Stage-Fall-Discharge 

Relations for 

Steady Flow in 

Prismatic Channels

By WILLIAM D. MITCHELL

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1164

Laboratory studies of effect of variable 

backwater on stage-discharge relations 

at stream-gaging stations

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1954
UNITED STATES DEPARTMENT OF THE INTERIOR 

Douglas McKay, Secretary

GEOLOGICAL SURVEY 

W. E. Wrather, Director

For sale by the Superintendent of Documents, U. S. Government Printing Office 

Washington 25, D. C. - Price 50 cents (paper cover)
CONTENTS

*LAbstract.......................................................................................................................... \

Introduction.................................................................................................................... " 2

Purpose................................................................................................................... 2

Scope.......... ............................................................................................................. 3

Acknowledgments.................................................................................................. 5

Theory of steady flow in open prismatic channels................................................... 5

Total energy of a cross section.......................................................................... 6

Comparison from point to point.................................................................... 6

Velocity-distribution coefficient................................................................. 7

Specific energy.............................................................................................. 9

Uniform flow........................................................................................................... H

Comparison from section to section............................................................ 11

Uniform depth................................................................................................. 13

Computations for uniform flow..................................................................... 13

Classifications of flow......................................................................................... 14

Nonuniform flow..................................................................................................... 15

General considerations................................................................................. 15

Geometry of the reach Ax............................................................................. 16

Coordinate systems and conventions.......................................................... 18

Integration of the varied-flow equation...................................................... 19

Direct integration.................................................................................. 22

The step method.................................................................................... 23

The laboratory data....................................................................................................... 25

Description of the data......................................................................................... 26

Analysis of the data.............................................................................................. 35

Preliminary considerations.......................................................................... 35

Conventions and use of terms.............................................................. 35

Revisions to stationing......................................................................... 36

Evaluation of velocity-distribution coefficients............................... 37

Computation of hydraulic radius.......................................................... 39

Values of Manning's n.......................................................................... 44

Direct method of analysis............................................................................ 44

Indirect methods of analysis........................................................................ 51

The step method.................................................................................... 52

Direct-integration methods................................................................... 55

Comparison of results........................................................................... 55

Tables of profiles by the step method................................................................ 71

Stage-fall-discharge relations for the laboratory channels...................................... 78

Determination of discharge under backwater..................................................... 78

Resume of current methods.................................................................................. 81

The stage-fall-discharge diagram............................................................... 81

The Boyer method....................................................................................... 84

Stage-ratio and submergence methods........................................................ 92

Miscellaneous methods..................................................................................... 98

Possible combinations of dimension!ess ratios....................................... 98

Results of various ratio combinations....................................................... 103

Proposed methods................................................................................................. 108

Recapitulation...................................................................................... 108

Three-dimensional Boyer diagram............................................................... 109

Fixed-backwater ratings............................................................................... 119

Analytical approach to fixed-backwater ratings ............................................ 135

Application to field problems...................................................................................... 142

Symbols.........................................................................................-...........-- -. 147

References..................................................................................................................... 150

III
IV CONTENTS

Page

Supplementary tables......................................................................................................... 151

Characteristics of the laboratory channels............................................................ 152

Areas of flood-plain cross section.................................................................. 152

Values of k.... ...................................................................................................... 152

Computed falls in 400-foot reach for selected values of stage and discharge.. 155

Index.................................................................................................................................... 161

ILLUSTRATIONS

Page

Figure 1. Energy in a cross section.............................................................................. 6

2. Specific-energy diagram.................................................................................. 9

3. Variable deptn at a section............................................................................ 11

4. Longitudinal section of uniform flow............................................................ 12

5. Longitudinal section of a case of gradually varied flow.......................... 15

6. Gradually varied flow in a very short reach................................................ 17

7. Geometry of the step method.......................................................................... 23

8. Analytical orientation of profile.................................................................... 35

9. Subdivisions of flow in cross section 5...................................................... 37

10. Hydraulic radii................................................................................................ 40

11. Subdivision of flood-plain cross section for computing R'........................ 42

12. Values of Manning's n for the various cross sections.............................. 43

13. Friction slope in cross section 2.................................................................. 48

14. Relationship between V/k and friction slope, cross section 4................ 50

15. Comparison of computation methods, cross section 2................................ 56

16. Comparison of computation methods, cross section 5................................ 57

17. Variations of water surface under fixed discharge.................................... 78

18. Stage-fall-discharge diagram, cross section 2, 400-foot reach................ 82

19. Stage-fall-discharge diagram, cross section 5, 400-foot reach................ 83

20. Boyer diagram (all profiles) cross sections 2 (left) and 5 (right)............ 85

21. Boyer diagram for various-length reaches, cross sections 2 (left) and

5 (right), with velocity-head correction.................................................... 87

22. Submergence diagrams, cross section 2, 400-foot reach (lefl) and

120-foot reach (right)...............................................................,.................. 89

23. Submergence diagrams, cross section 5, 400-foot reach (left) and

120-foot reach (right).................................................................................. 99

24. y /y versus F /y , cross section 2, 400-foot reach (left) and 

o m mm

120-foot reach (right).................................................................................. 100

25. y AT versus F /y , cross section 5, 400-foot reach (left) and 

o tn mm

120-foot reach (right).................................................................................. 104

26. F /y versus F /y , cross section 2, 400-foot reach (left) and mo mm

120-foot reach (right).................................................................................. 105

27. F /y versus F /y , cross section 5, 400-foot reach (left) and mo mm

120-foot reach (right).................................................................................. 106

28. Three-dimensional Boyer diagram, cross section 2.................................... 107

29. Three-dimensional Boyer diagram, cross section 3.................................... 110

30. Three-dimensional Boyer diagram, cross section 4.................................... Ill

31. Three-dimensional Boyer diagram, cross section 5 (v 2.5 through

0.7).................................................................................?...........-..............-.. 112

32. Three-dimensional Boyer diagram, cross section 5 (y 1.1 through

0.4)................................................................................. ............-............... H3

33. Three-dimensional Boyer diagram, cross section 6 (y 2.5 through

0.9).......................................!.......................................*............................ 114

34. Three-dimensional Boyer diagram, cross section 6 (y 1.1 through 0.5)........................................;........................................!?............................. us

35. Three-dimensional Boyer plot, developed from smoothed profiles,

cross sections 2 (above) and 5 (below).................................................-. 116
CONTENTS V

Page 

Figure 36. Stage, discharge, and fall for a fixed-backwater condition, cross

section 2...................................................................................................... H8

37. Boyer diagram for cross section 2, to accompany figure 36.................... 121

38. Stage, discharge, and fall for a fixed-backwater condition, cross

section 4...................................................................................................... 123

39. Boyer diagram, for cross section 4, to accompany figure 38.................. 125

40. Stage, discharge, and fall for a fixed-backwater condition, cross

section 3...................................................................................................... 127

41. Boyer diagram, for cross section 3, to accompany figure 40.................. 129

42. Stage, discharge, and fall for a fixed-backwater condition, cross

section 5...................................................................................................... 131

43. Boyer diagram, for cross section 5, to accompany figure 42.................. 132

44. Boyer diagram for the rating O equals 0.99 Q , cross section 2.......... 133

45. Halation between observed fall and effective fall, 400-foot reach,

cross section 2.......................................................................................... 136

TABLES

Page

Table 1. Smoothed profiles, cross section 2.............................................................. 27

2. Smoothed profiles, cross section 3.............................................................. 29

3. Smoothed profiles, cross section 4.............................................................. 30

4. Smoothed profiles, cross section 5.............................................................. 32

5. Smoothed profiles, cross section 6.............................................................. 34

6. Stationing for the profiles............................................................................ 36

7. Computation of a velocity-distribution coefficient.................................... 38

8. Summary of values of c .............................................................................. 38 ' in

9. Friction slope in the rectangular channels................................................ 46

10. Friction slope in the flood-plain channels................................................ 47

11. Sample computation of profile by step method.......................................... 53

12. Summary of step-method computation of profiles, cross section 2.......... 59

13. Summary of step-method computation of profiles, cross section 3.......... 63

14. Summary of step-method computation of profiles, cross section 4.......... 67

15. Summary of step-method computation of profiles, cross section 5.......... 72

16. Summary of step-method computation of profiles, cross section 6.......... 76

17. Computation of stage-fall-discharge ratios, cross section 2, 400-foot

reach, yQ = 0.391 foot, Om = 3.90 cfs.................................................. 90

18. Computation of stage-fall-discharge ratios, cross section 2, 400-foot

reach, y = 1.840 feet, Qm = 40.39 cfs.................................................. 90

19. Computation of stage-fall-discharge ratios, cross section 2, 120-foot

reach, y = 0.391 foot, Om = 3.90 cfs.................................................... 91

20. Computation of stage-fall-discharge ratios, cross section 2, 120-foot

reach, yQ .= 1.840 feet, O^ = 40.39 cfs.................................................. 91

21. Fall ratios and discharge ratios, cross section 2, 400-foot reach.......... 93

22. Computation of stage-fall-discharge ratios, cross section 5, 400-foot

reach, yQ = 0.250 foot, <?ffl = 0.34 cfs.................................................... 94

23. Computation of stage-fall-discharge ratios, cross section 5, 400-foot

reach yQ = 2.525 feet, Qm = 31.28 cfs.................................................... 94

24. Computation of stage-fall-discharge ratios, cross section 5, 120-foot

reach, yQ = 0.250 foot, (?m = 0-34 cfs.................................................... 95

25. Computation of stage-fall-discharge ratios, cross section 5, 120-foot

reach. yn = 2.525 feet, O = 31.28 cfs.................................................. 95 ^ 0 m

26. Fall ratios and discharge ratios, cross section 5, 400-foot reach.......... 96

27. Area of flood-plain cross sections.............................................................. 152

28. Values of k, cross section 2....................................................................... 152

29. Values of k, cross section 5........................................ -. ...... - 153

30. Values of k, cross section 4.......................................................... 153
VI CONTENTS

	Page 

Table 31. Values of k, cross section 5....................................................................... 154

32. Values of k, cross section 6...................................................................... 154

33. Computed falls, cross section 2, 400-foot reach...................................... 155

34. Computed falls, cross section 3, 400-foot reach...................................... 156

35. Computed falls, cross section 4, 400-foot reach...................................... 157

36. Computed falls, cross section 5, 400-foot reach...................................... 158

37. Computed falls, cross section 6, 400-foot reach...................................... 159
ABSTRACT

In a prismatic channel, the shape of the surface profile remains fixed as long as the 

discharge, the slope, and the boundary conditions remain fixed. This "Principle of rigidity of the profile" was previously applied to the observation, under laboratory conditions, of 41 M-l type profiles for steady flow in prismatic channels. The same "Principle of rigidity of the profile" is now applied to the derivation, from these same profiles, 

of extensive data on relations among stage, fall, and discharge. These data are used to 

test the applicability of various methods of determining discharge under variable fall, 

under the particular conditions of boundary and slope which existed in the laboratory. 

Diagrams are presented to show that, using the uniform-flow ratings as a basis of comparisons, all the known curves of relation between stage, fall, and discharge become 

three-dimensional diagrams, rather than single curves. A new method of preparing such 

three-dimensional diagrams is proposed.

Other diagrams are presented to show that by introduction of an optimum amount of 

backwater and substitution of the resulting fixed-backwater rating in place of the uniformflow rating as a basis of comparisons, the correlation between discharge ratios and fall 

ratios is so improved that a single curve of relation will, under all conditions that were 

tested, lead to a close approximation of the true discharge.

Throughout the report emphasis is placed upon the fact that the findings and conclusions are solely with respect to the particular conditions of boundary and slope which 

were used in the laboratory. The applicability of these conclusions to field conditions 

has not been extensively explored. The final section of the report presents suggestions 

for use in connection with field problems.

In the early part of the report, and incidental to the main theme, there is presented the 

theory of flow in prismatic channels. Data are presented to show the extent to which the 

laboratory observations are in agreement with this theory.
2 STAGE-FAIL-DISCHARGE RELATIONS

INTRODUCTION

PURPOSE

Gradually varied flow may be observed in almost any natural water course. 

From one cross section to another, even though the discharge is the same, variations may be expected in the velocities. Although usually due to changes in the 

size or shape of the cross sections of the stream channel, or a combination of 

these changes, the velocity variations may occur in perfectly uniform channels 

also. The changes often are so gradual as to escape casual observation, but the 

hydraulic problems involved are nevertheless problems in gradually varied flow. 

The problems are of particular importance on streams for which the discharge 

cannot be determined by reference to stage alone, but which also require the additional factor of fall as determined from an auxiliary gage. In such cases readings from the base gage and from the auxiliary gage determine two points on a 

surface profile for gradually varied flow. Thus a better understanding of this type 

of flow will contribute to better understanding of the stage-fall-discharge relations at gaging stations affected by variable fall.

Many hydraulic engineers have only meager knowledge of gradually varied flow. 

Even among those who work exclusively with flow of water in open channels the 

subject has been greatly neglected. Some have not realized the importance of the 

problems; others have been discouraged by the complex nature of the mathematics 

that forms the usual basis of approach. Furthermore, study of the problem by observations on natural water courses is handicapped by inability to evaluate the 

pertinent hydraulic factors. Study in the laboratory, where such factors may be 

controlled and evaluated, has until recently been considered impractical, because 

of the excessively long channel that was thought to be necessary.

Foremost among the reasons for the lack of experimental data on gradually varied flow profiles lies the incongruity between the length of laboratory channels 

and the length of the curves to be investigated. Except for comparatively small 

discharges or comparatively steep slopes, or a combination of both, the investigator is likely to find that the curve he wishes to investigate is several times the 

length of the longest channel available for the study. To develop a curve 1,000 

feet long in a channel whose length is only 100 feet seems, at first glance, a bit 

irrational. Further study of the problem, however, indicates that, for prismatic 

channels, such studies can be made in a comparatively simple manner. A technique for this purpose was first suggested by Mononobe (1938) and later developed more fully by graduate students at the University of Illinois (Mitchell and 

Barron, 1946). After this work had been tested and reviewed, an agreement was 

reached between the University of Illinois and the United States Geological Survey to extend and enlarge the scope of the observations. The data obtained under 

this agreement have been published as Bulletin 381 of the University of Illinois 

Engineering Experiment Station (Lansford and Mitchell, 1949), hereafter referred 

to in this report as Bulletin 381.
INTRODUCTION «5

In Bulletin 381 the objective was to obtain and publish laboratory observations. No attempt was made to apply the data to the particular use of the Geological Survey: the determination of stage-fall-discharge relationships. These relationships, however, were considered to be the ultimate objective of the Survey's 

interest in the data, and a study was begun at the earliest opportunity. The present report presents the results of that study.

SCOPE

This report is considered to be a companion volume of Bulletin 381. There is 

little, if any, duplication of material in the two reports. They deal with the same 

data, but with different objectives. Bulletin 381 presents laboratory observations 

with only such analyses as were essential to a proper presentation of the data; 

the present work applies these data to development of stage-fall-discharge techniques. Enough basic information has been included in the present work to make 

it self-sufficient for its intended purpose. For example, the uniform depthdischarge ratings for the several channels, which appeared as a part of Bulletin 

381, have not been repeated here. In this report, however, for each of the several 

channels tables of depth versus k have been provided, and information has been 

provided for computing area for any depth in any channel. The factor k is defined 

as a function such that, when multiplied by area and the square root of the bed 

slope, the uniform discharge is obtained. Thus the uniform-depth-discharge ratings, although not tabulated in this report, may be obtained from information contained herein. Likewise, the detailed tables of profile observations which constitute almost 50 percent of Bulletin 381 have not been repeated here. Instead 

there has been condensed into a few pages a series of "smoothed profiles" differing from the original data only in the manner in which a completed rating table 

differs from the raw tabulation first read from a rating curve that is, the data 

have been slightly adjusted to obtain smoothly changing values of the first and 

second differences. This smoothing has been essential to certain portions of the 

subsequent analysis and as presented herein may save other students many hours 

of tedious effort. On the other hand, profiles drawn through these smoothed data 

will differ only imperceptibly from profiles drawn from the original data of Bulletin 381. The smoothed profiles, tables 1 to 5 should not be confused with the 

computed profiles, tables 12 to 16, which form the basis of analysis in the final 

sections of the report.

Limitations of funds and of personnel have prevented the complete analysis of 

all the profiles presented in Bulletin 381. To provide the greatest range of information intensity of the analysis has been varied in the following order:

1. The six profiles of cross section 2 a rectangular channel lined with material of the same roughness throughout the entire depth range.

2. The seven profiles of cross section 5 a flood-plain channel lined identically with that of cross section 2.
4 STAGE-FAIL-DISCHARGE RELATIONS

3. The seven profiles of cross section 4 a rectangular channel lined with material of the same roughness throughout the entire depth range, this material being 

much rougher than for cross section 2.

4. The five profiles of cross section 3 a rectangular channel in which the lining of the upper part was identical with that of cross section 4, and in which the 

lower part was the same as that of cross section 2, except for the deterioration of 

lining material during the 2 years which had elapsed.

5. The four profiles of cross section 6 a flood-plain channel in which the lining of the lower part was identical to that of cross section 5, and in which the 

upper part was unlined.

In general, a given method of analysis or correlation was tried first on the profiles of item 1 above. In cases where further tests appeared desirable, they were 

continued on the profiles of item 2. If conclusions were still in doubt, tests were 

continued on the other profiles in the order indicated above.

Most of the results of such tests have been included in the following pages. For 

many of the techniques tested the results, as applied to these laboratory data, 

have proved unsatisfactory. It does not necessarily follow that these same techniques are unsatisfactory for other channels in which different conditions may be 

more favorable to the particular technique. Whether the results of a given test 

were good or poor, however, they have been included to the end that the reader 

may see for himself the nature of the results and thus make his own decision as 

to whether the method should be applied to his particular problem or analysis.

It cannot be too strongly emphasized that the tests contained herein are for 

steady flow in prismatic channels. Furthermore, the channel slope, although 

"mild" in the technical classification of fluid mechanics, is much steeper than 

generally encountered in field problems of gradually varied flow this steepness 

of slope arises from practical limitations of the laboratory technique. Thus, the 

conclusions which may be drawn from the analysis of these data must not be 

blindly transferred to other problems on natural watercourses. Not only the probable flatter slopes, but also the irregularities of cross section of natural watercourses, and the phenomenon of changing discharge point toward untold complications. An ultimate objective to this study, of course, would be an answer to the 

question, "What change in the analysis of field problems is indicated by these 

investigations?" Much thought has been given to this question, and a number of 

suggestions have been made under the heading, "Application to field problems." 

(See pp. 142 to 146.)
THEORY OF STEADY FLOW IN OPEN PRISMATIC CHANNELS 5 

ACKNOWLEDGMENTS

This report is an analysis of the data resulting from an investigation in the 

Fluid Mechanics and Hydraulics Laboratory at the University of Illinois as a 

part of the work of the Engineering Experiment Station (Lansford and Mitchell, 

1949). In part it is similar to an earlier and less extensive analysis made in the 

department of Theoretical and Applied Mechanics of the same university (Mitchell and Barren, 1946). Except for the basic computations essential to the preparation of Bulletin 381 (Lansford and Mitchell, 1949), this analysis was made 

and the manuscript prepared by, or under the technical supervision of, William D. 

Mitchell, Hydraulic Engineer, working under the administrative direction of J. H. 

Morgan, District Engineer, Champaign, 111. Other members of the Survey staff 

have assisted with the computations. Mr. John R. Stipp, Hydraulic Engineer, was 

in direct charge of computations and the preparation of tables. Mr. Salvatore A. 

Rocci, Hydraulic Engineer, assisted with the computations, and was in direct 

charge of the preparation of illustrations. Grateful acknowledgment also is made 

for the helpful advice and criticism of the Washington staff of the Survey, particularly W. S. Eisenlohr, Jr., of the Technical Coordination Branch, and C. H. 

Ilardison, of the Surface Water Branch. The work was financed by Research and 

Development funds made available by C. G. Paulsen, Chief Hydraulic Engineer, 

through J. V. B. Wells, Chief, Surface Water Branch.

THEORY OF STEADY FLOW IN OPEN PRISMATIC CHANNELS

Steady flow is flow which is constant with respect to time; it is sufficiently 

accurate, for most channels, to say that, at a given point along the channel the 

discharge must be the same at all times within the period under observation. 

Such a condition rarely exists in the flow of natural streams, but often the condition is closely approached.

A prismatic channel is one for which all cross sections, perpendicular to the 

direction of general motion, are similar and parallel polygons which are equal in 

area. It should be noted that these characteristics apply to cross sections of the 

channel, but not necessarily to those portions of the cross sections which contain water.

There appears in the remaining articles of this section a presentation of the 

conventional concepts of flow under the limitations stated above. From many 

practical engineers, a purely mathematical discussion of this subject, including 

the varied-flow equation, will receive scant welcome. However, to avoid the necessity of involved or ambiguous statements during later sections of this report, it is imperative that certain mathematical relationships be recorded, and 

certain working formulas be explained. Most of these relations have been published many times before; they will not be new to workers familiar with variedflow theory. For these, as well as for others willing to accept working formulas 

without worry over derivation, it is suggested that this section be disregarded 

and that a study of this report be resumed with the next section (p. 25).
o STAGE-FALL-DISCHARGE RELATIONS

TOTAL ENERGY OF A. CROSS SECTION

COMPARISON FROM POINT TO POINT

At any point in a given flow, the total head (energy per unit weight of fluid) is 

composed of velocity head, pressure head, and elevation,

thus; £ = 2 + _ + _ 

in which E is total energy above the datum plane, Z is elevation above datum 

plane, p is intensity of pressure, V is velocity, all at the given point, y is specific weight of the fluid, and g is the acceleration of gravity. Thus for point 1, 

figure 1,

E -z 1+ !i+ .i

Energy line

Water surface

Stream bed

Datum plane

I [ 1111 M 1111111111111111 n 11111 III I M M 11III M11111111M m i H I M 111111111 nil 

Figure 1. -Energy in a. cross section.
THEORY OF STEADY FLOW IN OPEN PRISMATIC CHANNELS 7

Now if, in a given vertical plane, the pressure be hydro static ally distributed that 

is, varies in ratio to lineal distance below the free surfacethen pj = yh j,

in which h-^ is the elevation of pj below the free surface. As the point p is allowed to approach a fixed free surface, 2 increases by the amount that h decreases. Hence it follows that Zj + p^/y is a constant, and may be replaced, for 

convience, by the values for p2 in the free surface:

PI ?2 yh 2

Z, + = Z 9 + = Z 9 + = Z 9 + 0 = Z + y.

y y y

in which Z is elevation of channel bed above datum plane, and y is total depth 

of flow.

Now if it be assumed that, in any cross section, the velocity is the same at all 

points, the velocity head V 2/"2g will be constant,

V*

so that E = Z + y+ = a constant (1)

2g

for all points in a given cross section.

Since this is the form of the basic equation of flow, both uniform and gradually 

varied, it will be apparent that two limitations have been imposed: (1) that pressure be hydrostatically distributed and (2) that the velocity be uniformly distributed throughout the cross section. The first limitation is observed by excluding 

from consideration all cases of rapidly varied flow that is, cases in which tire 

changes in flow occur, in such abrupt manner as to disturb appreciably the normal 

distribution of pressure. The second limitation must be violated in most practical 

problems, but errors due to this cause may be compensated by use of a velocitydistribution coefficient.

VELOCITY-DISTRIBUTION COEFFICIENT

In equation (1), let it be considered that E is the energy per unit weight of fluid 

as applied to the cross section as a whole. E might be obtained by dividing thetotal flux of energy per unit time, Et , by the mass discharge, y@. However, to ob* 

tain the total flux of energy per unit thine, it is necessary to obtain an integration, over the entire cross section, of the values at any point p. If V were truly a 

constant at all points in the cross section, this would be a simple algebraic operation, but because V is actually a variable, integration must be used, or, at 

least, the value of E should be computed for a very large number of points, each 

applied to its appropriate portion of the flow, and a summation obtained, that is,

n yQ
8 STAGE-FALL-DISCHARGE RELATIONS

or, since z± + 11-^ = ^ 2 + a 2 = ^3 + 13= Z + y

and sinc^ Cj + C 2 +....Cn = 1,

then, £ < [ V 

__=2 + y+ Cl __ + C 2 (a)

in which C is the portion of the flow to which any V (with appropriate subscript) 

is applicable. Now if, in place of these varying values of V, there is used the 

average V = Q/A (in which Q is total discharge and A is total area of the cross 

section),

2

If V truly varies from point to point, it will be obvious that the result of equation 

(a) must differ from that for equation (b), for in any series of unequal numbers the 

average of their squares is greater than the square of their average. For example, 

with the numbers 3 and 5, the average of the squares is (9 + 25) / 2 = 17, while 

the square of the average is 16. Thus if these values of 3 and 5 were used for 

Vi and V 2 in computing Et by equation (a), using C l = C 2 = 0.5, and then the 

value 7 = 4 was used to compute Et by equation (b), it would be found necessary 

to apply to the latter computation a coefficient of 17/16 to harmonize the results. 

It will be noted that the greater the variation in the values of V, the larger will 

be the necessary coefficient. Thus, if Vi = 1 and V 2 = 7, the average of the 

squares is 25, while the square of the average is 16, as before. It will also be 

noted that the required coefficient will vary with the relative values of Cj and 

Cp Using the principle of conservation of energy and the calculus, it can be 

demonstrated that

in which v is the velocity at any point, and a is the portion of the cross section 

to which v applies, and A and V are values for the entire cross section.

Using the principle of conservation of momentum and the calculus, it can be 

demonstrated that

in which a, v, A and V have the same significance as before.

There has been much heated discussion as to whether the ce or cm coefficient 

should be used. It might be pointed out that use of cm weights the various v's in 

accordance with their appropriate area, whereas use of c e weights them in accordance with their appropriate discharge. For some years, the use of cm has 

been standard practice in the Geological Survey. Use of cm as the velocity distribution coefficient will be continued in this report.
THEORY OF STEADY FLOW IN OPEN PRISMATIC CHANNELS

SPECIFIC ENERGY

Further consideration of the energy at a cross section provides a basis for approach to the problem of gradually varied flow. In a given cross section, let it be 

assumed that the discharge is to be held constant, but that means are provided 

for varying the depth at which flow will occur. This is readily imagined by assuming a channel which can be tilted, so as to allow the bed slope to vary. Under 

these circumstances different values of depth will result in different values of 

energy. The value of Z, being constant at a cross section, may now be disregarded, or in other words, the datum plane may be transferred to coincide with the bed 

of the channel. The energy with respect to this new datum is known as specific 

energy, or specific head, and denoted by the symbol H. Thus,

V 2

(2)

The manner in which H varies with y, depth, is illustrated by the accompanying 

figure 2, in which y has been plotted as ordinate and H as abscissa. Since y is

Figure 2.-Specific-energy diagram.
10 STAGE-FALL-DISCHARGE RELATIONS

one of the factors of which // is composed, this portion of H may be represented 

by the 45° line through the origin of coordinates. That is, for any depth (y 1 ), H 

is composed of an abcissa (y') plus an increment (V 2/1g) of velocity head. If y1 

is very great, the velocity will be very small, and the velocity head, involving 

the square of a very small number, will become negligible. Thus, at great depths, 

H approaches y asymptotically. If y1 is very small, the velocity will be very 

great, and the velocity head, involving the square of a large number, will become 

extremely large. Thus, as y approaches zero, H approaches infinity. At some intermediate point, // must have a minimum value, which may be obtained by equating to zero the first derivative of equation (2):

cftf dy_ ( d \ \\ 

dy dy

But 7 = 4

,. , , - r ,-,,*-, |0 2 ||-2||cM

so that .U 3 ](d*

but dA = b(dy)

JJLUL-U-0^. (3)

T, , dH dy Q 2b 1 heretore = - - , dy dy g^3

which for minimum specific energy must equal zero, so that

dy_ Q 2b 

dy gA A

or Q 2b = gA 3 (4)

Q 2b 

whence = gA

and = 

6 b

But A/b is the average depth,

1/2 y 

whence = 1£L (5)

2g 2

that is, R, the specific energy, is a minimum when the velocity head is equal to 

half the average depth. This value of the depth, y, for which H is a minimum, has 

great significance in open-channel flow, and is designated as y c , the critical 

depth.
THEORY OF STEADY FLOW IN OPEN PRISMATIC CHANNELS 11

Figure 3. -Variable depth at a section.

UNIFORM FLOW

COMPARISON FROM SECTION TO SECTION

In the preceding section consideration has been given to the evaluation of E, 

the energy at a cross section. Attention is now invited to a comparison of the 

values of E for different cross sections. Referring to figure 4, a longitudinal 

section of a prismatic channel, with finite value of bed slope, S0, it should first 

be mentioned that it is customary to measure depth as the vertical distance, y, 

rather than the distance y' normal to the stream bed. Obviously y 1 = y cos /3. 

Except for very steep slopes, however, cos /8 is so near unity as to make little 

difference whether y is measured in the vertical or normal direction. For example, 

the angle whose tangent is 0.003 (S 0 as used in the laboratory work) has for its 

cosine the value of 0.999996. Similarly x, the distance along the channel between 

any two cross sections, may be taken as the horizontal distance, rather than the 

distance parallel to the channel bottom.

Let figure 4 represent a reach of length x taken from a prismatic channel of 

infinite length, in which there is steady flow. At section 1, with depth y lf and 

with respect to the horizontal datum plane,

V = 2i + R

E = + y ^

296350 O 54-
12 STAGE-FALL-DISCHARGE RELATIONS

I } (2]

>*_^

Figure 4. -Longitudinal section of uniform flow.

Since this channel is the same for all cross sections, not only between sections 

1 and 2, but for any number of sections upstream and downstream, y± and y 2 must 

be equal. This follows from the fact that the controlling factors, whatever they 

may be, which lead to a depth yj at section 1, are identical to the factors which 

control the depth at section 2, and must therefore lead again to the depth yj. The 

water surface must therefore be parallel to the stream bed. It follows that the velocities, and therefore the specific energy, H, must be the same at sections 1 

and 2. Whence £j = Z-^ + 7/j and £ 2 = Z 2 + H\, so that Ej - E 2 = £j - %2' Bmding both members of this last equation by x discloses the loss per unit distance, 

or the rate of loss:

E i - E i) Z i - Z

But from figure 4, (Zj - Z 2)/x is the bed slope, SQ. Thus the slope of the energy 

line, as well as the slope of the water surface, is the same as the slope of the 

bed, S0. This is a distinguishing characteristic of uniform flow. Flow is said to 

be uniform when all the elements which characterize the flow remain constant 

from one section to another.
THEORY OF STEADY FLOW IN OPEN PRISMATIC CHANNELS 13 

UNIFORM DEPTH

Reference was made in the preceding article to "the controlling factors, whatever they may be, which lead to a depth yj." These factors now will be examined. The movement of water from one cross section to another constitutes work, 

which can be accomplished only by expenditure of energy. In uniform flow, this 

energy cannot come from the water itself, for the specific energy remains the 

same at all cross sections. The energy must be derived from change in elevation, 

and all energy derived from this source must be expended for this purpose. Now 

the amount of energy needed for moving a given mass of water varies somewhat 

with the boundary conditions of the channel, but primarily with the velocity. If, 

in figure 4, the depth of flow were to drop below y lt an increase in velocity 

would be required to pass the discharge through the decreased cross-sectional 

area. But this increased velocity would require the expenditure of greater energy, 

which is not available. Hence the water would not be entirely removed from the 

cross section, and the residual would immediately act to increase the depth. If 

the depth were to rise above y lf a lower velocity would occur, and excess energy 

would be accumulated. This excess energy would immediately act to increase the 

velocity, and thus restore the original depth yj. Thus it follows that, for a given 

discharge in a given prismatic channel, of very great length with a given bed 

slope, S0, there is one, and only one, depth at which the flow will occur. This 

depth is known as the uniform depth, and designated by the symbol y0 . The discharge corresponding to this depth is known as the uniform discharge, @ 0.

COMPUTATIONS FOR UNIFORM FLOW

Problems in uniform flow frequently involve the determination of discharge for 

flow at a given depth. Here the objective is to determine an average velocity for a 

cross section such that the rate of change of the energy line, S (often called the 

friction slope), is equal to the rate of change of elevation, S0. Defying rigorous 

mathematical analysis, the problem has led to empirical relationships, of which 

the Manning formula is perhaps the most popular:

in which n is a coefficient of roughness, dependent primarily upon the lining of 

the channel, R is the hydraulic radius (cross-sectional area divided by crosssectional wetted perimeter), and S is the bed slope. The formula has objectionable features, among them being the fact that the two-thirds power of the hydraulic radius is, in some instances, an inadequate measure of boundary effects. In 

spite of its shortcomings, the formula is generally used for lack of a better one.
14 STAGE-FALL DISCHARGE RELATIONS

Fortunately, in the laboratory investigations the use of such a formula was 

unnecessary. With a given steady flow in the laboratory channel, the regulating 

equipment was manipulated- until a uniform depth, y0 , was found to exist at all 

cross sections. The discharge, Q Q , was then measured, usually by current meter. 

Thus the values of y0 versus Q Q , or the uniform stage-discharge relations, were 

established without need of a formula. This was a great advantage in preparing 

the computed profiles, tables 12 to 16, in that it permitted the computations to 

be based on laboratory observations, and kept them free of the complications 

which attend the use of formulas such as Manning's.

CLASSIFICATIONS OF FLOW

Means now may be described for the classification of types of flow and of the 

channels in which they occur. In discussion of specific energy, there was proposed the concept of a tilting channel, permitting a variation of the bed slope, 

but otherwise holding fixed all the physical features, including roughness. Attention is now invited to uniform flow in such a channel. Assuming a given constant discharge and remembering that uniform depth is dependent on the establishment of a velocity which will demand a rate of energy expenditure, S, exactly equal to S 0 , the following facts become apparent: As the slope of the channel becomes very great the normal depth will become very small. Both the specific energy and the friction slope will be very great, but the specific energy is 

self-sustaining, and the friction slope is sustained by the high value of bed 

slope. As the lower end of the channel is raised, decreasing S 0, the value of 

normal depth, y0 , will increase. As long as the values of y are less than yc (see 

p. ID), this increase in y0 will be accompanied by a decrease in specific energy. Under these conditions, the channel is said to have a steep slope, and the 

state of the flow is said to be rapid, or shooting. As the slope of the channel is 

further decreased the condition is reached in which the normal depth will be exactly equal to the critical depth (each being equal to twice the velocity head) 

and the specific energy reaches a minimum value. Under these conditions, the 

channel is said to have critical slope, and the state of flow is said to be critical. 

As the slope of the channel is still further decreased, the normal depth will continue to increase, but since y 0 is now greater than yc , the increase in y0 will be 

accompanied by an increase in specific energy. Under these conditions, the 

channel is said to have a mild slope, and the state of the flow is said to be 

tranquil, or turbulent. Other channel slopes, such as adverse, are recognized, 

but these have no application within the scope of this report. Another state of 

flow (laminar) may occasionally occur, but a different sort of criterion must be 

used for its proper description. This subject will be mentioned again in a later 

section of the report. (See p. 49 .)
THEORY OF STEADY FLOW IN OPEN PRISMATIC CHANNELS 15

In the preceding: paragraph, the assumptions are given as a constant (steady) 

discharge and uniform depth. This second assumption is the same as assuming a 

prismatic channel of very great length (see p. 13). In a much shorter channel, 

conditions of uniform flow will not necessarily exist, but rather the depth at successive sections may vary. Thus, if a channel of mild slope be obstructed by a 

dam, the section immediately upstream will have a depth greater than the uniform 

depth; if this channel ends abruptly as a free fall, the section immediately upstream will have a depth less than the uniform depth. There are other conditions 

of discharge, such as under a sluice gate, which may result in a condition of flow 

in which depth may be less than critical depth, as well as less than uniform 

depth. Such possible conditions give rise to further classification of flow: when 

the depth is greater than both the normal and critical, the flow is said to be class 

1; when depth is greater than either of these values, but less than the other, the 

flow is said to be class 2; when depth is less than both values, flow is said to 

be class 3. In the measurement of river discharge, most of the problems which 

arise are in connection with class 1 flow in channels of mild slope or, as it is 

commonly expressed, the M-l profiles. Although the laboratory data included a 

few observations on M-2 profiles (see p. 26), the present analysis is limited to 

M-l conditions.

NONUNIFORM FLOW

GENERAL CONSIDERATIONS

Figure 5 represents a case of gradually varied flow. This example differs from 

uniform flow in several respects, the most obvious of which is that successive 

values of y are not equal but gradually increase from left to right. Since the chanFigure 5. -Longitudinal section of a case of gradually varied flow.
16 STAGE-FALL-DISCHARGE RELATIONS

nel cross section is the same at all points, and the discharge is constant, it follows that velocity (and velocity head) must decrease as y increases. As in uniform flow, the finite value of S0 produces variation, from section to section, in 

the value of Z. Thus all the factors which enter into the evaluation of E become 

variable quantities.

To study the variations of these factors in the reach x between the point at 

which y = yj and the point at which y = y 2 , recourse is made to a very short 

reach, Ax, such that the rates of variation within the reach are essentially the 

same as the rates of variation at any given point that is, the same as the slope 

of the various lines. This slope is obtained by taking the derivative, with respect to x, of equation (1), replacing d by the value A for a finite, but very short, 

reach:

GEOMETRY OF THE REACH, Av

The interrelation of these variables is illustrated in figure 6, page 17, which 

is a plot of actual data from Bulletin 381, table 1, page 39 (Lansford andMitchell, 

1949). Data not given in table 1 have been computed by principles set forth under 

"Total energy of a cross section," with the assumption that cm is unity. Datum 

plane has been taken as the horizontal plane which intersects the channel bed at 

station 0. Taking as Ax the reach from station 1212 to station 1182 and following the simple rule that change in any item is equal to the final value minus the 

initial value, the following observations become apparent:

(1) AE has the value 4.102 - 4.144 = -0.042, or the distance between 

points (1) and (2) in figure 6, the line through point (2) having been drawn 

parallel to the datum plane and through energy line at initial section.

(2) A2 has the value 3.546 - 3.636 = -0.090, or the distance between 

points (3) and (4), the line through point (4) having been drawn parallel to 

the datum plane and through bed elevation at initial section.

(3) Ay has the value of 0.523 - 0.466 - 0.057, or the distance between 

points (5) and (6), the line through point (6) having been drawn parallel to 

bed slope and through water surface at initial section.

(4) A(7 2/2£) has the value 4.060 - 4.069 = -0.009, or the distance between 

points (7) and (5), the line through point (7) having been drawn parallel to 

the energy line and through water surface at initial section.
THEORY OF STEADY FLOW IN OPEN PRISMATIC CHANNELS 17

CO 

CO

Elevation 

4.144

I

4.012

3.636

3.546

 0

1212 1197 

Ax1182

Figure 6. Gradually varied flow in a very short reach.
18 STAGE-FALL-DISCHARGE RELATIONS

(5) A# has the value of 4.060 - 4.012 = 0.048, or the distance between 

points (7) and (6), lines through these points having been drawn as already 

described; obviously A// represents the change of energy line with respect 

to the bed slope.

(6) Ax, of course, has the value 1182 - 1212 = -30, or the distance between 

y = 0.523 and y = 0.466.

Special attention is invited to the fact that the change in specific energy, A#, 

is equal to the algebraic sum of the change in depth, Ay, and the change in velocity head, A(7 2/2g).

COORDINATE SYSTEMS AND CONVENTIONS

Justification already has been made for the practice of measuring y and x in 

vertical and horizontal directions, rather than with respect to channel bed. It 

generally is convenient to establish the datum plane as was done in preceding 

article that is, the horizontal plane which intersects the channel bed at the 

point for which y has some predetermined value usually that for the downstream end of the reach of channel. (In the laboratory work, this value of y was 

invariably taken as 4.000 feet. For further discussion, see p. 35.)

In the use of equation (6) it is customary to adopt the convention that a sustaining slope that is, a channel which slopes downward in the direction of 

flow is a positive slope. \Vith the convention as used in the preceding article, 

it will be noted that both dZ and dx are negative, so that dZ/dx (the sustaining 

slope) is positive in accordance with usual custom.

As a check upon the consistency of these evaluations, the numerical values of 

the preceding article are now substituted in equation (6):

dE dZ dy d (V 2) ,,.

( '

-0.042 -0,090 +0.057 -0.009 

-30 -30 + -30 -30

0.0014 = 0.0030 - 0.0019 + 0.0003 

0.0014 « 0.0014

In working with a finite reach of channel, great caution must be observed to 

keep the reach, Ax, sufficiently short to avoid the effects of curvature.
THEORY OF STEADY FLOW IN OPEN PRISMATIC CHANNELS 19 

INTEGRATION OF THE VARIED-FLOW EQUATION

It has just been indicated that, under varied flow, computations for finite 

reaches are valid only if the reach, Ax, is very short. Thus for great accuracy, 

one is tempted to return to the infinitesimal reach, dx, and apply methods of integration to obtain values for a finite reach. This is the more desirable since, in 

most practical computations it is necessary to rearrange equation (6) into a form 

such that x may be computed from available data. In other words, it becomes desirable to combine as many as possible of the various factors, solve the resulting 

expression for dx, and integrate the final expression between appropriate limits 

of y. Unfortunately, the mathematics becomes extremely complicated for all except certain special cases.

Considering an infinitesimal reach:

dE dZ dy d (V 2 1 ,,.

---,-. _ -, ._ i _^^ j_ - j - -- i ID)

dx dx dx dx \2g\

To appreciate the significance of this equation it is convenient to recall (see 

p. 9) that y + V 2/2g = H, so that equation (6) may be written

dE_ = d£ d£ (6a) 

dx dx dx

which usually is interpreted to indicate that the rate of change (with respect to 

x) of energy is equal to the rate of change of bed elevation plus the rate of change 

of specific energy. In theoretical mechanics it is customary to compute change as 

the final value of a quantity minus the initial value of the quantity. But, as flow 

always involves a loss of energy, such computations here result in undesirable 

negative values for dE. In this report, this situation is remedied by use of a dE 

which is the negative of the above, or initial value minus final value, and which, 

to avoid confusion, is termed loss rather than change. l

It is entirely appropriate that equation (6a) be interpreted to indicate that the 

rate of loss of energy is equal to the rate of loss of bed elevation plus the rate of 

loss of specific energy. It is not proper, however, to regard the rate of loss of 

energy as equal to the rate of lass of bed elevation plus the rate of change of 

specific energy. If, in equation (6) or (6a), one term is regarded as a loss, then 

all terms should be regarded as loss, or initial value minus final value.

The author is indebted to Carl E. Kindsvater, Professor of Civil Engineering, Georgia Institute 

of Technology, for suggesting the use of this distinction between loss and change,
20 STAGE-FALL-DISCHARGE RELATIONS

Considering figure 6 now to represent an infinitesimal reach, and using subscripts (1) and (2) to indicate the initial section and final section, respectively, 

dE/dx = (E l - E 2)/dx = S, and dZ/dx = (Z l - Z2)/dx = S0, in which both dE and 

dZ are regarded as loss. Making these substitutions in equation (6) gives

dy _d_| 

dx dx (7)

a form of expression which is common to all the methods which are subsequently 

discussed. The differences in the methods lie in the differences in treatment of 

the last term of the righthand member that is, the small, but sometimes significant effect of the change in kinetic energy. From equation (3), page 10,

d (V2} -Q 2b

Making this substitution in equation (7),

dy dy <?*& 

gA 3 dx

dx

dz 

dy

1-

s- sr

-dx (8)

In the interpretation of equation (8) it should be remembered that, as S and S0 

were expressed as rates of loss, dy also must be regarded as a loss, or dy = 

y\~ y?r Some authors have regarded dy as the change, y 2 ' ^i» ^ n which case the 

minus sign vanishes from the left member of the above equation. However such a 

mixture, in one equation, of the concepts of loss and change frequently has been 

a source of confusion, and of error in computations.
THEORY OF STEADY FLOW IN OPEN PRISMATIC CHANNELS 21

For a fixed discharge in a channel of known geometric characteristics, dx in 

formula (8) might be computed for two given values of y, provided S were known. 

Problems involved in computing S for uniform flow were discussed on page 13. 

In nonuniform flow it must be assumed that S is the same as for uniform flow under the given conditions of discharge and depth in other words, discharge 

through a given cross section is a unique function of depth and friction slope. 

An expression for such an S may be obtained by use of the fundamental equation 

0 = AV, and the substitution for V of one of the usual formulas for uniform flow. 

Using the Manning formula (see p. 13),

1.486 a/3 ya 

0 " n 0

Here, for a given depth of flow, y, the expression

, 1.486 a/

is dependent only on the cross section. Bakhmeteff (1932) has suggested that, 

for simplicity, this expression may be replaced by a single symbol, K, called 

conveyance. Letting K Q be that particular value of conveyance in which S = S 0, 

or the flow is uniform,

Qo " K 0 s o'/J 

or, -s

Now if the flow be at the same depth, y, but be gradually varied instead of uniform, K must be the same as before, since it is dependent only on the cross section which is unchanged. However, for gradually varied flow, 0 no longer is Q0, 

nor is S equal to S 0, so the above equation becomes

0 2

S = ^_ (10) 

K 2 ^0

Similarity, the discharge Q Q , when flowing at a depth y other than y0 will be related to a specific value of S and to a K dependent on the depth y, so that

K 2
22 STAGE-FAIL-DISCHARGE RELATIONS

Dividing equation (11) by equation (9),

Furthermore,

Srv o - " S ° = - S°o 1-

and formula (8) becomes

1-

- dx

i -

K2

.dy (12)

which, except for the sign of dx, is a common form of the differential equation 

for gradually varied flow.

DIRECT INTEGRATION

Bahkmeteff has observed that K, in general and between wide limits, is an exponential function of y. With that assumption, plus the ratio between S 0 and S c , 

and making use of a table which he has computed, substitutions may be made in 

a rearrangement of equation (12) so that the distance, x, between two given and 

widely different values of y may be directly computed. In some instances, however, the ratio S0/SC may vary with y, so that it may be necessary to take comparatively small increments of y. Used with care, the method will yield results 

in satisfactory agreement with the results of other reliable methods.

Von Seggern (1950) made use of Bakhmeteff's observation that K was an exponential function of y, and added the thought that M = A\jA/b also is an exponential function of y, and that (M C /M) 2 may be substituted in equation (12)in 

place of Q 2b/gA^. With these assumptions, a table similar to Bakhmeteff's, and 

a second table to be used in connection with the new assumption, substitutions may be made in another rearrangement of equation (12) so that the distance, x, between two widely different values of y may be directly computed. 

In many instances the method will yield results which are in satisfactory agreement with those of other reliable methods.

A detailed comparison of the Bakhmeteff and Von Seggern methods will be 

found among the published discussions of Von Seggern's paper. A comparison 

between the results of these two methods and the laboratory observations will 

be found on page 55.
THEORY OF STEADY FLOW IN OPEN PRISMATIC CHANNELS 

THE STEP METHOD

23

After noting the difficulties which attend any attempt at complete integration, 

one is inclined to look with greater favor upon the step method the method in 

which average values at the ends of a reach are assumed to apply throughout the 

reach. This favorable attitude is the more pronounced in view of the extreme simplicity both of the required formula and of the method of its derivation. Figure 7

Figure 7. -Geometry of the step method.

reproduces certain pertinent features of figure 6. Point O is the water surface at 

the upstream end of the reach. Points A, B, and C are the points shown in figure 

6 as 5, 6, and 7, respectively. Point D is the intersection of a line through A, B, 

and C with a horizontal line through point 0.

It is obvious that

and

(D-B) = (D-C) + (C-B).

(D-B) = Ax S0 

(D-C) = Ax S

(13)

The significance of (C-B) is not so apparent. It must first be noted that both (D-B) 

and (D-C) represent a loss, or initial value minus final value, and must be distinguished from change, which was defined (p. 16 ) as the final value minus the 

initial value. For consistency, as well as convenience, the final term in equation 

(13) should be viewed also as a loss. (C-B), however, is not the loss in H between the initial and final sections; rather it is the gain in H or, as is generally 

called, the change in H. At either section, the value of // is represented by the 

elevation of the S line minus the elevation of the S0 line. At the initial section, 

H, as depicted by figure 7, is zero; at the final section, H is represented by 

(C-B). Hence the loss in H from the initial to the final section is

Aff = 0 - (C-B)
24 STAGE-FALL-DISCHARGE RELATIONS

whence (C-B) = -AH 

Making these substitutions in equation (13),

AxS 0 = AxS-AH 

whence Ax = -M/(S0-S), (14)

in which AH is defined as a loss, or initial value minus final value, consistent 

with the computation of S and of S0 .

For those who prefer an analytical approach, this same relation might have 

been derived directly from equation (8). In that equation, the numerator of the 

right-hand term might have been written: dy - Q 2bdy/gA 3 . As pointed out in the 

development of that equation,

-Q zbdy/gA 3 = d( 

Remembering that

dy + d(V 2/%) = dH, 

equation (8) becomes:

SQ - S

the same as in the graphical development, except for the change between d and 

A, representing the difference between an infinitesimal and a finite reach. Since 

S varies from point to point throughout a finite reach, formula (14) should be 

used only for short reaches. Furthermore, S should be computed for each end of 

the reach, and the average value, S, should be used as being more representative of the proper value of S. Thus the proper equation for step-method computation becomes:

Ax = -AH/(S0 - S). (15)

This formula has been used as the basis of extensive computations in the 

following section of this report. (See pp. 52-55.)
STAGE-FALL-DISCHARGE RELATIONS 25

THE LABORATORY DATA

In the introduction to this report (see p. 3) reference was made to University 

of Illinois Engineering Experiment Station Bulletin 381 (Lansford and Mitchell, 

1949) which presents the basic experimental data that form the foundation for the 

present report. In arranging with the university for the collection of the data, the 

primary objective of the Geological Survey was the acquisition of a reservoir of 

reliable data on which might be based detailed studies of stage-fall-discharge 

relations. For practical reasons, these detailed studies could not be immediately 

completed. Thus, Bulletin 381 was designed and published with the particular 

objective of preserving the base data and of making it immediately available to 

such parties as might care to make their own analysis. This report is designed 

to present an analysis of those data, but does not duplicate the tables of observations previously published.

In its field association with problems of stage-fall-discharge relations, the 

Geological Survey is almost always limited to cases of the M-l type of profile: 

the profile in which the actual observed depths are greater than the normal depth 

and the normal depth is greater than the critical depth. In the design of the experiment, a basic consideration was that the resulting profiles should be of this 

type that is, the channel should be one of mild slope. An exceedingly flat bed 

slope would perhaps more nearly have approached the slopes of natural streams, 

but it would have made the curves of such great length as to add greatly to the 

experimental work required and, for the length of channel available, might require greater accuracy than would be possible, even in the laboratory. A steeper 

slope would decrease the length of the curves, but would require that the channel 

roughness be greater to keep the depths greater than critical.

It appeared feasible to line the channel with a roughness material which would 

provide a value of n of about 0.02. The Manning formula indicated that, with this 

value of n, a slope of S 0 = 0.003 would provide normal depths well above the 

critical. With an effective channel length of about 135 feet, this bed slope provided a total change in bed elevation of about 0.40 foot, which appeared to be 

satisfactory from the standpoint of the probable accuracy with which the observations could be made. Later it was found feasible to vary the channel with 

respect to roughness, and with respect to shape of cross section. However, the 

amount of labor involved in varying the bed slope would have been so great as to 

make that change prohibitive. Thus, throughout the series of experimental observations, channel conditions were made to conform with conditions which might 

be met in the field, but only within the limitations imposed by practical laboratory requirements.
26 STAGE-FALL-DISCHARGE RELATIONS

DESCRIPTION OF THE DATA

With respect to material which is useful in the present analysis, and in contrast with other material which is primarily informative, Bulletin 381 contains 

the following data:

1. Details of the cross sections for which profiles were observed (See fig. 

4, p. 14.)

2. Normal depth-discharge ratings for cross sections 2-6, inclusive. (See 

Fig 6, pp. 18-19 and tables 42-46, pp. 81-85.)

3. Observed surface profiles: for the seven cross sections, there are a total 

of 41 profiles. (See figs. 8-12, pp. 27-31, and tables 1-41, pp. 39-80.)

4. Typical velocity-distribution diagrams for representative cross sections. 

(See figs. 13-15, pp. 32, 33 and fol. 34.)

For cross sections 5-7, the dimensions given in Bulletin 381 are nominal; a careful survey indicated that elevations for the break in side slopes should be at 

1.011 instead of 1.00, and 1.207 instead of 1.200. A table of areas, based on 

these refined elevations, appears in this report as table 27. (See p. 154^

A few words might be said concerning the observed surface profiles. For all 

cross sections, each profile was observed from the greatest depth which could 

be obtained (limited either by the depth of the laboratory channel, or in the case 

of low discharges, by the maximum depth which could be induced by extreme 

manipulation of the backwater gate) to that point near normal depth at which the 

water surface becomes so nearly parallel to the bed slope as to represent the 

practical limit of observation. For cross sections 1 to 4, inclusive, depths were 

observed at points along the channel which were 15 feet apart. For cross sections 5 to 7, inclusive, depths were observed at 10-foot intervals, except for 

very great depths of low discharge, for which 20-foot intervals were used. In 

cross section 3, yn = 1.512 feet (see Bull. 381, table 22, p. 60) a portion of the 

M-2 curve, approximately 300 feet in length, was included. Likewise for cross 

section 5, yn = 1.704 feet (see Bull. 381, table 35, p. 74) a portion of the M-2 

curve, approximately 230 feet in length has been included.

Owing to various causes, among which might be mentioned the personal error 

in observations and the effects of local disturbances in the channel, differences 

between depths for successive equal increments of length are sometimes somewhat irregular. In some phases of the analysis (such as substitutions in equation 

(19), p. 44) it has appeared advisable to smooth out these irregularities by 

applying small and.compensating changes to the observed values. Such a table, 

designated as a smoothed profile, has been prepared for each of the observed 

profiles of cross sections 2-6, inclusive. These smoothed profiles, not published 

in Bulletin 381, are found as tables 1-5 in this report.
THE LABORATORY DATA 

Table 1.--Smoothed profiles, cross section 2

27

y0»0.391

 

2.3280 

2.2830 

2.2380 

2. 1930 

2. 1480

2.1030 

2.0580 

2.0131 

1.9682 

1.9233

1.8784 

1.8335 

1.7886 

1.7438 

1.6990

y0=0.612

3.0030

2 01 sn

2

Q7 1 A

2 fl97n

2 7B30

2

£QCO

2.6513

2.6074

2

e^oc

2.5196

2.4757

2 4.318

2 1B7O

2.3440 

2.3002 

2.2564 

2.2126 

2.1688

2.1250 

2.0812 

2.0374 

1.9936 

1.9499

1.9062 

1.8625 

1.8188 

1.7752 

1.7316

y0=0.812

» 91 CA

3.0860

3.0430

3.0000

2 0^70

2.9140

2 a? in

2 C9fift

2 70?n

2

7 jl 1 A

2 £QQO

2.6561

2.6132

2 597A

2.4845

2.4416

2 ^OO7

2.3558 

2.3129 

2.2700 

2.2271 

2.1842

2.1413 

2.0984 

2.0556 

2.0128 

1.9701

1.9275 

1.8850 

1.8427 

1.8006 

1.7587

y0=i.!2l

3.5090

3.4668

3.4246

3 31(94.

3 3403

3 9009

3

OC £ 1

3.1300

3

AQO A

3.0460

3.0040

2

O£O 1

2 Q909

2

fi7ft3

2

O Q£l

2 7Q4,c

2

^c o^c

2 71ftft

2.6690

2

£9*?Q

2.5854

2.5436

2

C A 1 Q

2.4600

2.4183 

2.3767 

2.3352 

2.2939 

2.2528

2.2119 

2.1713 

2.1310 

2.0910 

2.0513

2.0120 

1.9731 

1.9346 

1.8965 

1.8589

y0=1.489

3

7AQ A

3.6670

3

£9£A

3 COC1

3 5442

3.5034

3 A ^C97

Q A991

3

OO 1 H

3 3412

 > Qf)f)Q

3

9/tfl7

3 O9 f\H

3

1 Q f\/i

3.1407

0 1 flflQ

3.0612

3

A9 1 ^

2 QO99

2

Q A O A

2.9040

2

f>£<i')

2

Ot)££

2 7OB9

2.7500 

27120

2.6743

2

XO£ft

2

CQOQ

2.5630

2.5266 

2.4906 

2.4550 

2.4198 

2.3850

2.3507 

2.3169 

2.2836 

2,2508 

2.2185

2.1867 

2.1554 

2.1246 

2.0943 

2.0645

y0= 1.840

Q 709?

3

£ £ QO

3 /I919

3

CQ£7

0 CJO t

3.5103

3 179A

3 4347

r> 7Q79

3 3599

3 799O

3 90^0

3 2492

3 919A

3.1767

3. 1409

3.1054

3.0702

3 0353

3.0007

2.9664

2 0^21

2 OQ07

2 or eo

2.8322 

2.7994

2.7669

2 714.7

2 7f\9ft

2 HI 1 9

2.6400 

2.6092 

2.5788 

2.5489 

2.5195

2.4907 

2.4625 

2.4350 

2.4082 

2.3821

2.3567 

2.3320 

2.3080 

2.2847 

2.2621
28 STAGE-FALL-DISCHARGE RELATIONS

Table 1.--Smoothed profiles, cross section 2 Continued

y0=0.391

1.6542

1.6094

1.5646

1.5199

1.4752

1.4305

1.3859

1.3413

1.2967

1.2522

1.2077

1.1633

1.1189

1.0746

1.0303

.9861

.9421

.8984

.8551

.8123

.7701

.7287

.6882

.6487

.6104

.5736

.5387

.5062

.4767

.4512

.4317

.4182

.4097

.4047

.4012

.3985

.3964

3O4.O 374O 

3O3£ . O7OD 

. 7Q97 *jy ft 1

. 3921

501 7 . **y 1. 1 

. 1Q1A *>7 A** 

 IQIO . jy ift 

QQ1 1 . *>7 1. 1.

y0=0.612

1.6880

1.6445

1.6010

1.5576

1.5143

1.4711

1.4280

1.3850

1.3422

1.2996

1.2572

1.2151

1.1733

1.1318

1.0907

1.0501

1.0101

.9709

.9326

.8954

.8595

.8252

.7927

.7623

.7344

.7095

.6881

.6702

.6558

.6444

.6355

.6287

.6238

.6202

.6175

.6155

.6141

£1 39 0132

£-1 rt<»

. 6127

£ 1 n A

. 0124 

jr 1 nn

. 0122 

d 1 91 . 0121

.6120

y o=0. 8 12

1.7170

1.6755

1.6343

1.5934

1.5528

1.5125

1.4725

1.4329

1.3937

1.3549

1.3166

1.2788

1.2415

1.2048

1.1689

1.1340

1.1003

1.0680

1.0373

1.0084

.9814

.9564

.9335

.9128

.8944

.8784

.8648

.8535

.8442

.8366

.8304

.8254

.8214

.8182

.8157

.8139

. 01 OlZ 971

. ni olZU on

0117 . oil ( 

fi 1 1 C. . olio

y0=1.121

1.8218

1.7852

1.7491

1.7136

1.6787

1.6444

1.6108

1.5779

1.5458

1.5145

1.4841

1.4546

1.4261

1.3987

1.3725

1.3476

1.3241

1.3021

1.2817

1.2630

1.2461

1.2310

1.2177

1.2061

1.1960

1.1872

1.1795

1.1727

1.1667

1.1614

1.1567

1.1525

1.1488

1.1455

1. 1426

1.1401

1 1 97Q A « i j c y

1.1360

1.1344

1. 1330

y 0=1.489

2.0352

2.0064

1.9781

1.9503

1.9230

1.8962

1.8699

1.8441

1.8188

1.7940

1.7697

1.7459

1.7226

1.6999

1.6783

1.6583

1.6403

1.6243

1.6103

1.5981

1.5874

1.5779

1.5694

1.5618

1.5550

1.5489

1.5433

1.5382

1.5335

1.5291

1.5250

1.5212

1.5177

1.5144

1.5113

1.5084

y0= 1.840

2.2402

2.2190

2.1985

2.1788

2.1599

2.1418

2.1245

2.1079

2.0920

2.0768

2.0623

2.0485

2.0353

2.0227

2.0107

1.9992

1.9882

1.9777

1.9677

1.9582

1.9491

1. 9404

1.9321

1.9242

1.9167

1.9096

1.9029

1.8966

1.8907

1.8852

1.8801

1.8754

1.8711

1.8671

1.8634

1.8600
THE LABORATORY DATA 

Table 2.--Smoothed profiles, cross section 3

29

y0=0.404

2.7890 

2.7442

2.6994 

2. 6546 

2.6098 

2.5650 

2. 5202

2.4754 

2. 4306 

2. 3858 

2.3410 

2.2962

2.2514 

29n££

2 1 £1 Q

2. 1170

2 AO7 A

1 OO9£

1 OQ-Jfl

1 QA DO

1.8034

1.7586

1 713Q

1. 6690

1 JCO4 3

y0=0. 606

3.0077

2 n£Q£

2 Q7CC

2.8315 

2.7875

2.7436 

2. 6997 

2.6558 

2. 6120 

2. 5682

2. 5244 

2.4807 

2. 4370 

2. 3933 

2. 3497

2. 3061

2 969C

2.1755

2 f\jlC9

1 O^QC

1. 8719

1 Q9Q7

1.7855

1. 7423

1 £QQO

y0=0. 773

3 1 QA £

3.1407

3.0969

3.0095

2 Q7QO

2.8353 

2.7919

2.7485 

2.7052 

2. 6619 

2.6187 

2.5755

2. 5324 

2.4893 

2.4463 

2.4033 

2.3604

2.3175

2 1OOO

2. 1465

2.0613

2 A1 fiQ

1.9764

1.9341

I fiOlO

1 QAOO

1

OA7Q

1. 7660

y0=i.07i

3.4637

3 37O9

3

33Cr

3.2507

3 9AQjl

3 1 ££.<)

3.1241

3.0402

2. 9567

2.8736 

2.8322

2.7909 

2. 7497 

2. 7087 

2. 6678 

2. 6270

2.5863 

2.5457 

2.5052 

2. 4649 

2.4247

2. 3847

2

e\£CC

2

c)O£O

2 1 AQ3

2.1096

2.0711

1. 9948

1.9571

1. 9197

1 QQ9£

1. 8459

y0=l-512

3.6699

3.6295

3.5490

3 ^nno

3

J £OQ

3 ino9

3. 1917

3

1 1 Oji

3.0744 

3.0356

2.9586 

2.9204

2. 8825 

2. 8450 

2.8080 

2.7716 

2. 7360

2.7015 

2. 6684 

2.6363 

2. 6048 

2.5736

2.5425 

2.5114

2 A Qf\A

2.4494

2 A 1 OC

2

OCXQ

2 90^

2

O£C 1

2. 1760

2. 1471

2

1 1 n£

y0=0.404

1.5796

1.4455

1.4008

1. 3561

1. 3114

1.2667

1.1775

1. 1330 

1rioQ£

1. 0004

O££c

.9130

.7851 

.7439

.7037 

.6647 

.6273 

.5918 

.5586

.5280 

.5002 

.4755 

.4541 

.4362

.4220 

.4117

.4055

.4026

y0=0. 606

1.6561

1.6130

1.5700

1. 5270

1.4840

1.4411

1 QOQO

1.3554

1.3127

1.2701

i.. ftft 1 1 

1.1855

1. 1436

1. 1021

1.0611

1.0207

Ofil 1

.9051 

.8691

.8345 

.8017 

.7709 

.7423 

.7161

.6926 

.6721 

.6551 

.6416 

.6316

.6246 

.6196

.6161

£.1 1£.

.6116

.6101 

.6090

An 09

.6076

.6071

.6067

.6064

.6062

.6061

y0=0.773

1. 6830

1.6418

1.6008

1. 5601

1. 5197

1.4796

1 JOQQ

1.4003

1. 3612

1.2463

1 onoo

1. 1720

1.1357

1. 1001 

1.0654

1.0318

.9996 

.9692

.9411 

.9157 

.8932 

.8735 

.8564

.8417 

.8292 

.8187 

.8100 

.8029

.7972 

.7927

7ftQ9

TQ£C

TO4C

y0=1.071

1.8096

1.7737

1 73ft9

1 7A^2

1.6687

1.6348

1.6014

1.5686

1. 5364

1. 5049

1. 4741

1.4147

1 QQXO

I

oeox

1. 3065

1 OQOO

1.2592 

1.2376

1.2174 

1. 1988 

1. 1819 

1. 1668 

1. 1535

1. 1418 

1. 1316 

1.1228 

1. 1153 

1. 1090

1. 1038 

1. 0996

1.0962

1.0934

y0=i-5i2

2 A one

2

A£OO

2.0355

2 AAQ£

1 QR91

1.9560

1.9303

1.9051

1. 8566

1. 8336

1.8117

1.7911

1.7718

1.7536

1.7363

1.7199

1. 6898 

1.6761

1. 6633 

1.6514 

1.6404 

1.6303 

1.6211

1. 6128 

1.6054 

1.5989 

1. 5933 

1.5886

1. 5847
30 STAGE-FALL-DISCHARGE RELATIONS 

Table 3. Smoothed profiles, cross section 4

y0=O.S28

2.7950

2.7501

2.7052

2.6603

2.6154

2.5705

2.5256

2.4807

2.4358

2.3909

2.3460

2.3011

2.2562

2.2113

2.1665

2.1217

2.0769

2.0321

1.9874

1.9427

1.8980

1.8534

1.8088

1.7643

1.7199

1.6756

1.6314

1.5873

1.5433

1.4994

y0=0.780

3.0157 

2Q79ft 71 &\J

2 * Q9B3 y Zoo

2

QQ J £

. oo4O

2

0AAQ

» o*«w 7

2.7972

2.7535

2.7098

2.6662

2.6226

2. 5790

2.5354

2.4919

2.4484

2.4049

2.3614

2.3179

2.2745

2.2311

2.1877

2.1443

2.1010

2.0577

2.0144

1.9711

1.9278

1.8846

1.8414

1.7982

1.7550

1.7119

1.6690

1.6264

1.5842

1.5424

y0=1.028

3 9799 <J ftmtfttt 

3 17Q1 J X 1 7 X 

31 3£1

 xoOx 

3 nooo J \ty Oft

3.0504

3.0076

2

Q£ JQ

. 7O47

2. 9223

2 Q7QD . o lyo 

2.8374

2.7950

2.7527

2.7105

2.6684

2.6264

2.5845

2.5427

2.5010

2.4594

2.4179

2.3765

2.3352

2.2940

2.2529

2.2119

2.1710

2. 1302

2.0896

2.0492

2.0090

1.9690

1.9293

1.8899

1.8508

1.8121

1.7738

1.7360

1.6987

1.6620

y0=l.369

3

A£n£

. 4O7Q

3 A9B9 . 4ZO4 

33Q£O . OOQ7 

r> 74,67 *> . <j*>u i

3

O f\ A £ 

. 3046

3

9£3£ . ZooO 

o 2227

*> . liffli ( 

31fl 1 O

. JLO 17

3. 1412

3.1006

3.0601 

r> nlQ7 *> . u 1.7 1

2.9794

2.9392 

2.8991

2.8591

2.8193

2.7797

2.7403

2.7011

2.6621

2.6232

2.5845

2. 5460

2.5077

2.4696

2.4317

2.3940

2.3565

2.3193

2.2824

2.2458

2.2095

2.1736

2.1381

2.1030

2.0684

2.0343

2.0008

1.9679

1.9357

1.9042

1.8735

1.8436

y0=l.877

3.5384

3.5009

3.4636

3.4265

3 QQO£

. OO7t> 

Q ICOn O. 39^7

3

0 1 £ A

. olt)4 

30OA1 . ZoU i

3. 2441

o 2HRA O . £UO9

3. 1730 Q 117Q O . 13 I 7

3.1031

3.0686

3.0344 

3.0005

2.9669

2.9336

2.9006

2.8680

2.8358

2.8040

2.7726

2.7416

2.7110

2.6808

2.6510

2.6216

2.5926

2.5641

2.5361

2.5086

2.4817

2.4554

2.4297

2.4047

2.3804

2.3568

2.3340

2.3120

2.2908

2.2703

2.2505

2.2314

2.2129

y0=2.335

3.6492

3.6142

3 . 3 C7OA 1 74

3.5448

3.5104

3

jl?£O 4 f OZ 

3 A A OO 44*4

3.4084

3 

77AD 3 (4o 

Q *> QJ1A J**i** 

3^h(*9 aUOZ 

o 97CO J * 4 I J A

o 94,oft J 44P&*)

3.2101

3 * 17ftA X i oU

3. 1462

3.1148

3.0838

3.0533

3.0234

2.9942

2.9658

2.9383

2.9117

2.8860

2.8612

2.8373

2.8143

2.7922

2.7709

2.7504

2.7307

2.7117

2.6934

2.6757

2.6586

2.6420

2. 6259

2.6103

2.5951

2.5803

2.5659

2.5519

2.5383

2.5251
THE LABORATORY DATA 

Table 3. Smoothed profiles, cross section 4 --Continued

31

y0=0.528

1.4556 

1.4119 

1.3683 

1. 3248 

1.2814

1.2381 

1.1949 

1.1519 

1.1091 

1.0666

1.0245 

.9830 

.9422 

.9023 

.8635

.8260 

.7900 

.7557 

.7233

.6930 

.6650

£ OQjC

.6171

* J y « y

. JO 1 1

.5575

.5490

. j * £*)

.5320

.5305

COQC

^Ofifi

^9fi3

y0 =0.780

1.5011 

1.4603 

1.4201 

1.3805 

1.3416

1.3034 

1.2660 

1.2294 

1.1937 

1.1589

1.1251 

1.0923 

1. 0606 

1.0301 

1.0009

.9732 

.9471 

.9228 

.9004

 ooUl

 oloU

.8011

.7972

.7940 

.7912

7Q £Q

7ft t:o

.7840

7Q31

7BO^

7Q91

7O1 O

. I Ol3

 ( o Lo 

701 1

.7810

y0=1.028

1.6259 

1.5905 

1.5558 

1.5218 

1.4886

1.4562 

1.4247 

1.3941 

1.3645 

1.3359

1.3084 

1.2821 

1.2571 

1.2334 

1.2111

1.1903 

1.1711 

1. 1535 

1.1376

1

1 OOJ

1.1109

1.1000 

1.0906

1* UoZ3

1.0755

1.0694 

1.0640

!.« v J 7 ft

1.0549 

1.0510

1.0475

1.0444

1.0416

I

f! 3£Q

1.0350

1.0334

1.0308

1 ftOQfi

1.0290

1.0280

y0=l.369

1.8146 

1.7865 

1.7593 

1.7330 

1.7076

1.6832 

1.6598 

1.6374 

1.6160 

1.5957

1.5765 

1.5584 

1.5414 

1.5255 

1.5106

1.4967 

1.4838 

1.4719 

1.4610

1.4511

1

A A OO

1

j o ,1 O

1.4272

1.4206

1.4144

1.4084 

1.4026

i . J y 1 v

1.3916

y0=i-877

2.1950 

2.1777 

2.1610 

2.1449 

2.1294

2. 1145 

2.1002 

2.0865 

2.0734 

2.0608

2.0487 

2.0371 

2.0260 

2.0154 

2.0053

1.9957 

1.9866 

1.9780 

1.9699

1 O£93

1.9486

1

QO 1 O

y0=2.335

2.5123 

2.4999 

2.4879 

2.4763 

2.4651

2.4543 

2.4439 

2.4339 

2.4243 

2.4151

2.4063 

2.3979 

2.3899 

2.3823 

2.3751

2.3683 

2.3619 

2.3559 

2.3503
32 STAGE-FALL-DISCHARGE RELATIONS 

Table 4.--Smoothed profiles, cross section 5

y0=0.456

2.4880 

2.4580 

2.4280

2.3980 

2.3680 

2.3380 

2.3080 

2.2780

2.2480 

2.2180 

2. 1880 

2.1580 

2.1280

2.0980 

2.0680 

2.0380 

2.0080 

1.9780

1.9480 

1.9180 

1.8880 

1.8581 

1.8282

1.7983 

1.7684 

1.7385 

1.7086 

1.6787

y0 =0.687

2

e XQA

9 «;73n

2 CAOA

2.4730 

2. 4430 

2.4130

2.3830 

2.3530 

2.3230 

2.2930 

2.2630

2.2330 

2.2031 

2.1732 

2.1433 

2.1134

2.0835 

2.0537 

2.0239 

1.9941 

1.9643

1.9346 

1.9049 

1.8752 

1.8456 

1.8160

1.7864 

1.7569 

1.7274 

1.6980 

1.6686

y0=0.905

2.6035

2 C7Q£

2

Cjt 07

2 ^1 3fi

2.4839 

2.4540 

2.4241

2.3943 

2. 364.5 

2.3347 

2.3049 

2.2751

2.2453 

2.2155 

2.1857 

2.1559 

2.1262

2.0965 

2.0668 

2.0371 

2.0074 

1.9777

1.9480 

1.9184 

1.8888 

1.8592 

1.8296

1.8001 

1.7706 

1.7411 

1.7117 

1.6823

y0=1.158

2 £Qf\f\

2

£F AQ

2

£<\f\£

 *jy\fy 

2C£ 1 9

2.5315

2.5018 

2.4721 

2.4424

2.4127 

2.3830 

2.3533 

2.3236 

2.2939

2.2642 

2.2345 

2.2048 

2.1751 

2. 1454

2.1157 

2.0860 

2.0564 

2.0268 

1.9972

1.9677 

1.9382 

1.9088 

1.8795 

1.8503

1.8212 

1.7922 

1.7633 

1.7345 

1.7059

y0=1.285

2

*7£QA

2 74q<

2.7040

2.6745

2.6450

2.6155

2

c Q£A

2.5565

2 C07A

2.4975 

2.4680 

2.4385

2.4090 

2.3796 

2.3502 

2.3208 

2.2914

2.2620 

2.2326 

2.2033 

2.1740 

2. 1447

2.1154 

2.0861 

2.0569 

2.0277 

1.9985

1.9693 

1.9402 

1.9111 

1.8821 

1.8532

1.8245 

1.7960 

1.7678 

1.7399 

1.7123

y0=1.704

3.1540

3. 1253

3.0966

3.0679

3 ft3Q3

3.0107

2 QP.91

9 o^t;

2 QOCA

2 QO£C

2 Q£Qf\

2

QO A£

2 0119

2 7Q9O

2.7545

2 79^9

2

£ AOA

2 ££QQ

2.6417

2 <:i 77

2 1010

a eeon

2.5303 

2.5028 

2.4755

2.4484 

2.4215 

2.3948 

2.3683 

2.3420

2.3159 

2.2901 

2.2646 

2.2394 

2.2145

2.1899 

2.1656 

2.1416 

2.1180 

2.0948

2.0720 

2.0496 

2.0277 

2.0063 

1.9855

1.9653 

1.9458 

1.9270 

1.9090 

1.8918

y 0=2.525

3.6740

3.6477

3 £91 7

3.5960

3.5706

3.5455

3 5207

3 A n£ *>

Q 4.79(1

3.4481

3.4245

a 1012

Q 7709

3 3555

3.3331

3.3110

3 9QQ9

3 9£77

3.2465

3 OOCX

3 2050

3.1847

3.1647

3.1450

3.1256

3.1065

3 A077

3 n<:o9

3.0510

3.0331

3.0155

2 QO01

2 O0HO

2 9639

2.9471

2 O9ni

2 014,1

2.8979 

2.8819 

2.8661

2.8505 

2.8351 

2.8200 

2.8052 

2.7907

2.7766 

2.7629 

2.7496 

2.7368 

2.7245

2.7127 

2.7015 

2.6909 

2.6808 

2.6712

2.6621 

2.6535 

2.6453 

2.6375 

2.6301

2.6231 

2.6165 

2.6103 

2. 6045 

2.5991
THE LABORATORY DATA 33 

Table 4.--Smoothed profiles, cross section 5 --Continued

Yo=0.456

1.6488 

1.6189 

1.5890 

1.5592 

1.5294

1.4996 

1.4699 

1.4403 

1.4108 

1.3814

1.3521 

1.3229 

1.2938 

1.2649 

1.2362

1.2077 

1.1795 

1.1516

1. 1241

1.0970

1.0704

1.0443

I

n 1 QO

.9924

.9666

.9409

QICi

Qofti

Q f Cl

.8404

Ql £1

7QOO

.7691

.7466

7O A O

.7037

£Q QJ!

.6639

£4 CO

^O7-r

.6108

CQ4Q

c,7qo

.5659

5529

.5409

5299

eioo

.5108

.5026

AQC9

Anon

AQ-l-t

J 700

.4740

.4703

A £.1*)

.4646

A XOC

.4608

Aeoc,

* e oc

A C TO

.4570

4C£Q

.4567

y0=0.687

1.6393 

1.6100 

1.5808 

1.5516 

1.5225

1.4934 

1.4644 

1.4355 

1.4067 

1.3780

1.3495 

1.3213 

1.2935 

1.2662 

1.2395

1.2136 

1.1886 

1.1646

1.1416

1.1196

i nooc

1 fl?n3

1 n^fio

1.0402 i (1921

1.0045

OQ74

Q7A0

O£ A d

Q3£fi

.9234

OftQ A

OQ9Q

07 ox

n£ CO

0^9A

OOQA

OO£Q

O 1 J X

oft 90

.7914

.7804

T^OD

7CQO

.7494

.7400

.7310

.7214

.7122

.7044

y0=0.905

1.6530 

1.6238 

1.5947 

1.5657 

1.5368

1.5081 

1.4796 

1.4513 

1.4233 

1.3956

1.3683 

1.3415 

1.3153 

1.2898 

1.2650

1.2410 

1.2179

1 1 QC.P.

1.1748

1.1550

1.1365

1.1194

1.1037

1 AQO A

1.0764

1.0646

1.0538

1.0438

1.0345 i no co

1.0176

1.0099

1.0027

OO^fi

ODQO

QQ01

.9716

.9663

O£1 Q

e\&££

Q^99

QA Q 1

QAA 1*

QA t\Q

O37£

0^17

QQOl

O9OQ

y0=l. 158

1.6775 

1.6493 

1.6213 

1.5935 

1.5660

1.5388 

1.5119 

1.4853 

1.4591 

1.4334

1.4083 

1.3839 

1.3603 

1.3376 

1.3159

1.2953 

1.2759

I

oe 70

1.2411

1.2004

1.1904

1 1 Q9C.

1.1766

1.1725

1 1 £OO

1.1684

1.1676

1.1671

1.1668

1. 1666

y0=1.285

1.6851 

1.6583 

1.6319 

1.6060 

1.5806

1.5557 

1.5314 

1.5077 

1.4847 

1.4625

1.4412 

1.4209 

1.4017 

1.3838 

1.3674

1.3528 

1.3403

1

3 1 £O

1 3 ftOfl

1.3077

1.3061

1.3049

1.3040

1 9QOQ

1.3024

1 ^091

1.3019

I Qftl Q

y0=1.704

1.8754 

1.8598 

1.8450 

1.8311 

1.8181

1.8060 

1.7948 

1.7845 

1.7751 

1.7666

1.7589 

1.7520 

1.7459 

1.7406 

1.7361

1.7323 

1.7292 

1.7267

1.7247

1 7O71

1.7219

1.7210

y0=2.525

2.5940 

2.5892 

2.5847 

2.5805 

2.5765

2.5727 

2.5691 

2.5657 

1.5624 

2.5592

2.5561 

2.5531 

2.5502

2.5474 

2.5447

2.5421 

2.5396
34 STAGE-FALL-DISCHARGE RELATIONS 

Table 5.--Snoothed profiles, cross section 6

y0=0.471

2

1QQA

2 icon

2 A9ftn

 oUoU

2

O 1 Q A

2

O 1 O A

 13OU

 U?oU

 U-JoU

 71 OU

1.9480

1. 7loU

 OOoU

 O3O1 

1 OOftO

y0=0.905

2.6070

2.5770

2.5470

2.5170

2.4570

2 A 97 A

. a? i U

2.3670

. J J 1 U

i *JW 1 V

2.2770

2.2171

. lo I ft

2. 1274

*. \iy i »i

2.0676

. U J i o

» UuoU 

1O7Q9

1.9484

X. y Lo i

. oO7 V

1 nCO4

y0=i.294

2.7660

2 TQX1

2 7 A^O

2

iC77O

2.6476

2

X1 QA

2 cope

2.5000

 *» I U J

2.4411

2.4117

 <JO^<J

2.2942

2

n £Af\

2

o?c<;

2.1770 

2.1477

 11 03

 Uo"3

2.0601

. \jj\jy

1.9727

1.9437

 ?14o

1 OC70

y0=2.431

3.6597

3

*r->o 1

3.6066

3.5801

3.5537

3.5010

3.4747 

3.4485

3 00^0

3

3 jt jt Q

3 9o^n

3 OX7 X

3

n ji n jt

3

1)11 it

3.1926

3.1680

3.1437

3.0960

3.0726

3 (\AQZ

3 (\tH1

3.0042

2 Ofi9H

2

AOQQ

2 Q1 77

2 flQ7n

2

Q*»£7

2

fiC^O

2 P.37C

2 fi1 Q^

 I O 17

2.7470

 i jUO

2.7140

 \J7O£>

 oozy

2.6400

2

£C\ £*9

* O 1 J 7

2.6016

 JO3O

 3 f O J

 00 ( (

2.5574

2.5476

ft . <J Ay tj

i f&H

2.5131 

2.5055

y0=0.471

1 7OQ1

1.7684

1.7385

1.7086

1.6787

I

f 1O O

1.6190

1 CQQ7

1.5597

1.5302

1.5008

1.4715

1.4423

1 A1 90

1

DQA O

1.3553

1.3265

1 *)O7fl

1

O£O*>

1 9AAQ

I

n i n/r

1.1847

1.1571

1.1299

1.1031

1.0768

1.0510

1.0257

1.0007

.9758

.9510

.9017 0770

a^9n

oo o £.

.8046

 7 C7X

 I J*»7

.7129

.6917 

.6715

££9^

.6349

£ 1 QQ

.6041

.5907

» 3 i o J

.5674

. j j e j

. «l «J7 1

.5174

.5105

.5037

.4970 

.4904

AR3Q

y 0=0.905

1 ft9O7

1.8001

1.7706

1.7411

1.7117

1

£Q AQ

1.6530

1

XOQO

1.5947

1.5657

1.5368

1.5081

1.4796

1.4513 l d?^

1.3956

1

Q £ QQ

1.3415

1.3153

1 9P.OQ.

1.2650

1.2410

1 9170

1 1 0CA

1.1748

1.1550

1.1365

1.1194

1.1037

I

ftBOA

1.0764

1.0646

I

n eo Q

1.0438

1.0345

1 A9CQ

1.0176

1.0099

1.0027

OQCfi

QftO^

OQ.31

.9772

Q7 1 £

.9663

O£l 9

AC XX

.9481

OQ7£

OOOfi

y0=1.294

1

OOQQ

1.8005

1.7725

1.7448

1.7174

1.6904

1

££QO

1.6376

1.6119

1.5867

1.5621

1.5381

1.5148

1.4922

1.4704

1.4495

1.4296

1.4108

i 90^9

1.3769

1.3620

1.3486

I

ooro

1

00^:7

1.3184

1.3119

1 1079

1.3039

1.3015

1.2996

1 9QD1

1 9OXO

1.2960

1 W\1

I

nn jt n

1.2944

1

OO1 1

I OQ70

y0=2.431

2 AQ&3

2.4914

2

AOAQ

2

17QC.

2 4,795

2.4668

2

J £ I O

2.4560

2

JCAO

2.4457

2.440?

2

1 o e Q
THE LABORATORY DATA 35

ANALYSIS OF THE DATA

PRELIMINARY CONSIDERATIONS

CONVENTIONS AND USE OF TERMS

The framework and general principles relating to conventions and the use of 

terms has been presented in connection with the theory of varied flow in prismatic channels (see pp. 15-18 ). A more detailed statement, moulded within that 

framework, is a prerequisite to the analysis of the data.

As already mentioned in consideration of the varied-flow equation (see p. 18) 

it is customary to adopt the convention that a downward slope (in the direction 

of flow) is positive. This is facilitated by assuming the channel stationing to 

increase in the upstream direction with the line of the bed slope passing through

Water surface

-5.0

1400 1200 1000 800 600 400 ZOO

Length, L, in feet

Figure 8. -Analytical orientation of profile.

the origin of coordinates (see fig. 8), and with the value of y = ymax coincident 

with the vertical axis. (This ymax is the maximum value of y in which Ihe investigator is interested. For example, if the profile were the result of an obstruction, 

such as a dam, in the channel, ymax would be the depth immediately upstream 

from the dam.) With respect to the laboratory data, consistency among the profiles was obtained by the invariable use of ymax as 4,000 feet. Now, if the 

horizontal axis of coordinates be taken as the datum plane, or level from which 

elevations are measured, it is apparent that at any point the elevation of the 

bed is S0L, and the elevation of the water surface is

Elev. = S 0L + y, (16)

L being the distance, in feet, measured from the point at which the channel bed 

intersects the datum plane, and also the distance from the point at which y is 

equal to ymax
36 STAGE-FALL-DISCHARGE RELATIONS 

Table 6. Stationing for the profiles

Cross 

section

2

2

2

2

2

2

3

3

3

3

3

4

4

4

4

4

4

4

5

5

5

5

5

5

5

6

6

6

6

Xo

0.391

.612

.812

1.121

1.489

1.840

.404

.606

.773

1.071

1.512

.528

.780

1.028

1.369

1.419

1.877

2.335

.456

.687

.905

1.158

1.285

1.704

2.525

.471

.905

1.294

2.431

Bulletin 381

Observed y

2.326

2.998

3.212

3.509

3.706

3.703

2.790

3.008

3.183

3.464

3.670

2.789

3.018

3.222

3.475

3. 512

3. *540

3.646

2.490

2.563

2.604

2.680

2.763

3.154

3.674

2.487

2.607

2.766

3.670

Published 

Stationing

560

333

263

166

101

109

404

332

277

185

125

405

332

265

185

175

178

171

503

479

465

439

413

291

135

503

463

412

131

Smoothed profiles

Smoothed y

2.3280

3.0030

3.2150

3.5090

3.7080

3.7025

2.7890

3.0077

3.1845

3.4637

3.6699

2.7950

3.0157

3.2222

3.4696

3.5384

3.6492

2.4880

2.5630

2.6035

2.6800

2.7630

3. 1540

3.6740

2.4880

2.6070

2.7660

3.6597

Revised 

Stationing

560

333

262

164

99

108

404

332

276

187

129

404

330

264

187

176

181

180

504

478

466

440

413

287

133

503

464

412

129

Correction to 

Bulletin 381

0

0

-1

-2

-2

-1

0

0

-1

+2

+4

-1

-2

-1

+2

+ 1

+3

+9

+1

-1

+1

+1

0

-4

-2

0

+ 1

0

-2

REVISIONS TO STATIONING

In Bulletin 381 (Lansford and Mitchell, 1949) it is stated (pp. 22, 25-26) that 

the stationing, or distance L, presented as a part of each profile table, is a 

result of computation rather than observation. The increments of distance, Ax, 

are of course a matter of observation, being the measured distance between the 

point gages (see p. 26). However since the value of xmax (4.000 feet) is beyond 

the depth which could be observed in the laboratory, the desired consistency of 

stationing among the various profiles could be obtained only by computing a 

stationing for some point on each profile. This was determined by matching the 

computed and observed data near the downstream end of the observed profiles. 

For each profile the computed data were plotted on the basis of depth versus L. 

Beginning with the greatest depth which was observed on two or more gages and 

extending stream-upward through 10 consecutive observed depths, the values of 

L corresponding to each observed depth were read from the curve the few slightly greater depths which were observed on only a single gage were not used in 

this determination, since there was no verification of their accuracy. The average 

differences were computed between the stations so determined and the temporary 

arbitrary stations which had been assigned to the observed depths. This difference was the correction then applied throughout the length of the profile to convert temporary stationing to stationing as shown in the profile tables.
THE LABORATORY DATA 37

As pointed out in Bulletin 381, these values were not required during the progress of observations; except as a matter of convenience, the stationing could 

have been neglected entirely until the time of the analysis. It also has been 

pointed out that the values of stationing published in Bulletin 381 might later be 

slightly revised as a result of more detailed analysis.

On the basis of the more complete analysis described in the following pages, 

a slight revision has been made to the stationing of some of the profiles. Although the changes are small, never more than a few feet, in the interest of accuracy they have been incorporated in the profiles shown in this report. Table 6, 

page 36, indicates the corrections to be applied to various tables in Bulletin 

381, and at the same time provides an index to the stationing which is applicable 

to the smoothed profiles presented as tables 1-5 in the preceding section of this 

report.

EVALUATION OF VELOCITY-DISTRIBUTION COEFFICIENTS

The necessity of a velocity-distribution coefficient has been discussed on 

page 7. This coefficient was determined by plotting a number of diagrams similar to those shown in Bulletin 381 as figures 13-15 (pp. 32-34). Each area between lines of equal velocity was planimetered, and computations were made as 

shown in right-hand section of table 7, page 38 of this report, under the subheading "Total cross section." These, with the two lines of computations at the 

foot of the table, are sufficient to determine the coefficient. The remainder of 

table 7, which is for the section shown as figure 15 in Bulletin 381, -has been 

presented to show a summary of conditions in this cross section for subdivisions 

as shown in figure 9.

Left flood plain

^ ___

Central 

section

y> I.OII

Central

section

y< I.OII

Right flood plain

_____ J

Figure 9. -Subdivisions of flow in cross section 5.
38 STAGE-FALL-DISCHARGE RELATIONS

 rt n

W 

U

s

60 < 

> U

< >

CTv 00 00 CS » 4 r ^ CO ^ ON t*- CO VO i-H

^f CT\ rf <O VO O CO NO VO VO ^i CO-H i ONi-Hooooi-4 in oo co co in ^i-iooin

00 CO VO CS CM ON -! f- in ^ CO CM ^H c^

O CM i-H i I i-J i I*' "* *£j

 i vo o vo cr\ CM in co * in i o> CM **

CM CO VO O\ Cfv r- VO O CO VO ON O ^> O

Tf i i oo vo vo in r in »* co CM CM CMC*

O*H ' ..... . . .^J

i-4 O i-l CfM-HCXVD CM CO *\O O O

ON CM CO *t t~ O\ (N O i I CM O 00 CO *» ONin^r Ox ^ OA in CM ON vo ON CM in

o * ** " ' ' *V

o or- vo co h- co co oo vo oco mi-i cor-icOi-Hin in ^t vo oo o i-iexo»«

CM ^ t CO ^J* i-H O i-H ON OO VO CM ON *^

i I CO CM CM* CM CM CO CM -H i-T r-T^H rH 00

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^1 OO O i I i-H ^i O * CM

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i-H in CM CM i-l i-H i-H O VO

O OO OO o ' ' ' * ' * ' i-l

ON ^t CM 00 O> 00 00 00 M»

CO ^ VO CO *tf" CO CO i-H W

VO VO 00 VO 4> -*OOfo m r-t *-t i-i i 11 i -H M

OOO O O O ' r-i

co in co -H in in co <n oo

CO CO CM CM i * CM ON CO C/i

OCM»-I^H^H rHOi-49t

OOOOO o'*'* '*.'*_

comcfv*t^H t i i co t 

CM « f-f-t- (-vOCOi-<

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oomvooin t- ^t CM o CM r-vo I-H co

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voo vo m ^

I-H co *J* o\ oo o in *t m ON I-H t- co o

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OOO

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vo « e*m in N i-H O

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CMO 0

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O O t~-

a\ oovo 1-1 vo co I-H

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t- O CO O

in I-H in co 

com ^ in 

o o ooo

oOi-H o r~- m vo vc r o CT\ invo^CM oinvoino * co o in h-intMCM^H f--* vo-*

i I^H^H^H^H O O O CM

i-in o Oco

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C<)O l~

O O 00 Cf\

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vom co i i o\ ON O O CM CO O '

CO CM ^J1 t*- ^J1 O > vo vo in in in -<t <

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S v

£ 2

m o in o c 

-H CM" ex co" *

VI VI VI VI VI

V V V V V

10 o in o

o i-i ex ex co

%

l^
THE LABORATORY DATA 39

Table 8, page 38, summarizes the values of cm for the various cross sections. 

It was found that appreciable variation occurred between one and another of the 

cross sections, but only in cross section 3 was there appreciable variation within the cross section itself. A more elaborate study might have disclosed further 

slight variations. However, it should be pointed out that the distribution coefficients were so near to unity as to have only a slight effect on the velocity 

head. (See fig. 16, p. 57 , for example, in which a profile has been plotted both 

with and without the correction for cm ). Because of the small magnitude of these 

effects, and also because the labor, both in observation and computation, is very 

great, a more elaborate determination of distribution coefficients appeared to be 

unjustifiable.

COMPUTATION OF HYDRAULIC RADIUS

Much of the analysis which follows is independent of hydraulic radius, so that 

the difficulties which arise from this troublesome factor are not everywhere a 

handicap to the work. However, in the evaluation of Manning's n, and in the attempts to correlate S and V with channel characteristics, there was definite need 

for this factor, and considerable attention was directed toward its proper computation. The results are presented here not only as a necessary prelude to discussion of Manning's n and the slope-velocity correlations, but also for information and guidance to those who may deal with other problems (such as slopearea determinations) wherein the geometry of the cross section approaches that 

of cross section 5.

For the rectangular section, the hydraulic radius, Rt was computed in the conventional manner,

K = A/w.p., (17)

in which A is the cross-sectional area in square feet and w.p. is the wetted 

perimeter, in feet. These values, when used in a formula such as Manning's, appear to give quite satisfactory results, except for small values of y. A curve of 

R as abscissa against y as ordinate will generally have the characteristic shape 

of the solid line farthest to the right in figure 10, page 40 . The curve passes 

through the origin at an angle of 45° to the axes, assumes a gradually increasing 

slope with increasing y, and finally becomes asymptotic to the line 2? = 6/2, in 

which 6 is the width of the channel.

As is generally realized, the use of equation (17) for a flood-plain cross section, such as cross section 5, leads to very unsatisfactory results. When computed by such a formula, R plotted against y becomes a curve as shown by the 

solid line farthest to the left in figure 10. Within the rectangular portion of the 

channel, the curve again passes through the origin at an angle of 45°, assumes a 

gradually increasing slope with increasing y, <and would eventually become asymptotic to the line R = 6/2 = 0.30 ft. At the depth y = 1.01, the value of R is 

0.3344 foot. 3ut as 6 is suddenly increased above this depth, a corresponding
40 STAGE-FALL-DISCHARGE RELATIONS

Figure 10. Hydraulic radii.
THE LABORATORY DATA 41

break to the left occurs in the values of R, -so that for y = 1.20, R - 0.2268 foot. 

After b again becomes a constant, the values of R begin once more to increase 

with y, but within practical limits of y remain substantially less than for the 

rectangular channel. The unsatisfactory feature of this relationship lies in the 

decrease of R with an increase of y. Velocity formulas conventionally assume 

that V varies directly (although perhaps exponentially) with R, and thus, <in the 

instance given above, would indicate a very pronounced decrease in average 

velocity between y = 1.01 and y = L2Q. Assuming,-as in the Manning formula, 

that velocity varies as the two-thirds power of R, -the indicated ratio of mean 

velocities would be (0.2268/0.3344) 2/3 = 0.77. The actual ratio, as taken from 

the data for cross section 5, as 1.925/2.069 = 0.93. In other words, -use of the 

Manning formula to compute the discharge for cross section 5 at depth y = L2Q 

would give a value about 17 percent low if R were computed as area divided by 

wetted perimeter.

Methods have been proposed to overcome this difficulty. A common procedure 

is to impose mythical channel boundaries by extending upward the vertical sides 

of the small rectangular channel and then computing the discharge separately for 

the resulting subdivisions of the channel. But the question immediately arises as 

to procedure for computing the wetted perimeter. To include the mythical boundaries would result in an overall lowering of the value of boundary roughness, with 

no way of computing the new value. Experience has demonstrated that, <for many 

problems, acceptable results are obtained by using the mythical boundaries only 

for subdividing the area, the hydraulic radius for each subdivision then being 

obtained by dividing each area by its actual wetted perimeter. For various values 

of y in cross section 5, such computations were made, and an overall value of 

hydraulic radius obtained by weighting the value for each subdivision by the ratio 

of its area to the total area. Results of these computations are shown, in figure 

10, by the dotted line labelled R°.

Within the range of depths required for these computations, this R° curve appears quite satisfactory; the objectionable break in the R curve has been greatly 

improved, and above a depth of about 1.6 feet the R curve and the R° curve are, 

for a considerable range, nearly coincident. Unfortunately, the method cannot be 

recommended for extremely large values of y. In the central section, as y increases the area increases proportionally, but the wetted perimeter is held constant. Hence the hydraulic radius for this section increases rapidly with depth 

and eventually becomes so great as to dominate the computation for weighting R 

and lead to weighted values of hydraulic radius much greater that the asymptotic 

limit, R =6/2. Thus, in principle at least, another method is desirable such that, 

for the central section, the area will be limited to values which are in keeping 

with the limitation upon wetted perimeter.

An examination of figure 15 (Bull. 381) suggests that the boundary effects of 

the small rectangular section are reflected upward and inward from reentering 

boundary angles (the breaks in cross section at which y = 1.011 feet). When reviewed in the light of a paper by Keulegan (1938), this fact suggests the use of a 

subdivision which is explained as follows: Bisectors, ci and fi, (see fig. 11) were 

drawn for the reentering angles, X and Y. These divided the cross section into
42 STAGE-FAIL-DISCHARGE RELATIONS

Figure 11. -Subdivision of Hood-plain cross section for computing R'.

three areas, A, B, and C. Their hydraulic radii were taken as being A/she, 

B/cdef, and C/fgh, respectively. The hydraulic radius of the entire cross section 

(hereafter designated as /?') was then obtained by weighting the hydraulic radii 

just computed by area, as follows:

A 2/abc + B 2/cdef + C 2/fgh 

A + B + C

(18)

Comparing the change in R' with change in 7, it was found that there was nearly 

a constant value of R* between the values of y = 1.01 and y = 1.20, in which 

range the value of 6 increased from LOO to 5.00. Plot of this relation appears as the 

dashed line (#') farthest to the left in figure 10. It will be noted that, for shallow 

depths on the flood plain, values for R' are nearly the same as for R°. At depth 

of 4 feet, R' is appreciably greater than either R or R°, but for extremely great 

depths (not shown in figure 10) R', like R, will become asymptotic to 6/2.

To determine whether this method of computing hydraulic radius is consistent, 

from a practical standpoint, with the conventional procedure (equation 17)> a 

corresponding computation was made for the rectangular section. For this, bisectors were drawn for the salient angles of the cross section (the 90° angles 

between bottom and sides,) thus dividing this cross section into three parts. 

Computations then were completed for R" by the method outlined above for /?'. 

Results are shown by the dashed line farthest to the right in figure 10. Since 

there is negligible difference in the results, it appears that the method of computing R' as used in the flood-plain cross section is consistent with conventional procedure for computing hydraulic radius.
THE LABORATORY DATA

OO16 0.018 0.020 0.022 0.024 0.026 0.028

Figure 12. Values of Manning's n for die various cross sections

296350<

43

0.030 0.032 0.034
44 STAGE-FALL-DISCHARGE RELATIONS

VALUES OF MANNING'S n

As was the case with hydraulic radius, most of the analysis which follows is 

independent of the value of Manning's n. However, a better understanding of the 

various channel ratings, as well as additional light on the applicability of the 

Manning formula, may be gained by an examination of the manner in which n 

varies between the several cross-sections, and between various depths within 

any cross section. This information is presented in figure 12, page 43. It will be 

noted that for low values of y, ail the cross sections show an apparent increase 

in the value of n as y decreases. If n be considered as a characteristic of the 

channel lining, which for any cross section is uniform for depths of less than 1.5 

feet, then the apparent increase of n with decrease of y can be accounted for 

only by the fact that the two-thirds power of the hydraulic radius inadequately 

reflects the effect of the shape of the channel. Particular attention is invited to 

the curves for the flood-plain cross sections 5 and 6. (For values of y less than 

1.5 feet the two cross sections are identical.) The continuous lines indicate the 

variation in n when R is computed in the conventional manner of equation (17). 

The dotted and dashed lines indicate the corresponding variation, for cross section 5, based on R° and R 1 , respectively. It will be noted that in the range of 

rapidly expanding width (between y = 1.01 and y = 1.20) and immediately above, 

the use of R°, or of R', results in much more stable values of n.

DIRECT METHOD OF ANALYSIS

In an earlier section of this publication, "Integration of the varied-flow equation 

by the step method, * it was demonstrated both graphically and analytically that

Ax =-A#/(S0 - S). (14)

"rtth appropriate knowledge of the velocity-distribution coefficient, and of the 

uniform-flow stage-discharge relation, A# may be readily determined for any two 

values of y. Bearing in mind that S0 is normally a known quantity, it will be obvious that the profile computations may be readily made if S can be properly evaluated. Thus it appeared most desirable that S should be computed for the laboratory data and correlations made between these values of S, the known discharges, 

and the channel characteristics. If a relation could be found which was applicable to all cross sections and all discharges, it would represent considerable 

progress in the evolution of a better formula for the computation of friction slope 

for gradually varied flow.

Introducing y and the velocity distribution coefficient into equation (14) and 

solving for S,

AXS0 + (7l + cm V^/W - (y 

Ax
THE LABORATORY DATA i 45

in which Ax = zj - x 2 is a positive quantity if the initial section (1) is upstream, 

or a negative quantity if the initial section is downstream from the final section 

(2). If y 2 and y\ be taken sufficiently close together,! the resulting value ofs 

may be considered as applicable to the point at which y = (y 2 + yi)/2, the average value of y. In computing S for the rectangular chamnel, Ax was taken as 15 

feet; for the flood-plain channel Ax was taken as 10 f£et. The values of y used 

in preliminary computations were those taken directly fibm the observed profiles. 

However, y had been observed only to thousands of a foot, and it was found that 

this was insufficient refinement to avoid "step breaks" in the computed values of 

S. Therefore the values of y as given in the observed profiles were smoothed and 

recomputed to ten-thousandths of a foot, giving the values shown in tables 1 to 

5, pages 27 to 34. Using these tables and formula (19), values were computed 

and large-scale plots were made of yavg as ordinate against S as abscissa for 

each of the smoothed profiles. !

Tables 9 and 10, pages 46, 47, containing the basic datja for the proposed correlation, were then prepared by reading from the large-scale plots the values of 

S corresponding to the indicated cross sections, depths, and discharges.

In the attempted correlations, each cross section wtos first considered separately. The first step was a plot, on logarithmic coordinates, of S as abscissa 

against V, the actual velocity under which the specific: S was observed, as ordinate. A common symbol was used to identify common values of y, but different 

symbols for different values of y. Each plotted point was labelled by its proper 

discharge. There resulted a great dispersion of points; but it appeared that, at 

least for the higher values of S and V, parallel straight lines might be passed 

through those points for which y was a constant. If this were the case, all the 

points could be brought to a common curve by dividing; V by some function of y.

In attempting to discover this unknown function of y, the straight lines were 

drawn as described above, and intercepts, i, read from; a common ordinate. (The 

ordinate S = 0.0004 was used as a matter of convenience.) These intercepts then 

were plotted on logarithmic coordinates against variops functions of y. Of the 

plots attempted, the one involving hydraulic radius was: most satisfactory, giving 

a relation, for example, in cross section 4, of R - 0.27 = .0002365 i 2. Unfortunately, the relation proved not to be constant for all cnoss sections, the relation 

for cross section 2, for example, being R - 0.15 = .OQ0141 i 2. The variation of 

the coefficient of i was to be expected and might haveibeen correlated with variation between the channel linings, but the variation in'the adjustment to R could 

not be logically interpreted. Thus, for lack of a common relation between R and 

i in the several cross sections, no further attempts w^re made to bring the data 

for all cross sections into a single plot.

The data for each individual cross section, however, now were recomputed. 

This was accomplished by dividing V by (R + c)°- 5, in which c was constant for 

a given cross section, and having, for example, the value of -0.15 for cross 

section 2. The resulting plot for cross section 2 is presented on page 48 as
46 STAGE-FALL-DISCHARGE RELATIONS

Table 9.--Friction slope and its exponent, 

rectangular channels

y

1.0 

1.0 

1.0 

1.0 

1.5

1.5 

1.5 

1.5 

1.5 

1.5

2.0 

2.0 

2.0 

2.0 

2.0

2.0 

2.0 

2.5 

2.5 

2.5

2.5 

2.5 

2.5 

2.5 

3.0

3.0 

3.0 

3.0 

3.0 

3.0

3.5 

3.5 

3.5 

3.5

Cross sectio

Q

3.90 

7.99 

12.38 

16.98 

3.90

7.99 

12.38 

20.13 

30.22 

30.53

3.90 

7.99 

12.38 

20.13 

30.22

40.39 

45.18 

7.99 

12.38 

20.13

30.22 

40.39 

60.83

7.99

12.38 

20.13 

30.22 

40.39 

77.45

20.13 

30.22 

40.39 

94.85

S

0.000113 

.000583 

.001536 

.003000 

.000037

.000191 

.000487 

.001283 

.002932 

.003000

.000012 

.000113 

.000223 

.000566 

.001381

. 002444 

.003000 

.000089 

.000176 

.000307

.000784 

.001344 

.003000

.000076

.000155 

.000258 

.000487 

.000881 

.003000

.000222 

.000362 

.000594 

.003000

n 2

b

0.4815 

.4913 

.4971 

.5000 

.4863

.4958 

.4992 

.4988 

.4998 

.5000

.4725 

.5102 

.4994 

.4966 

.5022

.5016 

.5000 

.5291 

.5202 

.4958

.5039 

.5012 

.5000

.5456

.5402 

.5146 

.5042 

.5054 

.5000

.5296 

.5109 

.5060 

.5000

Cross section 3

0

3.77 

7.37 

10.90 

16.30 

3.77

7.37 

10.90 

18.10 

29.55

3.77 

7.37 

10.90 

18.10 

29.80

40.28

3.77 

7.37 

10.90

18.10 

29.80 

51.37

7.37

10.90 

18.10 

29.80 

62.80

29.80 

74.51

S

0.000127 

.000541 

.001240 

.003000 

.000033

.000198 

.000441 

.001198 

.003000

.000023 

.000132 

.000232 

.000607 

.001485

.003000

.000015 

.000094 

.000156

.000384 

.001086 

.003000

.000068

.000107 

.000261 

.000547 

.003000

.000405 

.003000

b

0.4869 

.4917 

.4941 

.5000 

.4809

.5034 

.5050 

.5046 

.5000

.4938 

.5153 

.5033 

.5002 

.4923

.5000

.4965 

.5226 

.5082

.5019 

.5054 

.5000

.5260

.5092 

.5028 

.4860 

.5000

.4891 

.5000

Cri

Q

3.79 

7.62 

11.64

3.79

7.62 

12.19 

19.10 

21.85

3.79 

7.62 

12.19 

19.10 

29.91

32.58

3.79 

7.62 

12.19

19.10 

29.91 

39.98 

43.67 

7.62

12.19 

19.10 

29.91 

39.98 

55.10

29.91 

39.98 

66.81

>ss sgctic

o

0.000290 

.001253 

.003000

.000093

.000326 

.000926 

.002224 

.003000

.000025 

.000142 

.000396 

.000920 

.002460

.003000

.000013 

.000112 

.000259

.000537 

.001340 

.002299 

.003000 

.000092

.000174 

.000362 

.000848 

.001223 

.003000

.000580 

.000833 

.003000

>n 4

b

0.4943 

.4981 

.5000

.5017

.4930 

.4994 

.4975 

.5000

.4771 

.4918 

.4963 

.4919 

.4977

.5000

.4755 

.5111 

.5063

.4956 

.4963 

.4927 

.5000 

.5254

.5098 

.5003 

.4971 

.4809 

.5000

.4976 

.4820 

.5000
THE LABORATORY DATA 47

Table 10. Friction slope and its exponent, 

flood-plain channels

y

0.5 

0.5 

0.8 

0.8 

0.8

1.0 

1.0 

1.0 

1.0 

1.1

1.1 

1.1 

1.1 

1.2 

1.2

1.2 

1.2 

1.2 

1.5 

1.5

1.5 

1.5 

1.5 

1.5 

2.0

2.0 

2.0 

2.0 

2.0 

2.0 

2.5 

2.5 

2.5 

2.5 

2.5

3.0 

3.0 

3.0 

3.5 

3.5

Croaa aecti

Q

0.79 

0.89 

.79 

1.32 

1.59

.79 

1.32 

1.84 

2.07 

.79

1.32 

1.84 

2.39 

.79 

1.32

1.84 

2.71 

3.01 

.79 

1.32

1.84 

2.71 

3.87 

7.24 

1.32

1.84 

2.71 

3.87 

11.30 

18.01 

1.84 

2.71 

3.87 

11.30 

30.61

11.30 

31.28 

44.61 

31.28 

59.71

S

0.002378 

.003000 

.000725 

.001974 

.003000

.000464 

.001376 

.002379 

.003000 

.000400

.001138 

.001811 

.003000 

. 000240 

.000680

.001118 

.002228 

.003000 

.000036 

.000122

.000200 

.000452 

.000878 

.003000 

.000025

.000044 

.000061 

.000124 

.001157 

.003000 

.000011 

.000033 

.000066 

.000374 

.003000

.000192 

.001503 

.003000 

.000740 

. 003000

on 5

b

0.5005 

.5000 

.4985 

.4964 

.5000

.5039 

.5091 

.5003 

.5000 

.5128 

.5160 

.5014 

.5000 

.5090 

.5113

.4999 

.4929 

.5000 

.5003 

.5111

.5018 

.5048 

.5017 

.5000 

.5208

.5169 

.4945 

.4938 

.4985 

.5000 

.5007 

.5164 

.5166 

.4944 

.5000

.4998 

.5015 

.5000 

.4926 

.5000

Croaa aecti<

Q

0.83 

.89 

.83 

1.59

.83 

1.84 

2.07

.83

1.84 

2.39

.83 

1.84

3.01

.83 

1.84

3.98 

7.24

1.84

3.98 

18.40

3.98 

30.63 

32.77

30.63 

49.86

30.63 

69.21

S

0.002453 

.003000 

.000727 

.003000

.000563 

.002380 

. 003000

.000452

.001807 

.003000

.000277 

.001125

.003000

.000073 

.000200

.000945 

.003000

.000030

.000132 

.003000

.000065 

.002498 

.003000

.001065 

.003000

.000545 

.003000

n 6

b

0.4948 

.5000 

.4918 

.5000

.5104 

.5003 

.5000

.5145

.5013 

.5000

.5118 

.5004

.5000

.5323 

.5018 

.5030 

.5000

.4999

.4966 

.5000

.5199 

.4960 

.5000

.4955 

.5000

.4950 

.5000
48 STAGE-FALL-DISCHARGE RELATIONS
THE LABORATORY DATA 49

figure 13. It will be noted that, for values of S greater than about 0.0003, all the 

points lie very close to the curve S 0 - 5 = 0.0119 V/(R- 0.15)°* 5 . However, for 

lower values of S the points depart from this curve, and it will be noted that 

points of equal discharge tend to define independent curves branching away from 

the parent relationship. Of these branches, all except that for the lowest discharge trend in the same direction. A positive reason is not evident for this 

apparent discordance of the curve for lowest discharge, but it seems appropriate 

to point out that for this curve the values of S were computed from differentials 

of y which were extremely small and may therefore be subject to very considerable errors. Neglecting this one curve, all the others indicate that as S decreases, 

the slope of the curve, or the appropriate exponent of S, sharply increases and in 

fact appears to approach a 45° line which would represent an S with an exponent 

of 1. Such a value for the exponent of S would, of course, be indicative of laminar flow. Thus the plot may be interpreted to indicate that, under the extreme 

back-water conditions for which these low values of S are appropriate, turbulent 

flow was not completely developed.

It was next assumed that the relationship indicated by figure 13 should have 

the more general form

Sb = f(v,y)

and efforts were made to evaluate 6, and correlate these values. For uniform flow 

at any given depth, y, by equation (9),

(20)

a value depending only on channel characteristics. For nonuniform flow at the 

same depth, these channel characteristics remain unchanged, unless, indeed, 

roughness is a function of velocity, which is contrary to usual assumption. As* 

suming the channel characteristics remain unchanged, then K remains unchanged 

and

0 » or 0 = ^F = 5F-- (21)

Combining equations (20) and (21) and solving for 6,

6 = ^ lpg So + 1^1' log V - log Vp (92 

Values of 6 were computed by this formula for all the values of S which appear 

in tables 9 and 10, pages 46, 47. These values of b (see tables 9 and 10) then were 

plotted against promising characteristics of the flow, including the Froude Number, Reynolds Number, and various ratios between V, V 0, y, and y0 . The most
50 STAGE-FALL-DISCHARGE RELATIONS

m

g O + X .$-<]

8-

g

I

bp 
THE LABORATORY DATA 51

satisfactory plots were obtained with respect to the Froude Number, which was 

computed as F = cm V 2/gR, in which 1? is the conventional hydraulic radius, and 

for which /?', as described on page 42, was substituted when considering the 

flood-plain channels. These plots indicated a definite trend of increase in b with 

low values of F, but there was much scattering of the points, and the data for the 

lowest profiles were in contradiction to the general trend. Still the trend was 

sufficiently pronounced to suggest use of the relation in test computations of 

backwater profiles. Agreement between these computed profiles and observed 

profiles was not materially better than the agreement obtained from computations 

by less involved methods. Thus, because the improvements in results were not 

definite enough to justify the vast increase in the work of computation, and also 

because of the contradicting trend of the data for the profile of lowest discharge, 

no further use was made of the variable exponent of friction slope. Assuming, instead, that 6 is invariably 0.5, equation (21) reduces to

K0 = PM/S0 ' 5 (23) 

which, combined with equation (20) gives

S = V 2 (SQ/V0 h (24)

For a given channel, S 0 is constant and V 0 is a function of y. Thus it becomes 

convenient to introduce a factor It = Vo/$o°* 5» s° that

(25)

The extent to which equation (25) is appropriate is illustrated by figure 14, page 

50, in which, for cross section 4, values of V/k have been plotted as ordinates with 

values of S as abscissa. The curve lias been drawn to conform with equation 

(25). As in figure 13, there are some notable departures of particular discharges 

from the slope of the curve as drawn, but since the trend of the lowest discharge 

is opposite to the trend of the higher discharges, the use of the curve as drawn, 

involving the exponent 2 in equation (25) seems most appropriate. This equation 

has been used for computation of S in the remainder of this report. (See p. 52.)

INDIRECT METHODS OF ANALYSIS

In the direct method of analysis, attempts were made to evaluate S, the slope 

of a point on the energy gradient. Because this slope changes from point to 

point along the channel, it was necessary to work with reaches of channel of 

such short length that the change could be neglected. This in turn involved the 

use of such small values of Ay that depths observed to thousandths of a foot, 

even when smoothed to ten-thousandths of a foot, were comparatively rough
52 STAGE-FALL-DISCHARGE RELATIONS

approximations. On the other hand, the values of y might be regarded as highly 

accurate with respect to a method of analysis which would permit the use of long 

reaches of channel. Because of curvature in both the line of the energy gradient 

and the line of the water surface, use can be made of long reaches only by indirect methods; that is, it becomes necessary (1) to assume that a particular 

method of computation is appropriate, (2) to compute the surface profile by this 

method, and finally (3) to compare the computed profile with the observed profile, 

thereby ascertaining whether or not the particular method is truly appropriate.

Of two indirect methods which produce equally satisfactory results, one may 

be the more desirable from the practical viewpoint. If only the total length of 

reach between two widely different values of y is desired, a method which completely integrates the varied-flow equation may be a great saving in labor. On 

the other hand, if it is desired to define accurately the shape of the profile between these two widely different values of y, many additional points on the profile may need to be computed solely for this purpose, so that the total number of 

steps may be greatly increased. For this purpose the step method, because of 

its simplicity, flexibility, and widespread acceptance, may be regarded as the 

more desirable.

THE STEP METHOD

The basic formula of the step method has been presented:

Ax = -Atf/fS0 - $) (15)

S being the mean value of S at the two ends of the reach Ax, and S being computed by the method deemed most appropriate. It has been indicated in the preceding section, and it is now assumed, that the most appropriate method of computing S is

2 <25)

Values of k appear in this report as tables 28-32, pages!52 to 154.

In the development of equation (15), it was specified (see page 24 ) that Aff 

should be considered as the loss in H, or H at the initial section minus H at the 

final section. Thus all the terms in the right member of equation (15) represent 

loss in the reach Ax. Whether Ax itself should be regarded as a loss (initial 

value minus final value) or a change (final value minus initial value) is a question which may be viewed in the light of convenience, and the objective of the 

computations. In fact there are instances in which Ax is appropriately regarded 

merely as a length of reach. In the present study, it is both convenient and appropriate to regard Ax as a loss, so that, if Lj be the value of the stationing for 

the initial section (1) and L 2 be the corresponding value for the final section
THE LABORATORY DATA 53

<] en en co vo

IM (M (M (M (M IM CS (M

00 * 0\ u» in *

00 «-< \O

OD rH CO CO
54 STAGE-FALL-DISCHARGE RELATIONS

w

OOoOOOOOi !

 *«** *

t/}

I COiVO**UivbOf-IO\i I l-l O\ CO CO Ul 

37

r co iNcoc^o^Oo--^coeo^Hr eiin

ooooooooooooooo

OoOOOOOOOOOOOOO

_ o_______

loo

o o o o o o c 

o o o o o o c

__ o_________________

OO oooooooooo® OOOOOOOOOOOOOOO

OOCOr-IOrfiHirtcOOCOiHfOOVOOO 

OO OOOOOOOOOOOOOOO

o

o> rco co co co co <o ro co co co co co co co co co co

X

o\ cri vo £i r- *3 u5 p> £." m ^>

o o o o o «-t 

o o o o o o

E

c?

Q """ o in

00 t-l 

OOO^t-OOOOOOOOOOOO
THE LABORATORY DATA 55

(2), L! - L 2 = Ax, which, combined with equation (15), gives L 2 = Lj + [A#/ 

(So ' sH in which AH still is defined as the loss, H-^ - H z* This equation is 

appropriate to the computation of any and all types of profiles, and yields the 

same arrangement of the values for L irrespective of whether the initial section 

be chosen upstream or downstream from the final section.

Computations by the above formula are conveniently made in the form of table 

11, pages 53 and 54. In the first column are listed values of y, beginning with 

ymax as described on page 35. Successive values of y are those corresponding 

to intervals needed for properly defining the profile, or for mitigating the effects 

of curvature, whichever may be smaller.Values of A in column 2 are the computed product of y and width b, or for the flood-plain cross sections, are taken 

from the table of areas (see p. 152, table 27). V is obtained by dividing the known 

discharge of the profile by A. To compute the velocity head, cm F 2/2g, Cjn is 

taken from table 8, on page 38. H is the sum of y and velocity head, and k is 

taken from tables 28 to 32. Column 3 divided by column 7 gives column 8, V/k, 

which is squared to obtain S, column 9. For each reach, the downstream is considered as the initial section, so that AH is the remainder of H on any line minus 

H. on the following line. Values of S at each end of a reach are averaged to obtain S; this subtracted from the constant value (0.003) for S0 yields column 11 

which, when divided into A// from column 6, gives column 12. Value of the stationing for the final end of the reach (L 2 ) is then obtained by adding column 12 

to the value of the stationing for the initial end of the reach, the results being shown 

in the next-to-last column of the table. In the final column, elevation is the sum 

of y and SQ^* (See equation (16), p.35 .)

DIRECT-INTEGRATION METHODS

Of the several methods of direct integration which have been proposed in 

recent years, those which appeared most adaptable to the laboratory data (namely the Bakhmeteff method and the VonSeggern method) have been discussed in the 

preceding section of this report. (See pp. 21, 22.) As a check upon the reliability and extent of agreement of various methods, selected, representative profiles were computed not only by the step method, but also by the above methods 

of direct integration.

COMPARISON OF RESULTS

A comparison of results is most conveniently made by figures 15 and 16, pages 

56, 57. Figure 15 presents the curves for the highest and lowest discharges 

observed for cross section 2. The curves represent the results of step-method 

computations and are identical with the curves presented as part of figure 8 in 

Bulletin 381, (Lansford and Mitchell, 1949). It will be noted that for the high discharge there is no appreciable difference between the results of the various 

methods. For the low discharge, the direct-integration methods produce eleva-
56 STAGE-FALL-DISCHARGE RELATIONS
6.0 

5.6

5.2

"S3 

<$£

c

 ;-4.8

.0 

'*-

a 

a> 

LJ4.4

4.0 

3.6

3.2

3.0 

15

^

^

 -

^S

X ^

fe

N

^.

^

"^

^

V

«o

^-

s,^

S

K,

c«

?»

X"

\

^

^

\O '

~>r

V

^-

fts

~^-

N' -

t

^

s=«l

X

N

> «

S

 - 

5«s

"^

N *^

Nr

' Ste 

o Bak

X 

«

 ill 1

 

p method

hmeteff w

Bakhmeteff w

» Von Seggern

^

^

00 1400 1200 1000 800 600 

Distance "L",in feet 

Figure 16. Comparison of computation methods, cross se
58 STAGE-FALL-DISCHARGE RELATIONS

tions which, between stations 800 and 1300, are slightly higher than those for the 

step method. Comparison with the corresponding curve of Bulletin 381 indicates 

that the step-method results are in slightly better agreement with the observed 

data.

Figure 16 presents similar data for the highest and lowest discharges observed 

for cross section 5. Comparing this plot with figure 11 of Bulletin 381 and bearing in mind that the step-method results are identical in the two figures, it will 

be noted that, although the results of the direct integration methods differ only 

slightly from the results of the step method, these differences are generally in 

the opposite direction from the observed data.

Thus, while the work involved in step-method computations is slightly greater, 

the results appear to be slightly more consistent with observed data. Furthermore, in the following sections on stage-fall-discharge relations, it is necessary 

that the profiles be carefully defined throughout their entire length; thus a great 

number of subdivisions is required even by the direct-integration methods. For 

these reasons, the step method was adopted as the standard procedure for computing the profiles.

TABLES OF PROFILES BY THE STEP METHOD

Using the step method, profiles were computed for all the profiles which were 

observed in cross sections 2 to 6, inclusive. These are identical with the continuous lines of figures 8 to 12 in Bulletin 381. Summary of the computations 

appears as tables 12 to 16, inclusive, on pages 59 to 77. It will be noted that 

these computed profiles are based on the laboratory data in that the hydraulic 

properties of the cross sections under uniform-flow conditions were determined 

experimentally, the properties thus determined for a given depth being assumed 

to apply to conditions of gradually varied flow at the same depth. Furthermore, 

the computed profiles (except for the two lowest discharges for cross section 5) 

were plotted and carefully compared with their observed counterparts, thus verifying the validity of the computed profiles. The values of y0 an(^ Q given in the 

column headings are related to each other through the use of the stage-discharge 

relation for uniform flow as given in table 42 in Bulletin 381 or as can be determined without reference to that bulletin through use of tables 27 to 32 of this 

publication as described on page 3.
Table 12. Stmmary of step-method computation of p

y

4.00 

3.60 

3 40

3.20 

3.00

2.90

2.80 

2.70

2.60 

9 50

2.40

2

Q£

2

00

2

QA

2 

OP

2.24

2.20

2

1 £

2

10

2 10

2 

Aft

2.00 

1.98

1

O£

1.94

1.92

1.91

1 00

Qy0 =

L

0 

133.47

266.99

400.57

467.39

534.25

601.16

668.13

3.90 

0.391

Elev.

4.00000 

4. 00041

4.00097

4.00171

4.00217

4.00275

4.00348

4.00439

Qy0 

L

0 

133.91

268.01 

335.15

402.38

469.72

537.21

604.89

672.84

70* O5

7.99 

0.612

Elev.

4.00000 

4.00173

4.00403

4.00714

4.00916

4.01163

4.01467

4.01852

i. O9nft5

Qy0 =

L

0 

134.73

269.91

405.77

474.09

542.77

611.96

646.79

681.82

717 11

12.38 

0.812

Elev.

4.00000 

4.00419

4.00973

4.01319

4.01731

4.02227

4.02831

4.03588

4. 04037

4.04546

A 05111

Q = 

*o =

L

0 

137.10 

206.07

275.43

O A C. O7

415.74

487.03

559.42

596.18

633.38

671.15

709.62

7AR QO
60 STAGE-FALL-DISCHARGE RELATIONS
THE LABORATORY DATA 61

i co i r- CM

1 VO 60 t- VO i 01/5^)100 i CM t*1 r- o

i *J" *j" **> in

 4< CM O SO VO 

CO OCO ON 1" 

ON O VO SO i-H 

 41 O VO O VO 1/5 vo vo f- ff-VOCM o-*-* 

oso o 

co es in

OOON O

I

1

i * *-*'-* *'** *" *" * -^ Tt" in" i

i I-H r- ON tj" 

i CM CM in m

I O in r-l ON i ^ tft r- oo

t O O O O 

1 . - . _

CM CM in oo in co

«-H i-l

so ON

r-H t~^

CO *

O\ OX

1 00 i co

1 VO

1 VO

i in

1 i-H

i OX

I

i r i t1 O 

1 O

 ON i in i .

1 CO 

1 0

i OX

OO * O VO CM

t- co I-H vo in

ON CO CM ON O 

O CO VO ^ O 

 -H i-H <-t i-H CM

4.13314

oo

CO

*$* 

VO

1

1

1

1

1 

1

1 

1

1

1 

1 

1

1 

1

CO 1

i-H 1

VO 1 

CM 1

SO 1

VO 1

in i

VO I

in i

i-H 1

VO 1

in i

i-H 1

 * 1

CM 1 

If) 1 

. 1

r- i

ON 

in in

1/5

VO

so 

ON 

ON

i

i 

i

i 

i

i

ONCM tVO SO O

VD CM VO 

CM VO O 

CM CM CO

I 1 1 

1 1 1 

1 I 1 

1 I I 

1 I 1

1 I 1 

I 1 I

1 1 1

I 1 1

1 1 I 

1 1 1

1 1 1 

1 1 1 

1 1 I

I 1 I 

1 1 1

1 1 1

1 1 1 

1 1 1

1 

1

VO

i-H 

O

ti-H

CM 

t"-

VO 

O

1 1

1 1 

1 1

1 1

1 1 

1 1

OO i

ON i

VO 1 

00 I

i i i 

 * I

VO 1 

VO 1

U1° 1

co i

O 1

i in i i i

I VO 1 O

I vo i t10 10

i 1/5 i ri in i vo

1 OO 1 CO

1 VO II 

1 OO II

1-41 ii

f CM II 

1 VO II 

1 . II

1 i-H II

1 1 1 

1 1 1 

1 1 1 

1 1 1 

1 1 1

1 1 1

1 1 1 

1 1 1 

1 1 1

so t vo 

CM i in 

r- i CM

O 1 CO 

IM 1 CM

W ! *'

VO 1 CM f- i in 

 1 in i f- \a i r~

O 1 O

1 SO 1

1 I-H 1

i r- i

I 0* 1

I VO 1 

t 0 1

i ON i 

1 O 1

1 1 ON i i r~

i i r1 1 ON 

1 1

i i in

I 1 0

i

i

i i

i

so in CM I-H ON 

r- CM t- CM <* 

r- \n in oo

 *C VO OO r-l *

CM CM CM CO CO

-* Tt *-*-*

vo m *$* iv co

CM t CM O OO

ON i ' 1/5 o vo

CO O i-H CO <*

O I- < i-H i-H i-H

 * i co i rco i t- i f-

 *C 1 CM I CM 

OO 1 ON 1 O 

O 1 O 1 i-H

CO 1 r-l 1 O\ 

f- 1 ON 1 1/S

.1.1. 

 «* 1 O 1 t 

O 1 O 1 O

1 1 1 1 CO

1 1 1 1 "*

1 1 1 1 .

I 1 I I O 

I I 1 i O

1 1 1

1 1 1 

1 1 1 

1 1 1

1 1 1

1 1 1 

1 1 1 

1 1 1 

1 1 1

CO O VO CO i-H

in o co I-H CM 

ON * co O CO

r- o co r- r-i 

co «* ^ ^c in

 <*-* -^ -*-*

i-H O CM r-l f-

\f) O i-H ^ O

VO OO r-l VO VO 

VO f- ON O CM 

 -H i-H i-H CM CM

1 1 O 1 I 

1 1 0 1 I 

1 1 \f> 1 1

1 1 i-H 1 1 

1 1 i-H 1 1 

1 1 . 1 1

l 1 O 1 I 

1 1 0 I 1

i i in i i

1 I O I 1

1 1 i-H 1 1

i i i i i

i i i i i

i i i i i i i i i i

i i i i i

CM o oo r- vo
62 STAGE-FALL-DISCHARGE RELATIONS

Table step-method 12. of--Summary computati

CO CM

rot 1 

 oo

CM .

 HO 

II II

CM

t-^O

II II

» -H.

-

li-

-

li-

-

2

-3

ON in i 

in oo i

f- O I 

ODfH 1

in ON i 

in ON i

0404 1

0 

0

ro

0 

CO

o

pH 

pH

t t i i i I i 1 I i i it

I I I I I 1 I 

I 1 t I I I I

1 I I i i I I 

I I I I 1 1 I

1 I I I I I 1 

I i 1 i I I I 

I I I I i i I 

1 I 1 i 1 1 1 

i I I I i I I

co oo in in oo ** vo oo co O -^ in 

ON pH vo *H OO CN t>> O ^ "^ ON CO 

ON c*fl vo ^" in ^j1 ch rH f«- o in ON

pH rH rH *H CN O4 O4 CO CO ^J1 ^}* If)

-<t -*t ^t -^ ^- * ^ ^f * ^- * -<f

i-H VCU^invCOO CNVCrHOOO pH 

CO omoONO ONCOOOOOON ON

co ^inoor-ioo t^-omvopH co 

CN co^inr-oo OCNCOUJON o

^H pHpHpHrHpH O4O4CNC4CN CO

,H ^pHpHrHpH rHrHrHpHrH *H

IOI CO CNIOO lit- 1 It* 

1 CO t in "* 1 pH lii-H 1 IOI 

1 OO 1 rH in 1 O ItVO 1 If 

ICNI CO CO 1 -it ll^i 1 U 

I O I O OiO IIO 1 1C 1 . 1 . . 1 . II* 1 t I-4-I «* «* 1 -«f 1 1 -*t 1 1 *

IOI rH <*pl\O IION 1 IO 

|pH| ifj *HlO IICO 1 IO

I.I . . 1 . II* 1 1 

i vo i o in i o i i m i IP-

» t~ 9 ON OtO4 IICO I IU

1 1 1

9 pH in O CN "^ CO tH in O VO *$>

^ -*t oo oo ovooNinvo oo co \o

5OOO rHrHi-«pHi-« O4C4CO

h^"*^« ^ ^ ^ ^ ^ -«t ^ ^t

^OPHO ^-oo*Hr-rH oooin

HOOVOVO VCOOCOPHCN vo r- o

*5 VO OO ON O pH CO "^ in VO OO fH 

HpHrHpH CNO4CMCNO4 CNO4CO

04 04 «~*

pH pH O\ 

O OO ** 

oo 00 oo oo t-- Is- t t^- Is- vo \o \o so vo \o vo vo in in u"i -^ ^ ^ ^ ^ -^ ^ co
Table 13. Summary of step-method computations of profil

y

4.00 

3.60

3.20 

3.00

2.80 

2.70

2.60

2.40 

2 in

2.20 

2.10

2

A£

2.00 

1 oa

1.94

1.90

1

O£

1 09

1 7O

1

*7 A

1.70

1

£O

1.66

1.64

1

st)

1.60

Qy0 -

L

0 

133.57

267.21

400.94

467.85

534.80

601.80

7 Q£ f\O

803.28

3.77 

0.404

Elev.

4.00000 

4.00071

4.00163

4.00282

4.00355

4.00440

4.00540

4.00661

4.00806

4.00984

Qy« 

L

0 

134.26

268.76 

336.14

403.62

471.22

539.00

606.96

£. *7 C O 1

709.44

743.76

m 9fi

812.76

7.37 

0.606

Elev.

4.00000 

4.00278

4.00628

4

AAO A O

4.01086

4.01366

4.01700

4.02088

4.02563

4

AOQ OO

4

AO I OO

4.03460

4.03828

Qy0 

L

0 

135.39

271.34 

339 49

407.96

476.71

546.87

616.46 

651.46

686.70

7OO Aft

793.57

829.78

10.90 

0.773

Elev.

4.00000 

4.00617

4.01402 

4.01847

4.02388

4.03013

4.04061

4.04938 

4.05438

4.06010

4.06624

4.07370

4.08071

4.08934

y

L

0 

139

900 

279

ori 

423

495

570

607

645

684

 7 £3 

804

846

890
STAGE-FALL-DISCHARGE RELATIONS

-o

ntinu

d

1

ro 

d 

O H 

_>

<U

01

O t-l 

o

<u

 H

m

!-,

a

«*-!

o

§

 H 

4J

4->

f

O

~a 

J

_>

1

D

<fi 

<H

0 

£T

C/3 l

CO 

r-<

0) 

t 1 

-Q 

03

OCM

co*-« 

.in

ON .

CM.-H

II II

0 o

0-H

 o

r-lrH

II II

o .

VO

r- o

ro vo

O 0

O o

*

.

V

J

r>

V

J

r>

W

-

U

J

a

o «* <-tONtm ON vo * oo 

o r- vo r- <-> 

in in vo r- ON

in in m in in

O OO t CO ON 

in co ON CM co

OO VO OO CO ON

o co vo cim

"1 "I "I **.'*.

1 1 CO 1 1

i i r- i i

1 1 CO 1 1 

1 1 OO 1 1 

1 1 CM 1 1

i i * i i

i i ON i i

i i r- i i 

l 1 O 1 1

1 1 ON 1 1

i i i i i

i i i i i

i i i i i

i i i i i

t i i i i 

i i i i i

. . , , .

i i i i i

i i i i i

i i i i i

i i i i i 

i i i i i

i i i i i

i i i i i 

i i i i i

eo r- vo in** 

in in in in in

CM O 1 1

r-l CM 1 I

i-H ON 1 1

co in i i

i i in in

II-* 0 

1 1 CO VO

1 1 ON r-l

1 1 CM CO

i t in in

1 1 rH CO

i i vo in

1 1 CM *

1 1 1 CO 1

i l l O l 

1 1 1 i-H 1 

1 1 1 . 1

1 1 1 ^C 1

1 1 1 VO 1

1 1 1 ON 1

1 1 1 ^f 1 

1 1 1 O 1

' ' ' * '

i i i r- i

l 1 1 CO 1

i i i r- i i

1 1 I 0 1

1 1 1 CO 1 

1 1 1 00 1

CM

r-l

CO CM CM O CO

in in in m ^

i i i i 

i i i i

i i i i 

i i i i

! ! ' !

co ON CM rrH r- r~ ON 

** ^* oo vo 

co in t^ o

CO CO CO *

 * * <»' *

i-H CO tj" ON

r- ON CM ON

 ^ xC VO 00 

VO OO O CM 

ON ON O O

1 r-l 1 1 O

I'll. 1 * 1 1 *

1 tj" 1 1 CO 

1 O I I *

1 O 1 1 r-l

i m i i vo

i tf i i in

l O i l O

i * i i i*

i CM i i r1 00 1 I ON

1 CO 1 1 CM

1 i 1 1 1 r-l

1 O i 1 O

1 w 1 I **

i r- i o

1 00 1 1 ON

^> O VO CM O

1 t 1 

1 1 1

1 1 1 

l l 1

l 1 1

*4> i I-H vo in 

o i in CM >-i 

rH i co ON ON 

 * l 00 O W

 * i *j" in in

xC i * «* *

oo i r- CM in

CO 1 r 1 VO CO

m i oo ON I-H

0 1 0 O -H

1 VO I CM 1 

i tj" i in i

1.1.1

1 00 1 tj" 1

i in i r- i

1 1 t l O

1 1 1 1 VO 

1 1 l l O

1 1 1 1 OO

1 1 1 1 CO

1 1 1 1 ON

l l 1 l O

l l 1 l O

I i i I «*°

i i i i .

1 1 1 1 CO 

1 1 1 1 ON

CO VO ^ CM O 

CM CM CM CM CM

1 1 1 1 1

1 1 1 1 1 

1 1 1 1 1 

1 1 1 1 1 

1 1 1 1 1

1 1 1 1 1 

1 1 1 1 1

1 1 1 1 1

CM * t O VO

vo r- r- CM CM 

 ^ r- CM f- 03 t i r- o *J" 

in vo vo r- t-

 * * * * *

tj" 00 ON O CM

r-l CM f- CM ON

co in r ON o

i-H rH r 1 i-H CM

\o i in i i

VO 1 i-H 1 1

CO 1 tj" 1 1

.1.11 

 *J< 1 *) 1 1

.1.11

i-H 1 OO * 1 

ON 1 O 1 1

1 r-l 1 1 O

i in i i ON 

1 VO 1 1 VO 

I O I 1 O

t *f 1 1 ^

1 CO 1 1 r-l

1 CO 1 1 CO

1 ON 1 1 ON

I I I I I

1 1 1 1 1 

1 1 1 1 1 

1 1 1 1 1

1 1 1 1 1

CO VO ** CO CM

I 1 1 1 1

1 1 1 1 1

1 1 1 1 1 

1 1 1 1 1

1 1 1 1 1 

1 1 1 1 1 

1 1 1 1 1

1 1 1 1 1

O Or-l I-H 1 

rH ON CO CO 1

ON m o r- i

ON VO VO OO 1

r- oo ON O i

** *}> tJ1 in i

o o r- o i

ON in o in i 

CM m ON co i

CM CM CM CO 1

i co ii

1 CO II

1 VO II

i r- ii

1 r-l II

i ii

i ^i ii

i it

i in ii

1 CM II

1 r-l II

II 1 VO

ii i r~

II 1 O

ii i *

II 1 O

II 1 00

II 1 ON

1 CM II

i ON ii

1 O II 

1 «* II

1 VO II

1 . II

i r- ii

i ON ii

r~

r-l O ON CO OO 

I-H I-H O O O
06

04

02

00

QQ

96

94

92

90

QQ

Q £

84

P.O.

Q9

Ql

80 

79

78073

76

74

79

70

68

66

65

64

£O

fty

61206

60

56

ZA

52

50

48

1,007.05

1,041.89

1,077.44

1,106.72

1 191 OO

1

1 ) *7 M Q

1 1 ^3 79

1,171.08

1,180.35

1,190.19

i 9no, a 1;

1.212.72

4.02115

4.02567

4.03232

4

r\ o c QC

4.04016

4 n"i97Q

4.06116

4

AT Q O jl

4.08105

4.09057

4.10255

4. 11816

1,013.16

1 fl9Q 1Q

1,044.35

1,060.57

1,095.01

1,113.78

1,134.29

1,145.49 

1,157.62

1,171.04

1,186.59

1,205.40

1,216.67

1 9Qfi 9fi

1,247.46 

1 272 39

1,308.04

4.07948

4.08574

4.09305

4.10171

4

1 1 O 1 T

4.12503

4.14134

4

1 £1 OT

4.17647

4

1 noo £

4

0 1 o 1 O

4.23977

4.27620

4.30001

4.33060

4

0*70 oo

4. 4,^717

4.53618

1,062.69

1 083 13

1,105.51

1,117.73

1,130.95

1

1 £ 1 QQ

I

f Q 1 AQ

1 109 t,t.

1 205 23

1

<)nr\ f i

1,240.12

1

O£ T jl O

1,309.47

4.19077

4

OA OA7

4.22939

4

r>C £C O

4.27319

4

9Q9Q C

4 31647

4.34564

4

00097

4.40699

4 43569

4.47183

4.52036

4.70914
66 STAGE-FALL-DISCHARGE RELATIONS

«- *- i.

fl II

O o

V

1 1

J

 HCO in o t~»H r- o

Ov ^* CN 00

 * * * *

t~ i-t in co

\o * * \o 

«N co ^in

vom * co

vo t- t~ 

co oo vo 

vo O»H

CO i-4 ^

oa co co

 * * *

M Or-l 

i-t«N «N

CM O--H

r- O^H

d CO CO

o

00 

CM ^H O
Table 14.--Summary of step-method computation of profiles, 

y

4.00

S

on

3.60

3 

A f\

3 30

3.20

3.00 

2 on

2.80

2

*7/r

2 77

2.70

2

£Q

2

£ A

2.60

2

C£

2 

CO

2 ^0

2 AD

2 

A L.

2 

A A

2.42

2.41

2.40

2 3Q

2

00

2

O£

2

OCQOC

Qy 0 

L

0

133.64

267.38

334.30

401.26

468.28

535.37

3.79 

0.528

Elev.

4.00000

4.00092

4.00214

4.00290

4.00378

4.00484

4.00611

Q =

y 0 =

L

0

134.58

269.56

337.25

405.12

472.89

541. 6*

7.62 

0.780

Elev.

4.00000

4.00374

4.00868

4.01175

4.01536

4.01867

4.02498

Q.

y0 =

L

0

136.59

2f)S 93

274.19

343.54

413.39

483.91

555.31

12.19 

1.028

Elev.

4.00000

4.00977 

4.01569

4.02257

4.03062

4.04017

4.05173

4. 06593

Qy o =

L

0

141.68

O 1 O A f.

286.10

359.85

435.05

512.22

551.75

592.10

19.10 

1.369

Elev.

4.00000

4.02504 

4.04038

4.05830

4.07955

4.10515

4.13666

4.15525

4.17630

Q

y 0 

L

0

142

215

288

363

439

518

558

599
68 STAGE-FALL-DISCHARGE RELATIONS

o\t 

.CO

CM i-l

II II

>,

.'tf 

O 

II II

II II

i-HCNJ

.0 

CM . 

 Ir-l

II II

0,^0

0

VOtt-o

O* O

CO 

ON(N

CIO

II II

0 o

«

w

J

V

u

J

>

_,

w

_J

j

w

J

!»

J

t i co * co co o i t ^ vo ^i co rH ^i t~vomr-i 111 i 111 i rHir-ONr- ^i co i \o i i CMCMcvioNvo vomcomi 111 i 111 i

TflCOcOON vO^'COON in CM O ON r- ( VO t i ( CM 1 III 1 III 1 

VCIVOt"-t CO ON 1 O O rHCMCOCOin VOCOCMCOI III 1 III 1

 «t i «* «} «* ^^imm inminmin mmvovoi 111 i 111 i

t^lOCOVO O^ICOrH CV5VOOCOCO ^irHCOt^l III 1 III 1

1 ON 1 1 O 1 1 ON 1 1 1 1 Tf 1 Tf linilfO IOIVOI ON " 't rH

1 . 1 1 . 11.11 1 1 . 1 . 1 . 1 . I.I.I . 1 .

icoiio iicoii i i co i co im i i i o i CM i co i co r- i^iiin iiinii II^IGO ION ico ivoiini coico rH

1 Tf 1 1 CO IICOII 1 1 t-- 1 ON ICM l^1 IVOICOI rH 1 ^ t~ 

IVOIIVO ilf^ll lit It ICO 100 I CO 1 CO I ONION ON

ICOIICM IIONII liONIi lll ll O 1 t 1 1 1 ON 1 CO 1

imiivo jioii iivoii iiinii voicoio i «t i ON i

1 fO 1 1 1 IICMII llVOll III III CO 1 1/5 1 CO 1 O 1 CM 1

1 Tt 1 1 Tt ll^ll ll^ll 1 1 Tt 1 1 1 1 1 Tt 1 1 1 1 Tt 1

iiiivo iiinii i i co i ' lit it 11101 iiicoi iiiim ii^ii« iicoii IIONII IIIONI iiicoi

i i i i Tt i i Tt i i i i »t i i i i «* i i i i i ^t i i i i «t t

llllrH ll^ill lit II llrHII lltl/>l IllOOl

IIIICO illll I 1 1 1 1 lllll IIIONI lllOl 

IIIIVO lllll IIVOII Illil IIICMI IIIONI

tm

CMOCO^O VOCMOCOVO ^CMOCOVO ^CMOONCO VO^CMOOO VD^CMOOO

co co CM CM CM 1-1 rH 1-1 o o o o o ON o\ ON ON ON co co co co co co t~ r-r-r-c-vo
THE LABORATORY DATA 69

1 f- 1 <*

1 O

i in

I CTv

1

1 CO 

I O 

I O 

1 .

IN I 

i t 1 

OO 1 

IN 1

in i

^ i

O 1

VO I

in t

ON 1

1 1 

1 1

1 1

1 1

t 1 

l 1

IINVO <* IN i-4 00 IN ^CTvCO^VO ii im CTM-I oco * i co in ^ ^

IOi-4 CO^Oi-tO t^ 1"* VO CM ON 

1 CM vo oVOCOi-4VO 1 CO VO t~ i-4

i^1 ^ -^ ^3" ^" in in inininmin

l ^ CM OO ^i- t vo ** COCOVOCOC4 i o in CTV o vo «* 4 co i i co T* co

IOO (NCOVOOO IN CO 00 1- ON 

1 ^ VO CO O CO t~ CTv i-4 CO VO O in 

IOO o i-l i-4 i-l -H (NINCNCOCO

«n i in in t*- vo co

 -4 t ^> IN O VOO in i 1-4 «* i i CM i

t I CO VO O «* ON in i vo vo t t^t 

m i in u

0 1 ^H f

in i t »

oo i < i c

0\ I 0 C

1 1 f

1 i-H

1 OO 

1 i 4 

1 .

1 < 1 

1 VO 

1 00

1 VO 

1 vo

III'*

1 1 I VO 

I 1 I O

I 1 I IN 

1 1 I CO

1 1 1 i-4 

I CO

I 1 1 ^

1 1 

1 I

1 CM 

1 i in

1 O 

1 OO

1 ON Cvl t 4

f CO ^ t 

o in t^- ON

3 O OO

H i-4 i I i-4

in i oo i CM i co t

VO 1 O\ 1 

 -4 l IN 1

 *f , * I

 l 1 

CO l VO 1 

t*- 1 Ov 1

00 1 CO I

1 1 CO 

1 t VO i i r1 1 O

i i ^

i i f1 l 00

1 1 i-4

1 1 1 I

1 1 .

1 1 ^

i i co 

1 1 - 

1 1 O\ 

1 1 CO 

I 1 CO

in i PO i co

O\ 1 O\ l f1- 1 vo l CO 

 <J- 1 i-4 1 O 

CO 1 O\ 1 O

^ i «* i in

in i 1-4 i 1-4

VO 1 CO 1 O\

<M i in i co

i-H 1 i-4 1 i-4

00 1 1 1 CO 

CTv 1 « 1 CO 

CO 1 1 I O 

<M 1 1 1 * 

CM 1 1 1 IN

 * . I I « 

VO III ^*

^ 1 1 1 CO 

1-4 1 1 1 CO

CTv 1 1 1 CTv

1 1 1 1 1 

1 1 1 l 1 

1 1 1 l 1 

1 1 1 1 1

1 1 l 1 «

1 1 1 1 1 

l 1 1 1 1

1 1 1 1 1 

1 1 1 1 1

1 1 1 1 t 

1 1 1 1 1

1 1 1 1 l 

1 1 1 1 1

1 1 1 1 1 

1 1 1 1 1 

1 1 1 1 1 

1 1 1 1 1 

1 1 1 1 1

CTv

fVO

in

o 

in

i i

i 

i

!

[

j

i

1

;

;

'.

|

! : ! ;

CTv i-4 VO t~

co o r- tm IN CTv CTv 

O i-l 4 IN

in in in in

PO t CM CTv 

i-4 vo CMn

O CO VO CTv 

CM IN IN IN

 1 i-4 i-4 i-H

1 1 1 t1 I I CTv 

1 1 1 CO i i i in

1 1 1 CM

 , , «.

1 1 1 ON 

I 1 1 IN i i i in

1 1 l CTv

1 1 1 CTv i i i oo

1 I I CO 

1 1 1 0

III'

1 1 1 -H 

1 1 1 CO

1 1 1 CTv i i i oo

I I 1 IN

; ; : 

1 l 1 VO 

1 1 1 > 

t 1 1 CO 1 1 1 t^

1 1 I CO

1 1 

1 1

: ;

: 5

 -( VO 

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00 i-4 

CO CTv

^ in 

in in

t- CTv

^i o

1 1

1 l

1 i 

1 1 

1 1 

1 1

| |

1 l

l 1 

1 1

1 1 

1 l

1 1

; :

; ;

: :

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s :

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CTv Cv]

o in

CO O 

IN CO

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CO »* 

^" O

co in

CTV cy\

i i "

1 1 U 

1 I C

1

1

1

;

1 

1

.

,

1

1

i i i i i 

i i i i i

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in

CO 

CO

1 t~- Ov ^ 

i-4 O t 

0 ~4 * 

1- Ov i-4

 ^1 1 * «* «» 

00

CO 

i-H

o

1 I

3 1 

 5 1

; !

ii-* i

l 1 O 1

1 1 CO I 

1 1 <3\ 1

1 1 IN 1

II. 1 

1 1 * 1

1 1 > 1 

1 1 CO | 

IIO 1

1 1 O\ i

1 

1

1

o 

o

CO

o 

o

1 1 

in

i 

i

!

!

CO O «rt

CO t*- 1 4 ^ in t 

o o o

1 CO 1 

1 1-4 1 

1 CM 1 

1 i-4 1

1 CTv 1 

t IN 1

1 VO 1 

1 CTv I

1 1 i-4 

1 1 O\ 

1 1 CO 

1 1 IN 

1 1 O 

1 1 

1 1 *

1 l ON 

1 l 

1 1 CM 

1 1 «* 

1 1 O\

CO CO IN >-l O
70 STAGE-FALL-DISCHARGE RELATIONS

t-cr\cvirooo

i-l O vO «H rf OO

< eo o vo ^ o M in CN *

r- 1-1 us -* u> o\ * -* o

^»om\o vo vo r- GO o\

CO M

m 9\

o o

II II

0

CTvIN

COO 

II II

O O

MOO vooo^1 ooom

i 1-1 ^ oo 1-1

vo o

^ « 

incocMOOTf

r- o o 

vo o ev]

CTvCv) 

 * CTv 

00 CTv

o o

I \
THE LABORATORY DATA 71

o t-oo 

vo O\ i i I

CO O VO

PJ t- r-t * C<i PJ 

 ^ l~- 00 O OO i-l 

IM «* t^ W CO IN CM c-j (N co en in
Table 15. Summary of step method computation of profiles, 

y

4.00 

3.80 

3.60 

3.40 

3.30 

3.20 

3.10 

3.00 

2.96 

2.92 

2.88 

2.84 

2.80 

2.78 

2.76 

2.74 

2.72 

2.70 

2.69 

2.68 

2.67 

2.66 

2.65 

2.64 

2.63 

2.62 

2.61 

2.60 

2.59 

2.58 

2.57 

2.56 

2. 55025 

2.50 

2.40

0= 0.34 

y0 = 0.350

L

0

133. 34

266. 68

400. 02

466. 69

533.37

Elev.

4.00000

4.00002

4.00004

4.00006

4.00007

4.00011

Q= 0.56 

y0 = 0.350

L

0

133.34

266.68

400. 03

466. 71

533. 39

Elev.

4.00000

4.00002

4.00004

4.00009

4. 00013

4.00017

Q<= 0.79 

y0 = 0.456

L

0

133. 34

266. 69

400.06

466. 75

533.45

Elev.

4.00000

4.00002

4.00007

4.00018

4.00025

4.00035

Q= 1. 32 

y0 = 0.687

L

0

133.35

266. 74

400. 16

466. 89

533.65

Elev.

4.00000

4.00005

4.00022

4. 00048

4. 00067

4.00095

Q = 1. 84 

y0 = 0. 905

L

0

133.41

266. 86

333.60

400. 36

467. 15

533.99

Elev.

4.00000

4.00023

4.00058

4.00080

4.00108

4.00145

4.00197

0= 2. 7

y0 = 1.1

L

0

133.49 

200. 26

267.05

333.88

400. 75

467.69

501.20 

534.74

El

4.0

4. 0

4.0

4.0

4.0

4.0

4.0

4.0

4. 0
2.30 

2.20 

2.16 

2.12 

2.10 

2.08 

2.04 

2.00 

1.96 

1.92 

1.90 

1.88 

1.86 

1.84 

1.82 

1.80 

1.79 

1.78 

1.77 

1.76 

1.75 

1.74 

1.73 

1.72104 

1.72 

1.70 

1.68 

1.66 

1.64 

1.62 

1.60 

1.58 

1.56 

1.54 

1. P?. 

1.50 

1.48 

1.46 

1.44

1 A9

600.05

666.73

733.42

800. 13

833.51

4.00015

4.00019

4.00026

4.00039

4.00053

600.08

666.79

733.52

800.32

83L-.77

4.00024

4.00037

4.00056

4.00096

4.00131

600.16

666.90

733.69

800.62

834. 18

4.00048

4.00070

4.00107

4.00186

4. 00254

600.45

667.32

734. 34

801.74

835.73

4.00135

4.00196

4.00302

4. 00522

4.00719

567.44 

600.91

634.41

667.96

701.58

735.30

769.17

803.32

837.96

4.00232 

4.00273

4.00323

4.00388

4. 00474

4.00590

4.00751

4.00996

4.01388

568.32 

601.95

635.66

669.47

703.43

737.61

772. 16

786.12

800.20

814.44

828.88

S43.58

858.65

R^A 94.

4. 

4.

4.

4. 

4.

4.

4

4

4. 

4

4

4

4. 

A. 
Table 15.~Summary of step method computation of profiles, cros

y

1.40 

1.38 

1.36 

1.34 

1.33 

1.32 

1.31 

1.30 

1.29785 

1.28 

1.26 

1.24 

1.23 

1.22 

1.21 

1.20 

1.19 

1.18 

1.17 

1. 16958 

1.14 

1.10 

1.06 

1.04 

1.02 

1.01 

1.00 

.99 

.98 

.97 

.96 

.95 

.94 

.93 

.92

Q14.AS

Q = 0.34 

y0 = 0.250

L

866.91

900.37

913. 78

927.20

940. 64

954. 11 

967.61 

981. 15 

987.94 

994. 75

1,001.57

Elev.

4.00073

4.00111

4.00134

4.00160

4.00192

4.00233 

4. 00283 

4. 00345 

4.00382 

4. 00425

4. 00471

0 = 0.56 

y0 = 0.350

L

867.30

900.99

914.53

928.11

941.74

955.45 

969.25 

983.18 

990.20 

997.27

1,004.38

Elev.

4.00190

4. 00297

4. 00359

4. 00433

4. 00522

4. 00635 

4.00775 

4. 00954 

4.01060 

4.01181

4.01314

0= 0.79 

y0 = 0.456

L

867.90

901.95

915.70

929. 55

943.50

957.61 

971.93 

986. 53 

993.95 

1,001.47

1,009.09

Elev.

4.00370

4.00585

4.00710

4.00865

4.01050

4.01283 

4.01579 

4. 01959 

4.02185 

4. 02441

4. 02727

0= 1.32 

y0 = 0.687

L

870.18

905.61

942. 84

983.30 

1,001.26 

1,010.77 

1,020.76 

1,025.96 

1,031.23

1,053.12

1,076.59

Elev.

4.01054

4.01683

4.02852

4. 04990 

4. 06378 

4.07231 

4. 08228 

4. 08788 

4. 09369

4. 11936

4. 14977

0= 1.84 

y0 = 0.905

L

873.58

911.36

927.51

944.62

962.94

983.38 

1,007.36 

1,037.32 

1,055.78 

1,077.64 

1,090.28 

1,103.94 

1,118.58 

1,134.75 

1,152.89 

1,173.68 

1,198.20 

1,227.05 

1,264.67 

1,319.66

1 W* 18

Elev.

4.02074

4.03408

4. 04253

4. 05386

4.06882

4. 09014 

4.12208 

4. 17196 

4.20734 

4. 25292 

4. 28084 

4.31182 

4. 34574 

4. 38425 

4.42867 

4. 48104 

4. 54460 

4.62115 

4.72401 

4.87898

K r>33KQ
QQ

.84

.80

IK

nc

.74

11

.71

.70

CQOO/7

£>f\

C£

CO

.50

.48

.47

.46056

A^

4.0

4.9

.41

.40

O.Q,

Q.B

Q."7

Qc

OCOCfl

T^

O.Q.

Q.O

Q1

O.A,

OQ

OB

.27

OC

OCOCA

i ftQ^ RI

1,070.32

1,105.26

1,140.90 

1 1SS A 4

1

1 P7A OQ

1 177 7^

1 IQ1^ Q7

1

1 QQ OQ.

1

1QO 1 O

i 1Q7 n.^

1,201.01

1,205.01

1,209.08

1 O1 Q O1

1 217 41

1

OO1 C7

1 226 01

1 9Qft A7

I

ooe APT

1

OOQ £J1

1,255.26

1,261.15

1 9£7 ^Q

1 974. fin

1

OQQ QQ

I

QI A C7

I

ooo >i a

4. 00743

4.01096

4.01578

4 099 10

4

AOCOO

4. 03069

4

AOQO^

4.03611

4.03769

4. 04303 

4. 04503

4. 04724

4 04963

4

«t;ooo

4.05501

4. 05803

4.06141

4.06521

4. 06943

4.07416

4

AOCr7O

4. 11380

4. 12816 

4. 15014

4 9K"788

1,040.31

1,077.09

1,115.26

1,155.91

1

1 7Q OQ

1

1Q1 7^

1

OA1 £JA

1 OOQ OO

1 OOQ 1Q,

1,235.40 

1 94.1 Q4

1 9[\ P7 l\^

1

O(?C QQ

1 9 P7 [\ T?

1 9R7 4T

1 ^m 'i'i

1

OOA f\D

1

O/?c cr A

4.02093

4.03127

4. 04578

A nfivri

4.07984

4

AQCOC

4. 10480

4.11600

4

1 QQ££

4

, iqr7Cr7

4.14620 

4. 15582

4

1 C7Q&

4.18115-

4 19749

A OT7Q1

4 94941

4 9TQQQ

4. 35024

4.45000

1, 048. 10

1,089.23

1, 134. 19

1

1 Q7 Q^

1

0*1 o A a

1

C) A £ QC

1

OCO Cr?

1 OQO QB

1,300.54

1

OOC C A

1 Q7 Q AQ

 , -T

4. 04430

4. 06769

4.10257

4.16205

4.20038

4

OCQAO

4.30601

4

QOQQA

4

0 o 1 /?o

4

CQC QC

1 1fl9 98

1 1Q1 IB

1,165.30

1 1 H^ 11

1,210.06

1

0QQ QA

1 9^Q AS

1,281.46

1 Q19 Oft

1,360.59

1,407.72

4. 18684

4

OQOCJ

4. 29590 

i 001701

4

OQA1 O

4

C£* 1OO

4. 64678 

4.78177
76 STAGE-FALL-DISCHARGE RELATIONS

Table 16. Summary of step method computations of profiles, cross section 6

y

4.00 

3.80 

3.60 

3.40 

3.30

3.20 

3.10 

3.00 

2.90 

2.86

2.82 

2.80 

2.78 

2.74 

2.70

2.68 

2.66 

2.64 

2.62 

2.60

2.58 

2.57 

2.56 

2.55 

2.54

2.53 

2.52 

2.51 

2.50 

2.49

2.48 

2.47 

2.46 

2.45531 

2.40

2.30 

2.20 

2.10 

2.00 

1.90

1.80 

1.76 

1.72 

1.70 

1.68

1.64 

1.60 

1.56 

1.52

1 SO

Q = 0.83 

yo = 0.471

L

0

133.35

266.70

400.06

466.75

533.44

600.15

666.89

733.69

800.65

jnj. 93

Elev.

4.00000

4.00005

4.00010

4.00018

4.00025

4.00032

4.00045

4.00067

4.00107

4.00195

A nnofio

Q = 1.84 

yn = 0.905

L

0

133.38

266.79

400.26

467.03

533.84

600.73

667.76

735.10

768.97

803.12

fll7 If,

Elev.

4.00000

4.00014

4.00037

4.00078

4.00109

4.00152

4.00219

4.00328

4.00530

4.00691

4.00936

A m son

Q = 3.98 

y0 = 1.294

L

0

133.55

267.23

334.14

401.14

468.28

535.63

569.42 

603.33 

637.41 

671.72 

706.38

741.57 

755.85 

770.30

784.96

799.90 

815.20 

830.99 

847.45

Elev.

4.00000

4.00065

4.00169

4.00242

4.00342

4.00484

4.00689

4.00826 

4.00999 

4.01223 

4.01516 

4.01914

4.02471 

4.02755 

4.03090

4.03488

4.03970 

4.04560 

4.05297 

4.06235

Q = 30.63 

yo = 2.431

L

0 

73.08 

147.89 

225.26 

265.28

306.56 

349.51 

394.72 

443.17 

463.77

485.28

507.93 

531.98 

557.85

571.65 

586.14 

601.47 

617.82 

635.40

654.52 

664.79 

675.64 

687.13 

699.40

712.60 

726.97 

742.72 

759.28 

777.88

800.96 

828.70 

863.77 

884.65

Elev.

4.00000 

4.01924 

4.04367 

4.07578 

4.09584

4.11968 

4.14853 

4.18416 

4.22951 

4.25131

4.27584

4.30379 

4.33594 

4.37355

4.39495 

4.41842 

4.44441 

4.47346 

4.50620

4.54356 

4.56437 

4.58692 

4.61139 

4.63820

4.66780 

4.70091 

4.73816 

4.77784 

4.82364

4.88288 

4.95610 

5.05131 

5.10926
THE LABORATORY DATA 77

Table 16.--Summary of step method computation of profiles, 

cross section 6 Continued

y

l Aft

1.46

1.44

1.42

1.40 

l 4ft

1 ^<C

1 35

i 44

1.33

1 94

1 ^1

1.30694

1.30

1 9£

1.22 l ID

1.14 

1.10

1.06

1.04 

1.02

1.01

1.00

.99

Oft

.97

.96

.94

 7<J

.92

.91405

.90

an

.70

.60

.56

.51

.50

.49

AD

.47571

Q-( 

YO=<

L

868. 00

902.12

7 A J 7 X

7*7* OA

7%O» O%

958.04

972.48

7 O 1 . ££

^x**. 1 O

1,002.36

1,010.09

1 1 1/47 O O

A 1 V 7 ft « U&

1,138.76

1,195.75

1 995 1ft

1

9£C 1O

X,A|7.V&

I

nn'7 Q<;

1

499 £C

1

0 ^ri nC

1 9QJ 00

3.83 

).471

Elev.

4.00400

4.00636

4.00773

4. 00946 

4.01152

4.01412 

4.01744

4.02166

4.02419

4

no7 AO

 v Ov& i

4.04949 

4.07606

4 1 i £90

4

1 O7QC

4.23554

4.31557

4

9 A QC£

4

QOQQQ

4.45795

4.56585

4.65870

Q-]

y0 = (

L

873.38

911.16

y & i o L

nen * A

7Od 1O

1,007.16

1,037.10

1,055.69

1,077.55

1,090.19

1, Ivd. 00 

1 lift 10

1,134.66

1

1 Co on

1,173.59

1

1OQ 1 1

I

nnt no

1,264.51

1,373.02

L.84 

).905

Elev.

4.02014

4

rt oo AO

4.04193

4.05326

4

AJCO99

4.08954 

4191 Aft

4.17130

4.20707

4

OCQJCC

4

00AC7

4.31155 

4.34547

4

OOOQD

4.48077

4.62079

4.87850

5.03311

Q-; 

Vo = ]

L

864.89

m i Q

OOO OO

OCA 1 5

905.34

O 1 f "tfi

QOO 1 A

940.35

949.77

nen O7

974.74

994.12

1,002.51

!.98 

L.294

Blev.

4.07467

4

AOO9JC

4.09146

4.10245

4.11602

4

1 Q99O

4.15630

4.17105

4 1 ftO31

4

41 9£1

4 9AA90

4

OQOQ|T

4.31447
78 STAGE-FALL-DISCHARGE RELATIONS

STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS

DETERMINATION OF DISCHARGE UNDER BACKWATER

In the operation of most gaging stations, discharge is determined from stage, 

or elevation of the water surface. This is accomplished by use of rating curves, 

developed from an appropriate number of discharge measurements, for each of 

which both the discharge (the dependent variable) and the stage (the independent 

variable) have been observed. The curves presented as figure 6, pages 18 to 19, 

in Bulletin 381 (Lansford and Mitchell, 1949) become rating curves for the laboratory channels if the scale of depth, y, be considered as a scale of elevation.

The use of these curves assumes there is a fixed and invariable relation between stage and discharge. Such an assumption obviously is true for uniform flow 

(see pp. 11-13). It also is true for certain conditions of nonuniform flow, as, for 

example, those in which an M-l surface profile is produced by nonuniformity of 

the channel, so that for a given stage and discharge the profile always will be 

the same. The rating curve for a channel upstream from a fixed spillway (see fig. 

17 (left), below) is an example of such conditions. If a gage be located at any 

point such as A or B, the stage will be greater than for uniform flow; but as long 

as the discharge is constant, the stage at A or B will be constant and the simple 

stage-discharge relation effective. In discussions on stream gaging, it is quite 

common to designate this as the "normal" stage-discharge relation. It will be 

obvious that, although this relation may be fixed and invariable, it nevertheless 

may differ materially from the uniform-stage-discharge relation described on 

pages 11-14 , and which also is referred to by many writers as the "normal" 

stage-discharge relation. To avoid confusion between the two, the relationship 

determined by the conditions of figure 17 (left) will be designated as the fixedbackwater stage-discharge relation.

Fixed source of nonuniformfty Vorioble source of nonuniformity 

Figure 17. Variations of water surface under fixed discharge.
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 79

But the surface curve between gages A and B may result from other causes 

than a fixed spillway. Let it be supposed that the channel discharges into a reservoir, as in figure 17 (right), and that the elevation of the water surface in this 

reservoir is subject to variation throughout wide limits. Assuming that the discharge is the same as for figure 17 (left), and elevation 1 of the reservoir is such 

that the stage at D is the same as before, then the profile upstream from B will 

be the same as before. If the reservoir surface is lowered to elevation 2, then 

the point 6' on the first profile will be transferred to b* on the second profile. 

In fact, the profile between gages A and B will be identical in form with the 

curve a 1 6'. The stage will be reduced at both gages A and B, although the discharge remains constant. Hence, under such variable backwater, the stagedischarge relation at each point will be variable, rather than fixed. It will be 

noted, however, that the reduction in elevation at B will be greater than the corresponding change at A, so that the fall in the reach A-B will be increased. If 

the reservoir surface is further lowered to elevation 3 so that the depth at B is 

equal to y0 » l^e condition of uniform flow will be established. The depth at A 

also will be y0 , the water-surface slope will be parallel to the bed slope, and 

the uniform stage-discharge relation will be effective. This is the limiting condition of the problem as now considered. Lowering the reservoir surface to elevation 4, so the depth at B is less than y0> wiU result in an M-2 surface profile 

in the reach A-B a situation beyond the scope of the present study.

It was noted above that the variation in stage at points A and B was accompanied by a variation in fall, so that stage and fall may now be used as two independent variables for the determination of discharge. Reliable values of discharge could be more easily obtained by the use of stage and slope of the water 

surface, both at gage A. Unfortunately, the slope of the water surface at a point 

is not susceptible to observation under field conditions. It can only be approximated by use of the fall in a reach which, of course, may be transformed to the 

average slope in the reach by dividing by the length, in feet, between the gages. 

An examination of the difficulties which arise from this approximation, and of 

the methods for mitigating these difficulties represents the objective of the present study.

Referring again to figure 17 (right), it will be noted that the shifting of the 

profile a1 -6' to the position a" -i" is an application of the principle of rigidity, described in Bulletin 381 (p. 17) as follows:

In a prismatic channel the shape of the backwater profile remains fixed as long as the 

discharge, the boundary conditions and the slope are fixed. Any change in the depth at 

which the water is flowing does not produce a change in the shape of the curve; it merely 

transfers that particular portion of the curve to a different part of the channel. As the 

depth of water is increased, the curve is displaced, rigidly, in theupstream direction, and another portion of the curve, characterized by a slope which is more nearly 

horizontal, is brought into place; as the depth of water is decreased the curve is displaced, rigidly, in the downstream direction and another portion of the curve, characterized by a slope which is more nearly that of the channel bed, ia brought into place. In 

neither case has any portion of the curve been altered; only a different part of the same 

curve has been brought under observation.
80 STAGE-FALL-DISCHARGE RELATIONS

Thus, means are now at hand for determination of an unlimited amount of data 

for studies of stage-fall-discharge relations in the laboratory channels. It is 

necessary only to decide what length of reach is to be investigated and to consider two gages, separated always by this length of reach, to be moved at will 

to any location on the channel. Thus, if it be desired to study a reach of 100 

feet in length, the upper gage may be considered to be located at station 100 

and the lower gage at station 0. From any of the profile tables (tables 1-5 or 

12-16), the discharge and the elevation at the two stations may be determined. 

(Values in the profile tables are given at such intervals that straight line interpolation may be used when necessary.) Subtracting the elevation obtained on 

the lower gage from that obtained on the upper gage (tables L2-16), provides 

the value of the fall, and complete data are available which are equivalent to a 

field observation of stage, fall, and discharge. If, now, another set of data are 

desired under backwater conditions only slightly less severe, it may be considered that the upper gage is located at, say, station 130, and the lower gage 

at station 30. From the profile tables another pair of stages may be read, which 

result in slightly lower stage at the base gage and slightly more fall in the 

reach.

The technique just described can be applied either to the smoothed profiles, 

tables 1 to 5, or to the computed profiles, tables 12 to 16. Experience has 

demonstrated, however, that in the studies which are to follow, it is more desirable to use tables 12 to 16. Under great backwater effects the observed fall 

becomes extremely small, in many cases less than 0.0005 foot. (See, for example, the Fft, column of table 24, p. 95 .) Tables 1 to 5, it will be recalled, 

were prepared from data observed only to thousandths of a foot, so that not 

infrequently falls computed from these tables will be in error by more than 

0.0001 foot, thus leading to. possible errors of more than 20 percent in the 

computation of fall ratios. Such errors lead to considerable scatter in subsequent plottings and thus tend to obscure the conclusions. The scatter from 

this cause is practically eliminated by the use of tables 12 to 16, in which 

elevations are computed to five decimal places. From the fact that the computed profiles are essentially the same as the smoothed profiles (differing 

largely only in degree of refinement) it follows that studies based on tables 12 

to 16 will lead to essentially the same results as those based on tables 1 to 5. 

Thus, to prevent conclusions being obscured by scatter due to insufficient refinement in fall, the stqdies which follow have been based on the computed 

profiles, table 12 to 16.

In order that readers who wish to make further studies may have convenient 

access to data in usable form, selected simultaneous values of stage, fall, and 

discharge for a 400-foot reach of the computed profiles are presented as tables 

33 to 37, pages 155to 159. Appropriate values of y for each profile are listed, 

together with the discharge, in the tables. The following example will illustrate 

the computation of the associated falls. The fall, F , in cross section 2, as
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 81

shown in table 33 for a depth of 1.60 feet and a discharge of 3.90 cfs, was 

computed from table 12, page 59, using a value for y$ of 0.391 foot (Q of 3.90 

cfs). The station and elevation for y of 1.60 feet are found to be 802.38 and 

4.00714, respectively. Subtracting 400 feet, the length of the reach, from 802.38 

gives 402.38, the station of the downstream end of the reach. From column 2 of 

the table using 402.38 and straight-line interpolation, the elevation of this station is found to be 4.00172. Subtracting this water-surface elevation from that 

at the upstream end of the reach gives 0.00542, the listed fall.

In working with field observations, the data which fail to support a particular 

analysis, or method of procedure, sometimes are viewed with suspicion. Their 

recording may have been affected by wind, or ice, or poor measuring conditions, or by 

errors in reading a gage. But for the data which may now be computed from 

tables 12 to 16, these excuses cannot be made. Values of discharge are the 

result of many observations which have been crosschecked and correlated. 

Values of stage are computed to the fifth decimal place. The analysis of these 

data by any given method becomes a test, not of the data, but of the method, as 

it applies to the particular set of channel conditions that existed in the laboratory. If the method gives acceptable results under the severe conditions of this 

test, it may be considered as generally applicable to other stage-fall-discharge 

problems. If it does not give acceptable results, it must be regarded as a method 

suitable only for conditions which are less severe than those used herein. 

With these thoughts in mind, the data have been computed and plotted by the 

several methods which follow.

RESUME OF CURRENT METHODS 

THE STAGE-FALL-DISCHARGE DIAGRAM

This, perhaps the oldest of all the methods of analysis, is illustrated by 

figures 18 and 19, pages 82 and 83. The first of these is for cross section 2 

(rectangular); the other is for cross section 5 (flood plain). Each has been prepared for a reach 400 feet in length. Since the bed slope is 0.003, the nobackwater fall (or fall for uniform flow) is 1.20 feet. The right-hand curves, 

which represent the uniform stage-discharge relations of tables 42 and 45, Bulletin 381 (Lansford and Mitchell, 1949), have been so labeled. Other solid lines 

represent the positions which would be occupied by the stage-discharge curves 

if the fall were fixed at the values shown. Preparation of these diagrams is 

quite simple. For example, in plotting figure 19, the needed values of stage, 

fall, and discharge are abstracted from table 33, page 155, until enough are obtained to permit the drawing of lines of equal fall.

In practice, it sometimes is necessary to develop such diagrams from comparatively few points. To obtain a systematic spacing of the lines, it is sometimes 

assumed that lines of equal fall should vary with the square-root relation. 

Curves developed on this assumption and based on the no-backwater stage-
82 STAGE-FALL-DISCHARGE RELATIONS

UI '3&DIS
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 83
84 STAGE-FALL-DISCHARGE RELATIONS

discharge relation appear as dashed lines in figures 18 and 19. The percentage 

differences are enormous and must lead immediately to the conclusion that, in 

this case at least, the true discharge does not vary from the no-backwater 

stage-discharge relation in accordance with the square-root-of-the-fall relation. 

In practice, of course, many of the discharge measurements, if not all, might 

have been obtained under backwater, and the curves would have been adjusted 

to fit these measurements. Such adjustments, properly applied, would tend to 

bring the diagram to the position of the solid lines, and with sufficient measurements the diagram might become an adequate expression of the stage-falldischarge relations. Since, however, there is such sharp departure from the 

square-root-of-the-fall relation, caution must be used in the extrapolation of 

diagrams based on meager data.

THE BOYER METHOD

In U. S. Geological Survey Water-Supply Paper 888 (Corbett and others, 1943) 

it is pointed out (p. 132),

"if at a given stage two discharges Oj and (? 2 occur at different times with the 

corresponding energy slopes S , and S g2 their relation may be expressed by the 

equation [5]:

It then is explained that, for reasons as stated above, the fall in a reach is not 

a direct measure of the slope of the energy line, but

"the slope S a as expressed in equation (5) may be considered as a function of 

the fall in a reach, so that the equation of relation becomes

Q i/ Q 2 = function F l/F 2 (6)

in which the falls are the difference in the heights of the water surface at the 

ends of the reach. If the velocity-head increment aV 2/2g is large enough to 

warrant consideration, it should be added to each observed fall. "

Now if Qm and Fm be observed for a given backwater condition, and if Qo 

and FQ be observed for conditions of no backwater, a pair of ratios may be obtained ((?m /(?o anc* Fm ^Fo^' ^ t^le backwater condition be permitted to change, 

and another observation made of Qm and Fm , a second pair of ratios becomes 

available. This procedure may be repeated to obtain as many pairs of ratios as 

are desired. Points may then be plotted, using the discharge ratios as ordinate 

and the fall ratios as abscissa; and the nature of the function as referred to above 

may be examined. Such a plot is commonly spoken of as a "Boyer diagram" in 

recognition of the development work of Mr. M. C. Boyer, at the time a hydraulic 

engineer of the U. S. Geological Survey.
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 85
86 STAGE-FALL-DISCHARGE RELATIONS

Figure 20 (left) is a Boyer diagram developed for the laboratory data, cross 

section 2, using a length of reach of 400 feet. It will be noted that the points 

plot, not as a single curve, but as a family of curves: a separate curve for each 

value of y0 (or Qm ). It will a^ s° be noted that as the discharge (represented by 

value of y0) increases, the curve of relation approaches but does not reach the 

position which would be represented by the relation Om /(?o = V^ni/^o*

Figure 20 (right) presents a similar study of data for cross section 5. Again 

it is obvious that there is a curve for each value of yo(or Qm ). But here the 

pattern of the curves is much more complex than for the rectangular channel; in 

fact, because of the braiding of the curves a general statement is extremely 

difficult. For the higher values of the discharge ratio, however, it may be noted 

that, beginning with the lowest discharge, the succeeding curves of low discharge, as plotted, lie progressively toward the right (toward the square-rootof-the-fall position). Thus, reading from left to right, we find y Q values of 

0.250, 0.350, 0.156, and 0.687. The condition of near-bankfull discharge at uniform depth, represented by the curve y0 = 0.905, cuts erratically across the pattern, resembling the curves for high discharge at the upper end, but resembling 

the curves for low discharge at the lower end. The curve for the next higher 

value of y0 (1.158, with flow on the flood plain) is quite similar to the curve for 

y0 = 0.250, and succeedingly higher discharge curves (y 0 values of 1.285, 1.704 

and 2.525) fall progressively further toward the right. Thus, for yQ values which 

are entirely within the small rectangular channel, the pattern is similar to that 

for cross section 2, and for y0 values which are above the flood plain another 

pattern somewhat similar to that for cross section 2 is superimposed upon the 

first. The entire pattern, however, is much too complicated to be developed from 

meager data such as might be collected in the field.

It is to be expected that the variation of the function, as shown in the first 

Boyer r'iagram, is due in substantial part to neglect of the velocity-head increment. To what extent this is true is shown by figure 21. Since the curves for 

discharges of 3.90 and 40.39 second-feet (y0 - 0.391 and 1.840 ft) form an envelope for all the other curves developed for cross section 2, only these two have 

been shown in figure 21 (left). The curves shown in continuous lines show the 

range of variation of the function, with correction for velocity-head increment 

but still for a reach of 400 feet. Although the curves more nearly approach the 

souare-root-of-the-fall relation, the variation still is quite large. Further improvement may now be obtained by shortening the length of reach, as indicated 

by the plotting of the curves shown in dashed lines further to the right on figure 

21 (left). The length of reach has been taken as 120 feet, and correction has 

been made for velocity-head increment. It will be noted that these curves show 

less variation, and are still closer to the position represented by the squareroot-of-the-fall relation. In fact, this relation appears to be the position which 

all these curves, when corrected for change in velocity head, approach as a 

limit as the reach is made shorter. This is due to the fact that, as the reach 

becomes shorter, the variation of fall in the reach becomes more representative 

of variation in slope at the base gage. That the square-root relation applies to 

slope at the base gage is demonstrated by the curve farthest to the right in this
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 87
88 STAGE-FAIL-DISCHARGE RELATIONS

figure. This curve has been drawn so that (?m /(?o is equal to the square root of 

Fm /F0, and data for the 3.90 and the 40.39 second-foot discharges, for a very 

short reach, and corrected for change in velocity head, have been plotted thereon. The only differences between the curve and the plotted points appear to be 

those which arise from lack of refinement of the computations. Figure 21 (right) 

presents similar studies of cross section 5 data.

The computations from which these plots were made are summarized in the 

tables which follow. Table 17, page 90, presents the data for the upper enveloping curve for cross section 2, 400-foot reach, while table 18, page 90, 

presents data for the lower enveloping curve for the same cross section and 

reach. Tables 19 and 20, page 91, are corresponding data for the same cross 

section, 120 foot reach. In these tables columns 5 and 6" give the discharge 

ratios and fall ratios used in plotting the enveloping curves of figure 20 (left). 

(The adjustments for velocity-head increment that were used in plotting figure 

21 have not been shown in tables 17 to 20.) The data are derived in the following manner: For a given observed depth, ym , and profile, Qm , the values of fall 

for a 400-foot reach may be found in table 33, page 155. The computation of this 

fall was as follows: For ym = 2.00, y0 = 0.391 (Qm = .;9J), cclurcns 2and3of 

table 12 give L = 668.13 and elev. = 4.00439. The L is entered in coJurnn 2 

of table 17. Entering the same columns of table 12 with L = 668.13 - 400.00 = 

268.13, you find, by interpolation, a water-surface elevation of 4.00098 feet. 

The difference between the two elevations, or 0.00341 feet is Fm , the observed 

fall for a reach of 400 feet as given in table 33. This is entered in column 3 of 

table 17. The uniform discharge, @ 0 , for y = 2.00 ft, is found from table 42 of 

Bulletin 381 to be 45.18 cfs, which is entered in column 4 of table 17. Values 

of F 0 , y 0 and Qm are constant throughout table 17, and are as given in the 

table heading. Columns 5 to 10, inclusive, are computed as indicated at the top 

of the column. Computation of column 11, the final column, will be explained in 

a later section (see page 92). For the Boyer diagrams, only the results of columns 

5 and 6 are required. The other columns will be used in connection with other 

types of diagrams.

To obtain all the curves of figure 20 (left), it was necessary to compute 

tables similar to table 17 for each of the profiles. For the four intermediate 

profiles (those between the highest and lowest discharges) the only values that 

were used were those for discharge ratio and fall ratio. These have been condensed into a single table, and presented as table 21, page 93.

A similar set of computations, parallel in all respects to those just described, was required for the plotting of figure 20 (right), the Soyer diagram for 

cross section 3. These are presented as tables 22 to 25, pages 94-95, which 

are comparable in form and content to tables 17 to 20, and as table 26, page 

96-97, which is comparable to table 21.
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 89

g 8 ?

4146)314 3606 3AI4.33J.J.3 J.U93.l3d

 i ^ 1S

in h

= E

I

8 1 j 

I I

8 a 

°- (M
90 STAGE-FALL-DISCHARGE RELATIONS

Table 17. Computation of stage-fall-discharge ratios, cross section 2, 400-foot 

reach; yQ =0.391 ft

[F = 1.200 ft; Q =3.90cfs] 

o m

Ym

2.50000

2. 00000

1. 80000

1.60000

1.40000

1.20000

1. 10000

1.00000

.90000

. 80000

. 70000

. 60000

.50000

.46000

.42000

.41000

.40000

. 39491

*

L

500. 82

668. 13

735. 19

802. 38

869.77

937.52

971. 62

1,005.97

1,040.71

1,076.10

1, 112. 60

1,151.29

1,196.03

1,218.80

1,252.19

1,266.02

1,287.88

1,310.51 

limit

F m

0. 00215

. 00341

.00422

. 00542

.00712

. 00977

.01170

.01436

.01811

. 02377

. 03263

. 04792

.08110

. 10878

. 16788

. 19888

. 25355

.31530 

1.200

*,

60.83

45.18

39.20

33.38

27.72

22.23

19.57

16.98

14.48

12.09

9.84

7.74

5.81

5.08

4.38

4.21

4.05

3.96

3 « QAyu

Qm/Qo

0.0641

.0863

.0995

.1168

.1407

.1754

.1993

.2297

.2693

.3226

.3963

.5039

.6713

.7677

.8904

.9264

.9630

.9848 

1.0000

F /F m o

0.0018

.0028

.0035

.0045

.0059

.0081

.0098

.0120

.0151

.0198

.0272

.0399

.0676

.0906

.1399

.1657

.2113

.2628

1.0000

F./VO

0.0055

.0087

.0108

.0139

.0182

.0250

.0299

.0367

.0463

.0608

.0835

.1226

.2074

.2782

.4294

,5086

.6485

.8064

.0

w.

0. 0009

.0017

.0023

.0034

.0051

.0081

.0106

.0144

.0201

.0297

.0466

.0799

.1622

.2365

.3997

.4851

.6339

.7984

rj*.

2.0833

1.6667

1.5000

1.3333

1. 1667

1. 0000

.9167

.8333

.7500

.6667

.5833

.5000

.4167

.3833

.3500

.3417

.3333

.3291

vo/vm

0. 1564

.1955

.2172

.2444

.2793

.3258

.3555

.3910

.4344

.4888

.5586

.6517

.7820

.8500

.9310

.9537

.9775

.9901 

1.0000

Percent 

submergence

99.91

99.83

99.77

99.66

99.49

99.19

98,94

98.56

97.99

97.03

95.34

92.01

83.78

76.35

60.03

51.49

36.61

20.16

Table 18. Computation of stage-fall-discharge ratios, cross section 2, 400-foot 

reach; yQ = 1.840 ft

[F = 1.200 ft; Q 40.39 cfs]

o i

Ym

2.90000

2.50000

2.28000

2.20000

2. 08000

2. 00000

1.98000

1.91000

1.90000

1. 89000

1. 88000

1. 87000

1. 86000

1. 85840

1.840

L

433.06

620. 26

744.32

797.30

892. 28

975. 80

1,001.49

1,125.23

1,151.20

1,181.45

1,217.73

1,263.49

1,326.69

1,339.83

limit

Fm

0. 18789

. 28015

.36877

.41639

. 51335

.61120

.64251

. 79885

. 83069

.86672

.90810

.95661

1.01557

1.02664

1.200

<?o

74.06

60.83

53.82

51.32

47.62

45.18

44.58

42.47

42.17

41.87

41.58

41.28

40.98

40.93

40.39

V^o

0. 5454

.6640

.7505

.7870

.8482

.8940

.9060

.9510

.9578

.9647

.9714

.9784

.9856

.9868

1. 0000

Fm/Fo

0.1566

.2335

.3073

.3470

.4278

.5093

.5354

.6657

.6922

.7223

.7568

.7972

.8463

.8555

1. 0000

F m' /v 'o

0. 1021

.1523

.2004

.2263

.2790

.3322

.3492

.4342

.4515

.4710

.4935

.5199

.5519

.5580

.6522

FnAm

0. 0648

.1121

.1617

.1893

.2468

.3056

.3245

.4182

.4372

.4586

.4830

.5116

.5460

.5524

.6522

Ym/Fo

2.4167

2.0833

1.9000

1. 8333

1. 7333

1.6667

1.6500

1.5917

1.5833

1.5750

1.5667

1.5583

1.5500

1.5487

1. 5333

y<Am

0.6345

.7360

.8070

.8364

.8846

.9200

.9293

.9634

.9684

.9735

.9787

.9840

.9892

.9901

1. 0000

Percent 

submergence

93.52

88.79

83.83

81.07

75.32

69.44

67.55

58.18

56.28

54.14

51. 7C

48.84

45.40

44.76

34.78
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 91

Table 19. Computation of stage-fall discharge ratios, cross section 2, 120-foot

0.391 ft

[F0 = 0

*»

3.60000

2.50000

2.00000

1.80000

1.60000

1.40000

1.20000

1. 10000

1.00000

.90000

. 80000

.70000

.60000

.56000

.50000

.46000

.43000

.42000

.41000

.40000

.39491

.391

L

133.47

500. 82

668. 13

735. 19

802.38

869.77

937.52

971.62

1,005.97

1,040.71

1,076.10

1,112.60

1,151.29

1,168.04

1,196.03

1,218.80

1,241.73

1,252.19

1,266.02

1,287.88

1,310.51

limit

Fm

0. 00037

. 00086

.00149

.00190

.00250

.00340

.00498

.00619

.00790

.01046

.01449

. 02108

.03288

.04071

.05980

.08267

. 11436

. 13165

. 15674

.19971

.24371

.360

°o

98.42

60.83

45.18

39.20

33.38

27.72

22.23

19.57

16.98

14.48

12.09

9.84

7.74

6.95

5.81

5.08

4.56

4.38

4.21

4.05

3.96

3.90

9,

0.

e

t

.

9

9

^

t

m

s

B

<

.

i!

.360 ft; Q .3.90 cfs] m

i o

9396

9641

0863

0995

1168

1407

1754

1993

2297

2693

3226

3963

5039

5612

6713

7677

8553

8904

9264

9630

9848

0000

F /F m o

0. 0010

.0024

.0041

.0053

.0069

.0094

.0138

.0172

.0219

.0291

.0402

.0586

.0913

.1131

.1661

.2296

.3177

.3657

.4354

.5547

.6770

1. 0000

F m /y 7 o

0.0009

.0022

.0038

.0049

.0064

.0087

.0127

.0158

.0202

.0268

.0371

.0539

.0841

.1041

.1529

.2114

.2925

.3367

.4009

.5108

.6233

.9207

W

0.0001

.0003

.0007

.0011

.0016

.0024

.0042

.0056

.0079

.0116

.0181

.0301

.0548

.0727

.1196

.1797

.2660

.3135

.3823

.4993

.6172

.9207

r»*.

10.0000

6.9444

5.5556

5.0000

4.4444

3.8889

3.3333

3.0556

2.7778

2.5000

2. 2222

1.9444

1. 6667

1.5556

1.3889

1.2778

1.1944

1. 1667

1. 1389

1.1111

1.0970

1.0861

yo/y«i

0. 1086

.1564

.1955

.2172

.2444

.2793

.3258

.3555

.3910

.4344

.4888

.5586

.6517

.6982

.7820

.8500

.9093

.9310

.9537

.9775

.9901

1. 0000

Percent 

submergence

99.99

99.97

99.93

99.89

99.84

99.76

99.58

99.44

99.21

98.84

98.19

96.99

94.52

92.73

88.04

82.03

73.40

68.65

61.77

50.07

38.29

7.93

Table 20. Computation of stage-fall-discharge ratios, cross section 2, 120-foot 

rejch; yQ = 1.840 ft

[F0 i OJ360 ft; <?m s 40.39 cfs]

*m

3.60000

3.40000

2.90000

2.50000

2.40000

2.28000

2.20000

2.08000

2.00000

1.98000

1.91000

1.90000

1.89000

1. 88000

1. 87000

1.86000

1.85840

1.840

L

150.46

227.95

433.06

620.26

673.73

744.32

797.30

892.28

975. 80

1,001.49

1,125.23

1.151.20

1,181.45

1,217.73

1,263.49

1,326.69

1,339.83

limit

F m

0.04098

.04699

.07314

. 11087

. 12538

. 14759

. 16632

.20325

.23747

.24749

.29260

.30058

.30893

.31765

.32703

.33696

.33886

.360

°o

98.42

91.31

4.06

70.83

57.62

53.82

51.32

47.62

45.18

44.58

42.47

42.17

41.87

41.58

41.28

40.98

40.93

40.39

Q

0

m/0o

,4104

.4423

.5454

.6640

.7010

.7505

.7870

.8482

8940

9060

9510

9578

9647

.9714

.9784

.9856

.9868

1.0000

F /F m' o

0. 1138

.1305

.2032

.3080

.3483

.4100

.4620

.5646

.6596

.6875

.8128

.8349

.8581

.8824

.9084

.9360

.9413

1.0000

FnA0

0.0223

.0255

.0398

.0603

.0681

.0802

.0904

.1105

.1291

.1345

.1590

.1634

.1679

.1726

.1777

.1831

.1842

.1957

F rm/y m

0.0114

.0138

.0252

.0444

.0522

.0647

.0756

.0977

.1187

.1250

.1532

.1582

.1635

.1690

.1749

.1812

.1823

.1957

Vm/Fo

10. 0000

9.4444

8. 0556

6.9444

6.6667

6.3333

6.1111

5.7778

5.5556

5.5000

5.3056

5.2778

5.2500

5. 2222

5.1944

5. 1667

5. 1622

5.1111

yo/ym

0.5111

.5412

.6345

.7360

.7667

.8070

.8364

.8846

.9200

.9293

.9634

.9684

.9735

.9787

.9840

.9892

.9901

1.0000

Percent 

submei" 

gence

98.86

98.62

97.48

95.57

94.78

93.53

92.44

90.23

88.13

87.50

84.68

84.18

83.65

83.10

82.51

81.88

81.77

80.43

296350(
92 STAGE-FALL-DISCHARGE RELATIONS

STAGE-RATIO AND SUBMERGENCE METHODS

The stage-ratio method is described in Water-Supply Paper 888 (Corbett and 

others, 1943) as having been used with some success at gaging stations in the 

Ohio River and Tennessee River basins. In this method, the ratio of the gage 

heights (to the same datum) at each end of the reach for each discharge measurement is determined and designated as the gage ratio. At the base gage, a second 

ratio is determined by dividing the effective gage height by observed gage height. 

This is designated as the stage ratio (the no-backwater gage height, ^Q, divided 

by the observed gage height, ym ). Curves are developed of effective gage height 

plotted against discharge, and of stage ratio plotted against gage ratio. In use, 

the latter curve is entered with the gage ratio, and the stage ratio is found. With 

this and the known observed gage height, the effective gage height may be computed. Thereafter the discharge may be determined from the effective gage heightdischarge curve. In this method, it appears that the datum used will have considerable bearing on the sensitivity, at least, of the results. Obviously, if mean sealevel datum, for example, is used for all the gages, .all the ratios will be much 

nearer to unity and the sensitivity of the curves will be less than if the datum 

plane were near the channel bed.

In any event, the stage-ratio method appears to be in fact merely a less convenient version of the submergence method which has at times been used at several 

stations in Illinois. In this method, the ratio obtained by dividing the effective 

gage height by observed gage height (yo/ym ) and multiplying by 100 is designated 

as "percent effective gage height" and is plotted against "percent submergence" 

to give a curve of relation. The submergence is obtained from readings of the 

gages at each end of the reach by following formula:

Elev. of WS at lower gage - Elev. bed at upper gage 

Submergence = , 

Elev. of WS at upper gage- Elev. bed at upper gage

This formula has been applied to compute column 11, the final column, of tables 

17 to 20 and 22 to 25. Taking again as an example line 2 of table 17, it has been 

pointed out (page 88) that the water-surface elevation at the lower gage (station 

268.13) is 4.00098 ft, and at the upper gage (station 668.13) 4.00439 ft. The elevation of the bed at the upper gage is SQL = 0.003 x 668.13 ft = 2.00439 ft. Hence,

4.00098 - 2.00439 1.99659

submergence = = = 0.9983 

4.00439 - 2.00439 2.00000

or 99.83 percent, the value which has been entered in column 11 of table 17.

Figure 22 (left), page 89, is a submergence relation curve developed from the 

data for cross section 2, using a length of reach of 400 feet. It will be noted 

that the points again develop a family of curves, although for this reach the 

variation appears to be somewhat smaller than in the Boyer diagram for the
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 93

Table 21.--Fall ratios and discharge ratios, cross section 2, 400 foot reach 

[for profiles not shown in tables 17 and 18]

ym

2.80000

2.50000 

2.00000 

1.80000 

1.68000

1. 60000 

1.56000

1.54000

1.53000

1.52000

1.51000

1 <\n3ao

1.489

1.40000

1. 30000

1.24000

1.20000

1. 17000

1.15000

1. 14000

1 11991

1. 10000

1. 00000

.94000 

.90000

.87000

.85000

.84000

.83000

OOrt 1 O

D1 0

.80000 

.74000

.70000

.68000

.66000

. 65000

. 64000

.63000

.62000

£. 1 O 1 O

a i o

y0 =

F /F m' o

0.0075 

.0120 

.0150

.0191

.0253

.0353

.0431

.0540

.0704

.1580

1 ono

O 1 1 t

n (L 1 t

.3033

^789

.4046

1.0000

0.612

Q vm' /Q vo

0.1313 

.1768 

.2038

.2394

.3594

.4083

.4706

cci o

.6609

BI 9n

.8491

.9111

.9804

.9850

1.0000

y0 =

VFo

0.0183 

.0293 

.0367

.0474

.0640

.0927

.1174

.1564

i a<m

OQ£O

O O 3<C

3 £O£

.4196

.5090

1.0000

3.812

Vo

0.2035 

.2740 

.3158

.3709

.4466

f.f.f.Q

£39 £

.7291

.7997 

.8550

.9487

.9664

QfMfi

1.0000

y0 =

F /F m' o

0.0496 

.0816 

.1047

.1408

9*£ C 1

3232

C1MQ

CQOC

('('QQ

1.0000

1.121

Q ^m' /Q vo

0.3309 

.4456 

.5135

.6031

7O£O

QA^Q

DiCQO

.9055

.9632

Q7CQ

QQ^i;

1.0000

y0 =

VFo

O AOQ7

.1180 

.2104 

.2913

37£<C

.4736

£AO D

.6449

£Q D7

7 1 J Q

7775

1.0000

1.489

Q m' /Q vo

0.4274

.4968 

.6689 

.7709

D A £.1

.9053 

057^

QCA9

.9630

.9717

oan^

.9863

1.0000
94 STAGE-FALL-DISCHARGE RELATIONS

Table 22. Computation of stage-fall-discharge ratios, cross section 5, 400-foot 

reach; yQ = 0.250 ft

[F s 1.200 ft; Q = 0.34 cfs]

ym

2.50000

2. 00000

1. 80000

1. 60000

1.40000

1.20000

1. 10000

1. 00000

.90000

. 80000

. 70000

.60000

.50000

.40000

.30000

.28000

.27000

.26000

OCOCA

. £d£t>U

oen . zou

L

500.03

666.73

733.42

800. 13

866.91

933.92

967.61

1.001.57

1,035.81

1,070.32

1, 105. 26

1, 140. 90

1,177.75

1,217.41

1,267.53

1,282.72

1,293.38

1.310,67

1

OOQ AC

, OdO. 40

limit

m

0.00008

.00015

. 00021

. 00033

. 00066

.00165

.00270

.00456

. 00726

. 01077

.01555

.02243

.03291

. 05177

. 10185

. 12725

.14911

.19072

OGCA1 . ZDuUI

1.200

<?0

30.61

18.01

13.41

9.15

5.53

3.01

2.39

2.07

1.83

1.59

1.35

1.12

.89

.67

.45

.41

.38

.36

Q£ . "0

.34

Vo

0.0111

.0189

.0254

.0372

.0615

.1130

.1423

.1643

.1858

.2138

.2519

.3036

.3820

.5075

.7556

.8293

.8947

.9444

. Q714. «7 1 it

1.0000

F /F 

m o

0. 0001

.0001

.0002

.0003

.0006

.0014

.0022

.0038

.0060

.0090

.0130

.0187

.0274

.0431

.0849

.1060

.1243

.1589

. 9917 ££JL 1

1. 0000

F m /y 'o

0. 0003

.0006

.0008

.0013

.0026

.0066

.0108

.0182

.0290

.0431

.0622

.0897

.1316

.2071

.4074

.5090

.5964

.7629

1.0640

4. 8000

F m /y 7 m

0

.0001

.0001

.0002

.0005

.0014

.0025

.0046

.0081

.0135

.0222

.0374

.0658

.1294

.3395

.4545

.5523

.7335

1. 0535

4. 8000

ym/Fo

2.0833

1.6667

1.5000

1. 3333

1. 1667

1.0000

.9167

.8333

.7500

.6667

.5833

.5000

.4167

.3333

.2500

.2333

.2250

.2167

, 91(\A fiJ-V**

OAQQ AUOw

Vy»

0. 1000

.1250

.1389

.1562

.1786

.2083

.2273

.2500

.2778

.3125

.3571

.4167

.5000

.6250

.8333

.8929

.9259

.9615

QQA1 » yyuj.

1. 0000

Percent 

submergence

100. 00

99.99

99.99

99.98

99.95

99.86

99.75

99.54

99.19

98.65

97.78

96.26

93.42

87.06

66.05

54.55

44.77

26.65

Table 23. Computation of stage-fall-discharge ratios, cross section 5, 400-foot 

reach; yQ <» 2.525 ft

[FQ r 1.200 ft; Qm r 31.28 cfs]

ym

3. 00000

2.88000

2. 80000

2. 72000

2. 65000

2.61000

2.59000

2. 57000

2. 56000

2. 55025

2.525

L

441.59

514.79

571. 86

640. 94

721.06

784.78

827. 05

882. 75

919.69

966. 25

limit

m

0.30614

.36964

.42859

.51011

.61543

. 70469

.76315

. 83957

.88714

. 94268

1.200

Co

44.61

41.14

38.86

36.62

34.68

33.58

33.04

32.50

32.23

31.97

31.28

Q /<? m o

0.7012

.7603

.8049

.8542

.9020

.9315

.9467

.9625

.9705

.9784

1. 0000

F /F 

m o

0. 2551

.3080

.3572

.4251

.5129

.5872

.6360

.6996

.7393

.7856

1. 0000

F /y 

m o

0. 1212

.1464

.1697

.2020

.2437

.2791

.3022

.3325

.3513

.3733

.4752

Fm^m

0. 1020

.1283

.1531

.1875

.2322

.2700

.2947

.3267

.3465

.3696

.4752

ym/F0

2.5000

2.4000

2.3333

2.2667

2. 2083

2. 1750

2. 1583

2. 1417

2. 1333

2. 1252

2.1042

Vym

0.8417

.8767

.9018

.9283

.9528

.9674

.9749

.9825

.9863

.9901

1.0000

Percent 

submergence

89.80

87.17

84.69

81.25

76.78

73.00

70.53

67.33

65.35

63.04

52.48
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS

Table 24. Computation of stage-fall-discharge ratios, cross section 5, 120-foot

reach; yQ = 0.250 ft

[F0 : 0.360 ft; Q_ = 0.34 cfs]

95

Ym

2.80000

2. 50000

2.00000

1.80000

1.60000

1.40000

1.20000

1, 10000

1.00000

.90000

. 80000

. 70000

.60000

.50000

.45000

.40000

.35000

.30000

.28000

.27000

.26000

.25250

.250

L

400.02

500.03

666. 73

733.42

800. 13

866.91

933.92

967.61

1,001.57

1,035.81

1,070.32

1,105.26

1,140.90

1,177.75

1,197. 05

1,217.41

1,239.81

1,267.53

1,282.72

1,293.38

1,310.67

1,338.46

limit

Fm

0. 00002

. 00004

.00007

. 00010

.00019

.00044

.00131

. 00222

.00381

.00605

. 00875

.01211

.01646

. 02358

. 02926

.03753

. 05082

.07824

.09969

. 11838

. 1537.0

.21496

OCA

 ooU

°o

38.86

30.61

18.01

13.41

9.15

5.53

3.01

2.39

2.07

1.83

1.59

1.35

1.12

.89

.78

.67

.56

.45

.41

.38

.36

.35

.34

Q /Q m o

0.0087

.0111

.0189

.0254

.0372

.0615

.1130

.1423

.1643

.1858

.2138

.2519

.3036

.3820

.4359

.5075

.6071

.7556

.8293

.8947

.9444

.9714

1.0000

F /F 

m o

0.0001

.0001

.0002

.0003

.0005

.0012

.0036

.0062

.0106

.0168

.0243

.0336

.0457

.0655

.0813

.1042

.1412

.2173

.2769

.3288

.4269

.5971

1. 0000

F /y 

m o

0.0001

.0002

.0003

.0004

.0008

.0018

.0052

.0089

.0152

.0242

.0350

.0484

.0658

.0943

.1170

.1501

.2033

.3130

.3988

.4735

.6148

.8598

1 J-. 4400 *r*vv

F m /y'm

0

0

0

.0001

.0001

.0003

.0011

.0020

.0038

.0067

.0109

.0173

.0274

.0472

.0650

.0938

.1452

.2608

.3560

.4384

.5912

.8513

1 J-. 4400 *r*w

Ym/Fo

7.7778

6.9444

5.5556

5.0000

4.4444

3.8889

3.3333

3.0556

2.7778

2.5000

2. 2222

1.9444

1. 6667

1. 3889

1.2500

1.1111

.9722

.8333

.7778

.7500

.7222

.7014

6944

y0/Ym

0.0893

.1000

.1250

.1389

.1562

.1786

.2083

.2273

.2500

.2778

.3125

.3571

.4167

.5000

.5556

.6250

.7143

.8333

.8929

.9259

.9615

.9901

1.0000

Percent 

submergence

100. 00

100.00

100.00

99.99

99.99

99.97

99.89

99.80

99.62

99.33

98.91

98.27

97.26

95.28

93.50

90.62

85.48

73.92

64.40

56.16

40.88

14.87

Table 25. Computation of stage-fall-discharge ratios, cross section 5, 120-foot 

reach; yQ = 2.525 ft

[F0 r 0.360 ft; Qm =31.28 cfs]

ym

3.60000

3,30000

3.00000

2,88000

2. 80000

2,72000

2,65000

2.61000

3.60000

2.59000

2.58000

2.57000

2.56000

2.55025

2.525

L

159. 63

290. 19

441.59

514.79

571.86

640.94

721. 06

784. 78

804. 69

827.05

852. 60

882. 75

919.69

966.25

limit

Fm

0.06114

. 08469

. 12655

. 15357

. 17772

.20896

. 24600

.27325

. 28109

. 28942

. 29815

. 30752

.31749

. 32776

.360

<?o

62.80

53.57

44.61

41.14

38.86

36.62

34.68

33.58

33.31

33.04

32.77

32.50

32.23

31.97

31.28

Qm/Qo

0.4981

.5839

.7012

.7603

.8049

.8542

.9020

.9315

.9391

.9467

.9545

.9625

.9705

.9784

1.0000

Fm/F0

0. 1698

.2352

.3515

.4266

.4937

.5804

.6833

.7590

.7808

.8039

.8282

.8542

.8819

.9104

1.0000

Fm/Yo

0.0242

.0335

.0501

.0608

.0704

.0828

.0974

.1082

.1113

.1146

.1181

.1218

.1257

.1298

.1426

Fm/ym

0.0170

.0257

.0422

.0533

.0635

.0768

.0928

.1047

.1081

.1117

.1156

.1197

.1240

.1285

.1426

Ym/F0

10. 0000

9. 1667

8. 3333

8.0000

7.7778

7.5556

7.3611

7.2500

7.2222

7.1944

7. 1667

7. 1389

7.1111

7.0840

7.0139

^o/Ym

0.7014

.7652

.8417

.8767

.9018

.9283

.9528

.9674

.9712

.9749

.9787

.9825

.9863

.9901

1. 0000

Percent 

submergence

98.30

97.43

95.78

94.67

93.65

92.32

90.72

89.53

89.19

88.83

88.44

88.03

87.60

87.15

85.74
96 STAGE-FALL-DISCHARGE RELATIONS

Table 26. Fall ratios and discharge ratios, cross section 5, 400-foot reach 

[for profiles not shown in tables 22 and 23]

y

2.50000 

2.00000 

1. 80000 

1. 60000 

1. 40000

1.30000

1.20000 

1. 10000 

1.05000

1.00000 

. 96000

. 94000

.93000

QOAftfi

.91405

Qfi^

. 90000

. 80000

.75000

. 73000

.71000

. 70000

.69387

XQW

. 60000

. 50000

. 48000

.47000

.46056

.456

.44000

. 40000

OQAAA

.37000

.36000

.35350

.350

yo =

Fn/Fo

0.0001 

.0003 

.0004 

.0007 

.0015

.0038 

.0063

.0107

0917

.0377

.0559

.0865

.1207

.1630

.1999 O^AQ

OQQl

.3689

1.0000

0.350

V^O

0.0183 

.0311 

.0418 

.0612 

.1013

.1860 

.2343

.2705

.3060

.4148

.5000

.6292

7Q£Q

QQCQ

QQQO

QOOO

.9655

QQOC

1. 0000

yo =

Fn/Fo

0. 0002 

.0005 

.0008 

.0014 

.0029

.0077 

.0128

.0223

.0364 

.0557

.0846

.1336

OCin

.3677

.4820 

1. 0000

0.456

VQ0

0.0258 

.0439 

.0589 

.0863 

.1429

.2625 

.3305

.3816

A O 1 T

.4969

coco

.7054

OO*7 4C

.9294

.9634

.9875

1. 0000

yo =

Fn/Fo

0.0006 

.0014 

.0022 

.0040 

.0082

.0229 

.0405

.0767

1HQ?

.3486

.6199

7/\<ro

1.0000

0.687

V^O

0.0431 

.0733 

.0984 

.1443 

.2387

.4385 

.5523

.6377

QOAO

QAQA

.9635

O77Q

Qfl^l

1. 0000

yo = (

Fn/Fo

0.0013 

.0028 

.0042 

.0074 

.0160

AO£Q

.0493 

.0994

.2558

.5071

5071;

.7005

7QC1

1. 0000

).905

V^o

0.0601 

.1022 

.1372 

.2011 

.3327

4532

.6113 

.7699

OOJ? A

.8889

.9684

.9787

OQQO

1

f\f\e\f\
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 97 

Table 26. Fall ratios and discharge ratios, cross section 5, 400-foot reach Continued

y'm

2.80000

2.50000 

2.20000

2.00000 

1.90000

1.84000

1.80000 

1.76000

1.75000

1.74000

1.73000

1.72104

1.704

1.60000

I

conftft

1. 44000

1.40000

1.36000

1. 34000

i 39f>on

1.31000

1.30000

1 90701:

1

OQ r

1.26000

1.22000

1.20000

1..3000

1

0AAA

1. 17000

1.16958

1 15ft

yo =

F /F 

nr o

0.0027

.0060

.0093

.0164

.0366

.0571

AQQ£

1 £.4 <>

.1896

9RO3

i nnnn

1.158

V^o

0.0885

.1505

.2021

.2962

.4901

.6259

QCOO

.9003

.9249 

.9476

Q713

i nnnn

yo =

Wo

0.0057

.0125

.0194

.0346

.0469

.0684

f\Q£f)

.1149

.1721

.1991

.2410

OCQ9

1.0000

1.285

VQo

0. 1264

.2149

.2886

A OO A

xnrO

£OOQ

R^Q^

.9236

Q^^O

.9603

1.0000

?o =

nr o

O AOQO

.0492 

.0816

.1259 

.1674

OAQ1

.2489 

.3146

3390

.3699

.4105

.4710

1.0000

1.704

VQo

O OOAQ

.3692 

.4941

.6274 

.7211

7RO7

.8427 

.9018

O1 QA

9347

qcoo

A£QO

1. 0000
98 STAGE-FALL-DISCHARGE RELATIONS

same conditions (fig. 20, left). No effort has been made to correct this diagram 

for change in velocity head.

Any hope that this method of correlation might be improved by shortening the 

reach is shattered by the plotting of figure 22 (right), which is a similar computation for a reach of 120 feet. It will be noted that as the length of reach becomes shorter the variation in the curves of relation becomes greater.

Results of this technique as applied to data for cross section 5 are presented 

as figure 23 (400-ft and 120-ft reaches), on page 99 . It will be noted that the 

family of curves for this cross section has a much greater spread than for cross 

section 2.

MISCELLANEOUS METHODS 

POSSIBLE COMBINATIONS OF DIMENSIONLESS RATIOS

To the end that no meritorious method might go unexamined, a survey has 

been made of the possible uses of other combinations of factors for the curve 

of relation. The elements which may be used in various combinations are as 

follows:

Om . This is the actual discharge: the discharge which is measured, or 

would be obtained by a discharge measurement.

OQ. This is the uniform-flow discharge: the discharge which, for a given 

gage reading, would exist if no backwater were present.

ym . This is the actual depth.

y0. This is the so-called uniform depth: the depth which, for a given 

discharge, would exist if no backwater were present.

Fm . This is the actual fall in the reach, as indicated by the difference 

in reading of two gages set to same datum.

FQ. This is the so-called uniform fall: the fall which under conditions 

of no backwater will exist for a given gage reading or a given discharge, 

or both. In the laboratory channels it is independent of such variation and 

is equal to 0.003 times the length of the reach.

. The velocity head for discharge Qm flowing at depth y0. 

cm Vm 2/2g. The velocity head for discharge Om flowing at depth ym .

F' . The velocity head increment, or velocity head at the upper gage 

minus the velocity head at the lower gage. It is a correction to be added 

to the fall.
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 99

11461314 3AJJ.03JJ3 4.U30J9CJ

i 8

0> _«

E -S
100 STAGE-FALL-DISCHARGE RELATIONS
STAGE-FAIL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 101

Two other elements (Q , the "rating" discharge and F , the "rating" fall), both 

of which imply the presence of appreciable backwater, will be added later, in 

connection with fixed-backwater ratings. Expressions might also be written for 

specific head and its variation within the reach. They might also be written for 

the energy line and its variation within the reach but this, however, is pointless at this time. In fact, there is no merit, here, in writing expressions for 

velocity head and its variation in the reach except to take cognizance of the 

fact that they exist. In a search for a practical method of correcting for backwater effects, any method which involves use of the velocity head is to be 

avoided. If the velocity head is included, solutions may be made only by trial 

that is, a discharge must be assumed in order that the velocity head may be 

evaluated, and the velocity head must be evaluated before the discharge may be 

computed. Kence, for practical purposes, consideration may be limited now to 

the first six of the above elements.

For practical use, it appears that these elements should be combined in the 

form of dimensionless ratios. Since only the first two have the dimension of 

L 3 !"" 1 , these must be used together, while the remaining four (for each of which 

the dimension is L) may be used in any combination. Thus the following combinations become possible:

P_m La La Is* LO 10. F-m.

The reciprocals of these ratios may, of course, be used. Such use, however, 

will not result in the development of any new relations, but merely in a change 

in shape of the curves. In this study the above expressions have been inverted, 

if need be, to obtain the range of values which tends to lie below, rather than 

above, unity. The following arrangements have been found to be most usable:

It will be noted that yQ/F 0 has now been deleted. For any discharge this value 

is a constant, regardless of the magnitude of other ratios. Hence it is not suitable for use in these studies.

After values are computed for the six ratios listed above, curves of relation 

may be obtained by plotting any one of the ratios as ordinate against any other 

as abscissa. This would provide 15 methods of plotting. Practical considerations, however, lead to the immediate elimination of nearly half the number. In 

order that the plot may have practioal use, it is necessary that in one ratio both 

numerator and denominator be known and that in the second ratio one of the 

values be known and the other be either Qm or y0 . The values which are known 

(assuming that a curve or relation between uniform gage height and uniform
1°2 STAGE-FALL-DISCHARGE RELATIONS

discharge has been developed) are QQ , ym , Fm and F0 . Of the above ratios, the 

only ones containing a known value and either Qm or y0 are

the only ones made up of two known values are

F m F m y' m

Thus the number of combinations that can be considered in light of the above 

analysis is limited to nine, as follows

. (J) against 

(2) against 

, Fm . Fm

(3) against 

?0 F 0

0 . F 

(4) against 

n, (5) against 

(6) against '

OT » ym (t) ~ against 

m (8) against 

F y

. . m . 'n (9) against 
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 103

Data for the plotting of these relations, for the profiles of highest and lowest 

discharge in both cross section 2 and cross section 5, are contained in tables 

17 to 20 and 22 to 25, pages 90 to 91 and 94 to 95, derivation of which has 

been previously explained.

RESULTS OF VARIOUS RATIO COMBINATIONS

The first of the above combinations already has been discussed at length. 

This is the plot presented in figure 20 as the Boyer diagram.

A plot of the second combination, y$/ym against Fm /F0 , has been made, although the results are not shown herein. It may be demonstrated that if there is 

a simple exponential relation between y0 and Co there will also be a corresponding relation between (Cm /Co^ an(^ ^o/^m^ ^ n agreement with this principle, 

the plots of yQ/y m against F m /F0 were found to be much like the Boyer diagrams. There was a somewhat wider variation between the curves, and a greater 

departure from the square-root-of-the-fall relation. There appeared to be no advantage over the Boyer diagram.

A study of combination (3) above, Fm /y0 against Fm/F0, reveals a limitation 

in the practical use of the relations which was not covered in the preceding 

discussion. This combination is lacking in one of the essential requirements, 

in that neither ym nor Co appears in either of the ratios. Only the fall, and not 

the depth, is taken into account. The relations are insufficient to give a determinate result.

The fourth relation, Om /@o apainst Fm ^ym ' develops also into a family of 

curves. It offers so little promise that the plot has been omitted from this report. 

The curves are practically superimposed for low values of Om /(?o> but ^or high 

values of this ratio the curves disperse rapidly, as the ratios of Fm/ym approach 

their limits of F0/y0.

Somewhat the same criticism is applicable to the fifth plot, of yQ/yn against 

Fm /ym . In the limit, as Fm/ym approaches F0/y0, a range of dispersion occurs 

which is dependent only on the range of y 0. However, for a 400-foot reach of 

cross section 2, the maximum error found in determining discharge by use of a 

single mean curve to represent this relation was near 2 percent. Figure 24 (left), 

page 100 , presents for cross section 2 the enveloping positions of the true 

curves (of highest and lowest discharges used in these computations). Figure 25 

(left), page 104, shows similar curves for cross section 5. Unfortunately, the 

comparatively accurate results which might be obtained in the first instance 

appear not to hold under other circumstances. This is further evident from figures 

24 (right) and 25 (tight), prepared for reaches of 120 feet. It will be noted that 

the range of variation becomes considerably greater
104 STAGE-FALL-DISCHARGE RELATIONS

I
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 105

\

\

« ^ .6 u? fc,
106 STAGE-FALL-DISCHARGE RELATIONS

a? t^ ic

\S

to 04

£ 

|

I
STAGE-FAIL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 107

296350 O 54 8
108 STAGE-FALL-DISCHARGE RELATIONS

The sixth relation, Fm /y0 against Fm /ym , appears at first glance to offer a 

very acceptable plot. It has been presented for the 400-foot reach in cross section 2 as figure 26 (left), page 105 . Closer inspection discloses that it also 

develops family characteristics. It also will be apparent that, as a condition of 

no backwater is approached, both of the ratios approach the limit of F0/y0. 

Each member of the family thus terminates on a straight line through the origin 

at 43° slope. Even for extreme backwater conditions, the variation from such 

line is small. Since this is true, the lines must be drawn to a high degree of 

accuracy to attain satisfactory results in determination of discharge. Furthermore, as the length of the reach decreases, the range of variation becomes considerably greater. Figure 26 (right) shows the enveloping positions for the 120- 

foot reach in cross section 2. Corresponding curves for cross section 5 are 

shown as figure 27, page 106.

The three remaining relations (?m /(?o against ym^Qi ^o^m against ym/F0, 

and Fm /y0 against ym /F0 offer so little promise that no plots have been 

shown. In each case one of the factors is ym/F0. Since F0 is a constant for 

these conditions of uniform prismatic channel, the results are essentially the 

same as obtained by plotting directly against the observed depth.

It will be noted that, throughout these discussions, for any given length of 

reach F0 is constant. This is in keeping with the fundamental concept of the 

uniform-flow rating for the prismatic channels. In the event that a fixed-backwater 

rating (see p. 78 ) is considered, a variable Ff will replace the constant F0; and 

the relations discussed above should be reconsidered.

PROPOSED METHODS

RECAPITULATION

A comprehensive investigation of the possible combinations of stage, fall, 

and discharge in a prismatic channel fails to reveal any two-dimensional solution, based on the uniform-flow rating, which is universally acceptable. Any of 

the three-dimensional curves of relation which have been presented as figures 

IB to 27 will give accurate results to the extent to which the curves are accurately drawn and accurately read. It will be noted, however, that all except 

figures 18 and 19 require trial solutions. That is, when entering the diagram 

with the known ratio, it is necessary to assume a discharge before a value of 

the second ratio may be obtained. If the discharge as computed from this second ratio does not agree with that which was assumed, a second assumption 

must be made. Such solutions, at best, are a tedious procedure.
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 109

This difficulty obviously would not arise if, for the family of curves presented 

for each of the methods, a single curve could be substituted with acceptable 

accuracy. In this respect it should be noted that, for the Boyer method, as the 

length of reach is decreased, the family of curves more nearly approaches a 

single curve and also approaches the curve expressing the square-root-of-thefall relation. For the other three methods illustrated in figures 22 to 27 as the 

reach is lengthened each family more nearly approaches a single curve. Thus 

for those instances in which a single curve is used to represent the family 

group, it appears that accuracy should improve with the 3oyer method as the 

reach is made shorter. It appears possible that accuracy should improve with the 

other methods as the reach is made longer. This latter appearance, however, 

may be an illusion, since the convergence of the family may merely represent a 

loss of sensitivity. In any event, the fact remains that the use of a single curve 

to represent the family is only an approximation of the true relationship.

THREE-DIMENSIONAL BOYER DIAGRAM

The true relationships may be shown, and the difficulties of trial solutions 

still may be avoided, by a diagram such as fissures 28 to 34, nasre 107 and pa^e 0 

110 to I In. The values of the fall ratio as abscissa have been plotted against 

values of the discharge ratio as ordinate, but the third variable has been taken 

as observed stage. No assumptions are required. The diagram may be entered 

with a known fall ratio and known observed stage, and the value of Om /(?o 

directly determined.

Let it be required, for example, to determine the discharge for depth of 2.00 

feet in cross section 2, when the fall in 400 feet is 0.105 feet. The fall ratio, 

Fm /F0 , is 0.105/1.200, or 0.0875. Entering figure 28 with this value, and proceeding to the line ym = 2.00, we find Om /C?o equal to 0.460. From figure 6, 

page 19, in Bulletin 381 (Lansford and Mitchell, 1919) or from table 42, page 81 

in that bulletin, QQ is equal to 45.18 cfs. Hence Qm must equal 45.18 cfs x 

0.460, or 20.8 cfs.

Although the diagram may be prepared on any type of cross-section sheet, the 

use of logarithmic coordinates is strongly recommended. Two advantages will 

be obvious, thus: (1) In those ranges for which backwater effects are high and 

the values of the ratios become very small, there is sufficient expansion to permit accurate use. (2) Throughout most of their range, the lines of equal depth 

will be systematically spaced, nearly parallel to one another, and nearly parallel to aline representing the relation Om /0n = (Fm /F0) 0t5. AH curves must, of 

course, converge to pass through the point 1, 1. Thus when meager data are 

available, the preparation of the diagram will be expedited by keeping the 

curves in this form.
110 STAGE-FALL-DISCHARGE RELATIONS

CO

g

1

8

S

£

fr

M
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 111

m\

\

I

n
112 STAGE-FALL-DISCHARGE RELATIONS

I

pa 

13
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 113

\
114 STAGE-FALL-DISCHARGE RELATIONS

fr

«
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 115

8

fr

pa
116 STAGE-FALL-DISCHARGE RELATIONS

0

13

0.10 

0

1.0

0

1

a

0.10 

0

0.8

^01

^

^

1.2

IJC

1.1

0.5

0.

1.6

^

9

1.4

^

0

2

 ^

£

1.

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2 

1

1.0 IX

8

S^fe'

^^~

ut

0.9

\

2.0.^

^

^

1 

^

-^

1.6-

^

^

^

^

EXPLANATION 

Figures show observed depth, ym 

Dot serves both as decimal and

1 1 II

0.10

^

1.9

I.I

/

/- ^

1 ?

0.6 0.9

.0

^^~t&

l.8o7B

^

^

^,

^

^

^

S"

EXPLANATION 

Figures show observed depth, ym 

Dot serves both as decimal and 

as point location

1

^ ^

\.

^^

01 '0.10 1.

Figure 35. Three-dimensional Boyer plot, developed from smoothed profiles, cross sections

2 (above) and 5 (below).
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 117

It will be noted that for channels such as those represented by cross sections 

2, 3 and 4 the systematic spacing of the lines of equal depth is auite simple. Unfortunately, for channels such as those represented by cross sections 5 and 6, the 

spacing, although systematic, is subject to reversals in trend, so that certain sections of the diagrams are superimposed. (See figs. 31-34.) Beginning with the 

greatest value of ym (2.5), it will be noted that, for any riven fall ratio, the values of discharge ratio increase with decreasing values of ym uo\vn to observed 

depth of 1.2 feet the point below which, at the upper end of the reach, water no 

longer covers the entire width of the flood plain. As ym is decreased below this 

value, the values of discharge ratio for any given fall ratio progressively decrease 

to the point ym = 0.7. Here there occurs a second reversal in trend, so that as ym 

is further decreased, values of the discharge ratio again increase in a manner 

similar to that for the upper portion of the channel.

A completely generalized statement as to the circumstances under which discharge ratio decreases with decreasing values of ym is not warranted by available 

data. There appears to be a relation between this phenomenon and the fact that 

the water covers the flood plain throughout only a portion of the reach.

It will be recalled that figures 18 to 34 have been plotted from computations 

such as those presented in tables 17 to 26. These computations, in turn, were 

based on tables 12 to 16, which are tables of computed rather than observed profiles. An attempt was made to use the smoothed profiles based on the observed 

data, but the data available were found to be too meager to allow the complete 

development of the families of curves. Illustrations of the results to be obtained 

by use of the smoothed observations (tables 1 to 5) are to be found in figure 35, 

page 116. Although families of curves could not be accurately drawn from these 

data, the plotted points verify the fact that the same relations exist here as in 

figures 18 to 34. :

The third variable, observed depth, might also be used with other combinations 

of depth and fall ratios such as heretofore discussed. Its use with Qm /Qo and 

Fm /FQ, however, has the following definite advantages: (1) The terminal points 

of every curve are known, since each curve must start from the origin and end at 

the point 1,1. (2) The spacing and shape of the curves, as discussed above, are 

systematic. (3) In the event the method is used in those cases in which profile 

curvature and velocity-head increment are truly negligible, the, plot should tend to 

resolve into a single curve. And conversely, (4) the development for any channel 

oT""a_pIptsuch as shown in tn^££gggjing figures will

that the effects of profilg_curvature and_^j^cjjty.^]ie^djjicj^ejnent, ffloneor in combination, are indeed factors in the stage-iall-discharge relation.
118 STAGE-FALL-DISCHARGE RELATIONS

a 3 3 °

V 39V9 IV '038 13NNVH3 3AOBV '133J Nl 'NOI1WA3T3

Q 

<tlOlottl«ft| 

<q ni eq « o <4 ni

0 ONV B S30W9 IV '038 13NNVH3 3AOBW 'l33d Nl 'NOUVA313
STAGE-FAIL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 119

FIXED-BACKWATER RATINGS

As explained at the beginning of this section (see page 78), there are two 

types of channel conditions for which there is a fixed and invariable relation 

between stage and discharge and for which fall is an unnecessary element in 

the rating: the first, uniform flow; the second, a condition of nonuniform flow 

induced by some fixed nonuniformity of the channel.

 ^

The second condition is illustrated by figure 17, page 78, in which it is necessary for all flow to pass over the crest of a fixed spillway which rises an appreciable distance above the channel bed. In varying degree this second condition 

commonly is encountered in the operation of gaging stations. Occasionally the 

station may be upstream from and in the backwater of a high dam which, even for 

highest discharges, is not submerged. More often it may be upstream from a dam 

of lesser height, which, as the discharge increases, will become at least partially submerged. Still more often the station may be upstream from a natural 

outcropping of ledge rock, or a group of boulders or other obstructing material, 

which, at low discharges, is said to "control" the stage-discharge relation. To 

the extent that any of these channel obstructions fulfill the function of a controlthat is, determine the relation between stage and discharge the steady-flow 

stage-discharge relation will be a simple two-dimensional curve, unencumbered 

by association with fall. When, with increasing discharge, these channel obstructions become submerged, and lose their effectiveness as a control, fall may or 

may not become an essential element of the steady-flow rating. The circumstances under which fall will become an essential element will be discussed subsequently. But first it is desirable to consider the simple case that in which a dam 

is placed in an otherwise uniform channel and built to such height that it does 

not become submerged, even by the maximum discharge.

Let it be supposed that figure 17 (left) (page-9*0 represents a longitudinal 

section of the channel for cross section 2 and that, at any point in the channel, 

there be erected a dam of large spillway capacity whose crest is 1.2 feet above 

the channel bed. (Downstream from the dam, the channel will be assumed to have 

the same cross section as upstream from the dam.) Let a gage be installed just 

downstream from the dam, another just upstream from the dam (taking care to 

avoid drawdown effects), and a third 400 feet upstream from the second. Let the 

gage farthest upstream be designated as A, the second one as B, and the one 

immediately downstream from the dam as C. For simplicity, let it be assumed 

that the longitudinal distance between 5 and C is negligible, so that the bed 

elevations at B and C may be considered to be the same. Finally, let it be assumed that, by some appropriate means such as current-meter measurements, a 

rating is developed for gage B, and is as shown by curve B of figure 36, page 

118. (The fact that curve B has additional significance, as discussed on pages 130 

and 134,does not preclude its use in the following explanation of basic concepts.)
120 STAGE-FALL-DISCHARGE RELATIONS

Obviously, for any given rate of steady flow, an M-l surface profile, or backwater curve, will now exist in the reach A-B, and for any of the rates of discharge treated in this report, the curve will be some segment of the profiles presented in table 12, pages 59 to 62. The particular segment can be determined by 

a simple computation, and hinges upon the fact that, for any discharge, the value 

of the depth y is known at gage B. For example, for the discharge of 40.39 cfs, 

the depth at B, as taken from figure 36, is 2563 feet. Interpolating in table 12, 

the corresponding value of L is found to be 588.70. Thus the segment of profile 

which exists in the reach A-B is that shown between stations 588.70 and 988.70 

of figure 15, page56. The depth at gage A is computed by working with the nextto-last column and the first column of table 12 and interpolating a value of y 

equal to 1.990 for L = 988.70. A point (Q = 40.39, y = 1.990) may now be plotted 

on figure 36, the ordinate being with respect to scale of elevation above channel 

bed at upstream gage. This is a point on the two-dimensional steady-flow stagedischarge rating curve for gage A, under the fixed-backwater condition described 

above.

From the other backwater curves given in table 12, five additional points for 

defining the fixed-backwater rating at gage A may be determined. Another valuable point is obtained by noting that when the discharge becomes zero the water 

surface will be a horizontal line upstream from the crest of the dam and will 

intersect the channel bed at elevation zero on the upstream gage. From these 7 

points, plotted logarithmically, curve A of figure 36 is drawn to represent the 

steady-flow stage-discharge relation for gage A under the fixed-backwater condition created by the dam previously described.

The dashed line in figure 36 is the uniform flow rating at gage A. If the imaginary dam be lowered, curve B would approach curve C as a limit and curve A 

would approach the Q0 curve at gage A and the F curve (see below) would approach 

F0 = 1.20 as a limit. Thus any curves to the left of curve C will give a curve lying 

to the left of the Co curve at gage A.

Let it be repeated that there are two types of channel conditions for which fall 

is superfluous to the steady-flow rating. In the first of these, uniform flow, fall 

is unnecessary because it is a constant; it always is the same as the fall in the 

channel bed. In the second type, nonuniform flow induced by some fixed nonuniformity of the channel, the fall may vary with discharge, but for a given discharge the fall always is the same, since the nonuniformity of the channel always 

is the same. As the discharge here is a unique function of stage, it follows that, 

for any given stage, the fall always is the same. This leads to consideration of 

the stage-fall curve which, although not needed under conditions described above, 

can now be easily determined. And it is well to become familiar with it now, for 

it becomes a very important element in ratings yet to be described.
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 121
122 STAGE-FALL-DISCHARGE RELATIONS

The fall for any stage at gage A may be obtained by entering figure 36 with 

the desired stage, moving horizontally to curve A, then noting the vertical intercept between curves A and D. A curve based on values so obtained is shown as 

curve F at the left of figure 36, where fall in the reach A-B is plotted as abscissa against the same ordinate as used for the rating at gage A.

Let it be remembered that the present discussion has dealt thus far with only 

one type of channel condition: that in which a dam is placed in an otherwise 

uniform channel and built to such height that it does not become submerged, even 

by the maximum discharge. That this assumption is fulfilled is evidenced by the 

fact that, in figure 36, 'Curve C, the steady-uniform-flow rating for gage C, does 

not intersect curve B.

For next consideration, let the above conditions be supplemented by one additional assumption: although the dam never is submerged as the result of high 

discharge, it may become submerged as a result of some phenomenon farther 

downstream, such as backwater from a downstream confluence. As such submergence becomes appreciable, the rating for gage D begins to change, leading to 

the displacement, in the upstream direction, of the surface profile and thus to an 

increase in the stage at A, and a decrease in fall in the reach A-B. The steadyflow stage-discharge relation at A is no longer a simple two-dimensional relation 

but must be modified by consideration of the fall. For any given stage at A, and 

dependent upon the extent of submergence, the discharge and the fall in the 

reach A-B may be equal to or less than shown by figure 36. Thus it becomes 

convenient to allow the curves for gage A to become the base rating, or base for 

comparison of other conditions of fall and discharge. In other words, values from 

these curves may be used as the denominators in computing discharge ratios 

and fall ratios similar to those described on pages 84 to 118.

Values from this base rating cannot now be designated Q0 and F0, as they no 

longer are for conditions of uniform flow. They do represent conditions which may 

normally occur, and so might be designated Qn and Fa a practice which was 

followed by Corbett, in U. S. Geological Survey Water Supply Paper 808. It has 

been suggested by Eisenlohr (personal communication, 1946) that a more appropriate subscript is r, thus indicating these are values from the base rating! whatever it may be. This suggestion is in keeping with concepts to be developed in 

succeeding pages and therefore is adopted here. Computations for Qm /Qt and 

Fm /Fr may be made in the same manner, and based on the same principles, as 

those for Om /0o and Fm /F0. (See pp. 79 to 81.)

Under the last assumption that the rating for the dam may be affected by submergence as a result of some downstream phenomenon only a portion of each 

profile may be used for computing discharge ratios and fall ratios. For a given 

discharge no portion of a profile may be used which provides, at the upstream 

end, a depth less than the corresponding ordinate of curve A, figure 36. Thus the
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 123

3 ONV 8 S39V9 1W '039 13NNVH3 3A08V '133J Nl 'NOI1VA3T3
124 STAGE-FALL-DISCHARGE RELATIONS

maximum value of fall is Fr, and the maximum value of discharge is Qt; the maximum possible values of Om /Qt and Fm/Fr are unity. This limitation arises from 

the fact that all discharge must pass over the crest of the dam. If, for a given 

discharge, the depth of A were to become less than shown by figure 36, then the 

depth at B also would be less than shown by curve B (the profile having been 

displaced in the downstream direction), and there would be insufficient head to 

pass the given discharge over the dam. But let another assumption be made regarding the dam: that it be provided with sluice gages so that any discharge, 

however great, may be passed through the dam with a stage at B not appreciably 

greater than the stage, for the corresponding discharge, at gage C. Now with any 

given discharge, as the sluice gates are opened the stage at B will begintofall, 

and the backwater profile will be displaced, rigidly, in the downstream direction, 

leading to a stage at A which is less than shown by curve A. For these conditions both (?m /0r and Fm /Ff will become greater than unity. Under these circumstances the remaining portions of the profiles (those portions for which at 

the upstream end of the reach the depth is less than shown by curve A) may be 

used to compute values of the discharge ratio and the fall ratio. However, the 

ratio Qm/Qr cannot exceed the ratio Qo/Qt for as described above with curve B 

lowered to curve C, curve A can do no more than to lower to the Q0 position at 

gage A.

The ratios which may evolve from these last two assumptions (1) the drowning of the dam by some downstream phenomenon, so that backwater effects in 

the reach A-B are greater than usual, and (2) the opening of sluice gates, so 

that backwater effects in the reach A-B are less than usual have been computed for cross section 2, and plotted as the Boyer diagram, figure 37, page 121. 

Here, at last, is a diagram for which all the data lie within very reasonable 

limits of a single curve. It will be noted that the points define a curve which 

passes through the point 1, 1. Values less than unity represent conditions described under assumption 1 above the existence in reach A-B of backwater effects 

greater than those envisioned in the preparation of the curves of figure 36 

whereas values greater than unity represent conditions described under assumption 2 above, or the existence in reach A-B of backwater effects which are less 

than those accounted for in the curves of figure 36.

Attention now is invited to a similar treatment for cross section 4. Let a dam, 

identical to that which was used for cross section 2, be placed in the channel, 

and let gages A, B, and C be located as before. (See page 119.) The curves 

which result from these assumptions are presented as figure 38, page 123. Curve 

C is first drawn. It is the steady-uniform-flow stage-discharge relation curve. It 

is applicable at gage C because channel characteristics below C are the same 

as in the reach A-B. Curve B is next drawn. As this is the rating for the dam, 

and the dam is identical with that used in cross section 2, curve B should be 

the same as curve B of figure 36. But a new factor has been injected! Curve B,
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 125

I

N «> _ 0> O) 2

<f ""-si 8882

£ 1 "" !

g

Irt 53I
126 STAGE-FALL-DISCHARGE RELATIONS

drawn as before, is found to cross curve C at a stage of about 3 feet. The explanation, which is obvious, is the same as that found at many stream-gaging stations: because of the increased roughness of cross section 4 over cross section 

2, the channel capacity is much less, and above a stage of about 3 feet on gages 

Z? and C the discharge capacity of the channel is less than that of the dam. In 

other words, the dam is now subject to submergence at high discharge, even 

though only uniform channel conditions exist downstream from the dam; the dam 

becomes drowned by fixed channel control. The obvious procedure, well known 

to experienced rating-curve analysts, is to draw the transition curve D' between 

curves B and C. The curve D-D'-C thus becomes the rating for gage B.

\\ith the technique described for cross section 2 (page 120) and each of the 

known profiles for cross section 4 points on curve A are computed. With logarithmic plotting and knowledge concerning the point of zero flow, curve A, the 

OT curve, is drawn. Fall in the reach A-D, or the Fr curve, is then determined 

as before, and is shown as curve F, at the left of figure 38.

Because the dam, at high discharges, becomes drowned by fixed channel control, a particular requirement^jor_the F^curve must now be met. It will be noted

~that~tne transition curve B' merges with curve C at a discharge of about 67 cfs. 

Now curve C is the'jjnjiiiojT^-f^oj^ 

for discharges greater than about 67 cfs. As there is an unobstructed uniform

^i^n^"Hel"\'veerr"gage"s A and B, it follows that for discharges greater than about 

67 cfs uniform flow must occur throughout the reach A-B, and the fall in the reach 

must be equal to that of the channel bed, namely, 1.200 feet. It will be noted 

that the fall curve of figure 38 meets this requirement. If,the rating curves were 

to be extended above gage height 3.5 feet, uniform flow would prevail under 

conditions of no backwater, so that for backwater conditions no satisfactory 

curve of Qm /QT against Fm /Fr could be drawn; for above 3.5 feet Q T would equal 

OQ ar| d FT would equal F 0 , and the ratios would plot much the same as in figure 

30. If backwater is to be exg.ecteqLat high_stages, curve. B should be ,dcawa,.somewhat tothe Tef^T_ciirve C eyen_ at_ high stjges. The final position ofjyjrvel.,

"andmerefore curve F, should be detertmned_by trial^and error. TJie^^goji.tioiLjaf 

cTiiVe~&-arrd~cTTrve""F~Tirat results in the least scatter of points in the Boyer diagram (fig. 40) is the best position for any one gage.

It will be obvious that, in figure 38, as in figure 36, the Or curve (curve A) 

and the Fr curve (curve F) are drawn for fixed-backwater conditions as dictated 

by the positions of curves D and C. If the rating at gage B were always as indi- , 

cated by curve B-B'-C, the rating at gage A would be the simple two-dimensional 

relation of curve A. 3ut now, as in cross section 2, let us assume that the stagedischarge relation at gage B may be influenced by (1) some phenomenon further 

downstream, such as backwater from a downstream confluence; and (2) by the 

operation of sluice gates in the dam. Under the first condition backwater effects 

in the reach A-B will be greater than, and under the second condition less than,
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 12 7

0 ONV S S39VO IV '038 T3NNVHO 3A08V ' Nl 'NOI1VA3T3
128 STAGE-FALL-DISCHARGE RELATIONS

the fixed-backwater conditions described by figure 38. (For high stages when 

uniform flow prevails at B and C opening of sluice gates between gages D and C 

could not lower the gage height at B.) Again use may be made of the backwater 

profiles for computing ratios of Qm /Qr and the corresponding values of Fm /FfSuch ratios are presented as the Boyer diagram, figure 39, page 125. Again it will 

be noted that all the data fall within reasonable limits of a single curve. Given 

the depth at the upstream end of the reach, and the fall in the reach, computations based on factors from figures 38 and 39 will yield a close approximation 

to the true discharge.

Let attention now be directed to a similar treatment for cross section 3. ;Let a 

dam be erected in a channel of this cross section. Let it have characteristics 

identical with those described for other cross sections. Let gages A, B, and C 

be installed at locations as previously described.

The resulting curves for fixed-backwater conditions are shown in figure 40, 

page 127. Again curve C, the steady-uniform-flow stage-discharge relation curve, 

is drawn first. Curve D is next drawn, identical to curve 2 of figure 36, except 

that, as explained on page 126, it cannot cross curve C. As curve B approaches 

curve C, the former is bent slightly upward, the two curves gradually merging 

into one; the dam is completely drowned by fixed-channel control above the.point 

of the merger, Here it occurs beyond the range of well-defined data but is assumed to be at stage of about 3.9 feet on gages " and C, so that uniform flow 

must exist for discharges beyond about 83 cfs. 'Vith the technique described for 

cross section 2 (page 120), each of the known profiles for cross section 3, knowledge concerning the point of zero flow and the discharge beyond which uniform 

flow must exist, and plotting logarithmically, curve A, the Or curve, is drawn. 

Fall in the reach A-D, or the Fr curve, is then determined as for the cross sections already described; it appears at the left of figure 40 as curve F.

Using these curves of QT and FT for fixed-backwater conditions, and adding 

the assumptions that the stage-discharge relation at gage B may be influenced 

by (1) some phenomenon further downstream, such as backwater from a downstream confluence, and (2) by the operation of sluice gates in the dam, use is 

made of the backwater profiles for computing ratios of Cm /0r and Fm /Ff. These 

ratios are plotted as the Boyer diagram, figure 4L page 129. It is apparent that 

the plotted points show more dispersion than was the case of the points of figure 

37 and 39, but a mean curve would lie within about 4 percent of all the data. In 

other words, use of figures 40 and 41 will provide results within about 4 percent 

of the true discharges for cross section 3.

Consider now a similar reach of cross section 5, with the same obstruction as 

before, and gages A, B, and C located in the same positions. In figure 42, page 

131, curve C again presents the steady-uniform-flow stage-discharge relation. 

Curve B, drawn as before, now intersects curve C at a discharge of about 11 cfs, 

indicating that the dam is drowned by fixed channel control at this very low
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 129

-1* s

NTHS JF- ction

fl 

s fr

T
130 STAGE-FALL-DISCHARGE RELATIONS

value, and that uniform flow would exist in reach A-D for higher discharges (or 

for a stage greater than about 1.7 feet at the upper end of the reach). In view of 

the results to be expected when Qr equals QQ, it becomes necessary to shift 

curve D to the posiUon B' to obtain even reasonably satisfactory__ag£roach to a

(curve A) and the Fr curve (curve F) will appear as shown by the appropriate 

curves of figure 42. Values of Qm/QT against Fm/FT, computed as for the preceding cross sections, now plot as shown in the Boyer diagram, figure 43, page 

132. It will be noted that dispersion in this plot is somewhat greater than in 

similar plots for other cross sections, but use of a single mean curve will result 

in errors of no more than about 6 percent. The shift of curve B to the position 

B' is the result of trial and error computations. Further discussion as to the 

propriety of trial and error solutions is given later in this report.

It is obvious that curve £' of figure 42 cannot reasonably be assumed to be a 

rating for fixed-backwater conditions at gage B unless those conditions are fixed 

by something other than the dam which previously has been described. Consider 

now the significance of the dam. Although it was assumed to be soundly built 

and carefully rated, and the rating was used in developmenf of the curves of 

figures 36, 38, and 40 it was repeatedly subjected to degradation; first it was 

drowned by backwater from points farther downstream, and then it was riddled 

with sluice gates. Only under certain conditions was it allowed to control the 

stage-discharge relation, even at gage D; at other times, for a given discharge, 

the stage at B was sometimes higher, sometimes lower, than the dam alone would 

dictate. Except for the very important fact that it has provided the concepts 

which led to the curves of figures 36 to 41, the dam just as appropriately might 

have been a fisherman's net, or never have existed at all. But even though it 

were agreed that the dam did not exist at all, figures 36 to 43 still might be used 

for reliable computations of discharge under all combinations of stage and fall 

defined by the base data. (For stages higher than those defined by base data, 

additional adjustments to curve A, and hence to curves D and F, might be necessary to obtain h'ghly reliable results.)

The purpose of the dam, which may now be considered to have been imaginary, 

was to give substance to the concept, in a uniform ch'annel, of a base rating 

which differs from the uniform-flow rating. If a channel is indeed uniform, any 

rating other than the uniform-flow rating must be a very elusive phenomenon. It 

can be ssdd^only^ that, for a given discharge, there will be a given depth at the 

base gage, provided the Fall has a properly related jvalue. The^ assumpjioji .ol.the 

imaginary Jam^serves as a basis for correlation of the factors. The ;__ imjjortarit 

^act_to_be_derived isjhis: In the channels winch have been studied, there may be 

found^a stage-discharge^ reTatipn (the Qr curve) whichjjjaken in_conxunctioin_with_ 

its associated fall relation (the Ft curve) w|U_giye cjpse apprpximatioris_to_the_ 

^ru^jd^^h^e^iu^der^ajl^jip s^bJLe^coffii»i»atioas. jaf . stage and fall, by the apj>Li-__ 

cation of a single-curve relation Qm /Qt vs. Fm /F f
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 131

V 39V9 IV 038 H3NNVHO 3A08V I33J Nl 'NOI1VA3H3

0 QNtf 8 S39V9 IV '038 H3NNVHO 3A08tf '133J Nl 'NOIJ.VA3H3
132 STAGE-FALL-DISCHARGE RELATIONS
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 133

csr

=3

I 

T
134 STAGE-FALL-DISCHARGE RELATIONS

It should not be assumed that just any stage-discharge relation may be used 

for the Qr curve, even though the associated values of Fr are properly determined. For a given reach of a given channel there is an optimum position of the 

Or curve, (and the associated position of the Fr curve) which will give a minimum 

of dispersion of computed values of Qm/Qf when plotted against computed values 

of Fjjj/F,* Many trials were required before the curves of figures 36, 38, 40 and 

42 were evolved, and even these (particularly the curves for cross sections 3 

and 4) may be subject to further improvement.

It has been suggested by Eisenlohr (personal cormiunication) that the position 

of the Qr curve might be obtained by multiplying the values from the 00 curve by 

some constant slightly less than unity, such as 0.99, 0.95, or even 0.90. Experience indicates that, for some channels (particularly a 400-foot reach of cross 

section 2), such technique is quite adequate, provided a particular optimum 

value of the constant is chosen. In fact, curve D (fig. 36) actually was drawn in 

such manner that the Qf curve (curve A) would be in agreement with the relation 

O f = 0.90 QQ. It has been found, however, that a change in the shape of the cross 

section, or the length of the reach, or even the channel roughness, may result in 

a change in the optimum relation between Q r and Q 0. For cross section 4, the 

re] ation Qr - 0.95 Q 0 was found to give better results than Qr = 0.90 Q0 » but 

neither was as satisfactory as the Qr curve which has been shown as part of 

figure 3C and which results from the assumed position of curve B. For cross section 5, also, no coefficient of £> 0 could be found which provided a satisfactory 

position of the Q f curve.

To further illustrate the variations which may occur as a result of a change in 

the position of the Or curve, it was assumed that, for a 400-foot reach of cross 

section 2, Qr was equal to 0.99 OQ- The associated F r curve was computed from 

the backwater profiles by the same principles that were used for computing the 

Fr curves of figures 36, 38, 40, and 42. Ratios of Qm /Qr and Fm /Fr were then 

computed and were plotted as the Boyer diagram, figure 44, page 133. Comparison 

of this plot with the Boyer diagram of figure 37 gives evidence of the fact that, 

in a 400-foot reach of cross section 2, Qr = 0.99 (? 0 places the Qr curve further 

from the optimum position than does Q r = 0.90 Q 0 .

It appears that, for cross section 2, the relation Qr - 0.90 O0 is adequate because of the fact that, by happy coincidence, it also fulfills another and apparently more important requirement for the position of the Q r curve. The nature of 

this other requirement can be described only by a more analytical approach to 

the whole problem, and will therefore be reserved for discussion in the following 

subsection.
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 135

ANALYTICAL APPROACH TO FIXED-BACKWATER RATINGS

The preceding article, presenting fixed-backwater ratings, has been designed 

to provide, both to the layman and to those who are constantly engaged in the 

analysis of si ope-affected stream-flow stations, a simple picture and a simple 

solution of a very complex problem. In fact, in our present state of knowledge of 

the problem, it may represent the best picture and the best solution which can now 

be provided. However, as an approach to the question "Why are these things 

true?" some worth-while bits of information may be provided from an analytical 

viewpoint.

In an earlier paragraph (p. 86 ) it was explained that, for a very short reach, 

if the correction were made for change in velocity head, the relation

became an adequate correlation throughout the entire range of discharge and of 

backwater effect. In other words, if the slope of the energy line, S, at the base 

gage could be used in place of the fall, F, in a finite reach, discharge would 

vary with the square-root relation of the slopes. But as field investigations are 

necessarily based on fall in a finite reach, it is impractical to determine S. If a 

method could be evolved by which this observed fall in a finite reach could be 

corrected to become representative of the energy slope at the base gage, much of 

the problem would be solved.

It becomes appropriate, therefore, to look at the relation between these two 

factors. It necessarily is complex, depending at least on the cross section of the 

channel (including roughness), the length of reach, the discharge, and the amount 

of backwater. But, for any gaging station, the channel cross section and the 

length of reach are fixed; thus it is appropriate that they be assigned fixed values for this study. Furthermore, for a given discharge, the amount of backwater 

is a unique function of the stage at the base gage. Thus, for a 400-foot reach of 

cross section 2, the ratio, t, between effective fall and observed fall (4QQs/Fm ) 

will appear as in figure 45, page 136. In these computations S was computed from 

the relation, S = S 0(@/00) 2 (see p. 21) and Fm was computed through use of 

table 12. It will be noted that "t" is the ratio of effective fall to measured fall, 

if effective fall is considered to be the fall that is the product of the length of 

reach and the slope of the energy line at the base gage and therefore the fall 

which divided by the length of reach would give the correct slope of the energy 

line at the base gage.

Attention should first be directed to the major family of curves, drawn in heavy 

lines. It will be noted that there is a different curve for each discharge and that 

each curve intercepts the y axis at the point y = y0. It also will be noted that, 

for each of these curves, t increases rapidly with a slight increase in depth 

(introduction of a small amount of backwater), but soon reaches a maximum value 

beyond which, as depth increases, t decreases. Most notable of all is the fact
136 STAGE-FALL-DISCHARGE RELATIONS

TENS

Figure 45. Relation between observed fall and effective fall, 400-foot reach, cross section 2.
STAGE-FAIL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 137

that, as depth increases, all the curves tend to merge into a single bounding 

curve, which has been labelled tb . For very shallow depths the position of this 

curve has been estimated.

Attention now is invited to the second family of curves, drawn in light dashed 

line. Each of these lines indicates the variation between t and y, not for a given 

discharge, but for a given value of the discharge ratio, Qm/Qo- (For example, to 

compute the point common to the curves Qm/Q0 = 0.80 and Qm = 3.90 cfs: r» 0 - 

3.90/0.80, or 4.875 cfs, and, interpolating from table 42, Bulletin 381, y0 = 0.448 

foot. Interpolating from table 12, page 39,-by the method described on page81, 

Fm = 0.12221 foot. From formula (24), page 51, S varies as the square of Q^/QQ, 

so s = 0.8 2 x 0.003 = 0.00192; t = 400S/Fm = 6.2£>4.) It will be obvious that the 

line for Qm /Qo = LOO will be coincident with the y axis. As this <?m /(?o rati° 

is made smaller, the O^/Oo curves move rapidly to the right, so that for (?m /(>o = 

0.60, the curve is quite near the position of the tb curve. For practical purposes, 

it may be considered that, for values of Qm/Qo of about 0.4 or less, all the 

Qm/Qo curves will be coincident with the tb curve.

By definition, t is the coefficient to be applied to the observed fall to obtain 

effective fall at the base gage. After this correction has been applied, the relation <?m/(?o = V^m/'P'o wiM ke valid. F°r those cases, then, in which Qm /Q0 

does not exceed about 0.4, (remembering that here t = tb ),

(26)

Consider now the case in which (?m /(?o * s much larger, say 0.90. Now t is 

much smaller than tb. Assume a condition such that, even though the proper 

Qm /Qo to use, may vary slightly with depth, t will be proportional to tb , or t = 

fb tb, fb being a constant less than unity, say 0.60. Then

It must be remembered that, for any observed depth, ym, @ 0 is the discharge 

which would flow at ym provided Fm were equal to F0, and that Qm is a variable 

discharge depending upon both ym and Fm . For any particular ym , let it be assumed that Qm has a specific value, Qft for which the associated value of Fm 

will be the specific value Ff. Then, in place of equation (27), it may be written 

that

tb F/F0 . (28)

If, now, in addition to fb being a constant, it were possible at the same time 

that Qf/Qo were also a constant, such as /', then

>. (29)
138 STAGE-FALL-DISCHARGE RELATIONS

Another look at figure 45 will now be profitable. Comparing the family of curves 

of Qm /Q0 with the tb curve, it will be noted that at neither limit of the family of 

curves is there a proportional relation to (meaning, on log-log paper, equidistant 

from) tb . For high values of Qm /0o these curves diverge from tb with decreasing 

depth, while for Cm/C*o ~ 0*^0 the curve converges on tb as depth decreases. Between these limits, one curve will be found which, more nearly than any other, 

is proportional to tb . If one such curve be truly proportional, then both of the 

above requirements that the discharge ratio, /', be constant and that ffe be constant are fulfilled, and Fr will vary only with tb .

In the development of figure 36 conditions were chosen such that curve A actually conforms to a constant ratio of 0.90 for Q/QQ. And since QT is merely a 

specific case of Qm , it follows that the rating of figure 36 is in conformance with 

the curve of figure 45 which is labelled (?m/(?o = 0.90. OD this figure has been 

added in heavy continuous line the curve t = 0.59 tb . It will be noted that the two 

curves are in close agreement throughout the range of well-defined data. Since 

the lines are so near together, for practical purposes they may be regarded as 

the same, hereafter referred to as the /-line. It now is possible to make some 

numerical substitutions in equation (29), using / = 0.90, fb = 0.59, and remembering that, in this case, F 0 = 0.003 x 400 = 1.2) ft. Hence Ff « 1.65/fb . Using 

this equation and values of tb from figure 45, it will be found that, up to the 

higher limit of well-defined data, computed values of Ff are in very good agreement with values from curve F (the Ff curve) of figure 36.

It should be further noted that, dividing equation (26) by equation (28),

Qa/Q, =

As the validity of equation (26) was limited to discharge ratios of about 0.4or 

less, a similar limitation must be imposed upon equation (30). Using tb = 0.59, 

equation (30) becomes:

which is, indeed, the equation of the lower end of the curve (discharge ratios 

less than about 0.4) of figure 37.

The above limitation on equation (30) may be removed by substitution in place 

of the constant fb another factor, f, which will vary primarily with the discharge 

ratio. The nature of this parameter, f, may be examined by referring again to 

figure 45. In general, for any given value of ym , f is the value of t from the /-line 

divided by the value of t from the appropriate Qm /QQ curve. Thus tb is merely a 

special case of f, being appropriate for all values of (?m /(?o l ess than about 

0.4, as these all lie on a common f-curve (tb ) which is proportional to the /-line. 

But as values of Qm /QQ increase to 0.6 and above, the value of f increases, so
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 139

that, by the time C*m /Co becomes equal to Q/Qg, the value of i is unity. Thus 

when Qm and Qr are equal, equation (30) becomes Qm /Qr = \lFm/Ffi which merely means that the curve of relation between discharge ratio and fall ratio, such 

as figure 37, will pass through the point 1,1. As Om /0o becomes greater than 

Q/QQ the value of f becomes greater than unity, reaching as a limit the value of 

t from the ;-line when Qm /QQ is equal to unity.

To illustrate by example, let Qm = 3.90 cfs and Fm « 1.20 feet. These are 

conditions of uniform flow, so (?m /0o wi^ be equal to unity, and ym (see table 

42, Bull. 381, Lansford and Mitchell, 1949) will be 0.391 ft. From figure 36, 

Qr = 3.50 cfs, and Fr = 0.165 foot. From figure 45, / = 5.8. Required, to check 

Om by substitution in equation (30), using f = 5.8 in place of ib .

<?m /3.50 = (1/V5.8) (VL20/0.165)

Cm = 3.50 x 0.415 x 2.70 = 3.50 x 1.12 = 3.92 cfe,

which is well within acceptable limits of the known value of 3.90 cfs and demonstrates the applicability of the value off. But at this high limit of Qm /Qr (that 

is; Qm /Qo = 1), it is obvious that f will vary also with ym, because of the great 

divergence between the /-line and the line (?m/(?o = L To illustrate further, let 

Qm = 40.39 cfs and Fm = 1.20 feet. Again these are conditions of uniform flow, 

so that Qm /Q0 - 1, and ym (see table 42, Bull. 381, Lansford and Mitchell, 1949) 

will be 1.840 feet. From figure 37, Q r = 36.3 cfs, and F f - 0.60 foot. From figure 

45, f = 1.62. Checking Qm by substitution in equation (30),

0M /36.3 « (1/VL62) (VI.20/0.60)

Qm = 36.3 x 0.79 x 1.41 = 36.3 x 1.11 = 40.4 cfs,

which agrees with the known value of 40.39 cfs and again demonstrates the 

applicability of the value of /. But here f has the value of 162, compared to 5.8 

for the first example. Obviously f here has become largely a function of ym . In 

fact, there is considerable evidence that, when Qm /Q0 = 1, f is inversely proportional to Ff> which is, of course, a function of ym . It is for this reason that 

(l.v/7)(\/Fm /Fr)remains fairly constant under uniform-flow conditions, as evidenced 

by the value of 1.12 in the first example and 1.11 in the second. The curve in 

figure 37 could be drawn to make Qm/QT equal 1.11 for all values of Fm /Fr 

greater than 2, for we could be assured that under these conditions Qm should 

very nearly equal @0 .

The variation of f with ym might seriously cloud the whole picture except for 

one saving circumstance: the value of f does not vary appreciably with depth 

except when the Qm /Q Q ratio is very near to unity. In other words, the two important factors affecting /operate, largely, one at a time, rather than simultaneously.

296350 0 54 10
140 STAGE-FALL-DISCHARGE RELATIONS

When the amount of backwater is great (small values of Qm /Qr), f is dominated 

by the discharge ratio; when the amount of backwater is small (values of Qm /Qt 

greater than unity and approaching the reciprocal of Q/Qj, i is dominated by 

ym . In some instances the two factors affecting f may operate simultaneously, 

and with some lack of harmony.

The foregoing observations are of course insufficient for the complete preparation of a rating analysis and are particularly handicapped by lack of a convenient 

method of computing tb . They do however provide an insight into the problem of 

using observed fall in a reach as a measure of the friction slope at a gage and 

thus provide a firm basis for the development of stage-fall-discharge relations 

underfield conditions. The following are the conclusions basecon these analyses:

(1) There is an optimum position of the Qf curve. The ideal position 

is that in which, for a plot similar to figure 45, the /'-line will be for all 

values of ym a constant ratio to the tb line and at the same time will 

have a constant value, /', of Qm /Q0 . If both these requirements can be 

entirely satisfied, Qr =jQ 0. But in certain channels, such as cross section 3, both requirements cannot be met simultaneously. In such channels- some dispersion must be expected in the plots of Qm/Qr against 

Fm /Ff, as described under (4) below. Experience indicates that such 

dispersion will be minimized by holding to the requirement that the 

/-line maintain a constant ratio to the tb curve and selecting that ratio 

such that j varies as little as possible from a constant value of Cm/Co* 

The dispersion in the plotting of the values of Qm/Qf against Fm/Ff, 

as evidenced by figures 39, 41, and 43, is due largely to the fact that 

for cross sections 3, 4, and 3 the two requirements mentioned above 

cannot be simultaneously satisfied.

(2) A forthright application of the principle that discharge varies as 

the square root of the friction slope would indicate that a correction 

factor should be applied to observed fall to transform it to friction slope 

(at the base gage). This, however, would lead to a complicated system 

of corrections. Instead, it is possible to approximate these corrections 

closely by combining most of them into a single parameter, the rating 

fall, having the form

F, = ; 2 V('b *«,) (29)

If the optimum position of the Qf curve has been otherwise determined, 

values of Ff may perhaps be more conveniently computed from other 

data such as backwater profiles; but results should be substantially the 

same as from equation (29).
STAGE-FALL-DISCHARGE RELATIONS FOR THE LABORATORY CHANNELS 141

(3) If the Q curve has been located in the optimum position, and if 

the Fr curve has been drawn to agree, reasonable conformance to the 

relation

(31)

may be expected, in which f is a further correction factor needed in the 

adjustment of F^ to effective fall. As f is largely a function of Qm /Qr 

it can be incorporated in the Qm /Qr against Fm /Ff curve. For low 

values of the discharge ratio, f will have the constant value ib (t from 

the /-line divided by t from the tb curve); when Qm is equal to Qt, t will 

be equal to unity; as values of Qm/Qt approach Qm /Qo = 1 as a limit 

(Qm/Qf = 1.11 for the example shown in figure 45), / will vary inversely with Ff so as to make Qm/Qf = (l/VrJF)(v/ Fm/Fr) a constant for 

all stages. In the range of Qm /Qr from about 0.60 to the maximum value 

of Qm /Qr, f will not be entirely a function of Qm /Qf but will be affected 

somewhat by variation in stage. The slight dispersion evidenced by 

figure 37 is largely the result of being unable to correct for this effect.

(4) The above paragraphs have considered the nearly ideal conditions, as in cross section 2, which makes possible the establishment of 

a /-line that is, a curve, (?m /0o plotted against ym , which will be in 

substantial agreement with another curve, fb tb plotted against ym . 

Analyses of other cross sections reveal that, in many cases, a j-line 

cannot be established, but that the tb tb curve, regardless of the value 

assigned to fb , will cross several of the curves of Qm /QQ> As the reasons for this departure from ideal conditions lie in the relation of 

various combinations of channel shape and channel roughness future 

exploration along theoretical lines would be likely to have little practical value. Rather, as has been indicated earlier, trial and error solution may be the most practical approach. The analytical approach can 

be used to explain why certain procedures should be followed, but the 

final position of the three curves of relation, y against Qf; y against 

Ff] and Qm /Qf against Fm /Fr can best be determined by a series of 

trial computations each one of which should result in less scatter from 

a mean curve of Qm /Qr against Fm/Fr until the best solution is reached. 

In these successive trials parts of the Qf curve, the Ff curve, or the 

Qm /Qr curve should be relocated if such relocation will result in a more 

consistent solution.
142 STAGE-FALL-DISCHARGE RELATIONS

APPLICATION TO FIELD PROBLEMS

From those who are concerned with the analysis of rating curves for natural 

watercourses affected by variable fall questions may appropriately arise. What 

is here that will be helpful to us? Warning was given on page 4 that the discussions were to be confined to steady flow in prismatic channels, and that the 

irregularities of natural watercourses and the phenomenon of changing discharge 

point toward further complications. Furthermore, the amount of data provided for 

these analyses is very large, adequately covering the entire range of all combinations of stage and discharge; in contrast, field data often are meager in quantity and limited in range of conditions. How, then, can the foregoing analysis be 

useful to us in drawing our rating curves?

Let it be pointed out that the value of laboratory work lies largely in the 

establishment of general principles, rather than in the solution of specific problems. In complex and specialized situations laboratory results cannot take the 

place of engineering judgment, but familiarity with laboratory results should be 

expected to improve the quality of the engineering judgment that must be used. 

Such familiarity should be of assistance in identifying the significant factors of 

a complex problem and in providing qualitative evaluation of their effect. It is 

from this viewpoint that the following suggestions are made relative to analysis 

of field ratings.

In the first place, few channels in nature are uniform channels. Some may so 

nearly approach this condition that the techniques for uniform channels, as 

summarized in following paragraphs, will need to be employed. But many actually do have dams which are real, and which are only occasionally drowned by 

backwater or even affected by operation of gates. These are*the simple cases. 

I Here, without doubt, the Qf and Ff curves should be developed from field data 

\i observed at times when the dam is fully effective; and here, if the dam is only 

| subject to drowning by back water _ from a downstream confluence, all values of 

Offl /(?r and Fm /Fr will be less than unity. On the other hand, if the dam is not 

' subject to such drowning, ^t^jnlxjp_jn^i£iuj^d^,pjragjates, and if the Q f and 

s Ft curves have been developed for the particular case of the dam being fully 

v effective (that is, all gates closed), then all values of Qm /Qf and of Fm/Ff will 

be greater than unity.

At the other extreme are those cases in which the only control appears to be a 

long reach of channel of comparatively uniform cross section and slope, plus 

backwater effects induced by some downstream confluence. Here the analyst may 

have a choice between two or more procedures. It may be that he will wish to try 

more than one of the possible techniques, finally settling on the one which 

appears best to fit his data and still offers a reasonable basis for any necessary 

extrapolation outside the range of conditions covered by the data. The possible 

techniques may be divided into two general categories, or simple modifications 

thereof: uniform-flow ratings and fixed-backwater ratings.
APPLICATION TO HELD PROBLEMS 143

The uniform-flow rating, or any modification thereof, involves the assumption 

that the associated fall is a constant-that is, regardless of stage, the rating 

discharge will occur only when the fall has some value which is the same for all 

stages. Such analyses are sometimes referred to as "constant-fall ratings." Of 

these, the well-known unit-fall rating, expressed by the relation Qm /Qr - v^Fm , 

Fr being assumed to equal unity, is the simplest case. For stations that always 

operate under a high amount of backwater, such a rating may be quite appropriate. 

If discharge measurements cover the entire range of flow conditions, and if such 

measurements closely conform to a unit-fall rating, there is no need to use more 

complicated techniques.

Care must be exercised, however, that the unit-fall analysis is not used where 

the variation in backwater is very great. Here this method may deteriorate into a 

stage-fall-discharge diagram such as discussed on pages 81 to82, and illustrated 

by figures 18 and 19. Under such conditions of great variation in backwater, use 

of the constant-fall rating should be associated with three-dimensional plots of 

the Boyer diagram, as discussed on pages 109 to 117 and illustrated by figures 

32 to 34. For Boyer diagrams in which the points scatter (as the data scatter in 

figure 35 (above), page 116), the points may be labelled with their observed gage 

height, and an examination may be made to determine whether a family of curves 

of equal gage height, similar to those in figures 32 to 34, can justifiably be 

drawn. If so, further improvement in the plot might be obtained by computing backwater profiles for desired discharges in the slope reach and abstracting needed 

data on stage, fall, and discharge from the computations. In development of the 

rating, such computed data should of course be given less weight than is given 

to reliable field observations. For favorable results along these lines, it appears 

that the field problem should exhibit the following features:

L For a given gage height at the base (upstream) gage, there should 

be a wide variation of discharge. In other words, the station should be 

such that high g,age height may result at one time from high discharge, 

at another time primarily from high backwater effects.

2. The channel gradient should be comparatively constant throughout 

the slope reach and "mild" by the technical classification of fluid 

mechanics. Within this limitation, the steeper the gradient, the more 

pronounced will be the family tendencies of the curves of relation.

3. The channel cross section should be as uniform as possible within 

the slope reach that is, the stage-discharge relation at the base gage, 

when free of backwater, should be completely channel controlled. If, 

without backwater, the stage-discharge relation is materially affected 

by a section control, the rating fall should not be considered to be 

constant.
144 STAGE-FAIL-DISCHARGE RELATIONS

4. The reach should be free of backwater effects for such periods as 

to permit the development by usual stream-gaging methods of the twodimensional no-backwater relation between stage and discharge.

For stations which tend to meet the above four requirements and for which a large 

number of measurements under a wide range of backwater are available, it seems 

appropriate to plot diagrams similar to figure 28 directly from observed data.

\ As an alternative to the procedure of the preceding paragraph the analyst may 

wfsh to undertake the second of the general catagory of techniques namely, the 

development of a fixed-backwater rating. This involves the assumption that the 

associated fall is a variable that is, for any given stage the rating discharge 

will occur only when the fall has some value which is a unique function of the 

observed stage, but not necessarily the same for different stagesXln this case it 

would be desirable, from the analytical viewpoint, to place the Qf curve in such 

position that the associated Ff curve could be expressed as Ff = ; 2F0//b ffe . 

(See pp. 137, 140.) From the practical viewpoint, however, it cannot be hoped 

that data will be available from which to evaluate all the factors in the above 

relation. Furthermore, in many cases it may be impossible to develop the nobackwater rating from discharge measurements because of the persistence of 

backwater conditions. As a substitute, the following steps are suggested:

L Study the profiles of the floods of highest discharge. If these are 

found to be nearly parallel (having the same fall in the reach between 

the two gages), it is likely that this fall may be used as F Q, the fall 

that the maximum value of F, cannot exceed. If these profiles arpi fnnqd, 

Tiot to T;>e nearly parallel, study should be made of causes of backwater. 

Those profiles which appear to be least affected by backwater should 

be given greatest weight in estimating the maximum value of Fr2. Plot discharge measurements for which the fall is near to that 

found in step 1. Avoid use of measurements made under rapidly changing 

discharge. Using the plotted measurements as a guide, and assuming a 

constant fall curve (that is, the rating fall is the same for all stages), 

draw a tentative stage-discharge curve, which will be an approximate 

QQ curve. To obtain sufficient definition of this curve in the absence 

of suitable discharge measurements it may be desirable to evaluate the 

stage-conveyance curve from channel surveys.

3. Plot the remainder of the discharge measurements and make an 

auxiliary plot of stage against fall for each measurement. Working with 

the two plots together, draw the __Q __ __, _

the left of the tentative discharge curve at low ste^esTbut nearer to the 

tentative curve at highest stages. DrawHtfie F f curye^to^a^rjeJ^aHLg^ 

if theO^urve^^asses through a discharge measurement, the F r curve
APPLICATION TO FIELD PROBLEMS 145

should pass through the corresponding plotted fall; if the Qf curve passes to one side of a discharge measurement, the Fr curve should pass to 

the same side of the corresponding plotted fall.) The Ff curve may be 

expected to approach zero fall at low stages, since at low stages nearly 

all of the uniform flow discharge (QQ) can be obtained with a small observed fall. Obviously, when the discharge is zero, the fall must be 

zero; or, in event there is no water in a part of the slope reach, the 

concept of fall becomes meaningless.

4. Compute and plot Qm /Qr against Fm /Fr. Draw a curve among the 

points, causing it to pass through the point 1,1. If the curve extends 

above this value, it should rapidly become nearly parallel to the Fm /Ft 

axis. For low values of Qm /Qr (generally less than 0.50) the curve 

should approach one whose equation is Qm /0r = c\/Fm /Fr. Present 

experience indicates that for low ratios the value of c will be greater 

than unity and will depend upon the length of reach, the roughness of 

the channel, and perhaps the geometry of the cross section.

5. Assuming the Fr curve and the curve of relation between ratios to 

be correct, compute and plot the values of Qf corresponding to each dis- 

. charge measurement. The Qr curve may now be redrawn to give better 

agreement with the plotted values of Qr. Percentage variation between 

the adjusted value of each discharge measurement and the corresponding 

value from the Q f curve should now be computed. If these variations 

are found to be within acceptable limits, the analysis may be considered 

as closed, and the necessary tables may be prepared.

If variation between adjusted measurements and the QT curve is great, further 

improvement sometimes may be obtained by assuming that the new Q r curve is 

correct and by repeating appropriate portions of the above procedure, beginning 

with step 3. It may sometimes be helpful to compute the value of Fr that would 

make the scattering measurements plot on the Qr curve and then adjust the F r 

curve to average these values, keeping a smooth F r curve. Then repeat steps 4 

and 5. This is the procedure referred to earlier in this report as the trial-anderror method.

It is possible that with some modifications the procedure for developing fixed 

backwater ratings in uniform channels as outlined above may be equally applicable to reaches in which a section control is effective at low stages or even to 

reaches that include nonuniform-channel configurations. Laboratory verification 

of this presumption would be difficult in view of the many possible combinations 

of conditions that would need to be investigated. Perhaps the best verification 

would be through application to actual field conditions.
146 STAGE-FALL-DISCHARGE RELATIONS

It will be noted that only in a few instances do these suggested techniques 

for solution of field problems differ materially from methods that long have been 

in use in many Geological Survey offices. Items 1 and 2, immediately above, are 

somewhat more far-reaching than heretofore proposed, but should result in a better understanding of the basic hydraulics of the particular slope reach, an approximation to the no-backwater rating, and a firmer foundation on which to base the 

analysis. The most outstanding innovation is the suggestion, in connection with 

uniform-flow (or constant-fall) ratings, of the three-dimensional aspect of the 

Boyer diagram, ym being the third factor; but, if a constant-fall curve is assumed, 

the validity of the three-dimensional Boyer diagram is adequately documented. 

Beyond these points the conclusions of this report are largely confined to a 

clarification of the various types of slope-affected ratings, the conditions under 

which they may be expected to be appropriate, and to a verification by laboratory 

study of the fact that many of the methods of analysis now in use by the Geological Survey are, indeed, very appropriate.
STAGE-FALL-DISCHARGE RELATIONS 147

SYMBOLS

(Except as otherwise explained in the text, numbers used as subscripts refer 

to particular points in the channel for example, EI refers to total energy per 

pound of fluid at point 1, with respect to an established datum plane.)

A Cross-sectional area of the fluid, (ft 2)

6 Width of channel at free surface of fluid, (ft)

b, as superscript Variable exponent of the friction slope, (dimensionless)

6, as subscript See immediately following prefixed symbol without subscript.

c, as subscript

d, as prefix 

E

An empirical constant applied to hydraulic radius, (ft) 

Also used as a miscellaneous constant.

Indicates prefixed symbol applies specifically to conditions 

of critical flow.

Velocity-distribution coefficient, based on principle of energy, 

(dimensionless)

Velocity-distribution coefficient, based on principle of momentum, (dimensionless)

Indicates mathematical derivative of accompanying symbol.

Energy per pound of fluid, with respect to an established 

datum plane, (ft)

Total flux of energy per unit of time, (ft-lbs/sec)

Fall; difference in elevation of water surface between the two 

ends of a reach of channel, (ft)

Froude number; cm V 2/gR. (dimensionless)

The ratio of t for any value of ;' (see below) to the value of t 

for any Om /0o» ym being held constant, (dimensionless)

The value of /when t is equal to t^ (see below; dimensionless) 

Acceleration of gravity; used as 32.16 ft/sec 2.

Specific energy; energy per pound of fluid with respect to 

bottom line of cross section as datum plane, (ft)

Elevation of water surface with respect to a given point in 

the fluid, (ft)

The ratio of the rating discharge to the uniform-flow discharge; 0/0o- (dimensionless)

Conveyance of a cross section, (cfs)

Conveyance divided by area of the cross section, (ft/sec)
148 STAGE-FALL-DISCHARGE RELATIONS

14

ai, as subscript

n

o, as subscript

P 

Q 

R 

R'

r, as subscript

S 

S

So 

t

V

w.p.

Distance along the channel from point at which channel bed 

intersects datum plane, (ft)

The M-function; M = A^A/b. (ft 2 '/z)

Except when used with c, indicates prefixed symbol applies 

specifically to an actual (treasured) condition.

Coefficient of roughness in the Manning formula, (ft 1 /6 ?)

Except when used with S, indicates prefixed symbol applies 

specifically to conditions of uniform flow.

Pressure intensity per unit area. (Ibs/sq ft)

Discharge, (cfs)

Hydraulic radius; /4/w.p. (ft)

Weighted hydraulic radius, as used for flood-plain cross 

sections. (See p. 42.)

Weighted hydraulic radius, as computed for rectangular crosssections. (See p. 42.)

Indicates prefixed symbol applies specifically to a particular 

set of stage-fall-discharge conditions other than uniform 

flow that have been adopted as the base rating.

Slope of energy grade line; friction slope, (dimensionless)

Mean of values of S for the two ends of the reach Ax. 

(dimensionless)

Slope of the channel bed. (dimensionless)

The ratio of effective fall at the base gage to observed fall 

in the slope reach, (dimensionless)

The maximum value of t for a given value of ym . 

(dimensionless)

Mean velocity in a cross section; Q/A. (ft/sec)

Wetted perimeter of a cross section, (ft)

Distance along the channel between any two cross sections, (ft)

Depth of fluid; perpendicular distance between stream bed and 

water surface; for this report, may be considered as vertical 

distance between stream bed and water surface, (ft)
SYMBOLS 149

Z Elevation with respect to an established datum plane, (ft) 

7 Specific weight. (lbs/ft 3)

A Indicates change in a reach of channel of such short length 

that under conditions of nonuniform flow the rates of variation within the reach are essentially the same as the respective rates of variation at any particular point within 

the reach.
150 STAGE-FALL-DISCHARGE RELATIONS

REFERENCES

Bakhmeteff, Boris A., 1932, Hydraulics of open channels, New York, McGrawHill Book Co.

Corbett, D. M., and others, 1943, Stream-gaging procedure: U. S. Geol. Survey 

Water-Supply Paper 888.

Keulegan, G. H., 1938, Laws of turbulent flow in open channels: Nat. Bur. Standards Jour. Research, vol. 2L

Lansford, W. M., and Mitchell, W. D., 1949, An investigation of the backwater 

profile for steady flow in prismatic channels: Univ. of 111. Eng. Exper. Sta. 

Bull. 381.

Mitchell, W. D., and Barren, E. G., 1946. The backwater profile for steady flow 

in a rectangular channel and its significance in the stage-fall-discharge relation: Univ. of 111. graduate thesis. (Unpublished)

Mononobe, Nagaho, 1938, Backwater and drop-down curves for uniform channels: 

Amer. Soc. Civ. Eng. Trans., vol. 103.

Rouse, Hunter, 1938. Fluid mechanics for hydraulic engineers, New York, McGraw-Hill Book Co.

Von Seggern, M. E., 1950, Integrating the equation of nonuniform flow: Amer. 

Soc. Civ. Eng. Trans., vol. 115, p. 7L

Woodward, S. M., and Posey, C. J., 1941, Hydraulics of steady flow in open 

channels, New York, John Wiley & Sons, Inc.
SUPPLEMENTARY TABLES
152 STAGE-FALL-DISCHARGE RELATIONS

CHARACTERISTICS OF THE LABORATORY CHANNELS

AREAS OF FLOOD-PLAIN CROSS SECUON

Table 27. Area, in square feet, of flood-plain cross sections 

[Note: for depths less than 1.00, area is equal to depth]

y

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

.00

1.000

1.180

1.564

2.064

2.564

3.064

3.564

4.064

4.564

5.064

5.564

6.064

6.564

7.064

7.564

8.064

8.564

9.064

9.564

10.064

10.564

11.064

11.564

12.064

12.564

13.064

13.564

14.064

14.564

15.064

.01

1.010

1.209

1.614

2.114

2.614

3.114

3.614

4.114

4.614

5.114

5.614

6.114

6.614

7.114

7.614

8.114

8.614

9.114

9.614

10.114

10.614

11.114

11.614

12.114

12.614

13.114

13.614

14.114

14.614

15.114

.02

1.020

1.240

1.664

2.164

2.664

3.164

3.664

4.164

4.664

5.164

5.664

6.164

6.664

7.164

7.664

8.164

8.664

9.164

9.664

10.164

10.664

11.164

11.664

12.164

12.664

13.164

13.664

14.164

14.664

15.164

.03

1.032

1.273

1.714

2.214

2.714

3.214

3.714

4.214

4.714

5.214

5.714

6.214

6.714

7.214

7.714

8.214

8.714

9.214

9.714

10.214

10.714

11.214

11.714

12.214

12.714

13.214

13.714

14.214

14.714

15.214

.04

1.047

1.308

1.764

2.264

2.764

3.264

3.764

4.264

4.764

5.264

5.764

6.264

6.764

7.264

7.764

8.264

8.764

9.264

9.764

10.264

10.764

11.264

11.764

12.264

12.764

13.264

13.764

14.264

14.764

15.264

.05

1.064

1.346

1.814

2.314

2.814

3.314

3.814

4.314

4.814

5.314

5.814

6.314

6.814

7.314

7.814

8.314

8.814

9.314

9.814

10.314

10.814

11.314

11.814

12.314

12.814

13.314

13.814

14.314

14.814

15.314

.06

1.084

1.386

1.864

2.364

2.864

3.364

3.864

4.364

4.864

5.364

5.864

6.364

6.864

7.364

7.864

8.364

8.864

9.364

9.864

10.364

10.864

11.364

11.864

12.364

12.864

13.364

13.864

14.364

14.864

15.364

.07

1.105

1.427

1.914

2.414

2.914

3.414

3.914

4.414

4.914

5.414

5.914

6.414

6.914

7.414

7.914

8.414

8.914

9.414

9.914

10.414

10.914

11.414

11.914

12.414

12.914

13.414

13.914

14.414

14.914

15.414

.08

1.127

1.470

1.964

2.464

2.964

3.464

3.964

4.464

4.964

5.464

5.964

6.464

6.964

7.464

7.964

8.464

8.964

9.464

9.964

10.464

10.964

11.464

11.964

12.464

12.964

13.464

13.964

14.464

14.964

15.464

.09

1.152

1.516

2.014

2.514

3.014

3.514

4.014

4.514

5.014

5.514

,6.014

6.514

7.014

7.514

8.014

8.514

9.014

9.514

10.014

10.514

11.014

11.514

12.014

12.514

13.014

13.514

14.014

14.514

15.014

15.514

y

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

VALUES OF k= V0 /\lSo 

Table 28. Values of k, in feet per second, cross section 2

y

4.00

3.60

3.40

3.20

3.00

2.90

2.80

2.70

2.60

2.50

2.40

2.36

2.32

2.30

2.28

2.24

2.20

2.16

2.12

2.10

2.08

2.04

2.02

2.00

1.98

1.96

1.94

1.92

1.91

1.90

k

103.154

99.828

98.064

96.217

94.268

93.252

92.200

91.112

89.994

88.848

87.667

87.186

86.692

86.445

86.195

85.697

85.180

84.660

84.139

83.862

83.597

83.053

82.774

82.487

82.213

81.936

81.631

81.340

81.193

81.043

y

1.89

1.88

1.87

1.86

1.85840

1.84

1.80

1.76

1.72

1.70

1.68

1.66

1.64

1.62

1.60

1.58

1.56

1.55

1.54

1.53

1.52

1.51

1.50389

1.50

1.46

1.42

1.40

1.38

I oOlti 

1 i. ^4. * >**

k

80.893

80.760

80.606

80.451

80.422

80.154

79.522

78.901

78.251

77.906

77.572

77.231

76.904

76.546

76.179

75.827

75.463

75.292

75.093

74.892

74.711

74.529

74.395

74.320

73.555

72.721

72.299

71.892

71.445

71.014

y

1.32

1.30

1.28

1.26

1.24

1.22

1.20

1.18

1.17

1.16

1.15

1.14

1.13221

1.12

1.10

1.08

1.06

1.04

1.02

1.00

.98

.96

.94

.92

.90

.88

.87

.86

O£ . OJ

D A . U4

k

70.539

70.081

69.636

69.146

68.672

68.151

67.644

67.151

66.881

66.640

66.362

66.079

65.889

65.531

64.963

64.408

63.832

63.269

62.648

62.002

61.405

60.783

60.133

59.455

58.749

58.091

57.752

57.405

57.049

C £ 7OQ Do. IZo

y

0.83

.82012

.82

.80

.78

.76

.74

.72

.70

.68

.66

.65

.64

.63

.62

.61812

.60

.56

.52

.50

.48

.46

.44

.43

.42

.41

.40

.39491

k

56.355

55.966

55.975

55.183

54.398

53.668

52.897

52.136

51.329

50.529

49.682

49.268

48.839

48.455

47.999

47.909

47.104

45.317

43.396

42.430

41.384

40.325

39.253

38.722

38.079

37.495

36.971

36.615
SUPPLEMENTARY TABLES 153

Table 29. Values of k, in feet per second, cross section 3

y

4.00

3.60

3.40

3.20

3.00

2.80

2.70

2.60

2.50

2.40

2.30

2.20

2.10

2.06

2.02

2.00

1.98

1.94

1.90

1.86

1.82

1.80

1.78

1.74

1.70

1

£ Q

. DO

k .

78.908

77.979

77.486

76.966

76.438

75.885

75.598

75.319

75.031

74.749

74.459

74.158

73.848

73.722

73.608

73.541

73.492

73.369

73.241

73.128

72.989

72.927

72.865

72.736

7 O £f\ 1

1 i. GUI

72. 529

y

1.66

1.64

1.62

1.60

1.58

1.57

1.56

1.55

1.54

1.53

1.52712

1.52

1.50

1.48

1.44

1.40

1.36

1.32

1.30

1.28

1.26

1.24

1.22

1.20

1 1 fi . lo

1.16

k

72.480

72.407

72.354

72.277

72.221

72.170

72.140

72.111

72.082

72.051

71.978

71.996

71.934

71.549

70.773

69.926

69.082

68.188

67.720

67.268

66.798

66.285

65.787

65.270

64.768

£A n A O

t>4. Z4B

y

1.14

1.13

1.12

1.11

1.10

1.09

1.08171

1.08

1.06

1.04

1.02

1.00

.98

.96

.94

.92

.90

.88

.86

.84

.83

.82

.81

.80

.79

.78073

k

63.676

63.401

63.118

62.831

62.541

62.276

62.044

61.973

61.387

60.812

60.178

59.519

58.908

58.272

57.608

56.916

56.193

55.519

54.814

54.120

53.717

53.348

52.925

52.536

CO -I OQ DZ laO

51.869

y

0.76

.74

.72

.70

.68

.66

.65

.64

.63

.62

.61206

.60

.56

.54

.52

.50

.48

.46

.45

.44

.43

.42

.41

.40804

k

50.976

50.182

49.346

48.512

47.685

46. so:

46.346

45.928

45.499

45.054

44.689

44.123

42.253

41.249

40.307

39.290

38.264

37.229

36.677

36.100

35.580

35.036

34.466

34.368

Table 30.--Values of k, in feet per second, cross section 4

y

4.00

3.80

3.60

3.40

3.30

3.20

3.10

3.00

2.90

2.80

I' 16

'2.72

2.70

2.68

2.64

2.60

2.56

2.52

2.50

2.48

2.46

2.44

2.42

2.41

2.40

2.39

2.38

2.37

2.36

2.35835

2.32 

2. 30

'

k

71.879

71.060

70.169

69.217

68.714

68.180

67.635

67.065

66.470

65.843

65.581

65.323

65.186

65.058

64.786

64.505

64.230

63.930

63.784

63.636

63.485

63.348

63. 193

63.105

63.034

62.961

62.873

62.800

62.725

62.707

62.406

62*. 075

y

2.24

2.20

2.16

2.12

2.10

2.08

2.06

2.04

2.02

2.00

1.98

1.96

1.94

1.92

1.90

1.89577

1.88

1.86

1.84

1.82

1.80

1.78

1.76

1.74

1.72

1.70

1.68

1.66

1.64

1.62

1.60

1

CQ

 jo

1.56

k

61.732

61.378

61.027

60.646

60.458

60.268

60.072

59.892

59.689

59.483

59.291

59.075

58.875

58.652

58.444

58.400

58.230

58.011

57.768

57.540

57.308

57.069

56.826

56.576

56.322

56.061

55.795

55.543

55.261

54.997

54.703 

54 425

54! 117

y

1.54

1.52

1.50

1.49

1.48

1.47

1.46

1.45

1.44

1.43319

1.43

1.42

1.41

1.40

1.39

1.38269

1.36

1.32

1.30

1.28

1.26

1.24

1.22

1.20

1.18

1.16

1.14

1.12

1.10

1.09

1.08

1.07

1.06

k

53.825

53.500

53.189

53.032

52.872

52.711

52.547

52.380

52.211

52.103

52.039

51.866

51.690

51.511

51.331

51.207

50.772

50.014

49.604

49.209

48.802

48.382

47.948

47.500

47.068.

46.587

46.124

45.611

45.112

44.889

44.628

44.364

A A 1 OO 44. I/O

y

1.05

1.04

1.03828

1.02

1.00

0.98

.96

.94

.92

.90

.88

.86

.84

.83

.82

.81

.80

.79

.78780

.76

.72

.68

.66

.64

.62

.60

.58

.56

.55

.54

.53328

k

43.887

43.608

43.573

43.066

42.503

41.956

41.384

40.827

40.206

39.558

38.921

38.299

37.645

37.351

37.004

36.696

36.332

36.007

35.921

34.978

33.574

32.166

31.425

30.638

29.918

29.152

28.330

27.581

27.154

26.778

26.499
154 STAGE-FALL-DISCHARGE RELATIONS

Table 31.--Values of k, in feet per second, cross section 5

y

4.00

3.80

3.60

3.40

3.30

3.20

3.10

3.00

2.96

2.92

2.88

2.84

2.80

2.78

2.76

2.74

2.72

2.70

2.69

2.68

2.67

2.66

2.65

2.64

2.63

2.62

2.61

2.60

2.59

2.58

2.57

2.56

2.55025

2.50

2.40

2.30

2.20

2 . 16£ 

2.12

k

88.225

86.485

84.530

82.292

81.072

79.794

78.465

77.097

76.542

75.965

75.383

74.795

74.183

73.886

73.583

73.274

72.958

72.635

72.471

72.325

72.177

72.007

71.836

71.684

71.529

71.352

71.173

71.012

70.851

70.687

70.521

70.353

70.198

69.303

67.512

65.623

63.612

62. 800

61.905

y

2.10

2.08

2.04

2.00

1.96

1.92

1.90

1.88

1.86

1.84

1.82

1.80

1.79

1.78

1.77

1.76

1.75

1.74

1.73

1.72104

1..72

1.70

1.68

1.66

1.64

1.62

1.60

1.58

1.56

1.54

1.52

1.50

1.48

1.46

1.44

1.42

1.40

1 3fi 1 oo 

1 16. 1 <JO

k

61.451

61.011

60.087

59.097

58.101

57.064

56.496

55.979

55.440

54.842

54.256

53.644

53.348

53.047

52.736

52.421

52.097

51.767

51.427

51.106

51.081

50.405

49.742

49.045

48.311

47.586

46.872

46.118

45.371

44.638

43.854

43.140

42.379

41.627

40.887

40.091

39.378

38.603

^7 > 1 .7^1. O91

y

1.34

1.33

1.32

1.31

1.30

1.29785

1.28

1.26

1.24

1.23

1.22

1.21

1.20

1.19

1.18

1.17

1.16958

1.14

1.10

1.06

1.04

1.02

1.01

1.00

.99

.98

.97

.96

.95

.94

.93

.92

.91405

.90

.88

.84

.80

 7ft I O

.76

k

37.176

36.862

36.531

36.186

35.914

35.839

35.418

35.065

34.879

34.831

34.892

34.954

35.138

35.286

35.522

35.697

35.746

36.292

36.979

37.559

37.840

37.946

37.780

37.793

37,806

37.632

37.645

37.466

37.475

37.486

37.300

37.309

37.152

37.123

36.929

36.732

36.287

36.047

35.794

y

0.74

.73

.72

.71

.70

.69387

.60

.56

.52

.50

.49

.48

.47

.46056

.46

.45

.44

.43

.42

.41

.40

.39

.38

.37

.36

.35350

.35

.34

.33

.32

.31

.30

.29

.28

.27

.26

.25250

k

35.775

35.514

35.500

35.229

35.211

35.259

34.081

33.581

33.004

32.498

32.416

32.330

31.854

31.713

31.751

31.646

31.536

31.419

30.864

30.725

30.581

30.430

30.269

29.606

29.415

29.320

29.212

28.996

28.770

27.956

27.680

27.386

27.072

26.734

25.695

25.279

24.982

Table 32.--Values of k, in feet per second, cross section 6

y

4.00

3.80

3.60

3.40

3.30

3.20

3.10

3.00

2.90

2.86

2.82

2.80

2.78

2.74

2.70

2.68

2.66

2.64

2.62

2.60

2.58

2.57

2.56

k

105.398

102.168

98.623

94.745

92.695

90.577

88.399

86.171

83.885

82.976

82.030

81.552

81.101

80.152

79.160

78.700

78.207

77.725

77.211

76.705

76.230

75.989

75.744

y

2.55

2.54

2.53

2.52

2.51

2.50

2.49

2.48

2.47

2.46

2.45531

2.40

2.30

2.20

2.10

2.00

1.90

1.80

1.76

1.72

1.70

1.68

1.64

k

75.498

75.226

74.972

74.717

74.456

74.193

73.950

73.703

73.453

73.201

73.066

71.615

68.956

66.198

63.348

60.377

57.253

53.963

52.589

51.212

50.494

49.742

48.311

y

1.60

1.56

1.52

1.50

1.48

1.46

1.44

1.42

1.40

1.38

1.36

1.34

1.33

1.32

1.31

1.30694

1.30

1.26

1.22

1.18

1.14

1.10

1.06

k

46.872

45.371

43.854

43.140

42.379

41.627

40.887

40.091

39.378

38.603

37.921

37.176

36.862

36.531

36.186

36.097

35.914

35.065

34.892

35.522

36.292

36.979

37.559

y

1.04

1.02

1.00

.99

.98

.97

.96

.95

.94

.93

.92

.91405

.90

.80

.70

.60

.56

.52

.51

.50

.49

.48

.47571

k

37.840

37.946

37.793

37.806

37.632

37.645

37.466

37.475

37.486

37.300

37.309

37.152

37.123

36.287

35.211

34.081

33.581

33.004

32.577

32.498

32.416

32.330

32.239
SUPPLEMENTARY TABLES 155

COMPUTED FALLS IN 400-FOOT REACH FOR SELECTED VALUES OF STAGE

AND DISCHARGE

Table 33.~Computed falls, cross section 2, 400-foot reach, for selected values of stage

and discharge

ym

2.90

2.80

2.50

2.50

2.50

2.50

2.50

2.50

2.28

2.20

2.08

2.00

2.00

2.00

2.00

2.00

2.00

1.98

1.91

1.90

1.89

1.88

1.87

1.86

1.858

1.840

1.80

1.80

1.80

1.80

1.80

1.68

<?m

40.39

30.22

3.90

7.99

12.38

20.13

30.22

40.39

40.39

40.39

40.39

3.90

7.99

12.38

20.13

30.22

40.39

40.39

40.39

40.39

40.39

40.39

40.39

40.39

40.39

40.39

3.90

7.99

12.38

20.13

30.22

30.22

Fm

0.188

.108

.00215

. 00905

.0219

.0595

.142

.280

.369

.416

.513

. 00341

.0144

.0351

.0979

.252

.611

.643

.799

.831

.867

.908

.957

1.016

1.027

1.200

. 00422

.0179

.0440

.126

.350

.452

Vm

1.60

1.60

1.60

1.60

1.60

1.56

1.54

1.53

1.52

1.51

1.504

1.489

1.40

1.40

1.40

1.40

1.30

1.24

1.20

1.20

1.20

1.20

1.17

1.15

1.14

1.132

1.121

1.10

1.10

1.10

1.00

1.00

<?m

3.90

7.99

12.38

20.13

30.22

30.22

30.22

30.22

30.22

30.22

30.22

30.22

3.90

7.99

12.38

20.13

20.13

20.13

3.90

7.99

12.38

20.13

20.13

20.13

20.13

20.13

20.13

3.90

7.99

12.38

3.90

7.99

Fm

0. 00542

.0229

.0569

.169

.568

.663

.731

.774

.826

.894

.933

1.200

. 00712

.0304

.0768

.247

.318

.388

. 00977

.0424

.111

.460

.546

.638

.711

.797

1.200

.0117

.0517

.141

.0144

.0648

ym

1.00

.94

.90

.90

.90

.87

.85

.84

.83

.820

.812

.80

.80

.74

.70

.70

.68

.66

.65

.64

.63

.62

.618

.612

.60

.50

.46

.42

.41

.40

.395

.391

3m

12.38

12.38

3.90

7.99

12.38

12.38

12.38

12.38

12.38

12.38

12.38

3.90

7.99

7.99

3.90

7.99

7.99

7.99

7.99

7.99

7.99

7.99

7.99

7.99

3.90

3.90

3.90

3.90

3.90

3.90

3.90

3.90

Fm

0.188

.235

.0181

.0845

.283

.340

.399

.442

.504

.611

1.200

.0238

.119

.153

.0326

.190

.216

.254

.280

.314

.364

.454

.486

1.200

.0479

.0811

.109

.168

.199

.254

.315

1.200

296350 0 5.
156 STAGE-FALL-DISCHARGE RELATIONS

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SUPPLEMENTARY TABLES 157

Table 35. Computed falls, cross section 4, 400-foot reach, for selected values of stage

and discharge

ym

3.178

3.02

3.00

2.90

2.80

2.80

2.64

2.50

2.50

2.50

2.50

2.50

2.50

2.50

2.46

2.42

2.40

2.40

2.38

2.37

2.36

2.358

2.335

2.28

2.16

2.00

2.00

2.00

2.00

2.00

3m

39.98

29.91

39.98

29.91

29.91

39.98

39.98

3.79

7.62

12.19

19.10

20.15

29.91

39.98

39.98

39.98

29.91

39.98

39.98

39.98

39.98

39.98

39.98

29.91

29.91

3.79

7.62

12.19

19.10

20.15

Fm

0.378

.221

.441

.241

.262

.541

.660

. 00479

.0189

.0503

.128

.144

.355

.817

.881

.956

.399

1.001

1.051

1.080

1.110

1.115

1.200

.465

.558

. 00750

,0307

.0811

.218

.243

ym

2.00

1.96

1.92

1.90

1.896

1.877

1.80

1.80

1.80

1.80

1.80

1.60

1.60

1.60

1.60

1.60

1.56

1.54

1.52

1.48

1.48

1.44

1.44

1.433

1.42

1.419

1.41

1.40

1.40

1.40

<?m

29.91

29.91

29.91

29.91

29.91

29.91

3.79

7.62

12.19

19.10

20.15

3.79

7.62

12.19

19.10

20.15

20.15

19.10

20.15

19.10

20.15

19.10

20.15

20.15

19.10

20.15

19.10

3.79

7.62

12.19

Fm

0.768

.856

.977

1.060

1.081

1.200

. 00935

.0385

.103

.289

.327

.0120

.0500

.136

.422

.494

.552

.487

.629

.585

.744

.683

.944

1.002

.754

1.200

.798

.0160

.0681

.194

ym

1.40

1.39

1.383

1.369

1.20

1.20

1.20

1.10

1.10

1.10

1.08

1.06

1.04

1.028

1.00

1.00

.90

.90

.84

.80

.80

.788

.78

.70

.60

.57

.55

.54

.533

.528

<?m

19.10

19.10

19.10

19.10

3.79

7.62

12.19

3.79

7.62

12.19

12.19

12.19

12.19

12.19

3.79

7.62

3.79

7.62

7.62

3.79

7.62

7.62

7.62

3.79

3.79

3.79

3.79

3.79

3.79

3.79

Fm

0.852

.920

.985

1.200

.0226

.0987

.318

.0277

.125

.479

.542

.634

.809

1.200

.0350

.167

.0460

.245

.347

.0635

.508

.654

1.200

.0967

.175

.231

.301

.367

.494

1.200
158 STAGE-FALL-DISCHARGE RELATIONS

Table 36. Computed falls, cross section 5, 400-foot reach, for selected values of stage

and discharge

ym

3.00

2.88

2.80

2.80

2.72

2.65

2.61

2.59

2.57

2.56

2.55

2.525

2.50

2.50

2.50

2.50

2.50

2.50

2.50

2.50

2.20

2.00

2.00

2.00

2.00

2.00

2.00

2.00

2.00

1.90

1.84

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.80

1.76

1.75

1.74

1.73

1.721

1.704

1.60

1.60

1.60

Cm

31.28

31.28

11.30

31.28

31.28

31.28

31.28

31.28

31.28

31.28

31.28

31.28

.34

.56

.79

1.32

1.84

2.71

3.87

11.30

11.30

.34

.56

.79

1.32

1.84

2.71

3.87

11.30

11.30

11.30

.34

.56

.79

1.32

1.84

2.71

3.87

11.30

11.30

11.30

11.30

11.30

11.30

11.30

.34

.56

riQ

 ' &

Fm

0.306

.370

.0339

.429

.510

.615

.705

.763

.840

.887

.943

1.200

.00008

.00014

. 00028

. 00077

.00154

. 00325

.00680

.0590

.0979

.00015

. 00033

.00063

.00174

.00330

. 00724

.0150

.151

.201

.250

.00021

. 00049

. 00095

.00267

. 00509

.0112

.0232

.299

.378

.407

.444

.493

.565

1.200

. 00033

. nnno UvUo 1?f

.00168

*m

1.60

1.60

1.60

1.60

1.52

1.44

1.40

1.40

1.40

1.40

1.40

1.40

1.40

1.36

1.34

1.32

1.32

1.31

1.30

1.30

1.30

1.298

1.285

1.26

1.22

1.20

1.20

1.20

1.20

1.20

1.20

1.19

1.18

1.17

1.158

1.10

1.10

1.10

1.10

1.10

1.05

1.00

1.00

1.00

1.00

1.00

96

QA « *7^t

cm

1.32

1.84

2.71

3.87

3.87

3.87

.34

.56

.79

1.32

1.84

2.71

3.87

3.87

3.87

2.71

3.87

3.87

1.84

2.71

3.87

3.87

3.87

2.71

2.71

.34

.56

.79

1.32

1.84

2.71

2.71

2.71

2.71

2.71

.34

.56

.79

1.32

1.84

1.84

.34

.56

.79

1.32

1.84

1.84

1. 84

m

0. 00474

.00886

.0196

.0415

.0562

.0820

.00066

.00177

. 00345

. 00986

.0192

.0440

.103

.138

.165

.0685

.207

.239

.0323

.0783

.289

.310

1.200

.106

.155

.00165

. 00459

.00920

.0275

.0592

.197

.228

.271

.343

1.200

. 00270

.00754

.0154

.0486

.119

.186

. 00456

.0129

.0268

.0921

.307

.473

.608

ym

0.93

.92

.914

.905

.90

.90

.90

.90

.80

.80

.80

.80

.75

.73

.71

.70

.70

.70

.70

.694

.687

.60

.60

.60

.54

.50

.50

.50

.48

.47

.461

.456

.44

.40

.40

.38

.37

.36

.354

.350

.30

.28

.27

.26

.252

.250

Qm

1.84

1.84

1.84

1.84

.34

.56

.79

1.32

.34

.56

.79

1.32

1.32

1.32

1.32

.34

.56

.79

1.32

1.32

1.32

.34

.56

.79

.79

.34

.56

.79

.79

.79

.79

.79

.56

.34

.56

.56

.56

.56

.56

.56

.34

.34

.34

.34

.34

.34

F m

0.705

.841

.954

1.200

.00726

.0206

.0437

.166

.0108

.0309

.0669

.292

.418

.496

.628

.0156

.0453

.101

.744

.847

1.200

.0224

.0670

.160

.227

.0329

.104

.302

.376

.441

.578

1.200

.145

.0518

.196

.240

.277

.346

.443

1.200

.102

.127

.149

.191

.266

1.200
SUPPLEMENTARY TABLES 159

C2 t- <N W US^OOO

<N O <N CO OlCOrf

OCOrfW C- Oi N O

. . . US rH 1C O

t-rH 00 00 Tf 1C CO O

Or-lrHCO ^ICCD(N

OOCrH

O3CT5O3CT5 OOOOOOOO

COU5CT5 OrHlrttN

1C U5 CD O O O (N

(M « * O O O CD

OOOO 

1C 1C 1C U5

IN" <N' N <N'

1C rH

00 CD 1C CO OOOO 

OOOOO

(N (N (N (N (N (N (N i

INDEX

Page 

Abstract......................................... 1

Acknowledgments............................ 5

Analysis of data. See Data, analysis of. 

Applications to field problems ............ 142

Barren, Edgar G............................... 2, 5

Bakhmeteff, Boris A......................... 21

Bakhmeteff method........................ 22, 55

Boyer, M. C................................... 84

Boyer diagram............................... 86, 88

three-dimensional......................... 109

Boyer method.................................. 84

Channel, prismatic.......................... 5

Constant-fall ratings......................... 143

Conventions...................................18, 35

Conveyance.................................... 21

Coordinate systems........................... 18

Corbett, D. M....................... 84, 92, 122

Critical depth................................. 10

Data, analysis of, comparison of results... 55

analysis of, conventions and terms... 35

direct method........................... 44

indirect methods, direct-integration

methods................................ 55

step method.............................. 52

necessity for using...................... 51

description of............................... 26

laboratory................................... 25

Depth, critical ............................... 10

uniform...................................... 13

Diagram, stage-fall-discharge............ 81

Discharge, rating.........................101, 122

Eisenlohr, W. S. , Jr................ 5, 122, 134

Elevation of water surface.................. 35

Energy, conservation of..................... 8

loss of........................................ 19

specific...................................... 8

total, of a cross section.................. 6

Equation of flow.............................. 7

Fall, effective................................ 135

rating ....................................101, 122

Fixed-backwater ratings. See Ratings, 

fixed-backwater.

Fixed-backwater stage-discharge

relation...................................... 78

Flow, classification of...................... 14

equation of................................. 7

nonuniform................................. 15

steady........................................ 5

uniform...................................... 11

characteristics of....................... 12

Geometry of the reach. 16

Hardison, C. H............................... 5

Hydraulic radius, computation of........39-43

Page 

Integration of varied-flow equation,

direct integration......................... 22

general.........................................19-24

step method................................... 23

Introduction...................................... 2

Keulegan, G. H ................................ 41

Kindsvater, Carl E............................. 19

Laboratory data................................. 25

Lansford, W. M................................. 2

Manning formula ........................13, 25, 41

Manning's n ................................... 39, 44

Method, Bakhmeteff......................... 22, 55

Boyer........................................... 84

direct........................................... 44

stage-ratio.................................... 92

stefp............................................. 52

submergence.................................. 92

Von Seggern................................ 22, 55

Methods, current, resume of................ 81

direct-integration........................... 55

indirect........................................ 51

proposed....................................... 108

Mitchell, William D.......................... 2

Momentum, conservation of................. 8

Mononobe, Nagaho............................ 2

Morgan, J. H.....,,.............................. 5

Nonuniform flow. 15

Paulsen, C. G................................... 5

Principle of rigidity............................ 79

Prismatic channel.............................. 5

Profiles, by step method, tables of........ 58

computed ..................................... 80

observed....................................... 26

smoothed.................................... 26, 80

Purpose............................................ 2

Radius, hydraulic. See Hydraulic

radius. 

Rating discharge............................101, 122

Rating fall...................................101, 122

Ratings, constant-fall........................ 143

fixed-backwater, analytical

approach................................ 135

conditions for............................. 119

steps in use ................................ 144

unit-fall ...................................... 143

Ratio combinations, results of.............. 103

Ratios, dimensionless, possible

combinations ol............................. 98

Reach, geometry of........................... 16

Relation, fixed-backwater stagedischarge ......'............................ 78

uniform stage-discharge..................14, 81

Relations, stage-fall-discharge, for

laboratory channels......................... 78

161
162 INDEX

Relations Continued

stage-fall-discharge, miscellaneous 

methods..................................

Rigidity, principle of......................

Rocci, Salvatore A.........................

Page

Scope...........................................

Slope, bed....................................

classification of...........................

Stage-discharge relation, uniform ...... 14,

Stage-fall-discharge diagram.............

Stage-fall discharge relations,

miscellaneous methods..................

Stage-fall discharge relations for

laboratory channels......................

Stage-ratio method.........................

Stationing, revisions to....................

Steady flow...................................

Stipp, John R.................................

Subdivision of cross section...............

Submergence method.......................

Terms, use of.

Uniform depth. 

Uniform flow..

Page 

Unit-fall rating................................. 143

University of Illinois, Department of 

Theoretical and Applied 

Mechanics................................. 5

Engineering Experiment Station......... 5

Engineering Experiment Station,

Bulletin 381............................... 2

3, 16, 25, 26, 36, 37, 41, 55, 

58, 78, 79, 81, 109, 137, 139 

Fluid Mechanics and Hydraulics

Laboratory.................................. 5

technique developed at.................... 2

United States Geological Survey,

agreement with University of Illinois.. 2 

United States Geological Survey WaterSupply Paper 888....................84, 92, 122

Velocity-distribution coefficient,

use of.......................................... 7

Velocity-distribution coefficients,

evaluation of................................ 37

Von Seggern, M. E ........................... 22

Von Seggern method......................... 22, 55

Wells, J. V. B................................. 5

Wetted perimeter, computation of........ 41

O