Bedload Equation for 

Ripples and Dunes

GEOLOGICAL SURVEY PROFESSIONAL PAPER 462-H

Bedload Equation for 

Ripples and Dunes

By D. B. SIMONS, E. V. RICHARDSON, and C. F. NORDIN, JR. 

SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 462-H

An examination of conditions under which bedload transport rates may be determined from the 

dimensions and speed of shifting bed forms

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1965
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

Washington, D.C. 20402 - Price 20 cents (paper cover)
CONTENTS

Symbols. _______________________

Abstract __ __________ ___ ___

Introduction ____________________

Page

___________________ IV

.__._--_____._..___ HI

_______ _ 1

Bedload Transport __ ______________

Conclusions_______________ ________

References cited. ___________________

Page 

. _____ ___ __ HI

_________ _ ____ 7

_ __ ___________ 9

ILLUSTRATIONS

FIGURE 1. Definition sketch of dunes.

Page 

H2

2. Comparison of computed bedload with

observed bed-material load__________ 3

3. Comparison of computed and observed 

bedload.___________________________

FIGURES 4-7. Relation of 

4. V.to(V-V

. . Vs

5.

6. h to V-VQ g

Page 

H5

6

(V V\ 

7. qriOToV[~y~ )______ _______ 8

TABLE

TABLE 1. Variables considered.

Page 

H2

m
LIST OF SYMBOLS

C Chezy coefficient

Ci A constant

D Mean depth

g Accleration due to gravity

h Average ripple or dune amplitude

KI, K2) Ka , KI, K5 Constants

q_b Bedload, by volume or weight

qr Bed material load

S Slope of the energy gradient

t Time

V Mean velocity

VQ Noneroding mean velocity

Vs Average velocity of ripples or dunes in the direction of flow

x Distance parallel to the direction of flow

y Elevation of the bed above an arbitrary horizontal datum

7 Unit weight of the fluid

5 A transformation, 5=x Vst

TO The shear stress at the bed

X Porosity of the sand bed

d Partial derivative
SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

BEDLOAD EQUATION FOR RIPPLES AND DUNES

By D. B. SIMONS, E. V. RICHARDSON and C. F. NORDIN, Jr.

ABSTRACT

A simple bedload transport equation based on the mean 

forward velocity and mean height of ripples and dunes is applied 

to 101 observations from flume studies by using a recirculating 

flume and four different sands. Where average dune velocities 

and heights are determined accurately with sonic equipment, 

and where suspension of bed material is negligible, the computed 

and observed transport rates of bed material agree reasonably 

well. General guidelines for adapting the procedure to field 

applications are given.

INTRODUCTION

The bedload in most natural channels cannot be 

measured economically by presently available (1964) 

sampling techniques. However, where the size and 

shape of bed configurations the sand waves or ripples 

and dunes which form on the channel bed and the 

speed of shifting of the bed forms can be determined 

with reasonable accuracy, a simple relation based 

on the mean height and velocity of the sand waves 

can be used to determine bedload transport. It is 

the purpose of this paper to investigate the applicability 

and some of the limitations of such a relation.

The bedload equation presented herein was previously 

given by Richardson and others (1961), and presents 

no new ideas; in fact, the concept of determining 

bedload from the dimension and speed of the dunes 

and ridges in a channel dates from the 19th century 

(Hubbell, 1964). However, because of the recent 

technical advancements in electronic equipment 

(Richardson and others, 1961; Karaki and others, 

1961) which is both portable and reliable and which 

permits rapid and accurate determination of bed 

configuration over large areas of a channel, it seems 

desirable to evaluate the applicability of the equation 

to the practical problem of determining bedload discharge.

BEDLOAD TRANSPORT

When the velocity of flow in an alluvial channel is 

low, the channel bed configuration is ripples or dunes, 

or both, and a part of the transported material moves 

as bedload, in almost continuous contact with the 

bed. The bedload moves up the face of a ripple or a

dune to the crest; the coarser particles then avalanche 

down the slope to be deposited on the downstream face 

of the dune or in the trough, and the finer particles 

may be deposited on the faces of downstream dunes or 

swept temporarily into suspension. The ripples and 

dunes move downstream owing to erosion from their 

upstream face and deposition on their downstream 

face.

The differential equation of bedload transport for 

the ripple and dune bed configuration (see fig. 1) may 

be written

"X.. -i "X 

=0. (1) d* (1 X) dx 

Using the transformation

d=x-V,t,

and

Substituting these expressions in equation 1,

(2)

(3)

(4)

?=0.

Simplifying, 

from which

(5)

(6)

(7)

Assuming that the dunes and ripples have triangular 

shapes, the final equation for bedload transport becomes

(8)

Values of g6 may be converted from volume to weight 

basis by multiplying equation 8 by the unit weight of the

sediment.

Hi
H2 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

In the above,

g6=volume rate of bedload transport per unit

width per unit of time, 

X=porosity of the sand bed, 

Vs=aver age velocity of the ripples or dunes in the

direction of flow,

h=average amplitude of the ripples or dunes, 

y elevation of the sand bed above an arbitrary

horizontal datum, 

x distance parallel to the direction of flow,

FIGURE 1. Definition sketch of dunes.

The constant of integration (<7i) may be interpreted 

as that part of the bedload which does not enter into the 

propagation of dunes and ripples. Obviously, at the 

threshold of movement, <?i=0 and, so long as the bed is 

entirely covered with ripples and dunes, <?i will remain 

equal to 0 (assuming that bedload is defined as the 

material that moves in almost continuous contact with 

the bed). But as the velocity of flow increases, the bed 

configuration changes from dunes through a transition 

region of multiple bed forms to a plane bed. For the 

plane bed condition, h=0 and C\=qi>- For the transition region, Ci is indeterminate.

Equation 8 was applied to 101 observations of equilibrium flows in an 8 foot-wide recirculating flume, 

where the average heights (h) and velocities (V,) of the 

ripples and dunes were recorded (Simons and Richardson, 1961). Velocities and heights of dunes were determined by probing, by visual observation in the transparent wall of the flume, and by the use of the sonic 

depth sounder described by Karaki and others (1961). 

A description of the flume and of the general procedures 

used in collecting data has been presented (Simons and 

Richardson, 1962; Simons and others, 1961).

The bed-material median diameters, the number of 

observations for each bed material, and the range of 

unit discharge of water and of bed-material considered 

are summarized in table 1.

TABLE 1. Variables considered

Median 

diameter, in 

millimeters

0.19 

.28 

.46 

.93

Observations

17 

32 

31 

21

Range of discharge per foot of width of 

Water, in cubic 

feet per second

0.428-2.75 

.520-2.75 

.865-1.95 

.862-2.84

Bed material, in 

pounds per second

0. 0000276-0. 226 

.0000069-. 211 

. 000086 - . 350 

.00220 -.474

Figure 2 shows the computed bedload transport rate 

from equation 8 (#&) plotted against the observed bedmaterial transport rate (gr) for each of these observations. For the coarser material (the 0.93-mm median 

diameter sand), the agreement between the computed 

bedload and the observed bed-material load is excellent 

throughout the range of transport considered. The 

three points which deviate most from the equal line 

are for observations where the bed configurations were 

in a transition between dunes and a plane bed. Similarly, for the 0.46-mm sand, the agreement between 

computed bedload and observed bed-material load is 

reasonably good, except at the higher transport rates 

where an appreciable percentage of the bed-material 

load was in suspension, or where the bed was in a transition stage between a dune and a plane configuration.

For the finer sizes, the computed bedload is approximately equal to the observed bed-material load for the 

ripple-bed configurations. For the dune bed, the 

observed bed-material load is always greater than the 

computed bedload a fact that indicates that a large 

part of the total transport was in suspension.

In figure 3, the bedload transport computed from 

equation 8 is plotted against observed bedload. This 

observed bedload was determined as the difference 

between the observed total bed-material load and the 

observed suspended load, for the 0.19- and the 0.28-mm 

sand. Suspended load was determined from the concentration of integrated samples collected to within 

about 0.1 foot from the bed. For the ripple-bed formation, the relation is not improved, probably because of 

inherent difficulties in obtaining representative suspended-sediment samples at the extremely low transport 

rates. For the dune-bed configuration, however, the 

relation of figure 3 is improved appreciably over that of 

figure 2. Again, the points which deviate most from 

the equal line are for bed conditions in transition between a dune and plane bed.

From figures 2 and 3, it is concluded that equation 8 

is suitable for computing bedload transport rates for
BEDLOAD EQUATION FOR RIPPLES AND DUNES H3

1.0

0.1

0.01

0.001

0.0001

0.00001

o

o

X Dunes - 0.93-mm sand

A Ripples ^i _ .,. , A Dunes )°-46-mm sand

O Ripplesl. 

 Dunes J Ul 0.28-mm sand

O Ripples'1 , 

 Dunes j°-19-mm sand

0.00001 0.0001 0.001 0.01 0.1

qr , IN POUNDS PER SECOND PER FOOT OF WIDTH 

FIGURE 2. Comparison of computed bedload (g&) from equation 8 with observed bed-material load (gr).

1.0

the entire range of conditions considered, excluding 

those observations when the bed configurations were 

in a transition from dune to plane bed. For the coarser 

material, where the suspended load was negligible, 

the equation may be used to estimate the total bedmaterial discharge, at least for shallow depths, that is, 

below approximately 3 feet. At greater flow depths, 

the suspended load perhaps would not be negligible. 

Attempts have been made, notably by Russian 

investigators (KondrateV, 1962) to formulate relationships for estimating dune heights and dune velocities 

from the characteristics of the flow. Several empirical 

relations have been advanced. Znamenskaya (1962) 

proposed that V, could be evaluated from the equation

(9)

and Barekyan (1962) presented the following relations:

(V-V«

(10)

(11)

In the above,

Z?=mean depth of flow, 

V=mean velocity, 

Fo=noneroding mean velocity, 

g= acceleration due to gravity, 

KI, K%, and Kz are constants, 

<7=Chezy coefficient, V/-\lgDS, 

$= slope of the energy gradient,
H4 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

0.00001

0.0001

FIGURE 3. Comparison of computed and observed bedload for the 0.19- and 0.28-mm sand.

0.001 0.01 

MEASURED BEDLOAD, IN POUNDS PER SECOND PER FOOT OF WIDTH

0.1

When equations 8 and 9 are combined, the rate of 

bedload transport is found to be independent of dune 

height, and the transport relation of equation 8 simplifies to

V0) (12)

when KI is a constant. Equation 12 is quite similar 

to the expression for stream capacity used by G. K. 

Gilbert (1914).

The velocity of dune movement ( Vt) is plotted against

-£ (V V0) in figure 4. The solid line in the figure is

equation 9 with the constant KI, as given by Znemenskaya. Figure 4 shows that equation 9 is unsuited to

express dune velocities for the data considered, irrespective of the value of K. However, it should be noted 

that equation 9 was given for the speed of shifting 

of "meso-forms" (large-scale bars and ridges in natural 

channels), so the equation probably is not applicable 

to the ripples and dunes formed hi laboratory flumes.

V3 

In figure 5, where V» is plotted against -jy> a trend

line drawn through the plotted points could be represented by equation 10, but the constant K2 is approximately 0.5, compared with 0.92 given by Barekyan, and 

the scatter of points about the line does not lend confidence to the relation.
BEDLOAD EQUATION FOR RIPPLES AND DUNES H5

i.o 

0.1

0.01

0.001

0.0001

x Dunes - 0.93-mm sand

A Ripples! o.46-mmsand

A Dunes J

D Ripples"! . 00 .

 Dunes )0.28-mmsand

O Ripplesj Q19d

 Dunes J

I lilt

0.5 1.0 10

(V-VfcJ £, IN FEET PER SECOND h

FlOtTBE 4. Relation of V, to (F-Fo) -r-.

The relation of equation 11 for dune height (K) 

shown in figure 6 is somewhat better defined than the 

relation for dune velocity in figure 5, and the constant

=20.6 has the same value as the constant given by 

Barekyan. Iti 

Combining equations 10 and 11 with equation 8 yields

(13)

in which

TO=shear stress at the bed (yDS) 

7=unit weight of the fluid, 

K5 =& constant.

The product of shear stress and velocity, sometimes 

called "stream power," has been shown to be an important factor in sediment transport (Bagnold, 1960; 

Cook, 1936), but the correlation of transport to stream 

power is better for flow over a plane bed with antidunes 

and standing waves than for flow over a dune bed 

(Colby, 1964). 

Values of q? are plotted against r0V( y~J m

figure 7. The constant K5) defined from the trend line 

through the plotted points, is approximately 0.3. 

Equation 13 does not fit the data for the ripple-bed 

configuration, and the scatter of points is again too 

great to provide confidence in equations 10, 11, or 13. 

It must be concluded that equation 8 is best suited for 

determining bedload when the average values of Vs and 

h can be determined quite accurately, as with the sonic 

depth sounder. The relations for determining heights 

and velocities of bed forms from flow parameters (eqs. 

9-11) do not appear to yield reliable results, especially 

for the ripples.

It is important to note the limitations of equation 8 

and to recognize the factors which influence its accuracy. 

Critical assumptions in the application of equation 8 

were: (1) that the ripples and dunes are triangular 

in shape, (2) that the ripples and dunes are twodimensional that is, the average V, and h are assumed applicable across the entire width of the channel, 

and (3) that the porosity (7) is constant.

Simons and others (1961) have indicated that the bed 

with a dune configuration is softer than the bed with 

ripples; this difference suggests that the porosity is not 

constant, even for a given bed. material. However, 

the porosity probably varies within rather narrow 

limits for natural river sand, and-the introduction of a 

constant value should not introduce errors of appreciable magnitude.

According to Simons and Richardson (1962), dunes 

of the coarser material are more angular than are 

dunes of finer material. The influence of the difference 

in the shapes of the dunes is reflected in figure 2, where 

the relation of computed to measured bedload is some-
H6 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

10

1.0

0.1

a:

UJ

OL 

fc

UJ 

U_

Z

0.01

0.001

0.0001

0.01

o

x Dunes 0.93-mm sand

O Ripples'1 nin . 

 Dunes )0.19-mmsand

t ! I T I I I I i i I I i l l

0.1 1.0 10

, 

gD

., IN FEET PER SECOND

FJGITBE 5. Relation of V. to ;
BEDLOAD EQUATION FOR RIPPLES AND DUNES H7

x Dunes - 0.93-mm sand 

A Ripples'1

D RipplesT ,. ._ . 

 Dunes ) °-28"mm sand

o Ripples] O19.mm sand 

 Dunes J

0.01 -

0.001

0.0001

i. Relation of h to-^^° ^ D.

what better for the 0.93 mm sand than for the finer 

sizes of bed material.

Figures 2 and 3 show that the relation of computed 

to measured bedload is better for the dune formation 

than for ripples, probably because the dunes are more 

nearly two-dimensional. The equation does not give 

satisfactory results for bed configurations in a transition stage between dunes and a plane bed. The 

unsatisfactory results are due to changes in the shape 

of the dune, to three-dimensional flow and bed configurations, and to the fact that not all of the bedload 

is active in propagating dune movement. Where these 

conditions exist, estimates of bedload must be obtained 

by other methods, such as by sampling (Hubbell, 1964) 

or by computation. Unfortunately, neither current 

(1964) sampling techniques nor existing bedload equations yield satisfactory results.

CONCLUSIONS

Equation 8 should be useful in field applications when 

used in conjunction with a sonic depth sounder, which

provides rapid and accurate determinations of bed 

configurations. Two proposed methods for estimating 

velocities of the bed forms from hydraulic characteristics were found to be inadequate. The analysis of 

flume data in this study gives some indications of the 

limitations of equation 8, and provides general guidelines for field applications. It is concluded that:

1. The equation is more suited for a dune-bed configuration than for ripples.

2. The equation will give more reliable results for 

coarse bed material than for fine material, because 

the dunes of coarse sand are more nearly triangular 

in shape than are the dunes of fine material.

3. The equation is applicable where average velocities 

and heights of the bed forms may be accurately 

determined and where the ripples or dunes cover 

the entire width of the channel. These conditions 

are probably met in straight, uniform channels of 

small width- to- depth ratios, where the flow is 

generally two-dimensional.
H8

i.o

o.i

1

5

5 o.oi

z 0.001

0.0001

0.00001

SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

 i i i i i 111 i i r i i M 11 i i 

A 

A x Dunes 0.93-mm sand 

O Ripples'1-.-

 Dunes j°- 19-mm sand

i l i i i i i i I____ i i i i i i i i I i i i i i i i i i

0.0004 0.001 0.01 0.1

ToVf-it&J. IN POUNDS PER SECOND PER FOOT OF WIDTH

Q

FIGURE.?. Relation of QT to nV C ~ °^ .

1.0
BEDLOAD EQUATION FOR RIPPLES AND DUNES H9

4. Within the above limitations, the equation provides reasonably accurate values of bedload transport rates. For shallow flow over coarse material 

in dune formation, or for flow over fine sand in 

ripple formation, where suspension is negligible, 

equation 8 may be used to estimate total bed 

material transport.

REFERENCES CITED

Bagnold, R. A., 1960, Sediment discharge and stream power a 

preliminary announcement: U.S. Geol. Survey Circ. 421, 

23 p.

Barekyan, A., Sh., 1962, Discharge of channel forming sediments 

and elements of sand waves, in Soviet Hydrology: Selected 

Papers, Second Issue (Amer. Geophys. Union Russian 

Translations), p. 128-130.

Colby, B. R., 1964, Practical computations of bed-material 

discharge: Am. Soc. Civil Engineers Proc., v. 90, no. HY2, 

p. 217-246.

Cook, H. L., 1936, Outline of the energetics of stream-transportation of solids: Am. Geophys. Union Trans., pt. 21, 

p. 456-463.

Gilbert, G. K., 1914, Transportation of debris by running water: 

U.S. Geol. Survey Prof. Paper 86, 263 p.

Hubbell, D. W., 1964, Apparatus and techniques for measuring 

bedload: U.S. Geol. Survey Water-Supply Paper 1748, 

74 p.

Kondratev, N. E., ed., 1959, River flow and river channel formation: Main Adm. Hydrometeorological Service Council of 

Ministers USSR (Selected chapters from Ruslovoi Protsess, 

translated by Y. Prushanoky) PST 471, U.S. Dept. Commerce, Office Tech. Services, 1962.

Karaki, S. S., Gray, E. E., and Collins, J., 1961, Dual channel 

stream monitor: Am. Soc. Civil Engineers Proc., v. 87, 

no. HY6, p. 1-16.

Richardson, E. V., Simons, D. B., and Posakony, G. J., 1961, 

Sonic depth sounder for laboratory and field use: U.S. 

Geol. Survey Circ. 450, 7 p.

Simons, D. B., and Richardson, E. V., 1961, Studies of flow in 

alluvial channels, basic data from flume experiments: 

Fort Collins, Colorado Univ. CER61EVR31.

 1962, Resistance to flow in alluvial channels: Am. Soc. 

Civil Engineers Trans., v. 127, pt. 1, p. 927-1006.

Simons, D. B., Richardson, E. V., and Albertson, M. L., 1961, 

Flume studies using medium sand (0.45 mm): U.S. Geol. 

Survey Water-Supply paper 1498-A.

Znamenskaya, H. S., 1962, Calculation of dimensions and speed 

of shifting channel formations, in Soviet Hydrology: 

Selected Papers, Second Issue (Amer. Geophys. Union 

Russian Translation), p. 111-116.

o