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UNITED STATES DEPARTMENT OF THE INTERIOR 

Harold L. Ickes, Secretary 

GEOLOGICAL SURVEY 

W. C. Mendenhall,-Director 

Professional Paper 189-E 

THE FORCE ·REQUIRED TO MOVE PARTICLES 

ON A STREAM BED 

BY 

WILLIAl\ti W .. RUBEY 

Shorter contributions to general geology, 1937 

(Pages 121-141) 

UNiTED STATES 

GOVERNMENT PRINTING OFFICE 

WASHINGTON : 1938 

For sale by the Superintendent of Documt-nts, Washington, D. C. - - - ·- - - Price 10 cents 
CONTENTS 

Page 

Abstract-------- ----- ------- ------- -- ---------------------- ------- -------------- -------------------------- 121 

Introduction ___ ___ ___ _______ __ __ ______ ~- -------- ------ -------------------------------------------------------- 121 

Physical basis of alternative theories _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 122 

The "sixth-power law" ___ ___ ___ _______ .. ________________ ... _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 122 

The "critical tractive force" _______ __ ________ ___ ___ __ __ __ ___________ _________________________________________ 123 

"Hydraulic lift" caused by velocity gradient_____ _________ ____________________________________________________ 124 

Gilbert's experiments on competence _____ ____ ____ ___________________ ___ ___ __ _____ ___________________________ 7 _ ___ 124 

Grade of debris _______________________________________ _____ _____ ______ _________ ___ _________________________ 124 

Discharge, width, and depth ___________ ______________________ _______ ____ _______________ :..____________________ 124 

Slope of surface_____ _________ __ ____________________ __________________ ________ ____ _________________________ 124 

Hydraulic radius _____ ____ ____ _________________________ _______ -··---_________________________________________ 124 

11ean velocity___ ____ ____________________ __ __ _______ _______________________________________________________ 124 

Turbulence criterion _____ ___ ~ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 125 

"Bed" velocity, shearing stress at channel walls, and thickness of laminar film ___ __ ________ __ _____________ . _ _ _ _ _ _ _ _ 125 

Movement of debris, symbol, and page reference_______ ______________ ____________ __________ ___________________ 125 

Effect of mean velocity and depth-slope product on particle movement _____ _____ ____ __ _______________________ _:_______ 128 

"Bed" velocity__ ___ ________ _________ ____________________________________ ______________________________________ 129 

Coefficient of frictional re istance __ _____________________________ _______ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 130 

Estimation of "bed" velocity and thickness of laminar film__________________________________________________________ 131 

Effect of "bed'' velocity and shearing stress at channel walls on particle movement_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 133 

Departures from the "sixth-power law" for smaller particles------ --------- ---- ---------------------~------------- 134 

Possible application to natural streams _________________________________________________________________ ~ ______ .. __ 136 

Stream competence and stream capacity___ _______ ______ ________________ ______ ______________ __________ ____________ 138 

Summary and conclusions_____ __ __________ ________ _____ _________________________________________________________ 139 

Index ------ ----- ---- ------ -- -- ----- ------------------ -- -- ----------------------------------------- ------- 141 

ILLUSTRATIONS 

Page 

FIGURE 16. Impact of moving water against particle on a stream bed____________________________________________ ____ 122 

17. The "critical tractive force" ... proportional to the depth-slope product______________________ ___________ 123 

18. Typical vertical velocity curve of a stream______________________________________________________________ 124 

19. 11ean velocities, depth-slope products, and movement of debris of grade G___________ ______________________ 128 

20. 11ean vel'ocities and depth-slope products at which debris of grades B, D, E, F, G, and H begins to move______ 128 

21. Roughness ratios, inverse measures of "bed'' velocity and coefficient of resistance, and movement of debris of grade G _ 131 

22. "Bed" velocities, shearing stresses, and movement of debris of grades E, G, and H _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 132 

23. Portion of Hjulstrom's diagram of the fields of erosion, transportation, and sedimentation for well-sorted · sediment________ __ ___ _____ ____ ____ ________ ________ __ ___________________ __ ____________ __ ________ __ 133 

24. Dependence of coefficient in impact equation upon the particle-laminar film ratio__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 134 

25. Idealized velocity gradient very near :;t stream bed _____ _____ ·------------------ --··------- - --------··------ 135 

26. 11echanical composition cf samples collected in .1S81 from the bed of the Mississippi River at Hannibal, Mo_ _ _ 136 

II 
THE FORCE REQUIRED TO MOVE PARTICLES ON A STREAM BED 

By WILLIAM W. RuBEY 

.ABSTRACT 

The movement of particles on a stream bed has been explained 

in several ways-by the impact or momentum of water against 

the particle, by frictional drag upon its surface, and by differences 

in pressure at its top and bottom caused by the gradient of velocities. The familiar textbook law, that the weight of the largest 

particles moved by a stream varies as the sixth power of the 

velocity, is based upon the impact theory. Frictional drag is the 

basis of the so-called law of "critical tractive force", which river 

engineers prefer because it gi~es the maximum size of moving 

particles in terms of the readily measurable quantities,·depth of 

water and slope of a stream. The "hydraulic lift", due to differences in pressure above and below, has the support of several 

physicists. 

G. K. Gilbert's laboratory experiments afford data for a test of 

these rival theories. Properly qualified by using "bed" velocities 

near the particle instead of mean velocities of the entire stream, 

the ''sixth-power law" seems valid for coarse sand and gravel but 

not for fine sand and ~lt, which require much higher velocities 

to start movement than are indicated by this law. The evidence 

suggests that the smaller particles are protected by a laminar film 

of low velocity; but further data are needed to show what forces 

finally cause movement of these smaller grains. 

The equations based on laboratory data give reasonable estimates of the maximum size of pebbles moved by certain large 

natural streams. 

INTRODUCTION 

In the study of sedimentary rocks and of the geomorphic work of ancient streams it seems a matter of 

first importance to understand as thoroughly as possible 

the principles that govern the erosion and transportatimi of rock materia.ls by moving water. Some of these 

principles apparently are understood sufficiently for 

purposes of geologic interpretation; others, equally 

fundamental, are subject to considerable differences of 

op1n1on. Several alternative theories of the force 

required to move particles on a stream bed have long 

been in the literature, anQ. each has its adherents today. 

It seems worth while to compare some of these theories. 

with observational data. The more complex and diffi-. 

cult problem of the bottom currents that move coarse 

sand and keep rock surfaces bare at places on the sea 

floor, which has recently attracted attention, will probably not be solved till after that of movement on stream. 

beds is better understood. 

The so-called "sixth-power law"/ announced more 

than a century ago and since then rephrased in several 

1 Leslie, John, Elements of natural philosophy, pp. 426-427, 1829; cited by Gilbert, 

G. K., The transportation of debris by running water: U. S. Geol. Survey Prof. 

Paper 86, p . 16, 1914. Hopkins, William, On the transport of erratic blocks: Cambridge Philos. Soc. Trans., vol. 8, pp. 225-233, 1844. Shelford, W., On rivers flowing 

into tideless seas, illustrated by the River Tiber: Inst. Civil Eng. Proc., vd. 82, 

p. 25-26, (discussion by Wilfred Airy), 1885. Law, Henry, idem, pp. 29-31. 

forms, states that the weight or volume of the largest 

particles that can be moved by a stream varies as the 

sixth power of the stream velocity. I~ the development 

of geologic thought the formulation of this relationship 

has served effectively to focus attention upon the essential fact that the transporting and erosive power of a 

stream increases tremendously with increase of velocity. 

But beyond this, the classic "law" has been of very 

little practical use, because it refers not to the surface 

or the maximum or the average velocity of a stream, any 

one of which can be measured readily enough, but to the 

velocity very near a stream bed, and "bed" velocities are 

almost impossible to measure accurately. 

An alternative theory that has been found useful for 

estimating the sizes of particles actually moved by 

running water is in wide use today by river engineers 

and laboratory workers. According to this alternative 

law of "critical tractive force", force d'entrainement, or 

current drag, as it is variously called, 2 the diameter of 

the largest particle moved by a stream varies as the 

depth of water times slope of stream. 

A third theory is that the dislodgment of particles 

from a stream bed depends not upon the velocity nor 

the drag force but upon the lift induced by the velocity 

gradient or the rate of shear between adjacent fluid 

filaments. 3 

Actually the three theories are not so completely different as they may at first appear. Their predicted 

results are in rough qualitative agreement, because of 

the fact that the square of the mean velocity of a stream 

and also the velocity gradient near the stream bed tend 

to vary approximately as the depth times slope. Yet 

this is far from saying that the three theories are identical or can be used interchangeably to explain the 

familiar facts of stream behavior. The magnitude of 

2 Du Buat-Nanc;ay, L. G., Princip~s d'hydraulique, vol. 1, pp. 98-105, vol. 2, pp. 

91-98, Paris, 1786. Du Boys, P., Le Rhone et les rivieres a lit affouillable: Annales 

des ponts et chaussees, 5th ser., vol. 18, pp. 150-155, 1879. ·winkel, Richard, The 

limits of transferability of experimental results and model similarity in river hydraulic 

experiments, in Hydraulic laboratory practice, pp. 57-58, 1929. Marzolo, Francesco, 

Some considerations regarding hydraulic models: Idem, p. 755. Kramer, Hans, 

Modellgeschiebe und Schleppkraft: Mitt. preuss. Versuch. Wasserbau und ' 

Schiffbau, Heft 9, pp. 33-38, 1932; Sand mixtures and sand movement in fluvial 

models: Arn. Soc. Civil Eng. Trans., vol. 100, pp. 798-838 (with discussion, pp. 839-

878), 1935. 

s Hooker, E. H ., The suspension of solids in flowing water: Am. Soc. Civil Eng. 

Trans., vol. 36, pp. 246-251, 264, 1896. Jeffreys, Harold, On the transport of sediment 

by streams: Cambridge Philos. Soc. Proc., vel. '25, pp. 27'2-276, 1929. Havelock, 

T. H., The vertical force on a cylinder submerged in a uniform stream: Royal Soc. 

London Proc., ser. A, vol. 122, pp. 387-393, 1929. Richardson, E. G., The transport 

of silt by a stream: Philos. Mag., ser. 7, vol. 17, pp. 769-783, 1934. Hjulstrom, Filip, 

Studies of the morphological activity of rivers as illustrated by the River Fyris: Geol. 

Inst. Upsala Bull., vol. 25, pp. 267-270, 1935. 

121 
122 SHORTER CON'l'RIBUTIONS 'TO GlDNERAL GlDOLOGY, 1937 

the forces basic to the three theories differs greatly, and 

so also does their degree of dependence upon other factors in stream regimen that are known to vary widely. 

Hence a series of observations that extends over a wide 

range of velocities, depths, slopes, and grain sizes may 

reasonably be expected to indicate the conditions under 

which certain theories fit the facts better than others. 

Ideally, in order to afford a thoroughly satisfactory 

test, a series of observations on competent velocity or 

"critical tractive force" should be sufficiently comprehensive to represent adequately the endless variety and 

complexity of natural streams. Practically, however, 

that is out of the question, because data of such comprehensiveness are not now available and may never be. 

Numerous observations have been made, either in the 

laboratory or under essentially laboratory conditions, 

of the competent velocity or "critical tractive force" required to move particles of different sizes; but very few 

quantitative data of this sort have been gathered from 

natural streams. Still, if the objective is an analysis 

or the better understanding of some particular process 

rather than a synthesis of the combined effects of many 

different processes, the data from the laboratory may be 

more useful than data from natural streams, because 

under controlled conditions some at least of the many 

extraneous and confusing factors of nature are thereby 

eliminated or held constant. The behavior in natural 

streams, of course, remains the final criterion by which 

conclusions founded on the relatively simple conditions 

of the laboratory must ultimately be tested. 

--------------v--------------

1-------------- 8 7r R2 v -----------1 

had the investigation included still smaller debris. 

Nevertheless, no other set of data known to me compares with Gilbert's as a basis for at least preliminary 

testing of some of the conflicting theories of the force 

that moves particles on a stream bed. 

The results of this inquiry have not been as clear-cut 

as was hoped, yet the need for certain additional data 

stands out distinctly, and some conclusions of general 

interest are definitely indicated. If this analysis of the. 

problem should stimulate or provoke additional experimentation or a more critical examination of the available data, it will have accomplished one of its principal 

purposes. 

The substance of this paper was presented orally 

before the Geological Society of Washington 5 April 10, 

1935. As a result of generous discussion at that time 

and suggestions from many of my colleagues of the 

United States Geological Survey since then, the paper 

has been modified materially and expanded in its sr.ope. 

I am particularly grateful to James Gilluly, W. H. 

Bradley, P. G. Nutting, R. W. Davenport, and C. H. 

Pierce for their helpful criticism. 

PHYSICAL BASIS OF ALTERNATIVE THEORIES 

The "sixth-power law."-The essentials of the theorY 

that underlies various statements of the familiar "sixthpower law" may be given rather. briefly. 

Movement of a particle on a stream ·bed starts when 

the component of the force of water tending to lift the 

particle up a slope angle, a, becomes equal to the oppositely directed component of weight of the 

FIGURE 16.-lmpact of moving water against particle on a stream bed. Schematic diagram to 

particle. The dynamic pressure of the water or 

the moving force is taken as equal to the momentum or the rnass times velocity of the column 

of water that strikes the particle in a unit of time. 

This mass of moving water in any unit of time 

is the product of its density (pF), its velocity (v), 

and its area of cross section. The column of 

water that strikes a spherical particle has a crosssectional area of fhrR2, where (} is an empirical 

coefficient discussed below and R is the radius of 

illustrate volumes of particle and of water cylinder which causes movement. (See text.) , 

Of the available data, the results of the comprehensive laboratory investigations by G. K. Gilbert 4 seem 

particularly worthy of critical study. Gilbert's data 

apply to very closely sorted sands and gravels, thus 

simplifying interpretation and facilitating direct comparisons, which are exceedingly difficult for poorly 

sorted materials. Furthermore, the sizes used by him 

ranged all the way from diameters greater than 5 millimeters down into the critical field of diameters less than 

7~ millimeter. It is true that Gilbert's experiments 

were chiefly measurements of stream capacity, or the 

quantity of debris .transported, but numerous measurements were also made of stream competence, or the size 

of debris moved. Furthermore, his experiments on 

competence would have been much more significant 

the particle. The force of the water is thus represented 

by (prV·07rR2

) v (see fig. 16), and the component of this 

force acting up slope angle a is (prv·O·JrR2) v cos a. 

'The weight or the mass times 'acceleration of a 

spherical particle immersed in fluid is %1rR3(p8 - PF) g, 

where Psis the density of the particle and g is the acceleration due to gravity. The component of this weight 

that must be overcome by the force of moving w:ater is 

given approximately 6 by the sine of the angle a up 

· which the particle must be lifted. When this angle is 

very small-that is, when the particle rests on a smooth 

flat surface--the force required to start movement is 

obviously very small. 

s Rubey, W. W., The force required to move particles on a stream bed: Washington 

Acad. Sci. Jour., vol. 25, pp. 571-572 [abstract], 1935. 

' Gilbert, G. K., The transportation of debris by running water; U. s. Geol. Survey 6 Approximately because the general slope of the stream bed is relatively small and 

Prof. Paper 86, 1914. therefore neglected. 
'l'HE FORCE REQUIRED TO MOVE PAE,TICLES ON A STREAM BED 123 

When the component of current force is less than the 

component of particle weight, _ there is no movement of 

the particle. When the component of current force 

exceeds the component of particle weight, the particle 

moves, and the dynamics of the problem is then complicated by the velocity acquired by the particle and 

by the moment of force which acts upon it. But when 

the component of current force exactly equals the component of particle weight-that is, when the particle 

balances on the . verge . of moving-the physics of the 

problem is much simpler. When these two forces are 

equal, 

or (pF·v·On-R2)v=%-rr-R3 (ps-PF)g· tan a . 2 

R=% ·t_y_ _-.!!!:__ .~ ____________ (1) 

an a Ps-PF g 

so that 

If g is taken in c. g. s. units, then R is measured in em 

and v in em/sec. 

The empirical coefficient e requires some explanation. 

Its value depends partly upon the proportion of the 

particle that is exposed to the current and partly upon 

the proportion of the force of the directly impinging 

water that is actually expended on the particle. For 

example, if a sphere is shielded by adjacent particles so 

that only one-half of its cross-sectional area is exposed 

to the current . and if only one-fourth 0f tlu~ force of 

the impinging current 7 is expended on the sphere, then 

0= }~X X= Ys. A priori deduction will not give reliable estimates of the value of this coeffiGient,8 but, for 

a given degree of sorting and a given shape of particle, 

the coefficient probably remains statistically constant 

and thus can be determined empirically. 

The angle a, up which the particle must be lifted, 

depends ·upon the shape and the relative size of adjacent particles and thus it also is probably statistically 

constant for a given degree of sorting and a given 

shape of particles. 

For water of known density flowing over a sand 

mixture of given roundness, sorting, a.nd density, e, 

a, PF, Ps, and g may all be taken as statistically constant. Hence it follows from equation 1 that, other 

things being equal, the radius of particles barely moved 

by a stream varies as the square of the current velocity. 

This becomes the familiar "sixth-power law" when 

both sides of the equation are cubed. Thus 

R3 ocv6 

that is, the weight or volume of the largest particles 

moved varies as the sixth power of the velocity of the 

stream. On page 129 a sharper definition is attempted 

of just which particular velocity of the stream is meant. 

The coefficient [% · ta! a]' which depends partly upon 

7 For a discussion that indicates something of the uncertainties in this estiml'lte, 

see Rubey, W. W., Settling velocities of gravel, sand, and silt particles: Am. Jour. 

Sci., 5th ser., vol. 25, pp. 327-329, 1933. 8 Tho·1Iet, M. J., Experiences relatives a la vitesse des courants d'eau ou d'air 

susceptibles de maintenir en suspension des grains mineraux: Annales des mines, 

8th ser., vol. 5, pp. 523-524, pl. 16, fig. 14, 1884. 

the shapes and relative sizes of the different particles 

making up the stream bed and partly upon other 

'factors, is discussed more fully on pages 134-135. 

FIGURE 17.-In the "critical tractive force" theory, the drag on unit areas of a stream 

. bed is taken as proportional to the depth-slope product. (See text.) . 

The "critical tractive force."-A column of water 

exerts a force or weight upon unit areas of a stream 

bed proportional to its density (pF) and its depth (d). 

The down-slope component of this weight of water per 

unit area is taken to be the tractive force that tends 

to drag along downstream the particles that lie on a 

stream bed. Thus the unit tra<"tive force is equal to 

PF·g·d· sin {3= pF·g·d·tan f3· cos {3=pp·g·d·S· cos {3, where 

{3 is the angle of the general stream slope and s is 

tan {3 or the fall per unit distance. (See fig. 17.) For 

slope angles less than about 10°, the term cos {3 is so 

nearly unity that, by comparison with the accuracy 

of other measurements, it may be neglected. 

The surface area of a spherical particle on which 

this tractive force is exerted may be taken as 1/;1rR2

, 

where 1/; is a coefficient that depends upon how nearly 

completely the particle is exposed. According to this 

theory, a particle will begin to move when the component of unit tractive force times the exposed area of 

the particle e.quals the component of particle weight: 

prg·d·S· cos a·l/;1rR2=% 1rR3 (ps-PF)g· sin a 

or 

so that 

R=%· -t 1/1 ·~·d·s ________________ (2) · an a Ps-PF 

Inasmuch as 1/;, a, PF, and Ps may all be taken as 

statistically constant for a particular sand mixture, it 

follows that the radius of the largest particles moved 

varies as the product depth times slope. 

Here, as in the "sixth-power law", the coefficient 

[% · ta~ a] depends upon the shapes and relative sizes 

of the different particles making up the stream bed; 

and therefore, for any given degree of roundness and of 

sorting, the term should have a characteristic value. 
124 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY, 1937 

"Hydraulic lijt" caused by velocity gradient.-It is well 

known that the velocity of flow is not the same at all 

points in a vertical cross section of a stream. The point 

of maximum velocity commonly lies some distance 

below the watersurface, and from this point or poin'ts 

the velocity decreases toward the bed, the side wails, 

and the surface, the change per unit distance being 

relatively gradual until the bed and walls are ~pproachedclosely, and there the velocity gradient becomes 

very steep (fig. 18). The mean velocity of a stream 

(defined nwre explicitly below) is simply the average 

of all the many different velocities in a cross section. 

0:: 

w 

~ 

~ 

l:L 

0 

I 

....,. 

0.. 

w 

0 

Surface of stream 

\ 

\ 

I 

} 

I 

I ' / ------------:/7 Stream bed 

VELOCITIES 

FIGURE 18.-Typical vertical velocity curve of a stream. Steep velocity gradient 

near stream bed and parabolic distribution of velocities above. · 

A particle resting on the bed of a stream lies in the 

zone where the velocity gradient is steepest. At the 

base of the particle the velocity is zero; at its· top the 

velocity may be very high. Because of the diminution of pressure with increase of velocity, a pressure 

difference is set up which tends to lift the particle. 9 

If this lifting force exceeds the particle weight, . the 

particle is lifted off the stream bed and carried· up to 

a position in the stream where the velocity gradient is 

gentler. . 

Several writers have attempted to evaluate the 

effects of this force on bodies of various geometric forms, 

but the problem is complex, and no simple solution 

directly applicable to debris particles appears yet to 

have been found. 

GILBERT'S EXPERIMENTS ON COMPETENCE 

Gilbert's apparatus and methods of laboratory procedure are fully described in Professional Paper 86. 10 

The results of all his experiments on competence 11 are 

given in table 1, recalculated from feet to centimeters 

and accompanied by other pertinent data derived by 

computation from the fundamental observations. Computations were carried to the liml.t of reading on a 

20-inch slide rule but are here given to only three 

9 Jeffreys, Harold, op. cit., p. 272. Hjulstrom, Filip, op. cit., pp. 267-268. 

1o Gilbert, G. K., op. cit., pp. 17-26, 68-69. 

II The much more numerous experiments on capacity are not given here. 

significant figures. The various columns in the table 

require some explanation. 

Grade of debris.-The materials used were moderately 

well-rounded river sand and fine gravel (mean density, 

2.69) that had been closely sized by sieving. Photographs 12 of each size separate, or grade, show the degree 

of rounding. The maxima and minima radii given in 

table 1 are taken from the sieve openings recorded in 

unpublished notes of Gilbert and his assistant, E. C. 

Murphy, in the files of the United States Geological 

Survey. The mean radius given in the table is simply 

the geometric mean of the maximum and minimum 

radii. For well-sorted aggregates the geometric mean 

appears to be a simpler and more directly comparable 

average than the relatively complex number averages 

given by Gilbert, 13 which depend largely upon particle 

shapes and are greatly influenced by minor quantities 

of smaller particles. 14 

Discharge, width, and depth.-The quantity or 

volume of water passing through any cross section of a 

stream in a unit of time, such as 1 second, is called the 

discharge. In a trough of rectangular cross section, 

such as Gilbert used, the discharge equals the product, 

width times depth times mean velocity of flow 

(Q=wdv.\1 ). The discharges from a constant-level 

tank were measured with a calibrated valve. 

The depths given in table 1 were gage measurements. 

A few depths were also determined by the method of 

profiles/5 but as the differences from gage measurements were slight they have been omitted here. 

Slope of surface.-Most of the measurements of 

stream slope or gradient or fall per unit distance were 

made on. the stream bed. For relatively steep slopes 

this measurement was sufficiently accurate; but for 

very gentle slopes it was found inaccurate, and consequently most of the gentler slopes were measured 

on the water surface instead, or on both water surface 

and bed surface. Where both slopes were measured, . 

the two results have been averaged in preparing table 

1. The surfaces measured in each experiment are 

indicated by the words "bed" and ."water." 

Hydraulic radius.-The hydraulic radius of a stream 

is defined as the area of cross section, wd, divided by the 

length of the wetted perimeter, P. In a wide, shallow 

stream in which the height of the channel side walls is · 

negligible in comparison with the channel width, the 

hydraulic radius is essentially equal to the mean depth. 

In a trough of rectangular cross section the hydraulic 

d

. wd 

ra rus r= w+2i 

Mean velocity.-The values of mean velocity given 

in table 1 are calculated from the records of discharge, 

12 Gilbert, G. K., op. cit., pl. 2. 

13 Idem, p. 21. 

14 See Perrott, G. St. J., and Kinney, S. P., The meaning and measurement of 

average particle size: Am. Ceramic Soc. Jour., vol. 6, pp. 417-439, 1923, and Weigel, 

W. M., Size and character of grains of nonmetallic mineral fillers: U.S. Bur. Mines 

Tech. Paper 296, pp. 18-27, 1924. 

15 Gilbert, G. K., op. cit., pp. 25-26. 
THE FORCE REQUIRED TO MOVE PARTICLES ON A STREAM BED 125 

width, and depth by the defining equation vM= ~i 

Turbulence criterion.-At low velocities and through 

small openings a fluid moves by a smooth gliding of 

filaments past one another that is called laminar, 

viscous, or stream-line flow. At higher velocities and 

through larger openings. the motion of a fluid becomes 

irregular and distinctly eddying and is then called 

turbulent flow. Abundant investigation has shown 

that the transition from laminar to turbulent flow 

coincides with a change in the frictional resistance, 

which in laminar flow varies as the first power of the 

velocity and in turbulent flow varies essentially as the 

square of the velocity .

16 The flow through pipes and 

channels and past obstructions of various form changes 

rather abruptly from la:q1inar to turbulent at some fairly 

definite value of the dimensionless Reynolds number 

PFvMl (where PF=density of the fluid; vM=mean velocP. -

ity; l=some linear dimension of the channel-for 

example, the pipe diameter; and JL=coefficient of viscosity). This critical -value of the Reynolds number 

depends, of course, upon the particular linear dimension 

that is chosen; for similar geometric forms a corresponding dimension must be -used. 

The critical Reynolds number at which the flow 

changes from laminar to turbulent in open channels of 

rectangular cross section has been found 17 to be around 

1,400 to 2,000, in which l=r, the hydraulic radius. 

This means that for water with a density of 1.0 and a 

viscosity coefficient of 0.01 (corresponding to a temperature of 20° C.), the flow would be laminar if the 

product, mean velocity times hydraulic radius, were 

less than about 14 to 20 and turbulent if this product 

exceeded 14 to 20. 

It appears to be an open question whether or not 

sediment is ever transported during laminar flow. If 

debris transportation is caused solely by turbulence or 

"mass mixing"/8 then even a slight movement of 

u Stanton, T. E., Friction, pp. 24--47, London, 1923. Prandtl, L., in Ewald, 

P. P ., Posch!, T ., and Prandtl, L., The physics of solids and fluids, pp. 269-283, 289-290, 

297-302,314-319,1930. Dryden, H.-L., Murnaghan, F. D., and Bateman, H., Hydro· 

dynamics: Nat. Research Council Bull. 84, pp. 3- 11, 2D-24, 89-96, 335-339, 385-399, 

1932. 

17 Allen, J., Stream-line and turbulent flow in open channels: Philos. Mag., 7th ser., 

vol. 17, pp. 1094--1096, 1934. Falkner, F . H., Studies of river-bed materials and their 

movement, with special reference to the lower Mississippi River: U. S. Waterways 

Exper. Sta. Paper 17, p. 53, 1935. See, however, Horton, R. E., Leach, H. R., and 

Van Vliet, R., Laminar sheet flow: Am. Geophys. Union Trans. 15th Ann. Meet· 

ing, pt. 2, pp. 393-404, 1934. 

1s Akerblom, F ., Recherches sur les courants les plus bas de !'atmosphere au-dessus 

de Paris: R. Soc. Sci. Upsaliensis, Nova Acta, ser. 4, vol. 2, no. 2, pp. l-45, 1908. Taylor, G. I., Eddy motion in the atmosphere: Royal Soc. London Trans., ser. A, vol. 

215, pp. 1-26, 1915. Schmidt, Wilhelm, Der Massenaustausch bei der ungeordneten 

Stromung in freier Luft und seine Folgen: K. Akad. Wiss. Wien, Math.-nat. Kl., 

Sitzungsber., pt. 2a, vol. 126, pp. 757-804, 1917; Der Massenaustausch in freier Luft 

und verwandte Erscheinungen: Problema der kosmischen Physik, vol. 7, pp. 1-118, 

Hamburg, 1925. Leighly, J. B., Toward a theory of the morphologic significance of 

turbulence in the flow of water in streams: California Univ. Pubs. in Geography, 

vol. 6, pp. 1-22, 1932; Turbulence and the transportation of rock debris by streams: 

Geog. Rev., vol. 24, pp. 453-464, 1934. O'Brien, M . P., Review of the theory of turbulent flow and its relation to sediinent transportation: Am. Geophys. Union Trans. 

14th Ann. Meeting, pp. 487-491, 1933. Christiansen, J. E ., Distribution of silt in open 

chawels: Am. GeJphys. Union Trans. 16th Ann. Meeting, pp. 478-485, 1935. Hjul-

~trom, Filip, op. cit., pp. 270-291. 

debris would be evidence of turbulence, and, as son1e 

debris was moved during one of Gilbert's experiments 

in which the product vMr was only 9.0, the turbulence 

criterion should be less than 9 instead of 14 or 20. 

However, if other processes, such as rolling on the stream 

bed, saltation, or "hydraulic lift", are effective in 

debris movement, Hl it is possible that particles could be 

moved almost as readily by laminar flow as by turbulent flow. In fact, the "hydraulic lift" should be more 

effective in laminar flow, because the velocity gradient is 

steeper. 20 Until the possibility of transportation during 

laminar flow has been disproved by experimental 

studies, it appears unwise to accept transportation as 

an independent criterion of turbulence. 

W.nether or not transportation is possible during 

laminar flow, the flow of nearly all natural streams is 

turbulent; laminar flow of water in nature is thought 

to be virtually restricted to ground-water movement 

and the flow of thin surface sheets. For this reason, 

laboratory investigations of stream processes are usually _ designed to assure turbulent flow, either directly 

by observation of the movement of dyes or indirectly 

by recording for each experiment the Reynolds number 

or some comparable criterion of turbulence. Gilbert 

made no direct observations on turbulence, and, as he 

did not record temperatures of the water with which his 

experiments were made, exact values of Reynolds 

numbers cannot be calculated. However, it seems 

reasonable to take the product vMr as a turbulence 

criterion and to assume tentatively that a value between 

14 and 20, say about 17, marks approximately the 

transition from laminar to turbulent flow. By this 

criterion the flow was turbulent in all but 6 of Gilbert's 

105 experiments on competence. 

"Bed" velocity, shearing stress at channel walls, and 

thickness of laminar film.-The significance of these three 

quantities and the methods by which they have here 

been estimated are discussed on pages 130, 132, and 133. 

lvfovement of debris, symbol, and page reference.-

The notes on movement of debris are quoted from 

Professional Paper 86. 

The symbols assigned to each experiment in table 1 

are intended to summarize the information on movement of debris. The hollow circle ( 0) signifies no 

movement of debris. The cross (X) indicates very 

few, few, or occasional grains moving or dunes forming. 

The solid circle (e) indicates some, several, or many . . grains moving. _ 

The page reference gives the page in Professional 

Paper 86 from which the data for that experiment are 

taken. 

I9 Gilbert, G. K., op. cit., pp. 26-34, 223-233. Richardson, E. G., op. cit., pp. 770, 

777-779. Hjulstrom, Filip, op. cit., pp. 268-270. 

20 Prandtl, L ., op. cit., p. 281. Hjulstrom, Filip, op. cit., p. 331 . 
Grade of debris 

Discharge Width Depth R1dius (em) (Q) (w) (d) 

(cm3jsec) (ern) (em) 

Mini- Maxi- Mean mum mum 

- ---------------

B 0. 0155 0. 0200 0. 0176 20,800 30.5 22.4 

10,300 30.5 14.9 10,300 30.5 14. 5 10, 300 30.5 14. 5 10,300 30.5 13. 0 

c . 0200 . 0290 . 0241 10,300 30.5 13. 7 10,300 30.5 11.9 

D . 0290 . 0450 . 0361 10,300 30.5 12.7 10,300 30.5 11. 9 10,300 30.5 11. 4 10,300 30.5 10.9 10,300 30.5 14.2 

10,300 30.5 12. 7 20,800 30.5 22.6 20,800 30.5 20.1 

E . 0450 .105 . 0688 538 40.2 . 427 

538 40.2 . 549 283 30.5 . 457 283 20.1 . 670 538 40.2 . 457 

538 30.5 . 762 10,300 30.5 11.8 1,100 30.5 1. 61 10,300 30.5 10.9 20,800 30.5 18.8 10,300 30.5 10.9 538 20.1 1. 28 10,300 30.5 10. 1 20,800 30.5 18.8 20,800 30.5 17. 1 1, 640 30.5 1. 83 1, 100 4(1. 2 . 792 1,100 40.2 . 732 20,800 30.5 17. 1 10, 300 30.5 9. 57 10, 300 30.5 9. 57 2,120 30.5 1. 98 . 538 20.1 . 853 2, 630 30.5 2.13 1,100 20.1 1. 52 

F [. 135]1 .175 .154 2,120 20.1 2.16 1,100 20.1 1. 31 

2, 630 20.1 2. 80 4,140 30.5 2. 62 

4, 650 30.5 2. 83 

a · . 175 . 250 . 209 10,300 30.5 7. 80 10,300 30.5 7. 25 10,300 30.5 6. 86 5,160 30.5 3. 78 1, 640 20.1 1. 58 6,170 30.5 4. 30 2,120 20.1 2.13 2,630 30.5 1. 86 3,140 30.5 1. 80 4,140 20.1 3. 63 7, 700 40.2 3 .. 51 7,220 30.5 4. 66 2,630 20.1 2.68 

10,300 30. b 6.13 

3,140 30.5 2.04 4,650 20.1 a.84 4, 140 30.5 2.44 

8. 210 30.5 4.85 

TABLE 1.-Gilbert' s experiments on competence 

Calculated Calculated 

Hy- Shearing "bed" 

Mean Turbu- velocity at average 

draulic stress at thickness Slope of surface radius velocity, lence channel boundary of laminar (8) (r) (VM) criterion walls of laminar film 

(em) (em/sec) (VMT) (pp!JTS) film (a) (VB) (em) (em/sec) 

--- ------

0. 00012 Bed _________ _ 9. 07 30.5 276 1. 07 13.7 0. 128 . 00029 Bed+water ___ 7. 54 22.6 170 2.12 13. 9 . 0655 . 00029 Bed+water __ _ 7. 42 23.3 173 2.11 14. 1 . 0667 

. 00037 Bed+water ___ 7. 42 23.3 173 2. 69 15.0 . 0556 

. 00045 Bed+water ___ 7. 00 26.0 182 3. 09 16.3 . 0528 

. 00055 Bed +water. _ 7. 22 24.5 177 3. 89 19.3 . 0496 

. 00071 Bed+water __ 6. 69 28.3 189 4. 65 20.3 . 0436 

. 00037 Water ________ 6. 94 26.5 184 2. 51 15.2 . 0604 

. 00045 Bed+water._ 6. 70 28.2 189 2. 95 16.3 . 0552 

. 00077 Bed+water __ 6. 50 29.7 193 4. 91 18.9 . 0386 . 00078 Bed+water ___ 6. 35 31.0 197 4. 85 19.3 . 0398 . 00115 Bed+water ___ 7. 34 23.8 175 8. 26 19.4 . 0235 

. 0018 Bed+water ___ 6. 94 26.5 184 12.2 22.6 . 0184 . 00175 Bed+water ___ 9. 11 30.1 274 15.6 25.9 . 0166 . 00295 Bed+water ___ 8. 66 34.0 208 25.0 30.9 . 0123 

. 0100 Bed __________ . 418 31.3 13. 1 4. 10 Lami oar flow? . 0100 Bed ____ ___ ___ . 534 24.4 13.0 .5. 23 Laminar flow? . 0200 Bed_--------- . 444 20.3 9. 0 8. 70 Laminar flow? . 0200 Bed __________ . 628 21.0 13.2 12.3 Laminar flow? . 01\}5 Bed.--------- . 447 29.3 13. 1 8. 54 Laminar flow? 

. 0194 Bed __________ . 726 23.2 16.8 13.8 Laminar flow? . 000465 Bed+water ___ 6. 65 28.6 190 3. 03 16. 1 0. 0531 

. 0050 Bed __________ 1. 46 22.5 32. 8 7. 15 16.4 . 0230 

. 00046 Bed+water ___ 6. 35 31.0 197 2. 86 16.4 . 0575 

. 00030 Bed+water ... 8. 42 36.2 305 2. 48 17.4 . 0700 

. 00065 Water ........ 6. 34 31.0 197 4. 04 17.9 . 0445 

. 0113 Bed.--------- 1. 13 20.9 23.7 12.6 18.0 . 0143 

. 00068 Bed+water ___ (i. 06 33.5 203 4. 04 18.6 . 0461 

. 0004 Water ________ 8. 42 36.2 305 3. 30 18. 7 . 0565 

. 0007 Water ________ 8. 06 39.8 321 5. 52 19.0 . 0344 

. 0050 Bed.--------- 1. 63 29.5 48.2 8.00 19. 5 . 0243 

. 0100 Bed __________ . 762 34.7 26.4 7. 47 19.9 . 0267 

. 0100 ~ed. --------- . 706 37.5 26.5 6. 92 20.3 . 0293 

. 00052 Bed+water ___ 8. 06 39.8 321 4.11 20.6 . 0502 

. 00098 Bed+water ... 5. 88 35.2 207 5. 64 20. 7 . 0367 

. 0011 Water ________ 5. 88 35.2 207 6. 34 21.3 . 0337 

. 0050 Bed __________ 1. 75 35.2 61.6 8. 58 2l. 7 . 0253 

. 0200 Bed __________ . 786 31.3 24.6 15. 4 22.8 . 0148 

. 0050 Bed.--------- 1. 87 40.5 75.8 9.17 23.8 . 0259 

. 0113 Bed __________ 1. 32 36.0 47.7 14.7 24.8 . 0169 

. 0105 Bed.--------- 1. 78 48.8 86.9 18.3 29.7 . 0162 

. 0251 Bed __________ 1. 16 42.0 48.6 28.5 30.0 . 0105 . 

. 0105 Bed __________ 2.19 46.7 102 22.6 30.9 . 0137 

. 0100 Bed __________ 2. 24 51.8 116 21.9 32.4 . 0147 

. 0100 Bed __________ 2. 39 53.8 128 23.4 33.7 . 0144 

. 0031 Bed +water ___ 5.16 43.2 223 15. 7 28.0 . 0179 . 00345 Bed+water ___ 4. 91 46.5 229 16.6 29.4 . 0177 

. 0048 Bed+water ___ 4. 73 49.2 233 22.2 32.4 . 0146 

. 0100 Bed _________ _ . 3. 03 44.8 136 29.7 32.5 . 0109 

. 0250 Bed __________ 1. 37 51.6 70.5 33.5 34.4 . 0103 

. 0100 Bed __________ 3. 35 47.1 158 32.8 34.5 . 0105 

. 0205 Bed ....... _ .. 1. 76 49.5 87.2 35.4 34.7 . 00980 

. 0252 Bed .• :. ______ 1. 66 46.5 77.0 40.9 34.7 . 00848 

. 0190 Bed __________ 1. 61 57.4 92.4 30.0 35.6 . 0119 

. 0110 Bed __________ 2. 67 56.7 151 28.7 36.0 • 0125 

. 0105 Bed ________ ~_ 2. 99 54.6 163 30.7 36.2 . 0118 

. 0100 Bed __________ 3. 57 50.8 181 35.0 36.4 .0104 

.0205 Bed __________ 2.12 48.8 103 42.5 36.4 .00856 

. 00675 Bed+water __ 4. 37 55.0 241 28.9 36.4 .0126 

.0252 Bed.--------- 1. 80 50.5 91.0 44.5 37.1 . 00834 

.0110 Bed __________ 2. 78 60.2 167 29.9 37.5 .0125 

.0190 Bed __________ 2.10 55.7 117 39.1 38.1 . 00972 

.0100 Bed._-------- 3.68 55.6 204 36.0 38.4 .0107 

Movement of debris 

Zero load collected. Dunes forming _____________________ 

A very few grains moving. Small dunes forming _______ A very few grains moving. Some dunes forming _______ _ 

Zero load collected. Dunes forming-------------------- Many grains moving. Dunes forming ___________ _______ 

A very few grains moving in a few places ______________ __ Many grains moving. Dunes forming ___ ____________ ___ 

Zero load collected. Dunes forming _____________________ 

A very few grains moving. No dunes forming ______ ____ 

Some small grains moving ___ --------------------------- Severa~ small g!ains moving. No dunes forming ________ No grams movmg _______________________________________ An occasional grain moving ________ _____________________ 

Very few if any grains moving ___ __ --------------------- Many grains moving. Dunes forming_-----------------

No grains moving ___________ ---------------------------- ____ .do .. ____________ --- __ . _______ ----------------------- A few grains moving in a few places. Dunes forming ___ A few grains moving in a few places _____________________ 

Some grains moving; many moving as small channels are formed or surface becomes rough. Many grains moving. Dunes forming-------------~--. 

~ ~ ~~i~~~~~f~~:: = === = = == == := = = = = = = = = = = = = == = = = = = = = = == = No transportation ________ -----.. ------------------------ __ . .. do .. ______________ .•• ____ -------------------.-------

Zero load collected .. ------------------------------------ A few grains moving in a few places .. ------------------- Very few grains moving in lower half of slope _________ ___ Zero load collected. [Dunes] _____________________ _______ Trace collected. [Dunes]. ____________ .... ____________ ._ A few grai_ns movi_ng in a few places _____________________ 

Many grams movmg _____ ------------------------------- Several grains moving. Dunes forming ____ ___ __________ 

Some transportation in lower half of slope _______ ________ 

A few grains moving_----------------------------------· Trace collected . Dunes forming_----------------------- Several grains moving in several places __________________ Several grains moving in nearly all parts of trough. __ ___ Transportation __ ____ •• __ .... _ .. _ .. ------....... -.--.-.-. Many grains moving. Surface becoming rough _________ 

A few grains moving in a few places _____________________ 

An occasional grain moving. Several grains moving 

after a time. Several grains moving in some places ____________________ Few grains moving in some places _______________________ 

Several grains moving _________ ------------.. ----------.-

No grains moving _______ ------------------------ __ . ___ ._ 

____ .do._-----·------------------------------------------- A few ~rains m_oving at one place __________________ ______ No grams movmg _______________________________________ 

____ .do _______ ... __ ...••• --.. -------.. ------------------- No (or .very fe'Y) grains moving_------------------------ No grams movmg __________ ----------------------------- .. __ .do ___ .. ______ -. __ .-----..• -------------------------- ____ .do _______ ---------------------.--------------------- ____ .do. ____ --------------------------------------------- ____ .do ______ _ ----- __ ------------------------------------ Very few grains moving __________________________________ 

____ .do _____________ ------•• __ ...••.•• -•. --------.. ------- Very few if any grains moving __________________________ _ 

Very few !?,rains moving in part of trough ..•. ------------ Very few grains moving _________________________________ 

_____ do. ----- _____________ .. ___ .------------.--.----.. --. 

Many grains moving ______ ------------------------------

Sym_ll 

bol 

X? 

X 

X 

X? 

• 

X 

• 

X? 

X 

• • 0 

X 

X? 

• 

) Not! plot· { 

ted ! 

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0 

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0 

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X 

X 

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6,170 40.2 2. 29 . 0140 Bed_--------- 2. 05 5,160 20.1 4. 24 . 0110 Bed __________ 2. 98 5,160 30.5 2. 71 . 0145 Bed __________ 2. 30 2,120 20.1 1.65 . 0250 Bed __________ 1. 41 9, 260 40.2 3. 81 . 0105 Be<L _________ 3. 20 3,630 30.5 2. 26 . 0252 Bed __ -------- 1. 97 4,140 40.2 1. 68 . 0250 Bed __________ 1. 55 10,300 30.5 5. 61 .OORO Bed+water __ 4.10 6, 710 40.2 2. 41 . 0140 Bed __ -------- 2.15 9, 770 40.2 3. 90 .0105 Bed __________ 3. 27 3, 140 20.1 2. 68 • 0205 Bed _____ _____ 2. 12 20,800 30.5 11.0 . 0048 Bed+water __ 6. 38 5, 660 20.1 4. 36 . 0110 Bed __________ 3.03 2,630 20.1 2.19 .0250 Bed_--------- 1. 80 4, 650 40.2 1. 86 . 0250 Bed_--------- 1. 70 7, 220 40.2 2. 56 .0140 Bed __________ 2. 27 10,300 40.2 3. 96 . 0105 Bed_--------- 3. 31 6,170 20.1 4. 66 . 0110 Bed __________ 3. 18 7, 700 40.2 2. 74 . 0140 Bed __________ 2. 41 5,160 30.5 2. 74 . 0190 Bed __________ 2. 32 7,220 40.2 2. 91) . 0200 Bed __________ 2. 5g 5,160 40.2 2.10 . 0250 Bed __________ 1. 90 6,170 30.5 2. 99 . 0145 Bed __________ 2. 50 6,170 40.2 2. 41 .0200 Bed __________ 2. 15 8,210 40.2 3. 02 . 0140 Bed __________ 2. 62 

10,300 30.5 5. 27 . 00965 Bed+water __ 3. 92 6, 710 40.2 2. 74 . 0200 Bed __________ 2. 41 4, 650 20.1 3. 41 . 0158 Bed __________ 2. 55 4,140 20.1 3. 23 .0205 Bed __________ 2.44 

7, 220 30.5 3. 20 . 0145 Bed __________ 2.114 5, 160 20.1 3.17 . Oi58 Bed __________ 2. 41 

5, 660 20.1 3.17 . 0158 Bed __________ 2. 41 

. 250 [. 370] 1 . 304 5,160 20.1 4.02 . 0205 Bed __________ 2. 87 7, 700 20.1 4. 97 .0130 Bed_--------- 3. 33 15,400 20.1 12.2 .0110 Bed ____ ______ 5. 50 

10,300. 20.1 6.46 . 0110 Bed __________ 3. 94 8, 720 20.1 5. 21 .0130 Bed __________ 3. 43 

6,170 20.1 4. 17 . 0205 Bed __________ 2. 95 12, 900 20.1 7. 92 . 0110 Bed _______ ___ 4. 43 9, 770 20.1 5. 61 . 0130 Bed __________ 3. 60 7,220 20.1 4. 30 . 0205 Bed __________ 3. 01 18, 100 20.1 10.0 . 0110 Bed __________ 5.02 

' Sieve openings not recorded in manuscript notes but plotted in unpublished graphs. 

fi7. 2 138 28.1 60.5 180 32.1 62.4 144 32.7 64.2 90.8 34.6 60.4 193 32.9 52.8 104 48.5 61.4 94.9 37.9 60.2 247 32.1 69.3 149 29.5 62.3 203 33.6 58.3 123 42.5 62.2 397 30.0 64.6 196 32.7 59.7 107 44.1 62.1 106 41.7 70.1 159 31.2 64.5 214 34.0 65.9 210 34.3 69.8 169 33. 1 61.7 143 43.2 60.7 156 50.5 60.9 J1.6 46.6 67.8 169 35.5 63.8 137 42.1 67.6 177 36.0 

63.9 251 37.1 60.8 147 4.7. 3 67.6 172 39.5 63. 6 155 49.1 

74.1 196 37.6 80.8 195 37.3 

88. 8 214 31.3 

6~. 7 183 57.7 77.0 256 42.4 63.1 347 59.3 

79.0 311 42.4 83.2 286 43 8 

73.5 217 59.2 80.7 358 47.8 86.6 312 45.9 83. 6 251 60.4 89.8 451 54.1 

38.4 . 0137 Very few grains moving near center of trough ___________ 38.5 . 0120 A few grains moving near middle of trough ______________ 38.7 . 0118 Very few small grains moving ________ ______________ __ ___ 38.8 . 0112 A few grains moving ____________________________________ 

38.8 .0118 Very few grains moving _________________________________ 38.9 . 00802 A few grains moving except near head of trough _________ 38.9 . 0103 Very few grains moving. __ ___ ___________________________ 

39.0 . 0121 An occasioml grain moving ___________________ ~ ___ : _____ 

39.6 . 0134 None· moving except at middle of trough_--------------- 39.7 . 0118 A few grains moving ____________________________________ 39.8 . 00935 _____ do __ __ _______________________ ____ ____________________ 

39.8 . 0133 A few grains moving in places ___________________________ 40.0 . 0122 

~!~}J~~~~~~~~~~~~============================~====== 40.3 . 00914 40.4 . 009!38 40.5 .C130 Very few moving except at middle of trough _____________ 40.5 .0119 Several grains moving ___________________________________ 40.9 . 0119 Several grains moving in lower half of trough."---------- 41.2 . 0124 A few grains moving ____________________________________ 

41.3 . 00955 Several grains moving ___________________ ---------- ___ --- 41.3 . 00818 -- ___ do ___________________________________________________ 

41..4 . 00888 _____ do ____________ ~ ______________________________________ 

41.4 . 0117 ~. ___ do ________________ --- ________________________________ 

41.5 . 00986 A few grains moving ____________________________________ 

41.6 . 0116 Several moving in middle and lower parts; very few in 

upper. 41.6 . 0112 Several grains; moving in center half of slope _____________ 42.0 . 00888 Several grains mo>•ing in middle part of trough ___ _______ 42.5 . 0108 Some grains moving in lower half of trough _____ __ _______ 43.5 . 00885 Several grains moving in middlP of trough, many in lower 

44.0 . 0117 part. c.utting o,f grade. Many grams rnovmg ____________________________________ 

45.7 . 0122 Several grains moving in lower half of trough but very few in. upper half. · 47.9 . 0128 Many grains moving in lower half but few in upper halL_ 

44.7 . 00774 No grains moving ___________________ " ___________________ 

45.9 . 0108 Occasionally one grain moving __________________________ 46.4 . 00782 Several grains moving in lower two-thirds of trough, and occasionally a grain in upper third. 46.9 . 0111 Occasionally a grain moving ________ . _______________ . ___ ___ 48.1 . 0110 Some grains moving below middle; 1 to 3 grains in a cross section. 48.2 . 00813 A few grains moving in lower three-fourths of trough ____ 49.2 . 0103 A few grains moving in ·lower half d.f trough _____________ 49.8 . 0109 2 to 5 grains moving in a ·cross section_·------------------ 51.9 . 00860 Many grains moving in lower three-fourths of trough ___ _ 53.8 . 00994 Several grains mo:ving in rniqdle third of-trough; a few in 

upper and lower parts. 

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-l 
128 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY, 1937 

.07 I I I . 1 I 1 

EXPLANATION 1- · Grade G • - GRAINS MOVING 

0 None 

.06 r-- X Very few to few • -

• Several to many w X 0.. 

0 - X X • • - _j • 

CJ) X ·~ ~.05- • • - L • XX 0 0 · .. x X X • • z - X - ~ 

. o? 0 

I X • XX • 1- • •• X? >tf . 0...04- 0 0 X X - w X 

0 0 

0 X 1- - 0 X X X 

.031- -

l ,.. I I I I I . 

:40 45 50 55 60 65 70 75 

MEAN VELOCITY (C~ec) 

FIGURE 19.-Mean . velocities, depth-slope products, and movement of debris of grade G. Oblique line separating points of no movement from those of slight 

movement indicates conditions at which movement of particles begins. · 

.10 

.09 

.08 

.07 

g: .06 

0 

_] 

(/) 

X 

~ u.os 

z 

I 

1--

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04 

.03 

.02 

.01 

\ 

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'?F ' 

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20 30 40 50 60 

MtAN YELOCITY(C~ec) 

\ 

\ H 

\ 

\ 

\ ' 

\ 

\ 

\ 

.-

70 BO 

FIGURE :20.-Mean velocities &rid depth-slope products at which debris of grades B, 

D, E, F, G, and H begins to move on a stream bed. Oblique lines separate the 

experiments in which thete_·•was no movement of debris of the indicated grade 

size from experiments in which there was ~li~ht to much movement. 

EFFECT OF MEAN VELOCITY AND DEPTH-SLOPE 

PRODUCT ON PARTICLE MOVEMENT 

Probably the most direct way to illustrate the extent 

to which the "sixth-power" and the "critical tractive 

force" theories agree or fail to agree with observationaJ 

data is to plot mean velocities against depth-slope 

products and to record thereon the observations on 

debris movement. 

Figure 19 shows the data for grade G plotted in tills 

manner. If the critical force required to start particle 

movement depends solely upon the mean velocity, the 

line separating the hollow circles (no movement) from 

the crosses and solid circles (slight and much movement) 

should be vertical. If the critical force depends solely 

upon the depth-slope product, the line should be horizontal. Actually the line is inclined, thus indicating 

that neither mean velocity nor depth-slope product 

alone accounts for the particle -movement. At relatively low velocities the particles start moving if the · 

depth-slope product is sufficiently great. At relatively 

high velocities the particles start moving at a much 

smaller depth-slope product. 

Gilbert's observations on the other grades of debris 

were much less extensive than those on grade G. Nevertheless, the data are sufficient to show the approximate 

position and slope of the lines that separate the symbols 

indicating no movement from those indicating slight 

to much movement. These lines are plotted to the 

same scale in figure 20, the dashed and queried lines 

indicating those based on the fewest observations. 

Several facts are evident from this diagram. As would 

be expected, the larger particles require higher velocities 

or higher depth-slope products before movement starts. 

At a depth-slope product of 0.03, grade F particles 

(diameters 2.7 to 3.5 mm) start moving at a mean 
THE FORCE REQUIRED To :Mov:l!f PARTict:Es 6:N A STREAM :B:Efi i29 

velocity of about 42 em/sec.; whereas at the same depth- From the boundaries toward the center the velocity inslope product grade G particles (diameters 3.5 to 5.0 ·creases, very rapidly at first through a thin marginal 

mm) do not · start moving until the mean velocity layer and then more slowly toward the center (fig. 18). 

reaches 62 cmfsec. At a mean velocity of 48 em/sec., Just which particular velocity in such a continuou·s 

grade F particles start moving at a depth-slope product series of velocities is the one to be designated and 

of about 0.023, but at this same velocity grade G par- measured as the "bed" velocity? 

ticles do not start moving until the depth-slope product Investigation has shown that in the zone of turbulent 

reaches 0.048. flow that makes up the body of a stream the velocity 

Another and perhaps more significant fact is shown gradient is relatively gentle a1id closely follows a parahclearly in figure 20. The lines that separate the sym- ola.22 But in the thin marginal layer or film where 

bols indicating no movement from those indicating :. the flow is dominantly laminar the velocity gradient is 

slight to much movement slope more steeply as the · steep and approximately uniform._23 At the boundary 

size of particles increases. This means that the larger between these two zones there.is a transitional rounding 

particles, such as grade H, start moving at a mean off of the velocity curve, which makes exact measurevelocity that is relatively independent of the depth- ment difficult (fig. 18). Nevertheless, a distinct signifslope product, and the smaller particles, grades B, D, icance attaches to the marginal film of laminar flow 

and E, start moving· at a depth-slope product which is with approximately uniform velocity gradient and to 

relatively independent of the mean velocity. In other the velocity that marks the transitional boundary 

words, the "sixth..:power law" appears to hold more between this film and the interior zone of turbulent flo:W 

nearly for gravel, and the theory of "critical tractive with parabolic distribution of velocities. Inasmuch as 

force" for finer sa.nds. the average velocity at this boundary between the 

The relationships brought out by figures 19 and 20 · laminar and turbulent zones can be estimated 24 from 

seem to show that neither the "sixth-power law" (at least other observational data on a stre-am, it is here tentaas customarily stated in terms of an unspecified velocity) · tively adopted as the partiptilar "bed" velocity w~ch 

nor the theory of "critical tractive force" adequately fits seems most useful for interpretations of particle 

the facts of observation. ·Some sort of compromise· movement. 

between the two theories-one that approximates the . A method of estimating this velocity at the boundary 

"sixth-power law" for larger particles and the theory of of the laminar film may be deduced from general 

"critical tractive force" for smaller particles-would fit considerations. 

the observations better. 

"BED" VELOCITY 

The most obvious criticism that may be raised against 

the preceding test of alternative theories is that it compares particle movement with the mean velocity of the 

entire stream instead of with the velocity in the immediate vicinity of the particle on the stream bed. The· 

criticism is well founded. Although many writers have 

assumed, some implicitly and others explicitly, that 

velocities near a stream bed are proportional to me&n 

velocities of the entire stream, th~re appears to be no 

adequate basis for such an assumption. Other writers, 

recognizing this element of uncertainty, have sought to 

compare particle movement with velocities at the streb.m• 

bed as determined either by actual measurement or by 

correction of surface or mean velocities. 21 . 

But this effort has been hampered by the difficulty' 

of defining precisely what is meant by "bed" velocity. At 

the actual boundaries of a stream the velocity is zero. 

21 For compilations of earlier experimental results on competent mean and "bed'' 

velocities see the following citations: 

Penck, Albrecht, Morphologie der Erdoberflache, vol. 1, pp. 283, 319-341, 1894. 

Grabau, A. W., Principles of stratigraphy, pp. 55-56, 247-251, 1913. Gilbert, G. K., 

op. cit., pp. 163-216, 1914. Schoklitsch, A ., Uber Schleppkraft und Geschiebebewe· 

gung, pp. 22-26, Leipzig and Berlin, Wilhelm Engelmann, 1914. Bucher, W . H., 

On ripples and related sedimentary surface forms and their paleogeographic interpretation: Am. Jour. Sci., 4th ser., vol. 47, p. 151, 1919. Twenhofel, W . H., and 

others, Treatise on sedimentation, pp. 31-32, 464, 1926. Fortier, S., and Scobey, F . 

C., Permissible canal velocities: Am. Soc. Civil Eng. Trans., vol. 89, pp. ·942, 947, 

951-953, 955, 1926. Hjulstrom, Filip, op. cit., p. 295, 1935. 

A stream of water flowing down its bed expends an amount of 

energy that is measured by the quantity of water and the vertical 

. distance through which it desce:O.ds. If there were no friction of 

the water upon its channel, the velocity of the current would 

continually increase; but if, as is the usual case, there is no 

increase of velocity, then the whole of the energy is consumed in 

friction.25 

In a unit of time, t, the energy generated by a t~eam \ 11 

in flowing downslope equals· the mass of water, PFQt, 

times the acceleration due to gravity, g, times the vertical distance of fall, vM s cos {3 t. The energy s:P ~nt 

in overcoming friction in the same unit of time is customarily 26 stated in terms of the mean velocity as the 

area of channel surface, Pv:Mt, times the mean fric~ional 

force per unit area, kMPFVM2

, times the distance through 

which the total frictional force acts, vMt, where kM is 

the coefficient of frictional resistance referred to the 

mean velocity and the other symbols have the same 

significance as before. For any short section of · .. 'ft:: i channel where bends in the &tream course and changes 

22 Stanton, T. E., Friction, pp. 28-31, Lona:On,1923. Merriman, Mansfield, Treatise 

on hydraulics, lOth ed., pp. 321a-322, reprinted 1931. 

23 Prandtl, L., op. cit., pp. 275-276, 281-282. Drydev, H. L., Murnaghan, F. D., 

and Bateman, H., op. cit., pp. 252-253, 342-343, 346, 386. 

u Jeffreys, Harold, The flow of water in an inclined channel of rectangular section: 

Philos. Mag., 6th ser., vol. 49, p. 795, 1925. 

2s Gilbert, G. K., Geology of the Henry Mountains, p. 106, U.S. Geog. and Geol. 

Survey Rocky Mtn. Region, 1877. 

26 See, for example, Merriman, Mansfield, op. cit., pp. 216, 275; Stanton, T. E . 

. op. cit., pp. 83-84; Prandtl, L ., op. cit., pp. 297-299; Dryden, H. L ., Murnaghan, 

F. D ., and Bateman, H., op. cit., p. 438. 
130 SHbRTER CbNTRlBUTlONS TO GENERAL GEOLOGY, 1937 

of mean velocity are relatively small, 7 these energies 

balance: 

Energy supplied=Energy consumed 

(ppQt)g(VMS. COS {3t) = (PvMt) (kMPFVM2

) (vMt) __ - (3) 

Simplifying, to obtain _ the mean frictional force per 

unit area, 

k 2 _pFQgvMs. cos {3 ( ) MPFVM- PvM2 PFgrs. cos !3----- 4 

Furthermore, this frictional force per unit area is the 

shearing stress at the channel walls, 28 or, in other 

words, the velocity gradient, vB/o, in the laminar film 

multiplied by the coefficient of viscosity, p,. That is, 

VB . ~-'8=pFgrs. cos !3---------------- (5) 

where o =thickness of laminar film and 

vB="bed" velocity at boundary of laminar film. 

Here, as above, the term cos {3 is nearly unity for 

slope angles less than 10°, and the shearing stress at 

the channel walls in all Gilbert's experiments is therefore measured closely by PFgrs. Values of this quantity given in table 1 were calculated by taking PF= 

1.00, and g=980 cmfsec2

• 

In the derivation of equation 3, the energy consumed in friction was stated in terms of the mean 

velocity. It may, however, by analogous reasoning 

be stated in terms of the "bed" velocity as the area of 

channel surface, PvBt, times the "bed" frictional force 

per unit area, kBPFVB2, times the distance through which 

the total frictional force acts, vBt, where kB is the 

analogous coefficient of frictional resistance referred 

to the "bed" velocity. Equating the energy supplied to 

tlie energy consumed, as above, . (pFQt)g(VMS. COS {3t)=(PvBt)(kBPFVB2)(vBt) _______ (6) 

Simplifying, to determine the "bed" velocity, and for 

slope angles less than 10°, 

to show that the larger particles, but probably not the 

smaller particles, start moving when some critical "bed" 

velocity VB is reached. 

From equation 7, 

and thus it is evident that for any particular "bed" 

velocity 

..jkBcx.vM.Jrs __________________ (8) 

COEFFICIENT OF FRICTIONAL RESISTANCE 

The coefficient of "bed" resistance, kB, is by definition 

analogous to the coefficient of mean resistance, kM, of 

which numerous studies have been made to determine 

the controlling factors. The principal results of these 

studies may be summarized briefly. 

At low values of the Reynolds number, the observed 

kM agrees closely 29 with the theoretical kM computed 

, from formulas for laminar flow-that is, kM varies 

inversely as the Reynolds number, PFVMr. At high vall-' 

ues of the Reynolds number, under conditions of turbu-

. lent flow, kM is more nearly constant. A great many 

observations on turbulent flow of water, oil, and air 

through smooth pipes of different sizes 30 indicate that 

However, under conditions· of turbulent flow In 

rough pipes, kM is essentially independent of the Rey1 nolds number and depends almost solely on a roughness 

ratio (the mean height of irregularities in the pipe 

divided by the pipe radius). The relationship has been 

found to be approximately I .)' iJ 1 Jl 

I l 

QgvMs. cos {3 _ g 2 where e=mean height of irregularities. By substituPkB -kBvM rs:.. ___________ C7) ; tion in equation 4 and putting cos {3=1, this relationship 

~nasmuch as all terms except vB and kB are known for 

Gilbert's experiments, the "bed" velocities are determinable if kB, the coefficient of "bed" resistance, can be 

evaluated. 

By this theoretical derivation, VB varies with both 

vM and rs. It is significant to recall in this connection 

that the data plotted in figures 19 and 20 indicate 

empirically that particle movement depends in a somewhat similar manner upon mean velocity and the 

depth-slope product. ·That is, the empirical data seem 

~7 For a more complete statement of this derivation, see Rubey, W. W., Equilib· 

rium conditions in debris-laden streams: Am. Geophys. Union Trans. 14th Ann. 

~eating, pp_. 497-502, 1933. . 

2s Jeffreys, Harold, op. cit. (1925), p. 795. Dryden, H. L., Murnaghan, F. D., and 

· ~ateman, H., op. cit., p. 3&6. Lindquist, E.· a. W., On velocity formulas for open 

cpannels and pipes: lug. vetensk. akad. Hand!., nr. 130, p. 43, 1934. 

· gives 

[ r)~ ]>1 vMcx. g -; rs 

. The latter proportionality is equivalent to the Manning 

formula (p. 138) for mean velocity widely used by river 

engineers,31 in which a roughness factor, n, takes the 

. eYs (kM)Y2 · place of gY2; that is to say, n varies as rYs g · In a 

. later analysis of more extensive data on turbulent flow 

• 2D Stanton, T. E., op. cit., pp. 52-53. Prandtl, L., op. cit., p. 300, 1930. 

30 Lees, C. H., On the flow of viscous fluids through smooth circular pipes, Royal 

Soc. London Proc.,. vol. 91, A, pp. 46-53, 1915. Stanton, T. E., op. cit., pp. 55-56. 

·Prandtl, L., op. cit., pp. 299-300, 1930; Dryden, H. L., Murnaghan, F. D., and 

Bateman, H., op. cit., p. 337. 

3t Dryden, H. L., Murnaghan, F. D., and Batem!;m, H., op. cit., p. 441. Lindquist, 

E. G. W., op. cit., pp. 17-18, 41, 51-52, 55-57. Falkner, F. H., op. cit., pp. 25, 47-48. 
THE FORCE REQUIRED TO MOVE PARTICLES ON A STREAM BED -131 

in rough pipes Prandtl 32 derived a more generally applicable formula, · 

1 r .JkM ='Y log-;+~- ____________ (9) 

where 'Y and ~ are constants. However, it should be 

noted that recent laboratory investigations 33 indicate 

that the roughness term in Manning's formula depends 

quite as much upon the condition of the bed as upon the 

dimensions of particles and stream channel. After 

particle movement is well started, ripples or dunes are 

formed, and the roughness factor, n, is thereby 

increased.34 

In short, from previous studies of the factors that 

control frictional resistance, it would be expected that 

under conditions of turbulent flow in debris-floored 

open channels the roughness of the channel would be 

the dominant factor controlling the coefficient of resistance. For Gilbert's experiments on the force required to start particle movement, the expected change 

of channel roughness after the particles start moving 

would not be involved. For his experiments the channel roughness is probably to be measured by the ratio 

of particle to channel dimensions, Rfr . 

. 200 

.ISO X 0 

X 

~ • 0 Wt... • X 0 

J. .100 e X 

X ~ Xx 

6 .. • • •tx x 

f= ••••• X ~.070 •• xx X 

X o? (f) 

(f) • 

w.oso • X z 

I 

(9 

6.035 

X :0::: 

proportionality 8, ~kB varies as vM~rs. (c) For turbulent flow, kB probably depends upon a roughness 

ratio, Rjr. A plot of the three variables-vM~rs, Rjr, and 

the particle-movement data-affords a rigorous test of 

these supposed relationships. 

After several preliminary attempts, it was found that 

the actual relationships are shown most clearly if log 

RMEAN. 

-- I d . 1 F' h h Is p otte against ~· 1gure 21 s ows t e 

r VM-yrs 

plot of the data for grade G. The equation of the 

straight line that separates the field of hollow circles. (no 

movement) from the field of crosses and solid circles 

(slight to much movement) is 

I R 

1

-=-0.025log -+0.074 VM-yrs r 

r =0.025 log R+0.074 __ for grade G 

Gilbert's observations on other sizes of debris were 

less complete than those on grade G, but his data are 

sufficient to indicate the approximate equations of the 

lines separating the symbols indicating no movement 

Grade G 

0 

X 

EXPLANATION 

GRAINS MOVING 

0 None 

X Very few to few 

e Several to many 

0 0 

.020L---~-----L----~--~L----L----~--~L---~-----L----~----L---~----~----~--~ .05 .06 .07 .08 .09 .10 .II .12 .13 .14 .15 .16 .17 .18 .19 .20 

INVERSE MEASURE Of''BEO"VELOCITY AND RESISTANCE COEFFICIENT,_l_· DM.JTS 

FIGURE 21.-Roughness ratios, inverse measures of "bed" velocity and coefficient of resistance, and movement of debris cf grade G. Oblique line marks conditions at 

which movement of particles begins. 

ESTIMATION OF "BED" VELOCITY AND THICKNESS 

OF LAMINAR FILM 

Several conclusions thus far reached may now be 

assembled •and tested. (a) The data plotted in figures 

19 and 20 seem to indicate that the larger particles 

start moving when some critical "bed" velocity has been 

reached. (b) Fora given "bed" velocity, according to 

32 Prandtl, L., Neuere Ergebnisse der Turbulenzforschung: Ver. deutsch. Ing. 

Zeitschr., vol. 77, pp. 105-114, 1933. Lindquist, E. G. W., op. cit., pp. 4Q-42. 

33 Falkner, F. H., op. cit., pp. 47-48, pls. 24-38. 34 This observation that the formation of ripples increases the roughness of a stream 

bed is of interest not because it bears closely on the present problem but because it 

seems to contradict the widely held theory that ripples iri sand, like waves at the 

boundary between two fluids, decrease tbe frictional resistance to relative movement 

between two media .. 

153161-38--3 

from those indicating slight to much movement for two 

other sizes. 

1 r --p=O.lllog R-+0.32 ____ for grade E VM-yrs ~ · 

=0.016 log :R+0.048 ___ for grade H 

It is noteworthy that the right-hand men1bers of these 

equations have the same form as Prandtl's general 

formula (equation 9). The two constants in each equation decrease as the particles become larger, but the 

ratio between them is nearly the same for all three 

equations (2.91, 2.96, and 3.00). 
132 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY, 1937 

Inasmuch as -JkB is proportional to vMFs (proportionality 8) for the particular "bed" velocity at which 

particles of a given size begin to move, it appears justifiable to conclude that 

1 r 

1

k oc 0.025log -R--+0.074 

-y B MEAN 

and, by substitution in equation 7, 

vB2 de (0.025log R____!_+0.074) -Jg vM -JT8 __ (10) MEA N 

That is, the "bed" velocity increases with an increase of 

mean velocity, hydraulic radius, and slope and with a 

deere as~ of the roughness ratio (increase of channel 

smoothness). The relationship is such that the "bed" 

velocity is most sensitive to changes of the mean velocity and only very slightly sensitive to changes of the 

roughness ratio. 

70 I I I I 

![) 60- L 

'00 

u. 

c:),_ 

II 

(f) 

:l 50!-

~ _J 

w 

z 

z 401- <( 

:r: 

(.) 

~ (f) 30 1- 0 

(f) 

LL.I 

0:: 

f-

(f) X 

C) 20 1-

z 

in which the bracketed te~·m [1.05 log -R r +3.11] MEAN 

corresponds to -J~B in equation 7. · Needless to say, far 

more data than those just used would be required to 

establish the generality of such an equation, but it is 

here accepted provisionally as a basis for estimation. 

It is of interest to remark that the increase in coefficient of resistance that might be anticipated (p. 131) 

after particle movement starts is not large enough to be 

perceptible in these data. Three of the velocity 

gradients shown in Gilbert's figure 82 were measured 

when loads of 38, 53, and 194 gm/sec of grade C debris 

were being transported. through a trough 20.1 em wide. 

Yet the "bed" velocities under these conditions of considerable transportation fit the same equation (11) as 

the "bed" velocities in the experi;m.ents where there was 

no movement of grades A, D, and F particles. 

I I I I I I I I I 

.. --~1 • X • : I 

: I 

.·r .. .. ·· ··~ ·· ···· .. ·. I 

. : 

Grade H 

.·· I 

I 

X 

•• 

I X e e 

f 

Xe X 

0 X e 

X e 0 • , 

<2, ~Yx'x o 7 x >0x 01. X 

IGrade G 

I 

I 

· .. I X X e 

X 

• 

. . .... 

EXPLANATION 

GRAINS MOVING 

·"':'. 

-

-

-

0::: • Grade E • • . o 0 

<( .. ........ . 

~ 10 f 0 J X.. .~ ..... . . . . . 

0 None 

X Very few to few - • Several to many 

e ? x · .· · 

·. Oo 0 :X? •. ·· .·. 0 . . · 1 • • r · , . '( . . I I I l I I 

!5 20 25 ~ ~ 40 45 50 55 

VB ''BED" VELOCITY (Co/sec) 

FIGURE 22.- "Bed" velocities, shearing stresses, and movement of debris of grades E, G, and H. Vertical Jines mark "bed" velocities at which movement of particles 

begins. 

This relation may be tested further by some direct 

observations on velocity distribution made during a 

few of Gilbert's experiments.35 The velocity gradients 

observed during flow over beds of; paraffin, smooth 

board, and debris of grades A, C, D, and F have been 

fitted empttically to parabolic curves and the "bed" 

velocities then estimated from the individual curves. 

"Bed" velocitjes derived in this manner fit fairly well 

the equation 

Inasmuch as equation 11, which, is based on measured 

velocity gradients above floors of grades A, C, D, and 

F debris, accords closely with the relationships found 

by plotting data on the movement of grades E, G, and 

ll, and also with Prandtl's general . formula, it is here 

accepted as a basis for calculating the probable "bed" 

velocities in all of Gilbert's experiments on competence. 

The results of these calculations are given in table 1. 

-If the "bed" velocity at the boundary of the laminar 

filn1 and the velocity gradient within the laminar film 

as Gilbert, G. K., op cit., pp. 245-246, ti!ijs, 78, 80, 82. 
THE FORCE REQUIRED TO MOVE PARTICLES ON A STREAM BED 133 

are known, an a ppro:xirna.te average thickness of the 

laminar film, 8, can then be calculated from equation 5. 

This equation gives only an approximate thickness, 

because it assumes no rounding off of the velocity gradient at the transitional boundary between the zones of 

laminar and turbulent flow but instead a uniform 

gradient within the laminar film and an improbably 

abrupt change to turbulence. This calculation has 

been made for each of Gilbert's experiments on competence by assuming a coefficient of viscosity of 0.01 

(equivalent to a temperature of 20° C.). The calculated 

average thicknesses are given in table 1. So far as 

known to me, no direct measurements have been made 

of the thickness of laminar films in water flowing ·over 

sandy beds.36 

EFFECT OF "BED" VELOCITY AND SHEARING STRESS AT 

CHANNEL WALLS ON PARTICLE MOVEMENT 

With these estimates of the "bed" velocity, vB, and the 

shearing stress at the channel walls, PFgrs, for each of 

Gilber:t's experiments, the data are available for 

a revision of figures 19 and 20 (the plots of n1ean 

velocity against depth-slope product and particle 

movement). Figure 22 shows the "bed" velocities, shearing stresses, and extent of movement 

of particles of grades E, G, and H. One differenee between figures 20 and 22 stands out conspieuously. In figure 20 the lines that separate 

the symbols indicating no movement from those 

indicating slight to much movement slope 

obliquely down from left to right, thus indieating 

dependence upon both mean velocity and the 

depth-slope product. But in figure 22 the observ_ations on grades E, G, and H seem to show that 

these lines are vertical or nearly so, thus indicating an independence of the shearing stress or the 

r·s product and a dependence upon "bed" velocity 

alone. 

u 

(1) 

100 

90 

80 

70 

60 

50 

40 

30 

~ 20 ~ 

1-

z 

w 

a: 

a: 

:J 

·u 10 

~ 9 

>- 8 

!::: 7 

u 6 

g 

w 5 

> 4 

3 

2 

I 

l 

s 

-

"hydraulic lift") but to be closely dependent upon 

the "bed" velocity instead. 

The data in table 1 that bear directly on the critical 

"bed" velocities of the different sizes of debris are summarized in table 2. 

TABLE 2.-Critical "bed" velocity for different grades of debris 

Radius (em) Critical "bed" 

Gra(}e velocity at which movement starts· . Minimum Maximum (em/sec) 

B 0. 0155 0. 0200 13.7-

D 0. 0290 0. 0450 15.2-

E 0. 0450 0. 105 { 17. 9+ 

18.0-

F [0. 135] 0. 175 29.7-

G 0. 175 0. 250 { 36. 2+ 

36.4-

H 0. 250 [0. 370] { 44. 7+ 

45.9-

Gilbert noted that "when an experiment was begun 

with a velocity well below eompetence, and the velocity 

was gradually increased, the first movement detected 

Fine sand Medium sand Coarse sand Very coa~se sand Granules Pebbles 

/ 

/ 

/ 

/ 

/ / 

/ / /L' Erosion / / //xy / 

/ 

/ 

/XG 

-- / / - - - _.,..v/ ---- 1- - - - - - V..f\o~f\ 

vf\ 'I / / 

ufloar'i --- / 

t."'act bo -j-" 

------xE 

------ 1------ -xo v 

Xg / teO / ~,?l . o<:'- s s / ,,,;~'""Y'3 o< r'- / ,.,R rre 

"<~io~~ 

~tef / 

\3 / 

/ 

/ ..l 

4 

v Sedimentation 

/ 

/ 

! I 2 

DIAMETER (IN MM) OF PARTICLES 

4 8 

Unfortunately, Gilbert's experiments were not 

sufficient.ly numerous to establish or even to indic~te clearly whether the lines that separate the 

fields of no movement from those of slight to 

n1uch movement of the finer debris of grades B, 

C, and D are vertical or inclined. Additional 

data are needed on the conditions required to 

start movement of fine sand and silt. But so far 

as the available observations go, the force 

required to start movement of eoarse sand and 

gravel seems to be independent of the depth-slope 

product (the "critical tractive force") and of the 

shearing stress at the channel walls (the 

FIGURE 23.-Portion of Hjulstrom's diagram of the fields~of erosion,Etransportation, and sedimentation for well-sorted sediment. -j The calculated "bed" velocities which start movement 

of Gilbert's debris of grades B,jD, E,:F, G, and H:(shown by crosses))follow closely Hjulstrom's 

lower limit of the boundary between the fields of erosion and transportation. 

36 Since the above was written, my attention has been called to an entirely different 

method of calculating the thickness of the laminar film. In the notation of the present 

paper, Von Karman (Turbulence and skin friction; Jour. Aeronautical Sci., vol. 1, 

pp.lO, 19, 1934) finds that 15=11.5-( J.t.) 1 approximately. This equation, applied to PF grs I 2 

Gilbert's data, gives values of i5 that range from the same to two times as thick as those 

calculated by the method of the presep.t paper, 

would be the saltation of some small or light particle, 

and then the number of particles moving would gradually grow with the quiekening of current." 37 In a 

few of his experiments, b.~ recorded that only small 
134 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY, 1937 

gr3!ins were being moved.38 In the absence of information on differences in the specific gravity of individual 

·grains, ·it is here assumed-somewhat arbitrarily, 

perhaps-that the first particles to be moved in Gilbert's experiments were the smallest ones in the mixture. 

In figure 23 the critical "bed" velocities of each grade 

of debris are plotted on a portion of Hjulstrorn's general 

diagram of the relations between current velocity, grain 

size, and the fields of erosion, transportation, and sedimentation.39 This interesting compilation by . Hjulstrom shows that higher velocities are required to start 

movement of silt and clay than of sand; but that, once 

movement has been started, the fine-grained silt and 

grades B, D, and E confirm the flattening of the 

diameter-velocity curve. Gilbert's data, unfortunately, 

afford no information on the "bed" velocities required to 

start movement of particles smaller than grade B. The 

reversal of Hjulstrom's curve with smaller particles is 

based upon considerable laboratory and field evidence/1 

and later observations in the laboratory 42 seem to show 

the same reversal. However, even if this greater resistance of silt and clay is considered proved for clear water, 

it still remains questionable that the sarne would be true 

·for water carrying sharp sand grains as tools of erosion. 

The very interesting corollaries in ~ngineering, sedimentation, and physiography that may be drawn 43 

.sr---~r-,------~~----~~----~~~------r-----TT----~~------~~r-----o 

from this supposed reversal of· 

the curve make it highly desirable that additional studies 

be undertaken of the force 

.4-

. 3-

0 

0 

-

required to erode silt and clay . - 0 ........ ·x· o ·••····•·• X ?~ <Q> 0 DEPARTURES FROM THE 

"SIXTH-POWER LAW" FOR 

SMALLER PARTICLES 

·· ··· 0 ···;,··0········ X • f'O X ?~·x:··~ .. 

z 

0 

i= 

.2- .. -x-1·· ·· x. x • 7 •.. •· . . X 

X e X, 

e? • 

·taT 

• • • • • 

-

The analysis of Gilbert's 

data thus far has tended to 

substantiate the "sixth-power 

law" for coarse sand and gravel 

but to indicate significant departures from this law for 

smaller particles. A further 

examination of Gilbert's data 

tells something about the nature of these departures. 

<! 

::::> a ~ 

UJ ~i~ 

~ fl ~.10 ~ ~ ~ .09 r-

- II .oaf-

~<:I:> I~ .07- 1- ~ z ~ UJ c.-:>:~ .06-

u 

u:: 

LL. 

UJ 

0 

u 

.05-

.04,... 

.03 r-

·X? * I 0 

x? 

• X 

• 

• • • fl 

X 

X? 

• EXPLANATION 

GRAINS MOVING 

o - None 

X Very few to few 

• Several to many 

-

-

-

-

-

-

-

-

As was pointed out in the 

discussion of the "sixth-power 

law" the coefficient[%·-

0

-] ' tan a 

in equation 1 depends partly 

upon the shapes and relative 

sizes of the different particles 

making up a stream bed. Yet, 

-~~u-~----;~2-------o~~----~·----~;~------~~~--~~----~20------~;~0----I~oo · ev~il~edegreeofroundne~ 

Rfo 

PARTICLE- LAMINAR FILM RATIO 

and of sorting of the particles 

were nearly the same throughout a group of debris samples 

such as Gilbert used, it seems 

FIGURE 24.-Dependence of coefficient in impact equation upon the particle-laminar film ratio. The coefficient bracketed 

in the equation R= [ 3{ 't_!_ J ...3.!!.._, 11 B\ the particle-laminar film ratio, and movement of debris of all grade-sizes in· 74 an a ps-PF g 

vestigated by Gilbert. 

clay are kept moving by inuch lower current velocities 

than are required to transport sand. He attributes this 

greater resistance to erosion of particles less than about 

% mm in dian1eter to the "cohesion and adhesion" which 

unites smaller particles.40 It is noteworthy that the six 

points from Gilbert's experiments follow closely the 

lower limit of Hjulstrom's boundary between erosion 

and transportation and. especially that the points for 

38 Gilbert, G. K., op. cit., pp. 69, 70. 

3Q Hjulstrom, Filip, op. cit., p . 298. 

•o Idem, p. 299. See also Matthes, G. H., Floods and their economic importance: 

Am. Geophys. Union Trans. 15th Ann. Meeting, pp. 428-429, 1934, who attributes 

this resistance to compaction of the finer-grained deposits. 

likely that the value of this coefficient would still be influenced greatly by other factors, 

particularly by the relative dimensions of the particles and 

the laminar film. That is to say, the "bed" velocities at 

the upper boundary of the laminar film do not affect small 

particles lying on the stream bed ora.ffect them only to the 

extent that the particles protrude through the laminar film 

into the mass of turbulently flowing water. If the particles 

lieentirelywithin the thickness ofthelaminarfilm,no definite relation between "bed" velocities and particle move:.. 

•1 Hjulstrom, Filip, op . .cit., pp. 295, 299-300. 

•2 Falkner, F. H., op. cit., pp. 32-33, 34-35, 6D-86. 

•3 Hjulstrom, Filip, op. cit., pp. 324-325. 
THE FORCE REQUIRED TO MOVE PARTICLES ON A STR:EAM BED 135 

.tnent is to be expected. It is true that the laminar film 

probably tends to cling to and follow over the surface 

of larger particles that protrude into the main current, 

but the force impinging against such protuberances is 

significantly greater than that which acts upon small 

particles lying entirely immersed in the laminar film. 

. It thus seems probable that the coefficient [%·ta~ a] 

should show some relationship to the ratio, radius of 

particle divided by thickness of laminar film. 

In figure 24 the values of this coefficient, %· ta~ a= 

Ps- PF g R I d 1· · hm. II · · h · --.- ·v--;--2· ,.are p otte ogant ICa y against t e ratw 

PF B 

R/o (where R is the radius of the particles first moved) 

and the observations on debris movement of all gradesizes recorded thereon. This diagram seems to show 

that the coefficient depends at 

least in part upon the value of 

the ratio R/o. The calculated 

points are far from adequate to 

establish any precise relationship, 

but they indicate that, for ratios 

of 0.1 to 1.0, the coefficient increases rapidly from · about 0.13 

to 0.23; for ratios from 1.0 to 13.0, 

the coefficient increases more 0 

gradually from 0.23 to 0.28; at 

a ratio of 13, the coefficient appears to drop abruptly to 0.22; 

and then it decreases very gradually to 0.21 at a !atio of 31.44 

for all values of the ratio Rjo. The fact that the coefficient varies shows that "bed" velocity (at least, the 

one used here) is not the only factor in particle movement. Furthermore, the fact that the coefficient varies 

most with smaller values of the ratio Rjo suggests a 

possible explanation for the variation. The abnormally 

low values of the coefficient where the laminar film is 

thicker than the particle radius means that "bed" 

velocities higher than those indicated by the "sixthpower law" are required in order to start particle movement. This is what might be predicted either for 

surface drag or for hydraulic lift on particles entirely 

immersed in the laminar film; because, when thus immersed in a film of steep velocity gradient, the force 

acting upon a particle increases rapidly with the 

dimensions of the particle or the height to which it 

stands above the stream floor (fig. 25). Under these 

conditions of' complete immersion in the laminar film, 

8 

In other words, when the particles 

are relatively small COmpared FIGURE 25.-ldealized velocity gradient very near a stream bed. Within the film of laminar flow the velocity increases 

greatly with distance from the bed. Thus the forces acting on particle B to push, drag, and lift it are much greater 

with the thickness of laminar than those acting on particle A-perhaps enough greater to move particle B, despite its greater weight, without moving 

film, the foree of the current particle A. 

is less efficient, so that "bed" velocities higher than 

those indicated by the "sixth-power law" are required 

to start movement; when the particle radii are from 1 to 

13 times as great as the thickness of laminar film, the 

current is considerably more efficient; and when the 

particles are relatively large compared with the thickness of laminar film, the current is of intermediate 

efficiency. 45 

If "bed" velocity were the only factor in particle 

movement, the coefficient should be the same for all similarly shaped and sorted particles of whatever size and 

H IfiJ is estimated at 78 (p.123), the coefficient, 0.21, means that the angle a is approximately 24°. This may be compared with maximum, minimum, and average angles 

of 35°19', 19°22', and 22°03' up which a sphere that rests upon three other mutually 

touching spheres of the same size would have to be !ifted in order to dislodge it. 

'5 If Von Karman's thickness of the laminar film is used (see footnote 36, p.133), the 

relationship is almost exactly the same as that shown in figure 24, except that the 

abrupt decrease of the coefficient comes at a ratio of 8 instead of 13. 

the steepness of the velocity gradient rather than the 

"bed" velocity at the upper boundary of the laminar 

film would be the controlling factor in particle movement. Somewhat larger particles, protruding short distances into the mass of turbulently flowing water, would 

be acted upon by two or more forces-(1) the head-on 

impact of the current (the "sixth-power law") and (2) 

either the surface drag or the "hydraulic lift" within the 

laminar film or both. For still larger particles the headon impact would become relatively more effective and the 

surface drag or "hydraulic lift" relatively less effective. 

For very large ratios of particle to laminar film the 

effects of the laminar film would thus become negligibly 

small and the coefficient [%· ta! a] in the "sixth-power 

law" should remain constant. 
136 siiORTER CONTRIBUTIONS TO GENERAL GEOLOGY, i93i 

The apparent discontinuity in the coefficient curve 

shown in figure 24, where the particle radius is about 

13 times as great as the thickness of the laminar film, 

suggests an analogy with the somewhat similar abrupt 

change in the coefficient of frictional resistance when the 

flow changes from laminar to turbulent.46 The ratio 

at which this discontinuity appears may possibly represent a sort of Reynolds number, so to speak, at which 

the laminar film clinging to protuberances on the stream 

bed is broken up and torn away by the force of the main 

current. 

rze t erms a ft er w en t wor th (J our. G eo 1 ogy, vol.30,p.384, 1922 

0:: 

w Pebbles Granules Sand 

GJIOO 

2 

<( / 

o / go II / z 

w 

2:: 80 

Q 

<( 

z 70 /! 

I f!l$ I- -..; .t:: 

5 60 -? Ji (f) '1/ fal 0:: .;:, 0 

~50 "{" (; 

0 

I-

~~~ :-._ll.J E; 

25 40 Cf '1J (J 0 

w -::-._Ill t 

s 30 

~ !J~ . 0'b -<)0 

>- co II J~ t5 20 <o ~ . <( v .r_,,;• I- / z 

w 10 

0 

~ v 0::: L? w 

0.. 0 

64 32 I 16 8 4 2 I 0.50 

DIAMETER IN MILLIMETERS 

Calculated drameter of largest pebbles 

moved continuously by flood of Apr:29,1881 

0.25 

POSSIBLE APPLICATION TO NATURAL STREAMS 

The relationships considered and the equations proposed in this paper are based almost entirely upon 

laboratory evidence. It is a long jump from the effect 

of uniform discharge on uniform debris in small experimental troughs to that of variable discharge on natural 

aggregates of sediment in large stream channels. 

Nevertheless, it is the hope of understanding principles . 

and finding relationships that may be applicable to 

full-scale natural processes that leads to laboratory 

investigation, and the temptation is strong to · 

see how the equations derived here may work 

when applied to the immensely more complicated 

circumstances of nature. 

0.125 

As a matter of fact, when properly arid sufficiently qualified, the conclusions indicated by 

Gilbert's experiments should be applicable, at 

least qualitatively, to the movement of particles in natural streams. Obviously, very flat 

fragments of rock offer a greatly differe:p.t resistance to stream flow than approximately 

spherical grains of the same volume; therefore, 

equations for the movement of rounded grains 

are not applicable to very flat fragments. 

FIGURE 26.-Mechanical composition of samples collected in 1881 from the bed of the Mississippi River at Hannibal, Mo., compared with calculated diameter of largest pebbles continuously moved by the flood of April 29, 1881. 

Similarly, the large "erratic" pebbles and 

boulders present in minor quantities in many 

stream deposits may never have b.een carried 

freely by the streams. They probably reached 

their sites of deposition by rafting, by caving 

from undercut banks, by dumping from steep 

tributaries, by slow creeping on locally steep 

stream beds, etc.-processes quite different from 

those by which the bulk of sediment is moved. 

Until these "erratics" have been ground down 

to manageable dimensions by the blast of passing finer materials, they may perhaps be moved 

only very short distances on the stream bed.47 

Any relationships found for normal sediment 

transportation would certainly not apply for 

These departures from the "sixth-power law" for 

smaller particles, when R!o<1, need much additional 

study before they can be interpreted satisfactorily. 

Whatever their correct explanation, the variations in 

. the value of the coefficient [%· ta~ a] are relatively small 

for larger values of the ratio, and when Rfo> 13 equation 1 may be rewritten: 

R=0.21___!!_!.__ .vB 

,Ps-PF g 

2 

By substitution from equation 11, 

R=0.22[logR__!:__+2.96]·-·_p!'__.vMf's _______ (12) MEAN Ps-PF ..Jg 

these "erratics." 

Again, a very large part of the work done by natural 

streams is accomplished in the relatively brief periods 

of high floods. Consequently, we cannot expect to 

learn much about the average regimen of an ancient stream from a study of the land forms and 

sedimentary deposits it has left. At best we can 

only hope to learn something about the flood conditions of that stream. 

And so on through a wide range of many other necessary qualifications, such as gentle slopes and turbulent 

flow, which must not be forgotten in any attempt to 

apply the laboratory results to natural streams. The 

real question is to learn whether or not the laboratory 

46 Stanton, T. E., op. cit., pp. 52-53. Prandtl, L., in Ewald, P. P., Poschl, T., 47 Barrell, Joseph, Marine and terrestrial conglomerates: Geol. Soc. America Bull., 

and Prandtl, L., The physics of solids and liquids, p. 300, 1930. vol. 36, pp. 33Q-331, 335-336, 338, 1925. 
THE FORCE REQUIRED TO MOVE PARTICLES ON A STREAM BED 137 

results apply, even approximately, to those natural 

streams in which conditions seem to be roughly analogous; and this necessarily implies a considerable 

knowledge of the conditions in the natural streams 

chosen for comparison. 

I have made several calculations where sufficient 

information was available about a natural stream to 

afford a critical test. The examples given are typical 

of others, but they are chosen for illustration here 

because I happen to have some first-hand familiarity 

with the geology and physiography of the particular 

areas cited. 

One of the largest floods of the Mississippi River at 

Hannibal, Mo., for which cross-section and velocity 

measurements are available, occurred April 29, 1881. 

The river was then 2,535 feet wide, its area of cross 

section was 52,498 square feet, its mean velocity 5.19 

feet per second, and its discharge 272,609 cubic feet 

per second.48 The exact slope of the river at the line 

of measurement is not known, but inasmuch as the 

gradient is fairly uniform in this part of the Mississippi, 

the average slope above and below Hannibal, a fall of 

21 feet in the 42 miles from Quincy, Ill., to Louisiana, 

Mo./9 may be taken as representative. 

Mechanical analyses of 38 samples dredged from the 

river bed at Hannibal in 1881 are also available.50 The 

coarsest one of these samples was collected August 10 

(fig. 26). The analyses of the 38 samples collected in 

1881 agree fairly well with more detailed mechanical 

analyses of 11 samples collected near Hannibal in 1925 

by Lugn.51 

TABLE 3:---: M_ echf!ni9al compositi?n of samples from the bed of the M~ss~ss~pp~ Rwer at Hanmbal, Mo., collected in 1881 

[Percentage of particles by diameter in fractions of an inch] 

s:l >0 ..c:l ..c: ..... Cll N <:) <:) 0 

..c:l..c:i 0 .s .s 0 ""<:> 

J..f·s o-§ .... .... §.g 0 0 

~~ ~.s .8 0 ..cis:~ 

~0 >0 >0 .8 ;·~ 3 .,.. N .... 0) 0 

0 0 0 0 H 8 ------------------------

Coarsest sample______ _____________ 8 17 68 7 100 

Average of 38 samples_______________ 3 11 72 14 100 

Transposing these data from feet and inches to centimeters gives 

vM= 158 em/sec. 

Mean depth=631 em. 

r=622 em. 

8=0.000095. 

RMEAN=0.07 em [the median diameter of average 

bottom sample, estimated from graph, 

fig. 26]. 

's Results of discharge observations, Mississippi River and its tributaries and 

outlets, 1838- 1923, p. 47, Mississippi River Comm., 1925. , 

'9 Gannett, Henry, Profiles of rivers in the United States: U.S. Geol. Survey WaterSupply Paper 44, p. 39, 1901. 

50 Vogel, H. D., Sediment investigations on the Mississippi River and its tributaries prior to 1930: U.S. Waterways Exper. Sta. Paper H, pp. 83-84, 1930. 

51 Lugn, A. L., Sedimentation in the Mississippi River between Davenport, Iowa, 

and Cairo, Til.: Augustana Library Pub. 11, p. 49, 1927. 

Take 

RMAx=0.80 to 1.0 em [estimated from graphs of 

mechanical analyses, fig. 26]. 

PF=l.OO. 

Ps=2.66. 

g=980. 

J.t=0.011 (corresponding to a temperature of 

15° C.). 

Then, from equation 11 

vB=[1.05 log -R r +3.11J gi vMt (rs)* MEAN 

=[1.05 log ~~~+3.11 Jt (980)~ (158)! (622 X0.000095)i 

=2.69X5.60X12.57X0.493=93 em/sec. 

=the "bed" velocity. 

From equation 5 

0= f..LVB 0.011 X93.4 

PF grs l.OOX980X622 X 0.000095 

1.03 0 = 57.

9=0. 18 em 

=the average thickness of the laminar film. 

s· R 0·

80 to 1·

0 

mce 5= 44 t 56 h' h. . h 

0

.

018 = o , w 1c IS greater t an 13, 

then from equation 12 

RMAx=0.22. [log-r-+2.96]· ~. vM{!s RMEAN ps-PF -J g 

[ 

622 J 1.00 = 0.22 log 0.07 + 2.96 . 2.66-1.00. 

158 (622 X 0.000095)! 

(980)! 

= 0.22 X 6.91 X 0.602 X 1.227 

=1.1 em, 

which is to be compared with the maximum radius of 

0.80 to 1.0 em estimated from .the graph of the mechanical analyses, figure 26. 

. The data are much less complete for the other two 

examples, and so only the results of the calculations are 

given in table 4. The calculated radius of the largest 

particles moved by the Green River at Daniel, Wyo., 

agrees poorly with that estimated from direct observation. This lack of agreement may disclose fundamental 

weaknesses in the equations, or it may simply indicate 

that no very large flood has yet been measured at this 

point. In the Potomac and Mississippi examples, on 

the other hand, the calculations agree very closely with 

observations; and for all three rivers the equations give 

results that are at least of the correctorderofmagnitude. 

It thus appears that the equations based on Gilbert's 

laboratory experiments give reasonably close estimates 

of the maximum size of particles transported by some 

laFge natural streams. Needless to say, the equations 

need far more extensive testing and considerable modification before they can be applied to streams generally. It appears certain, for example, that more detailed testing will show significant variations in the 

numerical coefficient 0.22, depending upon the degree 

of roundness and of sorting of the particles on a stream 
138 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY, 1937 

bed and upon the local curvature of the stream channel. 

Also the maximum sjze of particles moved must vary 

from point to point across a stream, depending upon 

differences in mean velocity, depth of water, slope of 

water surface, and local roughness of the bed. The 

approximate agreements thus far found must be largely 

the result of compensating errors. Yet this very probability of compensating errors may mean that equations of this type will ultimately be found useful for 

average stream conditions. 

TABLE 4.-Calculated "bed" velocity, thickness of laminar film, and size of largest particles moved at definite points in the Green, 

Potomac, and Mississippi Rivers during certain floods 

Obs~rved and estimated from observation Calculations based on Gilbert's 

laboratory experiments 

River Locality Largest meas- ured floodt Hydraulic Mean ve- Discharge RMEAN RMAX VB 0 RMAX 

(cu. ft./sec.) Slope radius locity (em) (em) (em/ sec.) (em) (em) (em) (em/sec.) 

------

Green _________ ------ ____ Daniel, Wyo __________ June 16, 1918 8, 750 0. 0020 175 200 21.0 3 10 140 0. 0045 2. 7 

Potomac ________ -------· Chain Bridge, D. C ___ May 14,1932 165,000 0. 00087 412 340 2,10 3 5.0 210 .0066 5. 7 

Mississippi._----------- Hannibal, Mo. _______ _ Apr. 29, 1881 272,609 0. 000095 622 158 (07 '0.80 to 1.0 93 .018 1.1 

t Largest flood for which cross-section and velocity measurements are available. Data for the Green and Potomac Rivers from files of Geological Survey. 2 Rough estimate. s Estimate. 

' Estimated from mechanical analyses. 

In the form in which they have here been written, most favorable circumstances the equations may be 

the equations based on Gilbert's experiments are of found useful for interpreting past conditions of streams 

little direct use for interpreting the size and regimen of whose relationships to the present topography have not 

ancient streams. In order to estimate the dimensions been effaced-that is to say, for some Recent and 

of the largest particles moved by a stream, the mean Pleistocene and a few Tertiary stream deposits. 

STREAM COMPETENCE AND STREAM CAPACITY 

It may not be out of place to close this paper with a 

velocity, the hydraulic radius, the slope, and either the , 

mean grain size· or the ·roughness ratio must be known. 

However, one of these determining quantities may be 

eliminated by using the Manning formula 52 for mean 

velocity in open channels, · comparison, in very general terms, of two entirely 

different aspects of the transporting power of streams 

that are sometimes confused. 1.486 % ~ f f . VM=-· --r · 8 or t.-sec. units 

n 

or 4.642 % 31 f . vM=--r 8 or em-sec. units 

n • 

where the coefficient n (see also p. 130) is a roughness 

factor that may be taken from tables In engineering 

handbooks.53 

With this simplification, something of significance 

may be learned about the probable conditions of flood 

flow of certain ancient streams. If the maximum and 

mean grain sizes of the deposits of an ancient stream are 

known and if the original undeformed slope of th~ 

stream can be determined from terrace remnants or 

from other evidence, estimates can be made of the mean 

velocity and the mean depth (or, more strictly, the 

hydraulic radius) of the stream at times of flood~ and 

then, from a comparison with modern streams, inferences may be drawn about the probable flood run-off, 

climate, size and character of drainage basin, etc. But 

if the ancient deposits have been so deformed and 

eroded that the original ·stream slope cannot be ascertained, the problem appears indeterminate or, at best, 

determinate only within very broad limits. Under the 

u However, it is fair to mention in this connection that Gilbert's experiments, on 

which these equations are based, fit the Manning formula very indifferently. 

63 Merriman, Mansfield, op. cit., pp. 288-288a. 

According to Gilbert, 5

4 "A. current flowing over debris 

of various sizes transports the finer but cannot move 

the coarser; the fineness of the debris it can barely 

move is the measure of its competence." A later definition 55 emphasizes an essential point: "By competency 

is meant the ability of a stream to transport in terms of 

dimensions of particles." The present paper is therefore a discussion of stream competence. 

Gilbert 56 defined capacity as "the maximum load a 

stream can carry." Capacity is thus the amount of 

debris a stream can transport; it has nothing whatever 

to do with the maximum size of the particles transported. Elsewhere 57 I have proposed a method for 

evaluating stream capacity, and this method may be 

used to illustrate the distinction between capacity and 

competence. For laboratory streams flowing over 

tranS'portable del;>ris, it was found that approximately 

2~~ percent of the total energy of the stream was spent 

in transportation. For a few natural streams on which 

the necessary data were available, the proportion was 

found to be about 4 percent. Thus, in a general way, 

64 Gilbert, G. K.; op. cit., p. 35. 

65 Twenhofel, W. H., and others, op. cit., p. 30. 

56 Gilbert, G. K., op. cit., p. 35. 

57 Rubey, W. W., Equilibrium conditions in debris-laden streams: Am. Geophys. 

Union Trans. 14th Ann. Meeting, pp. 497-505, 1933. 
THE FORCE REQUIRED TO MOVE PARTICLES ON A STREAM BED 139 

where L=load or mass of debris passing a cross section 

of the stream in a unit of time (measured 

in gmjsec.) 

v3 =average settling velocity in quiet water of all 

the debris particles being transported 

(measured in em/sec.) 

and the other symbols follow the notation of the present 

paper. Dividing through by the wetted perimeter, 

this becom!3S 

L p ·V3 rxpF{}VM2rs, approx ___________ (13) 

Or,. (the load per unit width of stream bed) X (the 

average settling velocity of the debris transported) 

varies approximately as the product v~vlrs. 

By equation 7, 

For the purpose of this general comparison of competence and capacity, variations in the coefficient of 

"bed" resistance, kB, may be neglected as relatively minor 

second-order effects, and thus proportionality 13 will 

reduce to 

L -=p·vsrxvB\ approx ____________ (14) 

That is, the load per unit width of a strearn multiplied 

by the average settling velocity of the debris transported 

varies approximately as the fourth power of the "bed'' 

velocity of the stream. 

If the average settling velocity of debris transported 

should remain the same, then the weight of debris per 

unit width of stream carried past a given point would 

vary roughly as the fourth power of the "bed" velocity. 

But only under unusual circumstances would the average settling velocity remain the same in natural streams. 

A stream flowing in an alluvial channel of its own 

making has relatively easy access to unconsolidated 

debris of many sizes. As the "bed" velocity of such a 

. stream is increased, the average settling velocity of 

debris transported does not remain the same but increases because larger and larger particles are picked up 

and added to the load. 

Probably no natural stream ever carries exactly equal 

w:eights of all sizes of debris smaller t4an the largest 

particles it is able to move. Nevertheless, in a stream 

free to pick up larger and larger material as its "bed" 

velocity increases, the average settling velocity of all 

debris transported will be influenced greatly by the high 

settling velocity of the largest particles carried and 

hence if much of the larger material is picked up the 

average settling velocity will increase approximately in 

proportion to the settling velocity of these largest 

particles. 

For small grains of silt and clay, to which Stokes' 

law of settling velocities is applicable, the principles 

governing the maximum size of particles. transported 

are not yet known; and consequently no simplification 

of competence and capacity can be attempted here. 

But for larger grains of sand and gravel, to which the 

"sixth-power law" applies best, settling velocities vary 

closely as the square root of the particle radius. 58 

That is, ~~R for sand and gravel. 

But according to the "sixth-power law," 

So that 

v3rxvB, approx _____________ (15) 

That is to say, in a stream able to pick up considerable 

sand and gravel the average settling velocity of debris 

transported will tend to vary directly as the first power 

of the "bed" velocity of the stream. 

Combining this relation with proportionality 14, we 

find that, because of the simultaneous effects upon both 

competence and capacity, 

L p rxvB3, very approx ___________ (16) 

In a stream free to pick up much sand and gravel as its 

velocity is increased, the unit width load will vary 

roughly as the third power of the "bed" velocity. 

The purpose of this generalized discussion has been 

to show that quantita-tive relationships of stream capacity are not to be confused with the "sixth-power 

law" of stream competence, wbich for comparison may 

be restated here as follows: The weight of the largest 

debris particles moved by a stream varies as the sixth 

power of the "bed" velocity. 

SUMMARY AND CONCLUSIONS 

1. The force that moves a particle on a stream bed 

has been explained in several ways-by the impact or 

momentum of the water which strikes against it, by 

the frictional drag upon its surface, and by differences 

in pressure induced by differences in current velocity 

at the top and bottom of the particle. The familiar 

textbook "law" that the weight of the largest particles 

moved . by a stream varies as the sixth power of the 

velocity is based upon the impact theory. The theory 

of frictional drag is the basis of the so-called law of 

"critical tractive force," which river engineers prefer because it gives the maximum size of particles moved in 

terms of readily measurable quantities, the depth of 

water and the slope of a stream. The "hydraulic lift" 

caused by pressure differences is measured by the 

velocity gradient very near a stream bed. 

2. G. K. Gilbert's laboratory experiments show that 

the force required to start particle movement depends 

not only on the mean velocity of a stream but also on 

the depth-slope product, the large particles being most 

ss Rubey, W. W., Settling velocities of gravel, sand, and silt particles: Am. Jour. 

Sci., 5th ser., vol. 25, pp. 327-335, 1933. Wadell, Hakon, The coefficient of resistance 

as a function of Reynolds number for solids of various shapes: Franklin Inst. Jour., 

vol. 217, pp. 467-475, 1934. Christiansen, J. E., Distribution of silt in open channels: 

Am. Geopbys. Union Trans. 16th Ann. Meeting, pp. 480-481, 1935. 
140 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY, 1937 

sensitive to changes of velocity and the smaller particles 

most sensitive to changes of the depth-slope product. 

3. The velocity in the immediate vicinity of the particle on the stream bed is more significant for this inquiry 

than the mean velocity of the entire stream. The 

velocity at the boundary between the thin film of laminar flow on the stream bed and the main mass of turbulent water above it is here defined as the "bed" velocity 

and evaluated in terms of mean velocity, hydraulic 

radius, slope, and a coefficient of frictional resistance of 

the bed. 

4. Experimental studies on the frictional resistance 

to flow in rough pipes suggest and Gilbert's data show 

that this coefficient of frictional resistance depends 

upon the channel roughness or the size of particles 

making up the stream bed .. This relationship permits 

an estimate of the "bed" · velocity in each of Gilbert's 

experiments. 

5. From the shearing stress at the channel walls and 

the "bed" velocity, the thickness of the laminar film can 

then be estimated. With this information, a reexamination of Gilbert's results shows that movement of 

larger particles starts at some particular "bed" velocity 

and is independent of the velocity gradient or the depthslope product. That is, the movement of coarse sand 

and gravel follows the "sixth-power law", but the smaller 

particles require much higher velocities than are indicated by this law. 

6. A satisfactory theory ·of the force required to start 

movement of fine sand and silt must await add!,tional 

observations. However, Gilbert's experiments ~\lggest 

that the departures from the "sixth-power la\y!' for 

smaller particles become large enough to be signi1icant 

when the laminar film is thicker than the radius of the 

particle, thusindicatingthatfrictional drag or "hydraulic 

lift," rather than impact, is the force that starts movement of small particles. 

7. The equations based on laboratory data give 

reasonable estimates of the maximum size of pe_bbles 

moved by some large natural streams, but much :i-nformation is required about a stream in order to make the 

necessary calculations. After more extensive testing 

and modification, equations of the type propose<\,may 

prove useful for interpreting the size and regim~n of 

streams that laid down certain Recent, Pleistocene, 

and Tertiary deposits. 

8. Although both size of particles and total load are 

aspects of the power of transportation of a stream, the 

two are not to be confused. The "sixth-power law" 

measures only the size of larger particles moved, and 

has nothing to do with total 'load or amount of debris 

transported. 
INDEX 

Page 

Abstract_ __ _____ ---------- ______ --------- _- ------- ____ ------------ ___ ____ _____ 121 

Acknowledgments for aid ____ _____ ____ __ -------------------------------------- 122 

"Bed" resistance, coefficient oLe--- ___ --- _______ __ _______ ____ ______ __ _____ __ 130-131 

"Bed" velocity, calculated, at b<)undary oflaminar film---- ---- --- -~ 126-127 

calculated, for Potomac, Gr~~n, and Mississippi Rivers------------------- 138 

critical, for different grades of debris____ ___ ___ __ ___________ ___ ___ _____ __ __ 133 

definition and calculation oL ___ _______ ___ ___________ ____ _________ ____ __ 129-130 

effect of, on particle moven{ent- -- ---- -- -- --- ---- ---- --- ----- ~- ---- 133-134 

estimation oL __ ___ _____ _____________ _ ----------------- _________________ 131-133 

relation of, to movement of small particles _____ ____ ______ ___ ___ _____ __ _ ·_ 134-135 

Capacity of streams, definition S:nd calculation oL ________ __ ___ _____________ 138-139 

Chain Bridge, D. C., Potomac l~iver at, calculated "bed" velocity, thickness 

of laminar film, and size of largest particles moved in flood by____ 138 

Channel walls, friction oL---- =~------- -- -- ---- -- -- -- ----- -------- 129-130 

See also Shearing stress at ch;mnel walls. 

Competence of streams, definitio.n and calculation oL _______ ___ ____ ___ ___ __ 138-139 

Gilbert's experiments on __ ____ __ ____ -------------- ____ __ __ ____ _ --------- 124-128 

tabular exhibit oL ______ ____ ____ ----------------- _______ _________ ___ 126-127 

Conclusions, statement oL ___________ ---------------- ---------------------- 139-140 

"Critical tractive force," basis oL_ ___ _________ ____ ______ ____________ __ __ ______ 139 

definition oL _____ __ ____ ------ _- __ __ __________ ---- ___ -__ __ __ ___ ___ __ __ __ __ 121 

observations on ____ ___ __ ___ _____ --------- __ ___ ___ ____ ___ _ ----------------__ 122 

physical basis oL __ ___ __ _____ ___ ---- ___ - __ ---"------ ____ ------------ ___ _ __ 123 

relation of, to particle movement_ _____ ___ __ ____ ___ ___ ___ ____ ___ ______ __ __ 133 

theory of, lack of agreement of observational data with ____ _____ __ __ _____ 128-129 

Currents, bottom, on sea floor, study oL-------------------------------------- 121 

Daniel, Wyo., Green River at, calculated "bed" velocity, thickness oflaminar 

film, and size of largest particles moved in flood by______ ___ ____ __ 138 

Debris, grades oL _____ _________ ___ ____ __ __ _________________ _________ ___ 124, 126-i27 

Depth of stream, measurement oL__ _________________ ______ _____________ ____ _ 124 

Depth-slope product, relation of, to movement of particles __ ____________ 128-129, 133 

Discharge, measurement oL ___ ______ __ ___ _____________________ _________ 124,126-127 

Erosion of rock materials by. moving water, boundary between transportation 

and __ _____ ---------- __ ----------------------------- _____ ____ ____ 133, 134 

principles governing __ ________ -- ___ _ ------ ___ _____ -------------- ____ __ __ 121-122 

Experimental data, application of, to natural streams ___ __________ ______ ___ _ 136--138 

Floods, studies of, application otexperimental data to ____ ___ ___ _____ __ __ ___ 137-138 

Frictional resistance, coefficient oL ----------------------------------------- 129-131 

Gilbert, G. K., experiments by __ ____ ___ __ ______ _____ __ ___ ____ ___ _______ 122,124-127 

quoted _____ ---------- --------------------------------------------___ ___ __ 12!1 

Grades of debris, _-------------- 7 ----- - ----- -------- -------- -------- _____ 124, 126--127 

Green River at Daniel, Wyo., Calculated "bed" velocity, thickness of laminar 

film, and size oflarg; st particles moved in flood by- -----,--------- 138 

Hannibal, Mo., calcq.lated "beQ: ~~ velocity, thickness of laminar film, and size 

oflargest particles moved in flood by Mississippi River at_ ____ ___ 138 

mechanical analyses of samples from bed of Mississippi River at _______ _ 136, 137 

Hjulstrom, Filip, boundary between fields of erosion and transportation 

defined by ___ : __ -------------------------------------- ____ _____ _ 133-134 

"Hydraulic lift," measurement 6L --- ---- ------- ---- -- -- -- -- -- -- -- -- 139 relation of, to particle movement_ _____ ___ ____ __ __ ____ ___ __ ___ ______ 133,135,140 

to velocity gradient _________ ____ ______ ------ ____ __ ___ ______ ____ ____ _ 124, 125 

Hydraulic radius, calcp.lation oL ___ ________ ____ ___ __ ____________ _____ ___ ___ 126-127 

definition of._-------- __ ____ ___ ____ ______ _____ ----------______ ______ ______ 124 

Impact equation, coefficient in ____ _______ ______ ___ _________ _____ ____ 123,134,135,136 

Laboratory experiments, object oL _____ ______ __ ___ ______ ______ _____ ____ 122,125,136 

Laminar film, calculated "bed".y elocity at boundary of_ ____ ___ _______ _____ 126-127 

calculated thickness of, aver:~ge ___ ___ ________ ____________ ______ _____ ___ _ 126-127 

calculated thickness of, for Potomac, Green, and Mississippi Rivers_ _____ 138 

ratio of, to size of particle _____ ____ __ __ _____________ ______ _______ 134, 135, 136,140 

relation of, to frictional force_ ________ ____ ___ _____ __ ___________ ___ _______ __ 130 

to particle movement_ _____ ___ ____ __ ---------------- ______ ---------- 134-135 

thickness of, estimation of. •••• ----------------------------- 125, 129; 131-133, 140 

0 

Page 

Laminar flow, formulas for_ _____ ___ _____ ___ __ __ __ __________ ____ _____ __ _____ ___ 130 

possible transportation of sediment during-------- ----- c----- ------- 125 zone oL ___ _____________ __________________________ ____ ___ ____ ___________ 126, 133 · 

Load of a stream, relation of, to "bed" velocity _____ ____ c____ ______ ______ ____ _ 139 

Manning formula for mean velocity, use of_ _____ ____________ ____ ____ ____ ____ _ 138 

Mean velocity, calculation of. ___ ___ _______ ________________ ____ __ __ ____ ____ _ 126-127 

definition oL _______ _____ ______ ________ _____ ___ _________ ------------ ____ 124-125 

effect of, on particle movement_ ___ ________ __ ____ _____ __ ______ _____ ______ 128-129 

relation of, to qischarge ___ · ____________ -- __________ ---------------------- 124 

Mississippi River at Hannibal, Mo., calculated "bed" velocity, thickness of 

laminar film, and size of largest particles moved in flood by _-- ~ 138 

mechanical composition of samples from bed of. ____________ ___________ _ 136,137 

Particle-laminar :film ratio. See Laminar film, ratio of, to size of particle. 

Particle movement, effect of "bed" velocity and shearing stress at channel walls 

on_ __ ___ ___ ___ __ _________ _________ _____________________ ___ ____ __ 133-134 

effect of mean velocity and depth-slope product on ___ _________ __________ 128-129 

Pleistocene streams, ~pplication of experimental data to __ ______ ________ ____ _ 138, 140 

Potomac River at Chain Bridge, D . C., calculated "bed" velocity, thickness 

of laminar film, and size of largest particles moved in flood by_____ 138 

Prandtl's general formula for coefficient of resistance; statement of.___________ 131 

Pressure, differences of, at top and bottom of particles-- -- -~- ----------- 124 

Radius, hydraulic, calculation oL-·---------------------------------------- 126--127 

hydraulic, definition oL _ ----------------------------- 7 ---- - - - - --- ---- ___ _. 124 

Resistance, coefficient of, calculation oL------------------------------------ 130-131 

coefficient of, increase in _______ ------------------------------------------- 132 

Reynolds number, critical value oL ------------------------------------------ 125 

relation of, to coefficient of "bed" resistance ___ __________________________ 130, 136 

Ripples, effect of, upon roughness of channeL_____ ___ _____ ________ ____________ 131 

Roughness, coefficient oL------------------------------------ 7 - ---- 130,131, 132, 138 

Settling velocity of particles, variation in------------------------------------- 139 

Shearing stress at channel walls, estimates based on___ __ _______ _________ ______ 140 

effect of, on particle movement_ ____ ___ ____ _____ __ __ ____ ________ 130, 132,133- 134 

measurement oL _ --C ------------------------------------------- ---- 125, 126-127 

Shear, rate of, between adjacent fluid filaments, theory oL---- -~- ---- -- ----- 121 

''Sixth-power law", definition oL------------------------------------------- 121,139 

departures from, for smaller particles __ _____ ____ ___ _____ ____ __ ____ __ __ __ 134-136 

essentials of theory of_: __ __________ ___ _______ __ ____ ___ __ ________ __ __ ____ _ 122-123 

lack of agreement of, with observational data. -------------------------- 128-129 limitations of application oL _________ ___ _____ ___ ___ ____ ___ ______ _______ 139-140 

Slope of water surface, measurement oL ______ ___ ____________ _________ __ 124,126-127 

Stokes' law of settling :velocities, application oL- ----------------------------- 139 

Streams, natural, possibl~ application of experimental data to ___ ___ _____ __ __ 136--138 

Summary ___ ______________________ __ ____________ ---------------------------- 139-140 

Symbols indicating movement of debris, explanation oL - --------------------- 125 

Tertiary streams, application of experimental data to _____ ______________ ____ 138, 140 

Transportation of rock materials by moving water, boundary between erosion 

and:. ________ __ __ ___ ___ __ _______ __ _____ -------------------------- 133, 134 

effect of turbulence on______ _____ ____ ___ _________ ____ ___ _______ _____ ______ 125 

principles gov~rning _____ _____ ----------- ___ __ -------------------------- 121-122 

Turbulence criterion, measurement oL _______ _____ ____ ___ ____ ______ __ __ 125,126-127 

Turbulent flow, effects oL---------------------------------- 125,129, 130, 131, 133,136 

Velocity gradient, "hydraulic lift" caused by--------------------------------- 124 

parabolic distribu'tion oL __ ____ ------- _ ---------------------'------------ 129, 132 

theory of. ___ ______ _ ----------------------------------------------------- 121, 129 

Velocity, mean, definition oL----------------------------------------------- 124-125 

mean, relation of, to discharge ____________ ____ _______ __ __________ ___ ______ 124 

Viscosity, coefficient oL ___ __ ___ _________ ------------------------------------- 133 

Wetted perimeter, relation of hydraulic radius to_ ________ ______ ____ _______ ___ 124 

Width of stream, relation of, to discharge__ ____ ___ ______ ___ ______________ ___ __ 124 

141