GEOLOGICAL SURVEY PROFESSIONAL PAPER 1288

An Analytical Treatment of 

Channel-Morphology Relations

By W. R. OSTERKAMP, L. J. LANE, and G. R. FOSTER

GEOLOGICAL SURVEY PROFESSIONAL PAPER 1288

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1983
UNITED STATES DEPARTMENT OF THE INTERIOR 

WILLIAM P. CLARK, Secretary

GEOLOGICAL SURVEY 

Dallas L. Peck, Director

First printing 1983 

Second printing 1984

Library of Congress Cataloging in Publication Data

Osterkamp, W, R.

An analytical treatment of channel-morphology relations.

(Geological Survey Professional paper ; 1288)

Bibliography: p.

1. Channels (Hydraulic engineering) 2. Stream measurements. I. Lane, L. J. II. Foster, G. R. III. Title. IV. 

Series.

TC175.088 1983 551.3'5 83-600312

For sale by the Distribution Branchf U.S. Geological Survey, 

604 South Pickett Street, Alexandria, VA 22304
CONTENTS

Page

Abstract ___________________________ 1

Introduction ______________________________ 1

Previous investigations ___________________________ 2

Development of equations ________________________ 4

Assumptions_________________________ 4

Procedure _________________________________ 5

Calibration ____________________________________ 8

Empirically developed relations __________________ 8

Calibration results ___________________________ 9

Sensitivity to assumed values of z ___________________________ 12

Applications _________________________________ 13

Channel adjustment _______________________ 13

Width-discharge data from stable natural channels _________ 17

Discussion ____________________________________ 18

Conclusions ________________________________ 19

References cited ___________________________ 20

Appendix_________________________________ _ 21

ILLUSTRATIONS

FIGURE 1. Graph showing variation of normalized shear stress with proportional distance along half the perimeters of hypothetical

alluvial channels _________________________________________________________ 6

2. Graph showing variation of normalized maximum shear stress at the midline of a rectangular channel _____________ 7 

3-12. Graphs showing width-discharge data and power relations for groups of channels:

3. W/D < 8.0 __________________________________________________________ 10

4. 9.0< WID< 11.0 ______________________________________________________ 10

5. 14.0 < W/D < 18.0 ________________________________________________________ 10

6. 18.0 < W/D < 22.0 ________________________________________________________ 10

7. 22.0 < W/D < 27.0 ________________________________________________________ 10

8. 27.0 < W/D < 33.0 ________________________________________________________ 10

9. 36.0 < W/D < 44.0 _________________________________________________________ 11

10. 45.0 < W/D < 55.0 ________________________________________________________ 11

11. 56.0 < W/D < 80.0 ________________________________________________________ 11

12. 80.0 < WID< 140 _________________________________________________________ 11

13. Width-discharge relations for groups of stream channels with similar width-depth ratios ____________________ 13

14. Graph showing the variation of the width exponent, b, with width-depth ratio __________________________ 14

15. Graph showing change ofb,f,m, z, and y with change of width-depth ratio ___________________________ 14

16. Graph showing change of 6, f, m, and y, with variation in values of z for selected width-depth ratios ______________ 14 

17-22. Graphs showing relations of width to discharge for channels of the Western United States:

17. Graph showing relation of width to discharge for 14.0 < W/D < 18.0 ___________________________ 15

18. Graph showing relation of width to discharge for 18.0 < W/D < 22.Q ___________________________ 15

19. Graph showing relation of width to discharge for 22.0 < W/D < 27.0 ___________________________ 15

20. Graph showing relation of width to discharge for 27.0 < W/D < 33.0 ___________________________ 15

21. Graph showing relation of width to discharge for 36.0 < W/D < 44.0 ___________________________ 15

22. Graph showing relation of width to discharge for 45.0 < W/D < 55.0 ___________________________ 15

in
IV CONTENTS

TABLES

TABLE 1. Comparison of theoretically developed power-function exponents proposed by investigators 3

2. Power relations of width to discharge developed from field data of channels in the Western United States and grouped by

width-depth ratios ___ ^

3. Calculated and assumed values of d, jt through j6, b,f, m, z, and y, for width-depth ratios of 5 to 1,000 12

4. Geometry-discharge data from natural stream channels with stable discharge 16

SYMBOLS

C Coefficient to hydraulic radius in the Chezy equation

D Mean channel depth

G Channel gradient

P Wetted perimeter of a channel section

Q Mean discharge of the water-sediment mixture

Qi Instantaneous discharge of the water-sediment mixture

R Hydraulic radius

SC Silt-clay content of channel material

V Mean velocity of the water-sediment mixture

W Channel width

a Coefficient to discharge in the power relation between channel 

width and mean discharge

a' Coefficient to the independent variables in the power relation 

between channel width and channel gradient, the Manning 

roughness coefficient, and mean discharge

b Exponent to discharge in the power relation between channel 

width and mean discharge

c Coefficient to discharge in the power relation between mean 

channel depth and mean discharge

d Coefficient to the independent variables in the power relation 

between mean channel depth and channel gradient, the Manning roughness coefficient, and mean discharge

d Exponent to proportional distance in the assumed power relation between normalized shear stress and proportional 

distance along the wetted perimeter of a channel selection

(ho Sediment particle size at which 50 percent of the distribution is 

finer

/ Exponent to discharge in the power relation between mean 

channel depth and mean discharge

ji Simplified exponent, equal to (3d+3)/(8d+3), to a unitsconversion constant, to the Manning roughness coefficient, 

and to mean discharge; in the power relation between channel 

width and a units-conversion constant, the product of a shearstress coefficient and the specific weight of a water-sediment 

mixture, critical shear stress, channel gradient, the Manning 

roughness coefficient, and mean discharge

j2 Simplified exponent, equal to 5/(8d+3), to the product of a 

shear-stress coefficient and the specific weight of a watersediment mixture and to critical shear stress; in the power 

relation between channel width and a units-conversion constant, the product of a shear-stress coefficient and the specific 

weight of a water-sediment mixture, critical shear stress, 

channel gradient, the Manning roughness coefficient, and 

mean discharge

j3 Simplified exponent, equal to (7-3rf)/(16rf+6), to gradient in the 

power relation between channel width and a units-conversion 

constant, the product of a shear-stress coefficient and the 

specific weight of a water-sediment mixture, critical shear 

stress, channel gradient, the Manning roughness coefficient, 

and mean discharge

j4 Simplified exponent, equal to 3d/(8d+3), to a units-conversion 

constant, to the Manning roughness coefficient, and to mean 

discharge; in the power relation between mean channel depth 

and a units-conversion constant, the product of a shear-stress 

coefficient and the specific weight of a water-sediment mixture, critical shear stress, channel gradient, the Manning 

roughness coefficient, and mean discharge

js Simplified exponent, equal to 3/(8d+3), to the product of 

a shear-stress coefficient and the specific weight of a watersediment mixture, and to critical shear stress; in the power 

relation between mean channel depth and a units-conversion 

constant, the product of a shear-stress coefficient and the 

specific weight of a water-sediment mixture, critical shear 

stress, channel gradient, the Manning roughness coefficient, 

and mean discharge

js Simplified exponent, equal to (3d+3)/(16d+6), to gradient in the 

power relation between mean channel depth and a unitsconversion constant, the product of a shear-stress coefficient 

and the specific weight of a water-sediment mixture, critical 

shear stress, channel gradient, the Manning roughness coefficient, and mean discharge

k Coefficient to discharge in the power relation between mean 

velocity of the water-sediment mixture and mean discharge

k! Simplified coefficient, equal to 1.0, that is the product of the 

coefficients a, c, and k in the power relations of width, depth, 

and velocity with mean discharge

m Exponent to discharge in the power relation between mean 

velocity of the water-sediment mixture and mean discharge

n Roughness coefficient of the Manning equation

p Coefficient to proportional distance in the assumed power 

relation between normalized shear stress and proportional 

distance along the wetted perimeter of a channel section

r Coefficient to discharge in the power relation between the 

roughness coefficient of the Manning equation and mean 

discharge

t Coefficient to discharge in the power relation between channel 

gradient and mean discharge

x Distance along the wetted perimeter of a channel section

x* Proportional distance, equal to x/P

y Exponent to discharge in the power relation between the roughness coefficient of the Manning equation and mean discharge

2 Exponent to discharge in the power relation between the 

channel gradient and mean discharge

7 Specific weight of the water-sediment mixture

? Average shear stress, the product of the specific weight of the 

water-sediment mixture, channel gradient, and mean channel 

depth

Tm Maximum bed shear stress at the center of a channel section

TC Critical shear stress

$ Units-conversion constant in the Manning equation
AN ANALYTICAL TREATMENT OF CHANNEL-MORPHOLOGY

RELATIONS

By W. R. OSTERKAMP, L. J. LANE l , and G. R. FOSTER

ABSTRACT

For a specified flow rate, the properties of channel width, mean 

depth, mean flow velocity, gradient, and roughness often are related to 

discharge by empirically developed power functions. Equations were 

derived that provide an analytical, or semitheoretical, basis for the empirical relations. The equations, which were calibrated and tested using 

field data, differ from previously derived power functions by incorporating variable exponents dependent on the shear-stress distribution 

along the channel section. Variable exponents permit the consideration 

of the entire range of natural channel geometries, including those of 

very wide braided channels that otherwise could not be described adequately by power functions. The derivation of these exponents is based 

on the continuity equation, the Manning equation, and an assumed 

equation for shear-stress distribution expressed in terms of channel 

width-depth ratio. By this approach, width-depth ratios are employed 

as surrogates for the channel sediment characteristics and the shearstress distribution.

The equations were calibrated using channel data from the western 

half of the United States. The calibrated power equations were tested 

using data from relatively stable channels of the United States-in 

Ohio and southern Missouri-New Guinea, and Australia. Stable channels with steady discharge characteristics were selected for the testing 

to approximate the steady-state discharges assumed by the derivations. Results of the calibration and tests lead to generalizations concerning channel dynamics.

INTRODUCTION

The ancients observed that channels are molded by the 

water and sediment passing through them. In recent 

times attempts to quantify the channel changes that 

result from varying conditions of discharge have been 

made using concepts such as regime theory, hydraulic 

and channel geometry, and dynamic equilibrium.

Most efforts at quantification have been empirical and 

involve the use of power functions or similar simple 

equations to approximate the manner in which channel 

characteristics vary with discharge. The two sets of relations commonly developed in geomorphic studies have 

been (1) the variation of geometric and hydraulic 

variables (such as water surface or channel width and 

mean depth) with discharge or stage at a particular fixed 

channel cross section and (2) the variation of hydraulic 

(velocity, roughness) or geometric (width, depth, gradient) properties with discharge at different locations 

along a channel, measured on the basis of a common

1 U.S. Department of Agriculture

hydraulic or geomorphic reference level. The first set of 

relations generally is termed the "at-a-station" case, and 

the second, to which attention is limited in this paper, is 

the "downstream" case (Leopold and Maddock, 1953, p. 

4). The basic mathematics and assumptions on which the 

empirical studies depend- are given in numerous papers 

on the topic and, therefore, are given here in summary 

form only. Many of the papers cited here, however, provide background explanations of the hydraulic- and 

channel-geometry methods.

This paper provides a semitheoretical basis for the use 

of power functions with variable exponents to describe 

the shapes of natural alluvial stream channels. Many of 

the empirical studies have defined the "downstream" 

changes of channel width, mean depth, and mean flow 

velocity as power functions of a discharge characteristic; the exponent values for each of the three power 

functions commonly have fallen in a limited range. As a 

result, approaches to definition of theoretical exponent 

values for the three power functions have concentrated 

on yielding single values representative of the limited 

range of values that the empirical studies typically provide. Variation in the exponent values for these empirical power functions does occur, however, and it cannot be reasonably assumed that single values of the exponents describe the range of conditions found in 

natural alluvial stream channels. The wide ranges of 

possible exponent values become apparent when the 

geometries of unusually wide or narrow channels are 

considered. Unlike previous theoretical or semitheoretical analyses of geometry-discharge relations, therefore, 

the present paper treats the entire range of geometries 

found in natural alluvial channels and proposes power 

functions with variable exponents partially based on 

theoretical considerations.

The mathematical approach of this paper is termed 

"analytical" instead of theoretical because the equations 

were developed, in part, from empirical relations and 

were calibrated using field data. The geometry data 

used for the calibration, collected from the western half 

of the United States, are summarized in an empirical 

study proposing variable geometry-discharge relations 

(Osterkamp and Hedman, 1982); this study is the basis
ANALYTICAL TREATMENT OF CHANNEL-MORPHOLOGY RELATIONS

for the present analytical approach. The Western 

United States data, separated into groups for calibration (depending on channel width-depth ratio), yielded 

width-discharge power functions with exponents that increase as width-depth ratios increase. For some groups, 

however, the data were insufficient to define with confidence a unique empirical power relation. Consequently, 

the analytical power functions of this paper, calibrated 

with the field data from the Western United States, are 

supported further with field data from stable stream 

channels of southern Missouri and Ohio (in the United 

States), New Guinea, and Australia.

PREVIOUS INVESTIGATIONS

Hydraulic geometry, as defined by Leopold and Maddock (1953), is the description of the graphical relations 

between plot points of hydraulic characteristics. These 

characteristics include width and mean depth, measured 

at or from the water surface. Channel-geometry investigations evolved from the hydraulic-geometry concept as a method of indirectly estimating discharge 

characteristics at ungaged channel sites. In the downstream case, the channel-geometry technique differs 

from that of hydraulic geometry in that some channel 

properties are defined by measurements taken from a 

recognizable geomorphic reference level rather than 

relative to a particular stage. Because hydraulicgeometry data can be collected only at prevailing flow 

rates and stream stages, the downstream relations for a 

specified flow duration ordinarily must be determined 

from graphs of the at-a-site changes of width and depth 

with discharge. By the channel-geometry technique, the 

various geomorphic reference levels for most perennial 

stream channels can be related directly to limited ranges 

of flow duration or flood frequency. Thus, width data 

measured from the active-channel reference level 

(Osterkamp and Hedman, 1977, p. 256), which commonly approximates a stage representing mean discharge 

and a flow duration of roughly 15 percent, are typically 

related to values of mean discharge. Geometry data collected at the flood-plain level might be related to flood 

discharges with recurrence intervals of 2 years.

This difference between the hydraulic-geometry and 

channel-geometry techniques enhances the comparability of data from diverse sites, but normally requires that 

a relation for mean velocity (V) must be developed indirectly. In this paper, width (W), mean-depth (D), and 

gradient (G) data are referred to the active-channel 

reference level, which, for perennial and intermittent 

streams, is coincident with the lower limit of perennial 

vegetation. The resistance to flow of an active-channel 

section is treated as a variable that decreases in the 

downstream direction (Leopold and Maddock, 1953, p. 

27) and is characterized by the roughness coefficient, n,

of the Manning equation. The power equations typically 

defined by empirical studies of channel geometry are the 

following:

W=aQ» 

D = cQf 

V=kQm 

G = tQ* 

n=rQy

(1)

(2)

(3)

(4)

(5)

where a, c, k, t, and r are coefficients, and 6, / ra, z, and 

y are exponents. Because discharge (Q) is the product of 

width, mean depth, and mean velocity (the continuity 

equation for streamflow), the product of a, c, and k and 

the sum of b, f, and m must both equal unity (Leopold 

and Maddock, 1953).

A basic goal of most empirical hydraulic- and channelgeometry studies has been evaluation of the exponents 

of equations 1 through 5, particularly those of the first 

three: b, f, and m. Results have varied, but generally 

values have fallen in the range of 0.45 to 0.55 for b, 0.3 

to 0.4 for/, and 0 to 0.2 for m. A wide range of empirical 

values for z, mostly -0.1 to -0.5, have been reported. 

The value of y probably has received the least attention 

but is commonly thought to range from about 0 to 

-0.15.

There are several possible causes of the inconsistent 

results for identifying the exponents of equations 1 

through 5. Two of the principal causes are (1) the use of 

regional or local data sets that represent limited ranges 

of the variables that possibly complicate the geometrydischarge relations, and (2) the specific lack of a method, 

in most studies, to discriminate variability of bank 

cohesiveness or stability. In the analytical treatment of 

this paper, the characteristics of bed and bank sediment 

are not considered directly but are included in the 

derivation as width-depth ratios. It is assumed that for 

any set of size characteristics for sediment forming the 

bed and banks of a channel, a characteristic width-depth 

ratio for relatively stable or adjusted channel conditions 

can be identified. Through the remainder of this paper, 

therefore, consideration of width-depth ratios implies 

specified conditions of sediment forming the channel 

perimeter.

The likelihood that the size distribution and cohesiveness of material forming the channel perimeter affects 

the geometry-discharge relations has been recognized 

repeatedly (as examples, see Leopold and Maddock, 

1953; Schumm, 1960; Wolman and Brush, 1961; 

Knighton, 1974; Smith, 1974; Hedman and Kastner, 

1977; Williams, 1978; and Mosley, 1981). Nevertheless, 

this possible cause of variation has been neglected in 

most empirical studies, and a single value for each exponent generally was reported. The use of single values 

causes little difficulty for the majority of alluvial chan-
PREVIOUS INVESTIGATIONS

nels that are moderately cohesive, but empirical data, as 

well as theoretical considerations, suggest that use of 

single values leads to substantial error for channels 

formed of either highly cohesive or noncohesive material 

(Schumm, 1960; Knighton, 1974; and Williams, 1978).

The main purpose of this paper is to propose a derivation of the width, depth, and velocity exponents, b,f, and 

ra, and to suggest that the value of each varies with the 

material forming the channel perimeter. The approach 

used in this paper extends the utility of several previous 

derivations by demonstrating that ranges of values for 

b, f, and m are justified on other than a purely empirical 

basis. Thus, the relations presented here can be used to 

anticipate the geometry resulting from specified conditions of water and sediment discharge or, conversely, 

for estimating discharge characteristics if channel conditions are known.

Substantial literature concerning the various power 

functions (eqs. 1-5) has accumulated. Most of the papers 

provide empirical evaluations of one or more of the exponents, but they are too numerous to provide a 

thorough discussion here. Among those that either are 

generally regarded as particularly significant or are 

especially pertinent to the present study are papers by 

Kennedy (1895), Lindley (1919), Lacey (1930), Lane 

(1937, 1957), Leopold and Haddock (1953), Wolman 

(1955), Blench (1957), Schumm (1960), Brush (1961), 

Hedman and Kastner (1977), Osterkamp (1978), and 

Osterkamp and Hedman (1982). A comprehensive list of 

references pertaining to hydraulic and channel 

geometry is presented by the Task Committee of the 

American Society of Civil Engineers (1982) on relations 

between morphology of small streams and sediment 

yields.

A variety of approaches has been used to replicate 

theoretically or semitheoretically the experimental 

geometry-discharge relations. Although the various 

methods use a wide range of equations and assumptions 

to define channel morphology, they all rely on the continuity equation (Q = WDV) and one or more additional 

relations to yield simultaneous solutions. Most approaches assume that the variables in the continuity 

equation can be expressed as power functions (eqs. 1-3). 

The assumption seems justified for the ranges of data 

normally considered, but is poorly founded for very 

small discharges (Osterkamp and Hedman, 1982). Use of 

the assumption, however, provides a second equation 

because the expansion of the continuity equation to 

power form requires that the sum of the exponents equal 

unity:

6+/+m=l (6)

Table 1 provides comparisons of the theoretically 

developed power-function exponents (eqs. 1 to 5) proTABLE I.-Comparison of theoretically developed power-function exponents proposed by investigators

[Dashed entry means a value is not available; some values have been rounded to two significant figures]

Investigator

Width

exponent

6

Depth 

exponent

Velocity Gradient Roughness 

exponent exponent exponent

Leopold and

Langbein (1962) _ 0.55 0.36 0.09 -0.74 -0.22 

Langbein (1964) __ .53 .37 .10 -.73 -.45 

TouKuo-jen(1964)_ .56 .33 .11 

Engelund and

Hansen (1967) __ .52 .32 .16 -.21 

Brebner and

Wilson (1967)_ .47 .35 .18 -.12 

Li Ruh-ming (1974) .46 .46 .08 -.46 

Smith (1974) ___ .64 .27 .09 -.18 

Parker (1979) __ .50 .42 .08 -.41 

Lane and Foster

(1980)1 _____ .46 .46 .08 

1 Triangular cross section.

posed by Leopold and Langbein (1962), Langbein (1964), 

Tou Kuo-jen (1964), Engelund and Hansen (1967), 

Brebner and Wilson (1967), Li Ruh-ming (1974), Smith 

(1974), Parker (1979), and Lane and Foster (1980). In 

general, the values should be compared with caution because the assumptions, characteristic discharge, and 

type channel that led to the exponents are variable. 

Despite these problems, the values for b, the width exponent, range only from 0.46 to 0.64. The/and m values, 

for depth and velocity, derived from various theoretical 

approaches, also show relative consistency (table 1) and 

agree well with the majority of the empirically developed power functions. The values of z and y, for channel 

gradient and roughness, however, show substantial 

ranges in table 1. These exponents apparently are much 

more sensitive to the assumptions of a derivation, particularly the water- and sediment-flow characteristics 

used, than are the width, depth, and velocity exponents. 

A large range of empirical values for z, for example, has 

been reported. The smallest values, as low as 0.0, 

generally have been associated with low discharge rates, 

as represented by rill erosion (Lane and Foster, 1980) 

and controlled inflow in small channels (Ackers and 

Charlton, 1971). Values of z as great as -1.07 have been 

related to stable armored channels, such as Brandywine 

Creek in Pennsylvania, at the bankfull stage (Wolman, 

1955, p. 26).

The basic form of a time-dependent model that recognizes channel morphology as the result of all discharges 

transmitted by the channel was presented by Pickup and 

Rieger (1979). Although this model does not provide 

specific derivations of geometry power functions, it is 

pertinent to the present analysis because the model proposes that channel size and shape are not unique to a 

dominant discharge. Pickup and Rieger suggested that 

channel geometry tends to fluctuate about a mean condi-
4 ANALYTICAL TREATMENT OF CHANNEL-MORPHOLOGY RELATIONS

tion, and that the geometry is partly dependent on the 

recent flood history of the stream. This view is consistent with empirical data by Schumm and Lichty (1963), 

Burkham (1972), and Osterkamp and Hedman (1982).

DEVELOPMENT OF EQUATIONS

The previously cited investigators of theoretically 

developed power-function exponents employed a variety 

of techniques and assumptions to show that the empirically defined power equations for hydraulic and 

channel geometry are reasonably accurate. A nearly 

common deficiency of the previous analyses, however, 

has been a generalized treatment for all alluvial channels or else a specific treatment for a selected range of 

alluvial channels. In both cases the result has been single 

exponent values of limited utility. The derivations are 

well designed to describe geometry-discharge relations 

of stable, well-formed channels, but are poorly equipped 

to consider less common channel types, such as those of 

braided channels. The derivation presented here develops geometry-discharge relations applicable to the 

range of alluvial-channel conditions normally found in 

nature.

ASSUMPTIONS

All hydraulic- and channel-geometry relations are 

developed from the continuity equation for the instantaneous discharge (Q,) of the water-sediment mixture:

Qi=WDV, (7)

where D represents mean depth and V represents mean 

velocity. When numerous downstream sites are considered, the simplifying assumptions are made that (1) 

the rates of change of the logarithms of W, D, and V 

with the logarithm of Q, are constant (that is, plots of W, 

D, and V against Q, yield straight rather than curved 

lines on logarithmic coordinates), and (2) these rates of 

change can be expressed by a multiple power-function 

equation:

Qt =kQfQ/Qr, (8)

where the coefficient, K, is the product axcxk and is 

equal to 1.0, and the sum of the exponents, b, f, and m, 

also equals 1.0 (eqs. 1, 2, 3, and 6). This second assumption is reasonable only if Q, represents the same flow 

characteristic for all sites considered. In this paper, it is 

assumed that discharge, Q, is steady or varies through a 

small range so that Q,, Q, and mean discharge are essentially the same. Width and mean-depth values pertain to 

measurements taken at the water surface or the corresponding geomorphic channel feature at mean (steadystate) discharge. Gradient is measured at the channel 

section, and roughness is a complex function of many

factors including the channel geometry and the particlesize distribution of the material forming the channel 

perimeter (Limerinos, 1969, p. 6).

The derivation that follows results in expanded forms 

of equations 1 and 2, the power functions for width and 

depth. By the continuity equation, a power function for 

mean velocity (eq. 3) is also defined. Evaluation of the 

derived equations 1 and 2 must be made implicitly because the coefficients a and c incorporate a critical 

shear-stress term, as well as the coefficients t and r for 

the gradient (eq. 4) and roughness (eq. 5) relations. 

Calculation of the exponents of the width and depth relations requires values for the gradient and roughness exponents, z and y (eqs. 4 and 5). An iterative evaluation is 

possible by employing established (assumed) empirical 

width-discharge relations and an assumed relation between the roughness and depth exponents (y and./).

Other assumptions necessary to the variable-exponent 

derivation presented here are the following:

1. Flow-resistance equations (the Manning and Chezy 

eqs.) are accurate through the ranges of discharges 

and channel conditions considered.

2. Channel-roughness variation in the downstream directon for steady-state discharge (and a condition 

of constant particle-size distributions of the 

material forming the channel perimeter in the 

downstream direction) is dependent only on the 

geometry (width, mean depth, and gradient) of the 

channel. Bed-form roughness is assumed constant 

in the downstream direction; particle-size distributions of channel sediment are not necessarily 

assumed constant in the downstream direction.

3. For consideration of shear-stress distribution in a 

channel section, the channel width is sufficiently 

large relative to depth that hydraulic radius (R) can 

be treated as approximately equal to mean depth.

4. All channels are assumed to be rectangular in shape 

and are subject to a relation between maximum and 

mean shear stress of

Tm/T=l+(W/D)- (9)

where rm is the maximum bed shear stress at the 

center of the channel, and T is the average shear 

stress across the channel. This assigned equation 

approximates the relation between experimental 

shear-stress and geometry data of Chow (1959), 

Kartha and Leutheusser (1970), and Prasad and 

Alonso (1976). Justification for the use of equation 9 

is provided in a later section.

5. Particle sizes of the sediment forming the wetted 

perimeter of a rectangular channel section vary to a 

maximum at the channel center; these particle sizes 

are compatible with associated variations in the 

distributions of velocity and shear stress.
DEVELOPMENT OF EQUATIONS

6. An equilibrium channel geometry (neither aggrading 

nor degrading) occurs when the shear stress at the 

lower corners of the rectangular channel section 

equals the critical shear stress for the given 

channel-bank material. It is assumed, therefore, 

that for specified conditions of discharge and particle sizes of channel material, an equilibrium condition signifies a specific width-depth ratio. By designating discharge and geometry (width-depth ratio), 

the need to consider explicitly particle sizes of channel material is eliminated.

All of the above assumptions were made, either implicitly or explicitly, in one or more of the theoretical investigations listed in table 1. For example, the assumption of power relations with discharge generally is made 

for the geometry variables (eqs. 1 to 5). Power relations 

describing shear-stress distributions also were assumed 

by Rohlf and Meadows (1980) and by Lane and Foster 

(1980), although the forms of equations differ from that 

assumed here (eq. 9). Rectangularly shaped channels 

commonly are considered when analyzing shear-stress 

data; these channels are assumed in the models of 

Brebner and Wilson (1967) and Lane and Foster (1980). 

Empirical flow-resistance equations or coefficients are 

incorporated into most of the derivations summarized 

by table 1.

PROCEDURE

The power-function derivations presented here, one 

for width and one for depth, involve four steps: (1) combining the continuity and Manning equations into a 

single equation involving both width and depth; (2) solving the assumed shear-stress distribution equation of a 

channel section for depth as a function of width (or vice 

versa); (3) substituting the result into the combined 

continuity-Manning relation to obtain a multiple power 

function for downstream changes of width or depth; and 

(4) substituting the required exponent values for the 

gradient and roughness power functions iteratively into 

the multiple power-function equations for width and 

depth until results consistent with empirical gradient 

relations and the Chezy equation are obtained. In other 

words, downstream equations for width and depth are 

developed as functions of the discharge, critical shear 

stress, gradient, and roughness at a channel section. To 

express the equations in the form of equations 1 and 2, 

all terms except that for discharge are included in the 

coefficient. Thus, in the end (eqs. 1 and 2), discharge remains as the sole independent variable, but its value 

depends on the input values for shear-stress distribution 

and the gradient and roughness exponents of equations 

4 and 5.

Returning to the first step in greater detail, the 

discharge form of the Manning equation is simplified by

assuming that depth and hydraulic radius, R, of a channel section are nearly equal:

(10)

The value of the constant ^ depends on the system of 

units used (for metric units, ^=1.0; in U.S. customary 

units, \(/- 1.49); other symbols are as previously defined. 

Rearranging gives

W3'*D = (ID

In order to consider shear-stress distribution in the 

channel section, an assumed distribution is solved in 

terms of width and depth and then combined with equation 11. An assumed rectangular channel has steady discharge, a width W, and a depth D. The wetted 

perimeter, P, is then W+2D. Let x equal the distance 

from the water's edge to any point on the wetted 

perimeter up to the channel center. The proportional 

distance, as*, along the wetted perimeter is the ratio of 

x, the distance along the wetted perimeter, to the length 

of the wetted perimeter:

x. = xlP. (12)

The proportional distance, as,, thus ranges from 0 at the 

water's edge to 0.5 at the midline of the channel. The 

width-depth ratio of most natural stream channels exceeds 12, which permits the simplification that width 

nearly equals wetted perimeter.

The normalized shear stress, T<JC*)/T, at any point along 

the wetted perimeter is a function of the proportional 

distance:

T(X*)/T=F(X*). (13) 

If the function, F(x*) is assumed to be a power function:

where p is a coefficient, d is an exponent, and

In order to maintain a steady-state channel condition, 

the shear stresses along the banks must be less than, or 

equal to, the critical shear stress of the bank sediment, 

thereby assuring no bank erosion or channel widening. 

The shear stresses at points along the bed, however, 

must equal or exceed the critical shear stress for bed 

sediment in order that bed-load movement is maintained 

without causing aggradation. The transition between 

the cohesive material in the bank and the mobile particles on the bed occurs at the intersection of the vertical 

bank and the bed. At this point, which is equivalent to 

xf =D/W, the shear developed by the flow equals the 

critical shear, TC , regardless of the character and mor-
ANALYTICAL TREATMENT OF CHANNEL-MORPHOLOGY RELATIONS

phology of the bed material. Thus, at the intersection, 

substitution into equation 14 yields

rJyGD = p(D/W)d, (15)

where r equals yGD, and 7, the specific weight of the 

water-sediment mixture, is disregarded as being nearly 

equal to 1.0 grams per cubic centimeter. After solving 

equation 15 for depth and substituting the result into 

equation 11, equation 11 is solved for width:

(16)

3), and 

+ 6), equation 16 can be simplified to

W= \l/-Ji(py)i2Tc-JiGJmJiQii- (17)

W=

Letting j\ 

j3 = (7-3d)l(lQd

Similarly, expressing equation 15 in terms of width, 

substituting in equation 11 and solving for depth yields

D = (18)

where j4 = and

Having expressed equations 17 and 18 in terms of p 

and d, the coefficient and exponent, respectively, of the 

shear-stress power function (eq. 14), it is necessary to 

evaluate p and d before geometry relations can be computed. Because the shear-stress distribution is symmetrical about the channel center line (#, = 0.5), by 

definition:

1 f 05

(19)

Substitution for i{xt} from equation 14 and solving 

yields:

p = 2d(d + l). (20)

If rm is the maximum shear stress at the channel center 

(*. = 0.5),

(Tm/T)=p(0.5)«. (21)

Substituting for p from equation 20 and rearranging 

yields

d=(Tjr)-1.0. (22)

Considering the perimeter of half of a rectangular 

channel section, the normalized shear stress, T(X*)/T, 

must have a mean value of 1.0. Although specific to the 

width-depth ratios and limited by the assumptions made 

previously, representative normalized shear-stress 

distributions along the channel beds are shown in figure 

1. From equation 22, rjr must be equal to or greater 

than 1.0, and from analytical and experimental data on 

shear-stress distributions (Chow, 1959, p. 169; Replogle 

and Chow, 1966; Kartha and Leutheusser, 1970, p. 1475; 

and Prasad and Alonso, 1976), it should not range as 

high as 2.0. Data on the ratios between maximum and 

average shear stresses as functions of width-depth 

ratios up to 12.5 (from Chow, 1959, p. 169, Kartha and 

Leutheusser, 1970, and Prasad and Alonso, 1976) are 

plotted in figure 2. The data show that the normalized 

maximum shear stress has the greatest value when the 

width-depth ratio (for rectangular channels) is about 3.

0.1 0.2 0.3 

PROPORTIONAL DISTANCE, x*, ALONG CHANNEL BED (xlW)>

0.4 0.5

FIGURE 1. -Variation of normalized shear stress with proportional distance along half the perimeters of hypothetical alluvial channels.
DEVELOPMENT OF EQUATIONS

2.0r

1.5 -V6\ v(vi Kartha and Leutheusser (1970)

Assumed relation: (T m /r = 1+( W/0)-°

0 10 20 30 40 50 60 

WIDTH-DEPTH RATIO, W/D

FIGURE 2.-Variation of normalized maximum shear stress at the midline of a rectangular channel, with width-depth ratio (W/D) for 

analytical and experimental data, and for the assumed relation

Field data justifying equation 9 also have been published for channels with width-depth ratios exceeding 

12.5. For example, some of Bathurst's (1979) normalized 

shear-stress data for gravel-lined natural channel sections compare favorably with the curves of figure 2. 

Because the Bathurst data were collected from natural 

channels that do not necessarily conform to the assumptions of this paper, the results are variable across channel sections and are not plotted in figure 2.

With increasing values of width-depth ratio above 3.0, 

the normalized maximum shear stress decreases asymptotically toward 1.0. With decreasing values of widthdepth ratio below 3.0, the normalized maximum shear 

stress in other studies declines toward 0, whereas the 

assumed maximum shear stress in this paper approaches infinity. However, this portion of figure 2 is of 

minor consequence to the present discussion because the 

width-depth ratios of natural alluvial stream channels 

are usually much larger than 3.0 (for example, see data 

of Hedman and Kastner, 1977; and Osterkamp and Hedman, 1982). Thus, it is assumed that equation 9 leads to 

a reasonable description of the distribution of shear 

stresses across alluvial channels. From equations 9 and 

22, it follows that

d=(W/D)-° 60, (23)

which is a necessary relation to solve equations 17 and 

18. For the range of width-depth ratios expected in 

natural alluvial stream channels, values of d vary from 

nearly 0 to about 0.4, but values of d for most alluvial 

channels fall between 0.1 and 0.3 (corresponding to 

width-depth ratios of roughly 50 and 7, respectively).

In equations 17 and 18, $, p, y, and TC can be incorporated into the coefficients a and c of equations 1 and 2. 

Gradient (G) and roughness (n), however, vary with

width-discharge and depth-discharge relations, and 

these variables must be considered when evaluating the 

exponents 6 and/. In other words, channel gradient and 

roughness are two of the five (eqs. 1 to 5) acknowledged 

degrees of freedom of self-adjusting channels (Williams, 

1978, p. 2). 

By defining coefficients a' and d, respectively, equal to

simplified to

and

and \f/~J4py)~ s Tc s,

and

equations 17 and 18 are

KA^i (17a)

D = dG-J*n/ty*. (18a)

By inserting the discharge terms for gradient and 

roughness from equations 4 and 5, simplifying and defining a'iA-r'i and cT-vK, respectively, equal to a and c (eqs. 

1 and 2), equations 17a and 18a are converted to

(24)

D = CQ<W«4>. (25)

As suggested by the range of exponents shown in table 

1, an accepted theoretical value for z is not available. 

Empirical data from relatively stable alluvial channels 

(having well-formed banks), however, indicate a value 

for z of -0.25 regardless of the particle-size 

characteristics of the material forming the channel 

perimeter (Lane, 1957; Osterkamp, 1978). This exponent is based on data from natural channels over wide 

ranges of mean discharge and gradient; the data were 

grouped according to similar characteristics of the channel material. For the downstream condition of stable 

channel sections formed of unchanging particle sizes, 

therefore, z does not vary significantly from - 0.25. Instead, from one channel to another, differences in the 

gradient-discharge relation (eq. 4), due to differences in 

the particle-size distribution, are reflected in the coefficient, t (Osterkamp, 1978). For very wide, braided channels, however, both theoretical considerations (for example, Leopold and Langbein, 1962, p. 12; Langbein, 

1964, p. 309) and field evidence (Osterkamp and Hedman, 1982) suggest that z, as well as f, m, and y, must 

approach 0 as 6 approaches 1.0.

The bed-roughness exponent, y, is known, both from 

field investigations (Leopold and Haddock, 1953, p. 27; 

and Wolman, 1955, p. 27) and geometry relations (table 

1), to be a generally small negative number when particle sizes of channel material do not change in the downstream direction. Although influenced by the widthdepth ratio and its rate of change with discharge, increasing discharge in the downstream direction generally results in decreasing surface area of the channel per 

unit volume of discharge and stream length. In other 

words, discharge tends to increase in the downstream
8 ANALYTICAL TREATMENT OF CHANNEL-MORPHOLOGY RELATIONS

direction at a greater rate than does the wetted 

perimeter. Thus, the frictional losses per unit volume of 

discharge decrease downstream, and a negative value 

for y is required. If a channel is extremely wide relative 

to discharge (W/D > 1,000, for example), and depth and 

velocity do not increase measurably in the downstream 

direction (/"and m approach 0), the roughness exponent, 

y, also approaches 0. Note, however, that the exponents 

of equations 24 and 25 are defined as functions of widthdepth ratio (eq. 23). Implicit results in equations 24 and 

25 are that (1) particle sizes of channel material must 

change in the downstream direction in order to maintain 

the specified constant width-depth ratios, and (2) assumed z and y values based on unchanging particle sizes 

are not necessarily applicable.

The extent of the downstream change in bed roughness can be evaluated quantitatively in terms of depth 

from the Chezy equation:

P0.167

- , n(26)

where C is a coefficient known as Chezy C. If C is assumed constant and hydraulic radius is nearly equal to 

depth, the roughness and mean-depth terms are proportional. Substituting the discharge terms for depth from 

equation 2 and roughness from equation 5, and defining 

y as negative:

y=-0.167/. (27)

For channels of specified width-depth ratio (thereby 

known values of d and jt through j6) and by assuming a 

reasonable starting value for y, /in equation 25 can be 

evaluated (where z is designated and / equals 

J4-zj6 + yj4). From the calculated value of/and equation 

27, a new value of y is computed. If the y-value 

calculated from equation 27 differs significantly from 

the assumed value, the procedure is repeated using the 

newly computed y value as the assumed y value for a reevaluation of/in equation 25. In this manner, two or 

three iterations should be sufficient to provide an assumed y in equation 25 that is reasonably accurate by 

equation 27. After y has been determined, it is used to 

evaluate equation 24 for the width exponent, 6.

CALIBRATION

From equations 6, 24, 25, and 27, it is apparent that if 

any one of the five power-function exponents (b, / m, z, 

and y of eqs. 1 to 5) is known or can be reasonably assumed for channels of specified width-depth ratio, 

unique solutions for the other four exponents follow. For 

example, assume that the empirically developed value 

for z, -0.25, for channels of constant particle-size 

distribution is also appropriate for channels of specified 

width-depth ratio. From equation 23, d for any channel

with a width-depth ratio of 30 equals 0.13; from equations 17 and 18, values of j, through j6 , respectively, are 

0.84,1.24, 0.82, 0.10, 0.74, and 0.79. (Values of j2 and js, 

are required only if the coefficients a and c are calculated by eqs. 17 and 18.) Using equations 25, 27, and 24 

as previously described, values of b, / and y are, respectively, 0.59, 0.29, and - 0.05. From equation 6, m is 0.12. 

Similar computations, all with an assumed z-value of 

- 0.25, yield a 6-value of 0.53 for channels with a widthdepth ratio of 5.0 and 0.60 for channels with a widthdepth ratio of 50. If an assumed z-value of 0 is applied to 

braided channels with width-depth ratios of 1,000, the 

calculated values of b, / and m, respectively, are 0.97, 

0.02, and 0.01.

The analytical results using a value for z of - 0.25 (or 0 

for braided channels) are consistent with empirical 

results for channels of relatively large width-depth 

ratios, but appear to overestimate b for deep, narrow 

channels. For the other exponents, field data usually are 

adequate only to define the exponent for the width-discharge relation. Channel depths generally are more 

difficult to measure with repeatable precision than are 

widths, and velocity and roughness data generally are 

either unavailable or have been estimated. Thus, empirically developed values of b are selected as the means 

to calibrate the other exponents of the channelgeometry relations.

EMPIRICALLY DEVELOPED RELATIONS

Several recent studies in the Western United States 

(Osterkamp, 1980; Hedman and Osterkamp, 1982; 

Osterkamp and Hedman, 1982) provide empirically 

developed 6-values for groups of stable channels with 

similar bed- and bank-sediment characteristics. The 

values range from 0.47 for channels formed largely of 

silt and clay to 0.62 for channels with sandy beds and 

banks; channels armored with gravel, cobbles, and 

boulders generally have 6-values ranging from 0.51 to 

0.55. Data from a small group of braided channels in the 

Sand Hills area of Nebraska yielded a 6-value of 1.0 

(Osterkamp, 1980, p. 193). Because large ranges of 

mean discharge are represented in each of the channelsediment groups, characteristic width-depth ratios cannot be identified for the various groups. Mean values of 

width-depth ratio, however, are smallest for the silt-clay 

channels and increase progressively as the value of b increases. Because the results described above represent 

conditions of unchanging channel-sediment properties 

rather than unchanging width-depth ratios in the 

downstream direction, these results cannot be used 

directly to calibrate the equations presented here. Like 

the empirical results for the gradient-discharge exponent, however, these results provide a basis for
CALIBRATION

evaluating the reasonableness of any assumed or 

calculated values of b and 2.

To construct a curve relating the width exponent, b, to 

width-depth ratio, geometry and discharge data from 

several hundred gage sites of the western half of the 

United States (Osterkamp and others, 1982) were divided into 10 groups based on width-depth ratio; 318 

channels were represented in groups: (1) WAD < 8.0, (2) 

9.0< W/D<11.0, (3) 14.0< W/D<18.0, (4) 

18.0<W/D<22.0, (5) 22.0<W/D<27.0, (6) 

27.0<W/D<33.0, (7) 36.0 < WID < 44.0, (8) 

45.0 <WID< 55.0, (9) 56.0 <W/D< 80.0, and (10) 

80.0 <WID< 14.0.

Data for each group were plotted on logarithmic coordinates and a power relation was determined graphically using the data representing the narrowest channels 

relative to discharge (figs. 3 to 12). Although all 

available data were plotted, most are presumed indicative of channels widened to varying degrees by 

erosive flow events. The narrowest channels relative to 

discharge, therefore, are judged most representative of 

stable, fully adjusted conditions, and it is the data from 

these channels (except for several that appear to have 

been measured improperly) that most closely approximate the assumptions previously given. Thus, the linear 

relations of figures 3 to 12 are envelope curves (power 

functions), describing the narrowest naturally occurring 

channels for specified ranges of width-depth ratio. 

Where data are insufficient to describe with reasonable 

confidence the slope of a lower envelope relation, the 

slope is based, in large degree, on statistically developed 

relations from previous studies (Osterkamp, 1980; 

Osterkamp and Hedman, 1982).

CALIBRATION RESULTS

The results of figures 3 through 12 and the data upon 

which they are based are summarized by the equations 

listed in table 2 and the graphical comparison of them in 

figure 13. Comparison of the equations shows a nearly 

consistent increase of both coefficients and exponents 

with width-depth ratios.

The b exponents of the various width-discharge power 

functions (table 2; figs. 3 to 12) were plotted in figure 14 

as a function of the mean value of width-depth ratio for 

each group. The plotted points are used to define a 

general curve (solid line) indicating the manner in which 

the b exponent for relatively stable alluvial stream channels varies with width-depth ratio. The solid line is extended as a dotted portion where data (table 2) are inadequate to provide reliable results. Also shown in 

figure 14 are similar points and a curve (dashed line) 

developed statistically from the width-discharge relations of Osterkamp and Hedman (1982) for groups of 

streams with similar channel-sediment characteristics.

In order that these data could be included and generalized in figure 14, mean values of width-depth ratio were 

determined for each channel-sediment group. Because 

several of the contrast groups show relatively large 

ranges of width-depth ratio, the mean values of widthdepth ratio for the sediment groups (crosses, fig. 14) do 

not necessarily provide an accurate relation with the b 

exponents. The dashed-line curve nevertheless provides 

a comparison that supports the solid-line curve used 

here (fig. 14) through the low to moderate width-depth 

ratios; the dashed-line curve provides increased accuracy for the higher ratios. The displacement between 

the two curves probably is caused by the difference in 

method used to develop them. The solid-line curve is 

based on data representing only narrow, presumably 

stable, channel sections; the dashed-line curve is based 

on all data representing channel-sediment groups, many 

of which do not approach steady-state conditions.

The solid-line curve given in figure 14 is used as the 

principal basis of calibration for evaluating equations 24 

and 25 because of (1) the similarity between it and the 

dashed-line curve of figure 14, (2) the general agreement 

of the 6-values, taken from figure 14, with numerous 

other empirical values as well as most theoretically 

developed values (table 1), and (3) the solid-line curve's 

general conformity to the theoretical requirement that b 

approach 1.0 as width-depth ratios become very large. 

The 6-exponent then corresponds to any width-depth 

ratio of the solid-line curve and is assumed accurate; the 

other four exponents (eqs. 2 to 5) are calculated accordingly.

Listed in table 3 for selected width-depth ratios of 5.0 

to 1,000 are exponent values (for eqs. 1 to 5) and the d 

and j values from which they were calculated. The b 

values, taken directly from the curve of figure 14, are 

those values corresponding to the selected width-depth 

ratios. Also listed are values of jz and j5 that are 

necessary to calculate the coefficients of equations 24 

and 25. The exponents are given to three significant 

figures in order to provide consistency, but there is no

TABLE 2.-Power relations of width to discharge developed from field 

data of channels in the Western United States and grouped by widthdepth ratios

____[W, channel width in meters; Q, mean discharge in cubic meters per second]____ 

Width-depth ratios

Group Figure No. Range Midpoint Equation

1 ______

2

3

4

5

6

7

8

9

10

3

4

5

6

7

8

9

10

11

12

<8.0 

9.0-11.0

14.0-18.0

> 18.0-22.0

>22.0-27.0

>27.0-33.0

36.0-44.0

45.0-55.0

56.0-80.0

>80.0-140

7.0 

10.0

16.0

20.0

24.5

30.0

40.0

50.0

68.0

110

JF=4.1Q° 48 

JF-4.6Q049

W-4.8Q0- 51

W=5.8Qa "

W-5.8Q0 - 5 '

TF=6.4Q053

TF=6.6Q056

TF=6.9Q° 58

TF=6.0Q°"

W-7.100- 82
10 ANALYTICAL TREATMENT OF CHANNEL-MORPHOLOGY RELATIONS

100

0.1' 

0.01 0.1 1.0 10 

DISCHARGE (Q), IN CUBIC METERS PER SECOND

100 0.1 1.0 10 100 

DISCHARGE (Q), IN CUBIC METERS PER SECOND

1000

FIGURE 3. -WID<8.0.

100

0.1 1.0 10 

DISCHARGE (Q), IN CUBIC METERS PER SECOND

FIGURE 4.-9.0 <WID< 11.0

W=4.8 Q 0.51

0.1 1.0 10 100 

DISCHARGE (Q), IN CUBIC METERS PER SECOND

FIGURE 6. -18.0 < W/D < 22.0.

1.0 10 100 1000 

DISCHARGE (Q), IN CUBIC METERS PER SECOND

FIGURE 7.-22.0<W7D<27.0.

W= 6.4 Q 0.53

1.0 10 100 1000 

DISCHARGE (Q), IN CUBIC METERS PER SECOND

FIGURE 8.-27.0< WD<33.0.

10,000

10,000

FIGURE 5. -14.0 < W/D < 18.0.

FIGURES 3-12.-Width-discharge data and power relations for groups of channels. The relation line is fitted to the
CALIBRATION 11

1.0

= 6.60056

1.0 10 

DISCHARGE (Q), IN CUBIC METERS PER SECOND

1.0

1.0 10 

DISCHARGE (O), IN CUBIC METERS PER SECOND

100

FIGURE 9.-36.0<W/Z)<44.0. FIGURE 11.- 56.0 <WID< 80.0.

lOOOr

1.0L

W = 6.9 Q 058

0.1 1.0 10 100

DISCHARGE (O), IN CUBIC METERS PER SECOND

FIGURE 10. - 45.0 <W/Z)< 55.0.

100

0.01

o o o W = 7.100.82

0.1 1.0 10 

DISCHARGE (Q), IN CUBIC METERS PER SECOND

FIGURE 12.-80.0<W/Z)<140.

100

0.1 0.2 0.3 

PROPORTIONAL DISTANCE, x*, ALONG CHANNEL BED (x/W)

0.4 0.5

data points representing the narrowest reliable width measurements, relative to mean discharge, that are available.
12 ANALYTICAL TREATMENT OF CHANNEL-MORPHOLOGY RELATIONS

implication intended that these results are accurate to 

three significant figures. The exponent data, as functions of width-depth ratio, are presented graphically in 

figure 15 and show the manner in which each of the different exponents changes relative to the other four.

The value of z can be considered a measure of the concavity of the channel profile to which it is applied; the 

largest negative value of z is associated with the most 

concave profile and a z-value of 0 indicates a linear profile. The previously described empirical results suggest a 

nearly fixed value for z, -0.25 for channels of constant 

channel-sediment characteristics. Those results are in 

general support of the larger negative values calculated 

here for channels of specified width-depth ratio. In both 

cases (specified width-depth ratio and specified channelsediment characteristics), the exponent shows little or 

no change for most channels-those with width-depth 

ratios ranging from 12 to 40. Larger negative values of z 

for channels of fixed width-depth ratio than for those of 

constant channel-sediment characteristics are consistent with numerous field observations. As mean 

discharge increases in the downstream direction, 

stream channels have a general tendency to increase in 

width-depth ratio and bed and banks content of finegrained, cohesive sediment. (See data of Osterkamp and 

others, 1982; Osterkamp and Hedman, 1982.) Without a 

downstream increase of silt-clay content, width-depth 

ratios necessarily must increase at a faster rate than is 

typical for alluvial stream channels. In figure 15, and 

thus in equations 24 and 25, increasing width-depth 

ratios are accompanied by decreasing negative values of 

z. Thus, the condition of constant particle sizes in the 

downstream direction, which generally signifies increasing width-depth ratios, also is suggestive of a less concave profile and lower negative values of z than is the 

condition of specified width-depth ratio. The absolute 

values of z calculated here (table 3 and fig. 15) are mostly about 30 percent larger than the value of -0.25 indicated by field studies (Lane, 1957; Osterkamp, 1978).

The general agreement of the - 0.25 value with the calculated z-values given here (table 3) provides independent qualitative confirmation of the calibration based on 

the assumed b values.

SENSITIVITY TO ASSUMED VALUES OF Z

Empirically developed values of b are more accurately 

known than those of z, largely because channel widths 

change at a much greater rate with mean discharge than 

do channel gradients. Therefore, if extensive gradientdischarge field data for channels of limited ranges of 

width-depth ratio were available, they probably would 

not permit as accurate a calibration as do the widthdischarge data. If it is accepted that the z-values for 

stable channels of specified width-depth ratio should be 

of moderately larger negative value than those values 

identified for channels of constant sediment 

characteristics (-0.25), then the calibration given here 

(table 3 and fig. 15), based on the empirically developed 

6-values (figs. 3-12), appears to be reasonably accurate.

Figure 16 illustrates the manner in which b, f, m, and y 

change with variations in the value of z for width-depth 

ratios of 10 and 30. The curves of figure 16, which were 

developed by use of equations 25, 27, 24, and 6, and a 

variety of assumed z-values, therefore represent the 

sensitivity of the other exponents to changes in z for 

width-depth ratios of a range of typical well-formed 

alluvial stream channels. The curves show that if the absolute values of z were less than -0.25, the resulting 

6-values would be larger than the range of 0.45 to 0.55 

commonly indicated by empirical studies as well as by 

several theoretical approaches (table 1). If z is of greater 

absolute value than -0.43, any value of b for width is 

necessarily smaller than that of/for depth; empirical 

studies and theoretical approaches have also suggested 

that 6 normally is of greater value than / for natural 

alluvial stream channels. Figure 16 suggests, therefore, 

that for a condition of constant width-depth ratio in the

TABLE 3. -Calculated and assumed values of A, ji through}^, b, f, m, z, and y for width-depth ratios of 5 to 1,000

Width-depth 

ratio (W/Dy

J3

7- 3d

3* 

3d

5.0

7.0

10

12

16

20

30

40

50

100

1,000

0.381

.311

.251

.225

.189

.166

.130

.109

.096

.063

.016

0.685

.717

.749

.766

.791

.808

.839

.859

.873

.910

.974

0.827

.911

.998

1.041

1.108

1.155

1.238

1.291

1.327

1.427

1.598

0.484

.553

.624

.659

.713

.751

.818

.862

.891

.972

1.111

0.189

.170

.150

.141

.126

.115

OQ7

.084

.076

.054

.015

0.496

.547

.599

.625

.665

.693

.743

.775

7Q6

.856

.959

0.591

.632

.674

.695

.728

.751

.791

.817

.834

.883

.967

0.446

.476

.500

.510

.516

.521

.529

.554

.616

.891

.974

0.413

.386

.365

.355

.351

.349

.347

.325

.276

.058

.015

0.141

.138

.135

.135

.133

.130

.124

.121

.108

.051

.021

-0.400

-.360

-.330

-.320

-.320

-.320

-.320

-.300

-.240

-.010

0

-0.069

-.064

-.059

-.059

^.059

-.059

-.056

-.054

-.048

-.010

-.001

1 For width-depth ratios of 5.0 through 100, values of z were assigned to yield an assumed b value conforming with that shown in figure 14; for a width-depth ratio of 1,000, a z value of 0, indicative of a linear channel profile, was assumed.
APPLICATIONS 13

100

10

Q

LU

1.0

I

o

EXPLANATION

1 WD<8.0

2 9.0<W/D<11.0

3 14.0<W/D<18.0

4 18.0<M//£><22.0

5 22.0<W/D<27.0

6 27.0<W/D<33.0

7 36.0<W/D<44.0

8 45.0<W/D<55.0

9 56.0<W/D<80.0 

10 80.0<W/D<140

0.1

0.01 0.1 1.0 10

DISCHARGE (Q), IN CUBIC METERS PER SECOND

100

FIGURE 13.-Width-discharge relations for groups of stream channels with similar width-depth ratios (WID).

1000

downstream direction, the ^-exponent (eq. 4) normally 

varies between -0.25 and -0.40. However, lower 

negative values occur for very wide stream channels 

relative to discharge (table 2).

Figure 16 also illustrates that the b, f, m, and y exponents show increasing sensitivity to changes in z with 

increasing width-depth ratio. The/and y exponents, for 

depth and bed roughness, show the greatest rates of 

change with variation of z, but the absolute effect on any 

?/-value by changes of z is minor owing to the small 

magnitude of y. The width exponent, b, shows limited 

change with z values of smaller magnitude than - 0.4; 

this suggests that, as pointed out earlier, of the five exponents (eqs. 1 to 5), b is well-suited to be the basis for 

calibrating equations 24 and 25.

APPLICATIONS

Equations 24 and 25, which are semitheoretical, and 

the equations of table 2, which provide both empirical 

support and a means of calibrating equations 24 and 25, 

demonstrate that geometry-discharge relations of 

alluvial channels cannot be readily generalized from 

single power equations. The recognition that the coefficients and exponents of equations 1 to 5 are variable, 

dependent at least in part on the mobile sediment supply 

of a channel system, leads to various assumptions and 

conclusions regarding fluvial processes. Several of these 

assumptions and conclusions are discussed in the following section.

CHANNEL ADJUSTMENT

Basic to the derivation leading to equations 24 and 25 

is that the mean shear stress in a channel is a function of 

the instantaneous water-sediment discharge:

-r=F(Q(). (28)

Among the assumptions of the derivation is the occurrence of approximate channel adjustment to the meandischarge conditions of the water-sediment mixture. As 

discharge in a channel section increases, the erosiveness 

(owing to greater velocities and turbulence) generally increases also. Thus, the assumption of equilibrium requires relatively uniform discharge conditions in which 

the erosive (shear) stresses at the channel perimeter are 

in balance through time with the gravitational and 

cohesive forces resisting movement of channel sediment.
14 ANALYTICAL TREATMENT OF CHANNEL-MORPHOLOGY RELATIONS

0.9

10 100 

WIDTH-DEPTH RATIO, IN METERS PER METER

200

FIGURE 14.-Variation of the width exponent, 6, with width-depth 

ratio. Circles and solid line represent data points and resulting curve 

(dotted where poorly defined) for channels grouped by similar widthdepth ratios (table 2); crosses and dashed line represent data points 

and resulting curve for channels grouped by sediment 

characteristics.

The geometry and other channel characteristics, such 

as bed configuration, the median particle size of bed 

material (d50), and the silt-clay content of the channel 

material (SC) are determined largely by the discharge of 

water and sediment. From equation 28, therefore,

[W, D, G, n, d50 , SC,...] -F(Q,); (29)

for self-formed alluvial channels, all the bracketed 

variables of equation 29 are associated with the distribution of shear stresses. The derivation resulting in equations 24 and 25 is based not only on relations in equations 28 and 29, but also on the assumption that Q{ is 

relatively uniform through time. Thus, the mean shear 

stress in the channel, T, also is assumed to have little 

variation through time.

Most natural alluvial stream channels do not have 

nearly constant discharge, but show variations of at 

least several orders of magnitude. A channel that is 

widened by the excessive shear stresses of an erosive 

flood, therefore, is not adjusted to the conditions of 

mean discharge following the flood. Generally, the channel requires an extended period of normal flow conditions and shear stresses before accretion and deposition 

of fine sediment are sufficient to affect channel narrowing and an essentially adjusted geometry. If the sedi10 100

WIDTH-DEPTH RATIO, IN METERS PER METER

FIGURE 15.-Change of b, f, m, z, and y with change of width-depth 

ratio. Values of b, f, and m are positive; values of z and y are 

negative.

0.01 0.1

FIGURE 16.-Change of b, f, m, and y with variation in values of z for 

selected width-depth ratios (W/D).

ment available for fluvial transport is principally of sand 

sizes, the rate of narrowing may be slow owing to a lack 

of fine cohesive material to form a stable channel section. If abundant fine sediment is available to a stream 

channel, narrowing toward a geometry approaching 

equilibrium following an erosive flood may be relatively
APPLICATIONS 15

0.04 0.1 1.0 10 100 1000 4000 

DISCHARGE (Q), IN CUBIC METERS PER SECOND

FIGURE 17. - Relation of width to discharge 

for 14.0 <WID< 18.0.

1.0 1 

0.1

1000

1.0 10 

DISCHARGE (Q), IN CUBIC METERS PER SECOND

FIGURE 18.-Relation of width to discharge for 

18.0 <WID< 22.0.

1 °00^ m i 

C/J EC

LU 

LU5 100-

2 ~

| :

B

10 100 1000 

DISCHARGE (Q), IN CUBIC METERS PER SECOND

4000

-6.4 Q °53

1.0 - 

0.1

I I II I I I ' I I I II I I i

t.O 10 100 

DISCHARGE (Q), IN CUBIC METERS PER SECOND

FIGURE 20.-Relation of width to discharge for 

27.0 <WID< 33.0.

1000

1000

100

,,68

10 1 

1.0

W = 6.60°

10 100 

DISCHARGE (Q), IN CUBIC METERS PER SECOND

FIGURE 21.-Relation of width to discharge for 

36.0 <WID< 44.0.

1000

1000

100

69,

 10 100 

DISCHARGE (Q), IN CUBIC METERS PER SECOND

1000

FIGURE 22. - Relation of width to discharge for 

45.0 <WID< 55.0.

FIGURE 19.-Relation of width to discharge for

22.0 <WID* 27.0.

FIGURES 17-22. -Relations of width to discharge for channels of the Western United States. Numbers refer to comparative data in table 4 from 

the United States-in Ohio and southern Missouri-New Guinea, and Australia; dashed segments are extrapolations of the relation lines 

beyond the limits defined by the Western United States data.
16 ANALYTICAL TREATMENT OF CHANNEL-MORPHOLOGY RELATIONS 

TABLE 4. -Geometry-discharge data from natural stream channels with stable discharge

Stream and location

Cheasapeake Spring at

Chesapeake, Mo.

Jones Spring near Springfield, Mo. ____

Markham Spring near

Williamsville, Mo.

Bartlett Mill Spring near

Waynesville, Mo.

Boze Mill Spring near Bardley, Mo. ___

Roaring River Spring near

Cassville, Mo.

Round Spring at Round Spring, Mo. ___

Montauk Spring at

Montauk State Park, Mo.

Blue Spring near Alton, Mo.

Alley Spring at Alley Spring, Mo.

Greer Spring near Greer, Mo.

Big Spring near Van Buren, Mo.

Upper Twin Creek at McGaw, Ohio ____

Wolf Creek at Trotwood, Ohio

West Branch Mahoning River

near Ravenna, Ohio

Middle Branch Nimishillen

Creek at Canton, Ohio

Massies Creek at Wilberforce, Ohio _

Sandusky River near Bucyrus, Ohio ___

Mahoning River at Alliance, Ohio

Clear Creek near Rockbridge, Ohio ____

Eagle Creek at Phalanx Station, Ohio _

Little Miami River near Oldtown, Ohio _

Pymatuning Creek at Kinsman, Ohio _

Ottawa River at Allentown, Ohio

Tinkers Creek at Bedford, Ohio

Short Creek near Dillonvale, Ohio

North Fork Licking

River at Utica, Ohio

Mad River near Urbana, Ohio

Olentangy River at Claridon, Ohio ____

Mill Creek near Bellepoint, Ohio

Wakatomika Creek near

Frazeysburg, Ohio

Yellow Creek near

Hammondsville, Ohio _____________

Bean Creek at Powers, Ohio _________

Captina Creek at Armstrongs

Mills, Ohio

Tymochtee Creek at Crawford, Ohio _

Greenville Creek near Bradford, Ohio _

Shade River near Chester, Ohio

Nimishillen Creek at

North Industry, Ohio

Twin Creek near Ingomar, Ohio

Cuyahoga River at Hiram Rapids, Ohio

Kokosing River at Mount Vernon, Ohio

Paint Creek near Greenfield, Ohio ____

Deer Creek at Mount Sterling, Ohio __

Sandusky River near

Upper Sandusky, Ohio

Blanchard River near Findlay, Ohio ___

Little Muskingum River at

Bloomfield, Ohio

Conneaut Creek at Conneaut, Ohio ____

Sandy Creek at Waynesburg, Ohio ____

Mad River near Eagle City, Ohio

Huron River at Milan, Ohio

Tiffin River at Strucker, Ohio ________

Data-set

No.;

Fig. No. 1

1;19

2;19

3;20

4;20

5;21

6;22

7;21

8;22

9;22

10;21

11;22

12;22

13;22

14;19

15;19

16;19

17;20

18;22

19;22

20;21

21;20

22;22

23;19

24;23

25;20

26;20

27;23

28;23

29;22

30;22

31;23

32;21

33;21

34;23

35;19

36;21

37;21

38;22

39;22

40;20

41;23

42;21

43;23

44;24

45;23

46; 19

47;21

48;22

49;22

50;23

51;21

Width

in

meters

1.98

3.05

4.42

8.08

7.92

13.6

11.6

14.6

16.5

14.5

27.7

31.1

11.9

8.84

7.32

9.75

11.3

15.2

13.7

16.8

11.3

19.8

8.23

17.7

10.7

13.1

20.7

15.5

23.5

25.0

20.7

18.9

14.6

25.9

18.6

18.3

17.7

17.7

25.6

21.0

23.8

22.3

25.9

30.2

21.6

20.7

18.0

22.9

24.7

33.5

30.5

Mean

depth in

meters

0.12

.183

.216

.442

.341

.46

.46

.521

.560

.558

.92

.95

.396

.518

.488

.579

.518

.549

.457

.701

.549

.732

.579

.427

.549

.701

.488

.427

.762

.792

.549

.701

.579

.671

1.07

.701

.671

.610

0.823

1.10

.640

.853

.640

.640

.518

1.22

.701

.792

.914

.914

1.13

Width-depth

ratio

16.5

16.7

20.4

18.3

23.2

29.7

25.2

28.1

29.5

26.0

30.1

32.7

30.1

17.1

15.0

16.8

21.8

27.7

30.0

24.0

20.6

27.0

14.2

41.5

19.4

18.7

42.4

36.3

30.8

31.6

37.7

27.0

25.2

38.6

17.4

26.1

26.4

29.0

31.1

19.1

37.2

26.1

40.5

47.2

41.7

17.0

25.7

28.9

27.0

36.7

27.0

Mean 

discharge

in cubic

meters

per second

0.066

.071

.214

.443

.653

.898

1.20

1.80

2.05

3.60

9.35

12.2

.408

.606

.782

.997

1.72

2.44

2.47

2.51

3.06

3.14

3.43

3.54

3.57

3.60

3.91

4.02

4.25

4.28

4.28

4.50

4.62

4.62

4.81

4.84

4.96

5.10

5.32

5.78

5.92

6.57

6.66

6.88

6.97

7.36

7.45

7.56

8.52

8.52

8.81

Source of

data2

1

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

Comments

Depth estimated; discharge

inaccurate

Discharge possibly inaccurate

Do.

Do.

Do.

Depth estimated; discharge

inaccurate

Depth estimated

Difficult geometry measurement

vegetation

Depth estimated

Do.

possibly

possibly

owing to
APPLICATIONS 17

TABLE 4. -Geometry-discharge data from natural stream channels with stable discharge- Continued

Stream and location

Data-set

No.; 

Fig. No. 1

Width

Mean 

discharge

Mean in cubic 

depth in Width-depth meters Source of

meters ratio per second data2 Comments

Little Miami River near

Spring Valley, Ohio _______________

Killbuck Creek at Killbuck, Ohio ______

Cuyahoga River at Old Portage, Ohio _ 

Tuscarawas River at Missillon, Ohio ___ 

Big Darby Creek at Darbyville, Ohio ___ 

Scioto River near Prospect, Ohio _____

Hocking River at Enterprise, Ohio ____

Great Miami River at Sidney, Ohio ____

Mad River near Springfield, Ohio _____

Little Beaver Creek near

East Liverpool, Ohio ______________

East Fork Little Miami River at

Perintown, Ohio _________________

Licking River near Newark, Ohio _____

Raccoon Creek at Adamsville, Ohio ___ 

Scioto River near Dublin, Ohio _______

Cuyahoga River at Independence, Ohio 

Paint Creek near Bourneville, Ohio ____

Mohican River at Greer, Ohio ________

Walhonding River near Nellie, Ohio ___ 

Scioto River at Chillicothe, Ohio ______

Muskingum River at

McConnelsville, Ohio _____________

Ok Tedi at Ningerum, New Guinea,

sec. 1 __________________________

Ok Tedi at Ningerum, New Guinea,

sec. 2 __________________________

Ok Tedi at Ningerum, New Guinea,

sec. 3 __________________________

Ok Tedi at Ningerum, New Guinea,

sec. 4 __________________________

Aure River near mouth, New Guinea _ 

Alice River at Kokonda,

New Guinea, sec. 1 _______________

Alice River at Kokonda,

New Guinea, sec. 2 _______________

Alice River at Kokonda,

New Guinea, sec. 3 _______________

Alice River at Kokonda,

New Guinea, sec. 4 _______________

Fly River at Kuambit, New Guinea ____

Purari River below Wabo Dam site,

New Guinea _____________________

Murrumbidgee River near

Maude, Australia ________________

Murrumbidgee River near Darlington

Point, Australia __________________

Murrumbidgee River near

Hay, Australia ___________________

Murrumbidgee River near

Narrandera, Australia ____________

Murrumbidgee River near

Wagga Wagga, Australia __________

52;23 

53;22 

54;22 

55;22 

56;22 

57;22 

58;20 

59;23 

60;22

62;24 

63;22 

64;19 

65;24 

66;19 

67;21 

68;23 

69;24 

70;22

72;23 

73;24 

74;24

75;22 

76;19

77;21 

78;21 

79;19

80;22 

81;21

83;19 

84;19 

85;19 

86;19 

87;19

30.2

29.0

31.1

29.9

39.0

40.8

27.7

29.0

32.3

61;23 38.4

39.3

51.5

27.7

57.0

27.4

39.0

55.5

79.6

92.0

71;23 151

150

170

175

150

110

185

170

100

175

260

82;19 250

50.3

67.1

74.7

75.0

83.2

.732

.945

1.01

1.07

1.31

1.49

1.34

.671

1.19

1.01

.792

1.71

1.62

1.19

1.52

1.55

1.31

1.65

2.80

3.84 

3.5 

3.5 

3.5

5.0 

7.5

7.0 

7.5 

6.0

6.0 

9.9

14.3

3.6 

4.0 

5.3 

4.5 

5.0

41.3

30.7

30.8

27.9

29.8

27.4

20.7

43.2

27.1

38.0

49.6

30.1

17.1

47.9

18.0

25.2

42.4

48.2

32.9

39.3

42.9

48.6

10.9

11.5

11.9

12.3

12.7

12.8

12.9

13.4

13.7

14.7

15.6

16.4

18.5

22.2

22.7

22.7

25.5

41.6

96.5

211

240

240

50.0 240

30.0 240

14.7 370

26.4 726

22.7 726

16.7 726

29.2 726

26.3 1780

17.5 2360

14.0 68.5

16.7 86.6

14.1 88.5

16.7 104

16.6 130

Below O'Shaughnessy Dam, regulated 

Below Mohawk Dam, regulated

Gage site

1030 m upstream of gage 

2250 m upstream of gage 

3310 m upstream of gage

4190 m downstream of gage 

2100 m downstream of gage 

Gage site 

2460 m upstream of gage

Width average of three sections; depth 

estimated

Depth estimated

Do.

Do.

Do.

Do.

1 First number is data number shown on the appropriate figure (graph), which is given by the second number 2.

2 1, Field investigations, W. R. Osterkamp; 2, E. E. Webber, written commun. 1981; Webber and Roberts, 1981; 3, Pickup (1977) and Pickup and others (1979); 4, Schumm (1968).

rapid-within a few years if erosive floods do not occur 

in that period (Osterkamp and Harrold, 1982).

WIDTH-DISCHARGE DATA FROM STABLE 

NATURAL CHANNELS

Because hydrologic conditions of the Western United 

States may be generally conducive to wide and poorly 

adjusted channel conditions, the relation lines of figures

3 to 12 were tested further using data from natural 

channels with relatively stable discharge (channels with 

low flood peaks relative to mean discharge). These data 

include widths, depths, and mean discharges from (1) 

spring-effluent channels of southern Missouri, (2) 

various-sized streams of Ohio (Webber and Roberts, 

1981; E. E. Webber, written commun., 1981), (3) large 

streams of southwestern New Guinea, an area of 

tropical rainforest (Pickup, 1977; Pickup, and others,
18 ANALYTICAL TREATMENT OF CHANNEL-MORPHOLOGY RELATIONS

1979), and (4) five sites along the Murrumbidgee River 

in New South Wales, Australia (Schumm, 1968). Comparisons of these data with the relations developed from 

those of the Western United States for width-depth 

ratios of 14 to 55 are given in figures 17 through 22. The 

data are listed in table 4. Data are insufficient for comparisons with higher or lower width-depth ratios.

The slopes (power-function exponents) of the comparison lines provided in figures 17 through 22 are those 

given in table 2 and figures 5 through 10. The calculated 

exponents of table 3 were not used in figures 17 through 

22 because the appropriate coefficients (intercepts) 

necessary to plot the power functions could not be 

calculated. The error introduced by this approximation, 

however, is probably small relative to the possible error 

inherent in the calibration technique. The measurement 

techniques used to collect the width and depth data of 

table 4 were consistent with the methods used for the 

Western United States data. Most of the mean 

discharges listed were based on streamflow records 

from gage sites at or near the measurement sites; the remainder were estimated using gage records from 

upstream or downstream sites.

Presumably because the banks are stabilized by large 

fractions of silt and clay sediment sizes (G. Pickup, written commun., 1981), the rivers of New Guinea conform 

well to the relations of inferred equilibrium (figs. 17 to 

22). The spring data from southern Missouri represent 

channels with relatively steady discharges. The springs 

transport almost no suspended sediment from which 

cohesive banks can form; but most of the spring channels are formed of sandy to gravelly material derived 

from the flood-plain alluvium of the rivers into which the 

springs flow. The lack of cohesive sediment is inferred to 

account for the slightly greater-than-predicted widths of 

equilibrium (figs. 19-22).

Within the limitations previously defined, equations 

24 and 25 indicate that for a specified discharge and 

width-depth ratio, only one set of conditions representing complete channel adjustment is possible. The data of 

table 4 are presented partly to support the power functions of figures 3 through 12 and partly to emphasize the 

unique values of the channel variables thereby defined. 

It is only the streams with sufficient suspended sediment to form and maintain stable, cohesive banks that 

closely approach a fully adjusted channel condition. If 

the discharge of a stream is very steady, a relatively 

small suspended-sediment load may be sufficient to provide approximate equilibrium; flashier streams require 

larger suspended-sediment concentrations to approximate adjusted channel conditions. The data of table 4 

suggest, as examples, that the sediment loads and 

regularity of discharges of the rain-forest rivers in New 

Guinea are adequate to produce relative channel equilibrium. Despite comparatively steady flow rates, the 

spring-effluent channels in southern Missouri are slightly wider than would otherwise be expected due to the 

lack of suspended sediment.

DISCUSSION

The importance of silt and clay sediment sizes in providing strength or cohesiveness to the material forming 

a soil or channel perimeter has been demonstrated by a 

variety of studies. Schumm (1960, p. 21) demonstrated a 

power relation between geometry, as width-depth ratio, 

and a weighted silt-clay content of channel material. An 

especially pertinent engineering approach is a study by 

Trask (1959), which shows that the principal variables of 

soil strength are water content, clay mineralogy, ratio 

of clay content to larger particle sizes, and the particle 

sizes of the sediment comprising the non-clay fraction. 

Numerous laboratory tests made for the study demonstrated that when other variables are held constant, the 

shear strength of a silica soil is increased steadily and 

significantly by increasing proportions of clay. Furthermore, the lowest soil strengths of a mixture of 20 percent kaolin and 80 percent variable sand sizes were 

associated with the largest sand sizes; between the sand 

sizes of 0.35 and 1.68 mm, however, a relatively small 

difference in soil strength was noted (Trask, 1959, p. 

573-577).

Studies such as those of Schumm (1960) and Trask 

(1959) suggest that the shear strengths of materials 

forming a channel perimeter vary with the particle sizes 

transported by a stream. Thus, such studies provide explanations for the inconsistent results of the table 4 data 

in figures 17 through 22. A fully adjusted condition 

(state of equilibrium) commonly is regarded as one in 

which a net balance occurs between the shear strength 

of the channel material and the shear stress of the 

water-sediment mixture. In a specified time interval, 

therefore, the sediment eroded equals the sediment 

deposited, and through a channel reach lacking 

tributary inflow, the mass of sediment inflow balances 

the outflow. An additional requirement proposed for a 

fully adjusted condition is that there must be a mass 

balance of the individual particle sizes in the sediment 

load (Osterkamp and Harrold, 1982).

The principal independent variables that shape a channel are the water and sediment discharges and the 

topography that partially determines channel gradient. 

When, as described earlier and suggested by equations 

28 and 29, a net balance occurs between the shear 

stresses caused by the independent variables and the 

shear strengths of the channel material, a unique set of 

adjusted channel characteristics is assumed. This 

description of fluvial dynamics implies that a channel ad-
CONCLUSIONS 19

justs its geometry to accommodate (1) the water and 

sediment discharges, (2) the gradient constraints imposed by topography, and (3) the fabric strength of the 

channel alluvium. In the derivation summarized by equations 28 and 29, these changes are viewed not simply as 

an adjustment of geometry to imposed conditions of 

shear stress, but also as an adjustment of shear stresses, 

at the wetted perimeter, toward a limited range that 

prevails for all stable alluvial channels. As indicated by 

figure 1 and subject to the assumptions of this paper, the 

shear stresses near the edges of a rectangular channel 

section rapidly approach 0 regardless of the width-depth 

ratio. This means that the distribution of shear stresses 

tends to be similar along the wetted perimeters of all 

stable alluvial channels that approximate the rectangular shape assumed, in this paper, for a condition of constant discharge. Variation in width-depth ratio, which 

appears to be dependent on the sizes and sorting of the 

sediment supply, serves to adjust the maximum shear 

stress and water velocity in parts of the channel sections 

other than the wetted perimeter. This conclusion is supported indirectly by data showing that fluvial-sorting 

processes of bank material in most relatively stable 

alluvial channels result in particle-size distributions that 

are largely independent of the size, shape, or gradient of 

the channels (Osterkamp, 1981; Osterkamp and Harrold, 1982).

In summary, the derivation of this paper is based on 

width-depth ratios that imply specific shear-stress 

distributions. Dependent on the sediment-discharge 

characteristics and topographic constraints, the channel 

gradient and roughness adjust to produce water 

velocities and turbulence that maintain the requirement 

of sediment mass balance. For example, a wide range of 

channel gradients was found for channels with widthdepth ratios of 20 (Osterkamp and others, 1982). The 

streams of high gradient are very turbulent owing to armoring and high channel roughness, whereas those of 

low gradient are commonly sand- or gravel-bedded channels. Both types are assumed adjusted to flow rates 

roughly appoximating mean discharge, although the armor of the steep channels is moved only during relatively high flows. However, should large sediment sizes become unavailable for transport by the high-gradient 

stream, continued armoring is not possible. The 

necessary adjustment to maintain sediment mass 

balance then would be channel widening, an increase of 

the width-depth ratio, and a reduction of the maximum 

shear stress (fig. 1).

CONCLUSIONS

The derivation of the width and depth exponents in 

equations 24 and 25 is based on the continuity equation,

the Manning equation, and an assumed relation describing the approximate manner in which shear-stress 

distributions of stable alluvial channels vary with channel geometry. The shapes of alluvial channels, however, 

are closely related to the sediment transported by 

streams and stored as material forming the bed and 

banks. Thus, the analytical approach of this paper indirectly considers the characteristics of channel sediment as an independent variable determining the downstream changes of channel morphology relative to water 

discharge. Unlike previous approaches that relate downstream changes in geometry and discharge as power 

functions with fixed exponents, the approach presented 

here results in variable power-function exponents (equations 1 to 5) that necessarily vary with the shape (and, 

therefore, also with the sediment characteristics) of 

alluvial channels.

Weaknesses of the approach described here include 

the assumptions necessary for the derivation of the exponents, such as a condition of constant discharge and a 

rectangular channel shape, and the need for iterative 

calibration by use of field data. Despite the weaknesses, 

however, the derived equations in this paper appear to 

relate the geometries of alluvial channels reasonably 

well to the mean discharges typical of those channels. 

The equations appear applicable to the entire range of 

natural alluvial channels, including those that are 

unusually wide or narrow relative to depth. The equations probably are not applicable to channels of 

ephemeral and intermittent streams or to channels that 

are not fully fluvial, such as those modified by tidal effects.

Although the assumptions necessary to the derivation 

of the exponents limit its application to natural stream 

channels, the forms of the equations derived in this 

paper are useful for understanding channel dynamics. 

Equation 17, for example, is based in part on the Manning equation, but perhaps it explains channel adjustments more effectively than the Manning equation. 

Because steady-state conditions are assumed, equation 

17 shows that a change in any one of the variables must 

result in adjustment by one or more of the other 

variables in order to reachieve balanced channel conditions. The equation suggests that the balance between 

geometry and discharge necessary to maintain stable 

channel conditions is the result of a similar balance between the distribution of shear stresses along the channel perimeter and the channel properties.

Equation 17, therefore, suggests that if, at constant 

discharge and roughness, the gradient of a channel is increased, a likely result is some combination of increased 

width and critical shear stress. Conversely, as a channel 

reach develops smaller and smaller gradients through
20 ANALYTICAL TREATMENT OF CHANNEL-MORPHOLOGY RELATIONS

time (years to decades) because of increased meandering 

and channel length, there is a tendency toward smaller 

widths, smaller median particle sizes of the transported 

sediment and, hence, lower critical shear stresses. If the 

discharge is increased, expected changes are increased 

width and critical shear stresses and decreased gradient. The manner in which roughness changes with 

changes in channel width or discharge cannot be determined from equation 17 without considering the values 

of the various j exponents.

Particularly for stream channels of high width-depth 

ratio, the data used for calibrating the derivation of the 

exponents presented in this paper are insufficient to 

yield trustworthy 6 values. With additional geometrydischarge data from stable alluvial channels, especially 

those of high width-depth ratio, the accuracy of the 

calibration curve of figure 14 may be improved considerably. Certainly, further testing with field data 

seems necessary before the results of this paper can be 

considered reasonably reliable.

Perhaps the principal value of the equations derived in 

this paper is that they explain why a channel reach can 

exhibit markedly different physical properties through 

time while maintaining an approximate balance between 

total inflow and outflow of sediment. By relating water 

discharge to ranges of possible channel properties, the 

equations identify the ranges of expected stable conditions and thereby predict the general channel changes 

that will occur if stable conditions are not upset.

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APPENDIX

[List of numbered equations provided in the text] 

Equation Equation

number

Page

1. W=aQb __________________________ 2

2. D = cQf ___________________________ 2

3. V = kQm ___________________________ 2

4. G = tQ* ___________________________ 2

5. n = rQy ___________________________ 2

6. 6 +/+ m = 1 ________________ 3

7. Q, = WDV _________________________ 4

8. Q, = fc'Q,6Q/Q.m ______________ 4

9. TJ'T=l + (W/D)-060 _________________ 4

10. Q = -*G1/2WD5/3 ____________________ 5

[

Vlf) T3/5 

^ I 5 ^1/2 J &

12. x* = x/P ___________________________ 5

13. T{X*)/T=F(X*) ______________________ 5

14. T(x*)/T=p(x*)d 5

15. TjyGD = p(D/W)d 6

/- / 5 \ ,-% /3a + 3\ 16 w= p£2"|(w73)r nQ_ ife) __ 6 

L-cJ L^1/2 J

17. W=t-ji(py)i*T-j*GJmjiQil 6 

17a. WWG'm'.Q', ______________ 7

18. D = t-J4py)-J*TcJ*G-J*rf<Q1* 6

18a. D^cfG-W'Q1* 7

1 C 05

19- T= 7rFJ° ^x*)dx* _________________ 6

U.o

20. p = 2d(d + 1) ________________ 6

21. (Tm/r)=p(0.5)d _______________ 6

22. d = (rjr)- 1.0 ________________ 6

23. d = (W/D)-° 60 7

24. TF=aQ<V*W______________ 7

25. D = cQ<VzV"4> ______________ 7

DO. 167

26. C~- _________________ 8

n

27. 2/=-0.167/ ________________ 8

28. T=F\Qt) 13

29. [W, D, G, n, d50 , SC,. . .]%F(Q,) ____ 14