The Hydraulic Geometry

*

of Stream Channels and 

Some Physiographic 

Implications

By LUNA B. LEOPOLD and THOMAS MADDOCK, JR.

GEOLOGICAL SURVEY PROFESSIONAL PAPER 252

Quantitative measurement of some of the hydraulic 

factors that help to determine the shape of natural 

stream channels: depth, width, velocity, and suspended load, and how they vary with discharge as 

simple power functions. Their interrelations are 

described by the term "hydraulic geometry"

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1953
UNITED STATES DEPARTMENT OF THE INTERIOR 

Oscar L. Chapman, Secretary

GEOLOGICAL SURVEY 

W. E. Wrather, Director

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

Washington 25, D. C. - Price 35 cents (paper cover)
PREFACE

This report was prepared by the Technical Coordination Branch of the 

Water Resources Division of the Geological Survey as a part of the investigation 

of water-utilization problems. The junior author, Mr. Maddock, is assistant 

chief of the Hydrology Division, Bureau of Reclamation.

The authors have benefited greatly from constructive suggestions made 

by many friends and colleagues. Particular acknowledgment is due J. Hoover 

Mackin and Walter B. Langbein whose counsel and criticism contributed greatly 

to whatever of value is contained in this report. Among the persons who read an 

early draft and whose suggestions are highly valued are A. N. Strahler, John P. 

Miller, E. W. Lane, L. C. Gottschalk, K. B. Schroeder, C. C. Nikiforoff, Charles 

E. Stearns, and W. M. Borland. The continuous encouragement of W. W. 

Rubey is appreciated.

L. B. L.

HI
GLOSSARY

Antidune. A transient form of ripple on the stream bed analogous to a sand dune; an antidune progressively 

moves upstream.

Control. The "control" of a stream-gaging station refers to the nature of the channel downstream from the measuring section which determines the stage-discharge relation.

Current meter. An instrument for measuring water speed; consists of cups which are rotated by the flowing 

water; rate of rotation is a function of water speed.

Cumulative frequency, or flow-duration, curve. Graph of rate of daily discharge in relation to cumulative frequency, indicating the percent of time a given discharge is equalled or exceeded.

Discharge. The amount of water flowing in a channel expressed as volume per unit of time.

i) 

Froude number. A measure of tranquility of flow defined as ~j==

Gage height. Th'e' water-surface elevation referred to some arbitrary datum.

Hydraulic gradient: The slope of the energy grade line; or slope of the line representing the sum of kinetic and 

• potential energy along the channel length. It is equal to the slope of the water surface in steady, uniform 

flow.

Hydraulic radius. A measure of water depth in a channel defined as cross-sectional area of flowing water divided 

by the length of wetted perimeter; approximates mean depth in a wide channel.

Manning formula. A formula including empiric constants relating open-channel water velocity to hydraulic 

gradient, hydraulic radius, and channel roughness.

Rating curve. The graph of discharge plotted against water-surface elevation for a gjiven cross sectior of a stream.

Sediment station. A river section where samples of suspended load are taken each day, or periodically.

Stage, or river stage. Synonymous with gage height.

Stream-gaging station. An installation including a mechanical continuous recorder of water-surface elevation; 

at the same cross section periodic velocity measurements are made, and are used to construct the rating 

curve for the station. These two types of data are used to furnish a continuous record of wate1* flow.

Suspended sediment-discharge curve. A graph of water discharge plotted against concomitant m easured suspended-sediment load.

SYMBOLS

a coefficient representing width at unit discharge

6 exponent in width-discharge relation

c coefficient representing depth at unit discharge

C Chezy coefficient in v=C-^Rs

d mean depth

D pipe diameter

/ exponent in depth-discharge relation

F a function of

<7 sediment-grain diameter

g acceleration due to gravity

j exponent in suspended load-discharge relation

k coefficient representing velocity at unit discharge

L suspended-sediment load, weight per unit of time

m exponent in velocity-discharge relation

n Manning roughness factor

p coefficient representing load at unit discharge

Q water discharge in cubic feet per second (cfs)

R hydraulic radius, approximately equal to mean

	depth in wide channels 

s slope of water surface in open channels 

v velocity 

w width
CONTENTS

A bstract _ __________________________________________

General statement- _________________________________

The hydraulic geometry of stream channels ____________

Concept of frequency of discharge._______________

Variation of hydraulic characteristics in a particular 

cross section of a river____---_-________________

Variation of hydraulic characteristics in a downstream direction ______________________________

Relation of channel shape to frequency of discharge _

The hydraulic geometry of stream channels in relation to

sediment load____________________________________

Relations of suspended sediment to discharge in a 

particular cross section of a river.______________

Relations of suspended sediment to discharge in the 

downstream direction _________________________

Interrelations of width, depth, velocity, and suspended load at a given discharge________________

Page 

1

1

2

2

16

19

19

21

23

Page

The hydraulic geometry of stream channels in relation to 

sediment load—Continued

Interrelations of width, depth, velocity, and suspended load to a variable discharge _____________ 24

Interrelations of width, depth, velocity, and bed load

at a given discharge---________________________ 28

Channel-shape adjustment during individual floods-- 30 

The role of channel roughness and slope in the adjustment of channel shape to sediment load_________ 35

Some physiographic implications-_____________________ 43

The stable irrigation canal—an analogy to a graded

river _ _______'_________----_----_------____--- 43

Sediment and the longitudinal profile._____________ 48

Summary and interpretation_________________________ 51

References.________________----_-___-__----------_- 53

Appendix___ _-_____----------_---------__-_--------. ' 54

Index _____________________________________________ 57

ILLUSTRATIONS

FIGURE 1. Cumulative frequency, or flow-duration, curve for Powder River at Arvada, WyO-____-_-___---_-_---------- 3

2. Comparison of different rates of discharge at a station and downstream._______________._______---_-_------ 4

3. Typical relations among width, depth, velocity, discharge, and gage height, Cheyenne River near Eagle Butte,

S. Dak_ ______________________________________________________________________________.._ 5

4. Relation of width, depth, and velocity to discharge, Powder River at Arvada, Wyo., and at Locate, Mont_____ 6

5. Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream, Powder

River and tributaries, Wyoming and Montana_______ ___________________________-___-____-.----.--------. 10

6. Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream, Bighorn

River and tributaries, Wyoming and Montana______-___-_________-_-_-_---_----_--_------------------ H

7. Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream, Arikaree,

Republican, Smoky Hill, and Kansas Rivers in the Kansas River system, Kansas and Nebraska------------ 12

8. Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream, main trunk

of Missouri and lower Mississippi Rivers.____________-_______________-_--__-------___-___------------ 13

9. Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream in various

river systems.___________________________________-______-___-____----_-_-_-_---___---------_------ 15

10. Width, depth, and velocity in relation to discharges representing two frequencies, Maumee and Seioto River

basins, Ohio.-__________________________________________-_______--_-______-----_--___-._----------- 17

11. Width, depth, and velocity in relation to discharge of various frequencies, Maumee and Seioto River basins, Ohio_ 18

12. Diagrammatic relations of width, depth, and velocity to discharge at a station and downstream._____________ 19

13. Relation of suspended-sediment load to discharge, Powder River at Arvada, Wyo___________________________ 20

14. Typical relations in the hydraulic geometry of natural stream channel systems._____________________________ 22

15. Average relation of suspended-sediment load to channel-shape factors at a constant discharge of 500 cfs (from

river data in appendix)___________________________________________________________-__--__'•__________ 23

16. Average relation of suspended-sediment load to channel-shape factors at four discharge rates (from river data

in appendix)______________!________________________________--_____-_________---_-_-_-------__----- 24

17. Example of width, depth, velocity in relation to discharge (constructed from fig. 16)________________________ 25

18. Relation of &,-p and j (constructed from fig. 16)_______________________________________________________ 25

19. Average hydraulic geometry of river channels___________________________________-__-__----_____------- 27

20. Relation of bed load to channel-shape factors at constant discharge, from Gilbert data_______________________ 29

21. Changes in channel-shape factors, San Juan River near Bluff, Utah, during flood of September-December 1941. 31

22. Channel cross sections during progress of flood, September—December 1941, San Juan River near Bluff, Utah__ 32

23. Changes in width, depth, velocity, water-surface elevation, and stream-bed elevation with discharge during flood

of December 1940-June 1941, Colorado River at Grand Canyon, Ariz__ ____________--_______-_-----_--__ 32

v
VI ILLUSTRATIONS

Page 

FIGURE 24. Channel cross sections during progress of flood, December 1940-June 1941, Colorado River at Grand Canyon,

Ariz________._._______._____________________________________________________ 34

25. Changes in channel shape factors during progress of flood, May-June 1948, Rio Grande near Bernalillo, N. Mex_ 35

26. Channel cross sections during progress of flood, May-June 1948, Rio Grande near Bernalillo, N. Mex________ 35

27. Relation of width, depth, velocity, water-surface elevation, and stream-bed elevation to discharge, Colorado

River at Yuma, Ariz__________________________________-___________-__-__-_-_-__--_-___-___-_-----_ 37

28. Progressive changes of Colorado River at Yuma, Ariz., resulting from the construction of engineering works

upstream-___-__-_______„___________________________.________-________-__-_-_-__---_--____-__-_-- 38

29. River profiles in reach of Colorado River near Yuma, Ariz_______________________-_---___-__-_-__--___-__ 39

30. Channel roughness (Manning n) in relation to discharge for various sizes of sediment (from flume data)______._ 41

31. Relation of width, depth, and velocity to discharge for stable (regime) irrigation canals in India (Lacey data) __ 44

32. Variation of velocity and depth downstream for three assumed longitudinal profiles_________________________ 49
THE HYDRAULIC GEOMETRY OF STREAM CHANNELS AND SOME

PHYSIOGRAPHIC IMPLICATIONS

By LUNA B. LEOPOLD and THOMAS MADDOCK, JR.

ABSTRACT

Some hydraulic characteristics of stream channels — depth, 

width, velocity, and suspended load — are measured quantitatively and vary with discharge as simple power functions at a 

given river cross section. Similar variations in relation to 

discharge exist among the cross sections along the length of a 

river under the condition that discharge at all points is equal in 

frequency of occurrence. The functions derived for a given cross 

section and among various cross sections along the river differ 

only in numerical values of coefficients and exponents. These 

functions are:

where

depth • 

veloeitjL ~ suspended-sediment load, in units of weight per unit time

Q = water discharge, in cubic feet per second (cfs); a, c, k, p, 

b, f, m, and j are numerical constants.

These relationships at a given channel cross section and downstream when plotted on graphs are greatly similar even for river 

systems very different in physiographic setting. The relationships are described by the term "hydraulic geometry/'

In the data studied it appeared that when discharges are of 

equal frequency at different points along a river, that is, equalled 

or exceeded the same percent of time, the velocity as well as the 

width and depth of flow, increases with discharge downstream. 

This increase of velocity downstream results from the fact that 

the increase in depth overcompensates for the decrease in slope. 

The tendency for velocity to increase downstream exists on most 

streams despite the decreasing particle size downstream. This 

indicates that the mean velocity in a given reach is not merely a 

function of the size of particles which must be transported but is 

governed by a more complex interaction among several factors.

Measurements of suspended-sediment load are available for a 

number of different river cross sections. The suspended load is 

shown to be an index to total load for purposes of explaining the 

observed average characteristics of natural channel systems.

An empiric quantitative relation among average measurements 

of width, depth, velocity, discharge, and suspended-sediment 

load is derived from data on natural rivers. This relation shows 

that depth and width, as well as velocity, are functions of the 

load transported in the channel.

In a simplified example, if the discharge is constant, the 

empiric relation is: At a given discharge, an increase in width 

at constant velocity is accompanied by a decrease in suspendedsediment load; conversely, at constant discharge and velocity, 

an increase in width is accompanied by an increase in bed load.

The writers, like many others, postulate that both discharge 

and sediment load are factors essentially independent of the 

stream channel and depend on the nature of the drainage basin. 

The typical relations between suspended-sediment load and 

water discharge are shown in quantitative terms. Specifically, 

at a given stream cross section, suspended-sediment load per 

unit of discharge characteristically increases rapidly with increased discharge. However, suspended load per unit volume 

of water tends to decrease slightly downstream.

These characteristics are important determinants of the shape 

of the cross section of a channel and the progressive changes in 

its shape downstream. The empiric relation between hydraulic 

characteristics of the channel and suspended load provides, in 

semiquantitative terms, a logical explanation of the observed 

channel shape.

The hydraulic geometry of river channels is presented for 

several river systems. It is shown that similar equations apply 

both to rivers and to stable ("regime") irrigation canals which 

neither scour nor aggrade their beds. The analogy demonstrates 

that the average river channel-system tends to develop in a way 

to produce an approximate equilibrium between the channel and 

the water and sediment it must transport. This approximate 

equilibrium appears to exist even in headward ungraded tributaries and in a given cross section for all discharges up to the 

bankful stage.

GENERAL STATEMENT

Geomorphology, that branch of geology dealing with 

land forms, their genesis and history, has been classically treated almost exclusively in a qualitative manner. 

William Morris Davis, for pedagogic reasons, invented 

words and contrived similes which were descriptive and 

easy to remember, but were supported by intricately 

developed general argument rather than by field data. 

The qualitative approach to geomorphology has indeed 

been constructive, but it would be desirable to analyze 

some of the concepts quantitatively.

There is available a mass of data on streamflow 

collected over a period of seventy years representing 

rivers all over the United States. The field procedure 

used by engineers to measure the rate of water discharge in a river furnishes concurrent measurements of 

mean velocity,width, shape and area of the cross section, as well as discharge. Moreover, in the last decade

1
THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

the systematic collection of data on suspended sediment 

has begun though the number of stations collecting is 

as yet few.

The present paper reports on analyses of some of 

these data, keeping in mind possible application to 

geomorphic problems. The available data are not 

necessarily the best type for an analysis of stream 

processes. The parameters measured are perhaps not 

the most meaningful for such analysis. Nevertheless, 

the available data do provide some generalized relations of interest to the geomorphologist.

The plan of the paper is as follows:

The characteristics of channel shape in certain river 

cross sections are examined. At a given cross section, 

the width and depth of the channel and, therefore, the 

velocity change with the amount of water flowing in 

the cross section. The changes in width, depth, and 

velocity in response to changes in discharge are observed to have certain characteristics which apply to 

many natural river cross sections, and these similarities 

are described in the first part of the paper.

These same characteristics are then examined downstream along the length of natural channels in order 

to determine how they change there. Changes in slope 

and roughness, which help determine the observed variations, are reserved for later discussion.

The data used are those collected at regular river 

measurement stations for purposes of determining 

river discharge. Such data are obtained for purposes 

other than a study of fluvial morphology and are, for 

the writers' purposes, incomplete. Particularly, data 

on river slope and bed roughness are meager but some 

useful ideas may be obtained even from the incomplete 

operational data.

In the next part of the paper, the river-channel characteristics including slope and roughness are discussed 

in relation to observational data on suspended-sediment 

load. It is recognized at the outset that the unmeasured bed load is of importance and that its omission 

from the available data is a limitation to the conclusions which may be drawn. Yet some interesting relations may be obtained by the use of data on suspended 

load alone. These relations provide some picture of 

the interaction of the hydraulic characteristics related 

to channel shape and a segment of the stream's sediment 

load and therefore seem to be potentially useful tools 

for the study of fluvial processes.

The relations between observed channel characteristics and suspended-sediment load involve the process 

of adjustment between channel shape and sediment 

load. Because the mechanism of adjustment between 

the capacity of a stream to carry a sediment load and 

available load is important to the understanding of the 

graded river, any advance in knowledge of this mechanism leads toward an improved understanding of river 

grading.

There are two possible levels at which this mechanism 

of adjustment of river-to-load may be discussed. The 

more detailed way would lie in the realm of hydrodynamics and would concern the mechanic? of sediment 

transport. A second and more general approach is 

through the consideration of river hydraulics but 

without a detailed treatment of the hydrodynamics. 

The second approach is the one used in the present 

paper, in part because it lends itself better to an 

understanding of the relation of hydraulics to geomorphology, and in part because it lies within the field 

of experience of the authors.

The last part of the paper is an attempt to explore in 

a preliminary way how the relations among the hydraulic variables bear on some geomorphic problems.

It is emphasized that the present study should be 

considered preliminary in character. Data on streamgaging and suspended load discussed here are clearly 

imperfect measures of the variables of specific interest 

to the geomorphologist. Moreover, the present analysis 

deals primarily with general trends in the relations 

among measured variables, and much further work is 

necessary to explain the deviations from these trends. 

And filially, the suspended-load data available are 

primarily derived from Western streams. Comparable 

data on streams in humid areas are most meager. 

These many restrictions are recognized by the authors 

who attempt continuously to keep before the mind of 

the reader the limitations of the data, while emphasizing 

the general trends rather than the innumerable individual variations.

Explanation of these individual variations and 

refinement of the general concepts must s.wait further 

work.

THE HYDRAULIC GEOMETRY OF STREAM CHANNELS 

CONCEPT OF FREQUENCY OF DISCHARGE

Before describing the channel characteristics at a 

given river cross section and how those characteristics 

change downstream, it appears desirable to discuss by 

way of definition, frequency of discharge and cumulative 

frequency, or flow-duration, curves. The hydrologist 

to whom these conceptions are everydsy tools will 

pardon the inclusion here of a brief discussion aimed at 

providing the geologist a picture of frequency relations 

which is essential to an understanding of what will 

follow.

The procedure used in making stream measurements 

is well described in the literature (Corbett, 1943; 

Linsley, Kohler, and Paulhus, 1949, pp. 182-198).

The daily mean discharge is the average of all rates 

for a day, and is expressed in cubic feet per second. 

The mean rate of discharge for each day of the year 

at measuring stations on many rivers is published by 

the U. S. Geological Survey in the annual WaterSupply Papers.
CONCEPT OF FREQUENCY OF DISCHARGE

The frequency of a daily mean discharge at a given 

river gaging station, is determined by counting the 

number of occurrences of this rate throughout a period 

of record. In hydrologic practice the number of 

occurrences is usually presented in the form of a cumulative frequency, or flow-duration, curve, an example 

of which is presented in figure 1 (after Hembree, Colby, 

Swenson, and Davis, 1952, fig. 30). To construct such 

a curve, a number of categories of rates of discharge 

are arbitrarily chosen. The number of days on which 

the measured daily mean discharge fell in each category 

is determined by tallying the data. The numbers in 

each category are then accumulated from the largest 

discharge category to the smallest, and each cumulative 

figure is divided by the total number of days of record. 

The resultant quotients represent the percent of time 

given discharges are equalled or exceeded at the station.

These figures representing percent of time are then 

plotted against discharge to provide the flow-duration 

curve of which figure 1 is a sample. On it a discharge 

of 1,000 cfs or more occurs about 8 percent of the time. 

This means that through a long period, 8 percent of 

the days, or on the average 8 days in every 100, experienced a discharge of 1,000 cfs or more.

The cumulative frequency, or duration, of a given 

daily discharge is expressed as the percent of time that 

this rate is equalled or exceeded. Discharge rates of the 

same cumulative frequency may differ, depending on 

the streams being considered. For example, a point

100,000 ^

10,000

1,000

100

10

O.I

MEAN ANNUAL DISCHARGE = 430 Cfs _ 

MEDIAN DISCHARGE = 160 cfs

0 20 40 60 80 100

Percent of time flow equalled or exceeded discharge indicated

FIGURE 1.—Cumulative frequency, or flow-duration, curve for Powder River at 

Arvada, Wyo., 1917-50.

219627—52———2

on a large trunk river and a point on a small tributary 

may have discharge rates of the same cumulative frequency—or frequency, as it is ordinarily expressed; but 

discharge in the tributary may be only a few cubic feet 

per second, and hi the trunk river it may be thousand? 

of cubic feet per second.

The median discharge is that discharge which ir 

equalled or exceeded 50 percent of the time; that is, 

half the days have flow greater than the median, and 

half less. At Arvada the median flow is about 160 cfs. 

The mean annual discharge, or the arithmetic mean 

of the flow of all individual days at Arvada is about 

430 cfs, or almost three tunes the median. Mean annual discharge is generally greater than median, and 

the mean rate of discharge is equalled or exceeded on 

somewhat less than half the number of days at most 

points on a river.

As a rough generalization, it may be stated that the 

mean annual rates of discharge at all points on a large 

number of rivers are equalled or exceeded about the 

same percent of time.

The values of average annual discharge at measuring 

stations on many rivers in the United States are readily 

available in the Water-Supply Papers of the U. S. Geological Survey. For example, the average discharge 

for the Powder River at Arvada, Wyo., for the period 

1916-47 is published with the description of the station 

in Water-Supply Paper 1086 (1950) on page 237.

For purposes of definition the conception of a varying 

discharge at a given river cross section and downstream 

will now be explained. In figure 2 cross sections A and 

B represent two points on a river at low discharge; 

C and D are the same sections at higher discharge. 

These sections are shown at the right, as A, B, C, and 

D, in their gelation to the full length of the river and to 

its watershed at both low and high discharge. Though 

the highest discharge may flow over bank, this paper 

is not concerned with discharges above bankful.

These diagrams illustrate the conception that the 

width and depth of the channel increase at a given 

river cross section with increase hi discharge. Variations in discharge follow a pattern in tune that is peculiar to the position of the cross section and to the 

river. The different discharges at a given cross section 

vary in frequency. Frequency is a function of the 

watershed and its hydrologic and physical characteristics.

Under the condition of low discharge represented by 

the diagram in the upper right corner of figure 2, it is 

postulated that every point along the river is experiencing a discharge that is small for that point; or, if all 

points along a river system are experiencing a relatively low discharge, the frequency of the rate of discharge at any one point is about the same as the frequency of the rate at any other point. Of course the rate 

of discharge, not its frequency, will generally be much
4 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

LOW DISCHARGE

( For example,flow equalled or 

exceeded 90 percent of the time)

LOW DISCHARGE

Constant frequency of 

discharge at each point 

(All points experiencing 

low discharge)

HIGH DISCHARGE

Infrequent condition 

[For example,flow equalled or 

exceeded I percent of the time)

HIGH DISCHARGE

Constant frequency of

discharge at each point

(All points experiencing

near-flood dischcrge)

FIGURE 2.—Comparison of different rates of discharge at a given river cross section and at points downstream.

greater near the mouth of a river draining a large area 

than at some headwater point. The same postulation 

is made for the condition of high discharge, as represented by the diagram in the lower right.

The two diagrams of the watershed are introduced to 

emphasize the points, first, that at a given cross section 

different discharges have different frequencies; and 

second, that at cross sections situated at various points 

along the stream the rates of discharge are usually different, one from another. Comparison of the various cross 

sections along a stream is made in this paper only under 

the assumed condition that they are experiencing equal 

frequency of discharge.

The term "at a station" is used in this paper to mean 

at a given cross section, and the term "in a downstream 

direction" refers to cross sections situated along the 

length of a stream. "Change of discharge downstream" 

means different discharges of the same frequency at. 

cross sections situated along the length of a stream.

The differences in discharge both at a station and in a

downstream direction are associated with differences in 

width, depth, and velocity, as will be shown later.

VARIATION OF HYDRAULIC CHARACTERISTICS IN A 

PARTICULAR CROSS SECTION OF A PIVER

Changes in velocity, depth, and width of flowing 

water take place as the discharge increases at a particular cross>section of a river. Because river cross sections 

tend to be roughly semielliptical, trapezoidal, or triangular, increasing discharge results in an increase of each 

of the other three factors. Typical relations of width, 

depth, and velocity to discharge will be shovm for various points on different rivers. • These relations will show 

that these three factors generally increase with increasing discharge as simple power functions.

In figure 3A a number of variables are er.ch plotted 

against river stage. Stage, or gage height, if the elevation of the water surface above some arbitrary datum, 

and is the quantity recorded by the water-level recorder 

of a stream gage. The discharge, width, and mean
VARIATION OF HYDRAULIC CHARACTERISTICS IN A GIVEN CROSS SECTION 5

velocity are plotted against stage as ordinate. For 

comparison, width, depth, and velocity are plotted on 

logarithmic scales against discharge in figure 3.5.

When velocity is measured by current meter at a gaging station for the purpose of computing discharge, the 

width of water surface and the depth and velocity at 

various positions in the cross section are recorded. Figure 3 and other similar figures to be discussed were constructed from current-meter measurements.

The current-meter data which include measurements 

of width of water surface, depth, and velocity at various 

points across the river measuring-section, are filed in the 

district offices of the U. S. Geological Survey and are 

available to the public for inspection and use. The 

original records must be used or copied in the Survey 

office, otherwise appropriate arrangements must be 

made to insure against loss of the originals.

The plot of stage (gage height) against discharge is, in 

engineering parlance, the rating curve of a station. 

Rating curves of all stream cross sections are similar in 

that the increase of discharge per unit increase of stage 

is greater for high than for low discharges. Thus the 

general form of the curve is typical of all open channels.

The width plotted in figure 3 is the width of the water 

surface at any given gage height.

In stream-gaging practice, the depth and velocity are 

measured at various distances from the bank. From 

these depth measurements and their spacing, the crosssectional area of flowing water is determined.

In this report, the depth represents the mean depth 

associated with a particular rate of discharge, and was 

determined by dividing the measured cross-sectional 

area of flowing water by the corresponding width of 

water surface. This provides, then, a mean depth equal 

to the depth of a rectangular channel having the same 

surface width as the actual channel at the given stage 

and having the same cross-sectional area.

Velocity discussed in this report is the quotient of 

discharge divided by the area of the cross section, and 

is the mean velocity of the cross section as used in 

hydraulic practice (see Linsley, Kohler, and Paulhus, 

1949, p. 197). This mean velocity is not the most 

meaningful velocity parameter for discussing sediment 

transport but it is the only measure of velocity for which 

a large volume of'data are available. Although the 

writers recognize its limitations, the mean velocity is 

used here in lieu of adequate data on a more meaningful 

parameter.

Figure 3A demonstrates that if width and velocity 

are plotted against river stage, there is considerable

"o 10,000 20,000 30,000 40,000 Discharge, in cfs 

01234 velocity, in fps 

o 200 400 600 800 width, in feet

1,000

12 200

0

^>*$

^^~° D D

;;^f °

^•'^ 

D°

° —

D 2.3 

S\op® ,^—-

X x I--' V

X

1 1 1

Xj

L-- 7 x

MM

x-"*"' xX X 

x —

if Ml

0 1,000 10,000 IOO.C

Discharge, in cfs

Relation to discharge

Relation to river stage

EXPLANATION

0 & 

Discharge, in cfs Width, in feet

x . a 

Velocity, in fps Depth, in feet

FIQT.-RE 3.— Typical relations among width, depth, velocity, discharge, and gage height at a gaging station, Cheyenne River near Eagle Butte, S. Dak.
6 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

500

200

100

50

20

TT

Width restricted 

f—— by bridg?. __

.t 0.5

O.I 1.0 10 100 

Discharge, in cfs

FEGURE 4, A.—Relation of width, depth, and velocity to discharge, Powder River at Arvada, Wyo.

1,000 10,000
VARIATION OP HYDRAULIC CHARACTERISTICS IN A GIVEN CROSS SECTION

1,000 rr

500

200

—. 100

10,000' 

FIGURE 4, B— Relation of width, depth, and velocity to discharge, Powder Biver at Locate, Mont.

100 1,000

Discharge, in cfs

100,000
THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

scatter of points and the curve defined is not easily 

expressed mathematically. If width, depth, and velocity, are plotted against discharge on logarithmic paper, 

their relations to discharge are expressed by nearly 

straight lines, as can be seen in figure 3B, and this may 

l>e simply expressed mathematically.

Figure 3B is typical of a large number of gaging 

stations for which similar curves were drawn. The 

slopes and intercepts of the lines varied somewhat 

between stations but straight lines on logarithmic paper 

typified the relations in the data studied.

Other examples of these curves are presented in 

figure 4 to indicate that through a considerable range 

of discharge this method of plotting provides essentially straight lines on logarithmic paper. There is 

considerable scatter of the points on these graphs. 

Part of the scatter apparently results from temporary 

scour and fill of the stream bed. At the lower discharges measured by wading rather than from a bridge 

or cable, part of the scatter of points may be attributed 

to the fact that successive measurements are not made 

at exactly the same cross section. For present purposes, the general positions of the lines representing 

mean relations are of interest and further work will be 

required to study the causes and import of the scattering 

of the data.

In the present study every available current-meter 

discharge measurement was plotted for stations having» 

short records. For stations having long records, of 

which figures 3 and 4 are typical, the measurements to 

be plotted were chosen at random from the available 

data. Enough measurements were obtained to define 

the curve through the whole range of discharge of the 

particular station.

A gaging station is installed at a river cross section 

where the relation of discharge to stage is as stable as 

possible. Specifically, a location is chosen if possible 

where discharge plots against gage height in a smooth 

curve. For this reason gages are installed either upstream from a "section control," where critical velocity 

(in the hydraulic sense) is attained over some outcrop 

or gravel bar, or in a reach of channel where the channel 

properties determine the stage-discharge relation. The 

latter, called a "channel control," merely means a 

relatively straight and uniform reach generally devoid 

of unusual riffles or bars.

In periods of low water, current-meter measurements 

are ordinarily made by wading. When wading is 

impossible, current-meter measurements are often made 

from bridges.

The cross section of a stream at a gaging station 

above a section control is not representative of an 

average reach of channel, particularly at high discharges. More representative sections are found at 

gaging stations having channel control. These stations 

are often equipped with a cable car from which the 

current meter may be operated.

In the following discussions, gaging stations having 

channel control and cross sections gaged from a cable 

car were chosen when possible. Many stations where 

measurements were made from bridges were used for 

lack of more representative data, but these measurements were utilized only for low stages where width 

was still increasing with gage height; that is, stages 

lower than those where bridge abutments constrict the 

flow.

Excluding the unnatural effects introduced by constriction such as bridge abutments, the data analyzed 

indicate that through ranges of discharge up to the 

bankful stage, the relations of width, depth, and 

velocity to discharge in a natural stream section are in 

the form of simple power functions and plot as straight 

lines on logarithmic paper. When the river overflows 

its banks, of course width increases rapidly and a new 

relation to discharge obtains which will not be considered here.

The artificial effects introduced by atypical locations 

of gaging stations do not appear to ir validate the 

conclusion just expressed.

It can be stated that the relation of discharge to other 

hydraulic factors in natural river cross^ sections can be 

described tby the expressions:

w=aQ1> 

d=cQf 

v=kQm

(1)

(2)

(3)

where Q, is discharge, w, d, and v represent water-surface 

width, mean depth, and mean velocity, respectively. 

The letters b,f, m, a, c, and k are numerical constants. 

Because width, depth, and velocity are each a function of discharge as described by the formulas above, 

the three equations can immediately be related to one 

another through the identity

Q== area X velocity, or, Q=wdv 

substituting from the above

or

It follows therefore that

b+f+m=1.0 

aXcXk=l.Q

and

In figures 3B and 4, the constants, b,f, and m represent the slopes of the three Lines on the graph, and the 

sum of these slopes must equal unity. The values of 

the slope of the lines are indicated in the figures and 

their sum is indeed unity.
VARIATION OF HYDRAULIC CHARACTERISTICS IN A DOWNSTREAM DIRECTION 9

The constants, a, c, and k represent the intercepts of 

the lines and are respectively, values of w, d, and v at 

discharge of unity. Because the values of discharge in 

the actual data are generally much larger than unity, 

the product of aXcXk is not computed at unit discharge 

but might better be considered in the following light. 

The product of the values of width, depth, and velocity 

at any given discharge must equal that discharge. For 

example, in figure 3B, at 1,000 cfs: w=245 ft, d=l.7 ft, 

and v=2A fps; their product w'Xd'Xv equals 1,000.

Because it has been stated that as a general rule 

equations (1), (2), and (3) characterize hydraulic 

relations for representative natural stream channel 

cross sections, it is logical to inquire how the slopes 

and intercepts represented by the numerical constants 

vary among streams, and what range of numerical 

values they represent. Width and depth for a given 

discharge vary widely from one cross section to another 

and, therefore, the intercept values, a, c, and k, will vary. 

Further work is necessary to determine the factors 

which govern these variations and to determine the 

extreme limits. However, there is some consistency 

in values of the slopes of the lines for different streams 

as will now be explained.

Average values of the exponents b, /, and m have 

been computed for 20 river cross sections representing 

a large variety of rivers in the Great Plains and the 

Southwest. Though these particular averages are biased 

toward semiarid conditions, they nevertheless provide 

some indication of the order of magnitude of typical 

values. These averages are

6-0.26 

/=0.40 

7n=0.34

The appendix includes a summary of some of the 

data from which the above values were derived. For 

each station, graphs similar to figure 3B were plotted 

from current-meter measurements, and a straight line 

was drawn by eye through the points. No statisticalfitting procedure was used in drawing the line because 

the use to which the data are to be put did not appear 

to justify such accuracy. The slope of each line was 

measured directly from the graph.

That the values of the exponents m, b, and / were 

averaged for the 20 river cross sections does not imply 

any expectation that the values should be closely similar. 

The mean values are presented so that the relative 

order of magnitude of the three exponents can be visualized. In a later section a partial and tentative explanation of the variability of these exponents among 

various rivers will be given.

For present purposes it is sufficient to note that at 

most river cross sections, depth increases with discharge 

somewhat faster than does width (/>&). The ratio

of width to depth decreases with increasing flow through 

the channel.

The relative rates of increase of width and depth are 

functions of channel shape. It can easily be shown thr.t 

a channel of triangular shape is the only channel fc~" 

which the width-to-depth ratio remains constant with 

changing discharge. Trapezoidal and elliptical char - 

nels are characterized by a decrease of the width-todepth ratio, and in rectangular channels the width-tc - 

depth ratio decreases with increasing discharge at a 

rate even faster than in trapezoidal sections.

In summary, relations of width, depth, and velocity 

to discharge in the natural river cross sections studied 

are in the form of simple power functions. The relative 

rates of increase of width, depth, and velocity are 

determined by the shape of the channel, the slope cf 

the water surface, and by the roughness of the wetted 

perimeter. It is emphasized again that the foregoing 

discussion is merely a description of natural river cross 

sections. The discussion of how width, depth, velccity, discharge, slope, and roughness interact to provide the observed relations is reserved until after the 

descriptive presentation is completed.

The aspect of the above discussion which is different 

from relations discussed in the voluminous literature 

on open-channel hydraulics is the plotting of widtt, 

depth, and velocity against discharge on logarithmic 

paper. A straight-line relation between gage height 

and discharge on logarithmic paper is widely known 

and is used in many offices of the Geological Survey in 

the construction of rating curves for river sections having a shifting control.

VARIATION OP HYDRAULIC CHARACTERISTICS H* 

A DOWNSTREAM DIRECTION

Frequency of discharge is especially important in an 

understanding of the hydraulic geometry of rive~ 

systems and how the width, depth, and velocity of 

natural rivers change in a downstream direction.

As already emphasized, comparison of various cros:^ 

sections along the length of a river is valid only under 

the condition of constant frequency of discharge at all 

cross sections.

The mean annual discharge is equalled or exceeded 

about the same percent of time on a large number of 

rivers. It represents roughly the discharge equalled 

or exceeded 1 day in every 4 over a long period. 

Though exactly the mean annual discharge may not 

occur at all points on any given day of record, it ir 

closely approximated for a large number of days eact 

year.

The data forming the basis of the discussion which ** 

follows are again current-meter measurements made

at gaging stations of the Geological Survey.

The width, depth, and velocity corresponding to the 

mean discharge are plotted against mean discharge ir 

figure 5 for all the gaging stations on all tributaries tc
10 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

200

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X^3

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'°I8

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_£--•£

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——

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to

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annual discharge,

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——

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EXPLANATION

1. Red Fork near Barnum, Wyo.

2. Middle Fork Powder River above Kaycee, Wyo.

3. Middle Fork Powder River near Kaycee, Wyo.

4. North Fork Powder River near Hazelnn, Wyo.

5. North Fork Powder River near Mayoworth, Wyo.

6. South Fork Powder River near Kaycee, Wyo.

7. Powder River at Sussex, Wyo.

8. Middle Fork Crazy Woman Creek near Greub, Wyo.

9. North Fork Crazy Woman Creek near Buffalo, Wyo.

10. North Fork Crazy Woman Creek near Greub, Wyo.

5- —— • —— —

—

—

12. Powder River at Arvada, Wyo.

13. North Fork Clear Creek near Buffalo, Wyo.

14. Clear Creek near Buffalo, Wyo.

15. South Fork Rock Creek near Buffalo, Wyo.

16. Rock Creek near Buffalo, Wyo.

17. South Piney Creek at Willow Park, Wyo.

C

5

500

^— -

—

—

18. Piney Creek at Kearney, Wyo.

19. Piney Creek at Ucross, Wyo.

20. Clear Creek near Arvada, .JVyo.

21. Little Powder River near Broadus, Mont.

22. Powder River at Moorhead, Mont.

23. Powder River near Locate, Mont.

1,000

in cfs

FIM-RE 5.—Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream, Powder River and tributaries, Wyoming and Montana.

the Powder River in Wyoming and Montana for which 

adequate data were available. Each gaging station 

provides one point in each of the plotted diagrams. 

Throughout the figure the abscissa value for each gaging 

station is its mean annual discharge.

For purposes of comparison, similar graphs representing the change of width, depth, and velocity with 

average annual discharge downstream are presented 

in figures 6-8. These represent the Bighorn River and 

tributaries hi Wyoming and Montana, the Arikaree, 

Republican, Smoky Hill, and Kansas Rivers in the 

Kansas-Nebraska region, and gaging stations along 

the main trunk of the Missouri and lower Mississippi 

Rivers.

Proceeding downstream in a given river, the discharge tends to increase because of the progressively 

increasing drainage area, though of course there are 

some streams in which discharge decreases downstream, 

particularly in arid areas. It can be seen that for all

the rivers plotted, the depth, width, and velocity tend 

to increase progressively as power functions of discharge. There is again considerable scatter of points 

in the graphs. There are reaches of the rivers represented in figures 5-8 on which either width, or depth, 

or velocity decreases downstream. The present paper, 

however, is concerned with the general trends and 

further work will be necessary to explain th°, details of 

deviations from these trends.

The graphs of figures 5-8 are referred to as the 

changes in width, depth, and velocity with increasing 

discharge in the downstream direction. It will be 

understood that a small tributary having a relatively 

low mean annual discharge may enter the main trunk far 

downstream, but its plotted position on the abscissa 

scale will be determined by its mean discharge.

Strictly speaking, the "downstream" grs.phs in figures 5-8 show how width, depth, and velocity change 

with discharges of equal frequency. Discharges of
VARIATION OF HYDRAULIC CHARACTERISTICS IN A DOWNSTREAM DIRECTION 11

500

200

100

19

100 1,000 

Mean annual discharge, in cfs

10,000

FIGUBE 6.—Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream, Bighorn Kiver and tributaries, Wyoming, and Montana,

and Yellowstone River, Montana.

1. Bighorn River at Manderson, Wyo.

2. Bighorn River at Thermopolis, Wyo.

3. Bighorn River at Kane, Wyo.

4. Bighorn River St. Xavier, Wyo.

5. Bighorn River near Ouster (Hardin), Mont.

6. Wind River near Dubois, Wyo.

7. Wind River near Burris, Wyo.

EXPLANATION

8. Wind River at Riverton, Wyo.

9. Wind River near Crowheart, Wyo.

10. Greybull River near Basin, Wyo.

11. Greybull River at Meeteetsee, Wyo.

12. Popo Agie River near Riverton, Wyo.

13. North Fork Owl Creek near Anchor, Wyo.

14. Owl Creek near Thermopolis, Wyo.

15. Medicine Lodge Creek near Hyattville, Wyo.

16. Gooseberry Creek near Grass Creek, Wyo.

17. Yellowstone River at Corwin Springs, Mont.

18. Yellowstone River at Billings, Mont.

19. Yellowstone River near Sidney, Mont.

219627—52-
12 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

1,000

500

100 1,000

Mean annual discharge, in cfs

10,000

FIGUKE 7.—Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream, Arikaree, Republican, Smoky Hill, and Kansas Rivers

hi the Kansas River system, Kansas and Nebraska.

1. Arikaree River at Haigler, Nebr.

2. Republican River at Culbertson, Nebr.

3. Republican River near Bloomington, Nebr.

4. Republican River at Clay Center, Kans.

5. Smoky Hill River at Elkader, Kans.

6. Smoky Hill River at Ellis, Kans.

EXPLANATION

7. Smoky Hill River at Russell, Kans.

8. Smoky Hill River at Ellsworth, Kans.

9. Smoky Hill River near Langley, Kans.

10. Smoky Hill River at Linsborg, Kans.

11. Smoky Hill River at Enterprise, Kans.

12. Kansas River at Ogden, Kans.

13. Kansas River at Wamego, Kans.

14. Kansas River at Topeka, Kans.

15. Kansas River at LecOmpton, Kans.

16. Kansas River at Bonner Springs, Kanf
VARIATION OF HYDRAULIC CHARACTERISTICS IN A DOWNSTREAM DIRECTION 13

5,000

2,000

1,000

-C

T3 500

200

Q. 

0>

Q

100

20

10

c: ~

o

.2 

o>

1,000 10,000 100,000

Mean annual discharge, in cfs

1,000,000

FIGUKE 8.—Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream, main tnmk of Missouri and lower Mississippi River?-

EXPLANATION

1. Missouri River at Bismarck, N. Dak,

2. Missouri River at Pierre, S. Dak.

3. Missouri River at St. Joseph, Mo.

4. Missouri River at Kansas City, Mo.

5. Missouri River at Hermann, Mo.

6. Mississippi River at Alton, 111.

7. Mississippi River at St. Louis, Mo.

8. Mississippi River at Memphis, Term.

9. Mississippi River near Vicksburg, Miss.
14 THE HYDKAULIC GEOMETRY OF STREAM CHANNELS

equal frequency tend to increase downstream with increasing drainage area. The phrase "in the downstream direction" is merely used as a short designation 

for this conception.

By use of the graph, tributaries can be compared 

directly with the trunk stream. On figure 5 the gaging 

stations on the main trunk of the Powder River (as an 

example, point 7) and a single large headwater fork 

(Middle Fork,, point 2) are designated by the solid 

triangles. Stations on a single tributary, Clear Creek 

(point 14), and its headwater tributary, North Fork of 

Clear Creek (point 13), are designated by solid circles. 

Each of these two sets of points describe a single river 

stem and are downstream graphs uncomplicated by 

tributaries. It can be seen that were lines to be drawn 

through those two sets of points, separately, there 

would be a difference from the lines on the graphs 

representing the Powder River basin as a whole. Yet 

the difference is astonishingly small.

The general alinement of points on the downstream 

graphs indicates that in a given river basin where all 

cross sections are experiencing the same frequency of 

discharge, the corresponding values of depth, width, 

and velocity at different cross sections having the same 

discharge tend to be similar, regardless of where in 

the watershed or on what tributary the cross sections 

may be.

The graphs presented are examples chosen from a 

large number drawn for various river systems in the 

United States. Figure 9, with the plotted points 

through which the mean lines were drawn eliminated, 

presents similar graphs for selected river basins. Rivers 

were chosen to represent a diversity of geographic location, and physiographic and geologic types. The basins 

are also considerably different in size. To indicate this 

diversity of size of drainage area, the stations used to 

represent the Tombigbee River in Alabama have mean 

annual discharges from 700 to 35,000 cfs, whereas the 

Belle Fourche basin in Wyoming has mean annual discharges ranging from 40 to 600 cfs.

The mean lines through the points representing individual gaging stations (figures 5-8 are examples) were 

fitted by eye. There is considerable scatter of points 

about the respective mean lines. Moreover, as discussed previously, the points on the graphs represent 

values of depth, width, and velocity at stream-gaging 

stations situated at positions not uniformly representative of the average river cross sections. Despite the 

possible variations that might be introduced by these 

considerations, there is a tendency toward parallelism 

of the lines on the graphs in figure 9. As a rough generalization, it may be stated that in a downstream 

direction the rates of increase in width, depth, and 

velocity relative to discharge are of the same order of 

magnitude for rivers of different sized drainage basins 

and of widely different physiographic settings.

Some river basins studied have too small a range of

mean annual discharge between small tributaries and 

the main stem to allow determination of the slope of 

the lines. A few river systems studied provide lines 

with slopes quite different from the majority. Figure 

9 includes one such anomaly, the Loup River system, 

in order to demonstrate the range of slopes of lines 

found in the data studied. In the example of the Loup 

River, the ground-water contribution from the sand 

hills at the headwaters is believed to be partl7 responsible for its anomalous behavior.

The streams chosen to be included in figure 9 differ 

markedly in physiographic setting. The Tombigbee 

River of Alabama is a coastal-plain stream, narrow and 

deep. At a discharge of 1,000 cfs its depth is about 7.5 

ft and its width 110 ft. The Republican and Kansas 

Rivers are typical streams of the Great Plain?. They 

are relatively wide and shallow. At 1,000 cfs the lines 

representing these streams indicate a mean depth of 

2.3 ft and a width of about 250 ft. In other words, the 

intercepts of the graphs on figure 9 differ greatly among 

rivers but the slopes of the lines are similar in magnitude. The rates of increase in depth, width, and 

velocity with discharge downstream may b^ similar 

among rivers despite marked differences in the width 

to depth ratio at any particular discharge.

From the data in figure 9 and other similar data it 

appears that the velocity tends to increase with mean 

annual discharge downstream in all the rivers studied. 

It must be remembered that this generalization is 

restricted to the situation in which discharge at all 

points along the river is of similar frequency.

Most geomorphologists are under the impression that 

the velocity of a stream is greater in the headwaters 

than in the lower reaches. The appeararce of & 

mountain stream, of course, gives the impression of 

greater Idneticity than that observed in a large river 

downstream. The impression of greater velocity upstream stems in part from a consideration of river 

slopes which obviously are steeper in the upper than in 

the lower reaches. It will be recalled, however, that 

velocity depends on depth as well as on slope, as shown 

in the Manning equation

v— 1.5 (4)

where depth, d, is approximately equal to hydraulic 

radius for natural river sections, v is mean velocity, n is 

channel roughness, and s is slope. It is recogrized that 

slope, s, in equation (4) is the slope of the energy grade 

line.* For practical purposes throughout mo^t of the 

discussion in the present paper, slope of the river profile 

sufficiently approximates the energy gradient for equation (4) to be used as if the slopes were identical.

The fact that velocity increases downstream with 

mean annual discharge in the rivers studied indicates 

that the increase in depth overcompensates for the 

decreasing river slope. The magnitude of this rate of
VARIATION OF HYDRAULIC CHARACTERISTICS IN A DOWNSTREAM DIRECTION

10 100

4,000

2,000

Mean annual discharge, in cfs

1,000 10,000 100,000 ,000,000

^ 2

•:-^£==£:==s:(8?© "'/ CD

10 100

®~ CD-'

(2D"

1,000 10,000 

Mean annual discharge, in cfs

100,000 1,000,000

STREAM AND LOCATION

TOMBIGBEE(ALA.) 

FRENCH BROAD (N.C.) 

BELLE FOURCHE(WYO.) 

YELLOWSTONE-BIGHORN (WYO )

I-T1-

REPUBLICAN-KANSASfKAN.)

•- LOUP(NEBR.)

MISSISSIPPI, MAIN STEM 

i -'——»- MADRAS IRRIGATION CANALS{INOIA)

FIGURE 9.—Width, depth, and velocity in relation to mean annual discharge as discharge increases downstream in various river systems.

change becomes clear by comparison of the exponents of 

depth and slope. In equation (4) the velocity depends 

on depth to the power f and on slope to the power ^. 

Increasing depth tends to increase velocity in a downstream direction. A variation of roughness in the 

downstream direction will also affect the rate of change 

of velocity and will be discussed later.

In summary, the rate of increase of depth dovrnstream tends to overcompensate for the decreasing 

slope and tends to provide a net increase of velocity at 

mean annual discharge in the downstream direction of 

a river.

The straight-line relations on logarithmic paper show, 

just as they do for a particular river cross section, tl H
16 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

width, depth, and velocity increase downstream with 

discharge in the form of simple power functions,

w^aQ* (1)

d=cQf (2)

v=kQm (3)

The coefficients and exponents are assigned the same 

letters as for a given cross section of a river, but the 

values will be different for points in a downstream 

direction from those for the given section. Average 

values of the exponents for river basins studied are

6=0.5 

/=0.4

To recapitulate, both at a given river cross section 

and in the downstream direction at mean annual discharge, width, depth, and velocity increase with discharge. The functions representing these relations are 

similar but involve different exponents and numerical 

constants.

RELATION OF CHANNEL SHAPE TO FREQUENCY 

OF DISCHARGE

Thus far the changes in hydraulic characteristics 

downstream in a river system have been considered only 

for a discharge equal to the mean annual rate at each 

point. Comparison of the data already presented will 

now be made with similar data for infrequent flows.

Computation of the frequencies of various discharges 

at a gaging station has been discussed. The computation is a somewhat laborious task. Complete cumulative frequency (flow-duration) curves for all streamgaging stations in a watershed are available for only a 

few localities; one of the best collections of such data is 

that by Cross and Bernhagen (1949) for Ohio.

It was desired to avoid choosing streams regulated 

by many reservoirs, and the Maumee and Scioto 

River basins were selected as being freer of regulation 

than most major river systems in Ohio. Duration 

(cumulative frequency) curves were available for 19 

gaging stations in these two watersheds. Though the 

Maumee River flows northeast into Lake Erie and the 

Scioto south into the Ohio River, the two basins have a 

common divide, and study of the data showed 110 

significant differences in the hydraulic characteristics 

for the purposes of this paper.

At each gaging station discharges that were equalled 

or exceeded 1, 4, 10, 30, and 50 percent of the time, as 

well as the mean annual discharge, were selected for 

study. For each of these discharges, the corresponding 

velocity, depth, and width were determined by analysis 

of current-meter data, as explained for figure 3. For 

each discharge frequency, curves were plotted representing the change of hydraulic factors with discharge

in the downstream direction. Figure 10 presents the 

individual plots of six curves in order to demonstrate 

the relative goodness of fit. It will be seen that the 

gaging stations available for the Maumee aid Scioto 

basins did not represent so large a range in discharges 

as did some of the rivers discussed earlier, and that the 

points did not fall in so good alinement as in some other 

basins studied. The width-to-discharge relations are 

the best; more scatter of points appears on the depth 

and velocity curves.

For the Maumee and Scioto basins combined the 

curves of discharge in relation to width, depth, and 

velocity for each of five frequencies are shown in 

figure 11.

There is a general parallelism within each family of 

curves, but this is somewhat more coiisistert in the 

width-to-discharge than in the velocity-to-discharge 

relation. In these data it appears that, with less frequent discharges (flood flows), the velocity increases 

somewhat faster with discharge downstream than it 

does in low flows. A similar study made for flood flows 

in some tributaries to the Yellowstone River in Wyoming showed that velocity remained nearly constant 

downstream despite a rapid decrease in slope.

It can be seen that there is a pattern of interrelationship of stream characteristics which include? relative 

frequency of discharge, and it is now possible to define 

important aspects of the hydraulic geometry of stream 

channels. An understanding of this geometry is basic 

to consideration of the adjustment of channel shape to 

cany the sediment load supplied to the streams. The 

relation of at-a-statioii characteristics to tie downstream characteristics will now be discussed.

In basin A of figure 12 the line A—C shows the increase 

of width with increase of discharge at an individual 

gaging station near the headwaters of a stream, and 

B—D shows the same relationship for a doyristream 

station. If the discharge A is of the same cumulative 

frequency (say 50 percent of the time) at the headwater 

gage as discharge B at the downstream gage, then the 

line A—B is the increase of width with discharge downstream, corresponding to a cumulative frequency of 

50 percent. Also if C and D have the same cumulative 

frequency (1 percent of the time), then line C—D is the 

increase of width downstream at a discharge frequency 

of 1 percent. Similarly then, E—G is the increase of 

depth with discharge at the upstream gage, and E—F is 

the increase of depth downstream for a frequency of 

50 percent. In the example under discussion, the 

curves are drawn so that at any frequency tire widthto-depth ratio remains constant downstream; that is, 

A—B is parallel to E—F.

It was noted earlier that because

Q = wdv

and w=a,Q*, d=cQf, and v=kQm, 

then 6-h/+m=1.0
RELATION OF CHANNEL SHAPE TO FREQUENCY OF DISCHARGE 17

IOO 1,000 10,000 100,000

500

200

•-C 100

Q. 

0) 

Q

T3

C.

o 

o 

<u

V)

k_ 

<U 

Q. 

•«— 

<U 

CD

100 1,000

Discharge, in cfs

10,000 100,000

EXPLANATION-PERCENT OF TIME DISCHARGE EQUALLED OR EXCEEDED: >; I. OPERCENT 

V 50.0 PERCENT

FIGURE 10.—Width, depth, and velocity in relation to discharges representing two frequencies, Maumee and Scioto River Basins, Ohio.

If the slope of the width-discharge curve for a given 

frequency is 0.5 (exponent 6=0.5), a typical value for 

many rivers, and if A—B parallels E—F (as in fig. 12, 

basin A), then the slope of the velocity-discharge curve 

J—K, corresponding to the discharge occurring 50 percent of the time,, must be zero (m=0). At that frequency the velocity does not change downstream. *T e 

line J—L is the increase of velocity with discharge at the 

upstream gage, and K—M at the downstream gage. It 

is clear that at the 1 percent frequency, the velocity is 

higher at both stations than at the more frequent discharge but does not change downstream.
18 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

1000

500

200

100

!00

Discharge, in cfs 

1,000 10,000 100,000

0.5

100 1,000

Discharge, in cfs

10,000 100,000

EXPLANATION 

PERCENT OF THE TIME FLOW IS EQUALLED OR EXCEEDED: —————1.0%', ——- 10%-, •30%: 50%.

-o MEAN ANNUAL DISCHARGE.

FIGURE 11.—Width, depth, and velocity in relation to discharge of various frequencies, Maumee and Scioto River basins, Ohio.

In figure 12, basin B, the width-to-depth ratio increases downstream at any frequency. This increase 

is shown by the fact that A'—B' slopes more steeply 

than E'—F'. If the slope of A'—B' is again 0.5, the 

steepening of the slope of 3'—K' makes the points 

J', K', L', and M' fall nearly in a straight line. Basin 

B is nearly identical in its geometry with the comparable curves for the Maumee-Scioto basin (fig. 11), and 

basin A .with the Bighorn basin, presented in figure 6.

The slopes of the width-discharge curves in a downstream direction were made the same in both diagrams.

This is in accord with the data which indicate that the 

slope of the width-discharge curve downstream is, 

between basins, somewhat more consistent than the 

other relations.

The channel characteristics of natural rivers are seen 

to constitute, then, an interdependent system which 

can be described by a series of graphs having simple 

geometric form. The geometric form of the graphs 

describing these interactions suggests the term "hydraulic geometry." Channel characteristics of a particular river system can be described in terns of the
SUSPENDED SEDIMENT AND DISCHARGE IN A PARTICULAR CROSS SECTION 19

Discharge - 

Basin A

Discharge 

Basin B

EXPLANATION

—— CHANGE DOWNSTREAM FOR DISCHARGE OF GIVEN FREQUENCY

•—— CHANGE AT GAGING STATION FOR DISCHARGES OF DIFFERENT FREQUENCIES

FIGURE 12.—Diagrammatic relations of width, depth, and velocity to discharge at a station and downstream.

slopes and intercepts of the lines in the geometric 

patterns discussed in figure 12. Though the listing of 

slopes and intercepts of these lines may not provide 

any visual picture of a river basin, comparison of the 

values of these factors among rivers has useful aspects 

which will be discussed.

Thus far, no consideration has been given to either 

river slope or bed roughness. The importance of these 

factors is recognized by the authors. However, discussion thus far is, of necessity, descriptive in nature. 

Because comparable data for slope and roughness are 

not available in so great a quantity as data on width, 

depth, velocity, and discharge, the particular data 

available for analysis make it desirable to organize 

the paper in its present form.

The observed relations of width, depth, and velocity 

to discharge in natural rivers have been described. In 

the next part the observed relation of width, depth, 

velocity, and discharge to measured suspended load 

will be presented. Then, the effect of slope and roughness will be discussed in an attempt to provide a 

partial explanation for the relations observed in the 

other quantities.

219627—52———4

THE HYDRAULIC GEOMETRY OF STREAM CHANNELS 

IN RELATION TO SEDIMENT LOAD

SUSPENDED SEDIMENT AND DISCHARGE IN / 

PARTICULAR CROSS SECTION

This second part of the present paper describes the 

interrelations of measured sediment with the other 

hydraulic variables. Following a description of t\e 

changes in suspended-sediment load at a station and 

then downstream, hydraulic and physiographic interpretation of the observed relationships will be explored.

Sediment transported by a stream may conveniently 

be divided into suspended load and bed load, but it 

must be understood that, depending on the flow conditions, sediment may at one time be part of the 

suspended load and at another time part of the b^d 

load. Moreover, the capacity of a channel to carry 

sediment of one size is at least partially independent 

of its capacity to carry another size. For this reason 

Einstein (1950, p. 4) prefers to differentiate between 

"wash load" consisting of particles finer than those 

making up the stream bed, and "bed-material load."

No simple and satisfactory method of sampling b^d 

load of a natural stream is in use at present. Any
20 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

sampler placed oil the stream bed provides an obstruction, to the natural flow and tends to set up turbulent 

eddies in the immediate vicinity which so change the 

transport near the bed that the resulting sample 

becomes nonrepresentative. The data available for 

analysis, therefore, consist of suspended sediment 

sampled by use of the improved depth-integrating 

samplers. The sampled load is made up of particles 

moving outside the bed layer. The measured suspended load does not present a fixed percentage of the 

total load in motion but varies with a number of factors, 

as proposed by Maddock and discussed by Lane and 

Borland (1951).

The student of fluvial processes may at this point 

seriously question the general applicability of relations 

derived only from data on suspended load. An attempt 

will now be made to demonstrate that relations between 

suspended-sediment load and channel characteristics 

which can be derived from the available quantitative 

data justify the use made here of the data on suspended 

load.

Again for purpose of definition, it is desirable to 

discuss a tool in wide use in hydrologic practice, the 

graph showing the relation of suspended sediment to 

water discharge.

The relation of suspended load to discharge (sometimes called the sediment rating curve) is usually shown 

as a plot on logarithmic paper of suspended sediment, 

in units of weight per unit of time, against discharge of 

the water-sediment mixture at a particular gaging 

station (see Linsley, Kohler, and Paulhus 1949, p. 331; 

Hembree, Colby, Swenson, and Davis 1952, p. 69). 

A typical relation of suspended load to discharge is 

presented in figure 13 (after Hembree, Colby, Swenson, 

and Davis 1952, fig. 33). The data for such a curve are 

obtained by daily, weekly, or other periodic sampling 

at the sediment station. Such stations are being 

installed and maintained by Federal and State agencies 

in increasing numbers in the United States. When 

their records encompass a long enough time, the growing 

network of stations promises a mass of basic data which 

will be of importance to the study of river morphology.

The sediment is sampled by a depth-integrating 

suspended-load sampler. No fewer than two samples 

are taken at various distances from the stream bank in 

a given river cross section. These samples are analysed 

in the laboratory to determine concentration of suspended sediment in parts per million by weight. The 

samples from a given cross section are combined to 

give a mean concentration which is then applied to the 

mean discharge Jor the day to provide an estimate of 

tons of suspended load passing the station on the 

given day. This figure is then plotted against the 

mean discharge for the day as a single point on the 

curve showing relation of suspended load to discharge. 

Each point on the curve of figure 13 was determined 

in this manner.

1,000,000^

100,000 —

10,000 —

— 1,000 =

IOO —

10 100

Discharge, in cfs

1,000 10,000

FIGUBE 13.—Relation of suspended-sediment load to discharge, Pcwder River 

at Arvada, Wyo.

The basic data on suspended-sediment load used in 

the present study are available to the public, as already 

explained, in association with current-meter measurements. Data on suspended sediment will be published 

by the Geological Survey, but as yet there is considerable delay between collection of data and their publication.

It will be noted in figure 13 that the curve showing 

relation of suspended load to discharge contains a wide 

scatter of points, as is characteristic. This scatter 

indicates that at a given rate of discharge, the amount 

of suspended sediment may vary considerably from one 

day to another. It will be shown later that s,t a given 

discharge, increased amounts of suspended load supplied 

to a reach tend to promote changes in the. channel 

characteristics leading to increased capacity to transport, particularly through adjustment of the ratio of 

velocity to depth. Thus, scatter of points above and 

below the mean line in the graph of suspended load and 

discharge implies corresponding compensatory arrangements of points about the mean curves showing relations 

of velocity to discharge and depth to discharge.

A straight line has been drawn through the points 011 

the curve in figure 13. The points are so scattered on 

some graphs of suspended load and discharge that a 

smooth curve drawn to represent them has little 

meaning. When the record is sufficiently long, as it
SUSPENDED SEDIMENT AND DISCHARGE IN A DOWNSTREAM DIRECTION 21

is in figure 13 and in the data for 20 stations analysed 

in this report and summarized in the appendix, a 

straight line on logarithmic paper is a usable approximation. Many graphs of suspended load and discharge may be represented by straight lines in the 

middle range of discharge, but by curves or lines of 

different slopes in the extremely low and high discharge 

ranges.

It is typical that the line approximating the data on a 

graph of this type slopes steeply, as in figure 13. At 

a station suspended load usually increases more rapidly 

than does water discharge. It will be noted that if 

water and suspended sediment increase at equal rates, 

the sediment concentration is constant. Constant concentration of suspended sediment would be represented 

in figure 13 by a line sloping upward to the right at 45°. 

The fact that the line slopes more steeply indicates 

that suspended-sediment concentration increases rapidly 

with discharge at a station.

This usual rapid increase of suspended load with 

discharge is interpreted by the authors as indicating 

that conditions of rainfall and run-off on the watershed 

combine to furnish to the major stream channels a large 

increment of debris for each increment of water. In 

other words, it is believed that the slope of the curve 

showing the relation of suspended sediment to discharge is the result of the process of sheet, rill, and gully 

erosion on the watershed and represents sediment imposed on the major stream channels which must carry 

it away.

No detailed hydraulic theory is available to explain 

why sediment increases at a faster rate than discharge 

at a given station. However, a number of widely 

known facts make this relation seem logical. Large 

discharges imply greater rainfalls than small ones. 

Great rains are characterized by a larger percentage of 

surface run-off than small rains, for many reasons. 

Three reasons are:

First, the initial rain must fill surface detention and 

depression storage before surface run-off can begin. 

Infiltration rate decreases rapidly during the initial 

period of a rainstorm.

Second, as the soil surface becomes wetted, areas 

poorly protected by vegetation are subjected to the 

churning effect of raindrop impact (Ellison, 1944) 

which increases with rainfall intensity. Puddling of 

the soil surface, if it occurs, probably increases with 

the duration and intensity of rainfall.

Third, moreover, large runoff rates imply greater 

depths and velocities of water flowing in eroding rills. 

These factors tend toward a greater efficiency of erosion 

with the increase in duration and intensity of rain. 

Thus, it might be expected that at a given river cross 

section sediment production increases faster than discharge.

It might be presumed that because the bed of an 

alluvial river tends to be scoured during high discharges,

the observed increase of suspended-sediment concentration with increased discharge results from the suspension of the additional material scoured out of the bed 

by concomitant high velocity.

It will be shown later that at certain river measuring 

stations deposition on the bed occurs under the conditions of high velocities accompanying the rapidly 

increasing suspended loads of flood rises. There 

velocities are higher than those accompanying scour 

of the bed during the decreasing suspended loads of 

flood recessions.

This occurrence is sufficiently common to support 

the contention that the changes in velocity and depth 

during a flood passage are caused by changes in concentration of suspended sediment introduced into the 

particular reach. Suspended-sediment concentration 

may generally be considered an independent variable, 

with velocity and depth dependent.

It is observed that the relation of suspended sediment to discharge at a given river cross section may be 

approximated, through at least a major part of the 

range of discharge, by a straight line on logarithmic 

paper. Such a line is denned by the expression

in which L is sediment load in tons per day, p and j are 

numerical constants. In the data analyzed, there is a 

tendency for the suspended sediment at a/ station to 

increase faster than discharge, or the slope, j, of tie 

suspended sediment-discharge curve is greater than 

unity. Values of j typically lie in the range 2.0 to 3.0.

In summary, it is typical at a given cross section of 

an alluvial stream for the suspended-sediment load to 

increase with increase in discharge, and at a more rapid 

rate. That is, the concentration of suspended sediment increases with discharge. Consideration of the 

physical characteristics of surface run-off and observations of bed scour lead to the tentative conclusion the t 

the observed increase in sediment concentration results 

primarily from erosion of the watershed rather than 

from scour of the bed of the main stream, in the reach 

where the measurement is made.

RELATIONS OP SUSPENDED SEDIMENT TO 

DISCHARGE IN A DOWNSTREAM DIRECTION

Stations where suspended sediment is sampled are 

still too few to provide data for a direct analysis of 

change in the concentration of suspended sediment in a 

downstream direction, as was done for the other hydraulic factors.

In long reaches of a river through more or less homogeneous topography the entrance of various tributaries 

does not appear to the eye to have any great effect on 

the turbidity of the water. Even over great distance** 

the changes in sediment concentration downstream are 

are not sufficient to be distinguished without actual 

sampling.
22 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

The change in sediment concentration downstream 

through homogeneous topography and lithology would 

seem to be small. However, the percentage of area of 

lesser surface slope increases with drainage area. Areas 

of little slope probably contribute less sediment per unit 

of water run-off than areas of steep slope. Groundwater contributes a larger percentage of flow to a stream 

draining a large area than to one draining a small area.

Inasmuch as the percent of land not contributing 

sediment and, therefore, the percent of sediment-free 

water tend to increase somewhat with drainage area, 

it would be logical to suppose that the average sediment 

concentration should decrease slightly in the downstream direction. "Rubey (1933, p. 505) reached the 

same general conclusion. It should be understood, 

however, that individual rivers may differ in this respect 

(Mackin, 1948, p. 480).

Strong support for this conclusion is provided by the 

observation that the average annual production oJ sediment per square mile of drainage area decreases with 

increasing drainage area. In other words, sediment 

production per unit of area is greater for small than for 

large drainage areas. This conclusion has been reached 

by the study of erosion of sample upland areas and 

valley cross sections in Illinois (Brune, 1950), by the 

analysis of sediment trapped in small reservoirs in 

Wyoming (Hadley, in preparation, and open file), and 

from similar data on reservoirs in Iowa and Nebraska 

(Glymph, 1951, p. 6). If sediment contribution per 

unit of area decreases with increasing area, on the average there should be a tendency toward smaller concentrations of suspended sediment downstream.

The relation of suspended sediment to discharge both 

at a station and downstream can now be considered in 

the hydraulic geometry of stream channels. In figure 

14 the curves of width, depth, and velocity in relation 

to discharge are similar to those discussed in figure 12. 

In the diagrammatic plotting of suspended-sediment 

load against discharge in figure 14, the dotted line 

A3—C3 is the curve showing the relation of suspended 

sediment to discharge for an upstream station, for 

which the at-a-statioii width, depth, and velocity 

curves are respectively A0—C0 , A!—Ci, and A2—C2 . 

The curves for a downstream station are, respectively, 

B0—DO, BI and DI, and B2—D2 ; and curve B3—D3 

shows the relation of suspended load to discharge.

The small number of suspended-sediment stations 

arranged along the length of a given river precludes the 

construction of curves representing the change of 

suspended-sediment load downstream, but such a curve 

is a necessary part of the hydraulic geometry. The 

general argument in the preceding paragraphs appears 

to support the hypothesis that concentration of suspended sediment usually decreases somewhat downstream with discharge of equal frequency. As in the 

suspended load-discharge curve of figure 13, equal 

concentration downstream would be represented by a

EXPLANATION

—— CHANGE DOWNSTREAM FOR 

DISCHARGE OF GIVEN FREQUENCY 

———— CHANGE AT GAGING STATION FOR 

DISCHARGES OF DIFFERENT FREQUENCIES

Discharge

NOTE; ALL SCALES ARE LOGARITHMIC

FIGURE 14.—Typical relations in the hydraulic geometry of natural channel 

systems, showing width, depth, velocity, and suspended-sedimert load platted 

against discharge both at a station and downstream.

line sloping upward to the right at 45°. Th? probable 

relation is therefore shown on figure 14 as the solid 

lines A3—B3 and C3—D3 which slope slightly less 

steeply than 45°.

A brief summary will explain the purpose of the next 

step in the analysis. The shape of the cross-section 

at any point on a river may be described by th^ observed 

increase of width, depth, and velocity with increase of 

discharge. Considering the river system from head-
HYDRAULIC FACTORS AND SUSPENDED LOAD AT GIVEN DISCHARGE 23

waters downstream, if discharge at all points is comparable in percentage of time this discharge is equalled 

or exceeded, the width, depth, and velocity are observed 

to increase downstream with increased discharge. 

These at-a-station and downstream relations can be 

expressed by simple graphs having a geometric pattern. 

This geometric pattern is an expression of the interrelationship of width, depth, velocity, and discharge 

expressed by the identity

Discharge = width X depth X velocity

This identity, however, must be only one of a number 

of equations which determine the observed relationships. 

There probably is a relation between the characteristics of the channel shape and the measured suspendedsediment load. The purpose of the next sections is to 

define, at least qualitatively, how the suspended load 

is observed to vary in different rivers having different 

width, depth, and velocity relationships. The analysis 

will first be attempted for a given discharge, and then 

will be extended to include different discharges.

WIDTH, DEPTH, VELOCITY, AND SUSPENDED SEDIMENT AT A GIVEN DISCHARGE

Certain aspects of the effect of suspended sediment 

on channel shape are best exemplified by consideration 

of the condition of constant discharge. The following 

discussion concerns the analysis of measured sediment 

load at various gaging stations when the discharge at 

the stations is equal.

Data for 20 sediment measuring stations were used 

in this study. The data consisted of curves showing 

relation of suspended sediment to discharge; these are 

summarized in tabular form in the appendix of this 

report. Sediment measurements are being made at 

many more stations but insufficient length of record or 

absence of simultaneous current-meter discharge measurements precluded their use. A few records were 

discarded because of unnatural cross sections. Particularly, data obtained at dams or in sections where the 

channel walls were confined were deemed unsuitable.

For each station, current-meter measurement data 

were used to construct curves relating discharge to 

width, depth, and velocity, as shown in figures 3 and 4. 

From these data and curves showing the general relation 

of suspended sediment and discharge, values of these 

four factors corresponding to a discharge of 500 cfs 

were tabulated.

For each station, width was plotted against suspended 

load in figure 15, and at each point the corresponding 

velocity was entered. Isopleths of velocity were then 

drawn; these are the lines slanting steeply downward to 

the right in the figure.

Because the graph represents data for a fixed discharge, every point in the width-velocity field defines 

a particular depth inasmuch as

Q=w d v= constant

Thus isopleths of depth can be drawn which are the 

lines slanting gently downward to the right.

Figure 15, therefore, is a field showing the relation of 

corresponding values of width, suspended load, velocity, 

and depth. The graph has many limitations but it is 

nevertheless a useful tool for some qualitative analyse?. 

Among its limitations are, first, the fact that tie 

velocity field is only roughly defined by the values in 

the available data. Second, only the suspended part 

of the load is being considered. Third, the scatter of 

points on any average curve showing relation of suspended sediment to discharge is large.

Nevertheless, certain general conclusions of apparert 

significance may be drawn from figure 15 and they will 

be tested against observed changes in the characterirtics of river channels.

If a constant discharge is assumed, at constant width 

of channel an increase in velocity is accompanied by 

an increase in suspended sediment. This requires, at 

constant discharge, a decrease in depth.

Conversely, if velocity is constant, an increase in 

width is accompanied by a decrease of suspended load. 

This is also associated with a decrease in depth.

500

NOTE: PLOTTED FIGURES REPRESENT — 

VELOCITY, IN FEET PER SECOND

100 500 1,000 5,000 10,000 50,000 100,000 500,000

Suspended-sediment load, in tons per day

FIGURE 15.—Average rektion of suspended sediment-load to channel-shape factors at a constant discharge of 500 cfs (from river data in'appendix).
24 THE HYDKAULIC GEOMETRY OF STREAM CHANNELS

In summary, decreasing width at constant velocity 

or increasing velocity at constant width each results in 

increased capacity for suspended load at constant discharge.

These relationships should be reviewed here in terms 

of the basic data from which they were derived. The 

basic data consist of 20 cross sections of different rivers 

(see appendix). The rivers include the Loup, Rio 

Grande, Bighorn, Smoky Hill, Powder, Moreau, 

Republican—in other words, rivers in the Great Plains, 

Colorado Plateaus, and Rocky Mountain areas. The 

data are therefore biased toward Western conditions, 

yet the rivers are rather diverse in general appearance. 

A station on each river was considered under conditions 

when 500 cfs was flowing in the channel. At that particular discharge, some of the river sections are wide, 

others narrow, some shallow, others deep. The 

observed suspended load at that discharge has been 

correlated with the concurrent width, depth, and 

velocity at the particular discharge.

The graphical relation shows that a wide river having 

a particular velocity is observed to carry a smaller 

suspended load than a narrow river having the same 

velocity and discharge.

It shows also that if two rivers have equal width at 

the same discharge, the one which is flowing at the 

higher velocity is carrying a larger suspended load.

These ,are empiric relations observed in the available 

river data. Nothing has been said regarding why one 

river is wide and another narrow, or one is deep and 

another shallow. Clearly these features depend on 

such factors as slope, roughness, and particle size which 

have not yet been considered.

In the analysis, the rivers were compared at equal 

discharge. It is now logical to examine the same 

relations at different discharges.

WIDTH, DEPTH, VELOCITY, AND SUSPENDED LOAD 

AT A VARIABLE DISCHARGE

Relations similar to those examined at constant discharge will now be discussed for three other discharge 

rates. Using the same 20 suspended-sediment sampling 

stations, measurements of depth, width, velocity, and 

suspended load were plotted for discharges of 250, 

1,000, and 2,000 cfs in a manner similar to that used for 

figure 15. The mean isopleths of velocity for the four 

discharges were replotted in figure 16, with the actual 

velocity values shown on figure 15 eliminated for the 

sake of clarity.

As in figure 15, when discharge, width, and velocity 

are specified, depth also is specified. In other words, a 

particular point in the graph of figure 16 specifies 

width, depth, and velocity for any of the four discharges.

Any arbitrary straight line drawn on figure 16 will 

cross the respective fields of velocity for the four discharges. Any point on that arbitrary line vdll represent simultaneous values of width and of velocity in one 

of the four velocity fields. The line may be drawn, 

then, to represent any chosen width-to-discharge relation and a chosen velocity-to-discharge relation. A 

corresponding suspended load-discharge relation becomes thereby determined.

An example will help clarify this idea. If one follows 

the horizontal line in figure 16 representing a width of 

100 ft to the 2-fps velocity line representing a discharge 

of 250 cfs, he will see that the point of intersection 

represents a suspended load of roughly 1,500 tons per 

day. The 2-fps velocity line for 500 cfs intersects the 

same horizontal line at a point equal to a load of 3,000 

tons per day. The 2-fps velocity line for 1,000 cfs 

intersects at 6,500 tons per day, and the same line for a 

discharge of 2,000 cfs at 15,000 tons per day. It will

400

200

100

1.5

100

DISCHARGE:

1,000 10,000 100,000

Suspended-sediment load, in tons per day

EXPLANATION 

————— 2,000CfS ; ————————I.OOOCfS; — — — — SOOCfS; ——-

1,000,000

Numbers indicate the velocity in fps isopleths

FIGURE 16.—Average relation of suspended-sediment load to channel-shape factors at four discharge rates (from river data in appendix).
HYDRAULIC FACTORS AND SUSPENDED LOAD AT VARIABLE DISCHARGE 25

be noted that these corresponding values of discharge, 

velocity, and suspended load could be used to define 

relations between discharge and the separate factors of 

velocity and suspended load. These relations have 

been plotted in figure 17. This suspended sedimentdischarge curve would represent the mean suspended 

sediment-discharge relation for a channel of constant 

width. In this hypothetical channel, various discharges would have the same mean velocity, 2 fps. 

Because for each of the four discharge points, velocity 

and width are specified, depth also would be specified. 

Therefore the relation of width, depth, and velocity to 

discharge can also be plotted, and these plots also are 

shown on figure 17.

500

10,000

1,000

100 1,000

Discharge, in cfs

10,000

FIGUEE 17.—Example of the relation of suspended-sediment load and water 

depth to discharge for specified relations of width to discharge and velocity to 

discharge (constructed from fig. 16).

In accordance with the procedure used in the example, 

any straight line drawn on the width-suspended loadvelocity field of figure 16 defines the rates of increase 

of two variables with discharge. The rate of increase 

of the other two variables with discharge must be given. 

In the previous example, the rate of increase of width 

with discharge and the rate of increase of velocity with 

discharge were specified. Then the rates of increase of 

depth and suspended load with discharge became known. 

The slopes of these lines can be measured and are the 

exponents 6,m,/,and j in the equations

w=aQ1>

where the symbols are as defined previously. In the 

graphs of figure 17 it will be noted that

6+/+m=0+1.0+0=1.0 

as is required by the identity

Q=wdv

By analogy to the example plotted in figure 17, it can 

be visualized that for any given assumption of 6 (the 

rate of increase of width with discharge), the slope j of 

the curve relating suspended load to discharge will depend on the rate of increase of velocity with discharge 

defined by the exponent m. And because &-t-m.+/—1, 

when b and m are specified, / is also specified. If the 

ratio of m to / is known and the value of b is knovoi 

then values of m and / are known. For example, 

assume arbitrary straight lines on the field of figure 16 

and find the slope j of the suspended load-discharze 

relation for various values of b and the ratio m to /. 

The results are plotted in figure 18. Each point in the 

figure represents a given straight line drawn in the 

graph of figure 16. These lines were drawn in such a 

way that various values of b were represented. The 

resulting relation of velocity and load with discharge 

were plotted just as in figure 17, the slopes of the lines 

representing values of m and/ were measured, and tl e 

ratio of m to / was computed. Values of j and the 

ratio of m to / were then plotted in figure 18.

From figure 18 a number of useful concepts may be 

inferred. The first concerns the interaction of velocity, 

depth, and the suspended-sediment load. For any 

given rate of increase of width with discharge—that is,

2.0

S» 0.8

0.4

0 I

Slope of the suspended-sediment rating curve//} where L=pQJ

FIGURE 18.—Relation of rates of increase of width, depth, velocity, and suspendei

iji load to increasing discharge, expressed by values of 6, y, and ,;' (constructed

from fig. 16).
26 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

for any given value of b—the diagram indicates that 

the value of j increases with an increase in the m to / 

ratio. That is, the rate of increase of suspended-sediment load with discharge is a function of the ratio:

rate of increase of velocity with discharge 

rate of increase of depth with discharge

Slopes of the lines representing depth, width, and 

velocity respectively, in relation to discharge are, according to the implications contained in figure 18, 

functions of the slope of the line representing the relation of suspended sediment to discharge. For a given 

value of 6, the steeper the slope of the suspended sediment-discharge line, the greater will be the slope of 

the velocity-discharge line.

It is also possible to observe in figure 18 the points, 

or zones, which represent the average river conditions. 

The average values of the rates of increase of width, 

depth, and velocity with discharge have been listed 

separately, and are summarized here.

At a station 

6=.26 

/=.40 

m=. 34

Downstream 

6=.5

/=.* 

m=. 1

The suspended sediment-discharge relations providing the basic data from which figure 18 was derived, 

represent a number of different stations. Earlier discussion of typical relations of suspended sediment to 

discharge brought out the fact that the slope, j, of 

typical curves is of the order of 2.0 to 3.0. As previously indicated, at a typical river cross section, width 

increases only slightly with discharge until the bankful 

stage is reached. This rate of increase is described 

by the average value b which is provided by the available data, 6=.26. For the average cross section

TfL TTl (at a station) -7-= .85. In figure 18 where -7.-=.85,and

6=.26, .7=2.3. Thus the value read on the curves 

of figure 18 compares favorably with the observed average slope of suspended sediment-discharge curves.

It was brought out in the discussion of the relations 

of suspended sediment to successive points in the downstream direction that suspended-sediment concentration should decrease slightly downstream. A value of 

.7 = 1.0 means that suspended load is increasing at the 

same rate as discharge and represents equal sediment 

concentration. It is to be expected, therefore, that 

the observed value for rivers should be somewhat less 

than 1.0. Referring to the mean values of m, /, and b

fYl in the downstream direction where 6=.5, and-7-=.25,

it can be seen on figure 18 that./=0.8. Thus figure 18 

indicates that concentration of suspended sediment 

decreases slightly in a downstream direction for a 

given frequency of discharge, as the earlier argument 

indicated.

Figure 18 appears to apply co average conditions 

both at a station and downstream. For any particular cross section of a given river or for any particular 

flood hydrograph, it should not be expected ttat figure

18 would define the relations because it will be recalled 

that figure 15, from which type of data figur?- 18 was 

derived, represents a series of smooth lines drawn 

through data having considerable scatter in the plot. 

Nevertheless, figure 18 is a useful guide in summarizing 

a large volume of data on suspended load of rivers.

It is presumed that as additional data on suspended 

load become available, it will be possible to refine the 

graph of figure 16 by adjustment of the slope and 

position of the velocity isopleths. From available 

data, the relations in figure 16 may be summarized as 

follows: When cross sections of different rivers having 

different channel characteristics are compared, the 

observed suspended-sediment load will vary directly 

as a function of the velocity, directly as a function 

of the depth, and inversely as a function of the width 

of water surface. The suspended load varies FS a large 

power of velocity, and as small powers of d°,pth and 

width.

The preceding discussion may now be summarized by 

indicating the relation which the curves of figure 18 

bear to the hydraulic geometry of river system? characterized by figure 14. To facilitate reading, figure 14 has 

been reproduced in essentially the same form in figure

19 with the addition of simple block diagrams to allow 

easy visualization of the at-a-station and dovoistream 

relations constituting the hydraulic geometry.

Figure 19 summarizes the average conditions observed 

in the numerous rivers studied. Cross section A represents the conditions at a headwater station during low 

flow. Its position will be noted in the uppe^ physiographic diagram of a watershed in which low flow 

prevails. Cross section C represents the same headwater station under conditions of high discharge. 

Similarly, B is a downstream section at low flow, and D 

the same section at high discharge.

The slopes of the lines in the graphs of figure 19 

represent the average conditions observed in rivers. 

In the width-discharge graph, line AO—B0 ha^ a slope 

upward to the right equal to 0.5, or in the expression 

w=aQ!) , 6=0.5, which is representative of the increase 

of width with discharge downstream.

Similarly, in the width-discharge graph, the line 

A0—C0 represents the increase of width with discharge 

at a station and has a slope 6==0.26.

In the depth-discharge graph line, AI—BI has a slope 

/=0.4, and A!—Ci has a slope/= 0.4. In the velocitydischarge graph, line m=0.1 for the downstream direction, and m=0.34 at a station.

Lines in the suspended load-discharge graphs also 

have slopes equal to average values for rivers, ^'=0.8 

downstream (line A3—B3 ) and j=2.5 at a station (line 

AS L< 3 ).
HYDRAULIC FACTORS AND SUSPENDED LOAD AT VARIABLE DISCHARGE 27

Discharge •

NOTE: ALL SCALES ARE LOGARITHMIC

EXPLANATION

—— CHANGE DOWNSTREAM FOR 

DISCHARGE OF GIVEN FREQUENCY 

———— CHANGE AT GAGING STATION FOR 

DISCHARGES OF DIFFERENT FREQUENCIES

FIGURE 19.—Average hydraulic geometry of river channels expressed by relations of width, depth, velocity, suspended-sediment load, roughness, and slope to discharge

at a station and downstream.
28 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

It is now possible to interpret the relation between

171 -ft b, and j shown in figure 18 and the diagrams in

figure 19. Figure 18 shows the observed relationship 

between suspended load in rivers and the channelshape factors. The figure shows that for a given value

Tfi of b, the value j increases with the ratio -y The river

data indicate that for given a rate of increase of width 

with discharge (a given slope of line A0 — B0 ), the steeper 

the slope of the suspended load-discharge line A3 — B3 , 

the steeper must be the slope of the velocity-discharge 

line A2 — B2 . This would require a smaller slope of the 

depth-discharge line AI — Bt in order to preserve the 

relation of slopes required by

Another way to look at the geometry of figure 19 

would be as follows: For a given width, and at a given 

discharge, an increase in suspended sediment requires an 

increase in velocity and a reduction in depth. Exactly 

this interpretation was drawn from figure 15.

This set of interrelationships provides a visual 

picture of how the channel system of a river is carved 

to carry the water and sediment provided by the 

drainage basin. The at-a-station and downstream 

relations between discharge and suspended load shown 

by the geometrical pattern AaBaCaDa may be considered 

essentially independent of the channel system, but are 

functions of the drainage basin. The relations of 

width, depth, and velocity to discharge tend to be 

adjusted to conform to the load-discharge function 

according to the principles just outlined.

The geometrical relations describe another typical 

relationship in rivers — that of the upstream and downstream station at a given discharge, represented along 

a given vertical line in figure 19. As an example, 

a particular discharge is represented by the dashed 

vertical line representing the discharge Q'. The A — C 

lines represent changes at the upstream station and the 

B — D lines at the downstream station. At the same 

discharge, Q', note that the suspended load at 

the downstream station (ordinate where B3 — D3 crosses 

the dashed line) is less than the suspended load at the 

upstream station (ordinate where A3 — C3 crosses the 

dashed line). This is reasonable because a given discharge at the downstream station may represent a 

frequent flow or a low flow condition, whereas to attain 

that same discharge the upstream station must experience a flood flow. A given discharge is equalled or 

exceeded more often at a downstream than at an upstream station, and common experience teaches that 

the frequent, or low, flow is likely to carry a small 

concentration of suspended sediment, and a flood flow 

a large concentration. The given discharge, then, 

seldom occurs in nature simultaneously at the headwater and downstream station.

It may now be asked how the channels at the various 

stations are shaped to promote the equilibrium required 

by this comparison. At that given discharge, the 

downstream station has a larger width and a lower 

velocity, as can be seen in the diagram. This is exactly 

what figure 15 suggests; that is, at a given discharge, 

a smaller suspended-sediment load requires fcr equilibrium a larger width or lower velocity or both.

Inasmuch as the diagrammatic relations in figure 19 

show the actual average slopes of the various lines 

derived from river data, the comparison at constant 

discharge again confirms the principles of channelshape adjustment postulated.

The diagram also includes observed average variations of roughness and slope which will be discussed 

later.

It is emphasized that these relations are those derived 

by comparing a large number of river data. The relations are those existing in natural rivers. Flumes 

could be constructed in which relations of slope, roughness, and the channel-shape factors are different from 

those existing in rivers. In such flumes the relations 

described above would not necessarily hold.

It is now desirable to observe what kinds of similar 

relations seem to hold for bed load.

INTERRELATIONS OP WIDTH, DEPTH, VFLOCITY, 

AND BED LOAD AT A GIVEN DISCHARGE

The experimental flume data of Gilbert (1914) have 

an advantage, for present purposes, over siiy other 

comparable experimental data. Gilbert used flumes 

of various widths under conditions of constant discharge. These data allow an analysis, therefore, of 

the relation of width, velocity, and load at constant 

discharge. The Gilbert sands were such that the load 

was carried primarily near the bed, and fo^ present 

purposes his results may be considered to represent 

primarily the transport of sediment as bed lor,d.

The Gilbert data used here were taken from the summary by Johnson (1943, pp. 59-63), in which the data 

are presented in centimeter-gram-second units1 .

For "sand B" of Gilbert, four widths of flume were 

used at a discharge of 15,433 cu cm per sec and five 

widths at a discharge of 5,154 cu cm per sec.

Figure 20 presents the relation of the bed load 

carried in the full width of the flume to width and 

observed mean velocity at constant discharge. Thus 

the graphs of figure 20 are for bed load and are analogous 

to figure 15 for measured suspended load in natural 

rivers. Isopleths of velocity are drawn through the 

observed velocity values appearing in the graph.

It is clear that the isopleths of velocity slope upward 

to the right in contrast to those in figure 15 which 

slope upward to the left. These graphs provide, then, 

the following tentative interpretation, which, if sustained by further work, may provide important assistance in the understanding of fluvial morphology.
HYDRAULIC FACTORS AND BED LOAD AT A GIVEN DISCHARGE 29

At constant discharge, an increased velocity at constant 

width is associated with an increase of both suspended load 

and bed load in transport. At constant velocity and discharge, an increase in width is associated with a decrease 

oj suspended load and an increase in bed load in transport.

It is unfortunate that the lack of data on bed load for 

rivers dictates reliance on flume observations. However, some general arguments and observation support 

the conclusions reached by analysis of flume data. 

Lane (1937 p. 138) observes,

If a channel is supplied with a hea vy bed load, in order to be stable 

it must move this load along. * * * To be stable, the channel 

carrying bed loads, therefore, should have a higher velocity along 

the bed, but the same velocity along the banks, and this could 

only occur with a wider, shallower section.

This statement by Lane is significant because it 

applies to the interpretation of river channels as they 

are observed to exist. That is, an alluvial river channel 

which is relatively wide and shallow probably carries a 

relatively large bed load.

Griffith (1927, p. 246) notes that the"* * * large 

rivers flowing through the alluvial plains of India, which 

in flood are heavily charged \vith silt, have in all cases a 

tendency to adopt a broad, shallow section * * *." 

Mackin (1948, p. 484) concludes from the British and 

Indian literature that "* * * the broad, shallow 

channel is the type of cross section best adapted for the 

transportation of heavy bed load."

Though these general observations are of interest in 

this paper, they imply rather than assert that a constant

c 100

10 100

Total load,in grams per second

500

FIGURE 20.—Relation of bed load to channel-shape factors at constant discharge, 

showing two discharges, same sediment material (sand B), from Gilbert data.

discharge is assumed. A constant discharge and constant velocity are specified in the present discussion of 

the effect of width on bed load and suspended load.

Further indication of the effect of width on bed load 

is provided by computations made using the Einstein 

(1950) equation. In a study for the Bureau of Reclamation, Lara and Miller (1951) computed by the Einstein equation the bed load which would be transported 

in a channel of variable width, under different conditions of discharge. The computations involved in tie 

equation require that a value be assumed for river slope 

and bed-material size. Computations were made ty 

Lara and Miller for different sized fractions of the load 

separately, and combined to provide the computed bed 

load having a size distribution characteristic of Rio 

Grande sediments. Lara and Miller assumed slope 

comparable to that of the Rio Grande.

The computed values of bed load for a constant dr>- 

charge of 500 cfs were plotted by the writers against 

width and mean velocity provided by the computation, 

in a manner similar to figures 15 and 20. In the graph, 

not included here, isopleths of velocity sloped upward 

to the ri^ht as in figure 20. Thus, bed load computed 

by the Einstein equation appears to increase with 

increased width at constant discharge and velocity—at 

least for the assumptions of slope and sediment size 

made by Lara and Miller.

In the preceding sections some relations have been 

developed among suspended load, bed load, and channel characteristics. By far the majority of data available on sediment load are restricted to measurements 

of suspended load only, and further detailed analysis 

of necessity depends on the measured fraction. Mackin 

(1948, p. 469) emphasizes that the size-distribution of 

the load of a given segment of a stream depends en 

"* * * the lithologic characteristics, relief, ard 

erosional processes in its drainage basin, and on processes in operation within the stream itself (as sorting.)" 

The sizes and quantities of available load determine, in 

part, whether the stream carries the load as suspended 

or as bed load.

A large part of the total sediment transported ty 

natural rivers is carried as suspended load. This is 

recognized by some geologists (Mackin, 1948, p. 468), 

but because bed load measurements are few, the relative proportions of bed and suspended load in natural 

rivers are not generally known:

Both suspended and bed load are being measured in 

the Niobrara River near Cody, Nebr. (Serr, 1950). 

The data already published show that for discharges 

between 200 and 1,000 cfs bed load constitutes an 

average of about 50 percent of the total load. Tl e 

percentage of the total carried as bed load varied from 

24 to 68 percent. No significant correlation existed 

between this percentage and either the water discharge 

or the rate of sediment transport (Serr, 1950, p. 15).
30 THE HYDKAULIC GEOMETKY OF STREAM CHANNELS

This comparison of bed load and suspended load in 

the measurements of the Niobrara River led the Bureau 

of Reclamation and the Geological Survey to begin a 

joint investigation for the purpose of extending this 

experimental work (Serr, 1951). The Middle Loup 

River has a sandy bed and the water appears clear and 

free of suspended load. The engineers of these agencies 

selected the Middle Loup River near Dunning as an 

example of a stream which carried most of its load as 

bed load. The junior author, at that time head of the 

Sedimentation Section, Bureau of Reclamation, represented that Bureau in the studies which led to the 

establishment of the measurement program on the 

Middle Loup River.

The measurements showed, however, that the bed 

load constituted an average of only 55 per cent of the 

total load even on the Middle Loup River (Vice and 

Serr, 1950, fig. 14).

Before regulation of the Colorado River by Boulder 

Dam, measurements were made in the overflow nappe 

on the spillway of Laguna Dam where presumably the 

total sediment would be caught by a suspended-load 

sampler. These measurements, when compared with 

suspended-load data at a normal section, indicated that 

before regulation and, at least at low flow, the Colorado 

River in the lower reaches transported almost all of its 

sediment load in suspension (Lane 1937, p. 190).

The 1948 survey of the sediment deposition in Lake 

Mead showed that 49 per cent by weight (978 million 

tons) of the total sediment deposited in the reservoir 

consisted of material fine enough to be transported in 

density currents (Gould, 1951, p. 48).

It might be argued, however, that despite a large suspended load in a river only the bed load is of real significance in fluvial morphology assuming it places the greatest tax on the energy of the stream.

It is possible that owing to the influence of the engineer, one may tend to place too much emphasis on bed 

load and to underrate the role of suspended load in 

determining the observed characteristics of natural 

streams. In the present discussion, an attempt is made 

to demonstrate that data on suspended load may provide an insight into the mechanisms by which river 

characteristics are determined.

The following detailed consideration of the effect of 

suspended load on river channel characteristics does not 

deny the importance of bed load. But if the measurements of suspended load allow the development of a 

meaningful picture of river channel characteristics, the 

importance of the suspended load in fluvial morphology 

can hardly be denied.

CHANNEL-SHAPE ADJUSTMENT DURING INDIVIDUAL 

FLOODS

It is known that during the passage of a flood, the 

channel of an alluvial stream is scoured and filled with 

considerable rapidity. Though such changes may be

fortuitous, the foregoing discussion implies that there 

might be a definite pattern of channel changes directly 

related to the suspended-sediment load provided to the 

river by the drainage basin. If this were true, these 

changes should be of the type required by the principles 

derived from a study of figures 15 and 16.

The gaging station on the San Juan River at Bluff, 

Utah, might first be considered. This river generally 

has a bed consisting of sand, silt, and some gravel, which 

characteristically shifts during a flood. At high stages 

the stream at the Bluff station is confined by ro°,k walls, 

but some changes in width are possible at lovr stages. 

The control at the station is a reach of gravel and 

boulders downstream from the gaging section, and this 

control is subject to shift under flood conditions. Current-meter measurements are made from a cable car.

Daily current-meter data and daily suspended load 

samples are available for a few years. These daily 

data are used in figure 21 to show the changes that 

occurred during a river rise between Septemb°r 9 and 

December 9, 1941. A measurement on a particular 

day is indicated in figure 21 by a point in each of six 

graphs. For example, during the first day of the rising 

stage the discharge was 635 cfs, width 171 ft, depth 

1.20 ft, velocity 3.08 fps, and suspended lor,d 2,140 

tons per day. To simplify the picture, an intermediate 

rise and fall in discharge between September 16 and 29 

are omitted.

The rising discharge was accompanied by a slight 

rise in width up to the peak discharge of 60,000 cfs, and 

the falling discharge retraces nearly the same path in 

the curve showing relation of width to discharge.

Suspended load, in contrast, increased at a very 

rapid rate as the discharge rose from 600 to 5,000 cfs, 

but then load increased at a less rapid rate while the 

discharge continued up to 60,000 cfs.

Depth rose uniformly with discharge until a discharge 

of about 5,000 cfs was reached. Then the slope of the 

depth-discharge curve increased as the discharge rose 

from 5,000 to 60,000 cfs. In the falling stage, a new 

slope of the depth-discharge curve was established, 

which was intermediate between the two slopes in the 

rising stage. At a discharge of 5,000 cfs, the depth was 

about 3.2 ft in the rising stage and 4.4 in the falling 

stage.

A corresponding adjustment in the velocity-discharge 

curve is apparent. In the rising stage at 5,000 cfs the 

velocity was 8.6 fps; at the same discharge in the falling 

stage it was about 6.0 fps.

The graph of stream-bed elevation in relation to 

discharge is shown in figure 21. Bed elevation, as 

plotted, is a relative value which was derived by subtracting the depth from the gage height (water-surface 

elevation). This difference is a satisfactory estimate 

of bed elevation in the examples discussed here because 

width varied but little with discharge. Detailed river 

cross sections at various intervals during the flood
CHANNEL-SHAPE ADJUSTMENT DURING INDIVIDUAL FLOODS 31

•5 30° 

>£!

c 200

<K 5

- Width •n-—i EXPLANATION

DISCHARGE 

Rising Falling 

Width + « 

Depth v f 

Velocity x « 

Load ° • 

Stage * «> 

Bed a •

500 1,000 5,000 10,000

Discharge, in cfs

50,000 100,000 500 1,000 5,000 10,000

Discharge, in cfs

50,000 100,000 cr

FIGDBE 21.—Changes in width, depth, velocity, water-surface elevation, and stream-bed elevation with discharge during flood of September-December 1941, San Ju<m

River near Bluff, Utah.
32 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

Distance, in feet

100 200 300

FIGURE 22—Channel cross sections during progress of flood, SeptemberDecember 1941, San Juan River near Bluff, Utah.

have been plotted from original current-meter measurements.

The bed elevation rose during the first part of the 

flood rise. Deposition of material on the stream bed 

at the measuring section during the early period of the 

flood can be seen by comparing the cross sections in 

figure 22 for September 9 (635 cfs) and September 15 

(6,560 cfs). This deposition continued progressively as 

long as the slope of the suspended sediment-discharge 

curve was steep. At a discharge of about 5,000 cfs 

the suspended sediment-discharge curve assumed a 

different, less steep slope, and this smaller slope of the 

line was maintained as the discharge continued to rise 

to the peak flow of almost 60,000 cfs.

At discharges between roughly 5,000 and 60,000 cfs 

the suspended-sediment concentration increased less 

rapidly per increment of discharge than in the range 

below 5,000 cfs. The smaller rate of increase of suspended-sediment concentration was associated with 

bed scour. After the peak flow, the falling discharge 

was accompanied by a stable bed elevation.

This flood is one in which the slope of the widthdischarge curve was about the same throughout the 

flood passage, and, therefore, the mutual adjustment 

among suspended load, velocity, and depth is quite 

evident. At 5,000 cfs the suspended load was about 

1,000,000 tons per day in the rising stage. In the 

falling stage at the same discharge and the same width

500

200

100

5,000,000

1,000,000

500,000

100,000

50,000

20.000

-5

-10

5,000 10,000

Rising 

Falling

50,000 100,000 200,000

Discharge, in cfs

EXPLANATION OF DISCHARGE 

Width Depth Velocity Load Stage Bed

FIGURE 23.—Changes in width, depth, velocity, water-surface elevation, and 

stream-bed elevation with discharge during flood of December 1940-June 1941, 

Colorado River at Grand Canyon, Ariz.
CHANNEL-SHAPE ADJUSTMENT DURING INDIVIDUAL FLOODS 33

the suspended load was only 100,000 tons per day. 

The lower suspended load was accompanied by a lower 

velocity and a correspondingly higher depth. This is 

indeed in accord with the relation already noted in 

figure 15; that is, at constant discharge and width, 

decrease in suspended-sediment load is accompanied 

by a decrease in velocity and an increase in depth.

The observed relations during this flood are in accordance with the ideas explained in association with 

figure 18. For the same value of b (slope of the widthdischarge line), a decrease in the value of j (slope of 

the suspended load-discharge line) is associated with a 

decrease in the m to / ratio; that is, the slope of the 

velocity-discharge line (m) decreased while the slope 

of the depth-discharge line (/) increased.

A second example of the day-to-day adjustments of 

channel shape and suspended load is the spring rise of 

1941 on the Colorado River at Grand Canyon (fig. 23).

In this flood passage, at a discharge of 20,000 cfs 

there is a break in the slope of the width-discharge 

curve, as well as in the load, velocity, and depth 

curves. At 20,000 cfs, in the rising stage, the load was 

about 1,000,000 tons per day, whereas in the falling 

stage the load was about 120,000 tons. This lowered 

load was accompanied by a definite increase in depth 

and lowered velocity while the width remained the 

same.

The bed elevation rose during the first part of the 

flood rise, then lowered as the discharge continued to 

increase toward the peak flow, it continued to lower 

during the falling stage. At 20,000 cfs, the bed elevation was 8.5 ft lower on the falling than on the rising 

stage. The gage height, plotted against discharge, 

showed that the water-surface elevation was the 

same in both rising and falling stages at each discharge. 

At the measuring section, therefore, the difference in 

channel shape was taken care of by changes in bed, 

but the control downstream, a bar of heavy gravel, 

did not shift.

The reality of deposition on the bed during the first 

part of the flood rise can be observed in the successive 

cross sections in figure 24. The river bed was lower 

on January 12, (5,210 cfs) than on March 6 (23,400 cfs). 

These cross sections demonstrate that the bed was 

filled during the first part of the flood rise and then 

was scoured as the discharge rose above 20,000 cfs.

A further example which illustrates another type of 

adjustment to changing suspended load is provided by 

the spring rise of 1948 on the Rio Grande at Bernalillo, 

presented in figure 25. There was some decrease in 

suspended load at a discharge of 11,000 cfs as the peak 

flow passed the station. The curve of bed elevation 

shows that the river scoured its bed during the rising 

stage when the sediment concentration was high and 

filled the bed during the falling stage when the concentrations were lower. The San Juan at Bluff and 

the Colorado at Grand Canyon deposited during a 

part of the rising stage, then scoured during the remainder of the rise and continued to scour during the 

falling stage.

Figure 26 presents three cross sections of the Rio 

Grande channel at the gaging station during the 

passage of the 1948 spring freshet. The scour of the 

bed was irregularly distributed across the section but 

on the whole, scour progressively occurred up to the 

peak discharge and deposition occurred during the 

falling stage.

The foregoing analyses of scour and fill of a river bed 

during flood demonstrate that the changes in the bei 

occurred simultaneously with changes in the rate of 

change of suspended-sediment concentration.

It is the thesis of this paper that the observed changes 

in the stream bed resulted from changes in the sedimer t 

load brought into the measuring reach from upstream. 

It is postulated that the hydrodynamic factors involved 

tend to promote a mutual adjustment between channel 

shape and the sediment load carried into the read . 

The change in sediment load which results in a change 

of channel shape involves both bed load and suspendei 

load. However, because only the suspended load is 

measured, it is necessary to use the data on the surpended fraction of the load as an index to how the totr.l 

sediment load interacts with the hydraulic variables. 

That the suspended fraction is a meaningful index is 

demonstrated by the fact that the relations between 

the channel-shape factors and suspended load, which are 

derived from measurements of a number of different 

rivers, appear to apply in principle to channel changes 

at a given station during an individual flood. Specifically, with no change of channel width, a decrease of 

suspended load at a given discharge was accompaniei 

by an increase in depth by bed scour that resulted in a 

decrease in velocity. This is the type of change indicated by figure 15. The decrease in velocity provides 

the adjustment of capacity for carrying the load of the 

particular size-distribution supplied by the watershed. 

In response to a decrease in load, the channel shape 

became adjusted through scour to the lower capacity 

required for quasi equilibrium.

In the hypothesis stated it was postulated that the 

changes in channel shape occurred in response to a 

change in sediment load brought into the reach. Thi« 

postulate requires proof that the observed change in 

suspended-sediment load was not the result of the 

observed change of velocity in the reach rather than 

the cause. This is just the question faced by Mackin 

(1948, p. 471) when he observed that

As set up in an equation * * * load is a function of 

velocity. In answer to a query as to which is the cause, and 

which is the effect, the average engineer will assert that velocity
34 THE HYDKATJLIC GEOMETRY OF STREAM CHANNELS

controls or determines the load carried by a stream. * * * 

[The equation as a] method of expression is inadequate for 

present purposes because the terms of an equation are transposable.

If the high suspended-sediment concentrations resulted from the scouring action of high velocities, it is 

implied that high velocity in a given reach scours the 

channel in that reach. The increase in sediment in 

transport resulting from the local bed scour should then 

account for the observed increase in sediment concentration. Under such an assumption, increasing velocity should be associated with bed scour and decreasing 

velocity with bed fill.

During the rising flood stage in both the San Juan 

River at Bluff (fig. 21) and in the Colorado River at 

Grand Canyon (fig. 23) the beds were filling during 

a period of rapidly increasing velocity concomitant 

with a rapidly increasing suspended-sediment load. 

The material deposited on the bed obviorisly came 

from upstream. How far upstream is unknown. It

100

Distance, in feet 

200 300 400

May 19, 1941 - 102,000 cfs

June 16,1941-57,400 cfs

- March 6, 1941 - 23,400 cfs

Jan. 12, 194!-5,210 cfs

£/

' .June 16

FIQUSE 24.—Channel cross sections during progress of flood, December 1940-June 1941, Colorado River at Grand Canyon, Ariz.
EFFECTS OF ROUGHNESS AND SLOPE IN ADJUSTMENT OF CHANNEL

o o

s?

.£ 8

* 85 

S fc i a

and average 

fee bed, in ream

Wa lev

100,000

10,000

.2 500 1,000

Rising 

Falling

5,000 10,000 50,000 

Discharge, in cfs

EXPLANATION OF DISCHARGE 

Width Depth Velocity Load Stage Bed

FIGURE 25.—Changes in width, depth, velocity, water-surface elevation, and 

stream-bed elevation with discharge during flood of May-June 1948, Rio 

Qrande near Bernalillo, N. Mex.

is presumed that the bulk of the sediment carried past 

Bluff and Grand Canyon during spring flood was derived 

originally from erosion in ephemeral gullies, rills, and 

by sheet wash rather than from the channels of the 

larger perennial tributaries and trunk streams.

Concerning the seasonal sequences of scour and fill, 

the three rivers discussed in the preceding examples

Distance, in feet 

100 200 300

10

-May 28,1948-12,400 cfs-

- — ••22, » - 4,920 - 

15, » - 1,540

FIGURE 26.—Channel cross sections during progress of flood, May-June 1948, 

Rio Grande near Bernalillo, N. Mex.

are typical of a number of streams in western United 

States. The spring floods are derived primarily froir 

snow melt in the mountains. Much sediment must be 

carried into the major stream channels by the melt 

water, but this load is augmented by the transport of 

readily available material deposited in the channe1 

system during the local summer floods.

The spring flood leaves the river bed of the trunk 

river at a lower elevation than had obtained before 

the flood, but the succession of summer floods tends tc 

aggrade the bed to the level observed preceding the 

spring snow melt. The succession of events appear? 

from available data to be approximately the same each 

year.

It can be said in summary that the scour and fill 

of the bed of the main stem of an alluvial river during 

flood appears to be an adjustment of channel shape 

in response to a varying sediment load. The adjustment takes place rapidly. This should not be interpreted as implying a complete absence of a time lag. 

In the long-continued flood passages typical of spring: 

snow-melt contribution to Western streams, the 

progressive adjustment of channel to load is easily 

observed from the data, as has been shown. However, 

similar graphs for small irregular changes in discharge 

even for the same stations show that the adjustment 

is not so nearly complete as this. A time lag measured 

in days and perhaps weeks is required for adjustment? 

to take place, as might be expected.

THE BOLE OF CHANNEL ROUGHNESS AND SLOPF 

IN THE ADJUSTMENT OF CHANNEL SHAPE TC 

SEDIMENT LOAD

It has been shown earlier that, at constant discharge, 

suspended-sediment load is related to the shape factors—width, velocity, and depth. It was noted that 

at constant width and discharge, increased suspendedsediment load would be associated with increased 

velocity. But because Q and w are constant, ther 

the product vXd must be constant. Any increase w 

velocity, therefore, must require a decrease of depth.

As the Manning formula states:

(4)

in which »=velocity

</—clepth = hydraulic radius 

s=slope of the energy grade line 

11=roughness parameter 

It is clear that an increase in velocity and a decrease

8*.

of depth requires an increase in the factor —'•> or, in other

terms, increased velocity and decreased depth would 

require an increase in channel slope or a decrease of 

roughness or both.
36 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

Under the condition of a varying discharge a similar 

result is obtained. The relations of depth and velocity 

to discharge have been expressed by the equations

fj — rQf (<?\ U, — ctjj \4J 

•3) —— Tf f)m ("^\ v — K- ^J ^O )

«!

71

(4)

The coefficients c and k are numerical constants for a

given set of conditions.

Thus

(n)

Under the condition that s and n do not change with 

discharge, then

(5)

or

where

and when

. .,, ,. , . — increases with discharge, ~T.>

— decreases with discharge, -F^TT

The average values of m and / for rivers studied were 

shown to be

At a station 

m=.34 

/=.40 

m

Downstream 

m=.l 

j— .4

y=.25

It appears, then, that for the rivers studied, the

s* 

factor — increases with discharge at a station and

Ti/

decreases downstream. In the following discussion 

the separate effects of slope and roughness will be 

considered. It will be shown that slope of the water 

surface tends to remain about constant at a station 

while the increase in the concentration of suspended 

load with discharge is associated with a decreasing

roughness. Thus, the at-a-station value of — tends

n

to increase with discharge, a fact which explains why

the mean at-a-station value of — exceeds —•

/ 3

It will further be shown that in the downstream 

direction the value of roughness, n, tends to be conservative, or remain about constant, while slope decreases. This leads to an explanation of the observed

, , m .,2 value of -r<-^- 

7 3

Because the present paper deals with the suspendedsediment load of streams, the general effect of sediment 

load on channel roughness will be considered. Vanoni 

(1941, p. 618-619; 1946, pp. 96 and 127) si owed that 

an increase in suspended load tends to decrease channel 

resistance and thus causes an increase in velocity. He 

explains this effect as a result of decreased turbulence. 

Buckley (1922) found similar results in measurements 

of the Nile River.

The effect of suspended-sediment concentration on 

channel roughness was discussed by Thomas (1946, 

pp. 43-45) who also concluded that increased concentration was associated with decreased values of Manning 

roughness factor n.

It should be expected, then, that when a given river 

cross section is considered, the large changes in suspended-sediment concentration which o°,cur with 

changes in discharge should be reflected in important 

changes in channel roughness. In a given cross section 

(at a station), an adjustment of water-surface slope 

does not appear adequate to account for tte observed 

variations in the velocity-depth relations; so, important 

changes in roughness must occur.

During the passage of a flood through a given river 

cross section, the slope of the water surface does change. 

In any given reach slope tends to be steeper during a 

flood rise than a recession. The steeper slope on the flood 

rise is due to the fact that the rise at a given cross 

section precedes the rise at any section downstream. It 

will be noted, however, that the normal shape of a flood 

hydrograph tends to make the most marked change 

of slope coincide closely with the passage of the flood 

crest in a given reach.

In the Colorado and San Juan River examples 

however, it is observed that there was a change in 

velocity-depth relations which coincided with a change 

in suspended load-discharge relations at a discharge 

well below the peak. There is no a priori reason that 

any change of the slope-discharge relation should have 

occurred at that discharge. These facts irdicate that 

such at-a-station adjustments in velocity-depth relations result primarily from changes in roughness 

associated with changes in sediment load, rather than 

from changes in slope of the water surface. Moreover, 

during the passage of a flood lasting several months, 

conditions approach steady flow and therefore changes 

in water-surface slope as a result of the passage of a 

flood wave are minimized.

The velocity-depth changes and their relation to 

suspended load observed at a single cross section in the 

Colorado River, Rio Grande, and San Juan River are 

typical of all the cross sections studied for which 

adequate data are available. This appears to support 

the idea that such changes are typical of fairly long
EFFECTS OF ROUGHNESS AND SLOPE IN ADJUSTMENT OF CHANNEL 37

reaches of channel, measured in tens of miles rather 

than in tens of yards.

Changes in velocity-depth relations typified by the 

individual floods measured at Grand Canyon, Bluff, and 

Bernalillo must primarily represent, therefore, changes 

in roughness rather than in slope. If observed changes 

in velocity-depth relations can be attributed to changes 

in roughness, then lower values of roughness are 

associated with larger suspended-sediment concentrations, and this is just the effect found by Vanoni, 

Buckley, and Thomas.

The changes which have occurred in the lower reaches 

of the Colorado Eiver after the construction of Hoover 

Dam, appear to confirm these observations, as will now 

be shown. When the sediment load of the Colorado 

River was stored in Lake Mead, clear water released 

from the reservoir caused degradation in the channel 

reach below Hoover Dam. The main degradation 

occurred in the reach of about 100 miles between 

Hoover and Parker Dams. The gaging station at 

Yuma, Ariz., is about 350 miles below Hoover Dam. 

A record of observations before and after the construction of Hoover Dam allows a comparison to be made of 

the channel characteristics at Yuma between a period 

when the river carried a large suspended load and a 

period when the load was markedly reduced. The 

current-meter measurements for that gaging station 

used in the present analysis are available in the offices 

of the Geological Survey.

In addition to the data on width, depth, velocity, 

and discharge included hi the current-meter data, summary data on river slopes and on suspended-sediment 

load at Yuma are contained in a detailed report by the 

Bureau of Reclamation (1949).l These data provide 

the basis for the present analysis.

The Yuma gaging records are obtained from a 

cablecar. The river at this point flows hi a flat alluvial 

valley bounded by low scarps probably representing 

pediment remnants. The channel conditions before 

the changes brought about by the construction of 

Hoover Dam were described by the Geological Survey 

as follows: 2

Channel at all stages composed of sand and silt. Brush and 

trees growing on both banks. * * * Channel is straight for 

about 1,400 ft above and a mile below the cable.

In figure 27 the width, depth, velocity, gage height, 

and stream-bed elevation are plotted against discharge 

for the Colorado River at Yuma. The figure includes 

graphs of these relations for the period 1927-34 before 

the construction of Hoover Dam and similar graphs 

for 1945-46 to represent measurements after the construction of the dam. It will be noted that the relations 

of these factors to discharge are similar to at-a-station

i The river-slope data used in the following paragraphs appeal in figures 5 and 9 in 

the Bureau of Reclamation report, and the suspended-load data are tabulated in 

table 8, p. 33.

3 Contained in description of the Yuma gaging station, dated June 3, 1938, in the 

files of the Geological Survey, Washington, D. C.

1,000

S 500

200

100 „___X-^__ ^' -v^'

• •» —

1,000

50C

20

10

500 1,000 10,000 

Discharge, in cfs

100,000

FIGURE 27.—Relation of width, depth, velocity, water-surface elevation, and 

stream-bed elevation to discharge, Colorado River at Yuma, Ariz. Sample 

periods before and after construction of Hoover Dam are shown.

curves presented for other river cross sections. The 

relations at Yuma changed, however, after construction 

of Hoover Dam.

Figure 28 presents the progressive change of the same, 

factors with time. The width, depth, velocity, gage 

height, bed elevation, and roughness representing the 

average values for each of five discharges are plotted 

against time, in years.

Figure 28 includes the annual suspended-sediment 

load and annual discharge at Yuma gaging station 

plotted against time. Also included are graphs of 

the median size of bed material and river slope. The
< Average

o for &

I (928-34

M 1936-38 

I? 1938-39

CD >~s

S- 1940-41 

^ 1941-42

II

3 I 1943-44

!>

| g- 1945-46

® i 1947-48

3 E

g- B- 1949-50

O Oq

| Average

ET for

I 1928-34

1936-38 

1938-39

1940-41

1941-42

1943-44 

1945-46 

1947-48 

1949-50

Velocity, in feet per 

second

Depth, in feet Width, in feet

Dischorge, in millions 

of acre-feet per year

Bed-material size, in 

millimeters

Slope, in feet per mile 

o — ro

Roughness n 

b o

Stream-bed elevation, in feet 
EFFECTS OF ROUGHNESS AND SLOPE IN ADJUSTMENT OF CHANNEL w

bed-material size was measured at Bureau of Reclamation section 7-S (1949, fig. 15), situated about 3 miles 

above Yuma. River slope was obtained as follows:

Successive measurements of particular cross sections 

were made by the Bureau of Reclamation to observe 

the degradation of the channel following building of 

the dam. These measurements permitted the construction of successive profiles of different reaches of 

the river. The profiles of the reach from Laguna Dam 

to the Mexican boundary, published by the Bureau of 

Reclamation (1949, fig. 5), are presented as figure 29 

of the present report. The average slope in the 7-mile 

reach (Reclamation sections 7-S to 10-S) which includes Yuma gaging station, is plotted against time in 

figure 28.

The conclusions to be drawn from figure 28 will now 

be presented. Following the construction of Hoover Dam 

the annual suspended load passing Yuma decreased 

from an average of nearly 150 million tons to less than 

25 million tons. The water discharge fluctuated from 

one period to another, depending on the releases from 

the dam and the contribution of tributaries entering 

the Colorado below the dam. In the period analyzed 

here there was, however, no large, consistent and sustained decrease in water passing Yuma after the dam 

was built.

This great decrease in suspended load supplied to 

the reach near Yuma caused the channel at Yuma to 

change markedly. Simultaneous with the reduction 

in suspended load, the channel at Yuma was degraded. 

The gage height, or water-surface elevation, decreased 

5 to 6 ft but the elevation of the river bed decreased 

about 9 ft. The difference represents the increase 

in mean depth, this increase was particularly great at 

high discharges. Despite a decrease in width from 510 

to 430 ft accompanying the degradation, the mean 

velocity decreased considerably. This decrease of 

velocity resulted from a considerable increase in roughness. The changes in slope were small and there was 

no progressive change during the period of record. 

The consistency in slope in the reach near Yuma can 

be seen in the river profiles of figure 29.

The strong increase in bed roughness was not caused 

by a corresponding, consistent change in the size of 

bed material, at least the correlation of the roughness 

and the median size of bed material is poor. It appears 

more likely that the increase in roughness was an 

effect of the decreased concentration of suspended 

sediment.

The explanation for the decrease in channel width 

at Yuma probably lies hi the cessation of large floods 

after construction of Hoover Dam. The channel was 

degraded. The large floods formerly characteristic of 

the reach had resulted in the development of a wide 

original channel to provide a balance among the requirements of width, depth, and velocity for transportation of the load which was supplied to the stream,

20 30

Miles below Laguna Dam

40

FIGURE 29.—River profiles in reach of Colorado River near Yuma, Ariz., comparing conditions in 1902 before the construction of engineering works upstream 

with conditions in 1940 and 1942 (after Bureau of Reclamation).

and the relative erodibility of bed and banks. Th°s 

new discharge regime lacked the high floods to provide 

such widening, and new width-depth-velocity requirements for transportation are balanced by the relative 

erodibility of bed and bank now existing.

The reduction in width implies a larger capacity for 

suspended load at a given velocity and discharge, and 

a smaller capacity for bed load. However, because 

velocity has decreased, the new channel can be considered to have a lowered capacity for both types cf 

load at a given discharge.

In summary, in this example for which considerable 

data are available, a decrease in sediment load—which 

originally, at least, was carried primarily in suspension— 

was accompanied by an important increase in bed 

roughness. Slope remained essentially constant. Undoubtedly, a different proportion of the load is carried 

as bed load in the present than in the original channe1 , 

and may be in part responsible for the observed increase 

of roughness.

It is possible that changes in slope which have 

occurred in upstream reaches near the dams may in 

time affect the Yuma reach. In the years since th°- 

construction upstream began, however, a quasi equilibrium has been established at Yuma by change in 

channel characteristics and not by changes in slope.

The example demonstrates that even during a'period 

of years adjustment of a channel reach in response to a 

change in suspended load may be made primarily by 

changes in channel shape through changes in roughness 

rather than in channel slope.

The longitudinal profile of most rivers is concave to 

the sky, or the slope tends to decrease downstream. 

The change in velocity-depth relations in the dowrstream direction, as discussed earlier, must be related, 

in part, to this progressive change of slope. The rota
40 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

of roughness, as compared with slope, in the maintenance of the observed -j ratio for the downstream

direction can now be analyzed.

At least three factors affect bed roughness: the 

particle size, the bed configuration, and the sediment 

load. The effect of sediment load on roughness is in 

itself partly determined by particle size and is intricately related to bed movement, as well as to the suspended fraction. The Einstein equation (1950) provides 

a method of determining bed roughness when certain 

other factors are given, particularly the slope and the 

size distribution of bed material. The Einstein equation, however, allows width of the channel to be 

assumed, and therefore does not specify what widthdepth ratio will be developed under natural conditions.

The present analysis is concerned with relations 

among width, depth, and velocity as they are observed 

to exist in natural rivers and is, furthermore, restricted 

in dealing with sediment load to its easily measured 

fraction, or suspended load.

The present discussion of the effect of the three 

factors of size, bed configuration, and suspended load 

on roughness aims at a general qualitative understanding of how the values of roughness observed in 

rivers appear to be related to the average characteristics 

of rivers.

A characteristic of most natural rivers is a decrease 

in particle size of the bed material in the downstream 

direction, due primarily to abrasion. The authors are 

in agreement with Mackin's (1948, p. 482) rejection of 

the oversimplified explanation that the downstream 

decrease in river slope is caused directly by the decreased caliber of load downstream.

A change in caliber of load must have an effect on bed 

roughness and, through roughness, may affect ozther 

stream characteristics. This effect will now be examined.

In the present paper a distinction is made between 

roughness resulting from ripples or dimes on the channel 

bed and roughness due to the individual particles of bed 

material. A nearly identical distinction was made by 

Einstein and Barbarossa (1951). It is assumed in the 

present discussion that the roughness due to channel 

irregularities includes not merely semipermanent obstructions such as rocks and bars but transitory 

conditions of bed ripples and waves which change with 

discharge.

The Manning roughness factor in pipes represents a 

more or less permanent characteristic of the boundary 

surface. This is also true for certain open channels, 

drainage canals, and even irrigation canals where the 

banks and bed are fixed. However, in most sedimentladen channels the roughness is not a permanent characteristic but must change with the configuration of the 

bed. It is logical to suppose that when the bed of an 

alluvial river is smooth, the roughness as described by

the Manning n is different from and probably lower than 

that which applies to the same reach at a different 

discharge when the bed is molded into dunes or antidunes. This idea was clearly stated by Rubey (1952, 

p. 131).

In figure 30 the relation of bed roughness measured by 

Manning's n is plotted against discharge for four sizes 

of sediment in transport. The data were collected by 

the U. S. Waterways Experiment Station and were summarized by Johnson (1943, pp. 80-84). From Johnson's 

summary, data for figure 30 were taken. The experiment concerned the transport of natural river sand by 

water in a flume.

It will be noted in figure 30 that a constant slope was 

maintained. When sand 1, the coarsest of the four 

sizes discussed here was used, an increase of discharge 

was accompanied by a slightly increasing roughness. 

At low discharge, the roughness associated with sand 3 

was only slightly higher than that for sand 1, but when 

the configuration of the bed changed from a "smooth" 

condition to one of "general riffles" (term^ used by 

Waterways Experiment Station), the roughness increased 

markedly. With further increase of discharge, roughness decreased gradually.

The importance of these relations may be summarized 

as follows. An increase in bed roughness is one type of 

change that will result in the decrease of velocity with 

respect to depth. Because changes in the velocitydepth relation constitute a part of the adjustment of 

channel shape to load, variations in roughness affect 

capacity for load.

Decreasing sediment size results in a decrease of 

roughness due to the individual particle. But in hydrodynamically rough channels, the roughness due to bed 

configuration may be much more important than that 

due to particle size, and sometimes decreased particle 

size may result in larger or more effective bed ripples or 

waves and thus in increased roughness.

It will be noted in figure 30 that the values of roughness representing "general waves" are markedly different among sizes of sand. Larger values of roughness are 

associated with smaller sand sizes. Thi« implies, 

therefore, that among the four experimental runs using 

different sands, even when the configuration of the bed 

looks similar, for a given discharge and slopi the sediment particles of smaller size may arrange themselves 

in such a manner that effective roughness is greater 

than for larger particles.

At a station with a specific bed confignration, an 

increase in discharge may be accompanied by either a 

slight decrease in roughness, as in sands 3, 6, and 7, or 

a slight increase. However, with increasing' discharge 

the bed may change successively through conditions 

described as smooth, riffles, bars, dunes, and finally 

antidunes. As indicated by Langbein (1942), antidunes 

form when the Froude number approaches a value of 

1.0, critical flow. Also, the finer the material, the lower
EFFECTS OF ROUGHNESS AND SLOPE IN ADJUSTMENT OF CHANNEL

.030

c

c 

o

en </> 

<D

c

Z3 

O

cr

.020

.010

Smaller sediment size

SAND 7

T ? I' T SAND 6

Larger sediment size 

.VM^* SAND

20 40 60 

0 {discharge),in thousands of cubic centimeters per second

EXPLANATION 

SLOPE .001

SANDS 

1367

0 I V D 

•9 X $

SMOOTH --------———

LOCAL RIFFLES OR WAVES —

GENERAL Rl FFLES OR WAVES - - — • + ? •

MEAN DIAMETER (MILLIMETERS)--.586 .525 .347 .310

FIGURE 30.—Channel roughness (Manning ri) in relation to discharge for various sizes of sediment (from flume data).

will be the value of the Froude number at which the 

transition occurs from a smooth bed to dunes and from 

dunes to antidunes.

At a given discharge, the change in bed configuration 

may increase the value of n more than does a considerable increase in discharge during which the bed configuration remains the same. *

Two general conclusions may be derived from figure 

30. First, for the same slope and discharge, the effect 

of decreasing particle size is a tendency to increase the 

roughness even when the condition of the bed is observed 

to remain the same. Second, with no change in particle

size, a change in bed configuration may violently change 

bed roughness and its effect may be of the same o^ 

greater order than the effect of differences in particle 

size.

Headwater tributaries flowing on gravel or boulders 

obviously move the large bed debris only in exceptional 

flows. The channel roughness in such streams must b^ 

determined by the sizes of the individual rocks or cobbles, because the bed material is not ordinarily in 

movement and is not thrown up into transitory ripples 

and waves in the sense used in describing the bed of an
42 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

alluvial stream. The downstream decrease in the size 

of the gravel or pebble bed therefore results generally in a 

downstream decrease in the value of channel roughness. 

This can be seen in the data tracing the change from a

headwater tributary down to the Powder River, 

Wyoming, listed below. The data apply to the conditions of velocity, depth, and width, prevailing at a 

discharge equal to the mean annual rate.

1

Downstream.

Station

Red Fork near Barnum _ _ _ Middle Fork above Kaycee _ _ Middle Fork near Kaycee _ _

Powder River near Arvada _ _ _

Mean 

annual discharge (cfs)

50

73

115

430

Slope

0. 0057

.0038

. 0019

. 0009

Roughness 

(n)

0. 058

. 087

.039

.017

^

Bed material

Gravel.

Gravel.

Sand and silt.

Silt.

The decrease in value of roughness n is interpreted as 

the result of decreasing particle size, but may also be 

influenced somewhat by a downstream increase in 

suspended-sediment concentration in that particular 

reach of river.

However, in the Arikaree, Republican, and Kansas 

Rivers in the Kansas River system in which the bed is

largely fine sand, there is a progressive dininution of 

slope but a nearly constant value of chanhe] roughness. 

Though the particle size decreases downstr3am in this 

river system, from coarse sand to fine sand r,nd silt, the 

values of roughness are sufficiently large for it to be 

assumed that bed configuration rather tl an size of 

individual particles controls the roughness.

Upstr 

Down

earn, 

stream.

Station

Arikaree River at Haigler _ _ _

Republican River at Culbertson_ . _ _ _ . _

Republican River at Bloomington _ _ - _ - _ . _ _

Republican River at Clay Center _ _______

Kansas River at Ogden_ __ ______

Kansas River at LeCompton _ ________ ._

Mean 

annual discharge (cfs)

29 

205 

670 

1,000 

2,300 

6,200

Slope

0 0031 

0015 

0010 

0004 

0004 

0004

Roughness 

(n)

0 040 

023 

036 

023 

035 

034

Though the slope decreased downstream, the roughness showed only a modest decrease that was nearly 

masked by its variability.

As a generalization, it may be said that the value of 

channel roughness is controlled by the size of individual 

particles when this size is large; that is, gravel, cobbles, 

or even coarse sand. When the size of sediment is 

large, channel roughness tends to decrease downstream 

as the size of the particles decreases. In streams whose 

beds are composed primarily of fine material, there is a 

tendency for the bed configuration (ripples, waves or 

dunes) to control the channel roughness. In streams 

like these roughness tends to change but little in the 

downstream direction when discharge of equal frequency 

obtains along the river length.

The average changes of roughness and slope at a 

station and downstream are part of the hydraulic 

geometry of the channel system. For alluvial streams 

in which roughness tends to be determined by bed configuration and, therefore, decreases downstream only 

slightly, figure 19 indicates the average relations with 

discharge. The line A4—B4 representing the downstream change of roughness for a rate of discharge that 

occurs frequently or for a low discharge, is nearly 

horizontal. For flood flow, the roughness is lower,

presumably in part because of the greater suspendedsediment concentration, as indicated by the lower 

ordinate position of line C4—D4.

The river slope decreases downstream as indicated by 

the slant downward to the right of line A5—B5 . In 

flood flow the river slope is probably very slightly 

greater than at low flow, hi part, because of the tendency 

for the main thread of water to eliminate eome of the 

minor contortions of the low water channel. Thus the 

line A5—C5 representing the at-a-station change of river 

slope is drawn so that it slopes slightly upward to the 

right.

In summary, the observed relations of velocity to 

depth in the downstream direction are considered to be 

those required for the transportation of the supplied 

load. It is postulated that these velocity-depth relations are provided by the concomitant change downstream of roughness and slope. Roughness is governed 

by the sediment size, the suspended-sediment concentration, and the bed configuration, all of which are independent of the channel or nearly so. That part of the 

required relation of velocity to depth not provided by 

the downstream change in roughness is provided by 

change in slope, because slope is the dependent 

factor which can be adjusted by stream processes.
SOME PHYSIOGRAPHIC IMPLICATIONS 43

The characteristically rapid increase of suspended 

load with discharge at a station requires for transportation of that load a relatively rapid increase of velocity 

in comparison with depth, or a relatively large value of

171 -7- For the cross sections studied this was equal to .85.

s^ 

This value, being larger than %, requires that -— increase

7X

with increasing discharge. This is accomplished primarily by the decrease of roughness, n, resulting from 

the increase of concentration of suspended load with 

discharge.

In the downstream direction, however, load increases 

not quite so fast as discharge; this means that the concentration of suspended sediment decreases slightly 

downstream. The transportation of this load requires 

that the depth increase with discharge much faster than

Tflr '

does the velocity, or the ~r ratio must be small. For 

the rivers studied its mean value was .25.

8*

To provide that velocity-depth relation, — must de7l>

crease downstream. Because of the nearly constant 

concentration of suspended sediment and the effect of 

bed configuration, roughness n tends to remain about

7YL

constant downstream. Thus the required -r ratio is 

provided primarily by the decrease of slope downstream.

SOME PHYSIOGRAPHIC IMPLICATIONS

THE STABLE IRRIGATION CANAL—AN 

TO A GRADED RIVER

ANALOGY

An irrigation canal constructed in alluvial material 

tends gradually to scour or fill its bed and banks until a 

cross section is achieved that remains in equilibrium 

with the water and sediment flowing through it. Engineers designing canals to carry sediment-laden water 

have faced many of the same problems that face the 

geologist attempting to understand the grading of rivers. 

A large literature exists, particularly in India, on the 

subject of "regime" channels; that is, channels that are 

neither scoured nor filled.

Early work by Kennedy (1895) and Lindley (1919) 

laid the foundation for the research of Lacey who in 

two papers (1930, 1939) presented a detailed analysis 

of stable canals studied in India. Important addition 

to this literature have been provided by Lane (1937), 

Das (1950), Blench (1951a, b), and others.

In figure 31 data 011 regime canals published by Lacey 

(1939, pp. 41, 49) have been plotted in the same manner 

as the river data shown in figures 5-8. There is indeed 

considerable similarity between the graphs for regime 

canals and those for rivers.

Lacey utilized measurements of channel characteristics made on reaches of Indian canals that, through 

time, had achieved a stable cross section. He proposed

a number of empirical equations that were derived from 

these data. From these equations, which he considered 

"basic," are derivable many other correlative relationships. Included in his basic equations (1939, p. 24) a.re

where P= wetted perimeter 

Q= discharge 

y=mean velocity 

.K=hydraulic radius = mean depth 

F=a, sediment factor

The equation relating wetted perimeter to discharge 

is identical in form to the relation found in the present 

study for river data, inasmuch as the wetted perimeter, 

P, is nearly equal to width, w, in wide channels. Tv e 

comparable equations are

Lace Present study

where 6=0.5 in downstream direction

Lacey recognired that his equation relating wetted 

perimeter to discharge applied to many natural streams, 

but having few river data he could not explore the 

limits of its applicability.

Lacey's data provided him with only a small range 

of intercepts, so that his width coefficients, ranging 

from 2.50 to 2.80 and averaging 2.6, included a much 

smaller range than corresponding coefficients for American rivers. Because of the small range of intercepts 

in the canal data and the fact that the few individual 

measurements on rivers also happened to plot with 

nearly the same intercept as the canal data, Lacey 

concluded that his width-discharge equation was fundamental and that it was independent of sediment size.

The second Lacey equation expresses the relation of 

velocity to depth,

Because R, the hydraulic radius, is essentially equal to 

mean depth, d, of wide rivers this equation is directly

comparable to the -j relation for rivers discussed earlier.

Equations (2) and (3) of the present study can be 

combined as follows.

and 

then

or

v=kQ

(2)

(3)
44 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

200

0 100 1,000

Discharge, in cfs

EXPLANATION: »——» PUNJAB REGIME SITES; *—-—* UPPER BARI DOAB-,

10,000 

MADRAS GODAVARI

FIGURE 31.—Eelation of width, depth, and velocity to discharge for stable (regime) irrigation canals in India (Lacey data).

In the downstream direction for river data -7-=0.25,

whereas the comparable Lacey exponent is 0.5. The 

difference between the numerical values of the exponent 

in these equations probably can be attributed to the 

difference in the downstream change in suspendedsediment concentration. It was shown in figure 18 

that for any given value of b (the rate of increase of 

width with discharge), the larger the value of the ratio

7YL -j the larger is the value of j. In terms of the comparison of the canal and river data, the regime canals 

show a larger -7 value and must experience a rate of 

increase of suspended-sediment load with discharge

along the canal length different from that of the average 

river in the downstream direction.

A canal system begins with the largest discharge at 

the intake where river water is diverted. Downstream 

the flow is gradually divided into successively smaller 

lateral canals. A regime canal is by definition one that 

neither scours nor fills its channel.

To preserve a condition of neither scour nor fill, as 

much sediment must be taken out of the system at the 

final points of diversion to the irrigated fields as was 

introduced at the head of the canal system. This is 

accomplished in practice by dividing, at each point of 

diversion, the sediment load in the same proportion as 

the water. That is, if at a given point of lateral 

diversion 10 percent of the flow of the mair canal is
SOME PHYSIOGRAPHIC IMPLICATIONS 45

diverted, then 10 percent of the sediment being carried 

in the canal is also diverted. Blench (1951b, p. 5) 

implies that this is the practice in handling regimecanal systems when he says:

One [canal] exhibited serious sediment trouble on one branch 

but not on another because engineers did not understand regulator designs that could divide sediment suitably.

Blench makes a similar statement elsewhere (1951a, 

paragraph 2.17):

* * * The Lower Shelum Canal gave trouble [by progressive 

filling of its bed] because the regulator for the two Branches 

divided sediment unfairly.

Equality of division of water and sediment is tantamount to the maintenance of a constant sediment 

concentration throughout the regime system.

If the suspended-sediment concentration remains 

equal at all discharges, then the slope of the line representing the relation of suspended load to discharge 

would equal 1.0. This slope is, by definition, the value 

j. Now, in figure 18, if in the downstream direction 

for both canals and rivers, the exponent b in the widthdischarge relation equals 0.5 as has been demonstrated,

Tfl then the -r ratio (corresponding to .7 = 1.0 in fig. 18)

Tfl should apply to regime canals. The -r ratio at the

value of 6=0.5 and j = 1.0 in figure 18 is seen to be 

about .43, or slightly less than 0.5. Tentatively this 

appears to be a satisfactory agreement.

Tfl As indicated previously, the smaller value of -r obtained from the river data analyzed here indicates that 

J<1.0, or that the sediment concentration hi the rivers 

studied decreases slightly in the downstream direction.

It can be seen that the general relation shown in 

figure 18 applies to regime canals, as well as to natural 

rivers. In fact, the Lacey equations for regime canals 

become a particular condition in the general relation

Tfl •

involving b, -j, and J. The particular condition is

merely that of constant suspended-sediment concentration in the downstream direction.

The comparison of regime canals and natural rivers 

is valid only in the downstream direction. Canals 

carrying water high in sediment concentration would not 

normally be expected to maintain the regime condition 

except at the discharge for which they were designed. 

The lack of a varying discharge at any particular canal 

cross section precludes a proper comparison with the 

at-a-station relation of rivers. Failure to recognize 

this would lead to fallacious reasoning, as in a discussion 

by Stevens (see Lane 1937, p. 147) in which he compared 

the width-discharge formula of a particular river gaging 

station with the Lacey width-discharge formula for 

canals.

Another fact that demonstrates the analogy between 

regime canals and rivers in the downstream direction

lies in a consideration of slope. In the design of canals, 

it has usually been found necessary to provide greater 

slope in the small laterals than in the large trunk canals. 

Thus canals in regime have a longitudinal profile convex 

upward because the flow is reversed as compared vdth 

natural rivers—that is, the discharge in a canal system 

decreases downstream. This is analogous except for 

the direction of flow to the condition in natural rivers, 

which are steeper hi the headwaters where the discharge 

is small than downstream where the flow is great.

The rate of increase of width with discharge is apparently independent of the size of the sediment particles. Not only is this rate similar for headwater reaches 

and lower reaches, but is is similar for rivers and canals 

which carry sediment of quite different sizes. Intercepts represented by a, c, and k, in equations (1) to (3) 

are probably related to sediment size. Lacey found so 

little variation of a in Indian regime canals that he 

considered it to be a constant for all streams. Inspection of the large differences in intercept of the lines in 

figure 9 representing natural rivers indicates that the 

value of a is not a constant. Das (1950, p. 160), reanalyzing the data on Indian canals, concluded that 

this coefficient is " * * * a function of the natur? of 

the periphery of the stream, i.e., it represents the nature 

of the soil surface forming the margin of the stream." 

He stated that values of a for canals and rivers in delta 

areas where sediment sizes are small are less than for 

canals and rivers where sediment sizes are larger. It 

will be recognized from common experience that sandy 

channels are wide compared with silty or clayey 

channels.

A regime canal is a system in equilibrium. By 

definition it is a canal which over a period of years 

neither scours nor fills. It is a system in which slope 

and channel characteristics are adjusted over a period 

of time to carry the load supplied to it at the upstream 

end. In this respect a regime canal fits perfectly the 

definition of a graded stream (Mackin, 1948, p. 471).

Mackin's definition is:

A graded stream is one in which, over a period of years, slope 

is delicately adjusted to provide, with available discharge and 

with prevailing channel characteristics, just the velocity required 

for the transportation of the load supplied from the drainage 

basin. The graded stream is a system in equilibrium; its diagnostic characteristic is that any change in any of the controUing 

factors will cause a displacement of the equilibrium in a direction 

that will tend to absorb the effect of the change.

The discharge is provided to the canal system by a 

source independent of the channel itself, just as Mackin 

visualizes a graded river. Load also is supplied by an 

exterior source. In the regime canal the water and sediment are diverted into the canal system from a river, 

whereas in a graded stream, the load and dicharge are 

supplied from the watershed. In both, however, the 

source is independent of the stream.

The regime canal is in equilibrium. Although local
46 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

or temporary scour and fill may occur to a minor degree, 

it is a diagnostic characteristic of regime that a change 

in any controlling factor causes a displacement of the 

equilibrium in a direction that will tend to absorb the 

effect of the change. Were this not true, local and 

temporary fill or scour would have an accumulative 

effect and the system would have, over a period of years, 

changed sufficiently for the term "regime" to be inapplicable.

It is logical to suppose, then, that the point in figure 

18 which describes regime canals, represents values of

Tft •

b, -y,ancl J which apply to a graded channel system in

the sense used by Mackui in his definition of a graded 

river.

It has been demonstrated by the concurrent changes 

in m, /, and j during individual floods that the general 

relations in figure 18 define, at least in general, the 

mutual adjustments which occur at a station when 

suspended-sediment concentration changes. Because 

figure 18 describes relations both at a station and downstream for average values of the factors observed in 

rivers, and approximately fits regime canals, any point 

in the graph is considered by the authors to represent 

concurrent values of m,f, andj required for equilibrium. 

It is understood, however, that the exact form and 

positions of the lines in figure 18 must be considered 

subject to adjustment as more data become available.

In practice, the suspended load-discharge relations 

vary within definite limits. At-a-station values of j 

probably vary between 2 and 3 and downstream values 

between 0.5 and 1.3. Values of b appear to remain 

close to 0.5 downstream and 0.1-0.3 at a station.

Therefore, actual values of the variables occur in 

only a limited part of the field of values shown hi 

figure 18.

The conception of adjustment of the cross section 

of a channel in response to changes in the load-discharge 

relationships has not been stressed in the geologic 

literature. Only Mackui (1948) and Rubey (1952) give 

this the prominence it deserves. As shown by the 

Yuma data, a change in the governing factors may result 

in changes in channel characteristics without a change in 

river slope.

The present analysis adds weight to the importance 

of adjustment in channel cross section, and the writers 

would support the alternate wording of the definition 

of a graded river which Mackin considered but, apparently for the sake of brevity, discarded. Mackin 

says (1948, p. 484):

The question arises * * * as to whether the foregoing 

definition [of a graded river] which describes the graded stream 

as one "in which slope is delicately adjusted," etc., should not 

be revised to read, "in which slope and channel characteristics 

are delicately adjusted," etc.

The latter phraseology would be preferred by the

writers, and in principle is the conclusion reached 

independently by Rubey (1952, p. 130).

Because the data in figure 31 represent canal sections 

chosen as examples of regime conditions, the figure 

may be considered to represent the relations of width, 

depth, and velocity to discharge in a channel system 

graded in the sense used by Mackin. The graded 

channel system represented in the graph is a specific one 

in that the graphs apply only to the conditions of 

sediment size and constitution of bed and bank existing 

in those canals. It has been, noted, however, that the 

intercepts of these lines are the factors which would 

change with the type of sediment.

The relative slopes of the depth-discharge and 

velocity-discharge lines are considered to be functions 

of suspended-sediment concentration at different discharge.

There might be some logic in postulating, that if 

data from a particular river plotted with as little 

scatter of points on these graphs as do the data on 

regime canals, the river could be considered in regime 

and therefore graded. River data discussed in this 

report plot in graphs similar to those for regime canals, 

and the differences in slope and intercept appear to be 

explicable by differences in sediment size and in 

sediment concentration.

In practical terms, however, a particular graded reach 

of a stream is generally too short to include a large 

enough number of gaging stations to allow a development of graphs adequate to describe in definitive manner 

the slope and intercepts of the downstream relations. 

Moreover, the slopes of the graphs depend for clear 

definition on a considerable range in discharge in the 

downstream direction. A stream segment may be 

graded when there is no increase in discharge downstream.

The downstream graphs presented in figures 5-8 do 

not necessarily represent river systems which are 

graded throughout their length. For example, if 

two graded reaches are separated by a canyon reach 

within which the river flows in a rock-lined gc^ge, the 

canyon section may be steep and may include waterfalls 

and would obviously not be called graded. If the 

gaging stations available for analysis were all located 

in the two graded reaches, it is possible that downstream 

graphs comparable to figure 5—8 could be plotted in 

which there might be a very good alinement of points. 

Figure 6 representing the Bighorn River system 

includes two canyon sections, the Wind River Canyon 

through the Owl Creek Mountain, in the reach between 

Riverton and Thermopolis, and the Bighorn Canyon 

between St. Xavier and Custer. Figure 6 shows a 

considerable scatter of points about the mean line 

and at least part of this scatter is due to the nongraded 

conditions of some of the reaches where gaging stations 

are located.
SOME PHYSIOGRAPHIC IMPLICATIONS 47

Figure 5 presents data for the Powder River system. 

The Powder Rivar, at least in the reach from Kaycee, 

Wyo., to Locate, Mont., has many of the characteristics 

used by Mackiii (1948, p. 472-474) to support his choice 

of specific reaches of certain streams as examples of 

graded reaches. Terrace remnants indicate that the 

Powder River once flowed at a high level. The terrace 

surfaces exhibit smooth, longitudinal profiles concaveupward, similar to the profile of the present flood-plain. 

As in Mackin's Columbia River example, downcutting 

during a period of planation was not prevented by 

bedrock resistance. The underlying material in the 

Powder River Valley is unconsolidated alluvium.

This graded reach is represented in figure 5 by the 

six solid triangles labeled points 2, 3, 7, 12, 22, and 23. 

These points have considerable alinement, and deviate 

from the mean lines drawn on the graph very little 

more than the scatter seen in the regime canal data of 

figure 31. But the recognition of the graded reach 

from the alinement of points would obviously be 

impossible.

It appears that the available data do not allow 

identification of a graded reach from reaches which are 

not definitely known to be graded. It follows that the 

gaging-station data taken at points distributed over a 

drainage system without regard to the existence of or 

lack of grade, define a change of width, depth, and 

velocity downstream with change in discharge as consistent as the changes observed in the same type of 

data within a graded reach. The same factors which 

tend to maintain equilibrium when a reach becomes 

graded are acting also on reaches not in complete equilibrium and this action is sufficiently potent to produce 

consistent patterns in the relation among the hydraulic 

factors of channel shape.

Stations on the upstream reaches and minor tributaries, many of which are probably not graded but have 

been downcut over a period of time, have width-depthvelocity relations similar to those of the graded reach 

of the same river system. One of the primary criteria 

for recognizing a graded reach is the lateral planation 

of a stream during a period when the longitudinal profile 

maintains stability (as discussed by Mackin, 1948, 

pp. 472-474). The same factors which operate to 

maintain grade are also acting during the period of 

downcutting.

When a stream in the process of downcutting has 

developed a smooth longitudinal profile through a given 

reach, the interaction of the hydraulic variables provides 

an approach to the condition of grade. That the overall longitudinal profile of a stream is in general concave 

upward and more or less smooth, even where only a 

few reaches are actually graded, is the result of the 

interaction of these variables. The tendency toward 

the condition of grade provides a tendency toward 

continuity in the change of width, depth, velocity, 

roughness, and slope in the downstream direction.

The data presented here lend quantitative support 

to the idea expressed by Mackin (1948, p. 506) tl *vt 

"* * * even in the aggrading stream the declivity 

of each segment is approximately adjusted to the load 

in transit through that segment." The quantitative 

data show that the approximate adjustment involves 

not only slope but width, depth, and velocity, as 

Mackin's stateni3nt obviously implies. The quantitative data not only demonstrate the existence of approximate adjustment but, more importantly, describe it.

Lest the reader conclude from the consistency in the 

hydraulic geometry defined by the available data between graded reaches and reaches not known to be 

graded, that there is no difference or that the graded 

condition does not exist in nature, it is emphasized 

hers that the present study is not aimed at the quantitative differentiation of graded from nongraded reaches. 

A graded stream by definition (Mackin, 1948) is one 

which is in equilibrium through a period of years. The 

hydraulic geometry as defined here does not demonstrate whether gradual changes are in process, though 

when a stream is sufficiently lacking in equilibrium to 

be suffering rapid degradation or aggradation, changes 

in the values of factors in hydraulic geometry can be 

measured, as was demonstrated for Yuma.

The present study is an attempt to define in quantitative terms, insofar as available measurements permit, how the factors operate which lead .to stream 

grading, which maintain the graded condition, and 

which cause adjustment when the equilibrium condition 

is upset. Mackin (1948) discussed these same factors 

in qualitative terms, and this study is an attempt to 

extend his reasoning by quantitative consideration of 

the hydraulic factors.

In summary, regime irrigation canals that over a 

period of years neither scour nor fill are observed to 

have a hydraulic geometry similar to that for natural 

rivers. Differences in the average slopes and intercepts 

of the lines representing width, depth, velocity, and 

discharge relations in canals and rivers can be qualitatively explained. One important difference lies in the 

load-discharge relations. The analogy of regime-canal 

data to those for rivers leads to the hypothesis that the 

m, f, b, and j relations in figure 18 represent requirements for equilibrium.

Graded reaches of a river are shown to have width, 

depth, velocity, and discharge relations similar to those 

of reaches not known to be graded. This similarity 

indicates that the same factors which operate to maintain equilibrium in a graded river are also sufficiently 

active in nongraded reaches to provide specific and consistent patterns in the hydraulic variables both at a 

station and in a downstream direction. Stream-gaging 

data, however, are not available in sufficient quantity, 

in a sufficiently close geographic network, and over a 

long enough period to allow differentiation between
48 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

graded and nongraded reaches by hydraulic geometry 

alone, though this might theoretically be possible.

The hydraulic geometry of rivers is useful in improving our understanding of the quantitative values of 

hydraulic factors which characterize rivers, and in 

understanding better the nature of the interaction of 

factors that lead to the condition of equilibrium or 

grade.

SEDIMENT AND THE LONGITUDINAL PROFILE

The longitudinal profile of a stream might be considered primarily a product of the change in size of bed 

material in a downstream direction. The bed of a 

mountain stream is usually covered with gravel or 

boulders and has a steep slope. Some distance downstream the bed may be sandy, and still closer to the 

mouth silt or clay may predominate. At the same 

time, the slope progressively decreases downstream. 

One tends to associate the particle size and slope in a 

direct cause-and-effect relation. Shulits (1941) derived an equation for the stream profile from a consideration of decrease of particle size as a result of abrasion. 

He made the assumption that the slope was proportional to the size of the bed-load material.

To assume that slope is proportional to size of bed 

material requires the operation of certain other hydraulic factors. It could be argued that a given slope 

is required to provide the particular velocity necessary 

to carry the particle size present at each point. This 

reasoning implies that the decreasing particle size 

downstream requires a decreasing velocity which, in 

turn, is provided by a decreasing slope in the down 

stream direction.

Mackin (1948, p. 482) argues correctly that a similarity in curves representing the longitudinal profile 

and the change hi particle size does not prove any direct 

cause-effect relationship. He (p. 481) believes the 

relationship to be as follows:

Graded profiles usually decrease in slope in a downvalley direction between tributary junctions, chiefly because of a downvalley decrease in caliber of load due to processes within the 

stream. The principle involved is that the velocity (and, other 

things being equal, the declivity) required for the transportation 

of the coarser fractions of the stream's load decreases with decrease in grain size, the total amount of the load remaining the 

same.

This explanation again implies a downstream decrease 

in velocity. The present study allows some further 

explanation of the mechanism implied in the phrase, 

"velocity required for the transportation of the coarser 

fractions of the stream's load." It has been pointed 

out that the velocity can be changed by a change in 

slope and a change in bed roughness or by either. One 

might suppose from Mackin's description that the effect 

of particle size on velocity is offset entirely by slope 

despite his qualification of "other things being equal." 

Examples of natural stream reaches can be found hi

which neither discharge nor load changes appreciably 

downstream. Mackin (1948, p. 482) cites a reach of 

the Greybull River as an example. He states that in 

this reach slope decreases downstream markedly and 

attributes this " * * * largely to a decrease in caliber 

of load in transit; pebbles of the valley floor and terrace 

gravel sheets decrease notably in size in a downvalley 

direction."

This observed decrease in particle size can be rssumed 

to result hi a decrease in channel roughness (value of 

Manning n) because the bed material being gravelly or 

pebbly is no doubt so large that the size of the individual particles rather than bed configuration determines 

the value of n. Consider the Manning equation

0=1.5

n

(4)

If load and discharge are assumed to remain eoiual, as 

Mackin specifies in his Greybull example, then according to the relations shown in figure 15 the velocitydepth relations required for transportation of the suspended part of the load remain constant. Let it be 

assumed that this is the condition which prevails 

through the reach of river cited. Such an assumption 

rests again on the premise that the suspended load is a 

usable index to the poorly understood load conditions 

which affect channel shape. One could argue that the 

diminishing size downstream of the channel gravel increases the percentage of load carried in suspension, 

but no data are available to support such a contention 

and it is not unreasonable to suppose that suspended 

load remains essentially constant in this reach, as does 

the total load. If constant velocity and deptb downstream are required to carry the constant suspended 

load, an observed decrease in slope downstream results 

from a decrease in roughness downstream caused by the 

reduction in particle size.

In a stream having relatively fine bed material, sand 

and silt, the roughness due to particle size usually does 

not control. The flow may be hydro dynamically 

"smooth"; with this kind of flow in which the thickness 

of its laminar film exceeds the roughness projections of 

the individual particles, as in some Indian canals 

(Blench, 1951a, fig. 4.1), or the effective roughness is 

controlled by bed configuration rather than particle 

size.

The analysis of figure 30 indicates that hi fine-grained 

materials, a decreasing particle size tends to increase 

bed roughness under conditions of constant slope and 

discharge. Roughness, where it is not controlled by the 

size of individual particles, is affected by the concentration of suspended sediment, larger concentrations beingassociated with smaller values of roughness, as 1-as been 

discussed. Bed load affects roughness but its effect 

is not distinguishable from that of suspended lor.d in the 

river data available.
SOME PHYSIOGRAPHIC IMPLICATIONS 49

The tendency for particle size to decrease downstream 

leads to a tendency for roughness to decrease downstream. As the particles in transit become small 

enough, the configuration of the bed finally tends to 

govern, and a further decrease in particle size promotes 

a tendency for increased roughness. This increased 

roughness is also partly caused by the tendency toward 

decreased concentration of suspended sediment downstream.

Though the relative importance of these three factors 

cannot be determined from the river data available, the 

net result is a remarkable conservatism hi the values of 

roughness. Particularly in streams handling finegrained material, the roughness varies between narrow 

limits in the downstream direction.

The relation between velocity and depth to slope, 

roughness remaining constant, can now be considered in 

the downstream direction.

It is obvious that streams having an infinite number 

of longitudinal channel profiles could carry the water 

furnished by the watershed. It will now be shown,

however, that different longitudinal profiles differ in 

then' rates of change of velocity and depth in a downstream direction.

For example, in a hypothetical stream in which sediment is not a factor, it is assumed that the only requirement of the channel system is that it cany a specified 

amount of water between two fixed altitudes. More 

specifically, it is assumed that the watershed is of such 

shape and hydrologic characteristics that, in the stream 

being considered, the discharge in the main channel 

increases linearly from zero at the headwater divide to 

10 cfs at a point 1,000 ft downstream from the divide.

Three completely arbitrary profiles of the stream 

thalweg having different concavity, as shown in figure 

32 lower right, are now assumed. Profile A is a straight 

line. Each profile is drawn between the headwater 

divide and a point of the channel 10 ft lower and 1,000 

ft distant horizontally.

It is desired to show the change of velocity and depth 

downstream for each of two arbitrarily assumed relations of width. The two conditions of width for which

Width increases 

downstream: w=2.75<?°-44

Width constant 

downstream

.05.5 

1.0 10

Al A2

.£ £ -- >•$

"

O.I 1.0 1.0

Bl

Discharge, in cfs

(INCREASING DOWNSTREAM)-

PROFILE 

A

PROFILE 

B

EXPLANATION : - DEPTH; - WIDTH;

Assumed rate of increase 

of discharge downstream

200 600 1000

Distance from headwater divide, in feet

\\ \ Assumed channel profiles

X

\

X

I T——————-J_I^

200 600 1000

Distance from headwater divide, in feet 

Note: Roughness n - 0.025

FIGURE 32.—Variation of velocity and depth downstream for three assumed longitudinal profiles; increase of discharge downstream is specified, and two rates of change

of width downstream are compared; channel roughness constant.
50 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

computations have been made are: (1) the widthdischarge relation is

and (2)

w=l

Diagram Al has an exponent i— 0.44, which is of the 

order of magnitude of the 0.5 already shown for natural 

rivers. In diagram A2 the width is assumed constant 

downstream. Because discharge is assumed to be 

directly proportional to length along the channel, 

diagram A2 has a width-discharge relation

It has been assumed in the example that the roughness, using Manning n as a measure, is constant throughout and has the value 0.025. At any position along the 

profile, the slope can be measured from the assumed 

profile curve. By assuming a depth, velocity can be 

computed. The discharge is given for each position 

by reading from the discharge curve. Width, being 

a function of discharge, is specified. Therefore, for 

each position along the stream length, two independent 

equations relating velocity to depth are available:

1.5

in which s and n are known. Also

Q=wdv

in which Q and w are known. The equations can be 

solved simultaneously to compute velocity and depth. 

Velocity, depth, and width have been plotted against 

discharge to show the changes in these factors in the 

downstream direction.

Inspection of the six sets of curves shows that, regardless of the manner in which width changes downstream, 

increase in concavity of the stream profile decreases 

the rate at which velocity increases downstream, and 

increases the rate of deepening downstream. The

m

profiles differ, therefore, primarily in the -j ratio.

A channel that does not increase in width downstream 

obviously must accommodate the increase in discharge 

by increase in velocity or depth.

It will be noted that in diagram Al, for which the 

profile is a straight line but in which the width increases 

downstream at approximately the natural rate, velocity increases downstream at a rate greater than for 

normal rivers. In diagram Bl of a profile of moderate 

concavity, the velocity decreases downstream. A 

concavity intermediate between profiles Al and Bl 

would provide a combination of depth and velocity 

curves that would approximate conditions in a natural 

stream having other characteristics similar to those 

assumed in the example.

The longitudinal profiles postulated in the example 

are but samples of an infinite number, each of which 

would be associated with particular channel characteristics necessary for fulfillment of the hydraulic laws. 

Any of these infinite number of combinations ^ould be 

quite capable of carrying the discharge of water postulated. However, it may be presumed that only one of 

the possible combinations would obtain in nature. The 

equilibrium profile differs from, other possible profiles in 

the relative rates of increase of velocity and depth in the 

downstream direction.

The longitudinal profile of a stream represents the 

simultaneous solution of eight equations involving eight 

unknowns. No attempt is made to indicate the nature 

of the relations or the hydrodynamics behird them. 

But, according to the reasoning in this paper the way 

in which the variables are dependent may be summarized as follows: 

The first equation represents continuity:

Q—wdv (The flow equation.) (A)

The second is independent of the stream channel itself 

and represents the independent characteristics of the 

drainage basin:

L=F(Q) (The load-discharge relation result; ng from 

the characteristics of the drainage area.) (B)

The next equations represent relations in which the 

values of the factors and the specific relations among 

them depend on processes or action in the stream; 

that is, processes not independent of the stream itself:

7j=F(Q) (The particle size changes downstream.) (C)

w=F(d) (A width-depth relation that invcVes 

the relative erodibility of bed and bank.) (D)

/' 7J7 \ 77?

j=F (j, bj (The relation of j> j, and 6.) (E)

n=F(d, <7, Q, s} (Where "g is the size of the sediment (F) 

particles; this relates sediment size to rougliness n.)

-i-ff=F (-j0 ) (There is a line on the roughness-discharge diagram corresponding to a particular rate 

of change of sediment concentration.) (G)

The final equation is purely hydraulic and represents 

the equality between energy losses in friction and the 

decrease of energy potential, or head:

£1 

n

(H)

In equation (H) slope is the dependent factor and is 

adjusted by the stream to satisfy the equation; velocity, 

depth, and roughness n were determined by equations

(C) to (G).
SUMMARY AND INTERPRETATION 51

The primary determinant of the slope of a stream is 

the past physiographic history which determines the 

differences in altitude and the distance between the 

mouth and the watershed divide. The problem discussed here concerns the form of the longitudinal profile 

which will be cut between these two points. It has 

been demonstrated that conditions of approximate 

equilibrium tend to be established in a reach of the 

stream as soon as a more or less smooth longitudinal 

profile has been established in that reach even though 

downcutting may continue. Within the limits of the 

conditions determined by the physiographic history, 

the interactions which determine the form of the longitudinal profile can be summarized:

Channels of different shapes have characteristic 

distributions of velocity in the cross section. With 

the same discharge a very shallow and wide channel 

tends to have a greater average shear in the vertical 

plane than in the horizontal. Conversely a very deep 

and narrow channel tends to have greater average 

shear on the bank than on the bed. The relative 

erodibility of bed and bank allows the channel shape 

to develop a velocity distribution which is in approximate equilibrium with the particular erodibility characteristics. In this manner the increase of width with 

discharge (value of 6) probably is determined. Why 

this exponent 6 should so consistently be nearly equal 

to 0.5 in the downstream direction for widely different 

rivers is not known and constitutes an important 

unsolved problem.

Different concavities of longitudinal profile are 

associated with different relations of velocity and depth 

to discharge, as required by the hydraulics of open 

channels. However, a particular rate of increase of 

both velocity and depth downstream is necessary for 

maintenance of approximate equilibrium in a channel, 

inasmuch as the drainage area produces sediment and 

water in a characteristic manner.

Koughness varies with sediment load and with 

particle size but the interaction of opposing tendencies 

results in a small range of values of roughness downstream in most rivers. The suspended load and its 

change downstream characteristic of natural rivers 

require a particular rate of increase of velocity and 

depth downstream. Under the conditions of a nearly 

constant roughness, to provide the required velocitydepth relations slope must generally decrease downstream ; it is for this reason that the longitudinal profile 

of nearly all natural streams carrying sediment is 

concave to the sky.

SUMMARY AND INTERPRETATION

Rivers play a major role in the geographic cycle. As 

explained by Davis (1899), the successive stages of 

geomorphic development of a region begin ideally with 

uplift. Surface rocks are attacked by weathering, and

rainfall washes the loosened waste down the initial 

slopes to incipient streams.

"The machinery of the destructive processes is thus 

put in motion," Davis says, "and the destructive development of the region is begun."

Larger rivers quickly deepen their valleys; the relatively brief period of youth is characterized by rapidly 

increasing relief. The mature stage that follows is the 

period of maximum relief and greatest variety of fc^m. 

After a transition period of slowly decreasing relief is 

an indefinitely long old age of faint relief in which 

further changes are exceedingly slow.

Throughout the cycle rivers are both the routes over 

which the erosion products are carried to the sea, and 

the mechanism for effecting the transportation.

Davis (1902) not only defined stages in the deveT opment of land forms through the cycle, but described a 

complete cycle of river life, consisting of successive 

stages of youth, adolescence, maturity, and old ago.

At the stage of maturity, a river attains the condition of grade which connotes the existence of equilibrium in the river system. Mackin (1948, pp. 471, 

484) says:

A graded stream is one in which, over a period of years, slope 

[and channel characteristics are] delicately adjusted to provide, 

with available discharge * * * just the velocity required 

for the transportation of the load supplied from the drainage 

basin.

Equilibrium, or grade, is attained by the larger 

streams relatively early in the geomorphic cycle, and 

by those of medium and smaller size at progressively 

later stages. But even in the youthful topography of 

the headlands where streams are gradually downcutting 

and are therefore not graded, the stream-channel characteristics and slope tend to be adjusted to the discharge 

and sediment load. Even the ungraded parts cf a 

channel system exhibit a neat pattern characterized by 

a tendency toward a regular increase of width and 

depth downstream with increasing discharge and vdth 

a concomitant decrease in slope.

The present paper describes in quantitative tennis 

the manner in which width, depth, and velocity change 

with discharge downstream and at a given river c~oss 

section. It is demonstrated that these factors tend 

to change progressively downstream in all natural 

stream channels studied in so nearly the same way as 

graded reaches change that quantitative data available 

from the existing network of stream gaging stations 

are not adequate in volume or geographic coverage to 

differentiate graded from ungraded reaches. This demonstrates the strong tendency toward establishment of 

a quasi equilibrium operating in channels carrying water 

and sediment.

The mechanism of the interaction which promotes 

the development of at least quasi equilibrium may be 

summarized as follows:

Load and discharge are characteristics of the hydrol-
52 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

ogy, geology, and physiognomy of the drainage basin. 

They are furnished to the trunk stream channels and 

are essentially independent of the trunk channel system. 

Flood flows are characterized by large sediment loads, 

and in terms of the suspended-load data available for 

analysis, are characterized by relatively large suspendedpediment concentrations. Low flows are characterized 

by low concentrations of suspended load.

Slope is the one hydraulic factor which can be adjusted over a period of time by processes within the 

stream and may be considered the factor which makes 

the final adjustment, as may be required for quasi 

equilibrium, after the interaction of all the other factors 

has resulted in establishment of mutually adjusted 

values of those factors.

The factors which are semidependent—that is, 

neither independent as are load and discharge, nor dependent as is slope—are width, depth, velocity, bed 

roughness, and size of sediment particles. The nature 

of the interdependence of these factors appears to be 

as follows:

The shape of the river cross section, roughly described by its width-to-depth ratio, determines the 

distribution of velocity and shear in the cross section. 

The mean velocity in an open channel usually is equalled 

at a position 0.6 of the distance from the surface to the 

bed. In two channels having the same mean velocity 

and discharge, the deeper one will have a smaller average 

shear between the bed and the position in the vertical 

where a velocity equal to the mean value exists. This 

may partly explain, at least qualitatively, why for the 

same discharge and mean velocity, a greater bed load 

characterizes a wide shallow section than a deep narrow 

one. Even less well understood is the observed fact that 

a wide, shallow stream carries less suspended load tl^an 

a deep narrow one of the same velocity and discharge. 

It may be that more of the total load will be carried as 

bed load in a wide shallow reach than in a deep narrow 

reach of a stream if the velocity and discharge are equal. 

In reality, however, changes in cross section in different 

reaches of the same stream are usually accompanied by 

some change in velocity even though the discharge may 

be equal.

The increased bed shear in a wide shallow reach having a given velocity promotes a tendency to scour. A 

narrow deep reach of the same velocity is apparently 

characterized by less shear on the bed but greater shear 

on the bank. Thus there is a tendency to adjust the 

cross-sectional shape in accordance with the relative 

erodibility of bed and bank.

For a given width and discharge however, the total 

load depends on the velocity, and any change in velocity 

requires a concomitant change in depth to satisfy 

continuity expressed by the equation

Thus for a given width and discharge, the total load 

depends oil the ratio of velocity to depth.

The velocity-depth relations required for tran^nortation of the supplied load are provided in part by adjustment of bed roughness which is a function .of the bed 

material size and the suspended-sediment concentration. With the roughness so determined, slope tends 

to be adjusted to provide the velocity-depth relations 

necessary to carry the load in quasi equilibrium.

During the passage of a flood through a given cross 

section of a river channel, a larger velocity is necessary 

to carry in quasi equilibrium the relatively great 

suspended-sediment concentration of the high discharge 

than the small concentrations of low flow. T'lis increase in velocity at a cross section is not provided by 

a marked change in slope because there is insufficient 

time for such an adjustment to be made, and because 

there must be some continuity of slope downstream in 

order that any cross section may generally be r,s close 

to equilibrium as any other.

The changes in velocity-depth relations required for 

quasi equilibrium with load are brought about primarily 

by changes in bed roughness associated with the 

changes in suspended-sediment concentration.

In the downstream direction, however, suspendedsediment concentration decreases only slightly with 

discharge. The average decrease in the particle size 

downstream tends to decrease bed roughness, bnt this 

is counteracted by the effect of decreasing sediment 

concentration and of changing configuration of the bed. 

The result is that roughness changes but little downstream. The particular rates of increase of velocity 

and depth with increase of discharge downstream required for transport of the load are, therefore, maintained by the downstream decrease of slope.

The interactions just summarized operate in both 

ungraded and graded reaches. They account for the 

maintenance of grade once the condition of equilibrium 

is achieved, but their influence is sufficiently potent, 

even in reaches not yet graded, to cause the drainage 

system to develop in a highly coordinated and systematic manner. Because of these interactions1 rivers 

tend to increase in width and depth downstream with 

increase of discharge in a remarkably consistent manner.

Each branch in the channel system of a river tends 

to occupy a valley proportioned to its size. This interrelationship is so similar in rivers widely different in 

physiographic setting that its importance to the understanding of land forms was recognized more than a 

century ago. Geomorphology cannot move ahead, 

however, if we remain content to describe processes of 

land sculpture only in qualitative terms. When viewed 

quantitatively, the interactions of the various hydraulic 

factors are found to be more complex than the qualitative analyses have led us to believe, but the unravelling 

of their complexities constitutes advance of knowledge.
EEFEEENCES 53

REFERENCES

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channels: Am. Soc. Civil Eng. Proc., vol. 77, separate 70, 

pp. 1-18.

BRUNE, G. M., 1950, Dynamic concept, of sediment sources: 

Am. Geophys. Union Trans., vol. 31, no. 4, pp. 587-594.

BUCKLEY, A. B., 1922-23, The influence of silt on the velocity of 

flowing water in open channels: Inst. Civil Eng. Proc., 

vol. 216, pp. 183-211.

CORBETT, D. M., and others, 1943, Stream-gaging procedure: 

U. S. Geol. Survey Water-Supply Paper 888.

CROSS, W. P., and BERNHAGEN, R. H., 1949, Ohio stream-flow 

characteristics, part 1, flow duration: Ohio Dept. of Natural Resources Bull. 10.

DAS, ISHAR, 1950, Theory of flow of water and universal hydraulic diagrams: Jour. Central Board of Irr. (India), 

vol. 7, no. 2, pp. 151-162.

DA vis, W. M., 1899, The geographical cycle, Geog. Jour. 14, 

pp. 481-504.

———— 1902, Base-level, grade and peneplain, Jour. Geol. 10,

pp. 77-111.

EINSTEIN, H. A., 1950, The bed-load function for sediment transportation in open channel flows: U. S. Dept. Agr. Tech.

Bull. 1026. 

EINSTEIN, H. A., and BARBAROSSA, N. L., 1951, River channel

roughness: Am. Soc. Civil Eng. Proc., vol. 77, 12 pp. 

ELLISON, W. D., 1945, Some effects of raindrops and surface-flow

on soil erosion and infiltration: Am. Geophys. Union Trans.,

vol. 26, no. 3, pp. 415-430. 

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

U. S. Geol. Survey Prof. Paper 86. 

GLTMPH, L. M., 1951, Relation of sedimentation to accelerated

erosion in the Missouri River Basin, U. S. Dept. Agr. Soil

Cons. Service, SCS-TP-120, 20 pp. 

COULD, HOWARD R., 1951, Some quantitative aspects of Lake

Mead turbidity currents: Soc. Econ. Paleontologists and

Mineralogists, special pub., no. 2, pp. 34-52. 

GRIFFITH, W. M., 1927, A theory of silt and scour, Inst. Civil

Eng. Proc., vol. 223. 

HADLET, R. F., The usefulness of small reservoirs in erosional

studies of minor drainage basin. U. S. Geol. Survey,

(open-file report).

———— Sources of sediment and sedimentation rates in small 

reservoirs, Cheyenne River basin above Angostura Dam: 

U. S. Geoi. Survey (in preparation).

HEMBREE, C. H., COLBY, B. R., SWENSON, H. A., and DAVIS, 

J. R., 1952, Progress report, sedimentation and chemical 

quality of water in the Powder River drainage basin, Wyo. 

and Mont.: U. S. Geol. Survey (in preparation).

HORTON, R. E., 1945, Erosional development of streams and their 

drainage basins—Hydro-physical approach to quantitative 

morphology: Geol. Soc. Am. Bull., vol. 56, no. 3, pp. 

275-370.

JOHNSON, J. W., 1943, Laboratory investigations on bed-load 

transportation and bed roughness: U. S. Dept. Agr. Soil 

Cons. Service, Sedimentation Sect., SCS-TP-50.

KENNEDY, R. G., 1895, Prevention of silting in irrigation canals: 

Inst. Civil Eng. Proc., vol. 119, pp. 281-290.

LACEY, GERALD, 1930, Stable channels in alluvium: Inst. Civil 

Eng. Proc., vol. 229, pt. 1, pp. 259-384.

———— 1939, Regime flow in incoherent alluvium: Central

Board Irr. (India) pub. 20, Simla. 

LANE, E. W., t 1937, Stable channels in erodible materials:

Am. Soc. Civil Eng. Trans., no. 102, pp. 123-194. 

LANE, E. W., and BORLAND, W. M., 1951, Estimating bed-load:

Am. Geophys. Union Trans., vol. 32, no. 1, pp. 121-123 

LANCBEIN, W. B., 1942, Hydraulic criteria for sand waves, Am.

Geophys. Union Trans., pt. 2, pp. 615-618.

LARA, J. M., and MILLER, C. R., 1951, Conveyance channelwidening study, Middle Rio Grande project: U. S. Bur.

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LINDLEY, E. S., 1919, Regime channels, Punjab Eng. Congress

Proc., vol. 7.

LlNSLEY, R. K., KoHLER, M. A., PAULHUS, J. L. H., 1949,

Applied hydrology, New York, McGraw-Hill Book Co.,

689 pp. 

MACKIN, J. Hoover, 1948, Concept of the graded river: Geol.

Soc. Am. Bull. 59, pp. 463-512. 

RUBEY, W. W., 1933, Equilibrium conditions in debris-laden

streams: Am. Geophys. Union Trans. pp. 497-505.

———— 1952, The geology and mineral resources of Hardin and

Brussels quadrangles, Illinois, U. S. Geol. Survey, Prof.

Paper 218. 

SERR 3d, E. F., 1950, Progress report, investigations of fluvial

sediments of the Niobrara River near Cody, Nebr., U. S.

Geol. Survey Circ. 67.

———— 1951, Measurement of bedload sediment, Am. Geophys.

Union Trans., vol. 32, pp. 123-126. 

SHULITS, S., 1941, Rational equation of river-bed profile: Am.

Geophys. Union Trans. pt. 3, pp. 622-629. 

STEVENS, J. C., 1937, Discussion of LANE, E. W., Stable channels

in erodible material: Am. Soc., Civil Eng. Trans., no. 102,

pp.145-149. 

THOMAS, A. R., 1946, Slope formulas for rivers and canals:

Jour. Central Board of Irr. (India), vol. 3, no. 1, pp. 40-49.

———— 1949, Analysis of hydraulic data of some boulder rivers: 

Jour. Central Board of Irr. (India), vol. 6, no. 1. pp. 67-71.

VANONI, V. A., 1941, Some experiments on the transportation of 

suspended load: Am. Geophys. Union Trans., pt. 3, pp. 

608-620.

———— 1946, Transportation of suspended sediment by water: 

Am. Soc. Civil Eng. Trans., no. Ill, pp. 67-133.

VICE, R. B., and SERR, E. F., 1950, Progress report, investigations of fluvial sediments, Middle Loup River near Dunning 

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Division, (open-file report).

U. S. Army, Corps of Engineers, 1935, Studies of river bed 

materials and their movement, with special references to 

the lower Mississippi River: U. S. Waterways Exper. Sta. 

Paper 17, 161 pp., Vicksburg, Miss.

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and investigations, Lower Colorado River Basin, calendar 

years 1948 and 1949: Boulder City, Nev., 124 pp. (oienfile report).
54 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

APPENDIX

APPENDIX A.—Channel-shape characteristics of rivers at stage corresponding to mean annual discharge

[Data on width, area, mean velocity, and mean depth represent approximations to the average and vary somewhat, depending on the particular section where measurements

are made]

Mobile River Basin

Tombigbee River at Aberdeen, Miss__________________

Buttahatchee River near Caledonia, Miss_________ _._____.

Tombigbee River at Columbus, Miss. __________________

Sipsey River near Elrod, Ala_-_-_____.____.________________

Tombigbee River near Coatopa, Ala.__ _.__-_.___._..____ .

Tombigbee River near Leroy, Ala_--__ _____________________

Scioto River Basin

Scioto River at LaRue, Ohio. _____________________ _______

Scioto River near Prospect, Ohio.________________________

Scioto River near Dublin, Ohio 1 .-.________________________

Scioto River at Columbus, Ohio. __________________________

Scioto River near Circleville, Ohio__ ______

Scioto River at Chillicothe, Ohio __________________________

Tennessee River Basin

French Broad River at Rosman, N. Ci___________________

French Broad River at Calvert, N. C______________________

Catheys Creek near Brevard, N. C_________________________

Davidson River near Brevard, N. C___________________

French Broad River at Bent Creek, N. C__________________

French Broad River at Hot Springs, N. C_ ________________

French Broad River near Newport, Tenn __________________

Watauga River near Sugar Grove, N. C____________________

Noland Creek near Bryson City, N. C.2____________________

Tennessee River at Knoxville, Tenn______________________

St. Lawrence River Basin

St. Joseph River near Blakeslee, Ohio___________________

Maumee River at Antwerp, Ohio.__ ___ _

Tiffin River at Stryker, Ohio..________________

Tiffin River near Brunersburg, Ohio_______ _______________

Auglaize River near Fort Jennings, Ohio. ________________

Ottawa River at Allentown, Ohio___________________ ______

Ottawa River at Kalida, Ohio._____________________________

Blanchard River near Findlay, Ohio____ __________________

Blanchard River at Glendorf, Ohio ________________________

Blanchard River near Dupont, Ohio __

Auglaize River near Defiance, Ohio ______________________

Maumee River near Defiance, Ohio ________________________

Maumee River at Waterville, Ohio_________________________

Yellowstone River Basin

Yellowstone River at Cprwin Springs, Mont_-----_________

Yellowstone River at Billings, Mont ______________________

Wind River near Dubois, Wyo ____________________________

Wind River near Burris, Wyo ___________________________

Wind River near Crowheart, Wyo__________________________

Wind River at Riverton, Wyo_ ___________________________

Popo Agie River near Riverton, Wyo_.--__-_-_____._______

Bighorn River at Thermopolis, Wyo_____________________

North Fork Owl Creek near Anchor, Wyo__________________

Owl Creek near Thermopolis, Wyo_________________________

Gooseberry Creek near Grass Creek, Wyo____._--__________

Bighorn River at Manderson, Wyo_________________________

Medicine Lodge Creek near HyattvUle, Wyo_____ _______

Greybull River at Meeteetse, Wyo__ — —--. — ---_ _________

Greybull River near Basin, Wyo__.______________________

Bighorn River at Kane, Wyo_____________________________

Bighorn River near St. Xavier, Mont_-_--------_-_____ ___.

Bighorn River near Hardin, Mont______-___________________

Little Bighorn River at Stateline, near Wyola, Mont..---.- 

Little Bighorn Rjver below Pass Creek, near Wyola, MontLittle Bighorn River near Crow Agency, Mont.-— —-------

Bighorn River near Custer, Mont _________________________

Yellowstone River near Sidney, Mont—--------------------

Red Fork near Barnum, Wyo _____________________________

Middle Fork Powder River above Kaycee, Wyo________-_.

Middle Fork Powder River near Kaycee, Wyo_________ ___

North Fork Powder River near Hazelton, Wyo. ____________

North Fork Powder River near Mayoworth, Wyo_-__--_-_- 

South Fork Powder River near Kaycee, Wyo.___._-_-___._ 

Powder River at Sussex, Wyo ___________________________

Middle Fork Crazy Woman Creek near Greub, Wyo ______

North Fork Crazy Woman Creek near Buffalo, Wyo_-__--. 

North Fork Crazy Woman Creek near Greub, Wyo _______

Crazy Woman Creek near Arvada, Wyo.--_-___---________

Powder River at Arvada, Wyo_ ________

North Fork Clear Creek near Buffalo, Wyo __.___-_.______

Clear Creek near Buffalo, Wyo.________ "__________________

South Fork Rock Creek near Buffalo, Wyo_________________

Rock Creek near Buffalo, Wyo.--_- _ __

South Piney Creek at Willow Park, Wyo_ _______ _________

Piney Creek at Kearney, Wyo______________ ____

Piney Creek at Ucross, Wyo __________________

Clear Creek near Arvada, Wyo_ _________________________

Little Powder River near Broadus, Mont _---_-__-_-_--_._ 

Powder River at Moorhead, Mont.._______________________

Powder River near Locate, Mont__________________________

See footnotes at end of table

Years of 

record

Mean annual

discharge

(cfs)

Width (feet)

2,863

1,066

5,791

715

21,450

26, 250

198

452

743

1,308

1,815

3,289

229

335

34

125

1,589

2,511

2,764

154

42.8

12,820

292

1,547

292

448

278

119

141

219

548

452

1,598

3,612

4,269

2,969

6,331

177

828

1,315

1, 106

683

1,908

10.2

45.5

20.8

I,950 

38

364

200 

2,383 

3,676 

4,535

154

202

294 

4,390

II,860

50

74 

157

13

39

48 

136

20

21

22

63

434

16

70

17

37

47

101

70

222

56

526

763

142

152

270

93

458

513

90

125

200

280

280

200

77

92

25.3

55

296

190

346

64

30.1

933

95

245

72

85

100

65

80

120

285

440

730

242

253

68

134

155

159

123

218

17

26

19.5

258

24

87

77

175

264

371

41

57

96

420

308

25

41

54

16

22

57

84

19

19

24

33

112

23

42

20

19

33

45

55

94

24

124

168

Area of cross

section 

(square feet)

1,633

1.973

3,220

650

10, 240

13, 360

342

300

860

1,093

3,030

1,840

151

168

22.7

88.0

828

1,115

1,833

115

57.2

13, 450

266

1,078

348

332

285

97

128

114

304

646

1,054

1,826

1,644

813

1,343

97

335

372

318

406

543

9.3

24.1

10.9

650

21

122

100

627

1,296

1,855

116

1,850

3,750

26

56

77

15

26

27

75

15

19

18

38

138

19

39

19

23

33

69

63

124

29

211

273

Mean velocity (fps)

1.8 

0.5 

1.8 

1.1

£ 6

1.5 

.9

1.2 

.6

1.8

1.5 

2.0 

1.5 

1.4 

1.9 

2.2 

1.5 

1.3 

.8 

1.0

1.1 

1.4

.8 

1.4 

1.0 

1.2 

1.1 

1.9 

1.8

.7 

1.5 

2.0 

2.6

3.6 

4.8 

1.8 

2.5 

3.5 

3.5 

1.7 

3.5 

1.7 

1.9 

1.9 

3.0 

1.8 

3.0 

2.0 

3.8 

2.8 

2.4

2.3

2.4

2.5 

2.4 

3.2 

1.9 

1.3 

2.0 

.9 

1.5 

1.8 

1.8 

1.3

1.1

1.2 

1.7 

3.2 

.8 

1.8 

1.0 

1.6

1.4

1.5 

1.2

1.8

1.9 

2.5 

2.8

11.5

13.0

11.9

7.0

22.3

26.0

3.8 

2.4 

4.3 

3.9 

10.8 

9.2

2.0 

1.8 

.9 

1.6 

2.8 

5.9 

5.3

1.8

1.9 

14.4

2.8 

4.4 

4.8 

3.9 

2.8 

1.6 

1.6 

1.8 

3.8 

5.4 

3.7 

4.2 

2.2

3.4 

5.3 

1.4 

2.S 

2.4 

2.0 

3.3 

2.5 

.6 

.9 

0.6 

2.5 

.9 

1.4 

1.3 

3.6

4.9

5.0 

1.6 

1.5 

1.2 

4.4 

11.9 

1.0 

1.4 

1.4 

.9 

1.2 

.5 

.9 

.8

1.0 

.7

1.1

1.2

.9 

1.2 

1.0 

1.5 

1.1 

1.3 

1.2 

1.7 

1.6

Drainage

area (square

miles)

2,210

823

4,490

515

15, 500

19,100

255

571

988

1,624

2,635

3,847

67.9

103

11.7

40.4

676

1, 567

1,858

90.8

13.8

8,934

369

2,049

444

766

333

168

315

343

643

749

2,329

5,530

6,314

2,630

11,180

233

1,220

1,920

2,320

2,010

8.080

58.2

484

155

11,900

86

690

1,130

15,900

20,700

199

429

1,190

142

450

980

25

69

1,150

3 000

83

52

174

956

6,050

29

120

44

60

29

106

267

1,110

8,030

12,900
APPENDIX 55

APPENDIX A.—Channel-shape characteristics of rivers at stage corresponding to mean annual discharge— Continued

IData on width, area, mean velocity, and mean depth represent approximations to the average and vary somewhat, depending on the particular section where measurements

are made]

Belle Fourche River Basin

Belle Fourche Biver at Hulett, Wyo...- . _.__

Belle Fourche River at Wyo.-S. Dak. State line 3 . . — ._.._

Belle Fourche River near Fruitdale, S. Dak _ .... _ ........ .....

Kansas River Basin 

Arikaree River at Haigler, Nebr.._______ _ _____________________ _ ..

Republican River at Clay Center, Kans . .

Smoky Hill River at Elkader, Kails. _._ _ . ..... _____ _ ______

Smoky Hill River near Ellis, Kans.

Smoky Hill River at Ellsworth, Kans._. __ _____ ___ _ ________

Smoky Hill River near Langley, Kans _

Smoky Hill River at Enterprise, Kans._--___._ _ __ _ _______ ...

Kansas River at Topeka, Kans. ......

Missouri River Basin

Missouri River at Pierre, S. Dak..

Missouri River at Hermann, Mo. .

Mississippi River Basin

Mississippi River at St. Louis, Mo

Years of 

record

7

12

2

3

16

15

16

18

30

5

5

4

34

4

17

13

30

28

30

17

30

21

20

20

20

20

12

14

13

16

Mean annual 

discharge 

(cfs)

79

69.5

150

250

500

447

31.7

226

724

1,093

36

66.3

223

215

362

292

1,419

2, 514

4,114

4,655

57,838

5,874

20,320

22,080

35,440

43, 710

69, 170

96, 670

166, 700

454,900

554,600

Width (feet)

39

50

59

66

134

112

45

120

230

272

41

60

104

90

92

80

102

342

525

481

728

563

1,202

962

863

1,093

1,578

1,750

1,586

1,950

2,610

A rea of cross 

section 

(square feet)

47

40

65

112

228

168

23.4'

101

390

544

24.6

48

114

126

193

192

736

1,300

2,150

2,210

3,352

3,320

7,323

8,718

9,927

12,750

22,950

32,500

44,440

99,400

105, 650

Mean veloc- - 

ity (fps;

1.7

1.8

2.3

2.2

2.2

2.7

1.4

2.2

1.8

2.0

1.5

1.4

2.0

1.7

1.9

1.5

1.9

1.9

1.9

2.1

2.3

1.8

2.9

2.5

3.6

3.4

3.0

3.0

3.8

4.6

5.3

Mean depth 

(feet)

1.2

.8

1.1

1.7

1.7

1.5

.5

.8

1.7

2.0

.6

.8

1.1

1.4

2.1

2.4

7.2

3.8

4.1

4.6

4.6

5.9

6.1

9.1

11.5

11.7

14.5

18.6

28.0

51.0

40.1

Drainage 

area (square 

mile?)

1,730

2,800

7,210

* 1,460

<10 600

<20 800

* 24, 570

3 555

5 630

6 965

7,580

7,857

8 110

19 200

45 240

55 240

5fi 710

58 420

59 890

186,400

243 500

424 300

489,200

528 200

171.500

701,000

932.800

1,144.500

' Located at a dam.

2 Concrete walls on side.

3 Poor records.

4 Includes some noncontributing area.

5 Long-term mean should be abont 5,200 cfs.
56 THE HYDRAULIC GEOMETRY OF STREAM CHANNELS

APPENDIX B.—Average channel characteristics and average suspended-sediment load at faced discharges

[Data on width, average depth, velocity, and suspended load represent approximations to mean values of their respective factors at the discharge specified. Exponents 6, /,

and m are defined by the equations on p. 8]

Station

Cheyenne River near Hot Springs, S. Dak......... — --- — .„.-] 250

500

1,000

2,000

Middle Loup River at St. Paul, Nebr_^-_-----..----------------- 250

500 

1,000 

2,000

Rio Grande at San Acacia, N. Mex.--_ _-__----.----------.----- 250

500

1,000

2,000

Rio Qrande at San Felipe, N. Mex_ ------------------------------ 250

500

1,000

2,000

Middle Loup River at Arcadia, Nebr---.---.------------.------- 250

500 

1,000 

2,000

Smoky Hill River near Ellis, Kans_-....... —.--------_-___.._-- 250

500

Powder River at Arvada, Wyo-- ----------------------------.-- 250

500 

1,000 

2,000

Moreau River near Faith, S. Dak- —------------- —— - — — ---- 250

500 

1,000 

2,000

White River near Oglala, S. Dak__ __.- ------------------------- 250

500

1,000

2,000

Belle Fourche River below Moorcroft, Wyo ______________ _ _ _ _ _ _ 250

500 

1,000 

______________________ 2,000

Virgin River at Virgin, Utah__._...._-_._..____._-_._.._-_..-. 250

500 

1,000 

____________________ 2,000

Rio Puerco at Rio Puerco, N. Mex_-_-_.—_.__-_-_-_-_-_-- 250

500 

1,000 

________________; 2,000

Rio Grande near Bernalillo, N. Mex.--___ — — — — _ — „— — — — 250

500 

1,000 

______________________ 2,000

Saline River near Russell, Kans..... — ...-...-.-.-—___-_-_-_ — ._ 250

500

1,000

2,000

Smoky Hill River at Ellsworth, Kans_.__„_„_„ —„_-_____—__ 250

500 

1,000 

_____ ______ 2,000

Republican Rivernear Bloomington, Nebr____._._................ 250

500 

1,000 

______________________ 2,000

Bighorn River at Kane, Wyo---—___-__—--------_---_-_-_-____- 250

500 

1,000 

2,000

Bighorn River at Thermopolis, Wyo.

Bighorn River at Manderson, Wyo_

Qrand River at Shadehill, S. DakDischarge 

(cfs)

500

1,000

2,000

250

500

1,000

2,000

250

500

1,000

2,000

Width 

(feet)

107

150

200

215

192

253

302

384

94

140

182

218

175

205

235

273

181

189

203

217

115

148

105

116

125

135

118

129

135

140

39

48

59

72

59

74

92

115

92

111

134

160

122

133

95

.156

260

265

66

85

112

145

92

100

111

122

92

135

200

210

152

160

168

175

179

195

215

130

150

179

210

105

125

140

Average 

depth 

(feet)

1.1 

1.3 

1.7 

2.5

0.8 

1.0 

1.3 

1.57

1.0

1.1 

1.4 

2.0

0.8 

1.1 

1.6 

2.3

0.9 

1.3 

1.9

2.7

1.2 

1.6

1.0 

1.5 

2.3 

3.4

1.3 

1.9 

2.8 

4.1

3.6 

5.6

2.1 

2.9 

4.0 

5.5

1.0 

1.2 

1.5 

1.8

1.2

1.4

1.5 

2.2

1.5

1.6

1.7 

2.31

1.8 

2.4 

3.1 

4.2

1.4 

2.2 

3.4 

5.1

1.8

1.9

2.0 

3.2

Velocity 

(fps)

2.2 

2.6 

3.1 

3.9

1.6 

2.0 

2.5 

3.3

2.6 

3.3 

3.8 

4.6

1.7 

2.1 

2.6 

3.2

1.5

2 0

2.6

3.3

1.8 

2.1

2.3 

2.8 

3.5 

4.3

1.6 

2.0 

2.6 

3.5

l.S 

l.H 

1.9 

2.1

2.0 

2.3 

2.7 

3.2

2.7 

3.7 

5.1

3.1 

4.1

5.4

1.7 

2.0 

2.3 

3.3

2.0 

2.4 

2.9 

3.4

1.9 

2.2 

2.6 

3.1

1.5 

2.0 

2.7 

2.9

1.3 

1.8 ! 2.5 

3.6

1.7 

2.1 

2.7

1.0 

1.4 

2.0 

2.8

2.0 

2.2 

2.8 

3.8

1.3 

1.7 

2.3 

3.2

1.6 

2.4 

3.5

1.9 

2.3 

2.8 

3.4

1.6 

2.2 

2.8 

3.4

Suspended 

load (tons 

per day)

3,100

13,000

46,000

155,000

120

500

3,100

22,000

3,000

10,000

30,000

80,000

400

1,400

5,500

21,000

150

650

4,800

26.000

1,000

3,900

8,000

26,000

90,000

320,000

1,200

3,400

9,000

24,000

2,600

6,000

13.200

30,000

1,700

6,500

28,000

110.000

15,000

90,000

400,000

1, 600, 000

57,000

140,000

360,000

1,000,000

200

1,000

4,500

15,000

1,350

6,100

15,100

29,200

1,620

4,700

12,400

32, 400

940

2,970

12,700

43, 700

300

2,600

14, 000

150

3,700

20,000

70

800

9,000

30,000

2,430

5,700

12,200

26,500

Rate of increase with discharge (O)

0.50

.32

.59

.36

.10

.10

.26

.37

.14

.06

.14

.22

.25

0.25

.13

.59

.16

.46

.38

.06

.50

0.25

.28

.30

Remarks

For Q belov 800 cfs.

For 0 belovr 700 cfs.

.16

.31

.25

.41

.25

.26

.28

.39

For O belov^ 400 cfs.

For O above 200 cfs.

For O above 400 cfs.

Curves poorly defined.

For Q above 1,000 cfs.

For Q below 1,000 cfs.
INDEX

Acknowledgments. _ _ 

At a station, defined.

Ill

4

B

Bed configuration, in relation to roughness_________.________________________ 40-42

Belle Fourche River._________________________________________________ 15

Bighorn River and tributaries, data._ ___________________________________ 11

Blench, Thomas, quoted...______________________________________________ 45

C

Canals, stable, similarity to graded streams.._______________________________ 43-48

slope of _----________-._.____________._______._____.____________ 45

Change of discharge downstream, defined.—________________ _ _ __________ 4

Channel control, defined_________________________________________ 8

Channel shape, relation to width-depth ratio.______________________________ 9

Channel shape factors, interrelations at given discharge________.____________ 28-30

Channel-shape adjustment, flood in Colorado River..______________________33,34,37

flood in Rio Grande. __________________________________________________ 33,34,37

flood in San Juan River. _________________________________________ 30-33,34,37

. Colorado River, adjustment of channel during flood _______________________ 33,34,37

suspended-load measurements _____________________________________ 30

Current-meter data...______________________________________________________ 5

Curve, cumulative frequency_____________________________________________ 2

flow-duration. See Curve, cumulative frequency.

D

Das, Ishar, quoted.... __ ______________________________ _ ________________

Davis, W. M., quoted______-_-__________________________ _ ________________

Depth, defined-- _. ________________________________________________________

Discharge, cumulative frequency _----___-____________________-..-__________

daily mean.. _ ____ _ ________________________________________________

frequency. _ _ _ .----____-_----______________________ _ ______________

given, in relation to channel-shape factors _ ___________________________ 23-24

in relation to depth (equation 2)_________-_______________________. ______ 8

in relation to suspended load at cross section _ ____ _ __________________ 19-21

in relation to suspended load in a downstream direction. ________________ 21-23

in relation to velocity (equations) 

in relation to wetted perimeter _ - 

in relation to width (equation !)_._ 

mean annual... _ __..._.________..

median. _-______----.-_________-___

records, method of selection _ _ ____

--______-__---____-__-____-.___ 8, 14

_____________ _ ______________ 43

_______________________________ 8

_____.___________....__________ 3

.___________-_._.______________ 3

_______________________________ 8

variable, in relation to channel-shape factors ____________________________ 24-28

Einstein equation.

Equation !.___.__.

2___ -_-___-__.

E

. — — _-___-._-.______ — .__.__._____-__._______.. 29

. ——— —— — - ——— — ——— —————— —— ————— 16

.________...-________________.__..._______.___._____ 16,43

3___-__---._-_._________----__________-_______-__.__.______-__-_________ 16,43

4- — — — —— — — — — —— — —— — — — ———— — — ————— — 35

5.______...--_______________-_._____._.________...._______.___________._ 36

Equations determining longitudinal profile of streams. _____________________ 50-51

Equilibrium, in channels, requisites for._____-..___________---____._-_______ 51

summarized ________.-----._______.___________..._________..______ 51

Erodibility of bank and bed. _________________._________..._______.._ 51

Exponents, b,f, m, average values...........................'........_...... 9,16

Flumes, experimentalFrench Broad River_..

G

Gage height. See Stage. 

Gaging station, locationof_-.________________

Gilbert, G. K., data on experimental flumes. 

Glossary of terms used____.________.._._._

Griffith, W. M., quoted......._..._-___..___

28 

IV

29

H-K 

Hoover Dam, effect on channel shape.______.--__.________..._____________ 37-39

In a downstream direction, defined. —— — ___--_ — ___________________ — — ___ 4

Kansas River system, data.--------_--_____.-_________--_.___-___--_____. 12

Page 

Lacey, Gerald, equations quoted...........---..____ ———_— — _ —— —— 43

Lane, E. W., quoted.----- — --. — -...----------------------.--—— 29

Load, bed, defined__---------__......._--_ — _-_-- — --__-------- —— —. 19

in relation to roughness._..__________-_______.____------_-___.. 48

measurements _-_—. —. — ____-__ — -- — — — -- — -- — — -—-—.. 29

significance-----. — ._—— — — ___________________-___- — - ——_ — — . 30

suspended, defined.__..-_____-------_.-- — --.--.----.--.- ——— 19

effect on channel roughness.____________________ ___—_—__—__—_ 36

in relation to roughness__..-.....--------..-.-.-- — --_..._... 36-42,48

method of sampling....__._. — ---.__________-__----_-- — ____--. 21

Loup River__..........,-. — _-..._________________________________ —. 15

M

Mackin, J.H., quoted———————— ——— ———— — — 29,33-34,45,46,47,48,51

Madras irrigation canals.-.. —...______-_--- — — --- — ._. —..._...... 15

Manning equation (equation 4)__.._.__.-_„ — . — —_ — _ — — -—— — _ — __ 14,35,48 

Mannhig roughness factor— — -. — -—- — — — — — — — __-_ — — — .—... 40,41

Maumee River basin....- — — —.—.. _ — _—...— ——— —___._ —.____—___- 16-18

Mean depth. See Depth.

Mean velocity. See Velocity.

Middle Loup River, bed-load measurements-.__________ — ... —— _._... 30

Mississippi River, lower, data—_____—-—. — _.__-- — ---_-______ — . 13

main stem_. ___..___——_.---. ——...--__-._________________ — — — 15

Missouri River, main trunk, data-.-.... ——— ________________ ........... 13

N

Niobrara River, bed-load measurementsPlan of report _________ —— _____

Powder River, data on tributaries _ 

Purpose of report....—— ———— —

29

R

Regime channels, defined—. — -.... —...-. -. — — _______-_.-____ —....... 43

Republican and Kansas Rivers—-- — — — — -____-_____ — — - — — — — ______ 15

Rio Grande, adjustment of channel during flood.— ._...__...._____. 33,34,37

Roughness, defined for this report.._———.. —_——___-____—_-_ — ——_... 40

factors affecting__ — _. —...—__________________ — _ — — — .________ 40

in relation to bed configuration._....——....__ ———.....„.—__.-_ 40-42

in relation to particle size.. ———_ — _ —--- — - — - — — — - — -- — -..._ 48 

in relation to sand sizes.-----.—--------——____________-----------_____ 40,42

in relation to suspended load. _._—______---_- — _--_-__-___.__ ——.... 36-40,42

Roughness factor. See Manning equation.

Rubey, W. W., cited. ———————————— — — —— - — — —— -— 46

San Juan River, adjustment of channel during flood ______ — _ — ..__ — 30-33,

Scioto River basin___.. ———. ——.—————... — .____ — ___ — _ ——....

Sediment rating curve, defined..........__.__-_____________---. ———..____

Sedimentation, of Lake Mead-.-----—-...-—--------.-.-.------- — --- ——.

Shear, characteristics.._...... ——...—....---------________-____-----. —..

Slope, effects summarized-_________________-_-__.-- — -. — -- — - —— —..

relation to velocity and depth in a downstream direction.-... — ........

Stage, defined______.. —————..... —.. —..— —.——_____.__.. —..

Streams, graded defined__—————._ —.. — ——... ———.. —.———..

method of identifying_.__ — __ — — - — — . —...--_- — _.. ——.

slope of.-..._-.--__----------------------- —— — — _____— — --

profiles, eight equations for determining longitudinal —- — — _ — — — .

in equilibrium and all others.---.--—-----------_--------- —._--

Summary and interpretation. __---.- — _— — __ —_______ — ---------- ———

Symbols explained._______ _ ___ _ ____________________ — — —... —.___

T-Y

Tombigbee River_____———. ——.—— —... 

Velocity.--.-----—_...—... ——— —— 

Yellowstone and Bighorn Rivers__————-— — 

Yuma, Ariz., gaging station. See Hoover Dam.

34,37

16-18

20

30

51

52

49-51 

4

45,51 

47 

45

50-51 

50

51-52 

IV

57

o