River Channel Bars 

and DunesTheory of Kinematic Waves

GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-L
River Channel Bars 

and DunesTheory of Kinematic Waves

By WALTER B. LANGBEIN and LUNA B. LEOPOLD 

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-L

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1968
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

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

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

Page Page

Abstract_________________________________________ LI Effect of rock spacing on rock movement in an ephemeral

Introduction.______________________________________ 1 stream._________________________________________ L9

Flux-concentration relations________________________ 2 Waves in bed form________________________________ 12

Relation of particle speed to spacing a flume experi- General features________________________________ 12

ment______-_____-____-____-_-____________--____. 4 Kinematic properties_._______-._________________ 15

Transport of sand in pipes and flumes_________________ 5 Gravel bars____________________________________ 17

Flux-concentration curve for pipes________________ 6 Summary__________________________________________ 19

Flume transport of sand_________________________ 7 References________________.______________________ 19

ILLUSTRATIONS

Page 

FIGURE 1. Flux-concentration curve for traffic___________________________________________________-_-_-__-___- L3

2. Sketch of flume___________._________________________________.__ 4

3. Graph showing relation between speed of beads and linear concentration_____________________________ 5

4, 5. Flux-concentration curves for 

4. Beads______________________ _ ________________________________ 5

5. Transport of sand in a 1-inch pipe________________________________________________________ 6

6. Graph showing relation between mean water velocity and mean particle speed________________________ 8

7. Flux-concentration curve for transport of sand.____________________________________________________ 9

8. Location map showing Arroyo de los Frijoles near Santa Fe, N. Mex___---___------------__---------_- 10

9, 10. Graphs showing 

9. Data on rock movement and percentage of rocks moved during two individual flows_____________ 11

10. Representative plots of the effect of spacing on the percentage of rocks moved by different discharges. ___________________________________________________________________________ 13

11. Generalized diagram showing discharge required to move all rocks of given size as a function of spacing. __ 14

12. Graphs showing field data for rock movement in Arroyo de los Frijoles and data for bead experiments in a

flume__-.__________________________________________________________ 15

13. Sketch of tilting flume used to observe movement of beads at different spacings..__________-__---___-__ 16

14-18. Graphs showing 

14. Variation in car density along a highway_____--________--___--____-_---_---__------- ___ 16

15. Generation of waves in a random model_________________-___________-____----__-_-_----__ 17

16. Length of sand waves in a flume in relation to mean water velocity, mean wave amplitude, and

depth of water___________________________________________________________________.__ 18

17. Suggested transport relations in tranquil flow_____________________________________________ 18

18. Suggested transport relations in rapid flow__----_---_----------------------------_--------- 18

19. Suggested flux-concentration curves for river gravel_..___________________-_-__-_-__-_-____-_---_-__-_ 19

TABLES

Page 

TABLE 1. Observations on 0.185-inch beads_______________________________.__________-___ L5

2. Data from Blatch's experiments on transport of 0.6-millimeter sand in a 1-inch pipe.____________-____---_--- 6

3. Data on flume transport of 0.19-millimeter sand_________________________ _-__ 8

4. Observations in miniature flume of slopes at which 25 percent of beads moved.___________---_-___---__----_- 14

in

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

RIVER CHANNEL BARS AND DUNES THEORY OF KINEMATIC WAVES

By WALTER B. LANGBEIN and LUNA B. LEOPOLD

ABSTRACT

A kinematic wave is a grouping cf moving objects in zones 

along a flow path and through which the objects pass. These 

concentrations may be characterized by a simple relation between the speed of the moving objects and their spacing as a 

result of interaction between them.

Vehicular traffic has long been known to have such properties. 

Data are introduced to show that beads carried by flowing water 

in a narrow flume behave in an analogous way. The flux or transport of objects in a single lane of traffic is greatest when the 

objects are spaced about two diameters apart; beads in a singlelane flume as well as highway traffic conform to this property.

By considering the sand in a pipe or flume to a depth affected 

by dune movement, it is shown that flux-concentration curves 

similar to the previously known cases can be constructed from 

experimental data. From the kinematic point of view, concentration of particles in dunes and other wave bed forms results 

when particles in transport become more numerous or closely 

spaced and interact to reduce the effectiveness of the ambient 

water to move them.

Field observations over a 5-year period are reported in which 

individual rocks were painted for identification and placed at 

various spacings on the bed of ephemeral stream in New Mexico, 

to study the effect of storm flows on rock movement. The data 

on about 14,000 rocks so observed show the effect of variable 

spacing which is quantitatively as well as qualitatively comparable to the spacing effect on small glass beads in a flume.

Dunes and gravel bars may be considered kinematic waves 

caused by particle interaction, and certain of their properties 

can be related to the characteristics of the flux-concentration 

curve.

INTRODUCTION

Natural river channels are neither smooth nor regular 

in form. As seen on a map or from an airplane, their 

sinuous curves convey only one aspect of their changing 

form. The streambed has a patterned or textured 

irregularity which is composed not only of the grains or 

cobbles making up the bed surface but also of wavelike 

undulations of larger magnitude. Ripples and dunes are 

characteristic of sand beds except for high flow. As for 

the majority of rivers which have beds composed of 

gravel of heterogeneous size, the alternation of deeps 

and shallows what we have called pools and riffles is 

ubiquitous. These forms result from the accumulation

of gravel in bars spaced along the stream at distances 

equal to five to seven channel widths.

Each of these bed forms ripples, dunes, and gravel 

bars is composed of groups of noncoherent particles 

piled up in some characteristic manner.

As compared with the surface texture presented by 

the bed grains themselves, these bed forms provide a 

larger scale bed rugosity and account for an important 

part of the total roughness. The character of the bed 

end the overall pattern of sinuosity of the channel as 

well as the hydraulics of the system are influenced by 

the undulatory forms assumed by bed particles.

These accumulations of grains, however, are by no 

means composed always of the same particles, for at 

those stages of flow when bed grains are in motion, 

there is a continual trading of particles as some are 

swept away only to be replaced by others. The form 

of ripple, dune, or bar exists independently of the 

grains which compose them and may move upstream, 

downstream, or remain stationary in the channel 

though the particles move into and out of it in their 

more rapid downstream progress. In a typical pool 

and riffle sequence of a gravel bed stream in eastern 

United States we painted individual cobbles on a 

riffle or bar at low flow. After a high discharge the 

painted rocks were gone, but the form and the position of the bar were unchanged. In flows of bankfull 

stage or within the banks, the trading process apparently involves only those particles lying at or very 

near the bar surface. In those rivers which we have 

studied in detail, the surface layer of gravel on a riffle 

bar will be in motion and participate in the trading 

process when the flow reaches about three-quarters 

bankfull stage.

In a fundamental paper, Lighthill and Whitham 

(1955a) direct attention to a class of wave motions 

which they distinguish as being different from the 

classical wave motions found in dynamic systems. 

These waves they call kinematic because their major 

properties may be described by the equation of continuLl
L2 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

ity and a velocity-relation. These define the association 

between the flux or transport of the objects or particles 

(quantity per unit of time) and the concentration 

(quantity per unit of space), which association will 

be called the flux-concentration relation. Although 

Lighthill and Whitham imply a contrast between 

these waves and dynamic waves, it is well to recognize, 

as they do, that dynamic considerations are involved 

in the flux-concentration relation, but that the latter 

relation may be defined experimentally without reference to the dynamics.

Automobile traffic is one of the most easily visualized 

examples of kinematic waves cited by Lighthill and 

Whitham (1955b), because it is experienced in everyday 

living. How exasperatingly often we encounter a slowly 

moving line of cars. Even on the open highway concentrations of cars occur, for it is a usual driving experience 

to overtake and finally progress through or pass a considerable number of cars that are close to one another, 

even after one has driven several miles without passing 

any. Such concentrations of cars are a form of kinematic 

waves, for through them move the objects (automobiles 

in this example) composing them.

These concentrations of automobiles are related to 

a well-known driving axiom that at high speeds a driver 

stays farther behind the car in front than he does when 

traffic is moving slowly. The spacing of cars, in other 

words, is dependent in part on the speed at which they 

are moving. It is an interaction between cars governed 

by the drivers' judgments of braking distance as well 

as other conditions along the road, such as likelihood 

of stopping.

The queue of persons waiting at a bus stop is also 

analogous. The queue may remain at the boarding point 

even though some passengers get on the bus, for other 

prospective passengers take a place at the end of the 

waiting line.

A flood wave in a river is a hydrologic example of 

kinematic waves. The concentration of water particles 

is a function of water depth, for the latter is proportional to the volume of water per unit area of channel. 

The flux-concentration curve for riverflow is the 

familiar discharge rating, as defined from current-meter 

measurements.

In addition to flood waves, other interesting wave 

phenomena in hydrology have been explained by the 

properties of a flux-concentration curve. Waves in 

glacial ice (Nye, 1958) are an example.

In the preceding two examples flood waves and 

waves in glacial ice flux or discharge increases with 

increase in concentration or depth. In these examples, 

as the depth increases, the velocity also increases. 

Their flux-concentration curves are monotonically increasing and do not show the bell shape that is characteristic of the flow of discrete noncoherent particles. In 

the flow of discrete particles in a given environment, as 

their number per unit of space increases, their mutual 

interference increases and their velocity decreases.

This paper, therefore, treats of the flow of noncoherent discrete particles. Several examples will be 

presented. The first is a simple experiment of the 

single-file movement of beads in a small flume. Others 

will deal with the transport of sand and gravel in 

pipes, flumes, and rivers. It is out thesis in the present 

paper that ripples, dunes, and bars on the bed of a 

river are kinematic waves composed of bed particles 

that concentrate temporarily in the bed form as in a 

queue and are later carried downstream.

FLUX-CONCENTRATION RELATIONS

Some properties of kinematic waves are readily 

noted by inspection of a graph in which flux is plotted 

in relation to linear concentration. These terms find 

easiest introduction in the single-lane traffic case. 

Flux refers to the number of cars passing a particular 

point on the route per unit of time. Linear concentration 

or linear density in a given zone is the average number 

of cars per unit distance of traffic lane. The reciprocal 

of linear density is the traffic spacing or average distance 

between cars. Flux is plotted in relation to linear 

concentration in figure 1. The attributes of the relation 

shown in figure 1 have been presented by Lighthill and 

Whitham (1955b).

If the cars on a highway are very far apart : that is, 

when linear concentration approaches zero it seems 

clear that the flux approaches zero, and so the curve 

goes through the origin of the graph. Experience 

indicates further that, as traffic density has increased 

to the bumper-to-bumper condition, flux again becomes 

zero. Thus the curve relating flux to density is of the 

shape shown in figure 1.

To illustrate more specifically, consider the movement 

of cars on the single-lane highway. Many data show that 

the mean speed, v, and mean linear concentration, k, 

follow a linear relationship (Highway Research Board, 

1950) that has been expressed as follows:

(1)

where «0 is the maximum car speed for the given highway and k' is the maximum linear density which equals 

the reciprocal of car length. This equation states that 

mean speed decreases linearly from VQ when a car is 

alone on the highway to zero when cars are bumperto-bumper, which exists when kfk'=l.O. 

Since the product of mean speed by mean linear
RIVER CHANNEL BARS AND DUNES THEORY OF KINEMATIC WAVES L3

3000

2000 -

1000 -

50 100 150 200 250 300 

TRAFFIC CONCENTRATION, IN CARS PER MILE 

FIGURE l. Flux-concentration curve for traffic.

concentration equals mean rate of transport, T, or cars 

per unit of time,

(2)

This equation, which is called a flux-concentration 

relationship, is graphed on figure 1. The equation 

defines a maximum transport rate, which occurs when 

the spatial concentration is half that when traffic is 

bumper-to-bumper, or in other words, when the cars 

are spaced two car lengths apart.

Haight (1963, p. 82) lists several theoretical and 

empirical relations, including the empirical equation 

2, which is his Case VI. The following theoretical 

equation, also included in Haight's list as Case III, is 

based on a car-following theory:

T=vjcln (j-\

Although this relation does not satisfy the condition 

that v=v0 when k is zero, the function indicates that 

transport is a maximum when linear concentration, 

k/k', is 0.37 or when average car spacing equals 2.7 

car lengths.

In either model, transport rates less than the maximum value can, theoretically, be carried at one of two 

traffic densities. Only one of these rates, however, is 

stable. Consider the effect of a perturbation at a point 

A on the rising limb of the curve (fig. 1). If the concentration or density should increase locally, the transport 

capacity would exceed the actual average transport 

rate, and the concentration would tend to return 

toward the average transport rate. Similarly, if the

concentration should decrease locally, the capacity 

would be less than the average transport rate, and 

concentration would tend to increase toward the 

average. Thus, the rising limb of the flux-concentration 

curve is stable. The highway is said to be uncrowded.

A similar analysis on the descending curve, say 

point D, will show that a perturbation, which would 

locally increase the concentration, would decrease the 

flux and so tend to ultimately increase the concentration to a bumper-to-bumper condition. Similarly, a 

perturbation, which would locally decrease concentration, would tend toward further decrease in concentration, and the concentration would decrease toward the 

rising limb. Thus, the descending limb is unstable. 

The highway is said to be crowded.

The instability characteristic of the descending limb 

of a flux-concentration curve is piquantly illustrated 

by the patronage of a honky-tonk bar. People who seek 

amusement of the sort offered are hardly attracted to 

the cold emptiness of Monday mid-morning. Rather, 

they want company. So, if an inspection from the door 

discloses a lively crowd, they go in and try to find a 

table; but if they find it virtually empty, they turn 

away and seek elsewhere. The more people a bar has 

gathered, the more inviting it appears to prospective 

customers, and thus it tends either to be crowded or 

empty. The proprietor, well aware of the effect if not 

the mathematics of the unstable queue, tries to dampen 

the oscillation by such devices as "ladies tables," 

"cocktail hour specials," or myriad mirrors to make 

the place look more crowded, or at least inviting, even 

when sparsely patronized.

Note that it is the interaction among individual units 

that causes changes in concentration and thus gives 

these concentrations the properties of kinematic waves. 

In the traffic case interaction between automobiles 

follows from driver reaction to the longer braking and 

maneuvering distance required at high speeds. In the 

example of patronage at a bar, it is the interaction of 

present and potential patrons that gives the relation 

its wave properties. The discussions of waves in the 

transport of grains by fluids will indicate how interaction among grains leads to analogous wave properties.

In figure 1, a line drawn from the origin to any point 

on the curve has a slope expressed by:

Number of cars . Number of cars __ Pistance__Q ' , 

Unit time ' Unit distance ~ Time ~~ ^ '

or the slope is equal to the average speed of cars at 

that place represented by the point on the curve.

In traffic flow there are, of course, local concentrations 

of vehicles. Behind and in front of such a concentration is 

a zone of lower density, and the alternation along the
L4 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

path may be considered to be a wave, in this case a 

kinematic wave. The zones of low density are troughs 

on either side of a wave peak. Since the spatial concentration in the trough is less than at the wave peak, 

transport or flux is also less than at the peak. Thus, 

there is a loss in volume between trough and peak, 

and the "upstream" part of the wave therefore decreases in volume and in height because of the net 

transport of cars away from the region. The opposite 

is true of the "downstream" part, which increases in 

volume by the exact amount lost by the "upstream" 

part. Thus, the wave form progresses at the rate

The slope of a chord connecting two points on the 

flux-concentration curve equals the celerity (that is, 

speed) of a kinematic wave whose peak and trough 

are represented by the two points, as for example, 

points B and A on figure 1. The derivation of this 

statement follows from the equation of continuity 

exactly parallels the derivation of the celerity of a 

flood wave as described by Seddon (1900).

A wave whose peak and trough correspond to B 

and A would move more slowly than the cars within 

the wave but in the same direction. In the example 

shown the wave celerity is 20 miles per hour. Car 

speed in the troughs (point A) is 44 miles per hour 

and at the peak (point B), 32 miles per hour.

A tangent at point D has a negative slope, and so 

the wave moves "upstream" as a shock. A tangent at 

point C is horizontal, and so the wave is stationary 

relative to the roadway.

RELATION OF PARTICLE SPEED TO SPACING 

A FLUME EXPERIMENT

The flume diagrammed on figure 2 was built only 

for determining whether the speed of discrete particles 

moved by water can be related to the distance between 

them and whether a flux-concentration curve can be 

developed for this kind of transport. Observations 

were made of spherical glass beads in a single-lane trough.

The flume was 2 feet long, 2 inches high, and 0.27 

inch wide and had one side of clear plastic. Glass 

spheres 0.185 inch in diameter were transported by the

water flow on a smooth, painted bed. Thus, the channel 

was wide en6ugh to permit easy passage of the glass 

spheres but not wide enough for the spheres to pass each 

other.

Water was fed into the upper end through a wirescreen baffle; the flow and tailgate positions were 

adjusted to obtain a uniform depth of about seveneighths inch. When so adjusted, the speed of a single 

bead on the bed was about 0.25 foot per second. Speeds 

were determined by timing over a 1-foot reach centrally 

located. The experiment was not designed to measure 

the relationship between water depth, velocity, or slope; 

so, no observations were made of these quantities. 

Water velocity, depth, and rate of flow remained constant throughout, as shown in headnote of table 1.

A number of beads were dropped in at the upper end 

of the flume at approximately equalintervals. A 

sufficient number of beads was used in a run to cover 

at least an 8-inch reach at the indicated spacing. A 

bead selected at random was timed over a 1-foot reach. 

Different beads moved with different speeds depending 

on the size of their group or as they left one group and 

moved ahead to overtake another. To obtain an average 

speed for a given average spacing each "run" was 

repeated several times. Table 1 lists the data obtained 

for beads of 0.185-inch diameter.

Figure 3 shows a plot of linear concentration, Ar/Ar', 

in relation to observed mean speed of beads. The points 

establish a well-defined line until the linear concentration reached about 0.5 bead per diameter. As will be 

explained, no observations could be obtained for closer 

spacings. The line is extended to zero speed observed 

when the beads would be in contact with each other. 

The data define the line V=DO [I (k/k')], which corresponds to the linear equation for highway traffic given 

previously.

The value of VQ , the speed of a single particle on the 

bed of a stream of flowing water, in relation to the 

hydraulic properties of the stream has been reported 

by Krumbein (1942) and Ippen and Verma (1953); so, 

no study was made of this relation.

The retarding action of close spacing may be viewed 

as being due to the shielding action of one particle

i. Water feed

Wire-screen baffle

\ / Funnel for feeding beads

%" clear 

plastic^

0.26" 

A" plywood

Cross section

FIQUBE 2 Sketch of flume.
RIVER CHANNEL BARS AND DUNES THEORY OF KINEMATIC WAVES L5

0.2 0.4 0.6 0.8 1.0 

LINEAR CONCENTRATION, IN BEADS PER DIAMETER

FIGURE 3. Relation between speed of beads and linear concentration.

TABLE 1. Summary of observations with 0.186-inch beads

[Hydraulic conditions: Discharge=0.0057cfs; width=0.022ft; depth=0.073ft; velocity 

=0.35 ft per sec]

Mean linear concentration, k 

(beads per inch)

(1)

2.6

2.5

2.2

2.0

1.67

1.67

1.25

1.25

.90

.83

.74

.39

.50

0

0

Spacing 

(inches)

(2)

0.39

.4

.46

.50

. 6

.60

.80

.80

1. 1

1.2

1.35

2.6

2.0

00

00

Speed of 

beads 

(ft per sec)

(3)

0. 14

.11

.13

.167

.18

. 185

.20

.20

.215

.22

.21

.25

.235

.26

.25

Transport 

rate or flux 

(beads per sec)

(4)

4.3

3.3

3.4

4.0

3.6

3.7

3.0

3.0

2.24

2.2

1.87

1. 15

1.41

0

0

Linear concentration, k/k' 

(beads per diam)

(5)

0.48

.46

.41

.37

.31

.31

.23

.23

.17

. 15

. 14

.07

.09

0

0

upon its neighbors downstream; that is, a particle 

tends to restrict the linkage of its neighbor with the 

water in which the particles are entrained. This 

shielding action was noted when two beads were placed 

in the flume about an inch apart. The upstream bead 

would move faster and overtake the downstream bead, 

and then both would move more slowly as a group.

The speed of a bead selected at random at the 

beginning of the 1-foot reach depended on whether 

the bead became a part of a group or whether it remained separate over the 1-foot measured course. 

The beads did not move uniformly but formed in 

groups rather quickly. Single beads sometimes left 

the front of one group and overtook the group ahead. 

As the rate of feed was increased, the size of the groups 

increased; and the average speed decreased so that 

ultimately a point was reached where a group became 

so large that a jam formed and all motion halted. The 

analogy to the traffic case is evident.

Figure 4 shows the results in the form of a fluxconcentration graph. This graph shows transport rate

SPACING OF BEADS, IN DIAMETERS 

4.0 2.0 1.33 1.0

2 3 

BEADS PER INCH

0 0.25 0.50 0.75 1 

BEADS PER DIAMETER

FIGURE 4. Flux-concentration curve for beads.

in beads per second in relation to the average linear 

concentration, in beads per inch, as calculated from 

the curve on figure 3. The transport rate (col. 4, 

table 1) equals the product of the linear concentration, 

k, and the speed (col. 3) converted to inches per second.

The peak of the graph on figure 4 represents the 

maximum rate of transport in the given flume under 

the given hydraulic factors. The descending branch of 

the curve represents only a kind of statistical average 

between cases when (1) concentrations of beads break 

up or become more open and transport takes place at 

the maximum rate and (2) the beads jam and transport 

is zero. The greater the spatial concentration, the 

greater the frequency of jamming; and when linear 

concentration is such that the beads are in continuous 

contact, transport is zero.

Note that the transport rate increases in a nearly 

linear fashion with density or concentration until the 

concentration reaches 0.7 to 1.0 bead per inch, which 

corresponds to a bead spacing of five to seven diameters. At closer spacing, the transport rate increases at 

less than the linear rate. It can be said, then, that the 

effect of close spacing does not become very large 

until the particles are as close as about seven diameters 

apart.

TRANSPORT OF SAND IN PIPES AND FLUMES

The traffic example and bead experiments serve to 

illustrate the nature of the flux-concentration curve for 

kinematic waves composed by discrete particles. As 

already stated, these flux-concentration curves differ

296-757 O -
L6 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

greatly from those of water waves. For example, the 

flux-concentration wave for water in a channel shows 

increasing velocity for increasing concentration (=stage 

or depth). The same is true of the flow of glacial ice 

(Nye, 1958).

In the traffic and bead examples, particles interact 

so as to reduce the mean speed with increasing concentration. As will be shown in this section, sand movement in pipes and flumes behaves similarly, even 

though as may be readily recognized, the movement of 

grains of sand is unlike that of cars on a single-lane 

road, or of beads in a single-lane flume. Sediment grains 

can readily pass one another; they start, stop, and 

exchange positions in rather complex fashion. The 

analogy is rather on bulk considerations. The interaction is considered not only between particular grains 

but between groups of grains. Grains grouped together 

move less readily than those widely or thinly spaced, 

and one type of group includes grains lying in a ripple 

or dune and thus temporarily covered by other grains.

The analogy suggests that similar flux-concentration 

curves will exist, provided that concentration is interpreted in terms of weight per unit of space, rather than 

particles per unit of space.

There is a further consideration, In the traffic and 

bead examples, environmental conditions remained unchanged ; as for example, in the bead experiment, the 

rate of flow and depth of water were held constant as 

transport was varied. However, the pipe and flume 

experiments were conducted for a different purpose: 

to define transport in relation to gradients, water 

velocity, and other hydraulic factors. In these experiments, therefore, constant environmental conditions 

could only be inferred from the data, by the methods 

explained.

FLUX-CONCENTRATION CURVE FOB PIPES

, Data on sediment transport in pipes are available 

from several experiments by Blatch (1906), Howard 

(1939), and Durand (1953). Blatch, whose experiments 

are among the best available and who perhaps ranks 

as the Gilbert among those who have studied sand 

movement in pipes, prepared a diagram (Blatch, 1906, 

p. 401) showing loss of head in a 1-inch pipe for various 

velocities and various percentages of sand. The "percent sand" is the volume of sand transported in ratio 

to the total mixture discharged. Figure 5 shows the 

bulk transport of sand (product of percent sand X 

mean velocity X cross-sectional area of pipe) in relation to concentration estimated as described in the 

following paragraph. Data plotted in figure 5 are 

listed in table 2, as read from Blatch's diagram for a 

constant head-loss rate of 30 feet per hundred, which

0.4

0.3

0-2 

o

tr

§5 0.1

Head loss, 30 ft 

per 100 ft

0.2 0.4 0.6 

LINEAR CONCENTRATION, IN POUNDS PER FOOT

0.8

FIGURE 5. Flux-concentration curve for transport of sand in a 1-inch pipe 

(data from Blatch, 1906).

is sufficiently great to insure that any deposition 

within the pipe was local and temporary.

In the single-lane flume, as on the highway, linear 

concentration meant the number of particles per unit 

length. In the transport of sand in pipes, one deals with 

bulk volumes of material, and so concentration will be 

defined in terms of weight of sand per unit length of 

pipe. As in other sediment examples discussed in the 

present paper, concentration is defined to include moving plus deposited sediment temporarily at rest. It is 

necessary to emphasize along with Guy and Simons 

(1964) that this definition of concentration is not the 

same as that usually ascribed to sediment in transport mass of sediment per unit mass of water 

sediment mixture.

TABLE 2. Data from Blatch's experiments on transport of 

0.6-millimeter sand in a 1-inch pipe for a head loss of SO feet 

per 100 feet of pipe

[From N. 8. Blatch (1908, p. 401)]

Velocity 

(ft per sec)

1

4

6.7

8.2

8.95

9.25

Percent of sand

16.5

19

15

10

5

0

Transport 

(cu ft per min)

0.06

.27

.36

.30

. 16

0

Linear concentration Ob per ft)

0.58

.43

.27

. 15

.06

0

NOTE. Pipe area 1.04 inch diam= 0.006 sq ft; spatial concentration

= r^c+0.7(l-p]o.9 ib per ft, 

where r'=9.25 ft per sec and7= percent of sand/100.
RIVER CHANNEL BARS AND DUNES THEORY OF KINEMATIC WAVES L7

The concentration in the terms used in the present 

discussion was not measured, but it can be estimated. 

When all said is carried in turbulent suspension, the 

concentration is cAw, where c is the measured proportion of sand in the effluent per unit volume of mixture, 

A is the area of pipe, and w is the unit weight of the 

sediment material. At the other extreme, when the 

pipe is clogged, sediment concentration is (1 i))Aw, 

the factor rj being the porosity, about 0.3 for sand 

particles. Within these limits the concentration is 

assumed to vary linearly with (l v/vf), where v is the 

reported mean velocity and v' is the mean velocity for 

the same rate of head loss for clear water. Other 

variations could have been assumed, but the results 

are not sensitive to this factor. When sand transport is 

light, v=v'; when the pipe is clogged, v=0. Hence, 

lineal concentration in the pipe has been estimated 

for medium sands by the following formula:

At low rates of transport,_w is near v', and the values 

of k approach the value of cAw. However, as v decreases 

relative to v', the value of k tends to exceed cAw and 

to approach 0.7Aw.

A check is available from a curve showing distribution of medium sand across a 4-inch pipe (Howard, 

1939, p. 1339). Integration indicates that the concentration, k, equals 0.28 Aw; value of the parameter, c, 

percentage of sands, was given as 14.4 percent and 

mean velocity of flow in pipe, as 8.88 feet per second. 

From the head-loss curves, the value of v' is about 11 

feet per second. Equation 4 gives a computed Aw value 

of 0.25 whereas the actual value is 0.28. Data on the 

cross-sectional distribution of solids compiled by 

Durand (1953, p. 101-102) provide a corresponding 

check.

Keferring again to figure 5, we note that the data 

define a typical flux-concentration curve, although the 

mechanics are different from the traffic or bead flume 

cases. According to Blatch (1906), the data representing 

the points on the ascending branch of this curve correspond to complete suspension of the transported sand. 

The descending branch indicates that sand is also being 

dragged along the bottom of the pipe. A layer of sand 

is built up on the bottom, and the sand is transported 

much in the same manner as in an open flume. As load 

increases, there is spasmodically a blockage, and transport exists only in an average sense. Transport ceases 

entirely as spatial concentration approaches its maximum, 0.63 pound per foot. Significantly, however, 

maximum transport occurs when concentration is at 

about 45 percent of its maximum value. Thus, the

pipe-transport example and the previous bead example two examples of the kinematic wave theory 

demonstrate that the average speed of particle movement decreases with an increase in concentration.

FLUME TRANSPORT OF SAND

Transport of noncoherent particles expressed in terms 

of kinetic theory, as derived from the traffic analogy 

and the bead experiments, is given in equation 2. In 

this equation v0 is the velocity of a single particle, k is 

the linear concentration (particles per unit length), 

and k' is the reciprocal of the linear dimension of a 

single particle.

To apply this formula to the transport of sand by 

flowing water in an open flume, T is defined in pounds 

of sand carried per second per foot of width and k, in 

terms of areal concentration pounds of sand in motion 

per square foot. Thus,

W (5)

where z>0 is the velocity of particles as concentration 

approaches zero. The value of v0 is a function of the 

velocity of water (depth and size of particles are 

assumed to be constant), and W is the areal concentration weight of sand per square foot of channel. 

According to the theory, the factor in parentheses of 

equation 5 decreases as concentration W increases, 

reaching zero as W=W where W is the areal concentration when transport ceases. The value of W conceptually corresponds to a state represented by dunes 

whose height is so great as to block the flow in the 

flume. The value of W should therefore vary with water 

depth, but it is assumed to be a constant in a set of 

flume runs in which depths vary over relatively conservative range.

As in the pipe example, concentration is the weight of 

moving plus deposited sediment temporarily at rest. 

Concentration includes sediment moving in suspension 

or in bed forms. In the flume moving bed forms account 

for nearly all transport. An estimate of W can be made 

from the reported height of dunes or sand waves on the 

flume bed (if suspended sediment is neglected). Inasmuch as the average depth of sand in motion is about 

half the dune height and the weight-volume ratio is 

about 100 pounds per cubic foot, therefore W in pounds 

per square foot numerically equals 50 times dune height 

in feet. That this estimate is correct may be judged 

from the experiments with radioactive tracer sands 

reported by Hubbell and Sayre (1964). They show that 

depth of zone of particle movement as determined by 

sampling the bed is closely the same as that computed 

from the bed forms. The validity of equation 5 can be
L8 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

tested using the data for 0.19 tested using the data for 

0.19-millimeter sand as reported by Guy, Simons, and 

Richardson (1966) and listed in table 3.

Transport is, in effect, the product of two factors 

mean particle velocity and areal concentration, W. 

Mean particle velocity, as previously stated, is a function of the water velocity and, as in the previous examples, of the concentration as well. Thus, there are 

two independent factors water velocity and concentration. Their effects can be separated by multiple 

regression analysis. The results of the regression 

analysis by a converging graphical process (Ezekiel 

and Fox, 1959) are shown in figures 6 and 7.

TABLE 3. Observed data for flume transport, 0.19-millimeter sand 

[Source: Guy, Simons, and Richardson (1966)]

Run

22a

25

20 

1

31

27

5 

23 

32

8 

28

33 

29

3 

11 

13

14

15

34 

12 

6 

7 

10 

9 

17

18

35

Velocity 

(ft per sec)

0.78

.87

1.11

.74

1.30

.93

1.54 

.85 

1.79 

1.99 

1.04

1.96 

1.13

1.18 

2.35 

3.09 

3.22

3.46

1.68 

2.69 

1.67 

1.78 

2.89 

2.10 

4.14

4.33

1.81

Mean 

depth 

(feet)

0.48

.93

1.00 

.58

1.02

.55

1.03 

.44 

.95 

.93 

.54

1.06 

.56

.55 

1.09 

.89 

.86

.79

.52 

1.02 

.61 

.68 

.51 

.49 

.67

.64

.52

Mean 

amplitude 

of sand 

waves (feet)

0.028

.034

.032 

.023

.038

.033

.045 

.031 

.096 

.18 

.038

.65 

.038

.035 

.39 

.13 

.17

.04

.044 

.32 

.18 

.31 

.10 

.20 

.10

.10

.13

Mean 

length of 

sand waves 

(feet)

0.57

.44

.62

.58

.53

.92 

.45 

3.97 

5.39 

.58

8.0 

.60

.70 

11.6 

13.4 

18.8

20.5

1.67 

17.7 

5.06 

7.4 

24.0 

5.19 

4.4

4.9

4.5

Speed of 

sand waves 

(ft per sec)

0.0004 

.000014

.0013 

.000049 

.004 

.003

.002

.00053 

.0035 

.0033

.0043 

.0033 

.0017 

.004 

.013 

.0025

.004

Transport T 

(Ibs per sec 

per ft;

0.

.000015

.00026 

.000026

.0024

.000125

.012 

.000046 

.03 

.06 

.0012

.11 

.0024

.0034 

.21 

.21 

.26

.34

.027 

.22 

.055 

.094 

.23 

.077 

.80

1.6

.059

Figure 6 shows the variation in v0) the imputed velocity of particles as concentration approaches zero, with 

mean water velocity. Values of v0 equal the quantity

==(!-:==,). It should be kept in mind that the curve of 

W W

figure 6 expresses a dynamic relation that applies only 

to the particular set of experiments with unigranular 

sands. Note that values of VQ are small relative to mean 

water velocity but somewhat greater than wave speeds 

listed in table 3.

Figure 6 applies where transport is in the bed forms 

and where suspended sediment represents saltation of 

grains from one position of rest to another. The data 

listed in table 3 include all those runs made by D. B. 

Simons and E. V. Richardson for which bed-form data 

were reported. This information was not given for runs 

made at steep slopes. Descriptions of these runs indicate that the bed was composed of "chutes and pools" 

and that sediment transport was in dispersed or suspended form. In these steep-slope runs, spatial concentration was very low, and VQ =V.

Figure 7 shows the values of T/v0 corresponding to 

various values of areal concentration, W. The curve 

corresponds to an imputed flux-concentration curve 

if water velocity in each run were the same (VQ =constant), but the runs differ in flux (transport) and in 

concentration (dune heights). Figure 7 is defined by 

multiple regression in that the values of v0 are read off 

the graph of figure 6. Thus, points in figure 7 correspond to different runs, each with different water 

velocities and therefore different conditions. Specifically, ordinate values of figure 7 represent the ratio of 

transport as measured in a given run to the value of

10 ~i i i i i i 11

> 0.5

o.i i i i i i 11 i i i i i i i i i i i i 11 11

.00001 .0001 .001 .01 

MEAN PARTICLE SPEED (vo), IN FEET PER SECOND

FIGUBE 6. Relation between mean water velocity and mean particle speed, 0.19-millimeter sand.

0.1 1.0
RIVER CHANNEL BARS AND DUNES THEORY OF KINEMATIC WAVES L9

15

10

0 10 20 30 40

AREAL CONCENTRATION (W), IN

POUNDS PER SQUARE FOOT

FIGURE 7. Flux-concentration curve for transport of sand.

v0 read from figure 6 for the mean water velocity 

measured in that run. In this way the observed transport data are adjusted for different water velocities as 

mentioned above.

The ordinate of figure 7 has the dimension of concentration as does the abscissa. The slope of the curve 

is 1 to 1 relative to the scales used at the origin. Thus, 

at the origin (concentration is low), mean particle 

speed equals v0 . As concentration increases, mean 

particle speeds decrease and thus the curve deviates 

from a 1 to 1 slope and flattens; in other words, the rate 

of increase in transport lessens as concentration 

increases.

There is a suggested maximum for a value of W 

of about 25 pounds per square foot, and W may be 

estimated at 50 pounds per square foot. The data 

shown in figure 7 follow the curve

"SO''

Hence transport in this set of flume experiments 

may be described by

_ ' ^

where v0 is defined by figure 6. The inference is that 

transport would be zero when W 0 and when W=50. 

This equation indicates that transport for a given 

depth and size of material varies with the water velocity 

in accord with previous findings by Colby (1964), and 

with areal concentration as well. Mean particle speed, 

T! W, decreases linearly with W to a value of zero when 

W=W.

However, in a sand-channel flume, unlike the bead 

flume, water velocities differ at troughs and crests of

dunes; therefore different flux-concentration curves 

apply at such points. Therefore the imputed fluxconcentration in figure 7 should not be used to define 

wave celerities. This matter is developed further in 

the section on "Waves in bed form."

THE EFFECT OF BOCK SPACING ON ROCK MOVEMENT 

IN AN EPHEMERAL STREAM »

Even before the experiments were set up to test the 

effect of spacing of beads in a small flume, we had conjectured from consideration of the high way traffic 

analysis that the interaction between individual rocks 

must be an important factor in the formation of gravel 

bars on a streambed. Observations of rock movement 

on a streambed, which were made before some of the 

other research discussed in this paper had been conducted, suggested the mode of investigation to be 

followed. The design of the field experiment perhaps, 

in retrospect, could have been made somewhat more 

efficient for present purposes. However, despite their 

limitations, the results are in keeping with the observations of the bead and flume experiments.

In streambeds that are covered by gravel of heterogeneous size, alternation of pools and riffles is an 

obvious characteristic. But the undulation of bed 

elevation is small or not apparent in those ephemeral 

channels that so commonly rise in the foothills in the 

semiarid countries and that either debouch in alluvial 

basins or reach master streams of a perennial character 

rising in nearby mountains. These sandy washes range 

in size from a few feet to several hundred feet in width 

and characteristically exhibit a bed profile of nearly 

uniform slope through reaches of moderate length. 

They have no deep hollows or deposited bars typical of 

gravelly streams of perennial flow. Walking along a 

typical sandy arroyo, one has the impression of walking 

on a nearly level surface free of bed undulations either 

along the length or across a main channel.

Such washes are ubiquitous in New Mexico in a broad 

area that is drained westward from foothills along the 

base of the Sangre de Cristo Range. One of these 

washes the Arroyo de los Frijoles, a tributary of the 

Santa Fe River and thereby of the Rio Grande was 

chosen for an intensive study. Its location, about 10 

miles northwest of Santa Fe, N. Mex., is shown in 

figure 8. It is in a semarid area having about 12 incnes 

of precipitation annually. Other investigations in the

i The fleldwork described in this section was initiated by L. B. Leopold with the 

help of the late John P. Miller and was pursued with the assistance of Robert M. 

Myrick, William W. Eramett, and personnel of the U.S. Geological Survey at Santa 

Fe, N. Mex.; all are sincerely thanked for this effort. The interest and the work of 

Wilbur L. Heckler, Leon A. Wiard, Leo Q. Stearns, Louis J. Reiland, Kyle D. Medina, and Charlie R. Siebet are particularly acknowledged for they not only helped 

with the painted rock experiments but made the field surveys and office computations of the flow discharges.Other aspects of the field experiment are discussed by 

Leopold {Leopold and others, 1966).
L10 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

same area are reported by Leopold and Miller (1956). 

The arroyo rises in low and rounded foothills that are 

underlain by Pleistocene fan gravels and the Santa 

Fe Formation, a poorly consolidated mixture of rounded 

gravel and sand. Despite the large amounts of gravel 

which tend to pave some of the hillslopes nearby, the 

ephemeral washes that drain the area are composed of 

sand and only minor amounts of gravel. Detailed 

planetable mapping shows gravel accumulations occurring hi a regular pattern along the channel length at a 

spacing comparable to that observed in the pool and 

riffle sequence of perennial streams. Furthermore, 

remapping at yearly intervals showed the location of 

these gravelly accumulations remained more or less 

constant from year to year.

A large number of holes were dug in the streambed, 

both on the gravel accumulations which we will here 

call bars and on the intergravel sandy areas. Nearly 

without exception it was found that the gravel occurred 

only as a thin skin, 2 to 4 cobble diameters in thickness, 

lying on top of the sand.

Flows which occur in these washes average in number 

about three per year and are caused by summer thunderstorm rainfall only. Snowmelt in spring produces 

no surface runoff. During a flow the channel typically 

scours to a depth averaging at least one-quarter of 

the maximum depth of the flowing water.

Individual rocks were painted and placed on the 

streambed in groups with varied spacing between the 

individual cobbles. The experiment was designed to 

determine the effect of spacing as well as of discharge 

and rock size on the ability of flowing water to move 

the rocks downstream. The rocks selected to be painted 

were obtained from the streambed itself. They were 

individually weighed and placed in groups of 24 rocks 

each. On each rock was painted a number that represented its weight in grams. Because the numbers 

generally consisted of four digits, the number also 

identified each rock.

Six categories of rock size were chosen, ranging from 

300 to 13,000 grams, and each individual group of 24 

rocks contained four of each of the six weight categories. The 24 rocks were arranged in a rectangle six 

rocks in the direction of the channel and four rocks 

in the cross-channel direction. The rocks within each 

group were arranged in a latin square so that in a given 

row two rocks of the same size did not occur and so 

that is each column of six rocks all the size classes were 

represented. Each group was arranged with one of 

three spacings; that is, in a given group the rocks were 

placed a distance apart equal to 0.5 foot, 1 foot, or 

2 feet. Typical rock groups placed on the streambed 

and rock movement resulting from a small flow are 

shown hi figure 152 of Professional Paper 352-G.

106°

i Area of map

NEW MEXICO

Lr,INDEX MAP

FIGURE 8. Arroyo de los Frljoles near Santa Fe, N. Mex.
RIVER CHANNEL BARS AND DUNES THEORY OF KINEMATIC WAVES Lll

The groups were placed in three reaches of the 

channel the most upstream reach was 9 feet wide, 

the intermediate reach was 35 feet, and the downstream 

reach was 100 feet. The number of rocks in place before 

a given flow averaged about 900 during the 5 years 

of experiment.

After each flow the streambed was searched the 

whole distance affected by that flow. The location of 

each rock found was recorded and related to its position before the flow occurred. After a small flow, 70 

90 percent of the rocks was usually recovered; and 

after a large flow, 50 to 80 percent recovery was usual. 

In one big flood, however, 98 percent was not found 

and was presumably carried out of the 7-mile study 

reach.

In the period from 1958 to 1962, a total of 14 flows 

occurred; but because of the scattered nature of thunderstorm rainfall, each flow did not necessarily affect 

all rock groups nor each of the three principal locations 

where the rock groups were placed. Discharges were 

measured by indirect or slope-area methods, the crosssectional area being determined by measurement of

100

scour depth registered by buried chains. Because of 

the difference in channel width at the various locations, 

discharge was tabulated for purposes of analysis as 

cubic feet per second per foot of channel width.

In the experiment, therefore, there were four variables discharge, rock size, rock spacing, and percentage 

of rocks moved by a given flow. Owing to the fact 

that even in a single cross section a given flow may not 

have a uniform effect on each rock group, the data are 

understandably scattered, and the problem in analysis was to choose a method of plotting which smoothed 

the scatter in some consistent and reasonable way. 

Smoothing of the data is justified because the total 

number of observations of painted rocks affected by 

a flow during the 5-year period was about 14,000.

The analysis began by treating each flow event 

separately and for a given flow plotting the percentage 

of rocks of a given size which moved from their original positions as a function of rock size and spacing. 

Two typical plots of this relationship are shown in 

figure 9 which shows that in a given flow a larger percentage of small rocks was moved than of large and

Symbol Spacing

 0.5ft

A 1.0ft

x 2.0ft

9 =1.8cfs per ft of width 

Total of 648 rocks

q = 1.2 cfs per ft of width 

Total of 648 rocks

100

200 500 1000 2000

PARTICLE WEIGHT, IN GRAMS

5000 10,000 20,000

FIGURE 9. Original data on rock movement plotted as percentage of rocks moved during two individual flows represented in the two separate graphs. Each graph represents data from 648 painted rocks affected by the flow.
L12 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

that rocks in groups having wider spacing (smaller 

value of linear concentration) moved more readily than 

rocks which were close together.

By considering all flows and only those rocks of a 

given size, we constructed plots from values read off 

the smooth curves of figure 9; the percentage of rocks 

moved was expressed as a function of discharge with 

rock spacing as a third variable. Examples for two rock 

sizes are presented in figure 10. The lines in figure 10 

were drawn as envelopes so that, other than groups 

where 100 percent of the rocks moved, at least 80 

percent of the individual points lies above the respective 

line. For example, for rocks averaging 1,000 grams, a 

line representing a spacing of one-half foot envelops 

or lies below all points except two.

The abscissa value, where a curve intersects the 

ordinate value of 100 percent of rocks moved, is an 

estimate of the discharge required to move all rocks 

of a given weight and at a given spacing.

Though the positions of these lines must perforce 

be subjective, a pattern that emerged in figure 11 was 

obtained from the intercepts on the discharge scale for 

100 percent of rocks moving.

The abscissa on the figure has two scales the 6-axis 

diameter and weight of particles. The ordinate also has 

two scales a scale of discharge in cubic feet per second 

per foot of width required to move 100 percent of the 

rocks and a scale of shear corresponding to the values 

of discharge using field measurements of stream slope. 

The family of lines represents spacing converted from 

feet to particle diameter. The graph shows that 100 

percent of the rocks of a particular size will be moved 

only by increasingly larger discharges as the spacing 

between the rocks decreases. In agreement with the 

data on glass beads, particle interaction is negligible 

for spacing greater than about eight diameters.

Another plot can be constructed from straight-line 

graphs exemplified by those in figure 11. For different 

values of particle size the discharge required for movement of 100 percent of the rocks was plotted in relation 

to rock concentration (rocks per diameter), as shown 

in figure 12A. If these lines are extrapolated to the 

point where each intersects the ordinate axis, the values 

then represent the theoretical discharge necessary to 

move all the rocks of a given size at zero concentration 

that is, for a single particle. The values of these discharges for zero concentration were used as the denominator in the ratio

____Discharge for the individual event____ 

Discharge for zero concentration (single particle)

which will be referred to as the discharge ratio. This 

discharge ratio could then be plotted against spacing

in diameters to result in the nondimensional diagram 

shown in figure 12.5.

Comparable date were obtained in a miniature tilting 

flume shown on figure 13 by using glass beads of 0.12- 

and 0.18-inch diameter. (See table 4.) Beads were 

placed at various spacings, and the flume was filled 

with flowing water sufficient to cover them by a uniform depth of at least three diameters. With discharge 

constant, the flume was then slowly tilted until the 

beads began to move. Readings were made of the 

slope at which 25 percent of the beads moved at 

various linear concentrations for a fixed discharge. 

A 25-percent value was used because the single-track 

flume could not be tilted sufficiently to obtain 100- 

percent movement when beads were closely spaced.

Values of slope representing zero concentration were 

estimated from, the data and used to compute values 

of a slope ratio, defined as

Slope of flume in individual experiment 

Slope for zero concentration (single particle)'

The data are plotted in relation to the square root of 

this ratio in figure 12B. The rationale for the square 

root is that discharge varies at constant depth as the 

square root of slope, as indicated in the Chezy equation. The ratio of the square roots of the slope is thus 

comparable to the discharge ratio in the arroyo. The 

comparison between movement of rocks in the arroyos 

and glass beads in the miniature flume is presented in 

figure 125.

The rough agreement between the comparable plots 

for these two very different experiments may be partly 

a matter of coincidence. But, two types of quantitative data rock movement in a sandy ephemeral 

wash and experimental findings concerning bead* 

moving in a small flume both show that, as the par 

tides are spaced more closely, the slippage is greater 

or the linkage is weaker between the flowing water 

and the grains. A larger shear must be administered 

to move individual particles closely spaced. The relation 

expresses a nearly linear tendency for particles to be 

moved as their spacing increases.

WAVES IN BED FORM

GENERAL FEATURES

Each of the three kinds of waves dunes, antidunes, 

and gravel bars that has been observed in flumes and 

rivers complies with the kinematic properties of the 

flux-concentration curves. Dunes can result from 

random variations in spatial concentrations which 

give rise to wave forms such as traveling queues or 

platoons on the highway. These dunes can be explained
RIVER CHANNEL BARS AND DUNES THEORY OF KINEMATIC WAVES L13

100

Particle weight 1,000 grams

Symbol Spacing

 0.5ft

A 1.0 ft

X 2.0ft

20

FlGDKE 10. I

5 10 

DISCHARGE, IN CUBIC FEET PER SECOND PER FOOT

on the percentage of rocks moved by different discharges. The re 

Data on these graphs read from smooth curves of which figure 9 shows two examples.
L14

30

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

PARTICLE SIZE (b AXIS), IN FEET 

0.23 0.30 0.39 0.51 0.64

3

200 500 1000 2000

PARTICLE WEIGHT, IN GRAMS

5000 10,000 20,000

FIGTTBE 11. Generalized diagram showing discharge required to move all rocks of given size as a function of spacing. Derived from values read from graphs typified

by figure 10 where lines intercept ordinate value of 100 percent.

TABLE 4. Observations in miniature flume of slopes at which 

25 percent of beads moved

[Discharge constant for each run, but not measured]

Linear concentration 

(beads per diameter)

Slope Slope ratio

Ban 1 (0.18-inch beads)

0 50 0. 053

32 044

20

13

034

0

023

02

0125

0115

4.6

3.8

2.0

1.75

1. 1

1.0

Ban 2 (0.12-inch beads)

0.475

.26

.053

0

0. 047

.025

.011

.0096

4.9

2.6

1. 15

1. 0

Ban 3 (0.18-inch beads)

0.90

.40

.03

0

0.10

.028

. 0093

.0085

12.0

3.3

1.1

1.0

Bun 4 (0.12-inch beads)

0.90 

.53

.014

0

0.087

.043

.0078

.0075

12.0

5.8

1.05

1.0

on the basis of kinematic relationships. As shown by 

field and laboratory evidence, the probability of a 

particle being moved increases as the spatial concentration decreases. Therefore, areas of low concentration 

(thin bed forms) tend to become thinner as areas of 

high concentration increase. Waves are formed that 

take on characteristic lengths and heights.

One of the well-known properties of highway traffic 

is the Poisson distribution of vehicular spacing, but 

even here waves or platoons, as they are called are 

formed. Figure 14, for example, shows "waves" in 

traffic along a 50-mile stretch of highway in Arizona. 

The example was obtained from a part of strip aerial 

photographs made along the highway from Tucson 

to Eloy. The vehicles measured were in a single lane 

in a divided four-lane highway. Such waves in highway 

traffic are common enough, but crossings, trucks, 

and towns may cause them to be very irregular.

Fairly regular waves can be generated by a random 

process, as illustrated by the following model. Consider 

initially a distribution of particles along a line with 

average concentration of k particles per unit length. 

These particles are subject to the condition that, where 

particles happen to be far apart, they are more easily 

moved than where particles are close. Thus, the movements will collect the particles into groups having wavelike form. The results of a trial are shown in figure 15. 

Particles that have an. average concentration k are.
RIVER CHANNEL BARS AND DUNES THEORY OF KINEMATIC WAVES L15

0 0.2 0.4 0.6 0.8 l.O 

LINEAR CONCENTRATION, IN PARTICLES PER DIAMETER 

A

 Q_ 

._ O LLl 8*1

-

p 1

Q <

Rocks in Arroyo de los Frijoles

0 0.2 0.4 0.6 0.8 1.0

LINEAR CONCENTRATION, IN PARTICLES PER DIAMETER

B

FIQUEE 12. A, Field data for rock movement In Arroyo de los Frijoles. B, Comparison of movement of rocks in arroyo with beads in flume.

arranged in random position in the first column (marked 

"Time interval 0")- Only one particle is permitted in 

a single position or box. Each column represents a successive unit of time, and the direction of motion is down 

the sheet. The diagram shown is merely the central part 

of a longer sheet on which a very long sequence of wave 

groups was developed. The whole suite of particles may 

be moving with some speed which does not concern this 

analysis of relative speeds and positions.

The following rule governs this relative movement. 

A particle cannot move if there is one in the space 

immediately ahead. A particle at the front of a group 

is free to move; and if it does so, it will move and overtake a particle in the group ahead. A particle free to 

move in a particluar time interval is selected to do so 

at random, its probability of selection being p. The 

process here of a particle taking off at random and 

overtaking the one ahead is not unlike that observed 

in the single-lane flume. (See p. L5.)

Figure 15 shows an experimental example where 

&=0.33 and p=0.50. After 10 tune intervals, a welldefined pattern of groups is established owing to the 

interaction between particles. The lengths of such 

wave patterns would be proportional to the mean 

concentration and speed. Figure 16 shows, for example, 

the relation of length of sand waves to water velocity 

and to the ratio of amplitude of waves to depth of 

water over dunes as observed in the set of experiments 

on 0.19-millimeter sand in a flume at Fort Collins, 

Colo., listed in table 3 (Guy and others, 1966). Because 

these relations are of a random nature rather than of 

dynamic origin like the unique relation between velocity 

and length of antidunes also shown in figure 16, precise 

relationships cannot be found.

KINEMATIC PROPERTIES

The flux-concentration curve, shown in figure 7, 

for transport of sand in a flume describes the relation 

between transport and average concentration along 

the flume for different runs. Consider a single run. 

Transport and concentration differ along the flume.

Unfortunately, flume experiments only report averages 

for each run, and so the nature of the flux-concentration 

curves along the flume can only be inferred as on figures 

17 and 18.

Flux-concentration curves, as suggested in figures 17 

and 18, can be useful in explaining the development of 

modes of transport, dunes, plane bed, or antidunes 

as usually observed in flume experiments. A perturbation in the bed creates a difference in depth of sediment, 

in velocity, and in rate of transport at that point. 

When velocities are subcritical (that is, less than vgD), 

the velocity over the top of a protuberance in the bed 

is greater than in a trough. One may infer a different 

flux-concentration curve for the crest and for the 

trough. However, the actual relationship is probably a 

loop as shown.

The relations for dunes are indicated by points 1 and 

2 in figure 17. At point 1, representing the crest, 

velocities are higher than at point 2, representing the 

trough, and transport is greater over the crest than 

in the trough. Thus, the action tends to decrease the
L16 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

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FIGTJEE 13. Tilting flume used to observe movement of beads at different spacing?.

20

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DISTANCE, IN MILES 

FIGURE 14. Variation in car density along a highway.

crest of the protuberance and to deposit particles in 

the trough. However, because of the forward motion, 

the cutting and filling action is continuous and leads 

to the formation of dunes in forward motion as previously explained. The rate of forward motion of the 

dunes equals the slope AT/A& of the chord 1-2 in 

figure 17, which is positive that is, downstreamward 

in this phase.

When transport is low, chord 1-2 is short; when 

transport increases, the chord moves up the graph and 

increases in steepness. Dune speed increases, but the 

dune height (the horizontal projection of chord 1-2) 

decreases. Ultimately, dune height decreases to zero, 

and the plane-bed phase is reached.

As velocity increases and approaches -^gD, standing 

hydraulic waves are formed. However, it should be 

noted that, although the lengths of antidunes or 

standing waves are controlled dynamically (Kennedy, 

1963), their speed can be inferred from the kinetic 

properties of the flux-concentration curve. These waves 

induce perturbations in the bed, as shown in figure 18. 

When velocities are supercritical (that is, greater than 

, the depth of water over the tops of the bed forms

is greater than the depth of water over the troughs; 

therefore, the water velocity over the tops is less than 

in the troughs. Hence, the slope of the chord connecting 

points 1 (crest) and 2 (trough) will be either flat 

(standing waves with zero speed) or negative (antidunes). The corresponding points are shown, in figure 

18. Transport is greater in the trough than at the 

peaks, and the sand waves grow in height, eventually 

become unstable and break, and throw large parts of the 

load into turbulent suspension. The slope of chord 1-2 

is negative; hence, wave movement is upstream, and 

the waves are therefore called antidunes.

As explained on page L4, the rate of movement 

of dunes equals the slope of a chord connecting points 

on the flux-concentration curves for troughs and the 

peaks. Dune speed, c, therefore equals the ratio

c=-

where the subscripts 1 and 2 refer to transport, T, and 

area! concentration, W, at the crest and trough, 

respectively. Since Wi W2=w(hi h2) and hi hz=h, 

mean amplitude of dunes, then

chw=Tl-T2

where w is the unit weight of the sand. For those runs 

in which wave speed, c, and mean height of dunes were 

observed, the left-hand side of the equation may be 

calculated, but data on TI and T2 are not available. 

It would be possible to report these quantities by 

observing changing transport rates as dunes move out 

of the flume.
RIVER CHANNEL BARS AND DUNES THEORY OF KINEMATIC WAVES LIT

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TIME INTERVAL 

FIGTJEE 15. Generation of waves in a random model.

Letting Ti = b T, where !Tis the mean rate of transport which is also assumed here to be the arithmetic 

mean of TI and T2 , then

where VP, the mean particle speed, equals TJW. The 

factor, b, can be calculated in the set of experiments 

cited in table 3 for those runs in which dune speed, c, 

was observed. It ranges from a negative quantity for 

those runs with low velocity and increases to over 0.7 

for those runs with high velocity. The significance of 

this range is that for low-velocity runs the interdune 

transport (net transport from one dune to the dune 

ahead) is zero or even upstream, but it increases as 

velocity increases. The interdune transport, for any 

given velocity also increases as dune height decreases. 

These results are in accord with the observation that as 

velocities increase the difference between crest and 

transport decreases and residence time in the dunes 

decreases.

GRAVEL BARS

Because of the limited supply of gravel, the spatial 

concentration of gravel is almost entirely in "wave" 

form. The "crests" of these waves are riffles or channel 

bars. The flux-concentration curve for the bar may 

be as suggested in figure 19. The flux is derived 

entirely by erosion of the bar deposits, where, in the 

kinetic terminology, spatial concentration decreases 

as river flow increases points 1, 2, and 3 in figure 19. 

The flux-concentration curve for low riverflow, zero 

gravel transport, coincides with the horizontal axis. 

On the other hand, the flux-concentration curve for 

the troughs (where concentration is near zero) virtually 

coincides with the vertical axis.

The flux is indicated to be the same in the troughs 

and the bars, and so the slopes of the chords 1-1, 2-2, 

and 3-3 are horizontal ; these slopes indicate that wave 

velocity is zoro. That is, the bars are fixed relative to 

the channel.

Mean particle velocity in the troughs is high; it is 

low in the bar. (Again, when we speak of mean particle 

velocity, it is the mean speed of all particles having 

concentration, k, even though not all are in motion 

at a particular time.)

A small-flume experiment confirmed earlier observations by Gilbert (1914, p. 243). A supply of sand formed 

dunelike deposits separated by bare reaches approximately equal in length. In the flume, particles moved 

rapidly from the toe of one dune to the heel of the one
L18 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

1 T

Parameter is ratio of mean 

amplitude of dunes to mean 

depth of water over dunes

0.15 1.0 5 

LENGTH OF SAND WAVES, IN FEET

FIOTTBE 16. Length of sand waves in a flume in relation to water velocity and to ratio of mean amplitude of dunes (0.19-mm sand) to mean depth of water over

dunes.

Flux-concentration curves Wave form (dunes) 

FIGUBE 17. Suggested transport relations in tranquil flow.

'<

\

\

\ \

Flux-concentration curves Wave form (antidunes)

FiGtJBE 18. Suggested transport relations in rapid flow.
RIVER CHANNEL BARS AND DUNES THEORY OF KINEMATIC WAVES L19

FIGUEE 19. Suggested flux-concentration curves for river gravel.

downstream. In this steady state of water and sediment 

input, the dunes moved slowly downstream. However, 

when the flow of water was increased, the dunes were 

slowly eroded away, but they remained fixed in position 

and did not then migrate down streamward.

As observed in rivers, gravel bars are spaced more or 

less irregularly at distances that average five to seven 

channel widths. The spacing of bars is seemingly controlled by the channel dimensions and remains fixed 

even though the velocity changes. To the extent that 

the theory relating to dunes applies, an increase in the 

velocity would tend to increase the spacing. However, 

the accompanying increase in depth of water and a 

decrease in height of the barjby scour has the opposite 

effect; therefore, the two conditions may combine so 

as to maintain bars at fixed distances apart.

SUMMARY

The theory and experimental evidence presented 

show that interaction among the individual units 

moving in a continuous flow pattern causes velocity of 

the units to vary with their distance apart. Concentrations of these units have certain wave properties 

encompassed under the term "kinematic waves." 

With these properties it is unlikely that such particles 

can remain uniformly distributed for any distance along 

the flow path, for random perturbations lead to the 

formation of groups or waves which take the form of 

the commonly observed dunes in sandy channels and 

the pool and riffle sequence in gravel river beds. The 

rationale may be summarized as follows.

Particles of sand or gravel which can be moved by 

the flowing water are seldom all in motion simultaneously unless their number is very small. There is some 

probability of one being set in motion instead of its 

neighbor. Once set in motion, the speed of a given 

particle is affected by its proximity to neighboring 

moving grains; the mean grain speed is slower when 

the grain density is higher. The result of this interaction 

is to intensify any initial random grouping whereas open 

spaces or places where grains are sparse tend to become

even more open have even smaller density of grains. 

Thus, any random influx of grains into a reach of 

channel will not continue to be random, for as the 

grains travel downstream their interaction will set up 

waves that is, groups of grains separated by relatively 

open spaces will tend to attain a more or less regular 

spacing along the direction of flow.

By such a process sand collects in dunes which are 

small compared with channel width. The same principle 

will apply in a transverse section as in the longitudinal 

profile just described. Transverse to the current the 

dunes will also alternate with open spaces or troughs. 

Thus, there will be built the pattern of dunes so characteristic of sandy channels, in which the whole surface 

is covered with dunes separated by troughs; the dunes 

will be arranged in a rather uniform and random 

pattern, with some dunes arranged in echelon but 

others appearing to be uniformly irregular.

In the situations where the particles composing the 

kinematic wave are only one to several diameters thick, 

as in beads in a single lane flume or a veneer of cobbles 

on the sandy bed of an arroyo, the linear concentration 

is easily visualized because nearly all the particles 

potentially in motion are on or near the surface and 

visible. Concentration can then be measured by the 

spacing of particles on the surface. In the case of 

dunes and riffle bars, the concentration is not so apparent at a glance because it is measured by the weight 

or number of particles on a unit area to a depth of the 

,sand or gravel above the plane representing the base of 

the moving dune that is, the depth down to which 

particles are participating in the motion as the dune or 

bar is eroded or moves.

With regard to pool and riffle sequences in gravel-bed 

streams, the kinematics suggest that spacing of the 

rimes is related in part to the thin veneer of gravel set 

in motion by the flow. The bar represents an interaction 

between two opposing factors: increasing water velocity, 

which tends to increase wavelength, and decreasing 

amplitude, due to erosion, which tends to decrease 

wavelength. Such a balance results in riffle bars which 

do not appreciably move downstream.

REFERENCES

Blatch, N. S., 1906, Works for the purification of the water

supply of Washington, D.C.: Am. Soc. Civil Engineers

Trans., v. 57, discussion, p. 400-408. 

Colby, B. R., 1964, Discharge of sands and mean-velocity

relationships in sand-bed streams: U.S. Geol. Survey Prof.

Paper 462-A, 47 p. 

Durand, R., 1953, Basic relationships of the transportation of

solids in pipes: Internat. Hydraulics Convention, St.

Anthony Falls Hydraulic Lab., Minneapolis, Minn., Proc., p.

89-103.
L20 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

Ezekiel, M. J., and Fox, K. A., 1959, Methods of correlation 

and regression analysis, linear and curvillinear [3d ed.]: 

New York, John Wiley & Sons, Inc., 548 p.

Gilbert, G. K., 1914, The transportation of debris by running 

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

Guy, H. P., and Simons, D. B., 1964, Dissimilarity between 

spatial and velocity-weighted sediment concentrations: 

U.S. Geol. Survey Prof. Paper 475-D, p. 134-137.

Guy, H. P., Simons, D. B., and Richardson, E. V., 1966, Summary of alluvial channel data from flume experiments, 

1956-1961: U.S. Geol. Survey Prof. Paper 462-1, 96 p.

Haight, F. A., 1963, Mathematical theories of traffic flow: New 

York, Academic Press, 242 p.

Highway Research Board, 1950, Highway capacity manual: 

U.S. Bur. of Public Roads, 147 p.

Howard, G. W., 1939, Transportation of sand and gravel in a 

4-inch pipe: Am. Soc. Civil Engineers Trans., v. 104, p. 

1334-1380.

Hubbell, D. W., and Sayre, W. W., 1964, Sand Transport studies 

with radioactive tracers: Jour. Hydraulics Div., Am. Soc. 

Civil Engineers Proc., v. 90, Hy 3, p. 39-68.

Ippen, A. T., and Verma, R. P., 1953, The motion of discrete 

particles along the bed of a turbulent stream: Internat.

Hydraulics Convention, St. Anthony Falls Hydraulic

Lab., Minneapolis, Minn., Proc., p. 7-20. 

Kennedy, J. F., 1963, Mechanics of dunes and antidunes in

erodible bed channels: Jour. Fluid Mechanics, p. 521-544 

Krumbein, W. C., 1942, Settling-velocity and flume-behavior of

non-spherical particles: Am. Geophys. Union Trans., pt. 2,

Nov., p. 621-632. 

Leopold, L. B., Emmett, W. W., and Myrick, R. M. 1966,

Channel and hillslope processes in a semiarid area, New

Mexico: U.S. Geol. Survey Prof., Paper 352-G, p. 193-253. 

Leopold, L. B., and Miller, J. P., 1956, Ephemeral streams 

hydraulic factors and their relation to the drainage net:

U.S. Geol. Survey Prof. Paper 282-A, 37 p. 

Lighthill, J. J., and Whitham, G, B., 1955a, On kinematic waves

I. Flood movement in long rivers: Royal Soc. [London]

Proc., v. 229A, p. 281-316. 

 1955b, On kinematic waves II. A theory of traffic flow

on long crowded roads: Royal Soc. [London] Proc., v.

229A, p. 317-345. 

Nye, J. F., 1958, Surges in glaciers: Nature, v. 181 May, p.

1450-1451 

Seddon, J., 1900, River hydraulics: Am Soc. Civil Engineers

Trans., v. 43, p. 217-229

U. S. GOVERNMENT PRINTING OFFICE : 1968 O - 296-757