River Meanders 

Theory of 

Minimum Variance

GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-H
River Meanders 

Theory of 

Minimum Variance

By WALTER B. LANGBEIN and LUNA B. LEOPOLD 

PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 422-H

The geometry of a meander is that of a random 

walk whose most frequent form minimizes the 

sum of the squares of the changes in direction 

in each unit length. Changes in direction 

closely approximate a sine function of channel 

distance. Depth, velocity, and slope are 

adjusted so as to decrease the variance of 

shear and the friction factor in a meander over 

that in an otherwise comparable straight reach 

of the same river

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1966
UNITED STATES DEPARTMENT OF THE INTERIOR 

WALTER J. HICKEL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

Library of Congress catalog-card No. 76-605539

First printing 1966 

Second printing 1970
CONTENTS

Page

Abstract_ ________________________________________ HI

Introduction _______________________________________ 1

Meander geometry__-_----____________-___________._ 2

Path of greatest probability between two points._._ 2

The sine-generated curve_____________________.__ 3

Analysis of some field examples__----_-_____-__-_- 4

Comparison of variances of different meander curves. 4

Meander length, sinuosity, and bend radius..______ 5

Page

Meanders compared with straight reaches.___-__--_--__ H8 

Field measurements.__-----_--_-_-___-_---_----.

Reduction of data..-.--..------.- ___________

Comparison of results---------------------------

Interpretive discussion-_________________-__--__--_.

References..__-__---____----_----------_-_--_-----.

9

9

11

13

15

ILLUSTRATIONS

Page 

FIGURE 1. Most frequent random walks________________________________________________________________________ H3

2. Plan view of channel and graph of direction angle 0 as a function of distance along a meander.________________ 3

3. Maps of meander reaches of several rivers and graphs showing relation between channel direction and channel

distance._________________________________________________________________________________________ 6

4. Map of laboratory meander and plot of thalweg direction against distance__________--_--___-____--_-__----- 8

5. Comparison of four symmetrical sinuous curves having equal wavelength and sinuosity...____________________ 9

6. Plan and profiles of straight and curved reaches, Baldwin Creek near Lander, Wyo__________________________ 10

7. Plan and profile of curved and straight reaches of Popo Agie River, ^ mile below Hudson, Wyo.______________ 11

8. Plan and profile of straight and curved reaches on Pole Creek at Clark's Ranch near Pinedale, Wyo----------- 12

9. Relation of sinuosity to variances of shear and friction factor__________________ _____________ _________ 14

10. Relations between velocity, depth, and slope in straight and curved reaches, Pole Creek, near Pinedale, Wyo 14

TABLES

Page 

TABLE 1. Field data for Pole Creek at Clark's Ranch near Pinedale, Wyo-_____________--__._____.__- _._.__ H13

2. Comparison between straight and curved reaches in five rivers____--__-_____------------------------------ 14

3. Comparison of coefficients of correlation__-___-________________--_-_--_-__-__--------------------------- 14

in
	SYMBOLS

a ratio of variances (table 3)

c a coefficient (equation 1)

k sinuosity, ratio of path distance to downvalley distance

R radius of curvature of a bend

r coefficient of correlation (table 3)

s unit distance along path

x a factor or variable

p probability of a particular direction (equation 1)

a2 variance of a particular factor, x

D water depth

8 energy slope

M total path distance hi a single wavelength

N number of measurements

v velocity

a a constant of integration (equation 2)

7 unit weight of water

X wavelength

if pi, ratio of circumference to radius

p radius of curvature of path

a standard deviation

0 angle that path at a given point makes with mean downpath direction

co maximum angle a path makes with mean downpath direction

IV
PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

RIVER MEANDERS THEORY OF MINIMUM VARIANCE

By WALTER B. LANGBEIN and LUNA B. LEOPOLD

ABSTRACT

Meanders are the result of erosion-deposition processes tending 

toward the most stable form in which the variability of certain 

essential properties is minimized. This minimization involves 

the adjustment of the planimetric geometry and the hydraulic 

factors of depth, velocity, and local slope.

The planimetric geometry of a meander is that of a random 

walk whose most frequent form minimizes the sum of the squares 

of the changes in direction in each successive unit length. The 

direction angles are then sine functions of channel distance. 

This yields a meander shape typically present in meandering 

rivers and has the characteristic that the ratio of meander length 

to average radius of curvature in the bend is 4.7.

Depth, velocity, and slope are shown by field observations to 

be adjusted so as to decrease the variance of shear and the friction 

factor in a meander curve over that in an otherwise comparable 

straight reach of the same river.

Since theory and observation indicate meanders achieve the 

minimum variance postulated, it follows that for channels in 

which alternating pools and riffles occur, meandering is the most 

probable form of channel geometry and thus is more stable 

geometry than a straight or nonmeandering alinement.

INTRODUCTION

So ubiquitous are curves in rivers and so common are 

smooth and regular meander forms, they have attracted 

the interest of investigators from many disciplines. 

The widespread geographic distribution of such forms 

and their occurrence in various settings had hardly been 

appreciated before air travel became common. Moreover, ah* travel has emphasized how commonly the form 

of valleys not merely the river channels within them 

assumes a regular sinuosity which is comparable to 

meanders in river channels. Also, investigations of the 

physical characteristics of glaciers and oceans as well as 

landfonns led to the recognition that analogous forms 

occur in melt-water channels developed on glaciers and 

even in the currents of the Gulf Stream.

The striking similarity in physical form of curves in 

these various settings is the result of certain geometric 

proportions apparently common to all, that is, a nearly 

constant ratio of radius of curvature to meander length 

and of radius of curvature to channel width (Leopold

and Wolman, 1960, p. 774). This leads to visual 

similarity regardless of scale. When one looks at a 

stream on a planimetric map without first glancing at 

map scale, it is not immediately obvious whether it is a 

large river or a small stream.

Explanation of river meanders have been varied. It 

has been suggested that meanders are caused by such 

processes as 

regular erosion and deposition; 

secondary circulation in the cross-sectional plane; 

seiche effect analogous to lake seiches. 

Attempts, however, to utilize these theories to calculate the forms of meanders have failed.

Although various phenomena including some of 

those mentioned above, such as cross circulation are 

intimately involved in the deviation of rivers from a 

straight course, the development of meanders is patently related to the superposition of many diverse 

effects. Although each of these individual effects is 

in itself completely deterministic, so many of them 

occur that they cannot be followed in detail. As 

postulated in this paper, such effects can be treated as 

if they were stochastic that is, as if they occurred in 

a random fashion.

This paper examines the consequences of this postulate in relation to (1) the planimetric geometry of 

meanders, and (2) the variations in such hydraulic 

properties as depth, velocity, and slope in meanders as 

contrasted with straight reaches.

The second problem required new data. These were 

obtained during the snowmelt season of high discharge 

in the years 1959-64 by the junior author with the 

invaluable and untiring assistance of William W. 

Emmett and Robert M. Myrick. Leon W. Wiard was 

with us for some of the work. To each of them, the 

authors are very grateful for their important contribution, measured in part by the internal consistency 

of the various field data, and the satisfactory closures 

of surveys made under difficult conditions.

HI
112 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

MEANDER GEOMETRY

Meandering in rivers can be considered in two contexts, the first involves the whole profile from any 

headwater tributary downstream through the main 

trunk that is, the longitudinal profile of the river system. The second context includes a meandering reach 

of a river in which the channel in its lateral migrations 

may take various planimetric forms or paths between 

two points in the valley.

The river network as a whole is an open system 

tending toward a steady state and within which several 

hydraulically related factors are mutually interacting 

and adjusting specifically, velocity, depth, width, 

hydraulic resistance, and slope. These adjust to 

accommodate the discharge and load contributed by 

the drainage basin. The adjustment takes place not 

only by erosion and deposition but also by variation 

in bed forms which affect hydraulic resistance and thus 

local competence to carry debris. Previous application of the theory of minimum variance to the whole 

river system shows that downstream change of slope 

(long profile) is intimately related to downstream 

changes in other hydraulic variables. Because discharge increases in the downstream direction, minimum 

variance of power expenditure per unit length leads 

toward greater concavity in the longitudinal profile. 

However, greater concavity is opposed by the tendency 

for uniform power expenditures on each unit of bed 

area.

In the context of the whole river system, a meandering 

segment, often but not always concentrated in downstream rather than upstream portions of the system, 

tends to provide greater concavity by lengthening the 

downstream portion of the profile. By increasing the 

concavity of the profile, the product of discharge and 

slope, or power per unit length becomes more uniform 

along a stream that increases in flow downstream. 

Thus the meander decreases the variance of power per 

unit length. Though the occurrence of meanders affects 

the stream length and thus the river profile, channel 

slope in the context of the whole river is one of the 

adjustable and dependent parameters, being determined 

by mutual interaction with the other dependent factors.

In the second context of any reach or segment of 

river length, then, average slope is given, and local 

changes of channel plan must maintain this average 

slope. Between any two points on the valley floor, 

however, a variety of paths are possible, any of which 

would maintain the same slope and thus the same 

length. A thesis of the present paper is that the 

meander form occupies a most probable among all 

possible paths. This path is defined by a random-walk 

model.

PATH OF GREATEST PROBABILITY BETWEEN TWO 

POINTS»

The model is a simple one; a river has a finite probability p to deviate by an angle d<j> from its previous 

direction hi progressing an elemental 'distance ds along 

its path. The probability distribution as a function 

of deviation angle may be assumed to be normal 

(Gaussian) which would be described by

(1)

where a is the standard deviation and where c is 

defined by the condition that J'dp^l.Q.

The actual meander path then corresponds to the 

most probable river path proceeding between two 

points A and B, if the direction of flow at the point A 

and the length of the path between A and B is fixed 

and the probability of a change in direction per unit 

river length is given by the probability distribution 

above. This formulation of the problem of river 

meanders is identical to that of a class of random-walk 

problems which have been studied by Von Schelling 

(1951, 1964). The Von Schelling solution demonstrated that the arc length s is defined by the following 

elliptic integral:

.-1 f 7=*= (2)

a. J V2(a cos ^)

where <j> is the direction angle measured from the line 

AB and a is a constant of integration. It is convenient to set

a=cos co (3)

in which co becomes the maximum angle the path makes 

from the origin with the mean direction. Curves for 

co 40°, 90°, 110°, which all show patterns characteristically seen in river meanders, are given in figure 1 

(after Von Schelling, 1964, p. 8). Von Schelling (1951) 

showed that a general condition for the most frequent 

path for a continuous curve of given length between 

the two points, A and B, is

. .

 =a minimum, (4)

where As is a unit distance along the path and p is the 

radius of curvature of the path in that unit distance. 

But since

/ A «\

(5)

where A<£ is the angle by which direction is changed 

in distance As, therefore,

 

AS = aminimum. (6)

i Acknowledgments are due A. E. Scheldegger for his assistance in clarifying the 

mathematical relationships of this section.
RIVER MEANDERS THEORY OF MINIMUM VARIANCE

Since the sum of all the directional changes is zero, or

0. (7)

113

equation 6 is the most probable condition in which the 

variance (mean square of deviations in direction) is 

minimum.

FIGURE 1. Examples of most frequent random-walk paths of given lengths between 

two specified points on a plane. Adapted from H. von Schelling.

THE SINE-GENERATED CURVE

For graphing meanders it is easier to make use of the 

approximation that

is a close approximation for

d$ J ds~ ff^

r 

 cos co)

(8a)

(8b)

where angles co and <£ are deviations from the central 

axis with the downstream direction as zero. With 

this simplification

/2(l COSco)

or

(9)

(10)

where M is the total path distance along a meander, 

and where w is the maximum angle the path makes 

from the mean downvalley direction. The maximum 

possible value of w is 2.2 radians, for at this angle 

meander limbs intersect. When w equals 2.2 radians, 

the pattern is a closed figure "8."

Comparison of equations 9 and 10 indicates that

(11)

"\ 2i \ 1 cos co)

all angles being in radians, as before.

DISTANCE ALONG CHANNEL

FIGURE 2. Theoretical meander in plan view (^4) and a plot of its direction angle <f> 

as a function of distance along the channel path (B).

Inasmuch as the ratio w/-\2(l cosw) is nearly constant ( = 1.05), over the range of possible values of u, 

meander path lengths, M, are inversely proportional to 

the standard deviation or; thus, M=Q.6/cr.

Equation 10 defines a curve in which the direction 4> 

is a sine function of the distance s along the meander. 

Figure 2 shows a theoretical meander developed from a 

sine function. The meander thus is not a sine curve, 

but will be referred to as a ''sine-generated curve."

Thus meander loops are generated by the sine curve 

defined by equation 10, and the amount of horseshoe 

looping depends on the value of w. This means that at

o

relative distance ^ equal to % and 1, <£ has a value of 

zero or the channel is locally directed in the mean

«

direction. At distance j£ equal to % and % the value

of <£ has its largest value w, as can be seen in figure 2. 

The graph on figure 2A has been constructed for a 

value of w=110°, and corresponds almost exactly to 

the 110° curve of figure 1 calculated by the exact 

equations of Von Schelling.

It will be noted that the plan view of the channel 

(fig. 2A) is not sinusoidal; only the channel direction 

changes as sinusoidal function of distance (fig. 2B). 

The meander itself is more rounded and has a relatively 

uniform radius of curvature in the bend portion. This 

can be noted in the fact that a sine curve has quite a 

straight segment as it crosses the x -axis.
H4 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

The tangent to the sine function in figure 2B at any 

point is A0/As which is the reciprocal of the local 

radius of curvature of the meander. The sine curve is 

nearly straight as it crosses the zero axis. Therefore, 

the radius of curvature is nearly constant in a meander 

bend over two portions covering fully a third of the 

length of each loop.

There are, of course, plenty of irregular meanders 

but those which are fairly uniform display a close accordance to a true sine curve when channel direction 

is plotted against distance along the channel, as will be 

shown.

ANALYSIS OF SOME FIELD EXAMPLES

Some field examples presented in figure 3 will now be 

discussed briefly with a view to demonstrating that 

among a variety of meander shapes the sine-generated 

curve fits the actual shape quite well and better than 

alternatives.

Figure 3A, upper part, shows the famous Greenville 

bends at Greenville, Miss., before the artificial cutoffs 

changed the pattern. The crosses in the lower diagram 

represent values of channel direction, <£ relative to a 

chosen zero azimuth, which for convenience is the mean 

downstream direction plotted against channel distance. The full curve in the lower diagram is a sine 

curve chosen in wavelength and amplitude to approximate the river data. A sine curve that was fitted to 

the crosses was used to generate a plot of channel 

direction against distance which has been superimposed 

on the map of the river as the dashed line in the upper 

diagram. It can be seen that there is reasonable agreement.

The same technique has been used in the other examples in figure 3 chosen to cover a range in shape from 

oxbows to flat sinuous form, and including a variety of 

sizes. Figure 3B shows a stream that not only is not 

as oxbowed as the Greenville bends but also represents 

a river of much smaller size, the Mississippi River 

being about three-quarters of a mile wide and Blackrock 

Creek about 50 feet wide.

Figure 3(7 shows the famous Paw Paw, W. Va., bends 

of the Potomac River, similar to and in the same area 

as the great meanders of the tributary Shenandoah 

River. The curves in these rivers are characterized by 

being exceptionally elongated in that the amplitude is 

unusually large for the wavelength and both are large 

for the river width, characteristics believed to be influenced by elongation along the direction of a fracture 

system in the rock (Hack and Young, 1959). Despite 

these peculiarities the sine-generated curve fits the 

river well.

Figures 3D and 3E are among the best known incised 

meanders in the western United States, where the 

Colorado River and some of its tributaries are a thousand or more feet below the lowest of the surrounding 

benches, the cliffs rising at very steep angles up from 

the river's very edge. Even these provide a plot of 

direction versus length which is closely approximated 

by a sine curve.

By far the most symmetrical and uniform meander 

reach we have ever seen in the field is the Popo Agie 

River near Hudson, Wyo., for which flow data will be 

discussed later. The relation of channel direction to 

distance is compared with a sine curve in the lower part 

of figure 3F.

The laboratory experiments conducted by Friedkin 

(1945) provide an example of near-ideal meanders. 

The meander shown in figure 4 is one of those that 

developed greatest sinuosity. The channel shown on 

figure 4A was developed in Mississippi River sand on a 

"valley" slope of 0.006, by rates of flow that varied 

from 0.05 to 0.24 cfs over an 18-hour period on a 

schedule that simulated the fluctuations of discharge 

on the Mississippi River. The sand bed was homogeneous and the meander mapped by Friedkin is more 

regular than those usually observed in natural streams.

The direction angles of the thalweg were measured at 

intervals of 2.5 feet. These direction angles are 

plotted on the graph, figure 4J5. Since each meander 

had a thalweg length of 50 feet, the data for the two 

meanders are shown on the same graph. Where the 

two bends had the same direction only one point is 

plotted. The data correspond closely to the sine 

curve shown.

COMPARISON OF VARIANCES OF DIFFERENT 

MEANDER CURVES

As a close approximation to the theoretical minimum, 

when a meander is such that the direction, 0, in a given 

unit length As is a sine function of the distance along 

the curve, then the sum of squares of changes in direction 

from the mean direction is less than for any other 

common curve. In figure 5, four curves are presented. 

One consists of two portions of circles which have 

been joined. Another is a pure sine curve. The 

third is a parabola. A fourth is sine-generated in 

that, as has been explained, the change of direction 

bears a sinusoidal relation to distance along the curve. 

All four have the same wavelength and sinuosity 

the ratio of curve length to straight line distance. 

However, the sums of squares of the changes in direction 

as measured in degrees over 10 equally spaced lengths 

along the curves differ greatly as follows:
RIVER MEANDERS THEORY OF MINIMUM VARIANCE H5

Curve 

Parabola._______________________ 5210

Sine curve______.___________________ 5200

Circular curve..._______________________ 4840

Sine-generated curve.___________________ 3940

The "sine-generated" curve has the least sum of squares.

The theoretical minimum curve is identical within

practical limits of drawing.

MEANDER LENGTH, SINUOSITY, AND BEND RADIUS

The planimetric geometry of river meanders has been 

defined as follows

<t>=w sin ^ 2ir (10)

where <£ equals the direction at location s, co is the maximum angle the meander takes relative to the general 

direction of the river, and M is the channel length of a 

meander. The values of co and M can be defined further 

as follows:

The angle co is a unique function of the sinuosity and 

is independent of the meander length. It ranges from 

zero for a straight line path of zero sinuosity to a maximum of 125° for "gooseneck" meanders at point of incipient crossing. The sinuosity, k, equals the average 

of the values of cos <f> over the range from 0=0 to <£=co. 

Thus a relationship can be defined between k and co. An 

approximate algebraic expression 2 is

firry

a (radians) =2.2-*/ = Y K

or

Sinuosity as measured by parameter k, as explained, is 

thought to be a consequence of the profile development 

which is only secondarily influenced by reaction from 

the meander development.

In the random-walk model, the standard deviation 

of changes in direction per unit of distance is a which

1 Relation of u to sinuosity, k.

where

2 COS 0AS

-r> 

M

With an assumed value of u>, values of <t> at 24 equally spaced intervals of 

s/Af were computed. The reciprocal of the average value of cos <t> equals 

the sinuosity, k. 

Values of« and k so computed were as follows:

(radians) 

0.5...... 1.06

(radians) 

1.5 .....

1.0....___..______........ 1.30 1.75...

The values of k approximate the folio wing 

*-4.84/(1.84-«»)

1.9ti 

2.67

w=2.2-Vs?

has the dimension of L~ l or the reciprocal of length. 

As previously shown, a is inversely proportional to 

meander length.

Bend radius is related to wave length and is virtually independent of sinuosity. Defined as before, 

as the average over the % of channel length for which <£ 

is nearly linearly related to channel distance, bend 

radius R is

Since $ ranges from +0.5co to 0.5co over this nearlinear range, A<£=co. Substituting for co its algebraic 

equivalent 3 in terms of sinuosity,

>=2.

and since M=k\, bend radius equals

Some typical values for bend radius in terms of 

sinuosity are

wavelength

k

1. 25.. --------._.

1. 5.. ..____..___.

2.0--_.-.-.--.--.

2.5--..-----...-.

Bend radius

(B)

._._. 0. 215X

..... .20X

..... .22X

..... .24X

Ratio- bend radius 

4.6 

5.0 

4.5 

4.2

These values agree very closely with those found 

by measurement of actual meanders. Leopold and 

Wolman (1960) measured the meander characteristics 

of 50 samples, including rivers from various parts of 

the United States, that range in width from a few 

feet to a mile. For this group, in which sinuosities 

were dominantly between 1.1 and 2.0, the average 

ratio of meander length to radius of curvature was 

4.7, which is equivalent to .213X. The agreement 

with values listed above from the theory is satisfactory.

o

The curve defined by <£=co sin j-j 360° is believed

to underlie the stable form of meanders. That actual 

meanders are often irregular is well known, but as 

observed above, those meanders that are regular in 

geometry conform to this equation. Deviations (or 

"noise"), it is surmised, are due to two principal 

causes: (1) shifts from unstable to a stable form caused 

by random actions and varying flow, and (2) nonhomogeneities such as rock outcrops, differences in 

alluvium, or even trees.

The irregularities are more to be observed in "free" 

(that is, "living") meanders than in incised meanders. 

During incision, irregularities apparently tend to be

' See footnote[2, this page.
H6 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

1 2 MILES

0 50 100 FEET

Riverbanks

^.MISSISSIPPI RIVER AT GREENVILLE, MISSISSIPPI

100

50

50

100

5 10 15 20 25 

DISTANCE ALONG CHANNEL, IN .MILES

30

C. POTOMAC RIVER AT PAW PAW, 

xx\ WEST VIRGINIA

DISTANCE ALONG CHANNEL, IN THOUSANDS Of- FtiT

6 + 00

12 + 00

160

100

1+00

2+00 x - -^10+00 

B. BLACKROCK CREEK AT TOGWOTEE PASS, WYOMING

100

160

T

5 + 00 10 + 00 

DISTANCE ALONG CHANNEL, IN FEET

30

D. INCISED COLORADO RIVER AT THE LOOP, UTAH

0 10 20 30 40 

DISTANCE ALONG CHANNEL, IN THOUSANDS OF FEFT

FIGURE 3. MAPS OF RIVER CHANNELS AND

A, Upper, Map of channel com pared to sine-generated curve (dashed) ; lower, plot of actual channel direction against distance (crosses) and a sine curve (full line). 

B. Upper, Map of channel compared to sine-generated curvo (dashed); lower, plot of actual channel direction against distance (crosses) and a sine curve (full line). 

C, Upper, Map of channel corn pared to sine-generated curve (dashed); Jowier, plot of actual channel direction against distance (crosses) and a sine curve (full line). 

D, Upper, Planimetric map of river; lower, plot of actual channel direction against distance (crosses) and a sine curve (full line).
RIVER MEANDERS THEORY OF MINIMUM VARIANCE H7

E. INCISED SAN JUAN RIVER 

AT THE GOOSENECKS, UTAH

0 10 20 30 

DISTANCE ALONG CHANNEL, IN THOUSANDS OF FEET

EXPLANATION 

Riffle

100 0

i__i i

From planetable map by Leopold and Wolman, August 1953

Circles show measured direction of 

channel at each hundred feet 

distance along channel

F. POPO AGIE RIVER NEAR HUDSON,

90 5 + 00 10 + 00 15+00 20+00 25 + 00 

STATIONING ALONG CHANNEL, IN FEET

30 + 00 35 + 00 40 + 00

PLOTS OF CHANNEL DIRECTION AGAINST DISTANCE

E, Eight, Planimetric map of river; left, plot of actual channel direction against distance (crosses) and a sine curve (full line). F, Upper, Planimetric map of river;

lower, plot of actual channel direction against distance (crosses) and a sine curve (full line).
H8 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

10 FEET

"^ ~-Distances along Thalweg, 

in feet

+ 100

-100

0 10 20 30 40

B DISTANCE ALONG CHANNEL, IN FEET

FIGURE 4. Map of laboratory meander and plot of thalweg direction against distance. 

A, Plan view; B, Comparison of direction angles with sine curve.

averaged out and only the regular form is preserved, 

and where these intrench homogeneous rocks, the 

result is often a beautiful series of meanders as in 

the San Juan River or the Colorado River.

MEANDERS COMPARED WITH STRAIGHT REACHES

Meanders are common features of rivers, whereas 

straight reaches of any length are rare. It may be 

inferred, therefore, that straightness is a temporary 

state. As the usual and more stable form, according 

to the thesis of this report, the meander should be 

characterized by lower variances of the hydraulic factors, a property shown by Maddock (Written commun., 

1965) to prevail hi self-formed channels.

The review of the characteristics of meanders and 

the theories which had been advanced to explain them 

showed that quantitative data were nearly nonexistent 

on many aspects of the hydraulics of flow during those 

high stages or discharges at which channel adjustment 

actively takes place.

Because there are certain similarities and certain 

differences between straight segments or reaches of 

river and meandering reaches, it also seemed logical to 

devote special attention to comparison of straight and 

curved segments during channel-altering flow. An 

initial attempt to obtain such data was made during 

1954-58 on a small river in Maryland, near enough to 

home and office to be reached on short notice during 

the storm flows of winter. Though something was 

learned, it became clear that rivers which rise and fall 

in stage quickly do not allow a two-man team with 

modest equipment to make the desired observations.

Attention was then devoted to the rivers which drain

into the Missouri and Colorado River systems from 

the Wind River Range in central Wyoming, where good 

weather for field work generally exists at the base of 

the mountains during the week or two in late spring 

or early summer when snowmelt runoff reaches its peak. 

Some of the smaller streams presented the difficulty 

that the area of study lay 15 to 20 miles from the farthest watershed divide a distance sufficient to delay 

the peak discharge by 9 to 12 hours. Maximum snowmelt in midafternoon caused a diurnal peak flow which 

reached the study area shortly after midnight and 

meant surveying and stream gaging by flashlight and 

lantern illumination of level rod and notebook.

The plan of field work involved the selection of one 

or more meanders of relatively uniform shape which 

occurred in the same vicinity with a straight reach of 

river at least long enough to include two riffles separated 

by a pool section. No tributaries would enter between 

the curved and straight reaches so that at any time 

discharge in the two would be identical. It was 

assumed and checked visually, that the bed and bank 

materials were identical in the two reaches as well. 

Later bed sampling showed that there were some 

differences in bed-material size which had not been 

apparent to the eye.

A curve introduces an additional form of energy dissipation not present in a comparable straight reach, 

an energy loss due to change of flow direction (Leopold 

and others, 1960). This additional loss is concentrated 

in the zone of greatest curvature which occurs midway 

between the riffle bars. The additional loss is at a 

location where, in a straight reach, energy dissipation 

is smaller than the average for the whole reach.

The introduction of river bends, then, tends to equalize the energy dissipation through each unit length, 

but does so at the expense of greater contrasts in bed 

elevation. Moreover, curvature introduces a certain 

additional organization into the distribution of channelbed features. The field measurements were devised to 

measure and examine the variations in these quantities.

Though more will be said about individual study 

reaches when the data are discussed, a salient feature 

was apparent which had not previously been noticed 

when the first set of such field data was assembled; this 

feature can be seen in figure 6. Where the profiles of 

bed and water surface are plotted on the same graph 

for comparison, it is apparent that at moderately high 

stage from three-fourths bankfull to bankfull the 

larger-scale undulation of the riverbed caused by pool 

and shallow has been drowned out, as the streamgager 

would say, or no longer causes an undulation in the 

water-surface profile of the meander. In contrast, at 

the same stage in the straight reach the flat and steep 

alternations caused by pool and riffle are still discernible.
RIVER MEANDERS THEORY OF MINIMUM VARIANCE H9

On the other hand, the profile revealed that the undulation of the bed along the stream length is of larger 

magnitude in the curved than in the straight reach.

Considering the features exemplified in figure 6, one 

could reason as follows. In a straight reach of channel 

the dunes, bars, pools, and riffles form more or less 

independently of the channel pattern owing to grain 

interaction. To the occurrence of pool and riffle the 

channel must adjust. The riffle causes a zone of greaterthan-average steepness which is also a zone of greaterthan-average energy expenditure. The pool, on the 

other hand, being relatively free of large gravel on the 

bed and of larger depth relative to bed roughness elements, offers less resistance to flow and there the energy 

expenditure is less than average. The result is a 

stepped water-surface profile flat over pool and steep 

over riffle.

Once pools and riffles form with their consequent 

variations in depth, width remaining relatively uniform, 

then slopes must vary. If slopes vary so that bed shear 

remains constant, then slope would vary inversely with 

depth. However, this result would require that the 

friction factor vary directly as the square of the depth. 

If, on the other hand, the friction factor remained uniform and the slope varied accordingly, then bed shear 

would vary inversely as the square of the depth. Uniformity in bed shear and the friction factor is incompatible, and the slope adjusts so as to accommodate both 

about equably and minimize their total variance.

FIELD MEASUREMENTS

The general program of field measurements has been 

described. The main difficulty experienced was that 

the combination of conditions desired occurs rarely. 

A straight reach in proximity to a regular meander

Sine

Parabola

"Sine generated" 

Circular

curve does not have a high probability of occurrence, 

and to find such a combination, airplane photographic 

traverses at about 1,000 feet above the ground were 

flown for some distance along the east front of the 

Wind River Range.4 Even after reaches were chosen 

from the photographs, ground inspection resulted in 

discard of significant proportions of possible sites.

When a satisfactory site was located, a continuous 

water-stage recorder was established; benchmarks and 

staff gage were installed and the curved and straight 

reaches mapped by plane table. Cross sections were 

staked at such a spacing that about seven would be 

included in a length equal to one pool-and-riffle 

sequence.

Water-surface profiles were run by leveling at one or 

more stages of flow. Distances between water-surface 

shots were equal for a given stream; 10-foot distances 

were used on small streams having widths of 10 to 20 

feet and 25-foot spacing used for streams 50 to 100 

feet wide. Usually two rods, one on each streambank, 

accompanied the instrument. Shots were taken opposite one another along the two banks. Watersurface elevations were read to 0.01 foot on larger 

rivers, and to 0.001 foot on small ones. For the small 

rivers an attachment was used on a surveying rod 

which in construction resembled a standard point gage 

used in laboratory hydraulic practice for measurement 

of water-surface elevation.

Velocity measurements by current meter were 

usually made at various points across the stream at 

each cross section or at alternate ones. In the small 

streams these current-meter measurements were made 

by wading, those in the larger rivers from a canoe. For 

canoe measurements a tagline had to be stretched 

across the river at the staked cross sections, an operation fraught with some difficulty where brush lined the 

banks and when the velocity was high at near-bankfull 

stage.

Figures 6, 7, and 8 show examples of the planimetric 

maps and bed and water-surface profiles. The sinusoidal change of channel direction along the stream 

length for figure 7 was presented in figure 3.

REDUCTION OF DATA

Mean depths and mean velocities were calculated 

for each cross section. The average slopes of the water 

surface between the cross sections were also computed 

from the longitudinal profile. To reduce these quantities to nondimensional form, they were expressed in 

ratio to the respective means over each reach. The 

variances of these ratios were computed by the usual 

formula

FIGUEE 5. Comparison of four symmetrical sinuous curves having equal wavelength and sinuosity.

* The authors acknowledge the assistance of Herbert E. Skibitzke 'and David E. 

Jones in the photo reconnaissance.
H10 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

N \ N

(11) As shown by Maddock (1965), the behavior of many 

movable-bed streams can be explained by a tendency 

where o-/ is the variance of the quantity X and N is toward least variation in bed shear and in the friction 

number of measurements in each reach. factor. Accordingly, the variances of the bed shear

Curved/ 

reach

Straight 

reach

100 0 100 200 300FEET 

i i i___I___i___i

Curved reach

B

100 200 300 

DISTANCE ALONG CHANNEL, IN FEET

400

FIGURE 6. Plan and proflles of straight and curved reaches, Baldwin Creek near Lander, Wyo. A, Planimetric map showing location of reaches; 

B, Proflles of centerline of bed and average of water-surface elevations on the two sides of the stream; curved reach is dashed line and straight reach 

is full line.
RIVER MEANDERS THEORY OF MINIMUM VARIANCE Hll

14

Stationing along channel Gravel bars as of stage=3.65

75 

1000 2000 3000 4000 5000 

DISTANCE ALONG CHANNEL, IN FEET

6000 7000

FIGURE 7. Plan and profile of curved and straight reaches of Popo Agie River, % mile below Hudson, Wyo.

and of the friction factor again considered as ratios to 

their respective means were also computed. Bed shear is 

equal to the product yDS, where y is the unit weight 

of water, D is the depth, and S is the slope of the energy 

profile. The unit weight of water is a constant that 

may be neglected in this analysis of variances. In this 

study water-surface slopes will be used in lieu of slopes 

of the energy profile. This involves an assumption 

that the velocity head is small relative to the accuracy 

of measurement of water surface. In any case, where 

the velocity-head corrections were applied, they were 

either small or illogical, and so were neglected.

Similarly, the variance of the quantity DS/v2, which 

is proportional to the Darcy-Weisbach friction factor, 

was computed.

Width is not included in the analysis because it is 

relatively uniform, and has no characteristic that distinguishes it between curved and straight reaches.

Table 1 shows a sample computation for Pole 

Creek, and table 2 lists a summary for the five streams 

surveyed.

COMPARISON OF RESULTS

The results are listed in table 2. It will be noticed 

that the variance of slope in each reach is larger than 

that of depth or velocity. The contrast, however, is 

less in the meandering than in the straight reach; this 

reduction may be noted by the figures in the column 

headed a, which is the ratio of the variance of slope to 

the sum of those of depth and velocity. Among the
H12 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OP RIVERS

From planetable map by Leopold and Emmett

0 100 200 300 FEET

10 + 00/

B

 - 4

I I I I I I I 

10 + 00 Stationing along straight reach

0 + 00 5 + 00 10 + 00 Stationing along curved reach 15 + 00 

DISTANCE ALONG CHANNEL. IN FEET

FIGURE 8. Plan and profile of straight and curved reaches on Pole Creek at Clark's Ranch near Pinedale, Wyo. A, Planimetric map showing location of 

reaches; B, Profiles of centerline of bed and average of water-surface elevations on the two sides of the stream; curved reach is dashed line and straight 

reach is a full line.

five rivers studied, this ratio is uniformly less in the 

meandering reaches than in the straight reaches.

The data listed in table 2 also show that the variance 

in bed shear and in the friction factor are uniformly 

lower in the curved than in the straight reach. The 

data on table 2 suggest that the decrease in the variance 

of bed shear and the friction factor is related to the 

sinuosity. As shown on figure 9, the greater the 

sinuosity the less the average of these two variances.

When one considers, for example, that the variances 

of depth and velocity are greater in the curved reach, 

and even that the variance of slope may be greater (as 

for example Pole Creek), the fact that the product DS 

and the ratio DS/V2 have consistently lower variances 

must reflect a higher correlation between these variables in the meandered than in the straight reach.

This improved correlation is shown in the following 

tabulation (table 3) of the respective coefficients of 

determination (= square of the coefficient of correlation.)

Figure 10 shows graphical plots for the Pole Creek 

data listed in table 2. The graph shows that in the 

meanders, depths, velocities, and slopes are more 

systematically organized than in straight reaches.

To summarize, the detailed comparisons of meandering and straight reaches confirm the hypothesis that in 

a meander curve the hydraulic parameters are so adjusted that greater uniformity (less variability) is established among them. If a river channel is considered in 

a steady state, then the form achieved should be such 

as to avoid concentrating variability in one aspect at the 

expense of another. Particularly a steady-state form 

minimizes, the variations in forces on the boundary 
RIVER MEANDERS THEORY OF MINIMUM VARIANCE H13

TABLE 1. Field data for Pole Creek at dark's Ranch near Pinedale, Wyo., June 1964

[Discharge =350 cfs, H bankfull]

Station (feet)

Depth

Feet Ratio to 

mean

Velocity (»)

Feet per 

sec.

Ratio to 

mean

Mean depth 

(D) in reach 

(ratio to 

mean)

Mean velocity 

(») in reach 

(ratio to 

mean)

Slope (S) in reach

Feet per 

1,000 ft.

Ratio to 

mean

DS (ratio 

to mean)

DS/»»(ratio 

to mean)

Straight reach

0__ ________ --___----_-__

150-------_-_-____-_____-__

300_ ____________________

450...... __________________

600---.-___. _______ ________

750-.--... ..._.._....._._.

900_--._......_.__. ._._..._

Mean. _______________

1. 52

2.26

1.92

2.01

1.79

1.69

2.29

1.92

0.79

1. 17

1. 00

1.05

.93

.88

1.19

019

3.67

3.26

2.44

2.44

3.52

2. 12

2.26

2.81

1.30

1. 12

.89

.87

1.25

.76

.80

. 042

0.98

1.08

1.02

.99

.90

1.04

1.21

1.01

.88

1.06

1.01

.78

1.67

2.14

1.14

.80

2.24

.30

1.38

1.20

1. 55

.83

.58

1. 62

.22

.26

1. 18

1.67

.85

.58

1. 46

.23

.25

0.81

1. 64

1. 10

. 52

1.43

.38

.21

Curved reach

0 _______________________

150-_-__--_-__._. __________

300-____________.__________

450_.________-_-___________

600_____________________.__

750--__-__-_____-_-__-__-_

900-__---_-______. ---_.-__

1050_-_-_-_-______________-

1200___________________.___

1350---____________-______-

Variance

2.68

1.96

1.44

2.37

3.77

1.95

1.25

1.25

2.34

2. 67

2. 17

1.23

.90

. 66

1.09

1.74

.90

.58

.58

1.08

1. 23

. 067

2.24

2.45

3.46

2.45

1. 63

2.62

4. 10

4. 02

2.20

1.98

2. 72

0.82

.90

1.27

.90

.60

.96

1. 50

1.48

.81

.73

.088

1.06

.78

.88

1.42

1.32

.74

.58

.83

1. 16

0.88

1.08

1.08

.75

.89

1.23

1.49

1. 15

.77

1.03

2.20

2.00

.70

1. 14

3.94

3.97

3.17

.27

2.05

0.50

1.07

.98

.34

. 56

1.92

1.94

1.55

. 13

.41

0.53

.84

.86

.48

. 74

1.42

1.13

1.29

. 15

. 15

0. 69

.72

.74

.85

1.22

.94

. 51

.98

.25

.07

that is, the steady-state form tends toward more uniform distribution of both bed stress and the friction 

factor.

INTERPRETIVE DISCUSSION

After a review of data and theories concerning river 

meanders Leopold and Wolman (1960) concluded that 

meander geometry is "related in some unknown manner 

to a more general mechanical principle" (p. 774). The 

status of knowledge suggested that the basic reason for 

meandering is related to energy expenditure, and they 

concluded (p. 788) as follows. "Perhaps abrupt discontinuities in the rate of energy expenditure in a reach 

of channel are less compatible with conditions of equilibrium than is a more or less continuous or uniform rate 

of energy loss."

Subsequent work resulted in the postulate that the 

behavior of rivers is such as to minimize the variations 

in their several properties (Leopold and Langbein, 

1963), and the present work shows how this postulate 

applies to river meanders as had been prognosticated.

Total variability cannot be zero. A reduction in the 

variability in one factor is usually accompanied by an 

increase in that of another. For example, in the meander, the sine-generated curve has greater variability in 

changes in direction than the circle. This greater variability in changes in direction is, however, such as to 

decrease the total angular change. The sine-generated 

curve, as an approximation to the theoretical curve, 

minimizes the total effect.

The meandering river has greater changes in bed 

contours than a straight reach of a river. However,
H14 PHYSIOGRAPHIC AND HYDRAULIC STUDIES OF RIVERS

TABLE 2. Sinuosity and variances for straight and curved reaches on five streams 

Sinuosity: ratio of path distance to downvalley distance. Variances: <?D, depth; a*, velocity; <r|> slope; 4a> bed shear; <?,, Darcy-Weisbach friction facton

Stream Sinuosity (K) 4 *\ 4 a "DS

2 »y

Straight reaches

Wind River near Dubois, Wyo _ _ __________

Baldwin Creek near Lander, Wyo __ ______

1.05

1. 12

1.0

1.02

1 O9

0.019

.03

.065

.026

.053

0.042

.017

.058

.017

0.26

. 16

.94

.70

.28

4.2

3.4

7.8

16.

2.6

0.25

. 14

.55

.46

.32

0. 21

. 11

.40

.30

Curved reaches

Popo Agie River near Hudson, Wyo _______ _

2. 1

2.9

2.0

1.65

1.65

0.067

.074

.048

.044

. 145

0.088

.032

.017

.029

0.41

. 13

.20

.38

.64

2.6

1.2

3. 1

5.2

2.2

0. 15

.06

. 15

.28

.21

0.07

.073

. 14

. 12

3.0

§ 2.0

1.0

Curved reaches

,Straight reaches

0.1 0.2 0.3 0.4 

VARIANCE OF SHEAR AND FRICTION FACTOR

0.5

FIGURE 0. Relation of meander sinuosity to average of variances of friction factor

and shear.

TABLE 3. Comparison of coefficients of determination

t coefficient of linear determination between depth and slope; rfs, coefficient of 

linear determination between velocity and slope]

Pole Creek _____ ..

Wind River __ . _ . _ ......... . .

Baldwin Creek _____ ... ___ ..

Popo Agie River. ____ ......

Mill Creek . -. 

Straight reach

fas

0.044 

.072 

.56 

.30 

0

rls

0.27 

.062 

.28 

.40

Curved reach

rls

0.72 

.55 

.42

.58

QQ

&

0.90 

.42 

.73 

.62

- 2

§ 1

u 

0

O

0 

r =C

Straig 

re

OO 

.52

tit 

ach

1

Curvec reach

r=0.95

1

WATER-SURFACE SLOPE, IN FEET PER HUNDRED FEET

x

I '

u 0 

r = -0

0 O 

.21

Straig 

re

0

0

it 

ach

= -0.85

Curved reach

01201234 

WATER-SURFACE SLOPE, IN FEET PER HUNDRED FEET

FIGURE 10. Relations between velocity, depth, and slope, in straight and curved 

reaches, Pole Creek, near Pinedale, Wyo. r=correlation coefficient.

these changes in depth produce changes in the slope of 

the water surface and so reduce the variability in bed 

shear and in the friction factor, as well as to lessen the 

contrast between the variances of depth and slope.
RIVER MEANDERS THEORY OF MINIMUM VARIANCE H15

These considerations lead to the inference that the 

meandering pattern is more stable than a straight reach 

in streams otherwise comparable. The meanders themselves shift continuously; the meandering behavior is 

stable through time.

This discussion concerns the ideal case of uniform 

lithology. Nature is never so uniform and there are 

changes in rock hardness, structural controls, and other 

heterogeneities related to earth history. Yet, despite 

these natural inhomogeneities, the theoretical forms 

show clearly.

This does not imply that a change from meandering 

to straight course does not occur in nature. The inference is that such reversals are likely to be less common than the maintenance of a meandering pattern. 

The adjustment toward this stable pattern is, as in 

other geomorphic processes, made by the mechanical 

effects of erosion and deposition. The theory of minimum variance adjustment describes the net river behavior, not the processes.

REFERENCES

Hack, J.T., and Young, R. S., 1959, Intrenched meanders of the 

North Fork of the Shenandoah River, Va.: U.S. Geol. 

Survey Prof. Paper 354-A, p. 1-10.

Langbein, W. B., 1964, Geometry of river channels: Jour. 

Hydro. Div., Am. Soc. Civil Engineers, p. 301-312.

Leopold, L. B., Bagnold, R. A., Wolman, M. G., and Brush, 

L. M., I960, Flow resistance in sinuous or irregular channels: U.S. Geol. Survey Prof. Paper 282-D, p. 111-134.

Leopold, L. B., and Langbein, W. B., 1962, The concept of 

entropy in landscape evolution: U.S. Geol. Survey Prof. 

Paper 500-A, 20 p.

Leopold, L. B., and Wolman, M. G., 1957, River channel patterns braided, meandering, and straight: U.S. Geol. Survey Prof. Paper 282-B, 84 p.

 1960, River meanders: Bull. Geol. Soc. Am., v. 71, p.

769-794. 

Von Schelling, Hermann, 1951, Most frequent particle paths in

a plane: Am. Geophys. Union Trans., v. 32, p. 222-226.

 1964, Most frequent random walks: Gen. Elec. Co. 

Rept. 64GL92, Schenectady, N.Y.

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