of a mixture 

is less than the capacity for the same component if transported separately. To this also 

there is a single apparent exception, but as it
FLUME TRACTION. 213

occurs among the data for traction over a gravel 

pavement it illustrates a feature of stream traction rather than flume traction.

The two principles may be illustrated together by saying that if a stream is carrying its 

full load of a grade narrowly limited in range 

of fineness, and a different grade of debris is 

added, the total load is thereby increased, but 

this increase is accompanied by a diminution 

of the quantity carried of the first-mentioned 

grade. In contrast with this is the law found 

for stream traction that the load of the initially transported grade is increased by the 

moderate addition of other debris, provided 

the added debris is relatively fine.

The difference between the two cases is 

thought to be connected with rolling. In 

flume traction over a smooth bed the path for 

rolling particles is roughened by the presence 

of smaller particles. In stream traction the 

pathway for larger particles is smoothed by the 

presence of smaller particles and rolling is 

promoted.

In stream traction the capacity for a mixture 

is determined chiefly by the capacity for its 

finer components, and as mean fineness also 

depends chiefly on the fineness of the finer 

components, mean fineness is a serviceable 

gage of capacity. In flume traction the relation is quite different. Because of the double 

ascent of the curve of capacity and fineness, 

it may readily occur that the capacity for a 

mixture is most nearly related to that for the 

coarsest component in fact, that is true of 

the three mixtures tested in our experiments 

and when that is the case there is no parallelism 

between capacity and mean fineness.

CAPACITY AND FORM RATIO.

The data bearing on the relations of capacity 

to the depth and width of current, and their 

ratio, are meager. Most of the experiments 

were conducted with a single trough width, 1 

foot. The only other width used was 1.91 

feet, and its use was associated with but four 

grades of debris and a single character'of 

channel bed the smoothest.

Depths of current were not in general measured during the passage of loads, because the 

surfaces of load-bearing currents were usually 

so rough as to make good determinations -impossible. Good. measurements were made of 

unloaded streams, and the results are -here 

tabulated. Attempts to measure depths ;of 

loaded streams yielded one result thought 

worthy of record. With a discharge Of.0.734 

ft.3/sec., a width of 1 foot, a slope of 4 per cent, 

and a full load of debris of grade (E3H2I3J2), 

the depth was 9 per cent greater than for the 

corresponding unloaded stream.

Table 76 compares the capacities found for a 

trough width of 1.91 feet with corresponding 

capacities for a width of 1 foot. By aid of 

Table 75 it brings capacities into relation also^ 

with depths.

TABLE 75. Depths and form ratios of unloaded streams, in. 

troughs of wood, planed and painted.

Q

0.303

.734

S

1

2

3

1

2

3

10 = 1.91

d

0.070

.002

.050

.119

.098

.080

R

0.040

.032

.029

.002

. 051

.045

w=1.00

d

0.120

.090

.082

.194

.154

.136

R

0.120

.096

.082

.194

.154

.13C

TABLE 76. Capacities for flume traction in troughs of different widths.

\

Q

0.363 

.734

.363 

.731

S

1.0 

2.0 

3.0 

1.0 

2.0 

3.0

1.0 

2.0 

3.0 

1.0 

2.0 

3.0

Grade (C).

w=1.91 10=1.00

Grade (E).

io=1.91 W=l,00

Grade (G).

10=1.91 io=1.00

Grade (H),

10=1.91 io=1.00

Values of C.

77 

224

i<)7 

459

93 

300

177 

548

69 

200 

370 

158 

415 

720

74 

205 

366 

145 

382 

665

85 

241 

415 

185 

460 

755

87 

234 

398 

151 

393 

668

275 

491

550 

915

254 

451

470 

830

Ci.sil

.... . ...._.. . .. Clofl

0.83 

.75

1.11

.89

0.93 

.98 

1.01 

1.09 

1.09 

1.08

0.98 

1.03 

1.04 

1.22 

1.17 

1.13

1.08 

1.08

1.17 

1.10
214 TRANSPORTATION OF "DEBRIS: -BY RUNNING WATER.

Simple combinations of the quantities, into 

which-it is not necessary to enter, show (1) 

that with constant depth, capacity-increases 

with .widtht.and more rapidly than width, and 

(2) that^with constant width, capacity increases 

with dep|h<and more rapidly than depth. The 

rate of increase with depth, if expressed as an 

e.xpqnent^.may he as low as 1.2 or as high as 

2.0. From these premises, a line of reasoning 

parallel to that :of Chapter IV shows that for 

flume traction^ as for stream traction, the 

function C=f(R} increases to a maximum 

and then decreases. The value of R corresponding to maximum capacity the optimum 

form ratio can not be determined from observations involving but two channel widths; but 

its limits can in some cases be indicated. For 

example, when the capacities. conditioned by 

the same slope and discharge are approximately 

the same for the two trough widths, it may be 

inferred that the optimum falls between the 

values of R associated with the two capacities. 

Thus for grades (E), coarse sand, with a slope 

of 2 per cent, the capacities are about the 

.same for the two widths, while the form ratios 

.are 0.032 and 0.096, or one-thirtieth and onetenth; and it is inferred that the optimum form 

:ratio falls between those fractions.

From inferences, of this sort, used in combination with the general principles of flume 

traction, a number of tentative conclusions 

have been drawn. They are of so hypothetic 

a character that the reasoning connected with 

them is not thought worthy of record, but the

 conclusions themselves may perhaps be of 

some service, until replaced by others of more 

.secure foundation. They are: The ratio of 

depth to width giving a current the highest

 efficiency for flume traction (1) is greater for 

gentle slopes than for steep, (2) is greater for 

small discharge than for large, (3) is greater for 

fine debris than for coarse, (4) is greater for 

rough than for smooth.channel.beds, and (5) 

is in general less than for stream traction. 

The first and second propositions apply also to 

stream traction. The third is the reverse of 

the relation determined for stream traction, 

&nd its applicability may be limited to conditions under which rolling is the dominant mode

 of transit. By aid of the fifth and fourth 

Table 31 may be roughly applied to practical 

problems of flume traction.

TROUGH?1 OF " SEMICIRCULAR CROSS SECTION.

A few experiments were made with a trough 

of galvanized sheet iron, 1 foot wide, having a 

semicircular section. It was given a slope of 

1 per cent, and in it were tested four grades of 

debris. The observational data appear in 

Table 77. The object of the experiments was 

to determine whether a channel with curved 

perimeter is more efficient or less efficient 

than one with rectangular section; the capacities obtained are compared, in the table, with 

those determined for a flat-bottomed trough 

of the same width. As the discharge used did 

not fill the semicylindric trough,. the' width of 

water surface was less than 1 foot. The 

width of channel bed occupied by the load 

ranged from one-fifth to one-half of the width 

of water surface. The medial depth of water 

was greater than in the rectangular channel, 

and the mean velocity was higher. The higher 

velocity is a factor .favorable to the development of capacity; the narrower field of traction 

is an unfavorable factor. The resultant of the 

two was unfavorable, the capacities for the 

semicylindric trough being only half as great 

as for the rectangular. The result is qualified 

by the fact that the troughs. compared were 

not of the same material, but .the disparity of 

capacities is too great to be ascribed to that 

factor. -

TABLE 77. Data on flume traction in a semicylindric iron 

trough of 0.5 foot radius; with comparative data for a rectangular wooden trough 1 foot -wide.

Slope 

(per cent).

1.00

Discharge 

(ft.s/sec.).

0.363 

.734

Grade of 

de'bris.

i

F

1 

G

Depth of unloaded/Q=0.363... 

stream (foot).. . ..... .\Q=0.731... 

Width of water surface,/Q= 0.363... 

unloaded stream ( foot )\Q= 0. 734. . . 

Width of be occupied by oad 

(foot).....-.....................-

Mean velocity of un-/Q=0.363... 

loaded stream(ft./sec.)\Q=0. 734...

Capacity (gm./sec.).

In semicylindric 

trough.

43 

33 

37 

64 

66 

74

0.181

.257 

.770 

.874

. 17 to . 37 

3.85 

4.60

In rectangular 

trough.

93

74

145

151

0.120 

.194 

'1.00 

1.00

1.00 

3.02 

3.78

The fact that the doubling of the discharge 

did not double the observed load indicates that 

the duty of water diminishes as discharge
FLUME TRACTION. 215

increases, but the data are too few to give 

confidence to the inference. The three capacities with a discharge of 0.363 ft.3/sec. indicate 

a minimum in the curve of C=f(F), thus supporting the generalizations already made from 

the results with rectangular troughs. .

The low efficiency here found for an open 

channel having a circular arc for its perimeter 

suggests that the cylindric form commonly 

given to closed conduits for the hydraulic conveyance of debris may not be the most efficient.

A second suggestion is connected with the 

fact that the semicylindric trough, while it narrows the field of traction, at the same time gives 

a high velocity to the water. It thus concentrates the available force and energy on the 

narrow field. Though the result is not favorable to capacity, it may be favorable to competence. When but a small load is to be trans-'. 

ported, the practical problem may be one of 

competent velocity; and such a trough appears 

well adapted to the production of competent 

velocity with economy of discharge and slope.;

SUMMARY.

In the transportation of debris in. flumes 

much of the movement is usually by rolling 

and sliding. This is especially true if the current is gentle or the debris coarse. With a 

very swift current or with fine debris the particles travel by a series of leaps, and with the 

finest d6bris the load is suspended. When the 

conditions are such that the principal movement is by rolling and sliding, the capacity of 

the current increases with the coarseness of the 

debris transported, this law holding good up 

to the limit of coarseness at which -the current 

is barely competent to start the particles. 

When the conditions are such that the principal movement is -by saltation, the capacity of 

the current increases with the fineness of the 

d6bris, the law holding good up to and probably beyond the critical fineness at which the 

current is competent to carry the debris in 

suspension. . Under all conditions the capacity is increased 

by steepening the slope, and the increase of capacity is more rapid than the increase of slope. 

The capacity may vary with a power of the 

slope as low as the 1.2 power, or with one higher 

than the second. A general average for the 

experimental determinations is the 1.5 power.

Under all conditions the capacity is increased 

by enlarging the discharge. It may be in-' 

creased in the same ratio, in a higher ratio, or 

in a somewhat lower ratio.

The highest capacity is associated with the 

smoothest, channel bed. Progressive increase 

of roughness reduces capacity progressively until the texture of the bed becomes coarser than 

the d6bris of the load. The mode of transportation then passes from flume traction to stream 

traction. Under like conditions of slope, discharge, and character of debris, flume traction 

gives higher capacities than stream traction.

Rectangular or box flumes have higher capacity than semicylindric flumes of similar 

width. Up to a limit, which varies with conditions, the capacity is enlarged by increasing 

the width of channel at the expense of depth of. 

current. The ratio of depth to width which 

gives highest efficiency has not been well covered by the experiments, but it is believed to 

be rarely greater than 1:10 and often as small 

as 1:30. For large operations the determination of width will usually represent a compromise between efficiency and the cost of construction and maintenance.

As most of the experiments -were made with 

sorted debris, each grade being narrowly 

limited as to range in the size .of-its particles, 

and as most practical work is with aggregations 

having great range in size, the loads and. 

capacities here reported need qualification. By 

experiments with mixtures of the laboratory 

grades it was found that the load carried of a 

mixture is greater than the load of any. pile -of 

its important components taken separately!'- 

It is in general true that the capacities for complex natural grades of debris are greater, than, 

the tabulated capacities- for the laboratory, 

grades they, most nearly resemble. .

COMPETENCE.

The experiments in flume traction were practically-limited in their range by phenomena of 

competence, and these limitations were'of use 

in determining values of a, K, and <£/ Jbu-t no 

effort was made to observe competence directly and precisely. There are, however, a 

few observations by others, which may properly 

be assembled here, although it is not practicable 

to use them as checks on our work. Our indefinite data pertain to slope and. discharge,.
216 TRANSPORTATION OF DEBRIS BY RUNNING WATER,

while the observations of others pertain to 

velocity.

The experiments of Dubuat - 1 (1783) have 

been assumed, both by him and by others, to 

pertain to stream traction, but his account of 

apparatus and methods makes it probable that 

what he really investigated was chiefly competence for flume traction. He used a trough 

of plank, with the grain lengthwise, and measured the velocity of the current by. observing 

the speed of balls slightly heavier than water 

as they were swept along the bottom. In a 

current of a particular velocity he placed 

successively various kinds of debris and noted 

their behavior, then changed the velocity and 

repeated. His results are as follows:

Competent bed velocity

(ft./sec.). 

Potter's clay. ..................... Between 0.27 and 0.*35

0.7 1.1

0.35

0.62

1.1

2.1 

3.2

0.53

0.7

1.55

3.2 

4.0

Coarse angular sand...............

River gravel:

Size of anise seed..............

Size of peas..................

Size of common beans.........

Rounded pebbles, 1 inch in diameter. ...........................

Angular flints, size of hen's eggs. .1

J. W. Bazalgette,. in discussing the flushing 

of sewers and therefore presumably cbnsidering flume traction rather than stream traction, 

quotes the following results of experiments by 

Robison: 2

Competent bed velocity 

(ft./sec.).

Fine sand....................................... 0. 5

Sand coarse as linseed........................... .67

Fine gravel...................................... 1. 0

Round pebbles, 1 inch in diameter................ 2. 0

Angular stones, size of eggs........................ 3. 0

In 1857 T. E. Blackwell 3 conducted elaborate experiments to determine the velocities 

necessary to move various materials in sewers. 

His channel was of rough-sawn plank, 60 feet 

long and 4 feet wide, with level bottom. Velocities were measured by a tachometer, but the 

relation of the velocity measurements to the 

bed is not stated. The tests were applied to 

natural debris of various kinds and also to types 

of artificial objects likely to enter sewers. The 

objects were treated singly and in aggregates, 

with the general result that an aggregation rei Dubuat-Nangay, L. G., Principes d'hydraulique, vol. 1, p. 100; vol. 

2, pp. 57, 79, 95, Paris, 1786. 

- 2 Inst. Civil Eng. Proc., vol. 24, pp..2S9-290, 1865.

s Accounts and papers [London], Sess. 2,1857; Metropolitan drainage, 

vol. 36, Appendix'IV, pp! 167-170, Pis. 1-5.

quires higher velocity to move it than does,a 

single object. It is evident that the experiments on single objects pertain to flume traction and some of those on aggregations, to 

stream traction. From his tabulated results 

the subjoined data are selected as representing 

or illustrating the velocities competent for 

natural debris, the column of mean diameter 

being added by me. He infers from the experiments that (1) for objects of the same 

character competent velocity increases with the 

mass; (2) for objects of the same size and form 

it increases with the specific gravity; (3) for 

objects of different form it is greater in proportion as they depart from the form of a sphere; 

and (4) for objects in motion the rate of travel 

increases with the velocity of the current.

TABLE 78. Observations by Blackwell on velocity competent 

for traction.

Material.

Single objects (illustrating flume traction) :

Do............................

Do...........................

Do...........................

Do.........;.................

Do...........................

Do...........................

Do...........................

Do...........................

Do...........................

Aggregations (illustrating stream traction):

Do...........................

Sand.............................

Volume.

Cubic 

inches. 

18.5

12.98

13.6

7.33

4.76

2.59

10.37

6.05

4.11

1.95

Mean diameter.

Feet. 

0.27

.24

.25

.20

.17

.14 

.22

.19

.16

.13

.20

.16

.13

.042

.021

Competent 

velocity.

Ft./sec.

2

7C 0 Art

2. 25-2. 50

2.00-2.25

2. 00-2. 25

2 OK 9 cn

1.75-2.00

3. 00-3. 25

2.00-2.25

2.25-2.50

2. 50-2. 75

1.50-1.75

2.25-2.50

2. 50-2. 75

1.25-1.50

0 -1.00

WORK OF OVERSTROM AND BLUE.

Certain experimental work on the capacity of 

currents for flume traction has for its specific 

purpose the determination of dimensions for 

launders, the flumes in which pulverized ore is 

conveyed. R. H. Richards's "Ore dressing" 4 

contains an abstract of results obtained by G. 

A. Overstrom, accompanied by the statement 

that the experimental data are extensive but. as 

yet unpublished. The troughs employed were 

flat-bottomed and probably of wood. For each 

slope, width of flume, and grade of transported 

material he found (1) that the duty of water 

varies with the discharge and that some particular discharge is associated with a maximum 

duty, so as to be the most economical; (2) that

 - < Vol. 3, pp. 1592-1594,1909.
FLUME TEAC.TION.- 217

the most economical discharge is sensibly proportional to the width, so that for each slope 

and grade of material there is a particular discharge per unit of width giving a maximum 

duty; and (3) that the most economical discharge is greater for low slopes than for high.

The first of these results is in fair accord with 

our own. Five of the nine values of the exponent o in Table 72 are less than unity. For 

the corresponding series the values of i3 range 

both below and above unity and the corresponding values of the variable exponent for 

duty in relation to discharge, iz 1, range below 

and above zero. The value zero evidently corresponds to a maximum value of duty. His 

third result is in strict accordance with ours; 

his second can not be compared without fuller 

details.

A diagram exhibiting his determinations for 

the traction of crushed quartz sized by 40-mesh 

and 150-mesh sieves shows for different discharges per unit width the variation of duty 

with slope. For the larger discharges duty 

varies as the first power of slope; for the smallest discharge with the second power. This corresponds to a variation of capacity with the 

second to third power of slope. Our most 

available data for comparison are those of 

grade (C), the capacity for which varies with 

the 1.66 power of slope (Table 70). As grade 

(C) was separated by 30-mesh and 40-mesh 

sieves, it is considerably coarser than the 

crushed quartz, and, being stream worn, it is 

less angular. The marked difference in tho 

observed laws of variation is evidently susceptible of more than one interpretation, but it is 

thought to be connected with difference in 

fineness, as more fully stated on a following 

page.

F. K. Blue 1 made a series of experiments in 

which the trough was of sheet iron, 50 feet long, 

5 inches deep, and 4 inches wide, the bottom 

being semicylindric with 2-inch radius. It was 

so mounted that it could be set to any slope 

up to 12 per cent. Two materials were used 

as load, the first a beach sand of 60-mesh average fineness, the other a sharp quartz sand of 

about 80-mesh fineness, containing about 10 

per cent of slime from a stamp mill. With each 

material the discharge and loado were varied; 

and for each combination of discharge and load

i Eng. and Min. Jour., vol. 84, pp. 530-539,1907.

the slope was adjusted to competence, and 

mean velocity was determined by means of a 

measurement of depth. Discharge and load 

were not measured directly but in certain combinations. Instead of discharge, the total volume of water and load was measured. This 

quantity was used chiefly in the computation 

of mean velocity, for which purpose it is better 

fitted than is discharge alone. Load was measured as a volume, the volume of the transported 

material as collected in a settling tank, and is 

reported only through a ratio, q, which is the 

quotient of the volume of load by the volume 

of discharge plus load. This is essentially a

duty but differs materially from duty ( U= ^ }

as defined in the present report. Representing 

by Wthe weight in grams of a cubic foot of 

debris, including voids, and by v the percentage 

of voids, it follows from the definitions that

=

From the discussion of his data Blue finds 

(1) that g varies as the square of the slope 

and (2) that it varies as the sixth power of the 

mean velocity. He does not specifically consider the relation of q to discharge, but examination of his tabulated data shows that q is 

but slightly sensitive to variations of discharge 

plus load.

As Blue's coarsest material, the beach sand, 

has approximately the fineness of our grade (A), 

while the finest we treated in flume traction is 

of grade (C), the most definite comparison of 

results can not be made, but there is nevertheless interest in such comparison as is possible. 

Computing values of U from his data for beach 

sand, and plotting them in relation to slope, I 

obtained UocS2 - 02 This gives for capacity and 

slope, C<xS3 -02 ; and the exponent 3.02 may be 

compared with values of 7X in Table 70, for the 

smoothest kind of trough bed. The exponent 

for grade (C) is 1.66, and the exponent has a 

minimum value of 1.30 for grade (G). In 

accordance with the generalization (p. 208) that 

the sensitiveness of, capacity to slope increases 

from a minimum toward both coarse grades and 

fine, we should expect for grade (A) an index 

of sensitiveness materially greater than 1.66. 

The data furnished by Blue thus tend to support the generalization, and additional support
218 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

is given by Overstrom's data, above cited. 

Such an inference is qualified as to Blue's data 

by the possibility that the exponent is affected 

in material degree by the form of the cross 

section of the trough. - . Blue's data on velocity are of such character 

as to warrant a fuller discussion than he gives. 

His comparison with duty (q) is made without 

regard to the accessory conditions of discharge, 

depth, and slope, but we have seen that in 

stream traction these conditions materially affect the control of capacity and duty by mean 

velocity. It is not practicable so to arrange 

his data as to obtain results for the condition 

that discharge, depth, or slope is constant, but 

moderate approximations to such conditions 

may be obtained by grouping. I have divided 

his observations on the traction of beach sand 

into three groups first, with reference to discharge ( + load); second, with reference to 

depth; and, third, with reference to slope 

and for each group have computed n on the 

assumption that capacity varies with the nth 

power of mean velocity. The resulting values 

are given in Table 79, and with them are placed 

the average of the corresponding exponents for 

stream traction, derived under the several conditions of constant discharge, constant depth, 

and constant slope (Table 53).

It appears (1) that in flume traction capacity 

is much more sensitive to variation of mean velocity than in stream traction; and (2) that in 

flume traction, as in stream traction, the order 

of sensitiveness as related to conditions is 

highest for constant discharge, intermediate for 

constant depth, and lowest for constant slope. 

In flume traction the sensitiveness appears to 

vary directly with discharge and depth and 

inversely with slope, while in stream traction 

it was found to vary inversely with discharge 

and slope, the variation with depth being indeterminate.

TABLE 79. Value of n in Coc Vmn , based on Blue's 

experiments on flume traction of beach sand.

The values of Q, d, or S being relatively 

\7 alue of n under condition that small 

range is given to 

Q

5.7 

6.2 

7.5

6.5

3.98

. d

5.7 

5.0 

7.0

5.9

3.68

S

3.6 

5.0 

3.3

4.0

3.21
CHAPTER XIII APPLICATION TO NATURAL STREAMS.

INTRODUCTION.

The flow of a river is a complex phenomenon. 

The transportation of debris by it involves 

intricate reactions. The quantity of d6bris 

transported depends on a variety of conditions, 

and these conditions interact one on another. 

Direct observation of what takes place at the 

base of the current is so difficult that the body 

of information thus obtained is small. In the. 

work of the Berkeley laboratory the attempt 

was made to study the influence of each condition separately, and to that end all the con-, 

ditions were subjected to control. This involved the substitution of the artificial for the 

natural; and while the principles discovered are 

such as must enter into the work of natural 

streams, their combinations there are different 

from the combinations of the laboratory. It is 

the province of the present chapter to consider 

the differences between the laboratory streams 

and natural streams, and in. view of those 

differences the applicability of the laboratory 

results to problems connected with natural 

streams.

FEATURES DISTINGUISHING NATURAL 

STREAMS.

KINDS OF STREAMS.

, Classification necessarily involves a purpose, 

or point of view, and there are in general as 

many scientific, or natural, or otherwise commendable classifications as there are functions 

to be subserved. The classification of streams 

here given has no other purpose than to afford 

a terminology convenient to the subject of 

debris transportation.

When the debris supplied to a stream is less 

than its capacity the stream erodes its bed, and 

if the condition is other than temporary the 

current reaches bedrock. The dragging of the 

load over the rock wears, or abrades, or corrades it. When the supply of de"bris equals or 

exceeds the capacity of the stream bedrock is 

not reached by the current, but the stream bed 

is constituted wholly of debris. Some streams 

with beds of debris have channel walls of rock,

which rigidly limit their width and otherwise 

restrain their development.. Most streams with 

beds of debris have one or both banks of previously deposited debris or alluvium, and these 

streams are able to shift their courses by 

eroding their banks. The several conditions 

thus outlined will be indicated by speaking of 

streams as corroding, or rock-walled, or alluvial. 

In strictness, these terms apply to local phases 

of stream habit rather than to entire streams. 

Most rivers and many creeks are corrading 

streams in parts of their courses and alluvial 

in other parts.

Whenever and wherever a stream's capacity 

is overtaxed by the supply of debris brought 

from points above a deposit is made, building 

up the bed. If the supply is less than the 

capacity, and if the bed is of debris, erosion 

results. Through these processes streams 

adjust their profiles to their supplies of debris. 

The process of adjustment'is called gradation; 

a stream which builds up its bed is said to 

aggrade and one which reduces it is said to. 

degrade.

An alluvial stream is usually an aggrading 

stream also; and when that is the case it is bordered by an alluvial plain, called a flood plain, 

over which the water spreads in time of flood.

If the general slope descended by an alluvial 

stream is relatively steep, its course is relatively 

direct and the bends to right and left are of 

small angular amount. If the general slope is 

relatively gentle, the stream winds in an intricate manner; part 6f its course may be in directions opposite to the general course, and some 

of its curves may swing through 180° or more. 

This distinction is embodied in the terms direct 

alluvial stream and meandering stream. The 

particular magnitude of general slope by which 

the two classes are separated is greater for sinall 

streams than for large. Because fineness is one 

of the conditions determining the general slope 

of an alluvial plain, and because the gentler 

slopes go with the finer alluvium, it is true in 

the mam that meandering streams are associated with fine alluvium.

219
220 TRANSPORTATION OF DEBRIS BY RUNNING WATER,

FEATURES CONNECTED WITH CURVATURE OF 

CHANNEL.

As nearly all the laboratory experiments were 

performed with straight channels, and as all 

natural channels are more or less curved, the 

features resulting from curvature constitute 

differences of which account must be taken in 

applying laboratory results. Some of these 

differences have been mentioned in connection 

with the short series of experiments with 

curved and bent channels, but a fuller account 

is desirable.

In a straight channel the current is swifter 

near the middle than near the sides and is 

swifter above mid-depth than below. On 

arriving at a bend the whole stream resists 

change of course, but the resistance is more 

effective for the swifter parts of the stream than 

for the slower. The upper central part is deflected least and projects itself against the outer 

bank. In so doing it displaces the slow-flowing 

water previously near that bank, and that water 

descends obliquely. The descending water displaces in turn the slow-flowing lower water, 

which is crowded toward the inner bank, while 

the water previously near that bank moves 

toward the middle as an upper layer. One 

general result is a twisting movement, the upper parts of the current tending toward the 

outer bank and the lower toward the inner. 1 

Another result is that the swiftest current is no 

longer medial, but is near the outer or concave 

bank. Connected with these two is a gradation 

of velocities across the bottom, the greater velocities being near the outer bank. The bed 

velocities near the outer bank are not only 

much greater than those near the inner bank, 

but they are greater than any bed velocities in 

a relatively straight part of the stream. They 

have therefore greater capacity for traction, 

and by increasing the fractional load they erode 

until an equilibrium is attained. On the other 

hand, the currents which, crossing the bed obliquely, approach the inner bank are slackening 

currents, and they deposit what they can no 

longer carry.

It results that the cross section on a. curve 

is asymmetric, the greatest depth being near

i The system of movements here described has been observed by 

many students of rivers. They were demonstrated by the aid of a model 

channel by J. Thomson, in connection with an explanation which differs somewhat from that of the present text. See Roy. Soc. London 

Proc., pp. 5-8,1876, and 356-357,1877;. also Inst. Mech. Eng. Proc., pp. 

456-460,1879.

the outer bank. As the winding stream 

changes the direction of its curvature from one 

side to the other, the twisting system of current 

filaments is reversed, and with it the system of 

depths, but the process of change includes a 

phase with more equable distribution of velocities, and this phase produces a shoal separating 

the two deeps. The shoal does not cross the 

channel in a direction at right angles to its 

sides but is somewhat oblique in position, 

tending to run from the inner bank of one 

curve to the inner bank of the other. In 

meandering streams it is usually narrow and 

is appropriately called a bar. In direct alluvial 

streams, where bends are apt to be separated 

by long, nearly straight reaches, it is usually 

broad and may for a distance occupy the 

entire width of the channel. In navigated 

rivers the locality of the bar is usually called a 

crossing, being the place where the thalweg, 

the line of strongest current, and the route of 

travel cross from side to side; and the name is 

often applied also to the bar itself.

The twisting current attacks the outer bank, 

being swifter at contact with that bank than 

in any other part of the wetted perimeter. If 

the bank consists of alluvium there is erosion, 

the amount being determined in part by 

resistances arising from roots, or adhesion 'of 

alluvial particles, or incipient cementation; 

and the eroded material, so far as it joins the 

bed load, helps to satisfy the bed current and 

limit downward erosion. In alluvial streams 

the erosion from concave banks offsets the 

deposition under convex banks, so that the 

channel may gradually shift its position without 

change of sectional area.

The sectional area may be either greater or 

less at a curve than on a reach, but the differences are normally l of small amount. Therefore the mean velocity does not vary greatly. 

The current in a curved channel, as compared 

to that in a straight channel, is characterized 

by diversity. Its bed velocities are .both 

higher and lower, and the same is true of 

velocities along the banks. This diversity is 

favorable to traction, because capacity for 

traction varies with a high power of velocity; 

but the advantage to traction is partly offset 

by the fact that increase of velocity affects a 

smaller portion of the wetted perimeter than

'That is to say, they are of small amount.when the system'of depths 

is adjusted to the discharge, as explained on a later page.
APPLICATION TO NATURAL STREAMS. 221

is affected by reduction of velocity. There is 

also diversity in the directions followed by 

elements of current, and this diversity includes 

not only the twisting movement but various, 

minor eddies and swirls. Diversified movements, by including upward movements, promote suspension, and in conjunction with 

diversified velocities they modify the partition 

of the load between traction and suspension. 

On the whole, suspension claims more in a 

diversified current, but it is also true that the 

line of separation between suspension and 

traction shifts to and fro in such a current. 

Much debris which is suspended in the swift 

water under the concave bank joins the bed 

load in passing the shoal between deeps, and 

the suspended load is still more restricted in 

passing the shoal of the convex bank. Deposition on the latter shoal includes both tractional 

and suspensional materials.

FEATURES CONNECTED WITH DIVEESITY OF 

DISCHARGE.

All streams vary in volume from season to 

season and from year to year. In a stream fed 

by springs the changes may be slight. At the 

opposite extreme are creeks and even rivers 

which exist only during storms. In most large 

streams the discharge at flood stage is many 

times greater than at low stage. Usually flood 

stages continue only for brief periods and in the 

aggregate occupy but a small fraction of the 

year.

It is broadly true that streams give shape to 

their own channels, and among alluvial streams 

there are few exceptions. It is broadly true 

also that the shapes of channels, including cross 

sections and plans, are the same .for large 

streams as for small. But the large stream 

requires and develops a larger channel broader, 

deeper, and winding in larger curves. Through 

variation of -discharge the same stream is alternately large and small, so that its needs are 

different at different stages. At each stage it 

tends to fit its channel to the needs of the particular discharge. The formative forces residing in the current are so much stronger with 

large discharge than with small that the greater 

features of channel are adjusted to large discharge, and this despite the fact that floods 

are of brief duration. The feebler forces of 

smaller discharges modify the flood-made forms 

but do not succeed in completing their work of

adjustment before it is interrupted by another 

flood. The deeps of high stage are pools at 

low stage and have currents too feeble for traction. As the reduced stream passes from pool 

to. pool it crosses the shoal formed at high 

stage with quickened current. The velocities 

are still diversified, but the greater and smaller 

velocities have exchanged places. The slope of 

water surface is more diversified than at high 

stage, being lower at the pools and higher between them. Traction is restricted to the 

shoals, and the loads are small. The load at 

each shoal is obtained from the shoal itself and 

is deposited in the next pool, and in this way 

shallow channels are developed from pool to 

pool.

In contrasting the features of high and low 

stages, it has been convenient to use the terms 

as if high stage and low stage were specific and 

definite phases of stream activity, thereby 

ignoring the actual diversity in fluctuations of 

discharge. Floods are of all magnitudes, and 

each flood presents not only a maximum discharge but a continuous series of changing discharges. At each instant the stream contains 

a system of currents of which the details 

depend not only on the discharge but on the 

shapes of channel created by the work of previous discharges. So long as the discharge 

continues, its currents are eroding and depositing in such way as to remodel the channel for 

its .own needs, and so long as the work of 

remodeling continues the loads and capacities 

at different cross sections are different.

With the changes in the values and distribution of velocities go changes in those values 

of competent fineness which on one side limit 

traction and on the other separate traction 

from suspension. With maximum discharge 

all the coarser grades of de'bris within the 

domain of the stream are in transit along the 

path of highest activity, and that path includes the deeps and the intervening shoals. 

With lessening discharge the coarsest material 

stops, but it stops chiefly in the deeps, because 

the change in bed velocity is there greatest. 

At the same time the coarsest of the suspended 

load escapes from the body of the stream and 

joins the bed load. By this double change the 

mean fineness of the tractional load is increased, 

and so also is the mean fineness of the suspended 

load. With continued reduction of discharge 

the tractional load in the deeps becomes gradu-
222 TRANSPORTATION OF DEBEIS BY RUNNING WATER.

ally finer and at last ceases to move, while the 

graduated deposit caused by its arrest receives 

a final contribution from the suspended load. 

The tractional load on the shoals changes less 

in mean fineness and may cease to change 

altogether when the supply from the deep is 

cut off. It is then derived wholly from the 

subjacent bed and is greatly reduced in quantity. Soon the derivation becomes selective, 

the finer part being carried on while the coarser 

remains, with the result that the shallow channels on the bars come to be paved with particles which the enfeebled currents can not 

move.

If the section of the alluvium underlying a 

shoal be afterwards exposed, it is seen to be in 

the main heterogeneous but veneered at the 

top by a layer of its coarser particles. The 

typical section of a deposit in a deep shows the 

coarsest debris below and the finest at top, 

with a gradual change.

With the return of large discharge the modeling work of smaller discharges is rapidly obliterated, and the debris deposited in the pools 

rejoins the tractional and suspended loads.

SECTIONS OF CHANNEL.

Kock-walled channels result from the aggradation of corraded channels. Often they are 

recurrent temporary conditions of corraded 

channels. Their widths have been developed 

in connection with the work of corrasion and 

are less than the widths of alluvial streams. In 

the fact that their sides are immobile they 

resemble the laboratory channels, and their 

types of cross section are illustrated by the 

experiments with crooked channels. The 

channels of all alluvial streams are strongly 

asymmetric at the bends, and in the meandering streams the bends constitute the greater 

part of the course. Departure from symmetry 

is less pronounced in the reaches of direct 

alluvial streams, but even there a close approximation to symmetry is exceptional.

Alluvial streams tend to broaden their 

channels by eroding one or both banks. The 

influence of vegetation opposes this tendency. 

Often, the erosion of the bank exposes roots, 

and some trees extend rootlets into the water. 

At low stages the bared parts of the flood 

channel are occupied by young plants. In 

these ways vegetation creates obstacles which 

retard the current at its contact with the bank

and thus oppose erosion. If the current is 

strong erosion is merely retarded, not prevented; if the current is weak deposition may 

be induced. As a meandering stream encroaches on its concave bank, the convex bank 

encroaches on the stream, and channel width 

is maintained. A large stream is less affected 

than - a small stream by the opposition of 

vegetation and maintains a channel of relatively small form ratio.

Some streams aggrade so rapidly that vegetation does not secure a foothold. By erosion 

of its banks such a stream broadens its channel 

and reduces its. depth until the 'slackened 

current clogs itself by deposition of its load. 

The built-up bed becomes higher than the 

adjacent alluvial plain, and the stream takes a 

new course. Before the assumption of the 

new course the banks are overtopped by 

shallow distributaries which deposit their loads 

on the banks, thus building them up, until the 

stream is made to flow on a sort of elevated 

conduit; and when the main body of water at 

last leaves this pathway, it is apt to start its 

new course with a steepened slope and scour 

for itself a relatively narrow channel.

The building up of the bank by deposition 

from overflow is more pronounced in the presence of vegetation. The ridge thus created is 

called a natural levee. Its crest separates the 

channel from the flood plain and delimits at 

flood stage two provinces in which the conditions affecting transportation are strongly contrasted. In both provinces the general slope 

of the water surface is the same, but the broad 

sheet covering the plain has so little depth that 

its currents are sluggish. Between the banks 

are the normal channel depths and currents, 

and transportation is active, alike by traction 

and suspension. Beyond them transportation 

is effected almost wholly by suspension, and 

the coarser particles of the suspended load are 

deposited. As the flood subsides the lateral 

sheets of water are returned to the main channel by a draining process which involves the 

making and maintenance of small channels 

within the plain.

When the channel of a river is fully adjusted 

to the discharge the same load is transported 

through each section. All sections are then 

equally adapted to transportation, though in 

different ways. The most symmetric lias a 

wide space at the bottom devoted to traction.
APPLICATION TO NATURAL STREAMS. 223

The least symmetric has a relatively narrow 

tractional space, but traction is there relatively 

active. The partition of the load between 

traction and suspension is not the same for the 

two sections, the tractional load having the 

greater range in the symmetric section and the 

suspended load in the asymmetric.

There is reason to believe that the sectional 

area is about the same in different parts of an 

adjusted channel. At low stages, when form 

is least adjusted to discharge, the sectional area 

is much larger for the asymmetric sections. 

At higher stages the contrast is less, and the 

greater area may be associated with either type 

of section. It is also true, if attention be restricted to the .channel proper and the expansions over flood plains be excluded, that the 

variations in width from point to point of an 

adjusted channel are not of large amount. If 

it were strictly true that both sectional areas 

and widths are equal in different parts of an 

adjusted channel, it would follow (1) that mean 

depths are equal, and (2) that form ratios are 

equal, provided form ratio be defined as the 

ratio of mean depth to width. Such a generalization, while crude and doubtless subject to 

important qualification, nevertheless warrants 

the selection of mean depth rather than maximum depth as the quantity to be used in applying the conception of form ratio to rivers.

Assuming the generalization as an approximation to the actual fact and connecting with 

it the fact that all sections of an adjusted river 

are equally efficient for transportation, we are 

able to make a general application of the 

laboratory results on optimum form ratio to 

rivers. The ratio of mean depth to width in 

alluvial rivers, as a class, is very much smaller 

than in the laboratory examples by means of 

which the optimum ratio was discussed in 

Chapter IV. It is so much smaller that tho 

range of form ratio for alluvial rivers overlaps 

but slightly the range observed in the laboratory. This disparity indicates, though without 

demonstrating, that the form ratios of the 

rivers are less than the optimum, and that their 

tractional capacities would be greater if they 

were narrower and deeper. As the optimum 

ratio is the one which enables a stream to 

transport its load with the least expenditure of 

head, it is probable that the slopes of most 

alluvial rivers can be lessened by artificially 

reducing their widths.

THE SUSPENDED JLOAD.

In speaking above of the transfer of load from 

traction to suspension no consideration was 

given to capacity for suspension. Certain students of rivers, comparing discharges or velocities with the percentage of suspended material 

and finding a rough correspondence, have inferred that suspended load is a function of 

velocity; others, giving attention to conspicuous examples of noncorrespondence, have inferred that the suspended load depends only 

on the supply of suitable material. There is a 

 measure of truth in both views, and their divergence is largely to be explained by considerations connected with competence.

The subject is illustrated by observations on 

,the suspended load of Yuba River. Its water 

was sampled during flood stages, in 1879 1 at 

Marysville, a load of 0.35 per cent, by weight, 

being found when the discharge was estimated 

at 26,000 ft. 3 /sec., and a load of 0.42 per cent 

when the discharge was 18,000 ft.3/sec. In 

1906, at a time when the discharge was estimated at 33,000 ft.3/sec., samples, were taken 

at the same place and the load was found to 

be only 0.065 per cent. At the time of the 

earlier samphngs hydraulic mining was in full 

operation in the basin of the Yuba, and the 

suspended load consisted chiefly of clay components of the auriferous gravels, artificially fed 

.to the stream. In 1906 there was little hydraulic mining and the suspended load consisted of material washed from the surface of 

the land by rain.

At low stages in 1906 the water at Marysville 

was clear, but in 1879 a sampling when the 

discharge was 510 ft.3/sec. gave a load of 0.8-6 

per cent. The fact that the river's load in 1879 

constituted a higher percentage at low stage 

than during flood is explained by the consideration that the turbid tributaries from the mines 

were less diluted by other tributaries at low 

stage than at high.

If we assume, first, that 0.065 per cent, observed at flood stage in 1906, represents the 

normal tribute from other sources than mining, 

and, second, that the mines contributed the 

same total amounts at low and high stages in 

1879 (discharges being 510 and 18,000 ft.3/sec.), 

and if we base on these assumptions a compu1 Manson, Marsden, Report of determinations of sediment held in suspension, etc.: California State Engineer Keport, 1880, Appendix B.
224 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

tation of the low stage percentage of load in 

1879, we obtain an estimate of 12.6 per cent, 

which is 14 times greater than the observed 

load, 0.86 per cent. The discrepancy is altogether too great to be accounted for by errors 

in the explicit assumptions, and its explanation involves the factor of competence. The 

stronger currents of flood stages were competent to suspend .heavier particles than were the 

feeble currents of low stages, and so a part of 

the load which at high stages reached the mouth 

of the river at Marysville was at low stages 

arrested on the way, being deposited in the 

low-water pools.

In these cases it is evident that the range of 

available fineness is determined, through velocity, by discharge, and that the load of debris 

within the range of adequate fineness is determined by supply. The load appears to bear 

no relation to capacity, and if the term capacity 

be used in the broad sense of a stream's ability 

to suspend material of unspecified fineness, then 

it is undoubtedly true, not merely of the Yuba 

but of all rivers, that the suspended load is less 

than the capacity and depends for its quantity 

on supply. If, however, capacity be considered 

with reference to particular degrees of fineness, 

the case is somewhat different, for a stream 

may carry a full load of that material for which 

it is barely competent and at the same time 

have less than a full load of finer material, and 

the matter is further complicated by interdependencies in virtue of which each element of 

load tends to limit the capacity for all other 

elements of load.

To show the basis for these statements and 

also to explain certain mutual relations between 

traction and suspension, it is necessary to give 

somewhat elementary consideration to the subject of capacity for suspension.

As already mentioned and implied in various 

connections, the process of suspension depends 

on the diversity in direction of the strands of 

the current. If the lines of flow were parallel 

to the stream bed, as is sometimes assumed for 

the sake of simplifying mathematical discussions, there would be no suspension.1 In the

1 There is a theory originating with Dupuit (Etudes sur le mouvement des eaux, 1848) that suspension is due, or might be due in the 

ideal case of parallel flow lines, to reactions between solid particles and 

contiguous threads of current having different velocities. Under the 

postulate that the solid tends to move faster than the liquid, it is shown 

that the path of least resistance treads obliquely toward the swifter of 

adjacent threads of current, and therefore obliquely upward. As this 

theory retains place in current hydraulic literature, the fact that it is

sinuous and swirling movements which characterize the flow of streams strands of current are 

continually passing upward and downward and 

are as continually dividing and blending. Particles of debris too light to resist the lower elements of the current are swept upward and are 

retained in the body of the stream through a 

process analogous to the stirring of the domestic pot. While thus incorporated they are 

impelled downward by gravity, and all but the

ignored in the text of the present paper may call for explanation. I do 

not accept the postulate and am of opinion also that the reasoning based 

on it ignores an essential factor. As a full statement and discussion of 

Dupuit's analysis would occupy much space, I will content myself 

with a statement of my own view. A good abstract of his theory, by 

E. H. Hooker, may be found in Am. Soc. Civil Eng. Trans., vol. 36, pp. 

246-247, 320-322.

In various discussions of the subject the velocities are treated as 

"absolute" that is, they are referred to the fixed walls and bed of the 

channel. As the only possible reactions between the solid particles and 

contiguous water are through relative velocities, it is better to focus 

attention on those by referring them to the center of the particle. Let 

us assume that the solid particle A, figure 70, is immersed in a current

FIGURE 70. Diagram of forces.

of which the parallel rectilinear filaments increase gradually in "absolute" velocity from below upward, and let us assume that at some 

instant it moves with the velocity and direction of the filament which 

is at the same level. Barring extraneous forces, it will continue indefinitely in the same direction and with the same velocity. The filament 

above moves, with reference to the particle, in a direction indicated by 

the arrow; the filament below moves in the opposite direction. Their 

relative velocities are the same, except for a possible difference of the 

second order of magnitude. The two filaments tend to draw the upper 

and lower parts of the particle in opposite directions, and the result is 

rotation. This is the only result dependent on the fact that the particle 

is solid. Now introduce the factor of density. The particle is denser 

than water. It is also part of a stream which is flowing, and the impulse 

it receives from gravity is greater than it would receive if it had the 

density of water. The component of gravity in the direction of flow, 

AB, acting on the excess of mass, draws the particle in the direction of 

flow. This component is proportional to the slope of the stream, which 

is a small fraction. At the same time the component of gravity normal 

to the direction of flow, A C, also draws the particle, which is equally 

free to move through the water in that direction. Its actual acceleration has the direction of their resultant, AO, which is vertically 

downward.

Dupuit's postulate was suggested and supported by the observed 

fact that a body floating down a stream moves faster than the visible 

current. Be"rard. demonstrated experimentally that the differential 

motion is due to the propulsion of the body by strands of current below 

the surface. See Annales des ponts et chausse*es, 6th ser., vol. 12, pp. 

830-S35,1886.
APPLICATION TO NATURAL STREAMS. 225

very finest actually move downward with reference to the surrounding water. From time to 

time they may touch the stream bed, but only 

to be lifted again by the next adequate rush.of 

water.

Three ways are known in which the velocities of a stream are affected by the suspended 

load. In the first place, the load adds its mass 

to the mass of the stream, and as the stream's 

energy is proportional to the product of mass 

by slope, and the stream's velocity has its 

source in this energy, the addition tends to 

increase velocity. Second, the suspended particles are continually impelled downward by 

gravity. Also, as the strands of current containing them have curved courses, the particles 

are subject to tangential force, and because of 

their higher density this force is greater than the 

force simultaneously developed by the containing water, so that they are impelled through the 

water. Motion through the water, caused by 

these two forces, involves work; and this work 

is a direct consequence of suspension. The 

energy expended is potential energy, or energy 

of position, given to the particles by the flowing water, and its source is the energy of flow. 

So the work is the measure of the work of the 

stream in suspending the particles. It may 

therefore be regarded as a tax on the stream's 

energy, resulting in reduction of velocity. 

Third, the imperfect liquid constituted by the 

combination of water and dSbris is more viscous 

and therefore flows more slowly than the water 

alone. The solid particles do not partake of 

the internal shearing involved in the differential movements of the current, and by their 

rigidity they restrain the shearing of water in 

their immediate vicinity. Moreover, each particle is surrounded, through molecular forces, 

by a sphere or shell of influence which still 

further interferes with the freedom of water 

movement.

The relative importance of these factors 

varies with conditions, and no simple statement 

is possible because the influence of each factor 

follows a law peculiar to itself. The most 

important conditions affecting the influence of 

the mass of the load are discharge and slope, 

while for the work of suspension and for 

viscosity the important condition is the degree 

of comminution of the load.

The mass, which the load contributes to the 

stream is equivalent, in relation to potential

20921° No. 86 14 15

energy, to an increase of discharge, and its 

product by the stream's slope is proportional 

to potential energy in the same sense in which 

the stream's energy is proportional to the product of discharge and slope. In a series of 

experiments with loadless streams of water 

flowing in straight troughs, the mean velocity 

was found to vary approximately with the 0.25 

power of the discharge and the 0.3 power of 

the slope. Under the particular conditions of 

these experiments

Vm = Q°- 25 S°-s X constant. 

Differentiating with reference to Q, we have

f) oc .O0.3

d V = x constant.

The increment to Q being interpreted for present purposes as the suspended load, we see that 

the corresponding increment to mean velocity 

has a magnitude which varies directly but 

slowly with the slope and inversely but more 

rapidly with the discharge.

Each particle in suspension is drawn downward through the surrounding water by 

gravity. It is impelled through the water in 

an ever-changing direction by tangential force. 

The average speed of the resultant motions, 

referred to the surrounding water, is greater 

than the constant rate of descent the particle 

would acquire if sinking in still water. Therefore the work of suspension, measured by ah1 

the motions through the water, is greater than 

the work of simple subsidence, a quantity as to 

which much is known. The measure of a 

particle's work of subsidence per unit time is 

the product of its mass, less the mass of an 

equal volume of water, by its fall in unit time 

by the acceleration of gravity. If we call the 

mass of the particle M and its velocity of 

subsidence Va , and assume its density to be 

2.7, the measure of the work of subsidence is

The coordinate measure of work for the 

stream's flow is the product of its mass by its 

fall in unit time by the acceleration of gravity, 

and the contribution which the particle, considered as a part of the stream, makes to the 

work of flow is therefore measured by the 

product of its mass by the fall of the stream in
226 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

unit time by the acceleration of gravity. As 

the fall of the stream in unit time is equal to 

the mean velocity multiplied by the slope, the 

measure of the particle's work of flow is Ms Vmg. 

This is a measure of the particle's contribution 

to the stream's energy, while 0.63 MVsg is a 

coordinate measure of that factor of its draft 

on the stream's energy which depends on the 

direct action of gravity. The other factor of 

.draft, the factor depending on tangential 

forces, varies with the violence of vertical 

movements and is of comparable importance 

only in the case of torrents. If we leave it out 

of account, the result of the comparison is that 

when 0.63 of the rate at which the particle is 

pulled through the water by gravity is greater 

than the rate at which it falls by reason of the 

general descent of the stream, its tax on the 

'stream's energy is greater than its contribution 

thereto. Any allowance for the neglected 

factor would be equivalent to increasing the 

fraction 0.63.

To illustrate this relation by a concrete 

example: Mississippi River between Cairo and 

its mouth has, at flood stage, such velocity and 

slope that any suspended particle of silt which 

would sink in still water faster than half an 

inch a minute retards the current more through 

the work of suspension than it accelerates the 

' current through the addition of its mass to the 

mass of the stream.

The velocity of subsidence has been elaborately studied. So far as river problems are 

concerned, it depends chiefly on the size of the 

suspended particles. For particles below a 

certain magnitude, which is controlled in part 

by impurities in solution, the velocity is zero. 

Between two critical diameters, which for 

quartz sand are about 0.02 and 0.5 millimeter 1 

(0.00007 and 0.00016 foot) the velocity varies 

with the square of the diameter. Below the 

lower diameter the variation is more rapid, 

and above the upper it is less rapid, becoming 

for large particles as the square root of the 

diameter.

It follows that the consumption of energy 

involved in the suspension of the suspended 

load is an increasing function of the size of the 

particles into which the load is divided; in 

other words, it is a decreasing function of the 

degree of comminution. On the other hand,

i Richards, R. H., Textbook on ore dressing, pp. 262-2C8,1909.

the contribution which it makes to energy by 

adding its mass to that of the water is independent of the degree of comminution.

The viscosity factor is not easily compared 

in a quantitative way with those just considered, but something may be said of the laws 

by which it is related to comminution. Attending first to that part which depends on 

interference with internal shearing of the water, 

let us conceive of a particle with center at C, 

figure 71, surrounded by water which is subjected to uniformly distributed shearing along 

planes parallel to A^AA^BJBB^, the direction 

of shearing being parallel to the line ACE. 

Conceive a right cylindroid figure tangent to 

the particle and parallel to AB, its bases being 

AJlA.fr and B1 EB2 F. Motions being referred to C as origin, the cylindroid body of 

water would assume after a time, but for the 

presence of the particle, the form of the oblique 

cylindroid with bases AlH1A2Gl and B^B^F^

FIGUEE 71. Interference by suspended particle with freedom of shearing.

Because of the obstruction by the rigid particle the simple shearing motions thus indicated are replaced by other motions, components of which are normal to the shearing 

planes. It is assumed that the sum of the 

transverse elements of motion measures the 

action occasioned by the presence of the particle. The actual movements caused in the 

water doubtless affect regions within and without the cylindroid, but their nature need not 

be considered. The necessary transverse movements are equivalent to the transfer of a lunate 

wedge of water, HA^Il^A^ to a symmetric 

position on the opposite side of the plane 

A^AyBJBi and a similar transfer of the wedge 

FB1 F1B2 . Linear dimensions of the firstnamed wedge, in the direction of rectangular 

coordinates, are A:A2, HA, and I1HV AtA2 

equals a diameter of the particle. HA equals 

a semidiameter. As the angle HAH1} being 

given by the general amount of shearing, is 

independent of the size of the particle,
APPLICATION TO NATURAL STREAMS. 227

equals AHxt&n. HAH1 and is proportional to 

a diameter of the particle. On the assumption 

that the diameters of the particle are equal it 

follows that the volume of the wedge is proportional to that of the particle, or to D3 . The 

distance of the center of gravity of the wedge 

from the plane A^AJBJB^ being a linear dimension of the wedge, is proportional to D, and 

the mean distance of transfer, which is double 

that distance, is also proportional to D. The 

quantity of motion normal to the shearing 

planes, occasioned by the particle, is measured 

by the wedge of water times the mean distance 

of transfer, and, the mass of the wedge being 

proportional to its volume, the quantity of 

motion is proportional to D3 xD = D*. The 

quantity of motion occasioned by the entire 

suspended load, all its particles being assumed 

to have the same size, is proportional to ND4 , 

where N is the number of particles; and if the 

mass of the total load remain unchanged while 

its degree of comminution is varied, it is evident 

that N varies inversely with D3 . Therefore the 

quantity of internal motion occasioned by a 

particular mass of suspended matter of uniform 

grain is proportional (since D*/D3 = D) to the

diameter of its particles. The fundamental 

assumption of the analysis is that this motion 

measures a resistance to the freedom of the 

water which is coordinate with viscosity and 

which may for practical purposes be considered 

as an addition to the resistance arising from 

the viscosity of the water.

That portion of the viscosity factor which 

depends on the molecular influence of the 

particle outside its boundary is still less susceptible of quantitative estimate but may yet 

be discussed with reference to the diameter of 

the suspended particle. In ignorance of the 

exact nature of the influence and also of the 

law by which it diminishes with distance outward from the boundary, I assume arbitrarily 

that at all distances from the particle less than 

I the freedom of water molecules is restricted, 

and that the amount of restriction is measured 

by the volume of the space in which the restriction is experienced. The imperfection of this 

assumption will of course affect any deduction

from it. That volume is n (D + 2Z) 3 - x D3 .

D D

is N times as great. For a load of invariable 

weight but variable comminution, N<x.j=^, and 

the total volume is proportional to

-^((V + UY- 

~~TJA

or, introducing a constant, /if, and reducing, we 

have

vi Volume .P.//P + 2ZY \ 

= A I ( ^ j 11

Evidently the influence of this factor varies 

inversely with D. When D is very large as 

compared to 2Z, it approaches zero ; when D is 

small as compared to 21, it is relatively very 

great. It is most sensitive to the control by D 

when the particles are very small.

The two divisions of the viscosity factor vaiy 

in their influence on velocity with the comminution of the load but in opposite ways, the 

influence of the first being greater as the 

particles are larger and that of the second as 

they are smaller. The laws of variation are 

such that their combination exhibits a minimum -that is, for some particular size of 

particle the influence on velocity is less than 

for particles either -larger or smaller.

Another mode of treating the viscosity 

factor assumes that, so far as the viscosity 

effect is concerned, the molecular influence is 

equivalent to an enlargement of each particle 

to the extent of I on all sides. Then, reasoning 

as before with reference to interference with 

shearing, we obtain

Resistance cc

For the entire suspended load, assumed to 

consist of N equal particles, the total volume

This expression is not only simple but has the 

advantage of giving definite indication of the 

position of the minimum. The resistance is 

least when D = 6l.

We may now bring together the qualitative 

results of analysis, and write

- vm +

This may be read: The mean velocity (Vms) of 

a stream carrying a load of suspended debris 

of diameter D equals the mean velocity (Vm) 

of the same stream when without load, plus a
-2-28 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

factor due to the weight (W) of the load and 

varying directly with the slope and inversely 

with the discharge, minus a factor due to the 

work of suspension and varying directly with 

the diameter of particles, minus a factor due 

to viscosity and varying inversely with the 

diameter if the particles are minute and 

directly with the diameter if they are larger.

It was convenient in the discussion to regard 

the whole suspended load as of one grade, but 

the result may be applied to any individual 

particle, For the actual stream with diversified load the equation might be written

= Vm + . .(Ill)

An investigation of the influence of suspended 

matter on viscosity has recently been made by 

Eugene C. Brigham and T. C. Durham. 1 The 

materials used by them were powders so fine 

that their tendency to settle in the water did 

not interfere with the conduct of the experiments. The work of suspension was therefore 

so small a factor as to be negligible. Various 

.mixtures of water and powder were allowed to 

flow through a vertical capillary tube, impelled 

by their own weight, and the time of transmission for a determinate volume was noted. 

The time for a mixture containing 100 parts 

of water to 2.15 parts, by weight, of clay was 

found to be 15 per cent greater than the time 

for clear water; or the average velocity, within 

a slender tube, was reduced 15 per cent by the 

addition of the clay. If we assume the flow 

lines to have been parallel, as was probable, 

the theoretic increment of velocity due to the 

weight added by the clay was 1.6 per cent; if 

the flow lines were sinuous, the theoretic increment was less. The loss of velocity due to 

increased viscosity was therefore somewhat 

greater than 15 per cent and may have been 

as much as 16.6 per cent. Had the experiment 

been so arranged as to involve sinuous or turbulent flow, it would apply more cogently to 

the phenomena of rivers, but without that 

adjustment it serves to show that when the 

suspended material is exceedingly fine the loss 

of velocity through added viscosity is very 

much greater than the gain of velocity because 

of added weight.

The size of the particles was not reported, 

but the fact that the rate of settling was ini Am. Chem. Jour., vol. 40, pp. 278-297, 1911.

appreciable suggests that their size may have 

fallen below that corresponding to a minimum 

influence on viscosity. Some light is throwti 

on this point by the results obtained with other 

fine powders. Interpolation from the reported 

data gives the following comparative estimates 

of the loss in velocity from the suspension of 

the same volumes of three substances: Infu^- 

sorial earth, 3 per cent; graphite, 12 per cent; 

clay, 15 per cent. Of these the infusorial earth 

was coarsest, remaining in suspension largely 

because differing little in density from the 

water, while the graphite was a commercial 

variety said to be permanently suspended. An 

independent determination by L. J. Briggs and 

Arthur Campbell 2 gave a loss in velocity of 

7 per cent, the material being a clay which 

"would not remain in a state of permanent 

suspension." The comparative data render it 

probable that the clay causing a retardation 

of 15 per cent was so finely divided as to give 

great effect to the molecular forces of the shell 

of influence.

The data from infusorial earth are useful in 

correlating the various factors, which modify 

the stream's velocity. Assuming that the diatom tests composing the earth sample were 

similar to those figured in geologic textbooks, 

I have estimated the velocity of subsidence of 

particles having the same size but as dense as 

ordinary river silt and find that it corresponds 

to the critical velocity computed for the Mississippi at flood stage. That is, a suspended 

silt so fine as to have a large viscosity effect, so 

that a charge of 2.15 per cent reduces velocity 

by 3 per cent, is at the same time able, through 

its work of suspension, to consume all the 

energy it contributes to the current through 

its addition of mass. A finer silt would retard 

more by increasing viscosity, and a coarser silt

would retard more through the work of suspension.

The available data are not fully demonstrative, but they render it highly probable that, \ 

under all conditions, streams are retarded by \ 

their suspended loads. If that be true, there ' 

is a capacity for suspension coordinate with 

capacity for traction. For each grade of suspended debris, and with any particular slope 

and discharge, it is possible by increasing 

the load so to retard the current that it is

2 Unpublished; communicated in letter.
APPLICATION TO NATUBAL STREAMS. 229

barely competent to suspend d6bris of that 

grade, and the stream is then loaded to its full 

capacity.

In a natural stream the suspended particles 

are of many sizes, and all tend to reduce velocity. Some of the particles are near the 

limit of competence, and any increase of load 

will so reduce velocity that these can no longer 

be upheld. As to these coarsest particles the 

stream is loaded to full capacity, but not as to 

finer material. The addition of fine material 

will cause the arrest of some of the coarser, 

but will increase the total load. An exactly 

parallel statement may be made as to the tractional load.

To recur to the laboratory determinations of 

retardation by suspended material, it is of interest to note that the loads tested in the experiments, while greater than those ordinarily 

found in rivers, are representative of flood 

conditions in the more turbid streams. The 

load of 2.15 per cent, which caused retardations 

of 3 to 15 per cent, is equaled by ordinary floods 

of the Colorado of the West and is exceeded 

by ordinary floods of the Rio Grande and the 

Pecos. For the Rio Grande there are several 

records in the neighborhood of 10 per cent,1 

and small streams in arid lands are liable to 

receive similar loads as a result of violent local 

storms. On the other hand, the Mississippi 

near its mouth carries an average load of only 

0.07 per cent, with a recorded maximum of 

0.8 per cent.2

An attempt was made to measure the retardation of the current by tractional load. It 

was assumed that the bed resistance of a loaded 

stream has two parts, one due to the texture 

of the bed, the other to the work of traction, 

and that the reduction of velocity by traction 

could be measured by comparing speeds of 

loaded and unloaded streams on beds of the 

same texture and slope. A series of experiments were accordingly performed with unloaded streams flowing over beds composed of 

fixed grains of debris, or debris pavements, 

and it was thought tliat their mean velocities 

would be materially higher than those observed 

with streams otherwise similar but bearing 

loads. When the comparison was made, how1 U. S. Geol. Survey Water-Supply Paper 274, pp. 102-104,1911.

2 The average is on authority of Humphreys and Abbot (Physics and 

hydraulics of the Mississippi); the maximum is from an observation 

reported by J. A. Seddon (Kept. Chief Eng. U. S. A., 1887, p. 3094) 

"231.9 grams per cubic foot."

ever, it was found that in 11 out of 16 experiments the observed velocities were higher with 

loaded than with unloaded streams; and the 

average of the 16 results was of the same 

tenor, ascribing a slight excess of velocity to 

the loaded streams. As the estimates of mean 

velocity had been based on observations of 

depth, and as the observations of depth were 

difficult in the case of loaded streams, it was 

thought possible that the result was affected 

by a systematic error in those depth measurements which had been made by means of the 

gage (p. 25). Examples were therefore sought 

in which the depth had been obtained by comparing the average height of the water surface 

during the run with the average height of the 

d6bris bed after the water had been drawn off. 

Thirteen instances were found in which such 

better-conditioned measures could be compared with measures of unloaded streams, and 

hi 11 of the 13 comparisons the unloaded 

streams gave the higher mean velocities. The 

average of the 13-velocities found for loaded 

streams was 10 per cent less than the corresponding average for unloaded streams. The 

best of the available data, therefore, give evidence of the consumption of stream energy by 

traction, but the evidence is not so consistent 

as to free the matter from doubt. In Table 80 

Vmt and Vmu represent the mean velocities of 

loaded and unloaded streams, and the various 

data are arranged according to the magnitude 

of the ratio^of these velocities. This arrangement brings out the apparent fact that the reduction of mean, velocity by traction is greater 

for small loads than for large, for gentle slopes 

than for steep, for low velocities than for high, 

and for large depths than for small. While it 

is entirely possible that such a result a result 

opposed to my preconceptions has been occasioned by a systematic error of observation, I 

am disposed to regard it rather as the expression of some physical law which has escaped 

my analysis. It is possibly connected with a 

fact brought out in the following chapter in 

the discussion of vertical velocity curves the 

fact that addition of load h#s a pronounced 

influence on the distribution of velocities, increasing the contrast between velocities near 

the bed and the mean velocity.

If the variations of the load effect be ignored 

and attention given only to the means of quantities compiled in the table, it appears that 10
230 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

per cent average reduction of velocity corre- cubic foot of discharge, or 0.52 per cent, by 

sponds to an average load of 148.5 gm./sec. per weight, of tractional load.

TABLE 80. Comparison of mean velocities of streams with and ^uithout tractional load.

[Width, 1.00 foot.]

Grade of 

debris.

D)

D)

D)

B)

D)

B)

D)

D)

B

B

15 \

(B)

(D)

(D)

Discharge.

Ft.ysec.

0.734

.734

.363

.363

.363

.363

.363

.545

.734

.734

.734

.363

.363

Slope.

Per cent.

0.37

.32

.43

.43

.58

.54

.35

.53

.38

.48

.54

2.11

1.94

Load.

Gm./sec.

21

12

12

21

23

25

12

30

38

52

80

258

229

Duty.

Gm.lsec.

28.6

16.4

33.1

58.0

63.4

69.0

33.1

55.0

51.7

70.8

109.0

711.0

632.0

148.5

Depth of 

unloaded 

stream.

Foot.

0.410

.411

.229

.193

.199

.176

.224

.240

.267

.256

.241

.098

.099

Vmt

Ft/sec.

1.79

1.79

1.97

1.88

1.82

2.03

1.62

2.27

2.75

2.87

3.04

3.70

3.66

Vmu

Ft./sec.

2.43

2.35

1.97

2.29

2.15

2.39

1.87

2.46

2.92

3.03

3.10

3.37

3.30

Y^L 

Vmu

0.74

.76

.80

.82

.85

.85

.87

.92

.94

.95

.98

1.10

1.11

.«0

THE TWO LOADS.

The reaction of tractional load on velocity 

affects primarily and chiefly the zone of saltation, but there is also a general retardation of 

the stream. The reaction of suspended load 

reduces all velocities, including those of the 

tractional zone. Thus the magnitude of each 

load affects capacity for the other load, and it 

also affects the conditions of competence for 

suspension and competence for traction.

As all parts of the load influence velocity, so 

all parts influence the general slope of an alluvial stream, which is automatically adjusted 

so as to give to the discharge the ability to 

transport all the material, coarse and fine, 

which is supplied. The adjustment is actually 

made through the coarser material, for the permanent deposits of the stream bed are from 

the tractional load.

Along with the adjustment of slope goes an 

automatic partition of the varied load into suspensional and tractional. If the lower Mississippi, for example, were to be supplied for the 

future with only that part of its load which is 

npw carried in suspension, it would so reduce 

its slope that the slackened current would drop 

a portion of that load and thereafter move it 

by traction. If, on the other hand, the river 

were to be deprived of the fine debris now carried in suspension, it would so quicken its current as to lift into suspension a portion of the 

debris now carried by traction and would 

adjust its slope in such way as to maintain the 

partition of load.

An exception to the general law of automatic 

partition is found when the load has only small 

range in fineness, and this was illustrated by 

the artificial conditions of the laboratory; but 

it is not known that natural streams illustrate 

the exceptional case.

This phase of river adjustment is well illustrated by Yuba River, which I studied in 

1904-1908. Where it issues from the mountains it carried a heavy load of coarse debris 

with which it was building up its bed at the 

edge of the Sacramento Valley. A dam thrown 

across it in the region of deposition arrested, 

the tractional load for a time and gave an 

opportunity, by the aid of measurements, to 

estimate its amount. There were also samplings of the water and measurements of suspended load. The tractional load consisted 

mainly of gravel, with coarse sand and many 

bowlders, and the suspended load during flood 

included sand and finer debris. Fifteen miles 

below, near the mouth of the river, the tractional load consisted of sand, with rare small 

pebbles, and only clay and silt were in suspension. Here, too, the load suspended at flood 

stage was estimated from a sampling of the 

water. A computation based on the various 

data indicated that at flood stages the suspended load was approximately equal to the 

' tractional load at each of the two localities.1 

While the data for this estimate were imperfect 

in many ways, they were nevertheless better 

than any other with which I am acquainted.

1 A somewhat fuller statement may be found in Geol. Soc. America 

Bull., vol. 18, pp. 657-658, 1898.
APPLICATION TO NATURAL STREAMS. 231

Information as to suspended loads is fairly 

abundant, but there are no satisfactory data 

as to the complementary tractional loads.

The character of the partition of load depends 

in each particular instance on the relative 

proportions of the various grades in the d6bris 

with which the stream is supplied. It varies 

from point to point along the course of the 

same stream. In the case of Yuba River the 

variation was due to the fact that exceptional 

conditions created by hydraulic mining had 

thrown the bed profile out of adjustment; and 

many other streams are dealing with new and 

man-made conditions; but natural streams 

also have strongly contrasted load conditions 

in different parts of their courses. These arise, 

first, from the diversity of detritus furnished 

by tributaries, and, second, from the gradual 

comminution >of the load as it is borne along.

The partition of load is greatly modified by 

variation of discharge and by temporary conditions of debris supply. A number of changes 

due to varying discharge have already been 

mentioned. The changes in velocity affect the 

grade of fineness marking the plane of partition. 

With reduction of discharge from a flood stage, 

the conditions of partition come to differ in 

pools and over shoals, and usually suspension 

ceases altogether, while traction is still continued on the shoals. With flood discharge, 

also, the partition is probably not quite the 

same for deep and shoal, a portion of the load 

traveling by suspension through the deeps but 

by traction over the shoals.

When a flood is occasioned by heavy rains 

the fine particles of soil are washed to the 

stream, with the result (1) that the suspended 

load is relatively large and (2) that the limiting grade is relatively fine. When an equal 

flood is caused by snow melting, without rain, 

the suspended load is smaller and the limiting 

grade coarser; in extreme cases there may be no 

suspension whatever. While a river is at low 

stage and without load, a local shower may wash 

to it a temporary supply of debris of which only 

the finer part will be immediately transported, 

giving suspension without traction.

The partition of load also varies greatly from 

point to point in the same cross section, competence for suspension and traction both 

responding to the control of velocity.

Despite these variable factors, it is quite 

possible that in the entire load of a stream

large enough to be called a river there is a 

fairly definite ratio between suspension and 

traction. Knowledge of that ratio would be of 

practical importance to the engineer and geologist, because the measurement of the tractional 

load is always difficult, while measurement of 

the suspended load is merely a matter of routine and patience. The estimate afforded by 

the Yuba is a rough approximation and is 

qualified by the fact that the stream had not 

an established regimen but was engaged in 

adjusting its slope to new conditions of load.

Of other estimates of tractional load, the one 

most often quoted is that of Humphreys and 

Abbot. They found that the bar across one of 

the mouths of the Mississippi increased its 

width at the top at the rate of 338 feet a year 

and that the material added was similar to that 

of the tractional load. Assuming that the 

deposit had an outward extent to the depth of 

100 feet and a transverse extent equal to the 

widths of all the mouths, they made a computation from which the tractional load was 

estimated at 11 per cent of the suspended 

load.1

An elaborate study of the loads of the Rhone 

was made by Adolphe GueYard,2 who measured 

the suspended material by sampling the water 

for two years and computed the total output 

from the soundings of the sea bed, the survey 

of which had been repeated after an interval 

of 31 years. He found the suspended load less 

than one-fourth of the whole. Various partial 

estimates have been based on the march of 

subaqueous dunes. Where the volume of a 

dune and its rate of progression are known, 

their product determines definitely a part of 

the load, and if the data cover a year they 

yield at once a minimum estimate' for the 

annual load. Some large dunes in various 

streams have been reported to 'persist from 

year to year, with progressive change of position, but Arthur Hider, who kept two tracts of 

the Mississippi channel under observation for 

about a year, found that the dunes were repeatedly readjusted in respect to size as the 

river stage changed, so that none could be 

identified through long periods, while the 

epochs of readjustment were characterized by 

general deposition or general scour, which 

could not be accounted for as a result of the

1 Hydraulics of Hie Mississippi, p. 149.

2 Inst. Civil F,ng. Proo., vol. 82, pp. 308-310, 1885.
232 TRANSPORTATION OF DEBRIS BY RUNNING .WATER,

dune transformations.1 A similar remodeling 

was observed by Partiot in the Loire.2 Hider 

estimated the whole movement in dunes as 

14,800 cubic yards in 24 hours, which corresponds to 2.2 per cent of the average suspended 

load reported by Humphreys and Abbot; but 

his judgment was that the entire movement of 

d6bris along the bed was at least ten times as 

great as the movement in dunes.

As a generalization from extensive observa- . 

tions of dunes in the river Loire, Sainjon formulates their rate of advance when the material is 

sand as a function of surface velocity, Vs. In 

his equation 3

Advance = 0.00013 ( Vs2 -0.il)

the units are metric. Substituting feet for 

meters, we have

Advance = 0.00004 ( Vs2 - 1.18)

The associated average height of dune crests 4 

is 0.77 meter, or 2.54 feet. As the mean height 

of a dune is approximately one-half the height 

of its crest, we may multiply the rate of advance 

by 2.54/2 and obtain

0.00005 ( F/-1.18)

as an expression for the Loire's load of sand 

carried in dunes, for each foot of channel width 

occupied by dunes. In a later discussion by 

Lechalas (see p. 193) it is shown that Sainjon's 

formula does not apply to surface velocities 

above 3.3 ft./sec., the rate of dune advance being 

then checked because part of the sand escapes 

into the body of the current and is not added 

to the downstream faces of the dunes; but the 

observations on dunes constitute the quantitative basis of the formulation of debris transportation by Lechalas, of which an account is 

given in Chapter X.

In this connection mention may be made of 

moving sand bodies of a different order of magnitude and probably of a different type. They 

are greater than the dunes of the same stream 

and are coordinate in size with the bars separating deeps but are distinguished from the

i Mississippi River Comm. Kept, for 1882, pp. 80-88. Observations 

by W. H. Powless, made at a different place and in another year, are of 

the same tenor. See idem for 1881, pp. 66-120.

n Annales des ponts et chausse'es, Me"m., 6th sen, vol. 1, p. 270,1871.

3 Quoted, with some of the data, by Partiot, idem, pp. 271-273. The 

coefficient is there erroneously given as 0.0013.

< Given by Lechalas in the same volume, pp. 387-388.

bars by their migration downstream. In some 

of the "regularized" streams of Europe, where 

the main channel is artificially restricted to 

curves of large radius, they are developed in 

systematic alternation at the two sides, and the 

thalweg winds between them. 5 In the Mississippi they sometimes appear in the reaches. 

Their progress downstream is accomplished by 

deposition on forward slopes and erosion of rear 

slopes, but the forward slopes are not steep, like 

those of dunes, and their material is not wholly 

derived from the tractional load. Blasius 6 regards them as essentially dunes, correlating 

them specifically with dunes of reticulated pattern. My own view, not necessarily inconsistent with his, connects them with the fixed bars 

separating the deeps of a meandering stream. 

A free stream does not tolerate a straight channel. If a straight channel of moderate width be 

given to a stream, the current swings rhythmically to right and left, and if the banks 

yield it develops meanders. The meanders 

then migrate, according to laws of their own, 

and the bars are fixed in relation to the meanders. If the banks do not yield, the system of 

shoals and deeps established by the swinging 

current migrates slowly downstream. It is 

evident that the migration of these shoals is 

one of the factors and may be an important 

factor in the work of transportation; and also 

that every measurement of the migration of a 

shoal is a partial measurement of load.

Pilots of Mississippi steamboats observe 

that the bars at crossings are built up by floods, 

and such changes have been measured by 

engineers. The generalization has sometimes 

been made that deposition is a specific function 

of floods, but a more satisfactory interpretation 

is given by McMath, 7 who maintains that the 

rising river scours from the deeps to deposit on. 

the shoals, and the falling river scours from 

the shoals to deposit in the deeps. The 

transfers are the joint work of traction and 

suspension. As such changes of the stream 

bed are measurable they afford quantitative 

data as to load, and it was from their observation that Hider, as previously quoted, inferred 

that the dune movement in the Mississippi 

includes but a small fraction of the tractional 

load.

& Engels, H., Zeitschr. Bauwesen, vol. 55, pp. 664-680,1905.

6 Idem, vol. CO, pp. 465-472, 1910.

7 Mississippi River Comm. Kept, for 1881, p. 252.
APPLICATION TO NATURAL STREAMS. 233

The dune movement, the migrations of 

greater bars, and the transfers of debris from 

deep to shoal and shoal to deep are all competent to give information as to tractional load, 

but the estimates they give are minimum 

estimates, to be supplemented by estimates of 

the material which at flood stages is swept 

steadily along without contributing to any of 

the temporary deposits in such way as to be 

accessible to measurement.

AVAILABILITY OF LABORATORY RESULTS.

THE SLOPE FACTOR.

We are now ready to inquire whether, in 

view of the diversities and complexities affecting traction by natural streams, the formula 

for tractional capacity derived under the 

comparatively simple conditions of the laboratory is of practical value in connection with 

natural streams. The four factors of the 

formula may first be considered separately.

The general slope of a stream is the quotient 

of fall by distance, the distance being taken 

along the stream's course. It is best measured 

at high stage, because the chief work of grading 

the channel is accomplished by floods.

With reference to variations in capacity at a 

single locality, slope does not enter, the variations being referred to discharge; but account 

must be taken of slope in comparing different 

divisions of the same stream and in comparing 

one stream with another.

In all such cases the stream's slope is as 

definite a quantity and is susceptible of as 

precise measurement as is the slope of the 

laboratory channel. It differs as to its representative character. The laboratory slope is 

connected with a single discharge and a single 

grade of debris of determinable fineness. The 

! slope of the natural stream does not represent 

\,n , the adjusting work of a determinable discharge 

' but is a compromise product of the work of 

many discharges, and it is usually true that 

the velocities associated with these discharges 

have determined equally diverse mean finenesses of d6bris. The work of the natural 

stream, moreover, has been characterized by 

greater diversity, from point to point, of the 

bed velocities, and its system of velocities has 

been regulated in part by suspended load.

These difficulties would prove insuperable if 

attempt were made to infer the capacity of a 

natural stream from that of a laboratory

stream, but they are not necessarily important 

in transferring a law of variation from a group 

of laboratory streams to an equally harmonious 

group of naturaLstreams. If the diversification 

of discharges and finenesses is of the same type 

for the examples of natural streams between 

which comparison is made, it may well be that 

the slopes are comparable, one with another, 

in the same sense in which they are comparable 

in laboratory work, and that their relations to 

capacity should follow the same law.

THE DISCHARGE FACTOR.

Discharge differences must be considered 

when the tractional capacities of. different 

streams are to be compared, and also in comparing the capacities- of the same stream at 

different tunes.

In making comparison between, different 

streams it is important that the discharges used 

be coordinate that is, that they represent 

equivalent phases of stream work. If coordination be not secured, allowance must be 

made in one stream or the other for the variation of capacity with stage. In case the problem is such that the choice of phase is optional, 

preference should be given to flood phases, 

because the general slope and the details of 

channel shape are approximately adjusted to 

such phases. The greatest known discharge is 

probably less representative of the channel 

conditions than is the mean of annual maxima 

of discharge. It is believed that with use of 

discharges that are both representative and coordinate the discharge factor of the empiric 

formula may be applied. The result of such 

application will be the more satisfactory in 

proportion as the streams compared are allied 

in type and will be relatively unsatisfactory for 

streams in different climatic provinces or for 

comparison of a direct alluvial stream with 

one which meanders.

It is to be observed that in all studies of alluvial streams the discharge of which ; account 

should be taken in connection with traction is 

the discharge flowing in the channel proper. 

That which passes the banks ceases to contribute of its energy to the work of traction, 

and the portion of load diverted with it is not 

tractional.

The case of variation of discharge in the 

same stream is complicated by simultaneous 

variations of fineness and competence. In the
234 TRANSPORTATION OF DEBRIS BY RUNNING WATER,

experiments with sieve-separated grades of 

de'bris fineness and competence were constants 

with reference to discharge; but in a natural 

stream, where the tractional load may have 

great range in fineness, the mean fineness of the 

load varies with discharge, and the reason of 

its variation is that competence varies.with 

discharge. The two competences which limit 

the range in fineness move up and down as 

discharge changes, and the mean fineness 

moves with them. Therefore the response of 

capacity to discharge can not be considered by 

itself.

For convenience in analyzing the conditions, 

let us assume first a discharge which is adjusted 

to the details of channel form, to the deeps and 

shoals. If, now, the discharge be increased, and 

with it the whole system of velocities, transportation will be everywhere stimulated, part 

of the tractional load will join the suspended, 

and the scouring of the deeps will bring into 

the tractional load a greater proportion of the 

coarser elements of the load. The mean fineness of that load will be reduced, and the capacity, while enlarged by increase of discharge, 

will be somewhat reduced by loss of fineness. 

The increase with discharge will be less than if 

the fineness were constant.

If, on the other hand, the discharge be reduced, some of the coarser material comes to 

rest, while finer de'bris is added from the 

suspended load. So the reduction of capacity 

from diminished discharge is qualified by the 

effect of increased fineness. But before the 

change in discharge has gone far the deeps 

become pockets for the reception of deposits, 

and traction is restricted to the intervening 

shoals, where it causes erosion. The erosion 

is selective, leaving an ever-increasing assemblage of residuary coarse material, which tends 

to protect the finer. The current on the shoals 

no longer obtains a full supply of the material 

for which it is competent, and the load and 

capacity part company. Or we may say that 

as the erosion of the shoals progresses the mean 

fineness of the accessible debris is reduced until 

a grade is reached for which the current is not 

competent. In either case the decadence of 

traction follows a law which is not well represented by the discharge term of the laboratory 

formula.

The above analysis postulates a wide range 

and somewhat equable distribution of fineness

in the de'bris of the stream bed, a condition not 

always found. It might not apply, for example, to a stream which drains a district of 

friable sandstone and is therefore supplied with 

nothing coarser than sand. Nor would it apply 

well to a stream supplied with very coarse and 

very fine de'bris but not well supplied with 

intermediate grades.

In most alluvial streams, and probably in all 

meandering streams, the work of traction which 

is accomplished on the shoals at low stage and 

midstage is almost negligible in comparison 

with the high-stage traction. Not only is the 

rate of traction slow, but the field of traction 

is restricted. If a single formula will not fit 

both low and high stages, the one adjusted to 

high-stage variations will have the greater 

practical value.

Yet another consideration enters here, and 

one of peculiar importance. When discharge is 

reduced, and the competence of the current for 

traction is thereby changed, the coarse material 

eliminated from the tractional range ceases to 

be transported; but when discharge is increased, 

and the competence of the current for suspension is thereby changed, the fine material 

eliminated from the tractional range continues 

to be transported. It is, in fact, transported 

more rapidly, so that a greater amount passes 

a given section each second. For most or all 

practical purposes the change in mode of transportation is of no moment, and those purposes 

would be served by a formula which should 

include the material shifted and ignore the 

change in mode. In the system of reactions 

set up by change of discharge the two modes 

of transportation are so interwoven, in fact, 

that the practical discrimination of the suspended and tractional loads is impossible. 

Even in the laboratory experiments devised

specially for the study of traction a certain 

amount of interplay was tolerated, for temporary suspension appeared over the crests of 

some of the antidunes and also in the bends of 

the crooked channels.

If the purely tractional point of view is to 

be exchanged for another, what shall be substituted ? One natural suggestion is to include 

in a single view the entire load, suspended and 

tractional; another to include along with the 

tractional only that part of the suspended load 

which for part of the time is tractional also. 

That which would be included in one view and
APPLICATION TO NATURAL STREAMS. 235

excluded from the other is the finer part of the 

suspended load, the part that does not sink to 

the bottom so long as the current is sufficiently 

active for traction. Being purely suspensional, 

its quantity is peculiarly a function of supply 

and is connected with discharge only through 

the association of discharge with ram. Wherever discharge is largely a matter of tribute 

from snowbanks, the suspended load is conspicuously independent of discharge.

If we exclude from view the purely suspensional'material, a natural criterion for inclusion 

is the finest debris which low-stage discharge 

moves by traction, and as low-stage traction is 

limited to the bars, or interpool shoals, it is 

the finest tractional debris of those shoals. If 

we consider a gradual increase of discharge from 

least to greatest, we have at first no traction. 

Then for a particular discharge, which may be 

called the competent discharge, traction begins 

on the shoals, only the finest of the debris being 

moved. Gradually coarser and coarser material is included, the range in fineness and the 

load increasing together; but in this phase of 

action the load is not necessarily the equivalent of the capacity, for it may be limited by 

the supply of debris of requisite fineness. 

After a time another critical discharge is attained, which initiates the loaning of debris 

from traction to suspension, and thereafter a 

constantly increasing share of the traveling 

debris is suspended. As the suspended particles travel faster than the saltatory, and as 

capacity is the ability of the stream, measured 

in grams per second, to move de"bris past a 

 sectional plane, the transfer from traction to 

suspension is an important factor in the enhancement of capacity.

The relation of capacity to discharge, contemplated from this viewpoint, has two elements in common with the discharge factor of 

the laboratory formula. It includes a competent discharge, corresponding to the zero of 

capacity, and it associates continuous increase 

of capacity with continuous increase of discharge. It differs, however, in important ways, 

and the possibility of expressing it by a definite 

formula is not evident. In the pool and rapid 

phase of activity the supply of debris suitable 

for traction is usually limited, and in many 

streams it is exhausted during each recurrence 

of the phase. In the phases of greater discharge, when traction occurs in the deeps as

well as on the shoals, the sequence of capacities 

depends not only on discharge but on the relative proportions of debris of different grades of 

fineness in the material of the load. It is probable that for most streams the load-discharge 

function is discontinuous at the limit of the 

pool and rapid phase.

Because of this presumable discontinuity, 

because the tractional work while the pools 

exist accomplishes only a local transfer of 

debris, and because the work performed is 

usually of negligible amount in comparison 

with the work of larger discharges, it is probably better to ignore altogether the pool and 

rapid phase in any attempt at general formulation. If that be left out of account and if 

the general features of the laboratory formula 

be retained, the constant K becomes the discharge which initiates traction in the deeps, 

and thus initiates through transportation of 

bottom load. If we accept that as a starting 

point, the material so fine as to be suspended 

by that discharge may be classed as purely suspensional, and other material suspended by 

larger discharges may be grouped with the 

tractional load. For the tractional load thus 

enhanced, or the amplitractional load, as it 

may conveniently be called, the rate of variation with discharge is evidently higher than 

the rates found for simple grades in the laboratory, and it may be much higher, for the debris 

diverted from traction to suspension, instead 

of lagging behind the lowest and slowest 

threads of the current, now speeds with the 

current's mean velocity.

It is possible that a practical formula for the 

fluctuations of an alluvial river's load may follow these lines, taking the form

where Ca is the capacity for amplitractional 

load, and /q is the smallest discharge competent 

to establish a continuous train of traction 

through deeps and shoals; but the suggestion 

as to form has no better basis than analogy, 

and no data are known tending to determine 

the magnitude of the important parameter o.

THE FINENESS FACTOR.

When the work of two natural streams is 

compared and the streams are of the same type, 

it is believed that the fineness factor of the
236 TRANSPORTATION OF DEBRIS BY RUNNING WATER,

laboratory formula is applicable. It is true 

that fineness enters in a relatively complex 

way into the determination of the loads of 

natural streams, but for the comparison indicated the elements of influence are severally 

represented by the experiments, and their totals should follow a law of the same type. For 

small discharges this inference is subject to 

certain qualifications, which will appear from 

what follows.

When the work of the same stream is compared under different discharges, a difference in 

fineness is developed under the laws of competence. With larger discharge the mean fineness is less than with small discharge, and the 

difference in fineness conspires with the difference in discharge to determine capacity. For 

reasons explained in the last section, however, 

capacity can not always be considered synonymous with load when the discharge is small.

THE FORM-RATIO FACTOR.

In the reaches of a direct alluvial stream there 

is approximate uniformity of depth at high 

stage, and the conditions involving form ratio 

are essentially like those realized in the laboratory. To such cases the principles developed 

in the laboratory studies should be applicable.

It is true in a general way, as already mentioned on page 223, that at a high stage of a 

natural stream the sectional area is about the 

same for the reaches as for the bends, and so 

too is the width. It follows that the mean 

depth is about the same, although the maximum depth may be very different. The highstage capacity is also the same at every section, after the channel form has been adjusted 

to the discharge. If these generalizations are 

correct, the principle involved in the form-ratio 

factor of the laboratory formula is applicable to 

curving streams, provided form ratio is interpreted as the ratio of mean depth to width, and 

not as the ratio of maximum depth to width.

In the analysis of conditions determining the 

relation of capacity to form ratio (Chapter IV) 

an important rdle was ascribed to the resistance 

of the banks; and the quantity of that resistance was represented in one of the'parameters 

of the formula, a. The optimum form ratio, 

p, was found to vary inversely with a and, therefore, to vary inversely with the resistance of the 

banks. The resistance afforded by river banks

is greater than that given by the smooth walls 

of laboratory channels, and this element tends 

to make the optimum form ratio relatively 

small for rivers. Its influence, however, is overshadowed by those of slope and discharge. As 

the optimim ratio varies inversely with slope, 

and as most rivers have lower slopes than the 

experimental streams, the general tendency of 

the slope element is to make the ratio large 

for rivers. As the optimum ratio varies inversely with discharge, and as the discharges of 

natural streams are relatively large, the tendency of this element is to make the ratio small 

for natural streams. The rates of variation 

being unknown, the net result of the three influences can not be inferred deductively. The 

data from Yuba River, cited in Chapter IV (p. 

135), show that for one case of a natural stream 

the optimum ratio is decidedly larger than that 

established by the stream in its alluvial phase 

and is of the order of magnitude of the determinations made in the laboratory.

THE FOUR FACTORS COLLECTIVELY.

The results of the preceding discussions admit of a certain amount of generalization. 

When different streams of the same type are 

compared, and especially when the type is alluvial, the law of their relative capacities at 

high stage may be expressed by the laboratory 

formula (109). The ability of that formula to 

express the variation of capacity with discharge 

in the same stream is problematic.

It has not been shown that the system of 

numerical parameters determined for laboratory 

conditions can be used in extending the application of the formula to natural streams. If 

the formula were rational, the result of an adequate mathematical treatment of the physical 

principles involved, the constants measured in 

the laboratory would be of universal application 

(with moderate qualification for the conditions 

imposed by the curvature of natural channels); 

but the constants of an empiric formula afford 

no basis for extensive extrapolation.

THE HYPOTHESIS OF SIMILAR STREAMS.

When the Berkeley experiments were planned 

it was assumed that the relations of capacity to 

various conditions would be found to be simple, and that the laboratory streams were representative of natural streams except as to the
APPLICATION TO NATUBAL STEEAMS. 237

characters associated with, bending channels. 

Because of the discovered complexity of the 

laws affecting capacity it is now apparent that 

the laboratory formulas can not be applied to 

streams in general. It is, however, probable 

that among the great variety of natural streams 

there is a more or less restricted group which 

is in such respect similar to the laboratory group 

that the empiric results of the laboratory or 

at least the results embodied in exponents 

may properly be applied to it.

The criteria of similarity between large and 

small have been discussed to some extent by 

others in connection with the investigation of 

hydraulic problems by means of models. 

William Froude inferred from theoretic considerations that if the speed of a ship and the 

speed of its miniature model "are proportional 

to the square roots of the dimensions, their resistances at those speeds will be as the cubes 

of their dimensions," 1 and he afterward verified this result by experiments. T. A. Hearson, in projecting a model river for the investigation of various hydraulic problems,...discussed separately the resistance to flow by the 

wetted perimeter, the influence of varying sectional area, and the influence of bends. He 

concluded that if the linear dimensions were 

kept in the same proportion, so that the river 

channel and its model were similar in the geometric sense, the velocities would be related as 

the square roots of the linear dimensions, and 

the discharges as the 2.5 powers of the linear dimensions. It would be necessary that the roughnesses of the channel surfaces have the same differences as the linear dimensions, and that the 

movable debris of the bed also follow the laws 

of linear dimensions.2 His deductions were not 

tested by the construction and use of a model, 

but they derive a large measure of support from 

the verification of Froude's analogous theorem. 

So far as I am aware, fJl the models actually 

constructed to represent rivers and tidal basins 

have been given an exaggerated vertical scale.3 

0. Keynolds 4 made a series of "models of tidal 

basins in which the scales of depth and of tidal

1 These words are quoted from Inst. Naval Arch. Trans., vol. 15, p. 151, 

1874. I have not seen Froude's original discussion of the subject.

2 Inst. Civil Eng. Proc., vol. 140, pp. 216-222, 1900-1901.

8 See Fargue, L., La forme du lit des rivieres a fond mobile, pp. 57,128, 

1908. Fargue recommended for a model river a vertical scale of 1:100 

and a horizontal scale of 1:20, from which he deduced a discharge ratio 

of 1:3,200 and a velocity ratio of 1:16.

< British Assoc. Adv. Sci. Repts. 1887, pp. 555-562; 1889, pp. 328-343; 

1890, pp. 512-534; 1891, pp. 386-404.

amplitude were greater than the scale of length, 

the ratios ranging from 31:1 to 105:1. No adjustment was made as to size of debris, the requirements of his investigation being met by 

any material fine enough to be moved by the 

currents. A tidal oscillation was communicated to water resting on a level bed of sand, 

with the result that the bed was gradually 

molded into shapes more or less characteristic 

of estuaries. From general considerations a 

"law of kinetic similarity" was deduced:

L-j ' constant

Ju

in which p is the tidal period, Ti the depth of 

water (proportional to the amplitude of the 

tide), and L the length of the estuary. Under 

this law the results were generally consistent, 

but there was found to be a limit to the range 

of suitable conditions, and this limit was formulated by

li*e = constant

in which e is the exaggeration of the vertical 

scale.

Eger, Dix, and Seifert,5 making a model of 

a portion of Weser River for the purpose of 

studying the effect of projected improvements, 

adopted 1:100 as the scale of horizontal dimensions and depths, and 1:6.7 as the scale of mean 

diameters of debris particles composing the 

channel bed. It was then a matter, first of 

theory and computation but finally of trial, to 

select scales for discharge and slope. The main 

condition to be satisfied was that for discharges 

corresponding to high and low stages the depths 

of water should be properly related, according 

to the scale of linear dimensions. For the scale 

of discharges 1:40,000 was finally adopted, and 

for slopes 650:1. The resulting ratio of velocities was 1:4; and this ratio, combined with the 

ratio of debris sizes, was found to give a time 

ratio (for the accomplishment of similar 

changes in the bed of the stream) of 1:360. 

The scale of velocities being only 1:4 while the 

scale of distances was 1:100, there was an 

exaggeration of velocities in the ratio of 25:1.°

The quantities of debris moved being in the 

ratio of 1:100 3 , the distances moved in the

6 Zeitschr. Bauwesen, vol. 56, pp. 323-344,1900.

0 So stated by the authors. An allowance for the general principle 

that velocities are proportional to the square root of the hydraulic mean 

depth, and therefore to the square root of linear dimensions, would 

indicate 1:10 as the normal ratio of velocities, and give 2.5:1 as the 

exaggeration.
238 TRANSPORTATION OF DEBRIS BY RUNNING WATER,

ratio of 1:100, and the times consumed in the 

ratio of 1:360, the ratio of loads (per second)

100 4 

was I'ocn" or 1:280,000. The results were

satisfactory; it was found that the successive 

forms given to the river bed by variations of 

discharge were repeated in the model.

The exaggerations of the vertical scale by 

Reynolds and of the slope by Eger, Dix, and 

Seifert had the important effect of shortening 

the time necessary to produce the desired remodeling of the mobile bed. The absolute 

proportionality adopted by Froude and recommended by Hearson would have entailed a 

prohibitive consumption of time and might 

have added a serious complication in connection with the use of very fine debris.1

The similarity controlled by Reynold's law 

was a relation between the wave periods and 

dimensions of tidal basins and is not closely 

related to similarity in the control of tractional load. The similarities obtained in constructing the model of the Weser are more in 

point, because they involve an average rate of 

movement of debris; but they throw no light 

on the laws of variation of debris movement* 

which is the important matter in bridging the 

interval between our experiment streams and 

natural streams. After attempting to use 

various suggestions which came from the 

adjustments of the Weser model, I have returned to the principles of geometric similarity 

employed by Froude and Hearson.

Let us assume, as possible or plausible, that 

the principles developed in the laboratory, 

together with all parameters which are of the 

nature of ratios, are independent of the scale 

of operations and may be applied to streams of 

far greater magnitude, provided all linear fac-' 

tors are magnified in equal degree. If width 

and depth are enlarged in the same ratio, the 

form ratio is unchanged. If longitudinal distance and loss of head are enlarged in the same 

ratio, the slope is unchanged. If the dimensions of the transported particles are enlarged 

in the same ratio as the linear elements of channel, the linear coarseness is increased, or the 

linear fineness is reduced in that ratio.

The natural streams which may be considered as similar to the experimental streams 

constitute a class of moderate size. The form

i Such considerations as these affected the selection of materials for the 

Berkeley experiments land prevented the employment of very gentle 

slopes.

ratio for rivers ranges lower than for experimental streams, but there is some overlap. 

The smaller of the form ratios of the laboratory 

are representative of a considerable number of 

rivers. The slopes of rivers range lower than 

for laboratory streams, but here again there is 

overlap, and the natural streams which are 

similar in respect to slope are in general such 

as have coarse debris, so that there may be 

correspondence in that regard also. The similar natural streams to which hypothesis extends 

the laboratory results are those of large form 

ratio and steep slope, carrying coarse de"bris.

The primary difference between a large 

stream and a small one being one of discharge, 

and our general inquiry being directed to the 

valuation of capacity for traction, let us seek 

an expression for the relation of capacity to 

discharge when similar streams of different size 

are compared.

The laboratory data determine control of 

capacity by slope, discharge, fineness, and form 

ratio. In similar streams slope and form ratio 

are. constant, and we need consider here only 

discharge and fineness. As we are comparing 

.the laboratory streams as a group' with similar 

natural streams, also taken as a group, it is 

advisable to employ a mode of formulation 

which lends itself to the use of averages, and 

the most convenient is that of the synthetic 

index. In

I3a and Zta are average values of the synthetic 

index and may be estimated, from data in 

Tables 32 and 43, as 1.32 and 0.77. Designating elements of the larger and smaller streams 

severally by subscripts and , , we have, from 

the above,

71 

F,

0.77

Q,

(113)

If we designate by L the ratio between a linear 

dimension of the larger stream and the corresponding dimension of the smaller.

Calling mean velocity V, bearing in mind that 

the hydraulic mean radius is a linear dimension 

of channel, and recalling that the Chezy formula
APPLICATION TO NATURAL STREAMS. 239

makes V vary as the square root of the hydraulic radius, we have

tf _ ro.s F, ~ L

Then, since discharge is the product of width, 

depth, and velocity,

Q, ~w, d, V, ~ 

whence

(Q»YM =1=^

\Qj L F,

W,

Substituting this value of the fineness factor in 

(113), and reducing, we have

-(115)

The result indicates that 

GnJitL

whence

C 

or, since ^= U, the tractional duty of water,

U,,= U,

That is, for similar streams the tractional duty of 

water is the same.

As the exponents connecting capacity with 

discharge, capacity with fineness, and mean 

velocity with hydraulic radius are all averages 

of low precision, the result is far from being so 

secure as might be inferred from equation (115). 

Its best support is really found in the plausibility of its conclusion. Our experience with a 

variety of physical laws makes it easy for us to 

believe that with suitable parity of conditions 

a unit of discharge will accomplish the same 

work as part of a large stream that it will accomplish as part of a small stream; and so the 

conclusion is plausible. The fact that an attempt to test the hypothesis of similar streams 

by combining it with experimental data has led 

to a plausible result is a fact favorable to the 

hypothesis.

Let us now assume the hypothetic law to be 

a real law and draw such inferences as may be 

warranted. The range of slopes which the experiments treat with confidence is from 0.5 to 

3.0 per cent. Direct application is limited to 

streams having slopes within that range. By 

postulate,

Substituting in equation (114), we have

Q, \D,

Q,,

That is to say, for similar streams, the ratio 7^5

may be regarded as constant. This relation 

affords a criterion for the discrimination of 

those natural streams which are similar to the 

laboratory streams, provided they are also 

similar in slope and form ratio. The following

limiting values for y=rf^ for different slopes are

all estimated on the assumption of a form ratio 

of 0.05: '

Limiting values of -^-

3,000,000-40,000,000 

2,000,000-30,000,000 

1,500,000-10,000,000 

500,000- 4,000,000

Slope 

(per cent).

0.5 

1.0 

2.0 

3.0

The form ratio 0.05 is considerably below the 

average of the ratios developed in the laboratory, and it is also much above the average 

for alluvial rivers at flood stage. Any allowance which might be made for this discrepancy 

would have the effect of increasing the estimate 

of limiting values of the ratios of Q to D2- 5 . 

Subject to this qualification the ratios indicate 

the types of natural streams which are "similar" 

to the laboratory streams and to which various 

laboratory results may be applied. The streams 

are in general either small creeks or else rivers 

transporting very coarse d6bris. As the slopes 

are determined by flood discharges, such discharges should be used in the^ classification.

For the streams thus classified as similar to 

laboratory streams the duty of water is of the 

same order of magnitude, and so are the rates 

of variation of duty with the several conditions 

of slope, discharge, and fineness. The rates of 

variation apply especially to comparisons of 

one stream with another. For the estimation 

of variation with discharge in the same stream 

something should be added to the laboratory 

rate to allow for the varying assistance which
240 TRANSPORTATION OF DEBEIS BY RUNNING WATER,

suspension gives to the work of traction (p. 234). 

For estimation of optimum form ratio something should be deducted from the laboratory 

indication to allow for the greater resistance 

of the channel walls.

SUMMARY.

X

Natural streams of alluvial type differ from 

the streams used in the laboratory in ways connected with the bendings of their courses and 

with variations of discharge. The differences 

affect forms of cross section, 'the distribution of 

velocities within the section, and the partition 

of the load between suspension and traction. 

The two portions of load are carried at the 

expense of the stream's energy, each reduces 

the velocity, and the reduction of velocity determines the limit of carrying power. The 

whole burden of the stream includes not only 

two divisions distinguished by mode of transportation but as many minor divisions as there 

are grades of debris, and the load carried of 

each grade reduces the capacity for all the 

grades.

These and other complexities make it dimcult to apply the laboratory results to natural 

streams. It is probable that the forms of the 

laboratory formulas are applicable, with limitations, to the comparison of one stream with 

another, but the availability of the exponents 

is problematic. There are special difficulties in 

attempting to use the formulas for the comparison of capacities of the same stream at different 

stages, and in such comparisons the tractional 

load can not be considered by itself, because 

much material "which is swept along the bed 

at lower stages is lifted by flood velocities into 

the body of the current.

It is thought that the laboratory formulas 

may be applied to natur-al streams which 

are geometrically similar to the laboratory 

streams that is, to streams having the same 

slopes and form ratios and carrying debris of 

proportionate size. The class of streams to 

which the formulas apply by reason of similarity is necessarily restricted, being characterized in the main by high slopes and coarse 

debris. It can include few large streams.

CONCLUSION.

It was a primary purpose of the Berkeley investigation to determine for rivers the relation

which the load swept along the bed.bears to the 

more important factors of control. As a means 

to that end it was proposed to study the mode of 

propulsion and learn empirically the laws con- 

.necting its output with each factor of control 

taken separately. The review of results in the 

present chapter shows that the primary purpose 

was not accomplished. In the direction of the 

secondary purpose much more was achieved, 

and a body of definite information is contributed to the general subject of stream work. A 

valuable outcome is the knowledge that the 

output in tractional load is related to the controlling conditions in a highly complex manner, 

the law of control for each condition being qualified by all other conditions.

With the aid of the Berkeley experience it 

would be possible to avoid certain errors of 

method and arrange experiments which should 

yield more accurate measurements of the same 

general class and it is natural that the experimenter should feel the desire to do his work over 

in a better way but I am by no means sure that 

adequate advantage would reward a continuance of work on the same or closely related lines. 

The complex interactions could be giv^n better 

numerical definition, but it may well be 

doubted whether their empiric definition would 

lead to their explanation. It is possible that 

the chasm between the laboratory and the river 

may be bridged only by an adequate theory, 

the work of the hydromechanist. It is possible 

also that it may be practically bridged by experiments which are more synthetic than ours, 

such experiments as may be made in the model 

rivers of certain German laboratories. (See 

p. 16.)

The practical applications for results from 

experiments in stream traction belong almost 

wholly to the field of river engineering. For 

the transportation of detritus and related materials by artificial currents, stream traction will 

rarely be used, because flume traction is more 

efficient. Our results in flume traction have 

therefore an immediate practical application, 

and as they were limited in range the advantage 

of extending them can hardly be questioned.

.The report on the Berkeley investigation 

projperly closes with this chapter, but there are 

several by-products which seem worthy of record. Some of them are presented in the following chapter, and others are contained in 

appendixes.
CHAPTER XIV. PROBLEMS ASSOCIATED WITH RHYTHM.

RHYTHM IN STREAM TRANSPORTATION.

This chapter is concerned with certain problems upon which the Berkeley investigation 

touched, but which were not seriously attacked.

The low precision of the observations on 

stream traction, a precision characterized by an 

average error of about 11 per cent and an. average probable error after adjustment between 2 

and 3 per cent, had for its chief cause the failure 

to eliminate from the experiments the influence of rhythm. The slope of the water surface, the slope of the channel bed, and the load 

of de'bris transported were all subject without 

intermission to rhythmic fluctuations. If materially better observations of the same sort are 

to be made, this difficulty must be successfully 

dealt with, and the first step toward mastering 

it is to understand it. The removal of a difficulty, however, is neither the sole nor the most 

important result to be expected from the study 

of tractional rhythms. Underlying them are 

physical principles which are of importance in 

the dynamics of rivers, and their study constitutes one of the available lines of approach 

to the broader subject.

For their empiric study the general plan of 

the Berkeley apparatus is well adapted, but our 

experience indicates that certain details should 

be modified. The use of a long trough is advisable, with contraction at the outfall, and 

with delivery of the load to a settling tank beyond the outfall. The appliances and methods 

should be such as to secure uniformity in discharge, in character of debris, and in rate of 

feed.

The apparatus, for regulating discharge, described on pages 20 and 257, was one of the most 

satisfactory parts of the Berkeley equipment. 

Its most important feature, as affecting precision, was the delivery of the water through an 

aperture under a considerable head.

Uniformity of debris can hardly be secured 

without the employment of an artificial, narrowly limited grade, and the available means 

of sorting is the sieve. It is to be observed, 

however, that after a grade has been separated

20921° No. 86 14 16

by sieves it is still subject to sorting by current, 

the current recognizing differences of form and 

density which the sieves ignore. When debris 

that has once been handled by the stream is to 

be used a second time, remixing may be advisable.

None of the devices we employed to feed 

debris to the current achieved uniformity. 

Those which depended on the flow of wet de'- 

bris through an aperture failed because the 

proportion of water could not be kept constant. 

The others depended on handwork and experienced the irregularity usual to handwork. 

An apparatus planned near the end of the experimental work, but never tried, is of such 

promise that its essential features are here 

described.

FIGURE 72. Suggested apparatus for automatic feed of de'bris.

A drum, A in figure 72, is turned slowly by 

power, its rate being regulated by clockwork. 

Its position is above the trough containing the 

experiment stream, B. The surface of the 

drum is uniformly roughened. Above it is a 

vertical rectangular shaft, filled with moist 

debris. The shaft does not touch the drum. 

The width of the separating space at D is controlled by some suitable device. As the drum 

turns, a debris'layer of uniform thickness is 

carried with it, and this falls into the stream. 

To prevent irregularities due to adhesion, a

241
242 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

detaching device is placed at some point E, 

the device possibly consisting of an open-rank 

comb of elastic wire. The shaft should be 

smooth and of uniform section, so that the 

particles of debris near the drum may be 

brought into actual contact with one another 

by pressure of the debris above. That the 

debris may not be caused to flow by excess of 

moisture, it 'should be thoroughly drained before use. In a rough construction designed to 

test the practicability of the apparatus the 

drum surface was roughened by covering with 

a wire screen, and this was found to secure the 

delivery of the debris.

Uniformity of feed having been provided, the 

rhythms of transportation may be observed as 

oscillations in the debris delivered at the outfall. Rhymths of slope may be studied, at 

least initially, by observing changes in the 

profile of the water surf ace.

In the record of the Berkeley observations it 

is not practicable fully to discriminate rhythmic inequalities from those occasioned by 

irregularities of debris feeding, but there is 

reason to believe that several rhythms of different period coexist. The shortest rhythms are 

those connected with the dunes and antidunes, 

and these are evidently associated with rhythms 

of the flow of water.

RHYTHM IN THE FLOW OF WATER.

Reynolds,1 treating of the flow of water 

through tubes, distinguishes two modes of flow 

as direct and sinuous. They are otherwise 

called steady and turbulent. In direct flow the 

filaments of current have simple lines, which 

are straight and parallel if the walls of the conduit are straight and parallel. In sinuous flow 

the filaments of current are neither simple nor 

parallel and may be intricately convoluted. 

The flow in tubes is direct for low velocities 

and sinuous for high, the critical velocity varying inversely with the diameter and the roughness of the tube and directly with the viscosity 

of the water. It would follow from his generalizations that the flow of such streams as were 

used in our experiments would be sinuous, and

i Reynolds, Osborne, An experimental investigation of the circumstances which determine whether the motion of water shall be direct or 

sinuous and of the law of resistance in parallel channels: Roy. Soc. 

Philos. Trans., London, vol. 174, pp. 935-982, 1883. Also, The two 

manners of motion of water: Roy. Inst. Great Britain Proc., 1884.

its actual sinuosity was a matter of observation. To a large extent the curvature of the 

flow lines was shown by the motions of minute 

suspended particles, and when this evidence 

was lacking it was still possible to infer diversity of current from continual changes in the 

configuration of the water surface.

It is probable that all the diversities of flow 

were rhythmic, for that is the nature of inequalities of motion developed by the interaction of constant forces; but in many cases 

the rhythms were so numerous and so related 

in period and other characters that their combination gave the impression of irregularity. 

In many other cases, however, some one 

rhythm was dominant, and these cases appear 

especially worthy of study.

The cycle of movements constituting a 

rhythm unit may be definitely related to space, 

or to time, or to both space and time. The 

pulsations of the water surface which obstructed our observations of water profile were 

manifestations of time factors. The regular 

sequence of dune crests was a manifestation 

of a space factor. The two phenomena coexisted. In the space interval from dune crest 

to dune crest certain elements of motion were 

constant. There was a large stationary vortex 

in the hollow between the crests (fig. 10, p. 31), 

and there might be stationary elements in the 

configuration of the water surface. At. the 

same time the flow lines of the water at all 

points swayed in direction and were affected 

by variations of velocity, and these changes 

were rhythmic with respect to time. Doubt^ 

less many of the fluctuations belonged to or 

were associated with vortices which traveled 

with the general current. It is conceivable, 

or even probable, that the stationary features 

and the traveling features were coordinated, so 

that within the rhythmic space unit corresponding to the dune interval there was a 

cycle of variations of motion which was 

rhythmic with respect to time.

The rhythmic dunes marched slowly down the 

trough and with them marched the intercrest 

vortices and associated motions. To that extent the features of the bed controlled the features of the current. Nevertheless the dunes 

were not essential to the existence of a water 

rhythm characterized by a definite space interval. There was positive evidence that the dune
PEOBLEMS ASSOCIATED WITH RHYTHM-. 243

interval was determined by a preexistent water 

rhythm. Not only was the dune interval a 

function of depth and velocity of current, 1 hut 

the creation of dunes by water rhythm was 

repeatedly observed. In certain experiments a 

slow current, moving over a bed of debris artificially smoothed and leveled, was gradually 

quickened until transportation began. The 

movement of particles did not begin at the same 

time all along the bed but was initiated in a 

series of spots separated by uniform intervals, 

and the first result of the transportation was the 

creation of a system of dunes.

In certain experiments on flume traction a 

slow current, moving over a smooth channel 

bed of wood, swept along a small quantity of 

sand. With increase of the load of sand local 

deposits were induced, which took the form of 

thin straggling patches, similar to one another 

in outline and separated by approximately 

equal bare spaces. These moved slowly downstream, the mode of progress being similar to 

that of dunes, and with further increase of load 

they acquired the typical profile of dunes.

In both groups of experiments it was evident 

that the primary rhythm pertained to the water 

and was independent of the work of transportation. In the second group, where sand swept 

steadily forward over the whole area of the bed, 

it was evident that the water rhythm did not 

involve reversed currents along the bed and 

therefore did not include such stationary vortices as accompany dunes.

In the experiments on stream traction the 

Development of dunes was conditioned by the 

three factors of velocity, depth, and load, besides an undetermined influence from fineness

i Measurements of dune interval in the laboratory were too few to demonstrate the factors of control, but comparison of laboratory data with 

data from other sources leaves little question that control is exercised by 

depth, velocity, and fineness. In the laboratory, where depths were a 

matter of inches, the du ne interval rarely exceeded 2 feet. In Mississippi 

River depths measured in scores of feet are associated with intervals 

measured in hundreds of feet. In tidal estuaries, where dunes of a 

special type are exposed at low tide, the depths of the formative currents are intermediate between those of the laboratory and those of the 

Mississippi, while the intervals are measured in feet or tens of feet. For 

data of the Mississippi see Johnson, J. B., Rept. Chief Eng. U. S. A., 

1879, pp. 1963-1967, and Eng. News, 18SS, pp. 68-71; Powless, W. H., 

Mississippi River Comm. Rept. for 1881, pp. 66-120; and especially 

Hider, Arthur, Mississippi River Comm. Rept. for 1882, pp. 83-88. For 

data on the dunes of tidal estuaries see Cornish, Vaughan, Geog. Jour., 

vol. 18, pp. 170-202,1901. Hider finds that the dune interval is greater 

at high stages of the river than at low, the depth and velocity both 

decreasing with the change from high to low. Cornish's observations 

show that the interval varies directly with depth of current under conditions which make it probable that the velocity varies inversely with 

depth.

of de"bris.2 It is probable that load and fineness enter only as factors of resistance, so that 

the essential conditions are velocity, depth, and 

bed resistance. Within certain rather, wide 

ranges of value for these controlling factqrs the 

bed is molded into dunes. When the limit is 

exceeded by increase of velocity or resistance, 

or by decrease of depth, the dunes disappear 

and the bed becomes smooth and plane. At 

the same time the oscillations' and other disturbances of the water surface are reduced; but 

as they do not altogether disappear it is to be 

inferred that the flow is still characterized by 

internal diversity.

With still further increase of velocity or 

resistance, or with further reduction of depth, 

another critical point is passed, and the process 

of traction becomes again rhythmic, but in an 

antithetic way. The bed is molded into antidunes, which travel upstream (pp. 31-34), 

and the water surface also is molded into 

waves, which copy the forms of the antidunes 

and move with them. The internal movement of the water is again characterized by a 

dominant rhythm, but the type of rhythm is 

different from that associated with dunes. 

The rhythm is also less stable, and its intensity 

exhibits a cycle of change. With low intensity 

the waves are nearly equal in height and 

length, but sooner or later inequalities-develop 

and the higher waves overtake the lower and 

absorb them. This process increases the wave 

length in the upstream part of the trough, and 

the influence of the change is in some way 

communicated to the downstream end, where 

the waves are first formed, with the result 

that larger and larger waves develop. Finally 

a master wave, with curling crest, rushes 

through the trough from end to end, and this 

has the effect of wiping out the irregularities 

and restoring the status of low intensity. 

Various phases of the cycle are illustrated by 

the profiles in figure 12 (p. 33).

When combinations of velocity, resistance, 

and depth similar to those causing antidunes 

are made for a stream flowing through a rigid 

straight channel, without movable debris, the 

water develops surface waves, and these travel

2 Eger, Dix, and Seifert (Zeitschr. Bauwesen, vol. 56, pp. 32.5-328) 

found that under certain conditions dunes were developed in a sand of 

uniform grade but not in a finer sand composed of several grades. In 

our experiments the smooth phase of transportation had greater range 

with mixed grades than with single grades.
244 TRANSPORTATION OP DEBRIS BY RUNNING WATER.

downstream faster than the current. They 

are initially rhythmic, but their period is 

unstable because their velocity of propagation 

varies with their size. A wave with slight 

advantage in size will overtake the one in 

front of it, and then the two will unite, making 

a wave with still higher velocity. Thus the 

system tends, for a time at least, toward 

reduction of the number of waves and increase 

of the wave interval. As the waves grow by 

composition, their fronts steepen and the 

culminating phase is that of a "roll wave," or 

bore.1 It may be noted, in passing, that the 

pulsations frequently observed in the overfall 

of a dam are probably rhythms of this type.

It may be assumed, at least tentatively, 

that all the dominant rhythms observed in 

straight conduits are initiated at the intake. 

Those associated with low velocity and an 

immobile bed appear to be stationary, but 

with a mobile bed they develop dunes, and 

they then travel downstream with the dunes. 

Those associated with high velocity and an 

immobile bed develop waves of translation 

which travel downstream. With a mobile bed 

antidunes are developed, and surface waves 

travel with them upstream. It is possible 

that the antidune waves coexist with the 

waves of translation, and that the cycle of 

intensity in the antidune phenomena results 

from the interactions of the two systems.

If there is warrant for the various correlations 

above indicated, the phenomena developed by 

our experiments in connection with modes of 

debris transportation afford a basis of classification for a considerable body of water 

rhythms. What we have called the smooth 

phase of traction marks a critical phase of 

water flow separating two types which are 

characterized by dominant rhythms, and the 

two dominant rhythms are in some way antithetic. The nature of their antithesis is not 

known to me, nor is the character of either 

rhythm; but it appears that the experimental 

and analytic study of the rhythms constitutes 

a field of research which is at the same time 

promising and important. I assume that the 

definite rhythms of water by which, when 

debris is present, the dunes and antidunes are 

caused are susceptible of analytic treatment,

1 See Cornish, Vaughan, Progressive waves in rivers: Geog. Jour. 

(London), vol. 29, pp. 23-31,1907.

and I believe that the experiment trough 

affords the means of preliminary delineation 

and ultimate verification.

THE VERTICAL VELOCITY CURVE.

No systematic study was made of the distribution of velocities within the streams of the 

laboratory, but incidentally, in connection with 

several different minor inquiries, the vertical 

distribution of velocities in the medial plane 

was recorded; and it happens that some of 

these records serve to illustrate the dependence of that distribution the vertical velocity 

curve on certain conditions. The observations were made with the Pitot-Darcy gage.

The first to be reported were made in connection with a comparison of the free outfall 

(fig. 1, p. 19) and the contracted outfall (fig. 3,

FIGURE 73. Modification, of vertical velocity curve by approach, to 

outfall.

p. 25). The trough used with free outfall had 

smooth sides and bed, was 1 foot wide, and 

was set level. The stream of water, 0.734 

ft.3/sec., carried no debris. Its surface slope 

gradually increased toward the outfall, and the 

mean velocity increased with the fall of the 

surface. Velocities were measured at three 

points, respectively, 5.5 feet, 2.3 feet, and 0.8 

foot from the outfall. Their curves, given in 

figure 73, show that the acceleration as well as 

the velocity increased as the outfall was approached, and that there was a coordinate 

modification of the shape of the curve. In the 

first curve it is evident, despite a discordance 

among the determined points, that the level of 

maximum velocity is somewhat above middepth; in the second the maximum is near middepth; and in the third it is below. The acceleration, for which an integrated expression 

appears in the horizontal space between curves,
PEOBLEMS ASSOCIATED WITH RHYTHM. 245

seems to have affected the lower part of the 

current more than the upper.

The trough used with contracted outfall was 

0.92 foot wide and was contracted, in the 

manner shown in figure 3 (p. 25) to 0.30 foot. 

The oblique walls producing the contraction

-{>

4-

Q_

V

0.2

.1

r J-5

. \ 0.7$

\ ! ° 2S

' 1 1

 /

,/ J

\

\

V

, )

Velocity

FiGUBE 74. Modification of vertical velocity curve by approach to 

contracted outfall. - - -

were 3 feet long. The first velocity station 

was 1.5 feet from the outfall, at a point where 

the width was 0.64 foot; the second and third 

were at 0.75 and 0.25 foot from the outfall, 

corresponding to widths of 0.47 and 0.35 foot. 

The velocity curves, given in figure 74, show 

the descent of the plane of maximum velocity 

from a position slightly above middepth to

FIGUKE 75. Plan of experiment trough with local contraction. The 

letters show stations at which velocities were observed, and are repeated in figures 70 and 77. Scale, 1 inch=2 feet.

near the bottom. The acceleration, in this 

case connected with contraction of channel, 

as well as with the release at outfall from the 

channel resistance, is a function of depth.

In the third arrangement a trough 0.92 foot 

wide, with a slope of 0.58 per cent, was contracted at one point in the manner indicated 

in figure 75. This gave to the stream such a

.FiG UKE 76. Profile of water surface in trough shown ua figure 75. Scale 

Iinch=2f3et.

profile as is -sketched in figure 76. Velocities 

were measured at four points, the distances 

from the point-of extreme contraction being 

2.0 feet, 1.0 foot, 0.5 foot, and 0, and the corresponding widths of current 0.92, 0.55, 0.37, 

and 0.19 foot. The velocity curves are plotted

.in figure 77. The level of maximum velocity 

is at or near the surface at the point of initial 

contraction, A, and then drops quickly, being 

below 0.9 depth in the narrowest strait,. D. 

In this case also the acceleration appears to 

increase in regular manner with distance from 

the surface, ........

To avoid the peculiarities observed near the 

outfall, as well as those appropriate to intake

A JJ

FIGUKE 77. Modification'of .vertical-velocity curve by local 'Contriifttlon of channel. The letters indicate positions of velocity stations in 

trough. (See fig. 75.) . - 

conditions, all other determinations of the 

curve were made near midlength of the trough, 

the ordinary distance from the outfall being 

17 feet and the least distance 13 feet.

In figure 78 are three curves illustrating 

the influence of slope of channel. The trough 

was 1.96 feet wide and its sides and bed 

were smooth; the discharge was 1.119ft.3/sec. 

Curve A shows velocities 17 feet from the outFIGUHE 78. Modification of vertical velocity curve^when mean velocity 

is increased by change of slope.

fall, with the trough level; B corresponds to a 

slope of 0.26 per cent and C to 0.56 per cent. 

With the trough level, maximum velocity 

occurs at about 0.75 depth, and with both 

inclinations of trough its position is indicated 

at from 0.2 depth to 0.3 depth. As curve 0 

differs in type from all others obtained, and as 

it may be made harmonious by rejecting a 

single observation, it is thought probable that 

the dotted line better expresses the fact. Its
246 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

substitution throws the maximum to the surface of the water and makes a consistent series 

'of the three curves.

In another series of observations the trough 

remained horizontal while the discharge was

FIGURE 79. Modification of vertical velocity curve when mean velocity 

is increased by change of discharge. The corresponding discharges 

are indicated in ft.3/sec.

Taried. The width was 0.66 foot, and the discharges 0.182, 0.363, 0.545, 0.734, and 0.923 

ft.3/sec. The resulting curves (fig. 79), with 

exception of that for the smallest discharge, 

form a consistent series, the level of maximum 

velocity rising slowly with increase of discharge. 

In this case also the discordant curve may be 

brought into harmony by the rejection of a

Velocity

.FIGURE SO. Modification of vertical velocity curve by changes in the 

roughness of the channel bed. Curve 1, paraflin; 2, roughness of debris 

grade (A); 3, roughness of grade (D); 4, roughness of grade (F).

single observation, and a substitute curve is 

suggested in the diagram by a broken line.1

In another series the texture of the channel 

bed was made to vary. Constant features were 

the discharge, 0.734 ft.3/sec.; the width, 0.92 

foot; the slope, 0.58 per cent; and the texture 

of the channel sides, which were planed and 

unpainted. The varieties of bed texture were 

(1) paraflin, coating a smooth board; (2) a

i In this instance, in the one before mentioned, and also in the case of 

three curves in figure 80 the aberrant point records a measurement which 

was made very near the surface of the water. In such positions the 

constant of the Pitot-Darcy gage has a special,value, and it is on the 

whole probable that the apparent errors of observation are occasioned 

by an error of the rating formula. The matter is discussed in 

Appendix A.

pavement of debris of grade (A); and (3 and 4) 

similar pavements with grades (D) and (F). 

The observed curves, given in figure 80, indicate that the resistance occasioned by a rough 

bed retards the whole current but retards the 

lower parts in greater degree than the upper, 

so that the level of maximum velocity is raised. 

The amount of retardation is greater as the 

texture of the bed is coarser.

In figure 81 are two curves illustrating the 

influence of load on the vertical distribution of 

velocity. One curve was observed in a stream 

without load, flowing over a smooth and horizontal bed, the other in the same stream when 

carrying a load of 17 gm./sec. on a self-ad justed

Velocity

FIGUKE 81. Modification of vertical velocity curve by addition of load 

to stream, with corresponding increase of slope. A, without load; B 

with load.

slope of 0.64 per cent. To judge from the data 

shown in figure 78 (p. 245), the effect of slope 

alone would be to double or nearly double the 

mean velocity, but the actual increase was only 

13 per cent. The acceleration due to slope was 

almost wholly neutralized by the retardation 

due to roughness of bed and to the work of 

traction. The retardation had also the effect 

of raising the level of maximum velocity.

FIGURE 82. Modification, of vertical velocity curve by addition and 

progressive increase of load.

Similar contrasts are shown in figure 82, 

where the curve for an unloaded stream is compared with curves for the same stream when 

transporting three different loads. The constant factors in this case are: Width of channel,
PKOBLEMS ASSOCIATED WITH RHYTHM. 247

0.66 foot; discharge, 0.545 ft.3/sec.; grade of 

debris, (C). The variables are as follows:

Load..........gm./sec.. 0 38 53 194

Slope of bed...per cent.. 0 0.38 0.56 1.14

Depth at 17 feet from outfall..............feet.. 0.353 0.313 0.303 0.226

Mean velocity...ft./sec.. 2.57 2.64 2.73 3.66

Level of maximum velocity; measured from the 

surface aa a fraction of 

depth................. 0.7 0.1 0 0

Letter indicating curve in 

figure 82............... A B C D

With the slope of 0.38 per cent the sand moved 

in dunes; with 0.56 per cent the phase of traction was transitional between the dune and the 

smooth; with 1.14 it was transitional between 

the smooth and the antidune. The conspicuous change associated with the addition of load 

is the raising of the level of maximum velocity, 

and this is correlated also with increase of slope, 

increase of mean velocity, decrease of depth, 

and modification of the mode of traction.

The plotted points for velocities near the 

channel bed are irregular when the observations 

were made above a bed of loose debris, and 

little use has been made of them in drawing the 

curves. As previously mentioned, the presence 

of the gage caused a deflection of the lines of 

flow and the formation of a hollow in the bed. 

Not only was it impossible to observe with 

accuracy the relation of the instrument to the 

normal position of the bed, but the velocity' 

observed was higher than that normally associated with the depth at which the instrument 

was placed. (See Appendix A.)

In most of the groups of curves the variations 

of form are associated with0 simultaneous variations of so many conditions that the nature of 

the control is not evident. For satisfactory 

interpretation a fuller series of observations 

seems to be required, but certain inferences may 

be drawn from those before us.

Many of the peculiarities of form are connected with the position of the level of maximum 

velocity. The movements of the maximum in 

relation to depth of current are of two kinds. It 

rises with increase of depth when that increase 

is caused by increase of discharge (fig. 79). It 

falls with increase of depth when that increase is 

independent of discharge (figs. 78, 81, and 82). 

Apparently depth, considered by itself, is not 

a factor of control. The maximum, rises with 

increase of mean velocity when that increase is

due to increase of discharge (fig. 79), or of 

slope (fig. 78), but falls with increase of mean 

velocity when the increase is due to lessened 

resistance of the channel bed (figs. 73, 80, 81, 

and 82). Apparently mean velocity, considered by itself, is not.a factor of control. If we 

give. attention to the three factors on which 

depth and mean velocity chiefly depend 

namely, discharge, slope, and bed resistance 

a more consistent relation is found. Variations 

of discharge affect only the group of curves in 

figure 79, and there the maximum rises with 

increase of discharge. Slope affects the groups 

in figures 78, 81, and 82, and in each case the 

maximum rises with increase of slope. Bed 

resistance affects the groups in figures 80, 81, 

and 82, and in each case the maximum rises 

with increase of resistance.

The lowering of the level of maximum velocity as the point of outfall is approached (fig. 73) 

is a harmonious feature, but in that case there 

is substituted for progressive reduction of bed 

resistance an abrupt cessation of all channel 

resistance. The resulting acceleration is propagated upstream, and its amount has a vertical 

distribution connected with pressure. In the 

case of contracted outfall (fig. 74), there is 

added an effect of convergence, which still further illustrates the graduation of acceleration 

in relation to pressure. The influence of contraction is important in other connections, but 

need not be further discussed in this place.

The influence of outfall may extend to a 

considerable number of the curves here figured. 

In most of the experiments made without 

contraction at outfall there was progressive 

decrease of depth and increase of mean 

velocity, from some point near the head of the 

trough to its end. This was most marked 

when the bed of the trough was horizontal 

(figs. 73, 7SA, 79, 81 A, and 82A). Reasoning 

from the observed fact that acceleration increases with depth, I think it probable that 

under such conditions the level of maximum 

velocity lies lower than it would with a uniform 

mean velocity.

Returning to the consideration of discharge, 

slope, and resistance, we may note that the 

variable resistance with which variations of 

the curve have been definitely correlated is bed 

resistance. In all the experiments the channel 

sides had the same texture, so that the side 

resistance was approximately proportional to
248 TRANSPORTATION OF DEBEIS BY RUNNING WATER.

the depth. The level of maximum velocity 

thus has the same relation to side resistance as 

to depth; it sometimes rises and sometimes 

falls when side resistance is increased. While 

a probability exists that side resistance influences the position of the maximum, the nature 

of its influence is not shown by the observations 

under consideration.

Discharge and slope, or the energy factors 

which they help to measure, urge the water 

forward, and their influence is applied to the 

whole stream. Bed resistance holds the stream 

back but is applied to its base only. The 

obvious tendency of these forces is to make the 

upper part of the stream move faster than the 

lower and produce a velocity curve with maximum at the water surface. This tendency is 

opposed by some other factor, unknown, which 

tends to depress the level of the maximum. 

Whatever that other factor may be, it loses in 

relative importance when discharge, slope, or 

bed resistance is increased.

A noteworthy feature of the curves is a 

tendency to change in character near the bed. 

The observations are not so precise nor so full 

as to afford a distinct characterization of the 

change, but there can be little question of its 

existence. It would appear that the peculiar 

conditions near the bed give great local importance to some factor of velocity control which 

is elsewhere of minor importance. Attention 

may also be directed to the fact that none of 

the curves resembles an ordinary parabola 

with horizontal axis. Had these been the 

vertical velocity curves to which mathematical 

formulas were first fitted, the equation of the 

parabola would not have been used.

The statement, above, that the factor 

tending to depress the level of maximum 

velocity below the water surface is unknown 

is perhaps rash, for several theories as to its 

nature have been advanced with confidence. 

To put the matter more cautiously it has 

seemed to me that each theory of which I have 

read was effectually disposed of by the discussion which it aroused. However that may be, 

there is certainly room for another suggestion, 

and this I proceed to offer.

Reynolds * arranged an experiment in which 

a liquid was made to flow over another denser 

liquid, the two being immiscible. Below a ceri Roy. Soc. London Philos. Trans., vol. 174, pp. 943-944, 1883.

tain velocity the surface of contact was smooth, 

but above the critical velocity the surface was 

occupied by a system of equal waves, which 

moved in the direction of flow but more slowly. 

With miscible liquids, or with two bodies of 

identical liquid, the waves are replaced by vortices. Under some conditions the vortices are 

as regularly spaced as the waves, but usually 

they are less regular, and various complications 

arise. The development of such vortices may 

readily be watched on a river surface wherever 

adjacent parts move with quite different velocities or move in opposite directions. Such vortices have vertical axes, and their direction of 

rotation is determined by the differential motion of the adjacent currents. If we conceive 

the water of a vortex as a body between parallel 

and opposed currents, then its direction of rotation is due to a mechanical couple contributed 

by the currents.

Transferring attention to the longitudinal 

vertical section of a stream, we find a couple of

FIGURE 83. Ideal longitudinal section of a stream, illustrating hypothesis to account for the subsurface position of the level of maximum 

velocity.

which one element is the general forward movement of the current and the other is the bed 

resistance. These tend to produce and maintain vortices with horizontal axes transverse 

to the channel and with forward rotation that 

is, with the rotation of a wheel rolling forward in 

the direction of flow. To visualize these features, figure 83 gives an ideal section of a stream, 

with flow from left to right, and the ovals A, B, C 

represent a system of forward-rolling vortices. 

The arrows within an oval show direction of 

rotation, and it is important to recognize that 

the motions they indicate are referred to the 

center of the vortex, or to the vortex as a 

wbole, and not to the fixed bed of the channel. 

With reference to the bed all parts of the vortex 

are moving toward the right, the lower part 

merely moving slower than the upper.

The tendency of vortices toward circular 

forms leaves certain tracts of the section unoccupied by the system of vortices. Consider the 

tract D, bounded below by the bed and above
PROBLEMS ASSOCIATED WITH EHYTHM. 249

by parts of vortices A and B, and give attention 

to the motions by which the water in the tract 

is surrounded and influenced, taking care to refer each motion to the middle of the tract-itself. 

Thus referred, the motion of the adjacent part 

of the rear vortex A is to the left and downward, that of the forward vortex B is to the 

left and upward, and that of the bed is to the 

left. The motions are indicated by arrows. 

The influences of the vortices tend to make the 

water about D rotate backward, while the influence of the bed tends to make it rotate forward. 

The result is not a priori evident, but may be 

assumed to be something different from simple 

rotation. Its possibilities will again be referred to.

Now consider the tract E, bounded below by 

parts of A and B and above by the water 

surface, and give attention to the motions by 

which its water is affected. Motions being 

referred as before to the tract itself, that of 

vortex A is to the right and downward, and 

that of vortex B is to the right and upward. 

Above is the motion of the air, which, in the 

absence of wind, is to the left, as indicated by 

an arrow. The three influences to which the 

tract of water is subject all tend to give a 

backward rotation, as indicated in a similar 

position at F. The vortex F is secondary to 

the ^4, B, 0 system and rotates in the opposite 

direction. If its motions be referred to the 

fixed stream bed, it is evident that the water 

in its upper part moves in the direction of the 

current less rapidly than the water in its lower 

part. The existence of such a vortex therefore 

tends to reduce the average velocity at the 

surface of the current and increase it at some 

lower level.

Abandoning now the specific and ideal, case, 

we may state the hypothesis in general terms. 

Among the important causes of vortical motion 

in a river or other stream is the mechanical 

couple occasioned by the general forward 

motion of the water in conjunction with the 

resistance of the bed. This tends to form 

vortices with horizontal axes and forward roll; 

and the tendency is probably strongest in the 

lower part of the stream. In a space adjacent 

to two forward-rolling vortices there exists a 

tendency toward the development of a secondary, back war d-rolUng vortex, but this tendency 

is apt to be nullified by other and adverse 

influences except in the upper part of the

stream. The free surface of the water does 

not oppose the development of such reversed 

vortices. Wherever reversed vortices abound 

the velocity at the surface (averaged with 

respect to time) is less than at some level below 

the surface.

The hypothesis as stated has no stationary 

element, but the phenomena of incipient dunes 

show that .in certain cases repetitive motions 

are associated with stationary space divisions. 

A supplementary .suggestion assigns these 

repetitive motions, or their initial phases, to 

the triangular space D in the ideal diagram, 

figure 83. The forces tending toward rotation 

in that space are antagonistic; and the suggestion is (1) that they produce some sort of 

alternating movement with regular periodicity, 

and (2) that the time interval of this movement, in combination with the forward movement of the major vortices A, B, 0, yields a 

stationary space interval.

An investigation based on this line of suggestion and designed to test it should lead also to 

an explanation of the observed changes in the 

method by which traction is accomplished. 

The dune method is associated with a depth 

which is large in relation to the mean velocity 

and with a moderate bed resistance. It is 

replaced by other methods in consequence of 

(1) a reduction of depth, or (2) an increase of 

velocity, or (3) an increase of resistance. 

Reduction of depth diminishes the space for 

development of the hypothetic vortices. Increase of velocity or of resistance tends to 

enlarge their pattern. In either case the 

coercion of the water- surface restricts the 

freedom of vortical movements and imposes 

conditions tending to modify their system.

THE MOVING FIELD.

The competent investigator is resourceful in 

the creation of new apparatus and methods as 

the need for them arises. While fully conscious 

of this fact, and of the further fact that no 

device is of assured value till it has been tried, 

I yet can not forbear to mention, for the benefit 

of others, a method which the Berkeley experience leads me to think valuable for the study of 

the internal details of a current of water.

In a current limited by the sides of a narrow, 

straight trough transverse movements are 

largely suppressed, so that most of the action 

can be learned from observation of what takes
250 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

place in vertical planes parallel to the axis. 

The movements in a vertical plane may be 

exhibited by giving to the trough a glass side 

and by giving to the water in that plane exclusive illumination. When one of our laboratory currents, being illuminated from above, was 

viewed from the side, small particles in suspension were seen to be conspicuous, so conspicuous, in fact, as to resemble the motes in a sunbeam. Some of the particles were shreds of 

wood fiber worn from the trough, and these 

could be more easily followed by the eye 

because of their distinctive forms. Impressed 

by this phenomenon, I am confident that water 

movements in a vertical plane can be effectively revealed by giving to the water a suitable 

amount of suitable suspended material and by 

giving to the selected plane a brilliant and exclusive illumination.

With the aid of such simple arrangements 

much may be seen, but measurement will be 

difficult; and vortical movements may not be 

easily discriminated from those which are 

merely sinuous. It is believed, however, that 

both these results may be achieved by aid of 

the moving field. In a very simple form this 

device was employed by us in the study of 

processes of traction (p. 27), and despite the 

crudity of the apparatus it was found to be 

highly efficient. An experiment trough having 

in its side a glass panel, A A in figure 4, bore a 

sliding screen, B, in which was an opening, 0. 

Moving de"bris was watched through the opening at the same time that the screen and opening were moved in the direction of the current. 

To the field of view was thus given a horizontal motion, and that motion was, in effect, 

subtracted from the motions of the objects 

observed. The apparent motions were motions 

in relation to the moving field, and not to the 

fixed trough. No provision was made for determining the velocity of the field, nor for 

measuring the apparent motions of the objects 

viewed; but even without these devices for 

quantitative work, the possibilities of the 

method of observation were sufficiently evident. With the necessary supplementary 

devices, the moving field promises to measure 

the horizontal and vertical components of 

motion in any part of the vertical section, and

it thus makes possible the complete delineation 

of velocities and directions of details of current, 

so far as those details may be exhibited in 

longitudinal vertical sections. By giving to 

the field a suitable velocity it should be possible to see a traveling vortex which rotates in a 

vertical plane, just as vortices of horizontal 

rotation are seen on the surface of a stream.

A notable defect in the Berkeley arrangement 

was the requirement that the observer move 

his head in unison with the moving field. This 

interfered with steadiness and also limited narrowly the space covered by an observation. It 

could be remedied by substituting for the slide 

a car which should carry both observer and 

peephole at a determined rate.

Another suggested arrangement places the 

eye of the observer at a fixed telescope and

FIGURE 84. Diagrammatic plan of suggested moving-field apparatus.

moves the field by means of a rotating mirror. 

This is illustrated by figure 84, where the 

trough T is shown in plan. The telescope E 

views the trough through the mirror, which is 

pivoted on a vertical axis at M. As the plane 

of the mirror rotates from a to & the field commanded by the telescope moves from A to B. 

It is evident that if the angular velocity of the 

mirror is constant the linear motion of the field 

along the trough will be relatively fast at A 

and B and relatively slow at G. The error thus 

arising may be avoided by some device of the 

nature of linkage. It will be corrected with 

sufficient approximation if the axis of the mirror be controlled by a rigidly attached arm 

MD, which is in turn controlled by an arm 

about one-third as long, the two having sliding 

contact at D and the short arm revolving uniformly about a vertical axis at F.
APPENDIX A. THE PITOT-DARCY GAGE.

SCOPE OF APPENDIX.

Many measurements of velocity were made 

with the Pitot-Darcy gage, but as only a 'few 

have been finally utilized in the preparation 

of the report, a discussion of the instrument 

seemed not appropriate to the main text. 

Certain phases of our experience, however, 

are thought worthy of record because they 

have practical bearing on the utility and the 

use of such gages, and these are the subject of 

the appendix.

FORM OF INSTRUMENT.

Darcy developed the Pitot tube by adding 

a second tube, differently related to the current, and by connecting the two above with a 

chamber from which the air was partly exhausted. The water columns in the two tubes 

were thus lifted from the vicinity of the water 

surface to a convenient position, where their 

difference in height could readily be measured. 

In the gages constructed for our use the 

aperture of one tube was directed upstream 

and that of the other downstream. The 

tubes were borings in a single piece of brass, 

which was shaped .on the outside in smooth 

contours, designed to interfere the least possible 

with the movement of the water. The form 

first given was afterward modified, and figure 

85 shows the third and last design, with which 

most of the work was done. The openings 

had a diameter of 0.1 inch. At the opposite 

or upper end of the brass piece were stopcocks, 

and above these connection was made with 

rubber tubes, which led to the complementary 

part of the apparatus, where the difference in 

height of the two water columns was observed. 

It is convenient to call the member exposed to 

the current the receiver, and the complementary member the comparator.

In the comparator were two glass tubes, 

straight and parallel, with internal diameters 

of about 0.8 inch. At the top they were 

connected by an arch, and at the summit of the 

arch was a branch tube, with a pet cock, used 

in regulating the amounts of air and water.

At the bottom they communicated with the 

rubber tubes through a brass piece, in which 

were two stopcocks, connected by gearing so 

as to open and close together. These parts 

were mounted on a board which also carried a 

scale of inches and decimals. A sliding index 

was arranged so that it could be set by the 

meniscus of a water column and its position 

then read on the scale; and there was a fixed 

mirror behind the tubes to aid in avoiding 

error from parallax. The board was supported in an inclined position, the slope given 

to the tubes and scale being that of 2^- horizontal to 1 vertical. This had the effect of

O.I 0.2 Foot

FIGURE 85. Longitudinal section of lower end of receiver of Pitot-Darcy 

gage No. 3, with transverse sections at three points.

making the movements of the columns 2.69 

times as great for the same change of pressure 

as they would be if the tubes were vertical.

In preparing for observation, the internal air 

pressure was so adjusted that the columns 

stood near the middle of the scale. The 

receiver was then held, by a suitable frame, in 

the selected part of the current, and the stopcocks were opened. In the glass tube connected with the receiver opening facing upstream the column rose; in the other tube it 

fell. When the full effect of the current had 

been realized, the stopcocks at the bottom of

251
252 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

the comparator were closed, and the heights 

of the columns were then read.

The use of the gage to measure velocities 

close to the bed of debris proved impracticable 

because the presence of the receiver modified 

the movement of the water and thereby modified the shape of the bed. (See pp. 26,155.) 

This effect could have been reduced by using a 

different form of receiver. Darcy and Bazin l 

bent the tubes at the bottom in such a way 

that one or both openings met the water at 

some distance upstream from the vertical part 

of the tubes, and it is probable that the adoption of their design would have diminished the 

difficulty, although it could not have removed 

it. With such a design, however, the practicable forms for the second opening relate it to 

the piezometer, and the advantage of the downstream opening is lost.

That part of the design of the comparator 

which consists in the inclination of tubes and 

scale is not to be recommended. It refines by 

magnifying the reading, but it introduces possibilities of error in other ways. If the tubes 

are not straight or are not equally inclined, an 

error is occasioned which does not enter if they 

and the scale are vertical. Evidence of such 

error was found in the fact that the still-water 

or zero-velocity readings of the two columns 

were not always identical, but no ready means 

of correction was discovered.

Another source of error was detected in inequalities of sectional area of the glass tubes of 

the comparator. To show the nature of this 

error, let us assume that the pressure of the 

current at the upstream opening of the receiver 

is exactly equal to the negative pressure, or 

suction, at the downstream opening. If the 

glass tubes are of uniform and equal bore, one 

column moves upward just as much as the other 

moves downward, and the volume of air above 

the columns is unchanged. Now, assume that 

the tube containing the rising column has the 

greater diameter. It is evident that equal 

movement of the two columns will displace 

more air in the one tube than it will provide 

space for in the other, and the pressure of the 

confined air will be thereby increased. The 

effect of the increased pressure will be to lower

i Bazin, F. A., Recherches experimentales sur 1'dcoulement de I'eau 

dans les canaux ddcouverts: Acad. sci. Paris Mdm. math, et phys., 

vol. 19, p. 49, PL IV, 1865. This memoir was published also as part of 

"Recherches hydrauliques," by H. Darcy and F. A. Bazin.

both columns. There is also a secondary effect 

of small amount connected with the fact that 

the pressure of the confined air plus the head 

of water between the tops of the columns and 

the surface of the stream, on the one hand, and 

the atmospheric pressure, on the other, are in 

equilibrium, but into this we need not here 

enter.

As a means for the discussion of these errors 

the tubes were calibrated, by Prof. J. N. Le 

Conte, in the following manner: The tubes 

being closed at the bottom, a weighed quantity 

of water was introduced into one and the height 

of its column was read. By i epeated additions 

of water and repeated readings the volume? of 

divisions of each tube were thus measured, the 

divisions being approximately 1.5 inches in 

length. In similar manner the volume was 

measured of the space above the straight tubes 

to the pet cock. The average sectional area of 

one tube was found to be 2.5 per cent greater 

than that of the other. The sectional area in 

the larger tube was found to vary through a 

range of at least 2.6 per cent, and the range for 

the smaller tube was 4.5 per cent.

A table of corrections to readings was 

computed- from the data of calibration, and 

this table was practically applied. In comparing the rise of one column with the associated fall of the other the greatest correction 

applied amounted to a little less than 1 per 

cent. In the determination of velocity the 

largest correction applicable for this reason was 

about 0.3 per cent. These corrections, however, pertain only to the discussions of the instrument in the following paget>. In the 

ordinary work of the gage they were not applied, because it was found that the errors of 

this class were practically eliminated when the 

same methods were employed in the preparation and in the use of rating formulas. The 

matter is mentioned here chiefly because errors 

from unequal tube caliber, which may sometimes prove important, appear not to have 

been allowed for in the discussions of PitotDarcy gages.

HATING METHODS.

The first and second gages were rated by the 

method of floats; the third was twice rated by 

the running-water method and several times' by 

the still-water method. The floats used weie
THE PITOT-DAECY GAGE. 253

vertical cylinders of wood so adjusted that the 

submerged depth was twice the distance below 

the water surface of the apertures of the receiver. Some of the still-water ratings were 

made at the Geological Survey's rating station 

at Los Angeles, where a car running on a track 

at the side of a reservoir drew the receiver 

through the water of the reservoir. The others 

were made later in the long trough of the Berkeley 

laboratory, the car in this case running above 

the water. In the application of the runningwater method a measured discharge was passed 

through, a rectangular trough and a survey of 

velocities throughout a cross section was made 

by means of the gage. By using in this survey 

the rating formula obtained by the still-water 

method, and then comparing the mean velocity 

thus computed with that computed from the 

discharge and sectional area, a correction is 

obtained which may be applied to the stillwater rating formula. Only a single comparison of this sort is practically available in connection with our instruments, and the terms of 

this are:

Ft./sec. 

Mean velocity by discharge and area.............. 1. 98

Mean velocity by gage (with still-water rating).2.08±0.07

The resulting correction to the still-water 

ratings is 5 per cent, but this determination 

has small value because of the large probable 

error of one of the compared determinations.

The fact that the apparent correction is 

small is in accord with a property of the gage 

independently observed. By reason of the 

sinuosity of flow lines in a stream, the directions of motion are not parallel to the axis of a 

straight channel. Therefore, a current meter 

which records the velocity in the direction of 

flow instead of the component of velocity parallel to the channel axis yields an overestimate 

of mean velocity. Some Pitot-Darcy gages 

have been found to overestimate velocity when 

placed obliquely to the direction of flow, and 

for such the correction would be large. It was 

found, however, that the Berkeley gage when 

placed somewhat obliquely to the current gave 

a lower reading than when facing it squarely, 

and throng]i this property it tended automatically to correct its readings for obliquity 

of current. If the correction were perfect, the 

still-water rating and running-water rating 

should be the same.

EATING FORMULA.

In the still-v/ater ratings velocities independently determined were compared with 

resulting changes in the water columns of the 

comparator. Starting from the same level, 

one column rose with increase of speed and the 

other fell. Except for the influence of modifying conditions the changes should be equal, 

the positive velocity head being of the same 

amount as the negative velocity head. To test 

this point various sets of observations were 

plotted on section paper, the readings of the 

rising column being taken as abscissas and the 

readings of the falling column as ordinates. In 

most cases the plotted points fell well into line, 

and there was no question that the line represented by them was straight. That is, the 

true ratio of the negative pressure at the downstream opening of the receiver to the positive 

pressure at the upstream opening was constant, 

under the conditions of this particular series of 

trials. The value of the ratio was found to vary 

with conditions, and the several values found 

are so near unity as to confirm the theoretic 

belief that unity is the normal value.

In considering the variations of value it is 

first to be noted that the gage with which the 

observations were made, No. 3, being symmetric, could have either opening turned upstream, and it was in fact used both ways, with 

record of its position. But its symmetry was 

only approximate and therefore the two positions gave different results. Used in one way 

it will be called No. 3a, and in the other way 

No. 3b. It also happened that between the 

date of the Los Angeles ratings and that of the 

Berkeley ratings the receiver was accidently 

marred at one of its openings, and though its 

form was afterward restored as nearly as 

possible, some difference remained which affected its constants. The values of the pressure 

ratio are accordingly arranged in four groups 

in Table 81. Two of these groups also are 

subdivided with reference to the position of 

the receiver in relation to the perimeter of the 

current.

The ratio was notably larger after the accident than before, and the change was greater 

for 3a than for 3b. The greatest value of the 

ratio was given by trials in which the receiver 

ran close to the side of the reservoir, which in
254 TRANSPORTATION OF DEBRIS BY RUNNING WATER,

that case was a plank trough. The interpreta- Substituting in (118), we have

tion of these results will be considered in

another connection. H H1 = 32.28 fc Vu

TABLE 81. Ratio of the suction at one opening of the Pilot- or, making K = 32.38 k, 

Darcy gage to the pressure at the other.

.(120)

Place of rating.

Los Angeles....

Do.........

Do.........

Do.........

Conditions.

Shallow, near side. ........

Ratio, suction to pressure.

Gage 3a.

0.993 ±0.015 

1.038± .007

1.052± .004

Gage 3b.

1.005±0.005 

1.010± .002 

1.016± .003 

1.014± .005

Let Ji represent the vertical space through 

which the column of water is raised by pressure 

from velocity V on the upstream opening of the 

receiver, and \ the simultaneous depression of 

the column connected with the downstream 

opening. Then^ each being assumed to equal 

the velocity head,

V2

-(116)

-(H7) 2?

In practice the full velocity head is not realized 

in instruments of the Pitot-Darcy type, and the2 coefficient determined by rating is less than ~-.

^ 

It is a common experience also in practical

application of hydraulic formulas to find that 

qualification is advisable in other respects. I 

therefore substituted tentatively for (117) the 

formula

(118)

and sought empiric values of u and &. The 

readings of the comparator corresponding to Ji 

and 7it may be called H and H^. The zero of 

the comparator scale being at its lower end, 

the difference between the readings corresponds to the sum of the spaces Ji and Jir The 

readings are in inches, while the unit used for 7i, 

Jil} and g is the foot. Moreover, from the inclination of the comparator, the space between 

the two columns, as read on the scale, is 2.69 

times the vertical space Ji + Ji^ Therefore, the 

product of 12 by 2.69 being 32.28,

The observed quantities being H, H^ and V, 

it was possible to plot on logarithmic section 

paper any series of values of H ^ in relation 

to the associated values of V, and thus compute 

graphically the corresponding values of u and 

K. The values were computed for all series of 

observations represented in Table 81, and they 

are given in Table 82. The mean of the seven 

values of u is 2.00, but their range is notable. 

The deviations from the normal may be ascribed 

in part to accidental errors. In the case of the 

third value, 1.90, and of the seventh, 1.94, the 

plotted positions are so scattered as to admit 

of considerable latitude in the drawing of the 

equation lines, but the control is much stronger 

for the values 2.09 and 2.12, arid these could 

not be greatly reduced without violence to the 

facts of observation. It seems clear that the 

exponent is not wholly free from the influence 

of special conditions.

TABLE 82. Values of K and u in H-H^KVu.

Place of rating..

Do.............

Do.............

Do.............

Do.............

Do.............

Conditions. Gage.

3a

3b

3a

3b

3b

3b

3a

K

0.70

.62

 .72

.52

.70

.46

.66

u

2.00

1 93

1 90

2.09

1 9J

2.12

1.94

For the practical purpose of rating the instrument, however, there is no advantage in 

departing from the normal exponent, and that 

was employed in the preparation of rating 

tables. The formula used for inferring velocities from readings is

(.122)

32.28 (119)

in which

The values of A graphically-computed, are 

given in Table 83, and these values were used 

in the computations of velocities.
THE PITOT-DAKCY GAGE. 

TABLE 83. Values of A in V=A-\/l-l J-J'l and -values of gage efficiency.

255

Place of rating.

Do.......................

Do.......................

Do.......................

Conditions.

Values of A.

GageSa.

1.22±0.02 

1.29± .04

1.26± .03

GageSb.

1.2G±0.01 

1.30± .02 

1.24± .01

1.37± .03

Efficiency of 

gage.

3a. .

O.G7 

.60

.03

3b.

0.63 

.59 

.65 

.53

Ail inspection, of the values of A with due 

attention to their probable errors serves to 

show that their differences are not to be regarded as wholly accidental, but must be ascribed in part to the variation of the instrumental constant with conditions. That the 

nature of the variations may be appreciated, 

the conditions will be more fully described. 

As already stated, gages 3a and 3b are the same 

symmetric instrument, but with opposite faces 

turned toward the current, while the instrument was modified to some minute extent by 

an injury and repair occurring between the Los 

Angeles and Berkeley ratings. Practically 

there were four gages, but so far as the mechanician could make them the four gages were 

identical in form. The reservoir at the Los 

Angeles rating station was broad and deep. 

The course followed by the receiver of the gage 

was several feet from the bank, at least 6 inches 

below the surface, and at least 1 foot above the 

bottom. The plank reservoir used in Berkeley 

was 1.96 feet wide, and the depth of water was 

0.44 foot. In the ratings tabulated as at" middepth" the opening of the receiver was central 

in the cross section of the water. In the " near 

bottom" rating the center of the opening (its 

diameter being 0.01 foot) was 0.02 foot from 

the bottom; in. the "near surface" rating the 

center was 0.03 foot below the surface of the 

water; and in the "near side" rating the center was 0.25 foot above the bottom and varied 

from 0.01 to 0.11 foot in its distance from the 

side, the course and the trough side not being 

quite parallel. The several relations of the 

opening to the water section are shown in figure 86.

There is netting novel in the variation of the 

instrumental constant with minute differences 

in the shape of the receiver. This has been 

observed by all critical users of such instruments, and the custom is well established of

giving a separate rating to each receiver. So 

far as I am aware the variation of the constant 

with the relation of the receiver to the walls of 

the conduit and to the water surface has not 

previously been recorded, and this is a matter 

of considerable moment, for various elaborate

FIGURE SO. Cross section of prism of water in trough, showing positions 

given to gage opening in various ratings.

discussions of the distribution of velocities 

within a cross section have assumed the constancy of the instrumental constant for all positions. As clearly indicated by the tabulated, 

results, the constant A increases when the free 

surface of the water is approached and diminishes when a rigid wall of the conduit is 

approached.

In the application of these results (and of 

cognate results with which it was thought best 

not to burden these pages) to our velocity determinations a graphic table was constructed 

to show the relation of the instrumental constant to the position of receiver in the conduit; 

and this table (fig. 87) supplied the constant 

to be used with each individual observation. 

This procedure had an important influence on 

the interpretation of the running-water rating, 

making the apparent correction for that rating 

less than it would be if the mid-depth constant 

were used exclusively. Its influence on the 

vertical velocity curves (figs. 74 to 82) was to 

give them less curvature near the water surface, 

greater curvature in approaching the channel 

bed, and (sometimes) a somewhat lower level of 

maximum velocity.

In equation (117) -^F2 equals twice the theoretic velocity head. If in equation (118) the 

exponent u be replaced by its mean value 2,
256 TRANSPORTATION OF DEBRIS BY RUNNING WATER.

we have k F2 as an expression for the double 

head actually produced by the gage. The

2 ratio of fc to measures the efficiency of the

instrument in realizing the theoretic head. 

Since A , . _. ,= I- = ^. and the measure of efficiency is

2 32.16 1 _ 0.996

32.28 A2 ' 2g "32.28 A2 ~ AThe values of the measure have been computed and are given in Table 83. The deepwater and mid-depth ratings being assumed to 

represent normal conditions, the mean of the 

corresponding efficiencies, namely, 0.62, may 

stand in a general way for the fraction of the 

theoretic head which is realized by this particular type of the Pitot-Darcy gage.

As the relation between velocity and 

observed head is that between agent and effect, 

it is evident that values of the efficiency ratio 

rather than those of the constant A should be 

compared in any attempt to explain the phenomena of variation. Restating from this 

viewpoint the results of comparative ratings, 

we have: The response of the head to changes 

of velocity is lessened when the receiver of the 

gage is brought near the free water surface and 

is increased when the receiver is brought near

a rigid part of the stream's perimeter. It may 

be surmised that the differences in head are 

connected with the facility with which the 

flow lines of the water are diverted in passing 

around the instrument, regarded as an obstruction. In midcurrent the diversion is resisted

Water surface

1.2 1.3 

Values of A

Bottom or side'

FIGURE 87. Graphic table for interpolating values of A, in V 

A *jH Hi, for observations made with gage 3b in different parts of a 

stream.

by the inertia of surrounding water. Near 

the surface the resistance of water is partly 

replaced by resistance of more mobile air. 

Near the conduit the resistance of water is 

partly replaced by the resistance of an immobile 

solid.
APPENDIX B. THE DISCHARGE-MEASURING GATE AND ITS RATING.

THE GATE.

The gate for measurement of discharge is 

mentioned at page 20, and its general relation 

to other apparatus is shown in figure 2. In a 

section on the measurement of discharge, page 

22, it is briefly described, and the history of its 

use is outlined. Figure 88 is designed to exhibit its relations more fully and shows, the 

arrangement of the general apparatus of the 

laboratory at the time of its calibration.

The gate controlled an opening in the side of 

a vertical wooden shaft, of which the internal 

horizontal section measured 3 feet by 0.5 foot. 

At the top the shaft communicated freely with 

a long, narrow tank, to which water was continuously delivered by'a pump. The surface 

level of water in the tank was regulated by 

means of an overflow weir and a valve above 

the pump and was observed in a glass tube or 

gage, AB, in figure 88, outside of the tank. 

This tube was given a slope of 1 in 10, so that

FIGURE 88. Arrangement of apparatus connected with tlie rating of the discharge-measuring gate.

the movements of its water column afforded a 

magnified indication of changes in the water 

level within the tank.

Details of the gate are represented to scale 

in figure 89. A brass plate, PP, attached by 

screws to the outer face of the shaft, was 

pierced by an opening of 10.5 by 2 inches, the 

longer dimension being horizontal. About 

three sides the edges of the opening were 

beveled outward, leaving a 45° edge at the

' 20921° No. 86 14 17

inner face of the plate. A sliding plate, G, 

rested against the inner face of the fixed plate. 

This was the gate proper, its function being to 

close either the whole or a definite portion of 

the opening and thus regulate the width of the 

issuing water jet. It overlapped by half an 

inch the margins of the opening, above and 

below, and was guided by two brass pieces, 

which appear in sections 1) and c. To the end 

adjacent to the water jet was given a chisel

257
258 TRANSPORTATION OF DEBRIS BY RUNNING WATER,

edge. The brass guides and the adjacent parts 

of the wooden shaft wall were shaped to a 45° 

bevel, which, if produced, would reach the edge 

of the opening in the fixed plate.

Movements of the slide were controlled from 

the outside. A brass rod, 0, firmly attached 

to it and running parallel to its axis (fig. 89dO, 

rested in a frame at the left of. the opening 

(fig. 89ttj c), where its motion was controlled by 

rack and pinion, the rack being cut on the 

upper side of the rod. At the left of the frame 

.and pinion the rod slid along a brass scale 

graduated to inches and tenths; and the gate 

was set for any desired width of aperture by 

bringing an engraved index mark on the rod 

 opposite the proper mark on the scale...

The operation of the gate was found to be 

satisfactory.

The head under which the jet issued, measured from the middle of the opening to the 

water surface in the upper tank, was 6.0 feet, 

plus or minus a small fraction observed by 

means of the gage above mentioned.

PLAN OF RATING.

The method .of rating was volumetric and 

empiric. The gate having been set at a particular graduation, and the discharge having 

been continued until the rate of flow through 

the stilling tank and experiment trough had 

become steady, the outflow of the trough was 

diverted for a measured time into a special

(ci) 

Section throughX-J?

Section through .C-D 

0 5 10

Section through E-F 

15 20 25 Inches

FIGURE 89. Elevation and sections of gate for the measurement of discharge.

reservoir where its volume was measured. The 

special reservoir is shown in figure 88 as "sump 

No. 2," and the diverting apparatus also is 

indicated. The "diverting trough" was pivoted at the remote end and could be turned

 quickly by hand. In one position it received 

the discharge from the experiment trough and 

delivered it to sump No. 1, containing the 

supply for the pump; in another it permitted 

the water to fall into sump No. 2, arranged for 

measurement. The work of rating was performed by J. A. Burgess, and a full report upon 

it constituted his graduation thesis in engineering at the University of California'. The

 details of apparatus and method were arranged

by him, and the f ollowing account of the rating 

is essentially an abridgment of hi) report.

CALIBRATION OF THE MEASURING RESERVOIR.

Sump No. 2, constructed of concrete, was 

approximately rectangular but was not quite 

regular in form. No attempt was made to 

base computations of volume on its linear 

dimensions, but a scale of volumes Was graduated directly. A hook gage was first installed, 

the hook being attached to a vertical rod, to 

which slow motion could be given. An index 

borne by the rod near its upper end followed a 

smooth surface which had been prepared to 

receive a graduation but initially was un-
THE DISCHABGE-MEASUBING GATE AND ITS EATING. 259

marked. The index was so arranged that by 

pressure it could be made to indent the prepared surface, thereby producing a mark of 

graduation. Starting with just enough water 

in the sump to permit the use of the hook, a 

lino of graduation was marked. Then a cubic 

foot of water was added, the hook was readjusted, and a second line was marked. By 

repeating this process a scale was given to the 

gage, the unit of the scale being 1 cubic foot. 

The added units of water were measured in a 

wooden bottle made in the form of a cube 

with the opening at one corner; and the volume 

of the bottle was so adjusted that it contained 

62.3 pounds of water. The capacity of the 

reservoir was about 38 cubic feet.

In the use of the scale thus provided, fractions of a cubic foot were read by means of a 

small free scale of equal parts applied obliquely 

to the space between two lines of graduation.

THE OBSERVATIONS.

The record of an observation included (1) 

the width of gate aperture, (2) the time interval, by stop watch, during which the water 

was delivered to sump No. 2, (3) two readings 

of the hook-gage scale, and their difference, 

giving the volume of water received by the 

sump, and (4) two gage readings of water 

level in the upper tank, one just before and one 

jiist'after the period of volume measurement.

The quotient of the volume of water by the 

time in seconds gave the discharge for the 

indicated width of gate opening. The readings 

of the high-tank water level gave data for a 

correction to the head, resulting in a small 

correction to the computed discharges.

It was found that repeated observations 

vdth the narrower gate openings gave, results 

nearly identical, while the results for wider 

openings showed more variation. The observations with wide openings were accordingly 

multiplied to increase the precision of the 

averages. The variation was ascribed to pulsations of the flowing water originating in the 

stilling tank. The general precision of the

accepted values of discharge, listed in Table 2 

(p. 23), is indicated by an average probable 

error of ±0.2 per cent. The largest computed 

probable error is ±0.4 per cent, being that of 

the discharge for a gate opening of 6 inches.

The general formula for discharge through a 

rectangular orifice when, as in the present case, 

the vertical dimension of the orifice is small in 

relation to the head, is

in which li is the head, measured from the 

middle of the orifice, Id the area of the orifice, 

and c the constant of discharge. The observations give the following values of c for different 

settings of the gate:

Width of gate 

opening.

Inches.

1

2

3

4

5

0

Mean. ....

Value of c.

0.704

. J03

-.667

.677

.677

.684

.686

For a " standard " orifice 2 inches square and 

a head of 6 feet Hamilton Smith's tables * give 

0.604 as the value of c. The inner surface 

about the standard orifice is vertical and plane, 

whereas the surfaces about the orifice of our 

gate were oblique. The oblique guiding surfaces served to increase the velocity at the 

orifice and thus enlarge the constant of discharge. The variation of the constant with 

width of opening is probably connected with 

the fact that the beveled surfaces were not 

symmetrically arranged about the opening. 

On three sides they made an angle of 45° with 

the vertical plane of the orifice, but the edge 

of the slide, constituting the fourth side, was 

beveled at a smaller angle.' The constants of 

the gate were thus affected by the conditions 

of its setting and a new rating would be necessary with a different setting.

i Hydraulics, p. 58.
INDEX.

A- Page. 

Accents, notation by............................................. 96

Acknowledgments............................................... 9

Adjustment of observations...................................... 55-95

Airy, Wilfred, on the law of stream traction..................... 16,102

Alluvium, natural, capacity for................................ 177-178

Antiduues, formation and movement of..................... 31-34,243

Apparatus, descriptions and figures of........................ 1,19,257

Arntzen, Waldemar, work of..................................... 9

B.

Bazalgette, J. W., on flushing sewers............................. 216

Bed, changes in roughness of, effect of, on velocity................ 246

character of, effect of, on flume traction........^............. 206

nature of, in streams and in flumes........................... 15

of ddbris, diagrammatic longitudinal section of.............. 57

profiles of...................................... .=.............. 58

stream, composed of ddbris grains, ideal profile of............. 155

contoured plot of......................................... 198

Berlin, laboratory of river engineering at......................... 16

Blackwell, T. E., observations by, on velocity competent for flume

traction.............................................. 216

JBlasius, H., on deposits resembling dunes........................ 232

on rhythmic features of river beds..................... footnote, 31

Blue, F. K., experiments by, on flume traction................ 217-218

Briggs, Lyman J., acknowledgments to.......................... 9

and Campbell, Arthur, work of, on viscosity as affected by

suspended matter.................................... 228

Brigham, Eugene C., and Durham, T. C., work of, on viscosity as

affected by suspended matter....................... 228

Burgess, J. A., work of............................................ 9

C.

Campbell, Arthur, and Briggs, L. J., work of, on viscosity as

affected by suspended matter....................... 228

Capacity, definition of ........................................... 10,35

for flume traction, in a semicy.lindric trough................. 214

in relation to discharge................................... 209

in relation to fineness.................................... 210

in relation to slope....................................... 208

of mixed grades.......................................... 212

tab) e of adjusted values of............................. 204-206

in relation to form ratio.................................. 213

for stream traction, in relation to depth.................... 164-168

in relation to discharge......................... 137-149,233-235

in relation to fineness ............................. 150-154,235

in lelation to form ratio........................... 124-136,236

in relation to slope ................................. 96-120,233

initiation to velocity......................... 155-163,193-195

maximum............................................. 124,130

of ii natural alluvium.................................. 177,180

of mixed grades............................... 113-115,169-185

review of controls of, by conditions .................... 186-193

s table of values of, adjusted in relation to slope........... 75-87

readjusted in relation to discharge................. 137-138

readjusted in relation to fineness .................... 151

for suspension............................................. 223-230

Channels, crooked, experiments with........................... 196-198

 curved, features of...................................... 198,220-221

form of....................................................... 10

of fixed width, relation of capacity to slope in............... 96-116

of similar section, relation of capacity to slope in............ 116-120

shaping of, by natural streams............................. 221,222

widths of..................................................... 22

See also Bed and Form ratio. 

Christy, S. B., acknowledgments to.............................. 9

Competen ce, definition of.................._..................... 35

in Hum traction, data bearing on..............._......... 215-216

simultaneous, for all controls of capacity .................... 187

Compoter.co constants, use of term............................... 187

Computation sheet, logarithmic, figure showing.................. 95

Contraction, local, effect of, on velocity........................... 245

Contractor, influence of.......................................... 57

outfall, description of......................................... 25

Controls of capacity, review of.................................. 186-195

Cornish, Vaughan, on antidunes............................footnote, 32

on progressive waves in rivers................................ 244

Cunningham, Allan, on the velocities of streams.................. 155

Current, nature of, at bends in channels.................... 193,220-221

See also Velocity.

D.

Darcy, H., and Bazin, F. A., modification of Pitot-Darcy gage by. 252 

formulas of, for velocities in conduits and rivers............ 193-194

Deacon, G. F., experiments by................................... 16

Dtfbris, collective movement of................................... 30-34

grades of, tables of.......................................... 21,199

mixed, causes of superior mobility of....................... 178-179

mixtures of grades of, evidence from experiments with..... 113-115

experiments with...................................... 169-185

table of observations with................................ 52-54

movement of particles of................................... 155-156

natural, capacity lor traction of............................. 177-178

table of observations with................................ 54

source and sizes of........................................ 21-22,152

table of observations with................................... 36-54

used in experiments, plate showing........................... 22

See also Fineness and Sand. 

Depth of water, adjusted values of, table of....................... 89-93

adjustment of observations on................................ 87-95

gage for measuring............................................ 21

in relation to slope, plot of observations 011................... 87

in unloaded streams in flumes, table of........................ 213

method of measuring......................................... 25-26

observations on, table of...................................... 38-54

relation of capacity to...................................... 164-168

relation of velocity to, ideal curves showing.................. 161

table of values of, for ddbris of grade (C), when the wid th is 0.66

foot and the slope is 1.0 per cent..................... 128

variation of, as related to variation of discharge............... 165

Dimensions of coefficients...................... 64,129,139,151,186,191

Discharge, change of, effect of, on velocity........................ 246,

competent, table of experimental data on.................... 70

constant, relation of capacity to depth under................. 164

relation of capacity to velocity under. '. .................. 157

control of constants by..................................... 132-133

definition of.................................................. 35

diversity of, in natural streams............................. 221-222

gate for measuring, description and rating of............... 257-259

measurement of.............................................. 22-23

mode of controlling........................................... 20

relation of, to a. .............................................. 66-67

relation of capacity to...................................... 137-149

values of, corresponding to gate readings...................... 23

variation of, in relation to variation of depth................. 165

Discharge factor, applicability of, to natural streams............ 233-235

influence of.................................................. 10

Dresden, laboratory of river engineering at....................... 16

Dubuat-Nangay, L. G., experiments of, on competent velocity. 193,216 

Dunes, definition of.............................................. 31

formation and movement of............................. 31,231-232

longitudinal section illustrating........................... 31

in relation to rhythm of current........................... 242,244

interval between, length of................................... 243

Dupuit, theory of, on suspension.......................... footnote, 224

Durham, T. C., and Brigham, Eugene C., work of, on viscosity as

affected by suspended matter........................ 228

Duty, constant for similar streams............................... 239

definition of....................................-.-.........:. 36,74

general formulation of....................................^ - - 192

in experiments of F. K. Blue ............................... 217

in relation to discharge..................................... 144,147

in relation to fineness........................................ 154

in relation to slope .......................................... 121

values of, corresponding to adjusted values of capacity, table of. 75-87

261
262 INDEX.

E. Page.

Efficiency, definition of.......................................... 36

general formulation of....................................... 192

in relation to discharge .................................... 144,148

in relation to fineness ....................................... 154

in relation to slope ........................................ 121-123

values of, corresponding to adjusted values of capacity, table of. 75-87

Efficiency and capacity, table comparing parameters in functions

Of........................................................ 123

Eger, Dix, and Seifert on scale of a model of Weser River.......... 237

Energy, effect of load on................ i................... 11,225-227

relation of, to competent slope............................... 64

Error, probable, for stream traction.... 56,73-74,94,113,142-143,151-153 

probable, for flume traction.................................. 206

Eshleman, L. E., work of........................................ 9

Experiments, method of making.............................. 10, 22-26

nature and scope of.......................................... 17-18

results of..................................................... 240

F.

Fargue, L., cited................................................. 197

on scale of model river................................. footnote, 237

Feeding of de'bris, suggested apparatus for........................ 241

influence of, on capacity...................................... 56

methods of.................................................. 23

Fineness, bulk, definition of.................................. 21,35,183

bulk, relation of, to values of a, table showing.......'.......'.. 68

competent................................................... 151

constant linear, variations of................................. 153

control of constants by....................................... 133

influence of, on the relation of capacity to form ratio, table of

quantities illustrating................................ 133

in natural and artificial grades of debris, range and distribution of, curve illustrating............................. 181

linear, definition of....................................... 21,35,183

mean, changes of, in natural streams.............:......... 221-222

definition and measurement of........................ 182-184

mixtures of different, experiments with.................... 169-185

of grades of de'bris .......................................... 21,199

of mixed grades and their components ....................... 180

relation of, to a. .............................................. 67-71

relation of capacity to...................................... 150-154

for natural grades......,.........................:..... 180-182

Fineness factor, applicability of, to natural streams............ 235-236

influence of.................................................. 10

Flow of water, rhythm in.............................. 242-244,248-249

Flume traction, definition of..................................... 15

discussion of............................................... 199-218

dune action in............................................... 202

experiments with, apparatus used in....................... 199-200

grades of de'bris used in, table of............................. 199

movement of de'bris in....................... '.............. 201-203

movement of particles in................................... :200-201

nature of..................................................... 11-12

rough surfaces used in experiments on, plate showing........ 200

scope of experiments on...................................... 18-19

speeds of coarse and fine de'bris in, table of.................... 200

table of observations on.................................... 202-203

Form factor, influence of......................................... 10

Form ratio, computation of...................................... 94

definition of.................................................. 35-36

discussion of, table of observations for........................ 51-52

for natural streams........................................... 223

influence of................................................ 116-117

of unloaded streams in flumes, table of....................... 213 .

optimum.................................................. 134-136

relation of capacity to ............................. 124-136,190-192

required to enable a given discharge, on a given slope, to

transport its maximum load, table of values of....... 135

Form-ratio factor, applicability of, to natural streams ............ 236

Formulation based on competence, discussion of............... 186-190

Froude, William, on proportionate resistances of ships........... 237

Grades of debris................................................ 21,199

See also De'bris. 

Graphic computation, accuracy of................................ 95

use of ................................-.-.-........-- ---.-..-- 94-95

Gue'rard, Adolphe, observations on the Rhone by................ 231

H. Page. 

Head of water, determination of.................................. 23

Hearson, T. A., on velocities in similar streams.................. 237

Hider, Arthur, on dunes in Mississippi River............ footnote, 31,231

on intervals between dunes........................... footnote, 243

Hooker, E. H., cited........................................... 195,224

Hopkins, William, on the law of stream traction.......'. ..'.... 16,162

Humphreys and Abbot cited.................................. 229,231

I.

Index of relative variation, nature of............................. 97-99

synthetic, nature of.......................................... 99

Index to symbols............................................... 13-14

Intake influences, nature of...................................... 56

Interpolation, procedure in....................................... 72-73

Interpolation formula, selection of................................ 60-65

J. 

Johnson, Willard D., acknowledgments to....................... 9

K.

Karlsruhe, laboratory of river engineering at..................

L.

16

Law, Henry, on the law of stream traction...................... 16,162

Lechalas, C., on advance of dunes................................ 232

Lechalas, formula of............................................ 192-195

Le Conte, J. N., acknowledgments to.............................. 9

calibration of Pitot-Darcy gage by........................... 252

Leighton, M. O., acknowledgments to............................ 9

Leslie, Sir John, on the law of stream traction.................... 16,162

Letters and symbols, index to.................................... 13-14

Levee, natural, formation of..............................'........ 222

Load, definition of............................................... 35

effect of, on the stream's energy......................... 11,225-227

effect of addition of, on velocity ........................... 246-247

effect of rhythm on........................................... 58

method of determining....................................... 24-25

observations on, table of..................................... 38-54

partition of................................................. 230-233

See also Capacity. 

Login, T., experiments by, on competent velocity................ 163

Loire River, observations on..................................... 232

subaqueous dunes of, table of data on ........................ 194

M.

McGee, W J, originator of term saltation ......................... 15

McMath, R. E., description by, of the movement of debris....... 156

on the building of bars....................................... 232

Manson, Marsden, on suspended load of Yuba River............ 223

Matthes, Francois E., acknowledgments to....................... 9

Mississippi River, dunes in ................................. 31,231-232

suspended load of ............................................ 229

Mobility of debris, causes affecting............................. 178-179

Moving field, apparatus for................................. 27,249-250

Movement of de'bris, collective............................... 30-34,201

modes of..................................................... 11,15

Movement of particles................................... 26-30,200-201

Murphy, E. C., on the movement of dunes and anticlunes........ 32-33

work of...................................................... 9

N. 

Notation............................'............................. 13-14

0.

Observational series, definition of................................ 55

Observations on load, slope, and depth, table of................. 38-54

on competent discharge, table of............................. 70

on competent slope, table of................................. 69

See also Experiments. 

Outfall, contracted, effect of approach to, on velocity............. 245

effect of approach to, on velocity........................... 244-245

Outfall influences, adjustments for............................... 56-57

Overstrom, G. A., experiments by, on capacity of launders. 210,216-217 

Owens, John S., experiments by, on competent velocity......... 163

on antid'unes............................................ footnote, 32.
INDEX. 263

P. Page. 

Particles, fineness of............................................. 10,11

modes of movement of ........................... 11,20-30,200-201

See also Fineness. 

Partiot, H. L., cited.............................................. 194

observation by, in the Loire.................................. 232

Pitot-Darcy gage, special form of, description of............... 251-252

difficulties in the use of....................................... 26

efficiency of.................................................. 255

rating of................................................... 252-256

ratios of suction at one opening of, to pressure at the other... 254 

variation of constant of, near boundaries of current....... 254-255

Plots, logarithmic, making £>nd study of.......................... 59-60

Poiseuille, J. L. M., theorem established by..................... 155

Power function and its logarithmic locus......................... 97-99

Powless, W. H., observation by................................. 232

Precision, approximation to, for stream traction.................. 56,

73-74,94,113,142-143,151-153 

approximation to, for fcume traction......................... 206

11.

Eankinc, W. J. M., formula of, for resistance of a wheel.......... 201

Reynolds, Osborne, on the flow of water through tubes.......... 242

experiments by, with one liquid flowing over another........ 248

on scales of model tidal basins................................ 237

Rhone River, observations on load of............................ 231

Rhythm, occurrence and influence of............................. 58-59

problems associated with................................... 241-250

Richards, R. H., on capacity of launders................... 210,216-217

cited .......................................................... 226

Riycr engineering, laboratories of................................. 16

Rivers. See Streams, natural.

Rolling of particles, features of................................... 26

observations on............................................ 200-201

space required for............................................ 211

S.

Sainjon, advance of clunks formulated by......................... 232

Saltation, analysis of.............................................. 26-30

comparison of, with rolling.............-'...................... 200

definition of.................................................. 15

zone of, curves of velocity as affected by, figure showing...... 161

Sand, method of collecting....................................... 24

mode of delivering........................................ 20,23-24

See also D<5bris. 

Sand arrester, description of...................................... . 20

influence of, on water slope, diagram illustrating............. 57

Scope of the investigation........................................ 10

Seddon, J. A., on suspended load of Mississippi River............ 229

Sensitiveness, definition of....................................... 188

Settling tank, description of...................................... 21

Sewers, experiments on traction in............................... 216

Shearing, interference by suspended particle with, diagram showing....................................;.............. 226

v, relation of, to discharge........................................ 66-67

relation of, to fineness........................................ 67-71

to width.................................................. 67

values of, adopted for interpolation formulas ................ 72

in formulas ior mixed grades of debris................... 169

in formulas for streams oi similar slope ................ ] 17-119

Sigma formula, use of............................................ 96-97

Sigma function, discussion of.................................... 99-100

Sizes of debris .................................................. 21,199

See also Ddbris. 

Sliding of particles, infrequency of............................... 26

observations on............................................ 200-201

Slope, capacity for llumo traction in relation to......... 203-206,208-209

capacity for str.jam traction in relation to......... 55,61-65,96-123

competent, definition of..................................... 35

in flume traction........................................ 203

observations on...........:.............................. 69

relation of, to a ........................'............. 64-70,187

definition of.................................................. 35

effect of chango of, on velocity............................. 245-246

for straight and crooked channels, table comparing........... 197

influence of, on the relation of capacity to form ratio......... 132

method of determining....................................... 21,25

observations en, adjustment of............................... 55-87

Page. 

Slope, observations on, in crooked channels ................... 196-197

observationsjwnj reduction of................................ 37

table of.................................................. 38-54

unit of, effect of changing.................................. 112-113

Slope an'd mean velocity, relative variation of.................. 158-159

Slope factor, applicability of, to natural streams.................. 233

influence of.................................................. 10

Smith, Hamilton, tables of discharge by......................... 259

Speeds of particles in fl ume traction, table of..................... 200

Streams, natural, application of laboratory results to........... 219-240

natural, kinds of ............................................ 219

forms of channel in..................................... 124,222

similar, hypothesis of...................................... 236-240

Subsidence, velocity of......................................... 225-226

Surfaces, rough, used in experiments on flume traction.:.... 200,206-208

Suspension, analysis of........................................ 223-230

definition of ................................................ 15

forces active in, diagram of................................... 224

in natural streams......................................... 223-230

relation of, to flume traction................................ 200,201

to traction in natural streams......................... 221-223

Symbols, index to .............................................. 13-14

T. 

Tables, list of ................................................... 5-6

Terms, definitions of............................................. 35-36

Thomson, J., studies by, on currents of streams.................. 220

Traction, definition and classification of.......................... 15

flume, modes of............................... ............ 200-202

observations on ....................................... 202-203

stream, foreign work on...................................... 15-16

modes of ................................................ 26-34

observations on......................................... 36-54

See also Flume traction. 

Transportation, modes of................................... 11,15,26-34

Trough, forms and sizes of........................................ 16

semicylindric, capacity for flume traction in.................. 214

width of..................................................... 22

relation of, to <r....................... ................... 67

with glass panels and sliding screen ......................... 27

with local contraction, velocities in .......................... 245

Troughs with bends, plans of..................................... 196

U. 

Units used....................................................... 34

University of California, acknowledgments to.................... 6

apparatus on campus of, plate showing............... Frontispiece.

V. 

Van Orstrand, C. E.,acknowledgments to........................ 9

Velocity, competent, experiments on.................... 69,70,162-163

influence of suspended load on............................. 225-230

influence of tractional load on............................. 229-230

maximum, hypothesis as to position of...................... 248

mean, and slope, relative variation of....................... 158-159

computation of........................................... 94

of streams with and without tractional load, table comparing. ............................................ 230

methods of measuring........................................ 26

relation of capacity to...................................... 155-163

vertical curve of, as modified by conditions............... 244-247

Viscosity, influence of.......................................... 226-229

Voids, percentage of, effect of mixed grades on................... 179

Von Wagner, on the velocities of streams......................... 155

W. 

Water, mode of handling......................................... 19-20

rhythm in the flow of...................................... 242-244

surface of, form of profile of................................... 57

profiles of, showing undulations associated with antidunes. 33 

relation of slope of, to capacity........................... 59

rhythmic fluctuations of................................. 58

slope of, as affected by feeding............................ 56

Water circuit, diagram ot........................................ 20

Water supply, quality of......................................... 19

Whirls, movement of sand by.................................... 32

Woodward, R. S.,acknowledgments to........................... 9

Y. 

Yuba River, observationsion........................... 223-224,230-231

o