Bibliography and Discussion of 

Flood-Routing Methods and 

Unsteady Flow in Channels

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1690

Prepared in cooperation with the 

Soil Conservation Service of the 

U.S. Department of Agriculture
Bibliography and Discussion of 

Flood-Routing Methods and 

Unsteady Flow in Channels

?y VUJICA M. YEVDJEVICH

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1690

^repared in cooperation with the 

^oil Conservation Service of the 

rJ.S. Department of Agriculture

NITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1964
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

The U.S. Geological Survey Library has cataloged this publication as follows:

Yevdjevich, Vujica M 1913-

Bibliography and discussion of flood-routing methods and1 

unsteady flow in channels. Washington, U.S. Govt. Print. 

Off., 1964.

iii, 235 p. 24 cm. (U.S. Geological Survey. Water-supply paper 

1690)

Prepared in cooperation with the Soil Conservation Service of the 

U.S. Dept. of Agriculture.

1. Flood control Bibl. 2. Flood control. 3. Floods. I.U.S. Soil Conservation Service. II. Title. (Series)

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

Washington, D.C., 20402
CONTENTS

	Page 

Introduction. _____________________________________________________ 1

Unsteady flow in channels and reservoirs----------------------------- 2

Classification of water waves ____-_________-____-_-_--_-----__- 3

Mathematical tools_________-_______-___________-_----------__- 7

Long waves-_________________-______-__-___-_-____----_-_- 7

Surges. ____-----_-_-_--_---_-_-__-_----_------_--_----__- 11

Intermediate waves-_--_______-____-_-_--___-____--.----_-_- 11

Flood routing_____________________________________________________ 12

Definitions and distinctions_--__-_--__--__-_____-----_-----__-_- 12

Problems solved by flood routing,_______________________________ 14

General approach____________________________________________ 14

Methods of flood routing___________________________________________ 15

Criteria for classification.______________________________________ 15

Characteristics of methods____________________________________ 15

Routing of flood waves by two partial differential equations-_______ 15

Analytical integration______________________________________ 16

Integration by finite-differences method (or method of instantaneous graphs)_-__________________________-______-______- 17

Integration by method of characteristics____________________ 18

Conclusions_____________________________________________ 19

Flood routing by storage differential equation.____________________ 20

Analytical integration_____________________.________________ 21

Numerical integration by finite differences-------------------- 21

Graphical integration by finite differences,___________________ 22

Coefficient methods-____-________-_____-__-___-___-__--_-__ 23

Devices and speedups_---_--_-___-__-_-_-__-_-_----_-___-__ 24

Hydraulic models________________________________________ 25

Electrical machines and analogs,____________________________ 26

Conclusions.-_ ____________________________________________ 27

Flood routing based on simplified partial differential equations-_____ 27

Empirical methods.___________________________________________ 28

Routing of surges and steep waves____________________________ 28

Additional remarks on unsteady flow and flood routing,________________ 29

Problems for study________________________________________________ 30

Bibliography_ ___________________________________________________ 33

List of symbols______________________________________________ 36

Chronological bibliography and abstracts_----__---_--___-_-__-_-_ 37

Author index.________________________________________________ 221

Subject index.________________________________________________ 226

m

BIBLIOGRAPHY AND DISCUSSION OF FLOOD-ROUTING 

METHODS AND UNSTEADY FLOW IN CHANNELS

By VTJJICA M. YEVDJEVICH

INTRODUCTION

Several methods of channel flood routing are in use; for example, 

about half a dozen methods are used by the Soil Conservation Service 

of the U.S. Department of Agriculture. Considerable disagreement 

on flood-routing methods exists between agencies, and much time 

is being spent in an attempt to improve these methods or to develop 

new ones. A survey of published literature on the subject is needed 

to eliminate loss of time and to increase efficiency of studies now in 

progress. To meet this need, a cooperative project was initiated 

between the Soil Conservation Service and the Geological Survey, 

U.S. Department of the Interior; funds were provided by the Soil 

Conservation Service, and the work was done by the Geological 

Survey.

The objective of this project was the preparation of a bibliography 

of literature on channel routing and a commentary on the various 

methods. No attempt was made to single out one particular method 

for universal use; however, the compilations and comments indicate 

the inutility of some methods and establishes the limits of application 

of those that can be used.

This report consists of a general discussion of unsteady flow in 

channels and the present status of flood-routing methods, plus a 

bibliography of domestic and foreign literature. The bibliography 

is arranged chronologically and contains abstracts of all papers that 

were obtainable (about 90 percent of them) within the time limits 

of the project; items are indexed by authors and by subjects.

Helpful background material was provided by a study of unsteady 

flow sponsored by the Corps of Engineers, U.S. Army, through the 

National Bureau of Standards and the Geological Survey. Coordination with the Soil Conservation Service was maintained through 

Harold O. Ogrosky, Chief, Hydrology Branch, Engineering Division. 

Special acknowledgment is due the staff of the Geological Survey 

library for their cooperation and extra effort in obtaining not readily 

available and foreign publications.

1
2 FLOOD-KOUTING METHODS AND UNSTEADY FLOW

UNSTEADY FLOW IN CHANNELS AND RESERVOIRS

In analytical studies, water flow in natural or artificial channels is 

generally classified as steady or unsteady. Flow of water in river 

channels, canals, reservoirs, lakes, pools and free surface flow in conduits or tunnels, where the velocity changes with time, is defined as 

unsteady flow (nonpermanent, nonstationary, or with time variable 

flow). Water flow in channels is almost always unsteady flow, 

but when the change of discharge with time is very gradual, unsteady 

flow approximates steady flow. The discharge hydrographs of flow 

in natural water courses are largely comprised of flood waves followed 

by their recession curves. Only those flows occurring during a 

prolonged drought and those occurring for short time intervals at 

the highest and lowest points of the hydrograph can be considered 

to be steady flow. In hydraulic problems it is important to know 

when an unsteady flow may properly be treated as a steady flow. 

For practical purposes, the answer is directed by judgment rather 

than by criteria that have been developed mathematically or experimentally.

The mathematical treatment of unsteady flow in any channel is 

considered to be among the most difficult problems in fluid mechanics. 

Basically, this difficulty exists because too many variables enter into 

the functional relationship and because developed differential equations cannot be integrated in closed forms except in very simple 

situations. Apart from these difficulties, it may be assumed that the 

velocity-distribution and friction-resistance factors in unsteady 

flow are generally approximated by those valid for steady flow, which 

are, in turn, approximations of the true relationships in steady flow. 

All formulas developed for unsteady flow, whether based on mathematics or experimentation, must be considered to be only approximations, and uses of the formulas should be cognizant of the basic assumptions and simplifications made in their derivation.

Because the velocity of flow in large reservoirs and lakes is much 

smaller than the velocity in channels, the approximations of unsteady 

flow usually differ. The influence of factors neglected in the approximations is much less in analyses pertaining to large bodies of water 

than for those pertaining to narrow sloping channels. Many cases 

fall between these two extremes, and it is often difficult to distinguish 

between a reservoir and a channel. The differentiation is usually 

based on judgment rather than on an application of objective criteria.

Waves are classified according to type to facilitate mathematical 

or experimental treatment. The meaning of "wave" must be understood in a broad sense, thus: a surface wave is any change with 

time in the surface shape of a body of water with the change being
UNSTEADY FLOW IN CHANNELS AND RESERVOIRS 3

propagated along the body of water. A standing wave is the limit

 of propagation waves.

The water wave in a river channel or reservoir is generally conceived as being any change (disturbance, intumescence) of discharge 

.and stage with time or any unsteady movement in or along the body of 

water. Unsteady water movement at sea, in the form of sea waves, is 

treated as surface-wave motion. Unsteady flow along channels 

is also analyzed as movement of surface waves, and the general 

theory of surface water waves applies to both phenomena. In 

physics, water waves are placed in the same category as the waves of 

other media, and the principles applicable to gas waves, capillarity 

waves, elastic waves, and others can be applied with moderate success 

to water waves in channels and reservoirs. However, the diversity 

of variables, especially those defining the boundary conditions, such 

as shape and roughness of channels, and the contact of two fluids, 

air and water, makes the treatment of channel water waves very 

specific. In practical application, the analogy of water waves with 

waves of other media is still further decreased in its importance.

Hydraulic research on waves in channels has been carried on by 

three basic methods: by theoretical (or analytical) studies, by experiments using small scale models, and by observations in nature. 

Combinations of these methods have given the most useful results.

Studies of unsteady flow in channels were started more than 150 

years ago with the work of the French mathematicians Laplace 

and Lagrange. The statement of the Lagrange celerity formula for 

small waves in shallow water provided the impetus for these studies. 

The chronological annotated bibliography, which is the last part 

of this report, shows the amount of knowledge that has been amassed 

since the beginning of research on unsteady flow. The bibliography 

also shows when the different methods of unsteady-flow analysis 

were popular and how these methods (theoretical, experimental, and

 observational) have been combined by individual workers.

Knowledge concerning unsteady flow in channels is summarized 

in numerous text books that present both general and specific treatment of the topic. This body of literature covers topics from the 

point of view of fluid mechanics or of general or applied hydraulics, 

and in terms of hydrology. The most important books in many 

languages are abstracted in the chronological bibliography.

CLASSIFICATION OF WATER WAVES

There is no unique classification of waves, but use of identifying 

criteria permits classification for study purposes.

The waves are generally classified as being: (1) orbital waves 

(nontranslatory), waves whose water particles describe closed tra-
4 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

jectories during wave movement, such as sea waves; and (2) translatory waves, whose water particles constantly progress during the 

wave movement. Unsteady flow, as applied to flood routing, is usually that of translatory wave movement, but may also be that of 

waves of small length moving in channels in which the water is at rest.

By comparing the direction of wave movement with the channel 

axis, waves may be classified as being either longitudinal or transverse. Although the transverse waves have hydraulic implications 

in some problems, unsteady free surface flow or flood routing is 

concerned only by longitudinal waves.

Waves are further classified as being deep-water, when only the 

surface layers of water are disturbed by the wave moment, or shallowwater waves, when the entire cross section of the body of water is 

disturbed by wave movement. The flood-routing problems are all 

of the shallow-wave type, except that the movement of floods along 

the lakes, reservoirs, and pools is influenced highly by density currents, which are caused either by differences in sediment load or 

differences in temperature.

Only two basic types of unsteady flow are generally considered 

important in hydraulics of surface flow in channels: discontinuous 

unsteady flow (progressing surges, bores, and depressions), and very 

gradually varying unsteady flow (progressing long waves in reservoirs, 

or long flood waves in rivers). These two basic types are generally 

differentiated in mathematical analysis and formulas involved, each 

being treated with somewhat different assumptions. Because all 

transitions between these two extremes exist in nature, the usual 

approach in analysis is to apply one of the two extremes to the case 

at hand, being mindful that the resulting evaluation of wave, as 

computed by corresponding formulas, is subject to error.

Waves are further classified according to the main forces that control wave movement and evaluation. In this respect they are classified as waves subject wholly or mainly to gravitational or momentum 

control (where friction losses are small and generally neglected, as 

for progressing steep surges, bores, and depressions), and waves subject largely to friction control as well as to the gravitational or momentum control (where friction losses are approximately of the same 

or greater influence as the gravity force, as is generally true for long 

waves, or current flood waves).

The waves in channels may be classified as being solitary waves 

or a train of waves. The solitary wave is considered to be a gradually 

varied wave followed theoretically on both extremes in channel by 

steady flow, with one rising limb, one peak, and one recession limb. 

For the most part, individual flood waves are treated as solitary 

translatory waves. The close sequence of several waves creates a
UNSTEADY FLOW IN CHANNELS AND RESERVOIRS 5

wave train. The characteristics of succeeding waves may be quite 

diiferent. This difference occurs when several flood waves follow 

in close sequence, and the second flood wave is superimposed on the 

recession of the first, the third is then superimposed on the recession 

limb of both previous waves, and so forth. Conversely, the waves 

may be nearly of the same shape, amplitude, and period, such as 

sea waves or tidal waves that penetrate estuaries and other channels, 

regular release waves that emit from powerplants and lock operations, and a group of incident and reflection waves. The close proximity of succeeding waves determines whether the individual wave 

of a wave train shall be treated as the solitary wave, or if the complete 

wave train should be treated as a whole. Sometimes the waves are 

considered as simple (solitary), multiple (several, but a restricted 

number of waves, or superimposed flood waves), and train wave 

(large number of individual waves, such as roll waves).

Direction of wave movement is a further criterion for wave classification. When the waves move along a channel that has a horizontal 

bottom and in which the water is at rest, the direction of the wave's 

movement is not important. If the direction of wave movement is 

the same as the direction of the slope of a channel, the wave direction 

is considered to be downstream. Wave movement that is opposite to 

the slope of channel has an upstream wave direction. Although the 

water usually moves downstream in river channels, wave movement 

may also create upstream water movement, such as a flood wave 

that moves upstream from the junction on a tributary or joining 

river and the tidal movement along estuaries. The upstream and 

downstream waves, therefore, correspond to the direction of slope of 

channel, and not to the direction of water flow. Generally, in flood 

routing in rivers, the direction of channel slope and of water flow 

coincide, but there are important exceptions.

When a wave moves along a channel that already contains steady 

flow (uniform or nonuniform) or a very gradual unsteady flow, the 

wave surface can be higher or lower, or occasionally both, than the 

level of the underlying regimen of flow. If the wave surface is higher, 

the wave is referred to as being positive; if the wave surface is lower, 

the wave is considered to be negative; and if the wave surface is both 

higher and lower than the level of flow, it is a combined positivenegative wave.

Waves are further classified as being single-faced (more commonly 

called "monoclinal") and two-faced (normal). A surge wave (positivebore or negative-depression wave), which starts with abrupt discontinuity in water level and discharge, passes either to a monoclinal 

wave (a negative wave always passes to monoclinal), or proceeds as 

an abrupt wave (bore).
6 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Normal waves also conceived as being composed of two monoclinai 

parts are classed as being symmetrical and asymmetrical (skew) wavesThe foregoing discussion is summarized in the following table:

Descriptive criteria and classification of translatory, longitudinal, shallow waves

Criterion Classification 

Rate of variation________________________ Surges

Intermediate waves 

Long waves 

Controlling force________________________ Gravity

Gravity and friction 

Frequency of occurrence________________ Simple solitary

Multiple 

Wave train 

Direction of movement_________________ Relative to channel bed slope

Downstream 

Upstream

Relative to underlying flow 

Downstream 

Upstream 

Wave surface elevation________________ Relative to level of underlying flow

Positive (higher) 

Negative (lower) 

Positive-negative . 

Form_ _______ ________________________ Single-faced

Two-faced

Symmetrical 

Asymmetrical

By this classification, it is implied that only translatory, longitudinal and shallow-water waves are applicable to those under study 

as flood waves in channels and conduits having free surface flowOrbital, transverse, and deep-water waves are not applicable.

Treatment of the different aspects of wave movement in channels 

is complex, due to the diversity of wave forms and the multiplicity 

of underlying factors which influence their evaluation.

For a sound approach to computing the transformations of theflood wave progressing along a body of water it is important that a 

clear definition first be made of the type of wave involved, its physical 

characteristics, and all possible forms to which it can change by wave 

translation along a channel subject to changing characteristics. The 

second step is the selection of the tools best adapted for treating that 

particular wave and channel complex. The third step is an evaluation 

of reliability of the results to be obtained; this evaluation takes into 

account the underlying assumptions in the selection of tools and in the 

method of interpretation and application of the tools, apart from their 

dependence on the accuracy of background data.
UNSTEADY FLOW IN CHANNELS AND RESERVOIRS 7

MATHEMATICAL TOOLS 

L.ONG WAVES

The basic partial differential equations for the gradually varied 

unsteady flow developed by De Saint- Venant in 1871, are the continuity (or mass-conservation) equation

and the momentum (or dynamic) equation

'-o »

where A=area of cross section, Q discharge, a?= channel length, 

£=time, g=lateral outflow (lateral inflow negative) per unit channel 

length; T7=mean water velocity (Q/A), g acceleration of gravity, 

S/= friction slope (determined by an appropriate formula, as Manning's or Chezy's), and He= distance of the energy line to a reference 

level, which is

*-H.+«g-H+.+«g (3)

where Hg= distance of water level to the reference level, H= water 

depth in channel, 0= distance of channel bottom to the reference 

level, and a. and ft are velocity-distribution coefficients of velocity 

distribution in the cross section. Putting Q=AV, and

in which 5=width at the water surface, and using equation 3, the 

two equations 1 and 2 become

and

ZH , ft dF V dF ^ ^ -^ -^rr+tt >:~='-> o o/ A

dx g dt g dx f gA J

in which S0= dz/dx.

The last term in equations 2 and 5 is usually neglected, assuming 

the momentum of lateral flows to be small.

Four terms in equation 4 when multiplied by (dxdt ) give dimension 

of volume. In the order of sequence, they have the following physical 

meanings: (1) storage caused by the rate of rise of level with time; 

(2) wedge storage, owing to the difference of depths at the beginning
8 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

and the end of the elementary reach dx; (3) prism storage; and (4) 

storage (positive or negative) due to lateral inflow or outflow. Six 

terms in equation 5, in the order of sequence, have the following 

physical meanings: (1) rate of change of depth (depth-taper or 

depth-change term), or the slope created by the change of depth 

along the channel: (2) acceleration term (ratio of accelerations, or 

ratio of the change of velocity with time and the acceleration of 

gravity, also called acceleration-head term, velocity-hydrograph 

inclination, localized acceleration gradient), or the slope created by 

the acceleration; (3) rate of change of velocity head (called also 

dynamic head, velocity-head term, energy grade line inclination, 

instantaneous energy gradient), or the slope created by the change 

of velocity head along the channel; (4) bottom-slope or bed-slope 

term; (5) friction slope, hydraulic-friction term, or hydraulic gradient; 

(6) part of the gradient on the energy line created by lateral outflow 

or inflow.

The equation for gradually varied unsteady flow is variously expressed in different papers: dimensionless as in equation 5, or with 

dimension of head, acceleration, momentum, energy, or other term.

The basic and general assumptions underlying the development 

and the applicability of equations 4 and 5 are:

1. The vertical acceleration can be neglected in comparison with the 

horizontal acceleration, because of the gradual change of depth 

and discharge with time and with distance.

2. The flow is gradually varied, or the vertical velocities are considered small in comparison with the longitudinal velocities.

3. The flow patterns are the same in vertical planes parallel to the 

longitudinal axis of the channel (in curvilinear channels the 

vertical cylindrical surfaces parallel to the longitudinal axis 

have the same flow pattern), or the influence of the channel 

sides and of its curvature on the flow patterns can be neglected.

4. The velocity distribution along a vertical, in unsteady flow, is the 

same as in steady flow, or the velocity-distribution coefficients 

a and ft in equation 5 are constants for given values of discharge, 

depth, and velocity, or the unsteady flow does not influence these 

coefficients (but because this assumption depends on the rate of 

change of velocities with time and distance, it is justified only 

when there is a small rate of change).

5. The friction resistance hi unsteady flow is the same as hi steady

flow. This assumption is justified only if the rate of change of

velocities with respect to time and distance is small.

Because no data exist in the literature to show the numerical effect

of these factors, either individually or as a group, on the computed

or observed waves along the river channel, evidence is lacking for
UNSTEADY FLOW IN CHANNELS AND RESERVOIRS 9

justification of these five assumptions in terms of the specific characteristics of a wave and of channel and lateral inflow or outflow.

The continuity equation involves the cross-section area, and the 

momentum equation is based on the rate of change of energy line, or 

of water surface position, plus the dynamic head. For irregular channels, having changing bottom slope and irregular cross-section shape 

and area, the bridge between these two partial differential equations 

introduces the first complexity in the mathematical analysis. The 

rate of change of cross-section characteristics, as related to the bottom 

position, and the rate of change of bottom slope with distance, when 

expressed in functional form, generally provide the bridge between the 

two equations. Some assumptions and simplifications for cross section and for bottom position are necessary to enable analytical treatment of equations 4 and 5.

The practical problems in following wave progress along a channel 

are those of wave deformation (attenuation and amplification), and 

the celerity with which either of the wave elements moves along the 

channel.

When the rates of change of acceleration and velocity along the 

channel are small in comparison with the rate of change of depth along 

the channel, or when the second and third terms in equation 5 are 

negligible in comparison with the first term, then only equation 4 or 

the continuity equation may be used. The volume of storage produced by a change of wave height then acts as the main factor, causing 

reduction of the peak of the wave as it progresses along the channel. 

This equation, generally called the water-storage (differential) equation, starts from the known-balance equation and implies that for a 

limited reach of channel the inflow minus the outflow for a given time 

interval is equal to the change of water storage in that reach. The 

two partial differential equations are thus reduced to a simple differential equation.

The few existant studies on this subject do not give sufficient evidence either of numerical and analytical criteria when a long wave 

may be treated by the simple water-storage differential equation, or 

of the errors caused by the neglect of momentum partial differential 

equation.

The celerity of propagation of long waves depends on the definition 

of magnitude of the property involved, such as the discharge, the 

depth, the center of gravity of the wave, the peak, or the same water 

volume before the propagating depth. Because the waves deform in 

travelling along the channel, the celerities of the individual discharges 

and depths differ, although they are related. Because the integration 

of equations 4 and 5, for V and H the dependent and x and t the independent variables, gives two functions: V=Fi (x, t} and H=F2
10 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

(x, t"), it can be seen that the celerities of any discharge and the depth 

at which this discharge passes are interrelated.

Studies of the motion of waves in a uniform rectangular channel 

have revealed that the wave propagates with the celerity

(6)

in which g the acceleration of gravity, where the celerity is referred 

to undisturbed water. This equation is valid when the wave length 

is great in comparison with the original water depth (Ho), or the wave 

is long, and when the wave height (h) is small in comparison to the 

initial depth (H0 ).

For both positive and negative waves, occurring in water moving 

along the channel with mean velocity V, the celerity is

(7)

where the first plus sign is for the wave moving downstream and the 

first minus sign is for the wave moving upstream, and where the 

second plus sign is for the positive wave of small height and the 

corresponding minus sign is for the negative wave of small height.

The literature has supplied many corrections of equations 6 and 7 

for the waves which are not small. Some of these corrections involve 

use of other second- or third-order terms under the radical sign, or 

introduce a coefficient n, which takes into account both the friction 

resistance and the higher order terms under the radical.

For long waves, where the friction force is of greater or same magnitude as the inertia and gravity forces, the general expression for the 

wave celerity (Graef, Kleitz, Seddon, Forchheimer) is

r_dQ__l dQ C~dA~BdH (8)

For a monoclinal wave of two depths, HI and H2 (with area AI and A2 ), 

and the corresponding discharges Qi and Q2 , the celerity is

_Qi-Qt_ Qi-Q2 = AQ .

If Chezy's formula for friction resistance is used for large rectangular 

channels, equation 8 gives C=3V/2; for Manning's formula, (7=517/3. 

Equation 8 states that the slope of tangent at the rating curve, with 

respect to the Q axis, divided by the channel width, is the celerity of 

wave for the corresponding discharge and height.
UNSTEADY FLOW IN CHANNELS AND RESERVOIRS 11 

SURGES

If the surges are small, positive or negative, equation 7 is generally 

'used for celerity. If the surge height (A) is not small, then the expression for surge celerity is

C=^sh(j--i)+gj (10)

where A=surge height, A=cross-section area before, and ^Lj=crosssection area after the surge passes, and where/= distance from the 

center of gravity of area AA=A Ai, or area AA=Ai A, to the 

lower level during the passage of the surge.

The expression for the mean velocity of accelerated water is

V=V0±U=V0± -- igh -l+g/ (11)

-where U is the velocity created by acceleration, plus or minus depending on which direction the surge of height (h) is moving in comparison 

with the direction of velocity (VQ ) of undisturbed fluid.

Many theoretical and experimental formulas relate the surge height 

to discharge change and channel characteristics and express celerity 

and velocity of water in surge movement.

INTERMEDIATE WAVES

Between the very long wave and a vertical surge are many intermediate conditions. One extreme is the infinitely long wave, with 

Ji/L=Q, or close to zero, where the flow is almost completely governed 

by the channel friction. Equations 4 and 5 give an adequate mathematical description of the physics of this wave. The other extreme 

Is the vertical surge, with h/L large, as in tidal bores, sudden water

 release waves, dam-breach waves, and so forth, where the flow is 

entirely influenced by gravity (momentum or dynamic effect). Equations 10 and 11 give the celerity and the velocity of this wave movement.

All channel waves occur in the range between these two extremes. 

'Therefore \=h/L and a h/H^ can be the basic characteristics of 

.transient-wave shapes. Very few theoretical studies treat this intermediate condition of wave flow, and there are no general equations 

.having X and a as parameters and equations 4 and 5, and 10 and 11 

.as the limit conditions.

Whenever a wave departs from the extreme conditions and when the 

equations for these intermediate conditions are used, an approximation is always involved; a discrepancy exists between the analytical 

treatment of wave and the true physics of the wave. This situation, 

among others, explains why there are so many approximate hydraulic 

procedures for computing the behavior of translation wave in channels.
12 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

FLOOD ROUTING 

DEFINITIONS AND DISTINCTIONS

Flood routing is a term given to the group of methods and procedures, based either on the theory of unsteady flow (long wave or 

surges) or on the simple water storage equation by which a flood 

hydrograph at a point of a channel is determined from the known 

hydrograph at some point upstream or downstream by using the known 

channel characteristics and the characteristics of lateral inflow or outflow between the two points.

Three groups of parameters or characteristics define a flood-wave 

movement:

1. The wave characteristics, which are determined by initial, 

intermediate, or final conditions, by means of stage hydrographs, 

discharge hydrographs, and wave profiles (the profiles are 

generally longitudinal surface profiles for given times; however, 

under specific circumstances, wave profiles may be defined by 

mathematical functions with appropriate parameters);

2. The characteristics of channel, such as its cross section, bottom 

slopes, channel storage, roughness coefficients, junctions, or 

singularities;

3. The characteristics of lateral inflow or outflow, either surface or 

underground, or both.

For short reaches of channel, flood routing may be used also for 

determining the channel characteristics for known flood-wave characteristics and known lateral inflow or outflow, or it may be used for 

obtaining lateral inflow or outflow for known wave and channel 

characteristics.

As it is commonly employed, flood routing is a way of determining 

the shape and timing of a flood wave as it progresses along a channel.

It is necessary to distinguish between flood routing and the theory 

of wave movement in channels, generally called the unsteady flow. 

The two subjects are based on the same physical principles and relationships ; they differ only in the way they are used in practical determination of wave movement, or in the degree of approximation used 

in each to solve the wave-movement problems. In some countries, 

there is no difference in terminology or any distinction made between 

physics of channel waves, which is part of hydrodynamics and hydraulics, and flood waves in natural channels, which is part of 

hydrology. The distinction that exists between the two may be 

compared to the distinction made with reference to the underground 

flow of water from the point of view of fluid dynamics or hydrology. 

Flood routing may be conceived, therefore, as being the hydraulics of 

water waves in channels, treated in terms of the complex conditions 

that generally exist in hydrology.
FLOOD ROUTING 13

Differential equations for wave movement along channels are 

solved analytically by the introduction of many assumptions (prismatic channel, or constant channel cross section with constant bottom 

slope, with confined water flow, with uniform roughness across the 

contour and along the channel) which can be applied strictly only to 

man-made conduits such as canals, tunnels, pipes, and galleries. The 

celerity and deformation of the wave as it progresses along a conduit 

are usually referred to as "unsteady free-surface flow."

The closer the approach made by theoretical treatment is to 

actual channel conditions, the more complex becomes the relationship 

between variables, and the greater is the necessity for more data 

describing the channel, in order to arrive at a good approximation of 

the true hydrograph at a point downstream or upstream from the 

point where a hydrograph is initially known. Inasmuch as the 

obtainable data concerning channel conditions are always limited by 

economics and time, the complex procedure in studies of flood waves 

based on the complete theoretical coverage of the problem becomes 

less justified with a decrease of basic data coverage.

On one hand, approximations of theoretical tools are required to 

reproduce actual wave movement in natural channels and reservoirs, 

in those instances where all characteristics (such as bottom slope, 

cross-section shape and area, hydraulic radius, roughness coefficient, 

channel storage, width for given depth, as well as lateral inflow or 

outflow) vary significantly from place to place along the river. On the 

other hand, practical methods of computing flood-wave movement 

along the channel must be derived. Together, these approximations 

of tools and their application comprise the concept of flood routing.

The theory of unsteady water flow in channels has lost its unity 

during the century and a half of its development, but surely it has 

gained in scope by broader application.

Inasmuch as wave theory imposes a logical framework of relationship, which requires more or less simplification of assumptions, the 

various theoretical results or results of applied theory do not agree 

with observations, and relationships between the two groups of results 

are not always clearly indicated. Therefore, some justification exists 

for maintaining for awhile a distinction between the concepts of unsteady flow in conduits and flood routing in channels and reservoirs. 

However, the two fields will tend more and more to become one, as 

further research, experience, and the generalizations derived therefrom widen the area of common application.

The interrelationship of unsteady water flow and flood routing 

should be considered with the aforementioned definitions and distinctions kept in mind.

728-245 64 2
14 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

PBOBLEMS SOLVED BY FLOOD BOUTING

Flood-routing procedures have been developed chiefly to provide a 

broad scope for the study of floods traveling through reservoirs, lakes, 

and river channels. The steady increase in number of pools and reservoirs being built and operated for water-resources development has 

heightened the need for knowledge concerning the unsteady movement 

of water in reservoirs, rivers, and artificial channels.

Waves that occur in rivers and reservoirs may be created by natural 

or artificial forces, and the wave shapes that may result are so varied 

that it is possible to find, in practice, all types of waves that are described in the literature. Therefore, acquaintance with the results of 

many studies of many different types of waves is necessary for practical 

application to problems of unsteady flow in channels.

Flood routing was first used for determination of the time and 

magnitude of flood peaks occurring at points along a river as a flood 

traveled downstream. This information was needed for forecasting 

purposes and for the design of structures that would confine the flood 

at its crest. Further developments in water use have revealed the 

need for information concerning the shape of the flood wave (stage 

profile, discharge or stage hydrograph) as the flood moves downstream. 

Knowledge of peak timing and of the changes of peak magnitude is no 

longer sufficient to accomplish all the purposes of flood routing.

Currently, flood routing is employed for solution of a wide variety 

of problems associated with water use. Some of these include: (1) 

evaluating of past floods, for which records are incomplete; (2) determining hydrographs of channel flow from hypothetical design floods 

on tributaries and upstream reaches of the main channels; (3) forecasting floods along the main stem of a river, by use of observed or 

predicted hydrographs at key inlet points in the channel net; (4) determining hydrographs modified by reservoir storage; and (5) studying the effects on the downstream flow conditions of any operation of 

storage reservoirs or pools.

GENERAL APPROACH

The type of problem involved, the economy of data survey that is 

imposed, and the accuracy of results required are the factors determining the amount of departure from the basic theory that will be 

necessary for derivation of a feasible flood-routing procedure. The 

selection of the best procedure, among those available, is often difficult, 

but it may be aided by consideration of the following questions, with 

regard to the particular situation at hand:

1. In view of the known factors influencing wave movement of a given 

flood wave complex, what parts of the basic theory, or what differ-
METHODS OP FLOOD ROUTING 15

ential equation or equations, would provide the most suitable basis 

for approach?

2. What method of integration is feasible for solution of the differential equation or equations?

3. What degree of accuracy of flood wave prediction can be expected 

by use of the selected equations and integration method? 

Having answered these questions, the next steps are operational: 

data collection and arrangement for routing, flood routing, and analysis of results obtained.

METHODS OF FLOOD ROUTING

CRITERIA FOB CLASSIFICATION

Among many possible criteria for classifying flood-routing methods, 

the following may be convenient as guides for selection of method to 

to be used in a systematic study: (1) physical principles and equations used as the theoretical basis for flood routing; (2) the procedures 

or methods used for integration of basic differential equations or 

equation; (3) some specific assumptions and approximations used in 

treating the flood-wave movement; and (4) problems to be solved.

The first two criteria for classification of flood-routing methods 

seem the most important and will be used here.

CHARACTERISTICS OF METHODS

Individual flood-routing methods will not be specifically described 

here. Many books and papers, referred to in the bibliography and 

particularly in the index by subjects, give detailed analysis of one or 

several flood-routing methods. This discussion will consider such 

generalities of method as: characteristics of principal technique, data 

necessary, type of work involved, speed, flexibility, expenses necessary 

to carry out the routing, and reliability. The purpose, here, is to give 

a general description of different groups of flood-routing methods.

ROUTING OF FLOOD WAVES BY TWO PARTIAL DIFFERENTIAL

EQUATIONS

Those flood-routing methods that are based on the two partial 

differential equations 4 and 5, usually expressed in different forms 

according to the two dependent variables selected and the simplications introduced, generally give the closest approximations of the 

actual flood movement through channels, if the basic conditions for 

applying the two equations are approximately satisfied. The most 

important condition is that of gradual variability of the flood; this 

condition is fulfilled in the majority of natural floods. Many integration procedures are used, and this group of flood-routing methods 

is best classified according to the integration procedures or methods 

used.
16 FLOOD-ROUTING METHODS AND UNSTEADY FLOW 

ANALYTICAL, INTEGRATION

The partial differential equations 4 and 5, with friction slope Sr 

proportional to square of velocity or to square of discharge, are nonlinear differential equations of hyperbolic type with changing coefficients. Due to inherent mathematical difficulties, there is no way to 

carry out the integration in closed form, unless many simplifications 

are introduced.

The classical approach, made first by De Saint-Venant, neglects 

the friction resistance and assumes the channel to be horizontal with 

rectangular cross sections. These assumptions deviate so much from 

reality for flood-wave movement in channels that the wave characteristics resulting from analytical integration are generally not comparable with the true wave characteristics.

This classical approach by means of analytical integration is an 

extreme; it may be considered to be a rough approximation, and, in 

accuracy, can be compared with some of the very simple integration 

procedures of routing that are based on the water storage differential 

equation.

Use of the method of anlytical integration makes it necessary to 

approximate both the initial wave conditions (hydrograph, wave 

profile) and the boundary conditions (channel cross sections, stagedischarge relationships, slope, and lateral inflow or outflow) by 

analytical expressions, which are to be used in equations 4 and 5.

The hydrographs and wave profiles of long waves progressing in a 

channel may be approximated by considering them to be either symmetrical or asymmetrical waves, with functions of bell-shaped curves 

(binomial, Poisson, Pearson-functions, and others). The channel 

conditions are represented by the cross-section area or width as 

functions of depth and distance along channel, with roughness coefficient usually being used as a constant, and bottom slope used either 

as constant or as a function of distance along channel. The lateral 

inflow and outflow are taken as constant or approximated as simple 

functions of channel and lateral flow characteristics and of time.

The great diversity in shape and roughness of natural channels and 

the complexity of pattern of the lateral inflow and outflows tend to 

complicate the analytical expressions that approximate these conditions to the extent that the analytical integration of two partial 

differential equations becomes impossible.

In summary, the two partial differential equations for unsteady 

flow can be integrated analytically, with expressions for wave evolution, by rather restrictive and very simplifying conditions, which 

generally are not acceptable for solution of current practical problems.

There are different mathematical approaches for the analytical 

integration of simplified partial differential equations. One approach
METHODS OF FLOOD ROUTING 17

introduces a new variable, $, which is used in equations Q AV=

and yl=d0/dx, where 0 is the volume of water traveling with the

current between two cross-sectional planes.

Analytical integration has practical application in those cases 

when wave height (h) of the long wave is relatively very small. 

Neglect of high-order terms, in expressions developed in power series 

form, can then be justified.

INTEGRATION BY ITNITE-DIFPERENCES METHOD (OR METHOD OF INSTANTANEOUS GRAPHS)

The two partial differential equations 4 and 5 are usually approximated by the two finite-differences equations, replacing the increments 

dx, dt, dQ, dV, dA, dB, dH, by the finite differences Ax, At, AQ, AV, 

AA, AB, AH. At the same time the partial derivatives are replaced 

by ratios of finite differences :

by AH/Ax, dF/d* by AVI At, 

ior example, equations 4 and 5 become:

and

Where Ax and At are selected in some way, AV and AH are changes 

ior these finite differences, and B, V, A, g, SQ , and Sf are the mean 

values for both At and Ax. For given values, Ax and At, the mean 

values of all variables, including assumed V and H at the end of Ax 

and At, are computed, and AV and AH are then determined from 

equations 12 and 13. If the assumed V and H at the end of Ax and 

At are good, the values V-+-AV, and H-\-AH should give the assumed 

values. The channel is divided into reaches Axi, Aa^, Az3 , and so forth 

(equal or unequal), and the corresponding time-intervals Ati, At2, At3, 

and so forth, are determined. The trend is now to use the relationship

Ax<C At (14)

where C is an approximation of celerity along the reach, for the time 

interval. As (7 is a function of wave and channel characteristics, it 

changes continually during flood routing, and the selection of minimum 

C is a practical problem that must be solved before applying the finitedifferences method of integration.

The basic characteristics of this approximation are: (1) All variables during time At, and along channel reach Ax, change linearly 

(or stay constant); (2) the accuracy of the method depends on the
18 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

selection and relationship of finite differences At and Ax; (3) the smaller 

are the values of Ax and At, the greater is the computational work, and 

also the greater is the accuracy; (4) the values Ax and At depend also 

on the amount and accuracy of background data, and on initial and 

boundary conditions; for a given status in terms of available data 

there are the minimum values of Ax and At which can be economically 

justified by the amount of computational work required; (5) the values 

of variables computed for the end of Ati and for the downstream end of 

Axi, become the initial values for At2 and the values for the upstream 

end of Ax2~, (6) an iterative procedure is commonly used in determining 

the dependent variables, V and H, or Q and H, or Q and A, or others, 

at the end of time At, and channel section Aaji.

Use of this time-consuming method of trial-and-error procedures, 

which are normally performed, is justified only when results with 

the high degree of accuracy are required.

However, with development of the digital computer, which, in 

time, will provide fast and inexpensive computations, the present 

drawbacks in economy of performing the operations of the finitedifferences method of integration will be largely eliminated. The 

method can be expected, then, to be highly favored, inasmuch as it 

is probably the most accurate of all practical methods of flood routing 

in channels. The fast advent of new numerical methods helps this 

trend.

INTEGRATION BY METHOD OF CHARACTERISTICS

The results of integration are given for two dependent variables as 

functions V Fi(x,t), or H=F2 (x,t). These two functions represent 

surfaces in the space (V, x, t] or (H, x, f). If there is any discontinuity 

in the four partial derivatives of equations 4 and 5, dF/df, dF/da?, 

d£?/5i, and dfl/daj, these discontinuities propagate along the channel, 

and the projection of the position of discontinuities at surfaces F\ 

and F2 in (x, t) plane produces curves that are called "characteristics," 

or "characteristic lines."

These lines are usually curves, but in application may be replaced 

by straight lines.

The characteristic lines are usually very simplified and given in 

the form

(15) 

and

=g(S0 -Sf)dt (16)

which equations are applicable for rectangular prismatic channels. 

It can be proved that equations 15 and 16 are equivalent to equations 

4 and 5, and therefore can replace them.
METHODS OF FLOOD EOUTING 19

For other channel cross-sectional shapes, that take into consideration velocity distribution coefficients a and 13, equations 15 and 16 

become more complex and show characteristics that generally are 

curved lines.

Equations 15 and 16 are usually integrated by replacing dx and dt 

with Ax and At, and d(V+2-^H) with A(F+2V0H)- Many 

graphical procedures have been developed for this purpose that use 

approximations in the form of straight-line characteristics, in the 

finite-differences form.

Certain features of the method of integration by characterises 

should be pointed out for consideration of the method applicability 

for particular cases in flood routing: (1) its use of finite differences Ax 

and At; (2) its consideration of the long wave as being composed of 

many elementary waves in the form of small surges, so that for the 

time At and the reach Ax, the velocity change AV and height change 

AH are considered as discontinuities traveling with celerity V±-JgH 

(providing only a rough approximation in the case of long flood waves, 

where the friction forces are not negligible); (3) its use of straight-line 

characteristics as approximations instead of curve-line characteristics; 

(4) its use of graphical integration of equations; (5) the length and 

tediousness of the integration; (6) the use of both planes (x, t) and 

(V, H) ; and (7) the complexity of its procedure when friction resistance, channel slope, sudden changes of cross section, bifurcations, 

junctions, and similar changes, are to be taken into consideration.

Few actual flood-routing cases in natural channels can justify use 

of the method of integration by characteristics in view of the amount 

of work required and also in view of the limitations imposed by 

properties of the method itself.

With the advent of digital computers and new numerical methods, 

the labor of integration of finite differences equations 15 and 16 and 

of partial differential equations 4 and 5 was eliminated. The comparative advantage of these two methods will bear scrutiny. From 

the standpoint of accuracy and labor required for the graphical or 

semigraphical part of the two methods there is a better choice. But 

considering work relief afforded by the digital computer, the finitedifferences method as applied to the two partial-differential equations 

is favored sometimes over the finite-differences method as applied to 

the four-characteristic equations.

CONCLUSIONS

All three methods analytical, numerical of finite differences applied 

to partial differential equations, and numerical of finite differences 

applied to characteristics (ordinary) differential equations when 

applicable give sufficiently accurate results, if enough data are available and the methods are pushed to their limits of accuracy.
20 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

These methods can be successfully applied to the analysis of particular waves which have been observed. The prediction of wave 

shapes and movement, however, requires a considerable amount of 

trial-and-error-work, especially when the net of channels is complex 

(tides, flood through river branches).

The mathematical difficulties of analytical integration of the two 

partial differential equations, the need for a large amount of data, and 

the accompanying drawbacks of time-consuming procedures and cost 

in applying the other approximate methods of integrating, have 

occasioned development of other simpler, but generally less accurate, 

flood-routing methods.

FLOOD ROUTING BY STORAGE DIFFERENTIAL EQUATION

The methods generally called "storage-routing methods" are those 

that are based only on the water-storage differential equation, or on 

the continuity or mass-conservation differential equation, equation 4, 

based on the principle that for any reservoir or channel, defined as 

total reservoir, or reaches of channels or reservoirs, the inflow minus 

the outflow is equal to the stored or depleted water in a given tune 

interval.

When the space is defined for which the inflow, outflow, and change 

in water storage are considered (total reservoir, a reach of channel), 

equation 4 can be expressed as

Pdt-Qdt=dW=AdH

where P=inflow discharge, Q outflow discharge, and W= stored 

volume of water, with dW= AdH, where A=area of reservoir or channel 

water surface, and H= depth or elevation of that area. Inflow is given 

as P j(t], the storage is generally W=j(H), or W=f(Hi, H2 and so 

forth) for some depths, and if Q=j!(H), then, by elimination, the function W=f(Q) can be determined. In that case, equation 17 has 

only two variables Q and t, in form of differential equation, whose 

integration gives Q=j(£).

Equation 17 serves generally for the computation of relations 

between five functions: (1) inflow hydrograph, P j\(t)', (2) outflow 

hydrograph, Q=J2 (t); (3) stage hydrograph, H fa(t}; (4) outflow 

rating curve, Q f^H); and (5) storage function, W=f5 (H), or area 

function A=J6 (H), with five variables: Q, P, H, W, and t.

When three of five functions with boundary conditions are given 

(and three variables can be excluded), equation 17 enables the computation of the relation between two remaining variables.

In the application of equation 17, the following curves are usually 

known quantities: (1) storage or area function (obtained by land 

survey); (2) outflow rating curve (obtained by gaging, hydraulic
METHODS OF FLOOD ROUTING 21

computation, model study, or other means); and (3) inflow or outflow 

hydrograph. Two other functions must be computed: (1) outflow 

or inflow hydrograph, and (2) stage hydrograph.

The basic conditions of applying equation 17 for flood routing are: 

(1) the storage space (reservoir, channel reach) responds in less time 

to any unsteady inflow or outflow, than the time unit generally used 

for integrating equation 17 by finite differences; (2) the wave is 

long, so that the change of discharge is very gradual; (3) the accuracy 

of basic data and the required accuracy of results do not justify any 

method that takes into consideration dynamic effects in unsteady 

flow; (4) the velocity and velocity changes along the reservoir or 

channel reach are relatively small, so that the dynamic effect is negligible in comparison with the storage effect during wave movements.

This approach has been the basis for nearly all flood-routing studies 

made in reservoirs, and for the majority of reservoirs built the storage 

capacity has been determined by this storage differential equation. 

The equation has then been applied to flood-routing studies in channels, 

with some adaptations for the more complex relationship of discharge 

storage or stage storage in a channel reach.

Many integration methods have been developed. They will be 

only briefly discussed here.

ANALYTICAL, INTEGRATION

By fitting mathematical expressions for P =/(£), Q j(H) and 

W=f(H), or W=f(Q), and using these relationships in equation 17, 

a differential equation with variables Q and t is obtained. The type 

of the functions fitted to inflow hydrograph, outflow rating curve, 

and storage curve determines the possibility of carrying out integration 

in closed form.

This integration method was used successfully with schematic 

inflow hydrographs, and simple linear relationships of storage and 

outflow discharge. The formulas are obtained on this basis for 

computing decrease of flash-flood peaks in a reservoir with free 

spillway and relatively small water-surface fluctuations.

The difficulties in fitting natural inflow hydrographs by tractable 

mathematical expressions, and the difficulties of analytical integration 

when these expressions become complex, limit this integration method 

to specific problems.

NUMERICAL INTEGRATION BY FINITE DIFFERENCES

Equation (17), as written in finite-differences form, is

(P-Q) At=A AH=AW (18)

where P, Q and A are mean values during the time interval At, and 

corresponding level difference AH. Taking PI and P?, Ql and Q2 ,
22 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

and Ai and A2 , the values at the beginning and the end of At, with a 

linear change during a sufficiently small At, then

(19)

For known Pi, P2 , Qi} Ai and At, and known relationship of W and 

Q, or A and fl", and Q and H, it is possible by trial-and-error method 

to determine the value Qz- Expressing equation 19 in different ways, 

especially by using storage factors, (W+QAt/2) and (W QAt/2), as 

functions of Q, a trial-and-error method may be replaced by a direct 

numerical integration procedure.

Personal differences in arranging equation 18 or equation 19 for 

the step-by-step computations may account for differences of the 

tabular numerical procedures. Procedures vary according to which 

difference factor (At, AH, AW, AQ) is given at the beginning, and 

which must be determined.

Accuracy of method depends on the accuracy of selection of the 

basic difference factor (generally At), apart from the general accuracy 

involved in the use of only one differential equation.

The main problems to be solved in using equations 18 and 19 for 

flood routing along a river channel are: 1) relationship of discharge 

and storage in a reach (or stage and storage), which may involve 

other variables (slope, level difference of the reach, celerity of wave, 

and so forth); 2) selection of reaches Ax, of their limits and length; 

and 3) selection of time interval At for integration. Both Ax and At 

may be constant, or may change to facilitate handling of changing 

factors (channel or wave shape):

This numerical integration is the most common method used in 

flood routing through reservoirs.

GRAPHICAL INTEGRATION BY FINITE DIFFERENCES

As the functions represented in equations 17, 18, and 19, and also 

the finite differences, may be arranged in many combinations to enable 

graphical integration of a simple differential equation, there are 

numerous graphical procedures, with different shortcuts, for computing an outflow hydrograph or solving other problems. However, 

evaluation of the advantages and disadvantages of each procedure 

and its shortcuts would entail lengthy comparisons made by application of each method to a group of selected routing examples. 

A simple narrative description of the various procedures and short 

cuts would not properly illuminate them for comparative purposes.

There are two general approaches in graphical integration: (1) a 

mass-curve procedure, that represents the given hydrograph in form 

of its summation or mass curve and obtains the routed hydrograph
METHODS OF FLOOD ROUTING 23

in the form of its mass curve; and (2) a procedure that uses the inflow 

or outflow directly for integration. Flash floods are not suitable for 

routing by the mass-curve procedure, because the error in the determined outflow hydrograph increases.

There are many semigraphical methods, which combine a numerical 

tabular procedure with partial graphical integration. Generally, 

At is selected as a constant, although AH, AW, or AQ may be also 

selected as constants. The accuracy of results, neglecting errors 

due to use of only one differential equation, is dependent upon the 

length of basic difference (usually At}, the graphical scale and graphical procedure used. If the graphical procedure is well applied to a 

body of water having low velocities and a relatively large storage 

capacity, the accuracy of results corresponds to the accuracy of basic 

data for flood wave, for body of water, and for lateral inflows and 

outflows.

This method is also commonly used for flood routing. It is based 

on the mean or total inflow and outflow during time At, and during 

changes of AH or AW. The accuracy of results depends on At, with 

better results to be expected for smaller values of At. As the amount 

of work increases with a decrease of At (an increase of computational 

steps, and thus of total time for flood routing), At must be a reasonable 

value, when there are many flood routing applications to be made. 

The graphical methods are generally restricted to simple problems, 

because for complex problems and many routing computations, the 

time involved becomes economically prohibitive. In flood forecasting, usefulness of this method may be impaired by the great time lag 

between the data supply and the occurred results.

COEFFICIENT METHODS

The coefficient methods comprise a group of procedures that approximate, by simple relationships, the complex relationships existing 

between the volume of water stored in a channel reach and the hydraulic magnitudes (inflow and outflow discharge, stages, slope, or 

others). The simple relationships, in the form of coefficients, give 

weight in a specific way to each variable involved, the simplest being 

the weight coefficients for inflow and outflow discharge and for time 

of travel of a wave moving through a reach. _

Although the coefficients are mostly empirical, being determined 

from flood wave movements that have actually occurred, they are 

useful in the sense that they have broadened the application of the 

storage differential equation to include channel flows having rather 

large mean velocities.

The empirical coefficients take care of the relatively great effect 

of changing inflow and outflow on the water stored in a reach. There
24 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

are still many possibilities for theoretical development of coefficients 

on the basis of wave and channel parameters.

This group of procedures may be considered an attempt to bridge 

in a simple way two methods, by using two partial differential equations, and by using the storage differential equation as the basic tool, 

but by taking care of dynamic conditions in a reach by use of empirical 

coefficients.

Improvements of these methods may be expected in the future.

The principal disadvantage of the coefficient methods is their dependence upon empirical coefficients, for which a large body of data 

is necessary to provide reliable determinations.

A practical rule for selecting the reach length Ax is that Ax<CM, 

with C determined either as a small disturbance movement, or better, 

as the celerity of a long wave controlled by friction forces.

Good reach division of a channel having singularities (such as 

junctions, slope drops-rapids, controls, beginning and end of converging and diverging channel sections, and bridges and places with reliable stage-discharge relationships), may enhance the overall accuracy 

of the method. This is especially true in the case of flood-routing 

procedures in channels, which are based on a single storage differential 

equation.

The storage-discharge (or storage-stage) relationship is determined 

by one of the following methods or combinations thereof: (1) by observed inflows and outflows, when topographic data are lacking; (2) 

by using stage-discharge relations; (3) by valley topography and observed cross sections; (4) by channel dimensions in the case of regular 

channel shapes.

When bank inflows and outflows occur in a reach, routing and unrouting (reverse routing) can reveal the lateral inflows and outflowsin the reach, if they are not a function of certain hydraulic magnitudes (stage, discharge, boundary area, channel length, or others). 

This procedure may detect on a hydrograph the time and the effect 

of a levee break, as well as a later occurrence of a reverse flow.

DEVICES AND SPEEDUPS

There is a group of devices and speedups apart from electrical 

machines and analogs, that is designed to make flood routing faster 

and cheaper, with procedures that are less laborious and tedious than 

are the previous methods of integration based on the simple storage 

differential equation.

Such devices and speedups are: nomographs, templates and transparent scales, slide rules (straight-line, and circular), polar diagrams, 

and mechanical integrators. (Practically speaking, all devices and 

speedups that are generally available for solving first-order differential equations are being applied for flood routing.)
METHODS OF FLOOD ROUTING 25

Very little empirical data exist that provide comparative analysis 

of the different devices and speedups, based on same typical examples, 

to enable making a reliable estimate of their limitations and advantages.

The inventors or zealous users of some of these devices or speedups 

are inclined to make biased judgments on basis of rather limited comparisons with other devices and speedups, especially as concerns their 

individual advantages and disadvantages under specific conditions 

of flood waves, channel characteristics, and lateral flows.

HYDRAULIC MODELS

Hydraulic models, used for studies of unsteady flow, are nothing 

more than integrating machines or devices. Achievement of close 

similitude between the prototype and the model is a difficult problem 

when a natural channel is to be reproduced. The model needs to 

be distorted, with two or three scales for geometrical dimensions (one 

for cross section, another for length; or one for depths, another for 

widths, and a third for lengths), so that calibration of the model, 

to assure exactness between model and prototype, especially in 

roughness, and in hydrographs of a series of floods reproduced at 

various points, involves work which sometimes becomes prohibitive 

economically.

It is generally agreed that the technique of flood routing by hydraulic 

models should be resorted to only under special conditions, such as 

when: (l) the model is built for other purposes, and flood routing is 

only an additional study that involves supplemental but not excessive costs; (2) the model has some educational and worthwhile purposes apart from hydraulic studies and flood routing; and (3) the 

channel net is so complex that the hydraulic model can compete 

economically with other methods of integration.

The model generally is adapted to past floods of a given shape. 

If the flood shapes are changed, some adjustments in roughness, 

geometrical elements, or lateral inflows may be necessary. The same 

hydrographs of past floods at different places along the rivers may 

be produced by combination of distorted channel shapes, of roughness 

and of local inflows or outflows.

As the model scales for long rivers with tributaries must be limited 

for economical reasons, accuracy of model results is generally of the 

same order of magnitude as that obtained by the methods of integration by finite differences or by characteristics, or even storage flood 

routings. The models are reliable for the range of discharges tested 

during the calibration of models. When there are many floodrouting studies to be made, use of the model is very time consuming. 

Hydraulic models are relatively expensive computers.
26 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

As long as the graphical or numerical processes of integration could 

be performed by manual procedures only, the hydraulic models, 

as a method of integration, were considered by some specialists to 

be competitive and even feasible under specific conditions. The use 

of digital computers and electrical analogs seems to be highly competitive with hydraulic models, if the models are built mainly for 

the purpose of flood routing. The maintenance of hydraulic models 

involves expenses regardless of the needs for flood routing.

The hydraulic models are included here under the flood routing by 

storage differential equation, because the calibration of model, model 

distortion, and other approximations give to this integration procedure 

the same accuracy, in general, as the accuracy of other methods based 

on storage equation.

ELECTRICAL MACHINES AND ANALOQS

There are three types of electrical machines used in flood routing: 

(1) high speed, general purpose electric analog computers, which 

are flexible, speedy and have variable computing speeds, but any 

change in flow conditions requires alteration of programming (a 

delaying operation, performed manually); (2) differential analysers, 

which have been found very suitable, especially when the floodrouting method is based on the simple storage differential equation 

(the change of flow conditions can be handled readily, and the accuracy 

is very high); and (3) digital computers, which are the best choice 

for justifiably high accuracy, and where the change in flow conditions 

can be programmed automatically (the computational speed is greater 

than for an analog, but the storage capacity of the machine restricts 

its use).

It is evident that availability of electrical machines, both digital 

computers and analogs, will tend to make the integration of differential equations in flood routing a much faster and more accurate operation. But integrating machines require expensive equipment, 

special recording graph papers or printing tables, and trained 

personnel.

The accuracy of results is not generally dependent on integrating 

procedures, but rather upon the accuracy of basic data. Inasmuch 

as the use of digital computers and of analogs is relatively expensive, 

their use should always be justified from the point of view of speed, 

economy, or of accuracy that is commensurate with that of the basic 

data and with the needs of the problem.

The electrical machines have a promising future in flood routings.
METHODS OF FLOOD ROUTING 27 

CONCLUSIONS

Although flood-routing methods based on simple storage differential 

equations have been used ever since routings of inflow hydrographs 

through reservoirs and flood prediction along the rivers were started, 

there are only a few studies which compare the flood-routing methods 

based on two partial differential equations to those based on the 

storage differential equation.

The literature contains only a small number of studies which 

compare accuracy, cost, advantages and disadvantages of various 

methods, devices, or speedups under different conditions of waves, 

channel and lateral flow.

FLOOD ROUTING BASED ON SIMPLIFIED PARTIAL DIFFERENTIAL

EQUATIONS

This is a group of methods which are transitions between those 

methods which use both the continuity and the momentum equation 

and those methods which use the storage (continuity) equation only. 

Some terms in the equations, mostly in the momentum equation, are 

neglected, or some other assumptions are made (that is, constant 

bottom slope, linear change of channel characteristics between two 

cross sections used, separate routings of storage component and 

translation component of a flood wave, and others). Examples of

these intermediate cases are: neglect of acceleration term, -r -; which (jt

is the second term in equation 5; neglect of velocity head term, V j >

OX

which is the third term in equation 5, or neglect of both acceleration 

and velocity terms; division of the total discharge in two parts, as 

discharge of steady flow plus a changing discharge caused by the 

unsteady flow, with some simplifications in the momentum equation.

The basic characteristic of most of these transition methods is the 

use of wave celerity, so that both the wave translation as well as the 

channel storage are taken into account.

A second characteristic is the lack of comparative studies between 

the transition methods and other methods. For certain conditions, 

it could be demonstrated that the transition methods would give 

more accurate results than methods based on the storage differential 

equation, at little increase in cost. Another comparative study 

might show under what conditions the transition method would give 

less accurate results at substantial savings in work time, compared 

with methods based on the two partial differential equations. There 

has been more effort in the past to invent or derive new methods of 

integrating the different types of equations in flood routing than to 

analyze the limitations and advantages of each method in particular 

and to compare all of them in general.
28 FLOOD-ROUTING METHODS AND UNSTEADY FLOW 

EMPIRICAL METHODS

The empirical method of flood routing is mostly based on the 

relationships developed from past floods. Graphs or equations are 

developed, which relate different wave characteristics at adjacent 

places of a river system. The most common method applied is the 

stage-relationship, covering two or many stations, on the main channel 

and on tributaries.

This empirical procedure is generally used as the first step in 

developing a forecasting service on a river, with a general trend to 

replace it by more accurate and reliable procedure, as soon as data 

are sufficiently supplied.

The accuracy of these empirical methods is less than the accuracy 

of other flood-routing methods.

ROUTING OP SURGES AND STEEP WAVES

The literature on surges is very abundant (see Bibliography with 

abstracts and indexes) concerning their height, celerity, and evolution 

as they progress along the channel.

The formulas for celerity, surge height, and velocity of accelerated 

water by surge movement, as equations 6 and 7, or others, are the 

basis for the computation of surge movement.

As the surges are discontinuities of partial derivatives, which 

propagate in the (x, t} plane along the characteristics lines, the most 

logical method of surge computation is the method of characteristics, 

whenever there are many surges mutually interfering or reflecting 

during their progress along the channel.

Nearly all surges (except some bores) change their shape by progressing to an intermediate form between a surge and long wave. 

Negative surges (depressions) usually take the shape of monoclinal 

waves.

When a wave is still steep, in a position between surge and long 

wave, either some adaptations of the two partial differential equations 

or the water-storage equation, are currently used, or the wave is 

approximated by a surge and treated as such.

This approximation of computation of rather steep waves (bHfox, 

dH/d£, dF/dx, and dF/di are not small for some parts of the wave) 

by flood-routing procedures for long waves, or by formulas for surges, 

is always a source of errors, apart from errors involved in the methods 

applied. With existing theories of unsteady flow, it can be stated 

that for waves between surges and long waves, the routing methods 

will not produce results as accurately as can be obtained for pure 

surges or long waves.
FLOOD-ROUTING METHODS AND UNSTEADY FLOW 29

ADDITIONAL, REMARKS ON UNSTEADY FLOW AND FLOOD

ROUTING

After surveying the literature on unsteady flow in channels and 

reservoirs and on flood-routing methods, during the work on abstracting references given in the second part of this study, the following 

general remarks and derivations can be made:

1. The literature on unsteady channel flow and flood routing increases 

by an exponential law; it was nearly doubled in the 18-year 

period 1941-59. Tajking into consideration war years, with 

the reduced research activities in this field, the rate of increase 

may be now considered as a growth doubling the number of 

papers, reports, and chapters in books in about 15 years. By 

1975 it might be expected that the actual literature, if not by 

substantial contributions, will be doubled at least by the amount 

of references.

2. The screening of literature for the current use in research and 

practice becomes important, especially as new methods of 

computing wave movement and flood routing are added every 

year to a rather large amount of existing methods and 

procedures.

3. Nearly all physical aspects of channel waves are studied, either 

theoretically, or experimentally and by observations in nature, 

and many specific problems were attacked together with the 

basic problems.

4. The references in chronological order show a great deal of repetitive 

research, where the authors of new methods or new development 

were unaware of and unacquainted with previous studies 

which produced similar or even the same results.

5. The studies and developments of new flood-routing methods 

are often undertaken with approximately the same approach 

as previously used.

6. The trend in the last three decades has been towards development 

of new methods of integrating the basic differential equations, 

rather than to attack the basic equations with new approaches.

7. The increase of the number of methods for flood routing has 

probably come to the point where there is more need to investigate the feasibility of available methods than to develop 

new methods. The new methods can be developed as well 

from the analysis of existing methods as from new ideas, 

independent of the properties of existing methods.

8. There is a rather limited exchange of ideas between different 

countries in this field; it is not unusual that a large study in 

this field carries only the references from one country, with 

rather limited or unimportant references from other countries.

728-245 64 3
30 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

9. Studies of errors resulting from approximations used in the 

different mathematical tools for the specific wave-, channel-, 

and lateral-flow conditions are lacking.

10. Studies of errors resulting from insufficient basic data and information are also lacking.

11. A great deal of comparisons of flood routing results, computed 

by a given method, with the experiments or observations under 

natural conditions are done by the authors for their method, 

usually under the specific conditions for waves and channels.

12. There is a lack of studies comparing the feasibility of many 

flood-routing methods under different conditions, taking into 

account the type of data available, errors in basic tools used, 

accuracy of results, and the economical aspects of the methods 

used.

13. The techniques which exist for treating flood wave evolution 

along channels are manifold, but there is a clear deficiency in 

the appraisal of the characteristics, errors, and economic 

feasibility encountered by using these techniques.

PROBLEMS FOR STUDY

There are many problems in the physics of unsteady water flow, 

and especially when applied to flood routings, which need a new 

scientific approach and treatment.

The complexity of fluid mechanics of waves hi channels, especially 

when boundary characteristics change along channels, with new 

variables involved, is the source of many potential studies and 

research in the future. The actual approximations can be pushed 

much nearer to the real physical aspects of the wave movement.

A solitary wave propagates without changing the shape in a frictionless horizontal channel, at least as proved by some mathematical 

analyses. If the friction force is added, and the channel has a slope 

which approximately compensates for the mean head loss during 

wave propagation, the question is, can a solitary wave be without 

change of shape, since the constant slope does not compensate equally 

for the friction loss in all parts of the wave?

What are the criteria when a wave starts to steepen, or what are 

conditions under which a wave is stable and in attenuation, or is 

unstable and in amplification, under different conditions of natural 

or artificial channels?

What are the friction losses in an unsteady flow, as function of rate 

of change of velocity along the channel, taking into account the 

changes in boundary layer, and when can they be assumed to be 

equal to losses hi steady flow, with the given tolerance error?
PROBLEMS FOE STUDY 31

What is the real influence of vertical acceleration of steep waves 

on the wave movement?

How does unsteady flow influence the velocity distribution coefficients?

There are many problems open for further study in the physics of 

channel unsteady flow, which, when solved, may throw light on the 

order of approximation of mathematical tools now used and also how 

to improve the tools.

When flood-routing methods are concerned, it is probably as 

important to appraise the methods now in existence, as it is to develop 

new methods. From the practical point of view, the appraisal o£ 

existing methods may seem to be much more useful.

The opposition sometimes encountered in advancing new research 

projects in flood routing may be proof that the sound appraisal of 

existing methods is necessary to find the best existing tools for practical 

use, and as a preliminary work for the advancement of new ideas and 

research projects which are justified by new approaches.

A possible approach to appraise the feasibility of existing or newly 

developed flood-routing methods may be designed in the following 

steps:

1. Several reaches on rivers and on artificial channels must be 

selected, which would cover a large variety of conditions (small 

river, medium river, large river, very large river, one or two 

artificial channels), and which should have the following 

characteristics:

(a) the boundary conditions (cross section, slopes, roughness, 

singularities) are known;

(b) lateral inflows and outflows are adjustable at will, or if they 

can not be adjusted, that they can be easily determined;

(c) a pool or reservoir upstream from the reach, with possibility 

to create any type of flood wave, and if this operation 

would cause downstream difficulties, that there is a pool or 

reservoir downstream from the reach to dampen the created 

waves; and

(d) facilities for obtaining observations, and conducting research 

work should be available.

2. The individual characteristics and advantages and disadvantages 

of existing flood-routing methods must be appraised (bibliographical part of this study can be considered as preliminary 

work for this step).

3. A research program must be designed on the basis of appraisal of 

flood-routing methods, and the available facilities of selected 

channel reaches.
32 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

4. Flood waves of different forms must be created and their progress 

observed at selected points along the reach.

5. After sufficient experience has been obtained in applying each 

method, the above-mentioned floods should be routed through 

the reach with different routing methods, different amounts of 

background data, and varied finite differences. The observed 

hydrographs for selected sections could then be compared with 

the flood hydrographs obtained with step-by-step routing. The 

cost of routing by different methods should be tabulated for 

study.

6. By using statistical methods, the observed and the routed flood 

hydrographs, or flood profiles for the reach, should be analyzed 

to obtain the errors in each specific case.

7. It is necessary to derive the relationships of accuracy of results in 

flood routing to the cost of routing, taking into account the 

basic cost of equipment and all other necessary tools, for applied 

flood-routing methods.

8. The feasibility of all selected flood-routing methods must be 

appraised from the point of view of accuracy, conditions under 

which a method supplies a given accuracy-cost relationship, 

and the conditions under which the basic investments start to 

be insignificant in the unit cost of routing.

9. Recommendations for practical use must be established. 

10. New approaches must be suggested or new problems to be attacked 

by new research projects in the field of unsteady channel flow 

and flood routing.

For given funds, time, and problem at hand, with the devices 

available, there is always the most accurate flood-routing method. 

Comparative studies of this kind do not exist, and the differences and 

controversies about the selection and use of flood-routing methods 

come from the lack of this comparison of methods. Currently, the 

the intuitive judgment and general experience are used more than the 

exact criteria in selecting a flood-routing method to be applied.

It seems that developments in the water-resources field have reached 

the point where routing of water waves along the natural and artificial 

channels has become, especially in discharge and stage forecasting, a 

widely prevalent need. By using the theory of errors, it would be 

possible to classify flood-routing method by objective criteria, and to 

restrict further the "sound engineering judgment," following thus the 

general trend in the contemporaneous scientific endeavor.

The development of methods of treating water waves along channels 

has now reached great breadth. The immediate need for an objective 

appraisal of methods is strongly stressed. With sufficient comparative studies and results, objective criteria could be established to
BIBLIOGRAPHY 33

enable selection of optimum methods for practical use under given 

conditions.

BIBLIOGRAPHY

The explanations and remarks contained in this introduction are 

made to facilitate use of the bibliography and also, it is hoped, to 

increase its worth to the reader.

Subject of bibliography. Basically, the subject of this bibliography 

is encompassed by one body of theory that is contained within one 

homogeneous area of research concerning unsteady flow of water in 

channels. However, practical application makes it advisable to treat 

unsteady flow in channels and reservoirs as being applicable to the 

field of hydraulics or fluid mechanics, and the derivative subject, 

flood routing, as being applicable to the field of hydrology. Both the 

hydraulic and hydrologic aspects of the reference studies are given, 

in this bibliography, in order to present abstracts of the literature 

according to applicability.

Many types of bodies of water are considered to be included in the 

category of reservoirs: artificial reservoirs; natural and artificiallycontrolled lakes; pools; and large channels with very small slopes, 

large cross-sectional areas, and small velocities. Similarly, the category of channels includes many types of conduits: river channels; 

torrent channels; estuaries; artificial canals; also, tunnels, pipes, 

culverts, and any elongated conduit with free water surface flow.

References on subjects devoted to the general theory of wave 

motion in channels as well as references describing practical methods 

and procedures for flood routing are included.

The range of method coverage is large, extending from simple 

empirical methods of problem solution to purely theoretical and 

mathematical treatment of subjects. Many references treat experimental work done in laboratories; others deal with observations 

made in nature or experiments conducted in actual river channels or 

canals. The characteristics of waves and channels and their interrelationships are subjects of many references and abstracts. References pertaining to the practical problems of flood-wave transformations and of predictions occupy a substantial part of the bibliography. 

The subject of tidal-wave movement in estuaries and channels is 

included, insofar as the references are concerned entirely or partly 

with unsteady flow in channels.

The bibliography comprises references to papers that are original 

reports of studies made, and also references to papers that are restatements, summarizing the results of previous studies.

The bibliography also includes reports on the physics of the reference subject. The resulting mathematical formulas are often given
34 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

in the abstract or clarifications are made for the approximations that 

have been introduced to describe the phenomena of wave movements. 

For their instructional value, abstracts of a few studies that reached 

erroneous conclusions are also included in the bibliography, and the 

errors are pointed out.

Purposes of bibliography and abstracts. In addition to the obvious 

purposes served by a comprehensive bibliography, this bibliography 

attempts to be a step toward systematization of the body of knowledge 

that concerns the hydraulic laws of unsteady flow in channels and 

reservoirs. It further attempts to provide the basis for a functional 

digest of flood-routing methods.

It is hoped that this bibliography, as an incidental chronicle by 

covering the literature in many languages, will also have some 

historical value or value for a future historical survey of the contributions to the sciences of hydraulics and hydrology made, over the 

years, by numerous scientists working in different countries, at 

different times, and frequently being unaware of an identical or 

closely related study performed elsewhere.

Format used for references and their abstracts. Each reference in 

the bibliography is identified by: number, author, year of publication, 

language (if other than English), title, journal or other organ of 

publication, volume, page or pages, and (for books) publisher's name 

and location. For those references concerning papers published in a 

language other than English, the title is given in the original language, 

followed by its English translation. Occasionally, particularly in 

regard to references taken from the older Proceedings of the French 

Academy of Sciences and other French journals, only the last name 

of the author is available, but fortunately, there is only one author 

by this surname so that no confusion of authorship results.

Abstracts, included herein, that are not restatements, are generally 

informative rather than being merely descriptive.

As far as possible, the text of each abstract has been kept within 

100 to 150 words. Some abstracts are necessarily longer but an 

effort has been made not to exceed a length of 500 words.

Sources of abstracts. The abstracts contained in this bibliography 

are from four principal sources:

1. Abstracts written by the compiler of this bibliography, comprising 

a large majority of the total number of abstracts. These 

abstracts do not show authorship and were compiled during the 

last half of 1959 or early in 1960, with some revisions, corrections, 

and additions in February 1962.

2. Author's own abstract (or summary, synopsis, introduction, or 

conclusions), used partly or in its entirety in this report, with 

author credit shown at the end of the abstract.
BIBLIOGRAPHY 35

3. Abstracts taken from published Transactions of American 

Geophysical Union, commencing with Part 4, 1938, p. 362. The 

name of the author of the abstract is shown at the end of the 

abstract.

4. Abstracts taken from "Bibliography on tidal hydraulics," compiled by the Committee on Tidal Hydraulics, Corps of Engineers, 

U.S. Army, as published in several reports and supplements. 

At the end of each abstract used from this source, the source is 

shown, together with the year of its publication.

Symbols used. For convenience, unified symbols have been used 

for magnitudes and terms commonly employed. All formulas in the 

original articles have been transliterated by using the unique set of 

symbols shown in "List of Symbols" (p. 36). Specific symbols in the 

original articles have been retained and their use explained in the 

abstracts. If a symbol used in the abstract is not clear to the reader, 

he will find its explanation in the list of symbols.

Chronological bibliography with abstracts. The references, together 

with their abstracts, are presented in chronological sequence, by year, 

according to date of first publication (and not according to republication date). Within a particular year, the sequence of references is by 

author's name, in alphabetical order. Thus arranged, references have 

been numbered consecutively. Each reference can therefore be 

identified by its number, author or authors, and year of publication. 

This chronological sequence provides a unique chronicle of theory 

development, showing how different aspects and problems of unsteady 

flow and of flood routing through reservoirs and channels were treated 

and solved during the last century and a half of active study.

Index by authors. The index by authors is useful in showing the 

whole array of an author's contribution to the subjects of this 

bibliography and indicates the finding number for each reference. 

This index is compiled in alphabetical order according to the author's 

name, and each of his reports (article, paper, book) is identified by two 

numbers, the first being the number of that reference in the bibliography's chronological listing, and the second being the year of 

publication.

Index by subjects. An index by subjects is also given, compiled in 

alphabetical order of subject, as determined from content of abstracts. 

Following each subject name are numbers referring to the references 

treating the subject, with these identifying numbers being taken from 

the bibliography's chronological list of references. The larger the 

number, the more recent is the reference. It is therefore possible to 

follow references on a subject in sequential order. The subject index 

does not attempt to be complete in covering all the ramifications of 

subject that might be inferred in the content of an abstract, but
36 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

rather attempts to orient the reader toward the major subjects treated 

by the references in the bibliography.

Last years. The coverage of years, 1959-61 is not as complete as 

it is for the previous years.

LIST OF SYMBOLS

These are the symbols most frequently used in the abstracts, which 

are not explained in the text. When these symbols are used differently 

than given here, they are explained in the corresponding abstract. 

All other symbols used are described in the text.

A Cross-sectional area

B Width of channel or reservoir

C Wave celerity (velocity of wave propagation)

c Resistance coefficient in Chezy's formula

FI, F2 Storage factors

/ Sign for a function, in general, as in /(x)

g Acceleration of gravity

H0 Depth in channel or reservoir for steady flow

H Depth in channel or reservoir for unsteady flow

h Wave height (H Ho), or wave amplitude

K. Coefficient in the routing formula

I Wavelength

L Length of reach or reservoir

n Resistance coefficient in Manning's formula

P Inflow discharge

Q Outflow discharge

q Outflow discharge per unit of width

R Hydraulic radius

S Longitudinal slope of water surface

S0 Bottom slope of channel

Sf Friction slope

t, At Time and time interval

U Mean velocity or mean velocity increment, created by 

acceleration due to the wave movement

V Mean velocity

W Storage volume

X A dimensionless constant in the coefficient method for 

flood routing

x, Ax Any length along the channel or reservoir

y Any wave height (h=y max)

z Elevation of water surface above the reference level.
BIBLIOGRAPHY 37 

CHRONOLOGICAL BIBLIOGRAPHY AND ABSTRACTS

1. Newton, Sir, I., 1685-86, Principia: Royal Soc. [London], Book 2, Propositions 44-46 [1687].

It is assumed that the waves can be compared to the oscillations of water 

in an inverted syphon. This leads to the conclusion that the propagation 

celerity of waves is proportional to the square root of the wavelength (from 

crest to crest), and that each wave travels that length in a time equal to 

the oscillation of a pendulum with the length of the double wavelength. 

This was not confirmed by further studies.

2. Laplace, P. S., 1775-76, Recherches sur quelques points du systeme du 

monde [Researches on some points of world system]: Acad. Sci. [Paris] 

M6m., (complete works, v. 9).

The differential equations of the water oscillations are given in the first 

approximation, starting from a fluid at rest. It is assumed that fluid 

disturbed from the state of equilibrium acquires surface shape that is 

trochoidal.

3. Lagrange, I. L., 1781, M6moire sur la theorie du mouvement des fluides 

[Memoir on the theory of movement of fluids]: Acad. Royal [Berlin] 

Mem., p. 151-198, [1783]; and p. 192-198 [1788].

This is the first development of the theory that the celerity of wave 

propagation along canals is C -\gH.

4. Lagrange, I. L., 1788, M6canique analytique [Analytical Mechanics]: 

Paris, Bertrand's ed., pt. 2, sec. 2, act. 2, p. 192; or Oeuvres [Works] 

sec. 11, act. 36.

First in the Memoirs of Berlin, then in the Analytical Mechanics, Lagrange treats the theory of waves in water of shallow constant depth. He 

stresses the analogy of wave and sound propagation celerities, which are 

independent of the original disturbance. The celerity is proportional to 

the square root of the fluid depth in a canal of constant width. A disturbance of the surface in any depth of fluid is transmitted a very small distance 

from the surface, and the celerity is proportional to the square root of that 

distance. This last conclusion is weakened by the uncertainty of 

distance determination due to the progressively diminishing velocity of 

particles.

5. Gerstner, Franz von, 1802, Theorie der Wallen [Theory of Waves]: Koen. 

Boehmischen Gesell. [Prague], Wiss. Abh.; and Gilbert's Annalen d. 

Phys., v. 32, 1809.

The concept of the relative movement of particles in circular oscillation 

in deep water sea waves is introduced by mathematical treatment. The 

particles form circles having fixed centers and decreasing radius in a geometric progression from the surface downward.

6. Young, Cf. T., 1813, A theory of tides: Nicholson's Jour., v. 35, 1813, 1823;

and Misc. Works, London, p. 262, 291, 1854.

Simple methods are shown for expressing several of the principal parts 

of the canal theory of tides.

7. Poisson, S. D., 1816, Memoire sur la th^orie des ondes [Memoir on the 

theory of waves]: Acad. sci. [Paris] Mem., v. 1, p. 71-186.
38 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Two differential equations of wave movement are given and integrated, 

assuming neglect of one horizontal dimension of fluid. An analysis is made 

of: (1) the propagation of surface waves in a vertical canal of constant 

width and great depth; (2) the propagation of fluid motion in the vertical 

direction, with the abstraction of one horizontal dimension. By employing 

all three dimensions of fluid, differential equations of the first order are 

integrated. By considering two horizontal dimensions, wave propagation 

is shown. By assuming that all three dimensions are known, analysis is 

made of the propagation of movement inside the fluid.

8. Bidone, George, 1824, Experiences sur la propagation du remous [Experiments on the wave propagation]: Reale Acad. sci. [Torino, Italy] 

Mem., v. 30, p. 195-292.

This is one of the earliest experimental studies on wave propagation. 

The experiments employ a small rectangular canal, 2 feet wide by 40 feet 

long. Study is made of the propagation of a negative wave produced 

when steady flow, having uniform depth H and velocity V, is suddenly 

stopped at the downstream end of a canal. By using the continuity 

equation, the propagation celerity Cis derived as Ch=VH, where A=wave 

height. The wave height is derived from the equation h=(2H+h)V2/2gh 

(shown by later studies to be incorrect). Other aspects of wave propagation 

are studied, including the propagation of tidal bores in estuaries. The 

propagation celerity found for h is approximately the same as that found 

in later experiments by Russell and Bazin.

9. Weber, Ernest, and Weber, William, 1825, Wellenlehre auf Experimente 

gegruendet, oder ueber die Wellen tropfbaren Fluessigkeiten mit 

Anwendung auf die Schall- und Licht-Wellen [The theory of waves 

based on the experiments, or the theory of sound and light waves 

applied to theory of liquid waves]: Leipzig, Gerhard Fleischer.

A compilation of studies made in the field of wave theory, from the time 

of Newton to 1825, that includes the works of Laplace, Poisson, Cauchy, and 

others, and constitutes a valuable history of wave research. Included 

also are the authors' experiments, principally on oscillatory waves in quiet 

water.

10. Cauchy, A. L., 1827, Theorie de la propagation des ondes & la surface 

d'un fluide pesant d'une propondeur indefinie [Theory of wave propagation on the surface of a heavy fluid with infinite depth]: Acad. 

sci. [Paris] Mem., v. 1, p. 3-312.

With surface shape of wave and wave producing forces being known, 

equations giving the initial conditions of fluid motion are developed. 

Then equations are developed for fluid motion and surface configuration 

during any time of motion. The general laws of wave movement are 

presented and constants of the development equations are determined. 

Use of the analytical formula is demonstrated in the concluding notes. 

This study constitutes an extensive analytical treatise on the general 

theories of waves. (Manuscript was submitted Sept. 1815.)

11. Green, G., 1837, On the motion of waves in a variable canal of small depth 

and width: Philos. Soc. [Cambridge] Trans., v. 6, pt. 3, p. 457-462 

[1938]. Reprinted in Mathematical Papers of George Green, London, 

MacMillan and Co., p. 223-230, 1871.
BIBLIOGRAPHY 39

By integration, using Bernoulli's studies and continuity equation, 

small wave propagation in a canal is shown to have the propagation celerity 

dxldt -\lgH. Wave height, velocity of the fluid particles, and wavelength 

are expressed as functions of the variable width and of the depth of the 

canal. (Read May 15, 1837.)

12. Russell, J. S. (Scott), 1837, Experimental researches into the laws of certain 

hydro-dynamic phenomena that accompany the motion of floating 

bodies, and have not previously been reduced into conformity with 

the known laws of the resistance of fluids: Royal Soc. [Edinburgh] 

Philos. Trans., v. 14, p. 47-109 [1840]. Published in French in Fonts 

et Chaussees Annales, sem. 2, p. 143-234, 1837.

In addition to treatment of title subjects, the paper also includes results 

of experiments made in 1834 and 1835 on wave-propagation celerities in

channels and canals, which later produced the formula C=V<7 (#<> + /&). 

(Read Apr. 3, 1837.)

13. Green, G., 1839, Note on the motion of waves in canals: Philos. Soc. [Cambridge] Trans. Reprinted in Mathematical Papers of George Green, 

London, MacMillan & Co., p. 273-280, 1871.

This study (a continuation of the author's theoretical studies of one of 

the subjects published in 1837) based on Russell's experiments and derivations, analyses the circular orbit of solitary wave movement as it occurs in 

still water in a channel. The wave celerity is analyzed. It is revealed 

that the propagation celerity of a long wave occurring in a triangular canal 

having one vertical side is the same as that occurring in a rectangular canal 

of half the depth. (Read Feb. 18, 1839.)

14. Kelland, P., 1839, On the theory of waves, Part I: Royal Soc. [Edinburgh]

Trans., v. 14, p. 497-545 [1840].

The celerity of a solitary wave is developed analytically in a complex, 

exponential form, involving canal depth, elevation of wave, and wavelength. 

An attempt was made to substantiate the formula by use of the experimental 

data of J. S. Russell. However, a discrepency was found in the experimental 

data. The author's analysis found the propagation celerity to be -T/gA/B, 

where A = area of the section of the canal and B=width of the canal at the 

water surface. (Read Apr. 1, 1839.)

15. Russell, J. S. (Scott), 1842-43, Report on waves: British Assoc. Adv. Sci.

Proc. for 1844, p. 311-390 [1845].

The wave is analyzed as a vehicle of energy. Water waves are classified 

as (1) waves of translation (solitary), (2) oscillating waves, (3) capillary 

waves, (4) corpuscular waves (solitary or sound waves). First-order waves 

(waves of translation), which are (1) positive or negative, (2) free or forced, 

are studied experimentally, the conclusions presented concern damping, 

celerity (empirical equation C i/gH or C=-^g(H0 -{-h'), wavelength, wave 

form, and velocities of water particles. Also, studies are made of oscillatory, capillary, and corpuscular waves. The wave-celerity formula developed is compared with the theoretical formulas of Kelland and Airy. (Reported to British Association for Advancement of Science at meetings 

in 1842-43.)

16. Airy, Sir George Biddle, 1845, Tides and waves: Encyclopedia Metropolitana, London, p. 241-396.
40 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

A study of long waves in canals with variable cross sections. Kelland's 

formula of wave celerity is redeveloped on a simplified basis. The theories 

of oscillatory waves and standing oscillations are treated fully. A thorough 

analysis of tidal wave is made. On the assumption that the horizontal 

motion of water is effectively the same, from top to bottom, tidal-wave 

calculation is presented in simplified form. It is shown that the front of 

the tidal wave becomes shorter and steeper as the wave progresses. Expressed for the first time is the theory that solitary wave without friction 

resistance is unstable and deforms as it progresses. (Doubt has been cast 

upon this theory by the subsequent investigations of Boussinesq, Lord 

Rayleigh, and De Saint-Venant.)

17. Stokes, G. G., 1846, Report on recent researches in hydrodynamics (chapter 

on waves): British Assoc. Adv. Sci., 16th Mtg., Southhampton, 1846. 

Proc.; also Report on the State of Science, [London], p. 1-20, [1847] 

Reprinted in Stokes Math, and Phys. Papers [Cambridge], p. 157- 

187, 1880.

An appraisal of the progress of science literature on wave motion up to 

the mid-19th century. The author wrongly supports Earnshaw's thesis 

that the change of wave shape is not chiefly due to friction but is an inherent 

characteristic of its motion.

18. Earnshaw, S., 1847, The mathematical theory of the two great solitary 

waves of the first order: Philos. Soc. [Cambridge] Trans., v. 8, pt. 3, 

p. 326-341.

A discussion of problems involved in the integration of differential equations for fluid motion in wave movement. The need for combined experimental and analytical research is stressed. Russell's experimental data 

and conclusions are discussed, together with a mathematical analysis of 

positive and negative solitary waves. A complex expression of celerity of 

small waves is derived that permits neglect of some terms in obtaining the 

approximate celerity -JgH. It is the author's conclusion (later proved to 

be^wrong) that wave degradation is due not only to friction but also to the 

consequence of initial motion of fluid particles. (Read Dec. 8, 1845.)

19. Stokes, G. G., 1847, On the theory of oscillatory waves: Philos. Soc. 

[Cambridge] Trans., v. 8, pt. 4, p. 411-455. Reprinted in Stokes 

Math, and Phys. Papers [Cambridge], p. 195-219, 1880. 

The author contends that, except for an infinite succession of oscillatory 

waves (discussed with an approximation), there can be no wave propagation without change of form or velocity. Second-order approximations 

are presented to indicate that, for small waves of finite height, wave celerity 

is independent of wave height. Expressions are derived for wave celerity, 

velocities of particles, and other characteristics of oscillatory waves.

20. Baumgarten, 1848, Jaugeages sur la Garonne [Flow measurements on

Garonne River]: Ponts et Chausse'es Annales [France], July-Aug., p. 45«

Observations on the Garonne River are presented indicating that during

a flood rise the velocities and discharges are greater for a given stage than

they are during a falling flood.

21. Caligny, Anatole de, 1848, Experiences sur une nouvelle espe"ce d'ondes 

liquides & double mouvement oscillatoire et orbitaire [Experiments on 

a new type of liquid waves with double motion, oscillatory and orbital]: 

Jour. Math. Pures et Appl. de M. Liouville [France], ser. 1, v. 13, p. 

91-110.
BIBLIOGRAPHY 41

Experiments with the waves of great height, and especially with solitary 

waves, are described. It is concluded that they can not be produced 

experimentally. The differences and similarities of flowing and solitary 

waves are discussed.

22. Partiot, H. L., 1858, Me"moire sur mascaret (Extrait par I'auter) [Author's 

abstract: Memoir on the tidal bore]: Acad. sci. [Paris] Comptes 

rendus, v. 47, p. 651-654.

This paper describes the tidal bore, with particular reference to two 

bores of the Seine (measuring 2.18 and 1.68 meters), which provide basis 

for the conclusions reached. Based on Virla's theory (Annales Fonts et 

ChaussSes, 1833). The memoir treats the analysis of the tidal bore.

23. Caligny, Anatole de, 1861, Experiences sur la ge'ne'ration des ondes liquides

dites courantes [Experiments on the creation of liquid waves called

flowing waves]: Acad. sci. [Paris] Comptes rendus, v. 52, p. 1309-1311.

This is a brief discussion of the paper "Cenni sul moto ondoso del mare

e sulle correnti di esso" [Remarks on the wave motion of the sea and on the

currents produced by it] by Cialdi, Academy of Sciences, no. 4, Rome, 1856.

24. Hagen, G., 1861, Ueber die Wellen auf Gewaessern von glechmaessiger 

Tiefe [On the wave in channels with uniform depth]: Abh. Kgl. Akad. 

der Wiss. [Berlin], Math. Abh. p. 1-79, [1862].

A study that is chiefly applicable to deep-water waves. The celerity 

of waves in the form C=-\/g\/ir is given (Gerstner's studies), and the 

velocity of particles in a wave movement is studied.

25. Partiot, H. L., 1861, Me"moire sur le mascaret [Memoir on the tidal bore]:

Fonts et ChaussSes Annales [French], sem. 1, no. 2, p. 17-48. 

The tidal bore on the Seine River is observed, and the results and explanations are given. The cosine line is applied to the shape of a tide, and 

the properties of tidal waves are discussed.

26. Partiot, H. L., 1861, Etude sur les mouvements des marges dans la partie 

maritime des fleuves [Study on the movements of tides in the maritime 

portion of rivers]: Paris.

27. Bazin, H., 1862, Experiences sur les ondes et la propagation des remous 

[Experiments with waves and the propagation of negative waves]: 

Acad. sci. [Paris] Comptes rendus, v. 55, p. 353-357, [1862], and v. 57, 

p. 302-312, [1863].

This study demonstrates that a positive wave, having a wave height h 

that is maintained unchanged for a long distance, travels with celerity 

C=^g(H0 -}-h). The negative wave, which cannot be maintained unchanged for as long a distance and which is always followed by a group of 

oscillatory waves, travels with celerity C=-\g(H0 K). In flowing water, 

with velocity V, the celerity is C=V±-\fg(H0 ±K), for both positive and 

negative waves. Considering the case of sudden stoppage of water movement (from V to 0), the celerity of the backwater is given both for the 

positive and the negative bore. Experiments reported in this paper are 

on a much greater scale than previous similar experiments made by Russell.

28. Rankine, W. J. M., 1863, On the exact form of waves near surface of deep

water: Philos. Mag. [London] (Rankine's Paper), p. 481. 

This report contains the complete solution for the trochoidal waves, as 

obtained by Gerstner in 1801 and by Rankine (independently) in 1862.
42 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

29. Cajand, N., 1864, Untersuchungen ueber die Wirkung der Ebbe and Flut 

in dem maritimen Teile der Fluesse [Investigations on the tidal effect 

in the maritime portion of rivers]: Allg. Bauzeitung.

30. Bazjn, H., 1865, Recherches exp^rimentales sur la propagation des ondes 

[Experimentalresearch on wave propagation]: Acad. sci. [Paris] Me"m., 

v, 19, p. 495-644. Also, Dunod, Paris, 1865, in second part of hydraulic researches of Darcy and Bazin.

Experiments constitute a confirmation of Russell's formula C=-\lg(Ho-}-K). 

The relative propagation celerity in moving water is shown to be C = 

T/g(H0+h)±V, where C=-\lg(H0 -\-h) + V for the positive wave moving in 

the current, and C=Vflf (H0 +70 ~~ V for the positive moving against the 

current, with the change of the sign of h for the negative wave. The positive and negative translation waves are studies by experiments which prove 

the above formulas. Experiments of the propagation celerity of steep 

"waves (surges and depressions) are made. These indicate that formulas 

approximately the same as the above can be used for rapidly changing 

unsteady flow. Study of the tidal bore (mascaret) is made.

31. Caligny, Anatole de, 1866, Experiences diverses sur les ondes en mer et 

dans les canaux, etc. [Different experiments on the sea and canal 

waves, etc.]: Jour. Math. Pures et Appl. de M. Liouville [France], 

s. 2, v. 11, p. 255-265.

The experiments with waves in a canal in Versaille (near Paris) are 

described. In the discussion of "flowing waves" and solitary waves, it is 

revealed that the author could not produce the solitary wave (although he 

could compute its celerity) but that he could produce flowing (current) 

waves by experiment. Studies made of the curves described by the course 

of a water particle during wave movement are discussed. The experiments 

on the formation of solitary waves are discussed extensively. At the end, 

remarks concerning waves produced by the emptying of navigation locks 

are given.

32. De Saint-Venant, Barre", 1870, Demonstration e'le'mentaire de la formule de 

propagation d'une onde ou d'une intumescence dans un canal prismatique; et remarques sur les propagations du son et de la lumiere, sur 

les ressauts, aussi que sur la distinction des rivieres et des torrents 

[Elementary demonstration of the propagation formula for a wave or a 

translatory wave in a prismatic channel; remarks on the propagation 

of sound and light, on hydraulic jumps; remarks on the distinction 

between rivers and torrents]: Acad. sci. [Paris] Comptes rendus, v. 71, 

p. 186-195.

Starting with Babinet's development of sound propagation in fluids, the 

celerity C=^gH is derived theoretically, thus supporting Lagrange's mathematical developments and Russell's and Bazjn's experiments, as well as 

Partiot's observations of tidal-wave propagations.

33. Boussinesq, Joseph, 1871, ThSorie de Pintumescence liquide appelee onde)

solitaire ou de translation, se propageant dans un canal rectangulaire

[Theory of the liquid intumescence, called a solitary wave or a wave of

translation, propagated in a channel of rectangular cross section]:

Acad. sci. [Paris] Comptes rendus, v. 72, p. 755-759.

The author presents a theoretical analysis of the celerity of a solitary

wave in an indefinitely long canal of constant depth containing still water.
BIBLIOGRAPHY 43

On the basis of the De Saint- Venant studies, a formula for celerity is derived as C -^g(H0 +3h/2) } where H0=depth before wave arrives, and H= 

wave height or the crest height above undisturbed water. The author 

supports the two Lagrange formulas for wave movement: C=-\fgH and 

UH=CH, where U= velocity of accelerated water. An equation is derived 

in exponential form for the profile of symmetrical waves.

34. De Saint- Venant, Barre, 1871, Theorie du movement non-permanent des 

eaux avec application aux crues des rivieres et £ 1'introduction des 

marees dans leur lit [Theory of unsteady water flow, with application 

to river floods and to propagation of tides in river channels]: Acad. sci. 

[Paris] Comptes rendus, v. 73, p. 148-154, 237-240. (Translated into 

English by U.S. Corps of Engineers, no. 49-g, Waterways Experiment 

Station, Vicksburg, Miss., 1949.)

The first part of this study, inspired by Partiot's studies, demonstrates 

the celerity and profile of the tidal wave in an estuary. An increase of discharge, and of depth, from H to y increases the water velocity, as shown by 

the continuity equation: U=2^/gy 2^gH,ii £7=0 for y=H, then the wave 

celerity C=3T/gy 2Tig~H. The application is discussed. The second part 

is the derivation of partial differential equations, now called "De SaintVenant equations" (continuity and momentum equations), for gradually 

varied, unsteady flow:

x=:Q (continuity)

(momentum)

where A = area, F=mean velocity, 2= position of water surface above 

reference level, S/=friction slope, aj=length along the rectangular prismatic 

canal, and t =time. <S/ is given in a general form. A comparison is made 

with Bazin's equations and some applications are shown.

35. Partiot, H. L., 1871, MSmoire sur le,s marges fluviales [Memoir on the tides

in rivers]: Acad. sci. [Paris] Comptes rendus, v. 73, p. 91.

On the basis of observation for formula for tidal wave, celerity C is given

as C=m Vflr^o+ V, with m=numerical coefficient representing the friction

and other factors, and T =the mean velocity at previous water depth. The

longitudinal-wave profile is given, starting with the sine-wave profile, at the

entrance of the river estuary. The profile or river bed is represented by

steps or by many horizontal reaches, and the velocity V takes the sign of C.

36. Boussinesq, Joseph, 1872, Theories des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communuiquant 

au liquide continu dans ce canal des vitesses sesiv lenient paralleles de la 

surface au fond [The theory of waves and backwaters which propagate 

along a rectangular horizontal canal, subject to continuous flow and 

having velocities in the canal that are approximately parallel to the 

bottom]: Jour. Math. Pures et Appl. de M. Liouville [France], s. 2, v. 

17, p. 55-108; additions, v. 18, 1873, p. 47-52.

Starting from the fundamental partial differential equations for unsteady 

flow, developed by De Saint- Venant (1871), the formula for velocities due 

to the acceleration by wave movement is developed, and wave celerity is 

shown to be

C2 = g[H0 + 3/i/2 + (H<?/
44 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

From this the formula for velocity is produced. The movement of the center of gravity of a gradually varied wave is analyzed analytically. The 

shape of solitary wave is derived. The moment of instability of waves is 

introduced and defined. Nonsolitary waves are also studied briefly.

37. Caligny, Anatole de, 1872, Rescherches sur les oscillations7[Researches on 

oscillations]: Paris, p. 1-258.

38. Boussinesq, Joseph, 1873, Theorie des ondes liquides periodiques [Theory 

of liquid periodic waves]: Acad. sci. [Paris] Memoires presentes par 

divers savants, v. 20, p. 509-615.

The paper discusses equations of continuous movement of any medium; 

the case of a heavy liquid; periodic waves of liquid, with fundamental relations; plane waves; waves propagated from any disturbance region, with 

first and second approximations; the special cases of the waves propagating 

in a canal, and of waves propagating in a infinite reservoir, starting from the 

same center of disturbance; the diffraction phenomena in liquid waves; the 

case (Note 1) when the terms of square and product orders are kept in the 

differential equations, with theory of sea waves; the case (Note 2) of the 

effort of internal resistance of fluids on wave phenomena; and cases (Notes 3 

and 4) that treat wave energy questions.

39. Graeflf, 1875, Memoire sur le movement des eaux dans les reservoirs & 

alimentation variable [Memoir on water movement in reservoirs 

with variable inflow]: Acad. sci. [Paris] Mem., v. 21, p. 393-538. 

The theoretical study of the modification of flows by routing through 

reservoirs, based on the equation: inflow minus outflow equals reservoir 

storage. Various functional relationships are studied: area of volume of 

reservoirs as functions of the depth, use of the outflow hydrographs to 

produce simple curves of storage volume versus depth and to produce 

outflow rating curves (representing use of valves or gates). Flood routing 

through a reservoir and the reservoir effects on floods is developed. The 

time required for partial emptying of reservoir storage is derived for 

simple cases. Some examples are given. This is probably the first published paper on flood routing through reservoirs.

40. Graeflf, 1875, Memoire sur 1'action de la digue de Pinay sur les crues de la 

Loire £ Roanne [Memoir on the effect of Pinay Dam on the floods of 

the Loire at Roanne]: Acad. sci. [Paris] Mem., v. 21, p. 539-626. 

The computation of the effect of a storage reservoir on flood waves is 

given. It is based on the general storage equation: inflow minus outflow 

for a given time unit is equal to the change of volume of stored water in 

the same time unit. The inflow and outflow hydrograph, the computational procedure, and other subjects (gaging of water, equations for uniform flow of water) are discussed. The propagation celerity of a flood 

wave is given as (7= (Pi Qz)/(Ai A2 ), where Qi and Qz are the discharges 

in two nearby sections, and AI and A% are the corresponding cross-sectional 

areas.

41. Graeflf, 1875, Sur 1'application des courbes de debits & 1'etude du regime 

des rivieres et au calcul de d'eflfet produit par un systeme multiple de 

reservoirs [On the application of the hydrograph to the river regime 

and to the computation of the effect produced by a system of multiple 

reservoirs]: Acad. sci. [Paris] Mem., v. 21, p. 627-674.
BIBLIOGRAPHY 45

The propagation celerity C of a wave with C (Q\ Qz)l(Ai A2) is 

discussed. The influence of a reservoir on the downstream flood is analyzed, 

showing that reservoir influence decreases with distance from the reservoir. The effects of many reservoirs on the downstream flow is discussed.

42. Herschel, Clemens, 1875, On waves of translation: Am. Soc. Civil Engineers Trans., v. 4, p. 185-200.

For a given water elevation in an upstream body of water, and a given 

area of the bottom gate opening, the wave thus created in the downstream 

channel is discussed. Based on Bazin-Darcy experiments and formulas, 

together with author's experiments, the surge and celerity in the channel 

are discussed, and an expression is developed as the function of the water 

surface elevation in a body of water and for an area of opening.

43. Rayleigh, Lord, 1876, On waves: Philos. Mag. [London], ser. 5, v. 1,

no. 4, p. 257-279.

Another method of approximation for solitary wave shape and propagation celerity is presented. By comparison with Boussinesq's paper 

(1871), this analysis produces an equation for surface shape that is similar 

to Boussinesq's and for velocity of propagation that is the same as Boussinesq's.

44. Russell, J. S. [Scott], 1876, On waves: Philos. Mag. and Jour. Sci. [London],

sec. 5, v. 1, p. 257-279.

The celerity of long waves with small heights in any cross section with 

area A and width B at the surface is redeveloped as C=i/gA/B. (Use of 

mean depth is introduced). The celerity of solitary wave is C=T/gy, 

where y is the distance from crest of wave to bottom of channel. The 

equation for the shape of solitary wave is derived. Periodic waves in 

deep water are analyzed. The concept of the moving coordinate system 

with constant celerity of solitary wave is introduced.

45. Boussinesq, Joseph, 1877, Essai sur la theorie des eaux courantes [Treatise 

on the theory of flowing water]: Acad. sci. [Paris] Mem., v. 23, p. 

261-529.

The third part (268 pages) constitues an extensive mathematical treatment of gradually varied unsteady flow. References are made to the experiments and observations of others. This treatise on fundamental 

theories of flowing water is of permanent value in this area of hydraulic 

investigations. Selected examples of the numerous analyses are given 

here; the second approximation formula for small-wave celerity in a frictionless horizontal rectangular canal is shown to be

C=

Center of gravity of a wave travels with celerity C=^/g(H0 +3y), where y 

is the distance from the center of gravity to the initial water level in channel; 

the wave energy E=2Wy (where W= volume of water) does not change in 

a wave, when the friction is neglected; long-wave celerity is expressed as 

C=T/gH0 + (l + 3ft/2Ho) ; differential equations of wave with great curvature 

of water filaments are developed, with the celerity

0 +70 +0.017F2 ,

which approximates C=V±^g(H0 -\-K); waves decrease in height by pro728-245 64 i
46 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

greasing, except in supercritical flow, where they can increase; the greater 

the velocity V of the permanent flow, the greater the decrease of maximum 

wave height by wave progression; increase of friction and canal slope 

decrease the celerity of all wave parts; Vm&J., Qmai, and #mai are rederived; 

by neglecting friction in a horizontal channel with changing cross section, 

the wave celerity is the same as in a rectangular section of the mean depth 

of changing section; the celerity of a given discharge is C=V±dQ/dA, 

which can be greater or lower than V. Some of the general problems 

treated are: (1) waves of small height in a channel that result in flow that 

differs little from the basic permanent flow; (2) the celerity of propagation 

of intumescence in waves of great curvature of surface; (3) the form, energy, 

moment of instability; and other aspects of the celerity of propagation of 

intumescence and on the mean velocity in a section; (4) the quasi-permanent 

regimen of floods in large rivers.

46. Boussinesq, Joseph, 1877, Additions et eclaircissements au me'moire 

intitule" : Essai sur la theorie des eaux courantes [Additions and clarifications for the memoir with the title: Treatise on the theory of flowing water]: Acad. Sci. [Paris] Mem., v. 24, s. 2, p. 51-58. 

Some clarifications of the analyses of unsteady flow as published by 

Acad. Sci. (Paris) Memoires, v. 23, p. 261-529 are presented.

47. Kleitz, ch., 1877, Sur la theorie du mouvement non-permanent des liquides 

et sur son application & la propagation des crues des rivieres [On the 

theory of unsteady flow of liquids, and on its application to the propagation of river floods]: Fonts et des Chausse'es Annales (France), 

sem. 2, no. 48, p. 133-196.

Unsteady flow is studied analytically with two basic differential equations 

developed. It is found that the maximum discharge propagates with 

celerity C=(l/B~)(dQ/dH), and any height of wave propagates with the 

celerity C=V±^g(H0 -\-y)(l \f/), where \f/ depends on the resistance and is 

positive, so that, with the increase of friction, the celerity C decreases. 

Other forms of celerity formulas are derived and discussed, including the 

furmula for the celerity of small disturbance which is given as C=V±

48. Raynolds, Osborne, 1877, On the rate of progression of groups of waves 

and the rate at which energy is transmitted by waves: Nature, [London 

and New York], v. 16, p. 343-344. Reprinted in Osborne Reynolds 

Papers on Mechanics and Phys. Subjects, v. 1, 1869-1882, p. 198-203, 

1900.

In waves of deep water the rate at which the energy is carried forward 

is half the energy of disturbance per unit of length multiplied by the rate 

of propagation. If this is true, the author concludes, then the celerity of 

a group of waves will be half the propagation celerity of the individual 

waves. The results are modified in shallow water, but are strictly in 

accordance with the theory. The particles have elliptic paths in shallow 

water and wave energy is transmitted by wave propagation. When paths 

approach straight lines, all energy is transmitted, and a group of waves 

propagates with the same celerity as the solitary wave. Therefore, Russell's 

solitary wave theory is possible.

49. Stokes, G. G., 1880, On the theory of oscillatory waves (Appendix) : Stokes's 

Math, and Philos. Papers [Cambridge], p. 219-229.
BIBLIOGRAPHY 47

In this supplement to his paper of 1847, the author carries his analysis 

one step farther in developing an approximation of an infinite train of 

oscillatory waves.

50. Comoy, M., 1881, Etude pratique sur les marees fluviales et le mascaret

{Practical study on the river tidal movement and the bore]: Paris. 

This paper studies the different aspects of tides in rivers, their movement 

and the bores created.

51. Graeff, 1883, Traite d'hydraulique [Hydraulics, chapter on flood forecast]:

Paris, p. 438-442.

The flood hydrograph from one station is routed to a downstream station 

(neglecting lateral inflows and friction resistance) by the use of the rating 

curve and the flood celerities, expressed as a function of stages H, or as 

duration curves of propagation celerity for a given reach.

52. Rippl, W., 1883, The capacity of storage-reservoirs for water-supply: Inst.

Civil Eng. [London] Proc., v. 71, no. 1864, p. 270-278. 

Mass curves of the discharge hydrograph are used in this study to determine the necessary storage capacity of reservoirs. This marks the introductory use of the mass curve for flood routing through reservoirs. The 

properties of mass curves are discussed.

53. Havestadt, A., 1884, Wasserbewegung im Flutgebiet [The water flow in 

tidal region]: Festschrift der Tech. Hochschule, Berlin [Germany].

54. Lechalas, M. C., 1884, Hydraulique fluviale [River hydraulics]: Annex, 

Chap. I, Theorie generale des reservoirs d'emmagasinement des crues 

[General theory of flood storage reservoirs]: Paris, Librairie Polytech., 

Annex, p. 434-460 (The Dayton-Morgan Engineering Company has a 

translation of this article, prepared by K. C. Grant; see Am. Soc. Civil 

Engineers Trans., v. 93, p. 741, 1929).

Comparison of reservoir inflow and outflow hydrographs is made, and 

flood modification by reservoirs is discussed. The conditions under which 

a reservoir is useful for flood control are studied. The question of the best 

distribution of flood control reservoirs in a river basin is approached. The 

author contends that there are no general rules that pertain to this matter. 

The flood having only one peak is considered. Noncontrolled and controlled outflows are discussed. The storage necessary for a given amount 

of flood reduction is analyzed. The Kleitz (1877) method and ideas are 

discussed.

55. Bazin, H., 1885, Experiences sur la propagation des ondes le long d'un 

cours d'eau torrentueux, et confirmation par ces experiences des formules donnees par Boussinesq, dans sa theorie du movement graduellement vane1 des fluides [Experiments on wave propagation in torrential 

flow and the confirmation of Boussinesq's equation for gradually varied 

flow of fluids]: Acad. sci. [Paris] Comptes rendus, v. 100, p. 1492-1494. 

The celerity of the solitary wave, as derived analytically by Boussinesq, is

 Vi 1+a8Y

1 + 26^L\ 2 / 1 + 26J '

where a, a2 and 6 are coefficients which depend on the nonuniformity of 

water velocities. Bazin gives the values 01, o2 , 6 (different symbols are used 

by Boussinesq and Bazin) for three experiments. He writes \2=V2/gH0-
48 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Bazin finds that C=V±kT/gH~0, where k depends on 6 and X. When 

X2> 0.202, k is greater than unity.

56. De Saint-Venant, Barre, 1885, Mouvement des molecules de 1'onde dite 

solitaire, propagee & la surface de 1'eau d'un canal [The movement of 

molecules of a solitary wave propagated on the water surface of a 

canal]: Acad. sci. [Paris] Comptes rendus, v. 101, p. 1101-1105. 

For a large rectangular channel with horizontal bottom, the vertical 

velocity in a wave movement is introduced, and the momentum equation 

takes the form

in which s=wave depth (H0 -\-Ji), A=wave height. With the above 

equation the celerity of a solitary wave is

57. Kelvin, Lord (Thomson, William), 1886, On stationary waves in flowing 

water: Philos. Mag., v. 22, p. 353-357, 445-452, 517-530 and v. 23, 

p. 52-58.

Two dimensional problems of stationary waves in flowing water are 

treated. The permanent steadiness of the motion in this case is analyzed, 

with velocity -JgH discussed as the criterion. Part 4 treats the stationary 

waves on the surface that are produced by equidistant ridges on the bottom58. Klunzinger, P., 1886, Ueber die Beziehungen der Flussregulirungssysteme 

zu dem Verlaufe der Hochwaesser [On the effects of river regulation on 

the flood flow]: Zeitschr. Oesterreich Ingenieuren u. Architekten-Ver. 

[Austria], v. 38, p. 10-19.

This is an analytical treatment of the attenuation of flood waves by 

storage reduction of tpeak discharges. The schematic waves are introduced 

and the water volume at peak of hydrograph is determined by integration.

59. Kelvin, Lord (Thomson, William), 1887, On the waves produced by a 

single impulse in water of any depth, or in a dispersive medium: Royal 

Soc. [London] Proc., sec. A., v. 42, p. 80-85.

The relation between the wave celerity and the wave length of an endless 

procession of periodic waves is studied. The formulas for the celerities in 

deep water, in water with depth H, and for various types of waves are 

derived.

60. Kelvin, Lord (Thomson, William), 1887, Stationary waves on the surface 

produced by equidistant ridges on the bottom: Philos. Mag., v. 28, p. 

52-87 or Math, and Phys. Papers, Cambridge Univ. Press, 1910, v. 84, 

p. 296-302.

A first-approximation study of waves due to ridges on the bottom that 

takes into account the effect of gravity is presented.

61. Basset, A B., 1888, A treatise on hydrodynamics, volume II, Chapter 

XVII, On liquid waves: Cambridge, Deighton Bell and Co., p. 144-187. 

A general review of the theory of waves up to 1888 is given in a condensed form. Assumptions and derivations are discussed. Theories of the 

propagation celerity and wave shape for the long wave in shallow water, for 

the solitary wave, and for other types of waves are reviewed.
BIBLIOGRAPHY 49

62. Ekdahl, 0. Z., 1888, Om berakningsmetoderna vid uppgorande af forslag 

till sjoars sankning och reglering [About the methods of computation 

in a suggestion for lowering and regulating a lake]: Lund [Sweden]. 

The study uses the storage equation for hydrograph routing through 

reservoirs. The storage factors (W/At) + (Q/2) and (W/&f)-(QJ2) are 

used in graphical and tabular form to facilitate the routing.

63. Flamant, A., 1889, Des ondes liquides non p^riodiques et, en particulier, de 

1'onde solitaire [About liquid nonperiodic waves, and particularly, 

about the solitary wave]: Fonts et Chauss^es Annales [France], s. 6, 

v. 18, sec. 2, p. 5-48.

This is a simplified treatment of problems of solitary wave, discussed 

more extensively by Boussinesq in his treatise on the theory of flowing 

water (1877). Wave celerity is defined as the velocity of a vertical cross 

section having all the time the same volume of the wave before it. The 

Boussinesq formula for wave celerity is rederived on the same assumption. 

The center of gravity of the wave is derived, and its celerity is given in the 

second approximation as i/g(H0 -\-3h/2). When friction is neglected, wave 

energy stays constant during its propagation, inasmuch as the acceleration 

of gravity is constant. The solitary wave is the most stable wave, because 

the moment of instability (introduced by Boussinesq) is the smallest in 

the solitary wave.

64. Francis, J. B., 1889, The effect of rapidly increasing supply of water to a 

stream on the flow below the point of supply: Am. Soc. Civil Engineers 

Trans., v. 21, p. 558.

The conditions in which gradually varied waves will become a bore are 

discussed. The explanation is given that in a torrent river and in an 

estuary the waves often build up to become a bore, and gradually varied 

waves become progressively steeper up to the breaking over into a tidal 

bore. Dam-breach waves are mentioned. T. P. Frizell discusses all three 

phemomena: bores in estuaries, bores in mountain rivers, and bores due to 

dam-breaches.

65. Massau, Junius, 1889, L'inte'gration graphique [Graphical integration]: 

Assoc. Inge'nieurs Sortis des Ecoles Spe'ciales de Gand [Belgium] 

Annales, v. 12, p. 435.

The principles of a new method for graphical integration of partial 

differential equations of unsteady flow are given. For the arbitrary curves, 

along which the derivatives exist, the elements are determined one after 

another. The discontinuities propagate according to the characteristics of 

Monge, and for unsteady flow the characteristics represent the movement 

of two infinitely small waves, the wave from upstream and the wave from 

downstream. The wave becomes finite for a bore.

66. Massau, Junius, 1889, Appendice au Me'moire sur I'inte'gration graphique 

[Appendix to Memoir on graphical integration]: Assoc. des inge'nieurs 

Sortis des Ecoles Spe'ciales de Gand [Belgium] Annales, v. 12, 

p. 185-444.

Explanations are made and procedures given for the graphical method 

of integrating partial differential equations for unsteady flow by methods of 

characteristics.
50 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

67. Auria, L. d'., 1890, A new theory of the propagation of waves in liquids:

Franklin Inst. Jour., v. 130, no. 6, p. 456-465.

Formulas derived for computing the delay between high and low waters. 

Discusses use of the formulas as applied to tidal rivers. Includes illustration based on the Delaware River.

[Abstract from Corps of Engineers, Bibliography on Tidal Hydraulics, 

Corps of Engineers, June, 1955.]

68. Mullins, J., 1890, Irrigation manual (section on reservoirs: The influence

of reservoirs as flood regulators): Madras Gov. [India], p. 215-223.

For flood routing purposes, the finite-difference method is shown. The

time interval, At, is used as an independent variable in the storage difference

equation, P Q = AdH/dt, where P=inflow, Q=outflow, A = area of

reservoir surface, and H height of reservoir surface. In computing the

outflow, Q, occuring in a given time, At, values are assumed for AH, and

by trial and error method the assumptions are corrected.

69. McCowan, J., 1891, On the solitary wave: Philos. Mag. [London], sec. 5,

v. 32, p. 45-58.

The solitary wave can be included in the general theory of long waves 

only as a rough approximation, because its celerity does not agree closely 

with that of the long wave, nor does it gradually increase in steepness in 

front as the long wave does. The change of a solitary wave is simply a 

decrease in height and consequent increase in length, which are caused 

by a dissipation of energy due to friction. It is derived theoretically, to 

the degree of approximation involved, that the wave consists solely of an 

elevation and that there cannot be a wave of depression capable of propagating itself unchanged with constant velocity (Russell's experimental 

results). The celerity is obtained as C2 = g(H0 + ho), where h0 is the function 

of wave heights. The theoretical approximate values of wavelength 

(where height is smaller than prescribed), volume of wave (is approximately 

2 aH0, where a=coefficient), energy of wave, limiting height of wave and 

breaking of wave (where Ho=0.8 T/W, where W volume of the wave), 

are given. It is concluded that a solitary positive wave can propagate 

with constant celerity without change of shape.

70. Partiot, H. L., 1892, Etude sur les rivieres & mare'e et sur les estuaires 

[Study on tidal rivers and estuaries]: Paris.

71. Ritter, A., 1892, Die Fortpflanzung der Wasserwellen [The propagation 

of water waves]: Ver. Deutsch Ingenieure Zeitschr. [Berlin], v. 36, 

pt. 2, no. 33, p. 947-954.

This study of wave propagation is based on the assumption of a horizontal 

prismatic canal, with the friction loss neglected and with uniform distribution of velocity in any cross section. If the water is suddenly accelerated to the velocity U (by forcing the water of by opening a gate), the 

celerity

C=Vj/Ho V(l + h/H0) (l + fc/2H0) with U=(hjH0)

Considering small h to be negligible, C=-^gH0 and U=h VflTHo, or C: U= 

H0 :h. If the water has already the velocity V, it must be added to Cor 

U. A continuous wavefront is analyzed as many finite discontinuous 

steps. The sudden removal of a gate in the canal shows the outflow velocity 

in that section to be U= (2/3)V0H0, *ne depth, y0= (4/9) H0 and the constant 

discharge .Q=8/27HoVffHo. The wave profile has equation y=H[(x-\-2\)f
BIBLIOGRAPHY 51

3X]2 where x=distance from the section, y=wave depth, and \=fa/gH0 

(f=time). The application is made for dam breaches, with the upper 

curve y=f(Ho, x, X) as the limiting curve. At the beginning of outflow 

the point 2/o = 4fl0/9 is a turning point where the first surface lines of the 

outflow wave appear. From a shape of concave upward it passes to a 

shape of convex upward, and the turning point shifts upstream.

72. Wheeler, W. H., 1893, Tidal rivers; Their hydraulics, improvement, navigation: London and New York, Longmans, Green & Co., p. 1-467. 

Chapter 1: The development of hydraulic science history, theories, 

formulas; tides, waves, and currents. Chapter 3: The motion of water 

in tidal rivers: details of flow, friction, disturbance, turbulence; directions 

and mixing of fresh and salt water; formulas. Chapter 5: The tides: 

terms, theories, tidal waves; Southern Ocean wave, ocean tidal wave, the 

coast tide, estuary tide, river tide, bores. Chapter 7: Bars at the moutha 

of tidal rivers, and littoral drift: theories; examples of bars in different 

locations and in different materials.

73. McCowan, J., 1894, On the highest wave of permanent type: Philos»

Mag. [London], sec. 5, v. 38, p. 351-358.

This paper, a supplement of the author's paper in 1891 on the solitary 

wave, investigates an approximation better adapted to the discussion of 

the extreme case of the wave at the breaking height.

74. Karteweg, D. J., and DeVries, G., 1895, On the change of form of long waves 

advancing in a rectangular canal, and on a new type of long stationary 

waves: Philos. Mag. [London], sec. 5, v. 39, p. 422-443. 

Lamb and Basset state that long waves in a rectangular canal necessarily 

change their shape as they advance, becoming steeper in front and less 

steep behind, even when friction is neglected. Investigations on solitary 

waves by Boussinesq, Lord Rayleigh, and De Saint-Venant have cast 

doubt on this theory. The authors define the cnoidal waves (so called in 

in analogy with the sinusoidal waves), under conditions that produce the 

solitary wave as a special case. In a frictionless liquid there may exist 

absolutely stationary waves. The form of their surface and the motion 

of the liquid below these waves may be expressed by a rapidly convergent 

wave series. The positive solitary wave, occurring as a stationary wave, 

has the known shape of sechyperbolic function. The stationary periodicwaves (cnoidal waves) are analyzed. The deformation of nonstationary 

waves is discussed. It is concluded that a solitary wave that is steeper 

than a stationary one, but has the same wave height, becomes less steep 

in front and steeper behind; the opposite occurs when the solitary wave isless steep than the stationary wave.

75. Polak, Ignatz, 1895, Die See-Retention. Graphische Darstellung derselben 

nach Prof. A. R. Harlacher [Lake-retentions. Their graphical representation by the method of Prof. A. R. Harlacker]: Zeitschr. der 

Oesterreichischen Ingenieur u. Architekien-Ver. [Austria], v. 47, no » 

50, p. 593-597.

Inflow and outflow hydrographs are used for the solution of different 

reservoir problems involving the use of storage differential equation. 

Graphical shortcuts are used for integration of the hydrographs for 

given time intervals.
52 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

76. Klunzinger, P., 1896, Weitere Studien ueber den Verlauf der Hochwasser 

[Further studies on flood-wave flow]: Zeitschr. der Oesterreiehischen 

Ingenieur u. Architekten-Ver. [Austria], v. 48, no. 4, p. 33-39; no. 5, 

p. 49-56.

The flood wave is approximated by a mathematical function. Using this 

function, the influence of storage on the flood transformation is derived 

analytically.

77. Boussinesq, Joseph, 1897, The"orie de 1'e'coulement tourbillonnant et 

tumultueux des liquides dans les lits rectilignes & grande section [Theory 

of turbulent motion of liquids in rectangular channels with large 

section]: Paris, Gautheir-Villards et Fils, p. 22-27. 

The momentum partial differential equation is given in somewhat 

different form, taking into consideration the velocity distributions in cross 

sections. The use of this equation for wave propagation is demonstrated. 

The celerity of waves is given in a more exact form, as

where

v» dA_ , (YlY dA

where v is the velocity at any point, and V the mean velocity.

78. Fantoli, G., 1897, Sul regime idraulico dei laghi [On the hydraulic regime 

of lakes]: Ulrico Hoepli, Milano [Italy], p. 1-349.

An analytical study of lake inflow and outflow for the case when lakelevel fluctuation is a sine function and outflow is a linear function of lake 

levels.

79. Bobylev, D., 1898, Ocherk teorii vodyanikh techeniy, vyrabo tannyy 

Bussineskom [Sketch of the theory on water movements, developed by 

Boussinesq]: SPC. [U.S.S.R.].

A summary of Boussinesq's works.

80. Bourdelles, 1898, Etude du regime de la marge dans le Canal de Suez [Study 

of tidal regime in the Suez Canal]: Fonts et Chausse"es Annales [France] 

trim. 3.

The propagation of tidal waves in the Suez Canal is studied.

81. LeVy, Maurice, 1898, Lecons sur la the"orie des marees [Lessons on the 

theory of tides]: Paris, Gauthier-Villars.

The theory of unsteady movement given by De Saint-Venant is verified. 

The tide in canals of both constant and variable width is treated, but with 

the additional consideration of the friction exerted upon the tidal wave. 

The height of wave decreases during its propagation with coefficient of 

extinction. The propagation celerity is the function of heights of tide 

and wave depth at half tide. The maximum current of water occurs 

before the highest and lowest levels of tide wave and decreases with an 

increase of the distance from the canal entrance. Taking the friction into 

account, this theory was closer to observations than Airy's or De Saintvenant's. In an attempt to apply his derived equations to the Suez Canal,
BIBLIOGRAPHY 53

the author found that the computed celerities were far less than the actual, 

and the attempt to use the equations for second approximation still did 

not give results that agreed completely with observations.

82. Roudzki, 1898, Ueber eine Classe hydrodynamischer Probleme mit besonderen Grenzbedingungen [On a class of hydrodynamic problems 

with special boundary conditions]: Math. Annalen [Germany], v. 50, 

p. 269.

The author considers that it is not known whether pressure is equally 

distributed on the water surface for any given surface shape.

83. Seddon, J. A., 1898, A mathematical analysis of the influence of reservoirs 

upon streamflow: Am. Soc. Civil Engineers Trans., v. 50, p. 401-427

In order to study the influence of storage on the hydrograph, the storage 

differential equation (inflow minus outflow is the change of storage in a 

given time increment) is integrated for a cycle inflow function (a sine 

hydrograph) and for an outflow rating curve, which is linear in respect to 

reservoir level. The function H=f(f), or reservoir level hydrograph is determined, where the area of the reservoir is either constant or changing with 

the reservoir level, and where H is the reservoir level and t, the time. The 

effect of storage on the amplitude of the crest flow and on phase shifting 

(the postponement of maximum Q) is derived analytically for one or more 

reservoirs.

84. Bourdelles, 1899, Etude du regime de la mare'e de La Manche [Study of the 

tidal regime in the English Channel]: Fonts et Chausse"es Annales 

[France], trim. 3.

The propagation of tidal waves in the English Channel is studied.

85. Haerens, E., 1899, Mouvement vari6 des eaux; propagation d'une crue et 

d'une mare'e fluviale [Unsteady flow; propagation of a flood and of a 

river tide]: Travaux Publiques de Belgique [Bruxelles], Annales s. 2, v. 

4, p. 1-19.

The momentum and continuity equations for unsteady flow are derived 

and shown in finite difference form in the equation of unsteady flow, for 

cross sections 0 and 1, that are Ax distance apart, and for time difference 

of At:

-s£a+! (#+£') ¥+» O+^>') f -

and

At

where A 2 is the drop of surface level for Ax and At, 6 is a coefficient; VQ and 

Vlt Q0 and Qi, A0 and Alt P0 and PI, are velocities, discharges, areas and 

wetted perimeters of two cross sections and of two times respectively. 

This is a trial-and-error method with finite differences. Two examples 

are given: (1) for a given discharge hydrograph, the hydrographs and 

wave profiles are computed downstream; (2) for a given tide shape at 

river confluence the waves upstream are determined.

86. Bourdelles, 1900, Etude sur le regime de la mare'e dans les estuaires et 

dans les fleuves [Study of tidal re'gime in estuaries and in rivers]: 

Fonts et Chausse'es Annales [France], trim. 2. 

A dissertation on the theory of the propagation of translation waves in
.54 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

canals, with the purpose of studying how closely the observed facts confirm the theory, is given.

87. Flamant, A., 1900, Hydraulique, Chap. 8, Mouvements ondulatoires [Hydraulics, Chap. 8, Undular movements]: Paris, p. 413-508. 

This extensive treatise on waves includes: translation waves (definition, 

celerity, celerity in function of wave shape, energy, and application); 

solitary waves (definition and equation, energy and stability, long intumescences, and wave propagation on moving liquid); sea waves; and tides. 

It is a comprehensive summary of the existent knowledge of waves at the 

turn of the century.

88. Gwyther, R. F., 1900, The classes of progressive long waves: Philos. Mag.

[London, Edinburgh, Dublin], ser. 5, v. 1, p. 213-216, 308-312. 

Starting from Lord Rayleigh's paper (1900) "On waves," the first approximation of the solution for long waves is obtained, and special cases 

are discussed, including the case of the solitary wave, which is emphasized 

in the discussion contained in the "Appendix". The study is analytical.

89. Gwyther, R. F., 1900, The general motion of long waves, with an examination of the direct reflexion of the solitary wave: Philos. Mag. [London, Edinburgh, Dublin], ser. 5, v. 1, p. 349-352. 

The long waves are studied analytically, giving a first approximation in 

the most general manner. The steady motion of long waves is dealt with 

by finding the differential equation which the velocity potential must 

approximately satisfy.

'90. Massau, Junius, 1900, Memoire sur I'inte'gration graphique des Equations 

aux derive'es parcielles [Graphical integration of partial differential 

equations]: Assoc. des Ingenieurs Sortis des Ecoles Sp6ciales de Gand 

[Belgium] Annales, v. 23, p.95-214.

This work is a fundamental treatise on the method of characteristics, 

used as a graphical method for integrating the two partial differential 

equations for unsteady flow in channels, through four ordinary differential equations, called "characteristic equations," and equivalent to 

the two partial differential equations of unsteady flow. The basic 

approach is first explained, then such problems are treated as: integration by elements following a variable; graphical integration by 

characteristics, covering the linear equations of first order, linear 

equations of second order, any equation of second order, and simultaneous linear equations; unsteady flow with the hydraulic jump; 

celerity of wave, with characteristics and their use; the propagation 

of finite wave; and the formation of finite wave in the continuous wave 

(tidal bore). (Translated into English by Putman, Rocky Mountain 

Hydraulic Laboratory.)

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

v. 43, p. 179-243.

The celerity of a flood wave on a river is developed as: C=(1/B) 

dQ/dH, where C= celerity, B=width of the river, and dQ/dH slope of 

the rating curve at the place where Q, H, B and C are related.

"92. Maillet, Edmond, 1901, Sur les graphiques et les formules d'annonces 

des crues [On the graphs and formulas for flood forecast]: Ecole 

Polytech. [Paris] Jour., ser. 2, no. 6, p. 148.
BIBLIOGRAPHY 55

Various procedures and formulas are given for purposes of flood 

forecasting of stage-discharge relationships at two or three stations in 

the river system.

93. Partiot, H. L., 1901, Recherches sur les rivieres & mar6e [Research on tidal 

rivers]: Paris, 33 p.

94. Forchheimer, Ph., 1903, Wasserbewegung in Wanderwellen [The motion 

of water in travelling waves]: Akad. Wiss. [Vienna] Minutes Proc., 

sec. 2-a, p. 1697-1720.

In this study of wave trains on a steep sloping canal the hypothesis 

is advanced that the discharge is uniformly constant, as shown by the 

equation

H(C- V) =

The celerity is computed as

where c= coefficient in Chezy formula, and S = slope of the canal. It 

is developed that C=3Q/2H=3V/g. For other formulas of steady 

flow the celerity C is also derived. Wave trains are briefly discussed 

and analyzed.

95. Maillet, Edmond, 1903, Sur les lois des mont6es de Belgrand et les formules 

du debit d'un cours d'eau [On Belgrand's laws of increase and the 

discharge formulas of a water course]: Ecole Polytech. [Paris] Jour., 

ser. 2, no. 8, p. 1.

For stage-forecast purposes, stage relationship (based on the equation 

Qi = Q2 + Qs) is developed between three stations, two on tributaries 

upstream from their confluence and the third below the confluence. 

The relationship is developed in the form of three rating curves: 

f\(Hi)=fz(H^)-}-fz(H^, with Hi, H2 , H3 as the corresponding stages, 

The law of stage increases is developed according to Belgrand's 

analysis. (See abstract of Maillet's paper of 1922.)

96. Chrystal, 1904, Some results in the mathematical theory of seiches: Royal 

Soc. [Edinburgh] Proc., v. 25, p. 328.

Progressive waves in a canal with varying sections are studied, and 

solvable cases for different cross sections are investigated.

97. Forehheimer, Philipp, 1904, Wasserbewegung und Wanderwellen [Water 

movement and translatory waves]: Gewaesserkunde Zeitschr., v. 6, 

6, p. 321-339.

The propagation celerity of a monoclinal wave with peak height HI 

(and Vi, or Qt) and with lower height H2 (and F2 , or <&) is C (Q\ Qs)f 

B(Hi H2), or C (1/B) dQ/dH, the same celerity as that derived by 

Graef or Seddon. If H2 is a maximum, then B(Hi H%) depends on 

HI. If Vz/Vi is between zero and unity, then it is found theoretically that C/Vi is between unity and 1.5. However, by the experiments CjVi is shown to be between 1.7 and 2.0.

98. Massau, Junius, 1905, L'integration graphique des Equations aux derivees 

parcielles [Graphical integration of partial differential equations]: 

Assoc. Ing&rieurs Sortis des Ecoles Spe"ciales de Gand [Belgium! 

Annales, v. 4, p. 65.
56 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

This is a complete development of the theory of the method of characteristics, which method is suitable for graphical integration of partial 

differential equations employed as the equations for unsteady flow when 

the boundary conditions have discontinuity.

99. Cornish, Vaughan, 1907, Progressive and stationary waves in rivers:

Engineer [London], v. 84, p. 118, 284, 347.

The paper describes roll waves (also called freshets and spates) on the 

rivers Tees and Ure (England), the Nile (Egypt), and the Nikko (Japan). 

It is stated that uniformity of depth in the cross section is essential for 

the persistence of such a wave. If the channel consists of a deep central 

groove with shelving banks, the waves will change direction and lose energy 

by breaking on the shores. A river consisting of alternate pools and 

shallows is favorable to the formation of roll waves. Small wedge-shaped 

solitary waves flowing down a precipitous cliff at Grindelwald are described; 

also waves in a 14-inch wide conduit having a slope of 1:2 at Territet (Lake 

of Geneva). Here the roll waves existed for low flows but not for large flows. 

Roll waves in steep open channels at Gruenbach and at Guntenbach are 

are also described. Other types of waves in rivers and estuaries are discussed. 

[Abstract by Herbert N. Eaton, Am. geophys. Union Trans., pt. 4, 1938.]

100. Levi-Civitfi,, Tullio, 1907, Sulle onde progressive di tipo permanente [On 

the progressive wave of the permanent type]: Reale Accad. Lincei 

[Rome] Atti., v. 16, p. 777-790.

The irrotational wave motion of an incompressible liquid in a prismatic 

canal with horizontal bottom is called permanent, if the surface seems to be 

progressing without change and with a constant celerity, or if the motion 

seems to be stationary, as observed by an observer moving at the rate of 

wave celerity. The motion will be permanent if the discharge observed 

by the moving observer is not affected by the time of observation.

101. Grunsky, C. E., 1908, Discussion on "The flood of March 1907 in the 

Sacramento and San Joaquin River Basins, California" (by W. B. 

Clapp): Am. Soc. Civil Engineers Trans., v. 61, p. 331-345. 

This is a general discussion of wave propagation and other flood phenomena. Mass curves are used as a graphical device for determining the maximum discharge and the flood-wave shape at the downstream point, using 

the upstream hydrograph. The continuity or the storage equation is used.

102. Maillet, Edmond, 1908, Sur les services hydrome'triques et d'annonces des

crues en France [On the services of hydrometry and flood forecast in

France]: Navigation Con., llth Saint Petersburg [Leningrad], 1908.

A general description of forecasting methods, which this paper classified

as being: methods of partial or total increase of stages or of discharges;

methods of comparison of stages or discharges; or a combination of these

methods.

103. Flamant, 1909, Hydraulique [Hydraulics]: 3d Ed., Paris, p. 386.

Waves are studied theoretically. Expressions are given for waves produced by moving a vertical plate horizontally in a rectangular channel.

104. Kabelac", Karl, 1909, Ueber ein neues Verfahren zur graphischen Loesung 

der See-Retentions-Aufgabe [On a new procedure for graphical solution of storage-retention problem]: Zeitschr. der Oesterreichischen 

Ingenieur u. Architekten-Ver. [Austria], v. 61, no. 22, p. 353-355.
BIBLIOGRAPHY 57

Starting with the simple storage equation, a graphical procedure is derived 

for the step-by-step integration. The use of the slope for tga = dH/dt= 

(P Q) IA as the development procedure is the characteristic of this method.

105. Cornish, Vaughan, 1910, Waves of the sea and other water waves: Chicago, 

Open Court Pub. Co. [or T. Fisher Onwin, London, 1910], p, 367.

This is a descriptive book on waves; it contains no analytical derivations. 

Its three parts consider: (1) the size and celerity of deep-sea waves; (2) the 

action of sea waves to transport shingle, sand, and mud; and (3) stationary 

and progressive waves in rivers.

106. Cisotti, Umberto, 1910-11, Integrate generale dei piccoli moti ondosi di 

tipo permanente in canal i molto profondi [The general integral solution 

for small wave movements of permanent type in very deep channels]: 

Atti. del Reale 1st. Veneto [Italy], v. 70, p. 33-47.

Using the velocity potential, the function of flow, and the complex 

variables, the unsteady flow differential equations are integrated, with a 

general solution in the approximate form. The equations for wave heights 

and for wave celerity are developed.

107. Rayleigh, Lord, 1911, Hydrodynamical notes: Philos. Mag. [London], ser. 

6, v. 21, p. 177-195.

Brief mathematical notes are presented on the following topics: potential 

and kinetic energies of wave-motion; waves moving into shallower water; 

concentrated initial disturbance with inclusion of capillarity; periodic 

waves in deep water advancing without change of type; tide races; rotational 

fluid motion in a corner; steady motion in a corner of viscous fluid. In 

each analysis the fluid is regarded as incompressible and the motion is 

assumed to take place in two dimensions. When there is no dispersion, 

the energy of a progressive wave of any form is half potential and half 

kinetic. Thus, in the case of a long wave in shallow water "if we suppose 

that initially the surface is displaced such that the particles have no 

velocity, we shall evidently obtain (as in the case of sound) two equal 

waves travelling in opposite directions whose total energies are equal, and 

together make up the potential energy of the original displacement. Now 

the elevation of the derived waves on behalf of that of the original displacement and, accordingly, the potential energy, is less than the ratio of 4:1. 

Since, therefore, the potential energy of each derived wave is one-quarter 

and the total energy one-half that of the original displacement, it follows 

that in the derived wave, the potential and kinetic energies are equal." 

[Abstract by F. T. Mavis, Am. Geophys. Union Trans. pt. 4, 1938.]

108. Bakhmetev, B. A., 1912, O neravnomernom dvizhenii zhidkosti v otkrytom 

rusle [On the unsteady movement of liquids in open channels): Chap. 5, 

[Russia].

The tabular integration of the storage equation, given as dt/dH=A/ 

(P Q), for routing floods through reservoirs and channels.

109. Cisotti, Umberto, 1912, Suirintumescenza del pelo libero nei canali a 

fondo accidentato [On the intumescence of free water surface in canals 

with an uneven bottom (a sill on the bottom)]: Rend, della Reale 

Accad. dei Lincei [Italy], v. 21, no. 8, p. 588-593.

A sill of height p in a horizontal canal is a resistance that creates an 

intumescence. The height of wave thus created as irV^/SgH2 , and the
58 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

profile of a wave is given as y = H+(irV2/8gH) sech (irx/2H) which is the 

profile of the solitary wave. The flow remains steady, but the standing 

wave that is created has the shape of a solitary wave. The parametric 

equations of the free surface, the heights of wave in exact form, the height 

of the sill (great or small) are developed or discussed.

110. Cisotti, Umberto, 1912, Sulle onde superficiali dovute a particolare conformazione del fondo [On the surface wave due to a particular configuration of the bottom]: Rend, della Reale Accad. dei Lincei [Italy], 

v. 21, p. 704-708.

Wave motion in a canal with transversal sills (ridges) of equal height 

and distance is studied. The problem is analyzed rigorously (referring to 

the approximate approach by Lord Kelvin, 1887), for the conditions in 

which it is legitimate to neglect the effect of gravity.

111. Cisotti, Umberto, 1912, Onde brevi causate da accidentally periodiche del 

fondo [The short waves caused by equidistant ridges on the bottom]: 

Rend, della Reale Accad. dei Lincei [Italy], v. 21, p. 760-764. 

This is a continuation study (author's paper 1912, p. 704 708) of the 

problem of surface standing waves in a steady flow over a bottom with 

equidistant ridges (vertical sills). Gravity can be neglected if V is of the 

order of fall velocity of height H/2Q. A new approximation is introduced, 

with results showing that the free surface has the profile of Stokes's waves 

of second approximation.

112. Ekdahl, O. Z., 1912, Ueber die Bewegung des Wassers in Kanaelen und 

natuerlichen Wasserlaeufen, und ueber die Wasserverhaeltnisse in 

Seen [On the movement of water in canals and channels, and on the 

water relationships in lakes]: Leipzig [Germany], Wilhelm Engelmann, 

p. 1-195.

The storage equation is used for flood routing through storage reservoirs. 

Two storage factors, (W/A t+Q/2) and (JF/A t Q/2), introduced in tabular 

and graphical forms, are used for flood routing and for the determination of 

necessary reservoir storage. This author is the first to use the storage 

factors in the above form.

113. Jacob, C. C., 1912, Computing the size of a reservoir spillway: Eng. NewsRec., v. 67, p. 1134.

The mass-curve method of flood routing is discussed in contrast to the 

numerical-integration method using the inflow hydrograph, as discussed by 

Palmer (1912).

114. Levi-Civita, Tullio, 1912, Sulle onde di canale [On the canal waves]: Rend.

della Reale Accad. dei Lincei [Italy], v. 21, p. 3-14. 

A theoretical study of wave motion that develops the concept of absolute 

and relative discharge. Complex variables are introduced. The total 

flow is integrated for a given time interval. The Stokes-Rayleigh's 

theorem is generalized, and a general equation between the global elements 

is developed.

115. Palmer, W. S., 1912, Methods of computing the size of spillway of a dam:

Eng. News-Rec., v. 67, p. 524-525.

The numerical integration of the differential equation for obtaining 

the linear increase of the area of reservoir surface, with level A=a-\-bH, 

and for a spillway with the rating curve Q = cLH3/2, where H=height
BIBLIOGRAPHY 59

above spillway crest, L=spillway length and c=coefficient is shown. 

The spillway effect upon the hydrograph and the storage-spillway discharges are discussed.

116. Cisotti, Umberto, 1913, Intumescenze e depressioni che dislivelli del letto 

determinano in un canale scoverto [The intumescences and depressions 

which the drops in channel levels determine in a free surface channel]: 

Rend, della Reale Accad. dei Lincei [Italy], v. 22, p. 417-422. 

The shape of a stationary wave in steady flow over a sill in channel 

bottom is studied analytically on the same basis as in the author's previous paper (1912, p. 588-593). The sill is not vertical, in this study, but 

is a ridge of triangular form, either positive (above the bottom) or negative 

below the bottom). The equations for shape of positive or negative standing 

waves are developed.

117. Cisotti, Umberto, 1913, Sulle onde semplici di tipo permanente e rotazionale 

[On the simple waves of permanent and rotational type]: Rend, 

del Reale Inst. [Italy], v. 46, p. 917-925. Reprinted in "Nuovo 

Cemento," [Italy], v. 7, p. 251-259, 1914.

The simple progressive waves having the sinusoidal profile of m re 

generalized form than Airy's waves are studied analytically.

118. Mueller, R., 1913, Ermittelung der Wirkung eines Ueberfalles bei einem 

Stauweiher [The determination of the spillway effect in a reservoir]: 

Allg. Bauzeitung [Austria], v. 78, p. 59.

Mass curves are used as the flood-routing procedure for determination 

of spillway effect on flood waves in a reservoir.

119. Fuller, W. E., 1914, Flood flows: Am. Soc. Civil Engineers Trans., v. 77,

p. 564-694.

In his analysis of flood frequency and development of a formula for 

computing the frequency interval (return period) the author demonstrates, 

for some simple hydrographs, that the influence of river basin storage upon 

flood attenuation is a function of the percentage of storage to total flood 

volume.

120. Karlson, 1914, Svallninger i oeppna kanaler [Waves in open channels]: 

Tekniske Tidskraft [Sweden], p. 137.

121. KoSeny, J., 1914, Ueber den Hochwasserverlauf in Fluessen und das 

Retentionsproblem [On the passage of flood wave in rivers, and the 

problem of storage retention]: Oestereich, Ingenieur u. Architeklen 

Zeitschr., [Vienna], v. 61, p. 261.

122. Massau, Junius, 1914, Meinoire sur 1'integration graphique des Equations 

aux derivees partielles [Memoir on graphical integration of partial 

differential equations]: Ed. du Centenaire par les soins du Comit6 

Nat. de M6canique, Mons [Belgium].

123. Mises, R. von, 1914, Elemente der technischen Hydromechanik [The

elements of technical hydromechanics]: Leipzig.

General outlines of a method for the study of gradual damping of translation waves under the effect of roughness are given in this book.

124. Rayleigh, Lord, 1914, On the theory of long waves and bores: Royal Soc. 

[London] Proc., ser. A, v. 90, p. 324-328.
60 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

A long wave of finite height can propagate in still water only with change 

of shape inasmuch as it is impossible to satisfy the requirement of a free 

surface for a stationary long wave, if h is the height of wave above the 

original level, unless h2 can be neglected. The force acting upon a free 

surface is determined, and conditions that must exist in order for a wave 

to become stationary are analyzed.

125. Running, T. R., 1914, Filling and emptying of reservoirs: Eng. NewsRec., v. 69, no. 3, p. 67-68.

Pilling and emptying of nonprismatic reservoirs are studied by using the 

storage equation and the graphical method of its integration \P Q = 

AdH/dt]. Since P=/(0, Q=f(H), A = /(H), then dH/dt=f(H, t). The 

family of curves of dH/dt~f(() for constant H can be plotted, when P, Q 

and A are known. Using the starting point B0 (Ho, to), and selecting the 

point BI on the line HI, so that the area under B0Bi is (Hi H0), the point 

h is obtained, and so on. The function H= f(t) is thus obtained.

126. Wedderburn, J., 1914, On long waves: Am. Jour, Math., p. 211.

Progressive waves in a canal with slowly varying cross section are studied 

and solvable cases are given, in which the elevation of water is inversely 

proportional to the square root of width and to the fourth root of the averagejiepth of cross section.

127. Cisotti, Umberto, 1914-15, Nuovi tipi de onde periodiche permanent! e 

rotazionali; Note la and Ila [The new types of periodical permanent 

and rotational waves, Notes la and Ila]: Rend, della Reale Accad. 

deiJLincei [Italy], v. 23, p. 556-561 and v. 24, p. 129-133. 

This paper is an analytical treatment of special waves which become 

periodic waves.

128. Bakhmetev, B. A., 1915, Vvedenie v izuchenie neustanovivshegosya 

dvizheniya [Introduction to the study of unsteady flow]: Leningrad.

129. Palatini, Attilio, 1915, Sulla influenza del fondo nella propagazione delle 

onde dovute a perturbazioni locali [On the influence of bottom on the 

propagation of wave caused by local disturbances]: Rend, del Circ. 

Matematico di Palermo [Italy], v. 39, sem. 1, p. 362-384. 

The case when channel depth is not small, but still has influence on the 

wave propagation, is studied analytically. Taking into account the boundary conditions of the water-free surface and of the bottom, the integration 

of wave equations is reduced to a unique partial differential equation of 

fourth order. The equation of free surface of water is expressed for each 

time of motion for a relative depth by a converging series. By further 

analytical treatment the influence of bottom is represented.

130. Palatini, Attilio, 1915, Sulla influenza del fondo nella propagazione delle 

onde dovute a perturbazioni locali; Studio asintotico del pelo libero 

[On the influence of bottom on the propagation of wave caused by 

local disturbances; the asymptotic study of free water surface]: Rend, 

del Circ. Matematico di Palermo [Italy], v. 40, sem. 2, p. 169-184. 

The Rieman method is used for integrating the influence of depth on 

wave movement, in order to determine the asymptotic value of an integral. 

The asymptotic study of the free-water surface is developed.

131. Fry, S. A., 1916, A graphical solution of the problem of storm flow through 

a reservoir: Eng. and Contracting, v. 45, p. 370-372.
BIBLIOGRAPHY 61

On the basis of the storage equation (inflow minus outflow equals the 

stored water) and using the records of the change of stage at the upstream 

gage to determine relationships of inflow, stage and time, graphs are 

developed to show time, in hours, required to fill the reservoir to any given 

elevation.

132. Jones, B. E., 1916, A method for correcting river discharge for a changing

stage: U.S. Geol. Survey, Water-Supply Paper 375-E, p. 117-130. 

For unsteady flow in a river, the ratio of two discharges Qi and Q2 at a 

river gaging station for the slopes Si and S2 is

where AH/At is the rate of change of stage and V is the surface velocity.

133. Alibrandi, M. P., 1917, Sur la theorie des ondes de crue [On the theory of 

flood waves]: Annali d'lngegneria e d'Architettura (Italy), v. 32, 

no. 10 and 12, and a large summary by Goupil, Ponts et Chaussees 

Annales [France], 1917, t. 40, v. 5, no. 29, p. 200-210. 

The analytical treatment of wave propagation gives C= F+ 3Vg (H0+y) 

 2-JgHo, and for y=o (beginning and end of wave), Cq=F+Vg#o, while 

for inside values, y=h, the celerity is Cb C0 ± 3-\/<7#o(Vl + h/Ho 1). 

The author compares this formula with Boussinesq's. By integration of 

partial differential equations for flood waves, it is concluded that the 

deformation of flood waves in rivers is more complex than that of the wave 

in prismatic channels without lateral inflow. The method followed in 

derivation is a combination for each variable of the value for steady flow 

and a changing value for unsteady flow neglecting the second- and thirdorder magnitudes of changing values.

134. Johnson, R. D., 1917, Surges in an open channel: Am. Soc. Civil Engineers

Trans., v. 81, p. 112-115.

The surge in a resistance-free rectangular channel is compared to the 

hydraulic jump, and two corresponding equations are found. Discussion 

of the author's conclusions and of earlier studies is provided by Church.

135. The Miami Valley Flood-Protection Work, 1917, Study of retarding-basin 

operation (the fourth of a series of five staff articles): Eng. News-Rec., 

v. 77, p. 186-188.

A short description of the flood-routing method using the storage equation 

is described. A demonstration is given of the trial-and-error method, 

which employs numerical integration by tabulation of factors (inflow, 

outflow, storage, water height, and so forth), that are enumerated, with 

reference to time, and are integrated for successive time intervals.

136. Schoklitsch, Armin, 1917, Ueber Dammbruchwellen [On waves created by 

dam breaches]: Akad. Wiss. [Vienna] Proc., v. 126, pt. 2A, p. 1489- 

1514.

The works of Ritter (1892) and Forchheimer (1914) are used as the 

starting basis for these experiments. Darcy's and Bazin's experimental 

data (1865) are used, as well as Zeitlinger's (1914) in small wooden channels 

for positive surges, with C=^fgH verified (for H=3.7 cm, C measured was 

91 percent of computed). The author's tests in a small laboratory channel 

and in an open ditch have shown: (1) the celerity is modified as the formula 

C=-<iJg(H-\-kQIV} where Q is the discharge creating the surge, V= velocity 

728-245 64 5
62 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

due to Q and k = 1.5 for smooth wavefront, and k = 1.0 for a breaking 

wavefront (many secondary waves); (2) the turning (rotation) point of 

 wave surface at a sudden gate removal is H/2 and not 4/z/9, as Bitter 

found by an analytical treatment; (3) the emptying of a reservoir (L-long, 

fl-deep) is (6 to 10)+L/V0fl seconds, if a side is suddenly removed. In 

the case of a sudden opening in the profile being smaller than the cross 

section, the height is given for the wave depth.

137. Thomas, H. A., 1917, Flood-retarding reservoir problem directly solved:

Eng. News-Rec., v. 79, no. 5, p. 226.

The storage equation P Q dWfdt is solved by mass curves. Given: 

Mass-curve of P, storage W=f(Q) computed from W=f(H) and Q=f(PT). 

An interval A£ is selected as fixed, inside of which all changes are considered 

linear. On the storage W for each outflow Q, }iQ&t is added and a new 

curve W+%QM =/(<?) is plotted. If the accumulated outflow is known 

for ti, and A£ added, in the middle of Af the value of SPSA£ is determined;

SP.-AiE-SQ.Af = W+ftQ&t,

and with this value from the curve W+$Q&t=f(Q) the mean value of Q 

in the interval A< is obtained. The outflow during the interval is QAt, and 

the new point of 2<?A£ becomes ~SQiAt-{-Q&t. In this manner, the integration is carried out.

138. Cisotti, Umbero, 1918, Equazione caratteristica dei piccoli moti ondosi in 

un canale di qualunque profondita-; Note la e I la [The characteristic 

equations for small wave movements in a channel of any depth; Notes 

la and Ila]: Rend, della Reale Accad. dei Lincei [Italy], v. 27, p. 

255-257, 312-316.

It is shown that the motion of small waves in a canal of finite depth can 

be represented by a unique function which is the characteristic equation of 

the wave motion. The second note shows how use of this characteristic 

equation produces a more feasible expression that eliminates the use of 

finite differences.

139. Cisotti, Umberto, 1918, Sulle onde superficiali progressive di tipo permanente [Progressive surface waves of permanent type]: Rend, del 

Reale Inst. Lombardo [Italy], v. 51, p. 85-94.

This is a study of the wave that propagates without altering its form and 

thereby appears to the observer, moving with the celerity C, to be steady 

flow.

140. Feifel, Eugen, 1918, Ueber die veraenderliche nicht-stationaere Stroemung 

in offenen Gerinnen, insbesondere ueber Schwingungen in TurbinenTriebskanaelen [On unsteady flow in open channels, expecially on the 

fluctuations in the canals of powerplants]: Forschungsarbeiten, Ver. 

Deutsch Ingenieure [Berlin], no. 205, p. 1-100.

The celerity formula C=T/gH is rederived. The height of a positive 

surge in upstream direction is given as A.H = C\H/~g is the final approximate 

form, but a better approximation is given as &H=H0 -}-\/Hoz -}-2H H0 , 

where H0 = V2/2g. The energy of the surge is determined. The watersurface profiles are given for constant celerity and for changing celerity 

along the canal. The positive surge due to increase of canal discharge is 

analyzed, showing results that are correspondingly similar to those of the 

previous case. The model experiments are described and the celerity
BIBLIOGRAPHY 63

formula is found to be a good approximation. The influence of the turbines' 

closing time on the surges created is studied experimentally, with practical 

formulas derived for maximum heights of surges. Experiments made in 

actual power-plant canals supplement the model experiments.

141. Horton, R. E. 1918, Determining the regulating effect of a storage reservoir 

(Differential equation for inflow, outflow and storage relations solved 

by using time interval as independent variable): Eng. News-Rec., v. 

81, no. 10, p. 455-458.

The storage differential equation in the form PAt=A A#+[(#i+(?2)/2]Af 

is discussed for integration (Qi and Q» at the beginning and at the end of 

At, A and H area and height of reservoir surface respectively, At=time 

interval). For given At, PI, A and Qi, and assumed AH, Qz can be determined, so thatAH and Qz can be obtained by successive approximations. 

In order to avoid this procedure, two functions are developed: Fi = AHi/ 

At Qt/2 and F2 = Qz/At+AHz/At, As Qi or Qz are functions of HI or H2 , 

Fi=f(Hi) and F2 =f(H2 ), and two curves can be computed that may be 

easily plotted. F2=P Fi, and for given FI and PI, the value F2 and H2 

(Qz) can be obtained, and so on.

142. Maillet, Edmond, 1918, Sur la propagation des crues [On the propagation 

of floods]: Fonts et Chaussees Annales [France], year 88, ser. 9, torn. 

46, v. 5, no. 20, p. 197-204.

The author derives analytically that the maximum discharge of a flood 

is a decreasing function with an increase of channel distance traveled, and 

that the minimum discharge of a recession curve following a rise becomes 

increasingly greater with the distance traveled.

143. Parsons, W. B., 1918, The Cape Cod Canal: Am. Soc. Civil Engineers

Trans., v. 78, p. 1-157.

The application of Levy's equations to Cape Cod Canal is given. Recognizing that the friction coefficient is not a constant, but varies slightly in 

an inverse ratio with the velocity, a new proposal is made that a friction 

coefficient, changing as a linear function of the mean velocity, will absorb 

the same total energy in the interval during which the velocity changes 

from zero to maximum.

144. Cisotti, Umberto, 1919, Sulle onde progressive di tipo permanente oscillatorie [On progressive waves of the permanent oscillatory type]: Rend, 

della Reale Accad. dei Lincei [Italy], v. 28, p. 174-178. 

The equation of Levi-Civita for the progressive waves of permanent 

type is further developed (in second approximation) and discussed.

145. Maillet, Edmond, 1919, Sur le mouvement graduellement vane" nonpermanent et la propagation des crues [On unsteady, graduallyvaried flow and the propagation of floods]: Fonts et Chaussees Annales 

(France), Partie Tech., (3) May-June, p. 289-330.

The author starts from a general momentum partial differential equation 

in Flament's Hydraulics (1909) and makes approximations by neglecting 

some of the five terms. He derives the formula for unsteady flow bV2/R= 

So \2>H/dx with the analysis of coefBcinet X and b under different conditions. The relation of hydraulic parameters of steady and unsteady flow 

is developed from the above expression. The orders of magnitude of the 

neglected terms in comparison with the term (l/S0)bH/bx are determined 

for many rivers. From the above formula it is concluded that all flood
64 FLOOD-ROUTING METHODS AND UNSTEADY PLOW

peaks attenuate and that sharp peaks attenuate most. The same conclusions are derived for discharges. The components of multipeak hydrographs are discussed, as are other aspects of unsteady flow, all derived 

from the above formula,

146. Nekrasova, A., 1919, O volne Stoksa [On the wave of Stocks]: Izv. IvanovoVoznesenkogo Politekhnicheskogo Inst. [U.S.S.R.], Petrograd [Leningrad], no. 1, p. 81-89.

The Stocks type of wave is studied analytically, and the angle between 

the tangents at the wave peak is proved to be 120°. The formulas for 

shape of wave, wave height (ft=0.165 L), and celerity of wave (approximately C= 1.12 -}-\fgLj2Tr with L=length of wave )are given.

147. Oeltjen, J., 1919, Ueber die Berechnung von Flutwellenlinien in einem

Tideflusse [On the computation of tidal wave profile in a tidal river]:

Zentralbl. der Bauverwaltung [Germany], year 39, no. 27, p. 137-139.

Two differential equations for unsteady flow are rederived and applied

to tidal-wave movement in a channel by the use of a simple finite-difference

method.

148. Pigeaud, 1919, Note sur la propagation des crues [Note on flood propagation]: Fonts et Chaussees Annales [France], v. 48, no. 4, p. 29-57. 

As the basis of this study, the author takes Graeflf's flood-forecasting 

method (1883), based on the twofold thesis that stage-discharge relations 

and wave celerities in a reach are functions of wave height. Although 

Graeflf's approximation is good, Pigeaud contests Graeflf's first thesis on 

the basis of Baumgarten's law (so-called in France), that for a particular 

stage, the discharge is greater during the flood rise than during the recession. 

The celerity given as C=dQ/dA is rederived, and Graeff's second thesis 

of wave translation celerity is discussed as giving higher peaks and lower 

troughs than actually occur in the rivers. Pigeaud replaces the celerity 

of Qmax, and a correction is applied to Graeflf's method by using either 

cPQ/dt2 or cPH/dP from the flood hydrograph. These factors are approximately proportional to (Qi 2Q2 +Q2) for the time of Q2, and real Qr= 

Q + Kd?Q/d&, where Q is the discharge from rating curve, with K a 

coefficient.

149. Cisotti, Umberto, 1920, Sull'integrazione dell'equazione caratteristica dei 

piccoli moti ondosi in un canale di qualunque profondita [On the 

integration of the characteristic equation for small wave movements 

in a channel of any depth]: Rend, della Reale Accad. dei Lincei (Italy), 

v. 29, p. 131-133, 175-180, 261-264.

This report is a continuation and further development of the author's 

(1918) study (p. 255-259, 312-316).

150. Forchheimer, Ph. and Ruemelin, Th., 1920, Hydraulische Folgerungen aus 

Beobachtungen in Trostberg [Hydraulic consequences from the observations in Trostberg]: Schweiz. Bauzeitung [Switzerland], v. 75, p. 

249; also v. 68, p. 21. 1916.

The experiments with surges (negative) in the canal Trostberg-Tacherling 

are described and analyzed. Starting from Q=35.45 m3 per sec flow was 

suddenly stopped, and the celerity of surge was computed as 4.17 m per sec, 

but was measured 4.02 m per sec.
BIBLIOGRAPHY 65

151. Maier, E. and Spaeth, K., 1920, Der Einfluss der Schiffschleusungen auf die 

Wasserkraftanlagen an dem zu kanalisierenden Neckar. Die 

Schleusungsversuche am Wasserkraftwerk Pappenweiler [The effect of 

manipulation of locks of canalized river on powerplants on the river 

Neckar. The experiments with lock manipulations at the water 

powerplant Pappenweiler]: Zentralbl. der Bauverwaltung, year^O, 

no. 77, p. 485-490.

Unsteady flow in a canal, caused by lock manipulations, is recorded. 

The records indicate clearly that the negative (depression) surges attenuate 

by progressing upstream. The attenuation of waves created by many 

manipulations is shown.

152. Woodward, S. M., 1920, Hydraulics of Miami flood control project (pt. 7, 

Routing floods through retarding basins): Miami Conservancy Dist. 

[Ohio] Tech. Repts. pt. 7, chap. 7, p. 156-200.

The word "routing" is used here, probably for the first time. According 

to the author, the expression "routing a flood through a retarding basin" 

refers to the operation of computing the probable effect which the retarding 

basin will have on the hydrograph of flood. For given P=f(t), W=f(H) 

and Q f(H), the Q=f(f) and H=f(f) are to be determined. Step methods 

and calculus methods are described. The step methods are: (1) trial-anderror methods (method with t assumed, method with t found by trial), and 

(2) direct graphical methods (using mass curve, and using inflow hydrograph). W=f(H) and Q=f(H) are usually combined in W=f(Q) by 

eliminating H. For step method with t assumed, the trial is made by 

selecting Q2 (end of A<) and recomputing Qz, and so on. Step method with 

t found by trial supposes a given Hz ($2), and t is determined by trial-and 

error method. The direct graphical methods use: (1) curve 

given by H. A. Thomas (1917), with mass curves, and (2) 

= W2 -{-%QAt together with inflow hydrograph (without mass curves). 

A five-sixth rule is developed as a rough approximation, showing that the 

average outflow discharge during a uniform inflow flood is approximately 

five-sixths of the maximum outflpw discharge.

153. Kobelt, Karl, 1921, Ueber eine kuenstlich erzeugte Hochwasserwelle in der

Aare [On an artificially created flood wave on the river Aar]: Mitt.

des Eidg. Amtes fuer Wasserwirtschaft, [Switzerland], no. 14, p. 1-33.

The flood wave artificially created on the Aar River on February 6, 1920,

is analyzed. The experiment is described, with description given of the

waves as recorded at the origin and at many stations along the river. The

celerity of wave i>s computed. The artificial wave is compared with a

natural flood wave. Waves created by the operation of hydroelectric

powerplants are discussed.

154. Krey, H., 1921, Die Wirkung der Schleusungen auf den Wasserstand und 

die Wasserbewegungen in den Haltungen [The effect of lock operations 

on the levels and water flow in the canals between locks]: Zeitschr. 

des Deutschen Wasserwirtschaft und Wasserkraft-Verbandes [Berlin], 

E. V., year 1, no. 5.

The wave movements created in canals by lock operations are studied.

155. Levi-CivitS,, Tullio, 1921, Les Ondes en los liquids, propagacion en los 

canals [The waves in liquids propagating in canals]: Instutut d'estudis
66 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Catalans [Barcelona, Spain], "Questiones deJ,Mecanica clasica y 

relativista," lectures made in Jan. 1921.

156. Nekrasova, A., 1921, O volnakh ustanovivshegosiya vida [On steady 

waves]: Izv. Ivanovo-Voznesenskogo Politekhnicheskogo Inst. [U.S. 

S.R.] Petrograd [Leningrad], no. 3, p. 52-65.

This analytical study treats the movement of steady waves on a horizontal 

bed having infinite water depth. The wave height is obtained. It is 

found analytically that the formation of the symmetrical, steady wave 

with double symmetry, propagating on the surface of infinitely deep water, 

also does not occur.

157. Pardoe, W. S., 1921, Method of computing waste-weir capacity and

reservoir rise: Eng. News-Rec., v. 86, no. 3, p. 114. 

The graphical integration of the equation dt = AdH/(P Q) for given 

Q=f(H) and A=f(H), and for given P=f(t) is shown.

158. Reineke, H., 1921, Die Berechung der Tidewelle im Tideflusse [Computation of tidal wave in a tidal river]: Jahrb. fuer die Gewaesserkunde 

Norddeutschlands [Berlin], Besondere Mitt., v. 3, no. 4, p. 1-22. 

The movement of a tidal wave in a channel is studied by the use of two 

partial differential equations, with the momentum equation being in the 

form given by Boussinesq. An approximate method for the computation 

of tide movement in the channel is shown. The factor for change of slope 

is used in finite-difference form, either with reference to time, as dS/dt, or 

to channel reach, as dS/dx.

159. Stevens, J. C., 1921, Computing reservoir outflow and height from inflow

and capacity: Eng. News-Rec., v. 87, no. 25, p. 1031-1032. 

In the integration of storage differential equation P Q dW/dt, the 

function W=f(Q) is replaced by dW=f(Q)dQ, where f(Q) = m is the slope 

of storage function and may be considered practically constant in certain 

limits of Q. The trial-and-error method is based on different Af and corresponding m. The general form of the storage differential equation for 

Q=f(PT) and W=f(H) as power functions of H is developed, but not integrated because of the difficulty to fit the hydrograph P f(t) by a mathematical expression.

160. U.S. Bureau of Reclamation, 1921, Graphical determination of hydraulics 

for detention reservoirs: U.S. Bur. Reclamation, Denver, Colo., Office 

Design Data, no. 40-C-235, Aug. 8, 1 p.

A short description is given of a graphical and numerical method for 

determining the effect of a detention reservoir, by use of a storage-outflow 

discharge relation, and mass curve of inflow discharge. The trial method 

of integration of the storage equation starts from equal storage increments; 

the corresponding times, reservoir levels, and outflow discharges are determined.

161. Taylor, G. I., 1921-22, Tides in the Bristol Channel: Philos. Soc. [Cambridge], Proc., v. 20, pt. 3, p. 320.

The propagation of tides is studied in a channel in which both width and 

depth vary in proportion to the distance from the upper end. The Bristol 

Channel was used for the approximation of this case. The spring tidal 

ranges, observed in natural occurrence at several places inside the estuary, 

are compared with the theoretical solution, which involves the first order
BIBLIOGRAPHY 67

of the Bessel function. The observed tidal ranges agree closely with the 

computed.

162. Bonneau, L., 1922, Etude sur les ondes stables dans les canaux et cours d'eau 

[Study of stable waves in canals and channels]: Ponts et Chaussees 

Annales [France], p. 273-279.

A study is made of fixed and traveling waves in canals where the wave 

length is generally not greater than the depth of water, except for the case 

of a solitary wave. The equation for a solitary wave is shown to be only 

a special case of the general equation for permanent waves. The assumption is made that the liquid portions of the waves are displaced parallel to 

themselves such that the particles trace homologous trajectories with regard 

to the bed. An expression is derived for permanent waves, with provision 

for the resistance of the bed and the excess of surface slope over bed slope.

H* , HjHo* I

H

where H= depth at any point, H0 =depth at upstream section, HI = depth 

at downstream section, x=distance from upstream section, and P0=total 

pressure against the upstream liquid cross section exerted in the downstream 

direction. For a solitary wave with a velocity of propagation C - 

the above equation reduces to z=2//(l + cosh n x), where n=] 

z (H H^ f=(Hi H0'), g=acceleration of gravity, and VQ= initial mean 

velocity. For a horizontal channel V0=0 and C=Vff#i, as found by 

Bazin. The impossibility of the occurrence of negative solitary waves is 

proven. Solitary waves depend upon only one parameter (height), while 

oscillating waves depend on two (height and length). Roll waves on water 

courses are very similar to surf waves of the ocean. Roll waves take up 

differences of energy between successive sections in the same stream having 

widely different characteristics. The surges of this nature are found upon 

the Congo River. Wind produces roll waves on the surface of water. 

This is more pronounced for quiet water than flowing water. 

[Abstract by C. A. Wright, Am. Geophys. Union Trans., pt. 4, 1938.]

163. Grover, N. C., 1922, Standing waves in rivers: Am. Soc. Civil Engineers

Trans., v. 85, p. 1400-1404.

Explanation of standing flood waves moving in a stream. The bottom 

friction is considered to be the essential cause.

164. Johnson, R. D., 1922, The correlation of momentum and energy changes in 

steady flow with varying velocity and the application of the former to 

problems of unsteady flow or surges in open channels: Engineers and 

Eng., v. 39, p. 233-240.

The difference between the laws of conservation of energy and momentum 

is pointed out. Pressure waves and suction waves are discussed, the former 

as standing waves set in motion relative to the earth. The suction wave 

has no counterpart in a standing wave and moves with constantly changing 

profile, and the top of the wave moves more rapidly than the bottom with 

continual lengthening of the wave front. By dividing the canal into short 

frictionless sections with a tiny drop in water level at the end of each section, P. Wahlman showed that the wave behavior can be analyzed without
68 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

significant error, it is stated. Wahlman's method is illustrated by an

example.

[Abstract by F. T. Mavis, Am. Geophys. Union Trans., pt. 4, 1938.]

165. Levi-Civita, Tullio, 1922, Risoluzione dell'equazione funzionale che caratterizza le onde periodiche in un canale molto profondo [The solution 

of functional equation which characterizes the periodic waves in a very 

deep canal]: Math. Annalen [Berlin], v. 85, p. 256-279. 

The periodic waves in deep channels are studied analytically by using the 

function of a complex variable. The works of Airy, Lord Rayleigh, Stokes, 

and others are discussed. Airy's celerity formula for deep water, given a» 

is verified.

166. Maillet, Edmond, 1922, Sur les procede"s d'annonces des crues fond^s in. 

en theorie [On the procedures of flood forecast, based on theory ]r 

Fonts et Chausse"es Annales [France], year 92, fasc. 1, v. 1, no. 8,. 

p. 145-156.

This paper is a criticism of Pigeaud's paper (1919) on flood forecast by 

stages and discharges. It is assumed that the rating curves and celerity 

graphs produce the same results when used according to Graeff's system. 

For the forecast between two stations, the approximate equation isHiKi=K(H2 JrK2'), where K, KI and K2 are constants. A similar relation. 

comes from three stations, two on the main river and one on the tributary. 

For the general type f(H)= aHn there is a relation ^Hl = (ozla\)kzn~l^H^ 

+ (aa/ai + (k3/k2') n-l ^H3, with Hi=kzHz and H3=k3H2 and with kz and k3. 

constants, which the author calls Belgrand's law of increases. It is found 

that the conical equation aiHin=a2H2 n -\-a3H3n can be replaced by somestraight lines of type Aff1 =c2Aff2 +C3Aff3, with two or three systems of c2 

and c5 values. Different formulas are developed for stage relations. These 

methods are all considered as first approximations.

167. Winkel, R., 1922, Aenderungen des Wasserstandes in den Haltungen infolgeSchiffsschleusungen [The changes of levels in canals between locks dueto lock operations]: Zeitschr. des Deutschen Wasserwirtschafts u.

Wasserkraft-Verbandes [Berlin], no. 13.

The wave movements created in canals by lock operations are studied.

168. Bonnet, L., 1922-23, Contribution & Pe"tude theorique des fleuves & marge- 

[Contribution to the study of tidal rivers]: Trauaux Publiques Belgique 

[Brusselles] Annales, 1922, fasc. 3, 4, 5, 6, and 1923, fasc. 1, 2, 3, 5. 

The propagation of a tidal wave is treated as a wave of translation continually modified by friction and by the shape of the bed. The author'* 

thesis that movement of the tide along the river creates a countercurrent 

permits an explanation of all the characteristics of a river tide.

169. Forchheimer, Ph., 1923, Durchfluss des Wassers durch Roehren und 

Graeben grosser Abmessungen [Discharge through pipes and canals 

of great dimensions]: Berlin, Springer.

The water velocity of a positive wave due to the release of water, wit 

flow resistance taken into account and using Chezy's formula, is given aa 

V=\RP-7 T/S dy/dx, where S is the initial slope of water surface and dy/dx 

is the slope of the line connecting the points cut by the wavefront and themiddle part of the wave. An expression for the height of steep wavefront. 

of positive surge is given as function of x, h, C, Q, S, and other parameters^
BIBLIOGRAPHY 69

170. Gibson, A. H., 1923, Hydraulics and its applications: 3d ed., Great Britain,

405 p.

The formula for the celerity of a long-wave crest, relative to the flow 

of the stream, C1 , is developed as C1 =T/2g (H0+h) 2/ (2H0 -f h) with H0= 

depth of stream before the arrival of the wave and h=height of wave 

crest above surface of stream.

171. Krey, H., 1923, Einfluss von Querschnitteinengungen auf die Sturmfluthoehe [The influence of cross-sectional constrictions on the flood-wave 

height]: Zen'iralbl. der Bauverwaltung [Germany], no. 67, 68.

172. Macy, F. H., 1923, Reservoir spillway discharge: Eng. News-Rec., v. 90,

no. 8, p. 361.

This is a presentation of the classical method of routing flood waves 

through reservoirs, using the storage equation and the mass curve, and is 

similar to the method presented by C. C. Jacob (1912).

173. Pacak, A., Smetana, J., and Till, J., 1923, Tests on the Vranany navigable 

canal at Horin: Permanent Internat. Assoc. Navigation Cong., 13th 

Gong., rept. no. 6.

Some experiments in a navigable channel are described. The paper is 

concerned principally with the utilization of navigable channels for powerplant purposes.

174. Sherman, L. K., 1923, Experiments on the effects of upper channel improvements upon the downstream flood heights: Western Soc. Engineers 

Jour., v. 28, no. 8, p. 293-318.

The effects of upstream channel improvements on downstream flood 

heights are studied experimentally in a flume 66 feet long. Use is made of 

of the celerity formula C=(QZ Q\)I(A 2 ^.1), where Q2 and A2 (area) are 

the initial values and Qi and AI are the final values for time interval Af. 

The celerity C computed by above formula is verified by experiments. 

The statement by the author and H. Thomas, that a monoclinal wave 

becomes stable after some time and for a given length travelled is questioned 

in discussion by S. M. Woodward. H. Thomas, in his discussion, derives 

the above formula for finite monoclinal wave with Qi and Qz, AI and A3 

thus replacing by finite values, the Forchheimer small-difference concept 

of both Q and A . Thomas' proof starts from the assumption that a monoclinal wave becomes stationary after a given channel length, taking the 

flow resistance into account. For a small increase in shape he finds that 

C=1.5 V.

175. Varet, 1923, Etude graphique des conditions d'exploitation d'un reservoir

de regularisation [Graphical study of operation of a storage reservoir]:

Fonts et Chausse"es Annales [France], v. 93, no. 4, paper 21, p. 61-77.

The use of mass curves for routing hydrographs through a reservoir is

shown, and the application is discussed.

176. Winkel, R., 1923, Hydromechanische Vorgaenge beim Schleusen eines 

Schiffes [Hydromechanical phenomena during the lock operation for 

a ship]: Die Bautechnik [Germany], no. 33.

177. Bonneau, 1924, The"orie de la propagation des crues [Theory of flood 

propagation]: Fonts et Chausse"es Annales [France], v. 94, no. 2, paper 

14, p. 282-325.
70 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Study is made assuming negligible difference between bed slope and 

friction slope in the second partial differential equation for unsteady flow. 

Some actual data are given to justify the assumption. The integration 

gives solutions that are practically the same as the characteristics. The 

propagation of discontinuities in partial derivatives is derived as the 

classical solution. For very gradual waves without critical points, the 

velocity is

and celerity is 

C= (3F/2) [1 - (H/35) (d2 #/&c2)/(d#/da;).] 

For y=fl"-#0 =2//(l + cosh px), C=3V/2 (approximately)

Study is also made of the general case, without the above assumption, 

showing the corresponding integration of partial differential equations. 

The expressions for the coefficient of attenuation, j8, and wavelength are 

given in function of other parameters, for the case hm&*<H0/5. The 

following problems are studied and discussed: the influence of river cuts 

on floods, the effect of the major river bed and of levees on floods, the 

effect of increase of bed level on floods, and so forth.

178. Bouasse, H., 1924, Houle, rides, seiches et mare'es [Waves, ripples, seiches,

and tides]: Paris, p. 1-516.

An extensive treatise on the waves, which is a summary of wave-knowledge that existed after the First World War. In chapter 9 progressive 

nonperiodic waves and solitary waves (p. 271-317) are studied. In chapter 

10 other progressive waves (bores, river floods, and so forth) are treated. 

In chapter 14 the propagation of tides in river channel is studied.

179. Egiazarov, I. V., 1924, Metod opredeleniya kolebaniy urovnya vody v

dlinnykh byefakh pri regulirovanii [Method of determination of the

water level fluctuations in the long canals due to pondage]: Nauchnomeliorativnyy Inst, [U.S.S.R.] Izv. (Trans.), no. 7, p. 1-75.

Many earlier methods for flood routing and wave progression are summarized. Special attention is given to the method which the author has

developed for the Volkhov powerplant canals. This method deals with

storage in a reach, and relates to water surface, in order to derive the

stored volume. Largely, this paper is devoted to analysis of the work

of other authors in the field of movement of waves and surges in channels.

180. Forchheimer, Ph., 1924, Wasserschwall und Wassersunk [Positive and

negative surges]: Leipzig and Vienna, Franz Deutike, p. 1-42. 

The equations for wave celerity, water velocity, length of travel, depth 

(Ho+2/), and discharge for positive surge wave occurring in both downstream and upstream directions, and for negative surgewave (depression), 

occurring also in both directions, are derived analytically. Numerical 

examples for computation are given.

181. Levi-Civita, Tullio, 1924, Fragen der klassischen und relativistischen 

Mechanik [The questions of classical and relativistic mechanics]: 

Published as four lectures, Berlin, p. 26-59.

Permanent wave motion, as defined in his previous paper (1907) is 

herein discussed in detail.

182. Levi-Civita, Tullio, 1924, Ueber die Transportgeschwindigkeit in einer 

stationaeren Wellenbewegung [On the celerity in a stationary wave
BIBLIOGRAPHY 71

movement]: Vortraege aus dem Gebiete der Hydro- und Aerodynamik, 

Lectures held in Innsbruck, 1922; Th. v. Karman and T. Levi-Civita, 

eds., Berlin, Springer, p. 85-96.

This is an analytical study of wave motion in a canal under simplified 

condition (rectangular channel, horizontal bottom, frictionless motion, 

with an observer traveling with celerity C and using a coordinate system 

traveling with him). The velocity potential and streamline function in 

the form of complex variables are introduced. The movement of a liquid 

particle during wave propagation is analyzed. The mean discharge through 

a vertical cross section is determined as Q=CH g, in which q is a relative 

discharge in a moving coordinate system. A proof of water movement 

due to wave translation is given and an expression for Q in integral form 

is developed. The velocity V=Q/H, and the application of results to the 

Airy waves is given. The discussion among Prandtl, Levi-Civita, and 

von Karman, concerning the use of a moving coordinate system, is 

summarized.

183. Samsioe, 1924, Dammopningars avbordning. Matningsmetod grundad pa 

vattenstandsvaxlingar i en narbelager holga [The discharge through 

dam openings. Method of computation based on changes in water 

elevations in nearby pool]: Tekniske Tidskraft [Sweden], 25 Oct.

184. Sen, N., 1924, On the equation of long waves in canals of varying section:

Philas. Mag. [London], ser. 6, v. 48, p. 65-78.

Green [Mathematical papers, p. 225] was the earliest investigator of this 

subject and showed that in the case of progressive waves in a canal with 

slowly varying sections the elevation of water is inversely proportional to 

the square root of the breadth and to the fourth root of the average depth 

at the section. Some special solutions, analogous to those corresponding 

to the motion of progressive waves, are discussed in the article, and the law 

of the variations of depth and breadth, for which such types of motion can 

exist, has been deduced. 

[Abstract by F. T. Mavis, Am. Geophys. Union Trans., pt. 4, 1938.]

185. Thijsse, J. Th., 1924, Berechnung der Gezeitenwellen mit betraechtlicher 

Reibung [The computation of tidal waves with significant friction 

resistance]: Vortraege aus dem Gebiete der Hydro-and Aerodynamik, 

Lectures held in Innsbruck 1922; T. V. Karman and T. Levi-Civita, 

eds., Berlin, Springer, p. 116-122.

Employing Lorentz's linearization of friction losses, the movement of 

tidal waves is studied, with friction term as kV instead of gV2/c2H. An 

expression for k is given, [k=8gVma.x(3ir czH], to simplify the formula, 

k/n=tg&, where t? is an auxiliary angle which is a characteristic of water 

movement in channels. By means of this linearization the continuity and 

dynamic equations are integrated. Expression for V and H are given in 

the general exponential form of complex variables, expressing constants 

in that solution as functions of #, among other given magnitudes. The 

celerity is given as C=-\/gH eostf/cosjt? and considering no friction, 

C= -\lglf. The friction factor decreases C, inasmuch as Vcost? is always 

smaller than cosjt?. Application to the Zuider Zee is discussed.

186. Winkel, R., 1924, Aufnahme der beim Schleusen in einer Kanalhaltung 

entstandenen Senkungswellen [The survey of a negative wave, created 

in a canal by lock operation]: Die Bautechnik [Germany], year 2, 

no. 25, p. 251-252.
72 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

The negative wave, created by filling the lock 1 in Niederfinow, is followed 

and surveyed. The wave heights based on the celerity formula are first 

computed (h=Q/BC, and C=-^gR, R=hydraulic radius), and then compared 

with the observed values, with good agreement. Using Krey's formula 

(1912), h is computed and compared to the observed values, also with good 

agreement.

187. Fantoli, G., 1925, Sul passaggio dell'onda di piena nella supposta rotta di 

un serbatoio [On the passage of flood wave along a supposed direction 

of a reservoir]: Ann. di Utilizzazione delle Acque [Italy], fasc. 1.

188. Jeffreys, Harold, 1925, The flow of water in an inclined channel of rectangular section: Philos. Mag. [London], sec. 6, v. 79, p. 793. 

A theory has been constructed to account for certain travelling waves 

observed by Vaughan Cornish in steeply inclined conduits. It appears 

that the uniform turbulent flow of a stream, with a plane free surface, 

becomes unstable when the mean slope exceeds 1 in 100 and that it is then 

replaced by a series of bores travelling faster than the water. Observational evidence is qualititive. Several points remain to be tested. 

[Last part of author's summary.]

189. Levi-Civita, Tullio, 1925, Determination rigoureuse des ondes permanentes 

d'ampleur finie [Rigorous determination of permanent waves with 

finite amplitude]: Math. Annalen [Germany], v. 93, p. 264-314. 

The periodic waves, irrotational and steady, which piopagate without 

alteration of form on the surface of a liquid infinitely deep, are studied 

analytically. The author finds that these waves are symmetrical in relation to the vertical lines through the peak and the lowest point but are 

not symmetrical in relation to the mean horizontal water level, because 

the height of the peak above this level is greater than the depression of 

lowest point below this mean level.

190. Lin, Ping-Yi, 1925, Prediction of floods: Iowa Univ., M. S. thesis, p. 202.

On the hypothesis that flood prediction is the most economical means 

of flood protection, the author considers the flood problem from several 

angles. Chapter 1 deals with the propagation of flood waves on rivers 

and presents mathematical derivations of formulas for determining the 

rate of propagation of flood waves, for computing the velocity of the flood 

wave, and for predicting the height of the wave. Chapter 2 develops 

equations for the profile and volume of a roll wave and a translatory wave. 

Chapter 3 deals with actual floods.

191. Proudman, J., 1925, Tides in a channel: Reprinted from the Philos. Mag.,

v. 49, p. 465-475.

The present investigation relates to the tidal dynamics of a channel 

whose section is uniform along the length but of varying depth from side 

to side. After making general deductions concerning a section of any 

shape, attention is concentrated on a parabolic section. One result is to 

show the degree of accuracy involved in the approximations of what may 

be called the "narrow sea theory," the theory which in recent years has been 

shown to account for many of the observed features of tides in gulfs, channels, and narrow seas. The essential characteristics of the narrow sea 

theory is that transverse currents are neglected, so that the motion is 

assumed to consist of longitudinal oscillations whose currents maintain 

oscillating transverse gradients in virtue of the earth's rotation. By
BIBLIOGRAPHY 73

introducing geometrical simplicity into the basin, it is possible to take 

account of the transverse currents and then to compare the results with 

those obtained on neglecting these currents. If the channel is not too 

wide or too shallow the degree of accuracy of the narrow sea theory is high, 

but this degree of accuracy decreases as the effect of the earth's rotation 

becomes important.

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

February 1954.]

192. Eisner, F., 1925-26, Proktisches Beispeil zur Berechnung des Stauschwalles 

in einem Obergraben bei Vorhandensein einer Heberentlastung [A 

practical example for the computation of surges in a headrace canal, 

where a spillway syphon exists]: Wasserkraftjahrbuch, Munchen 

[Germany].

193. Bernadskiy, N. M., 1926, Teoriya i raschet rechnogo povodka s primenyeniyem k raschety sutochogo regulirovaniya r. Volkhova [Theory 

and computation of river flood waves with application to the computation of daily regulations of river Volkhov]: Materialy po Issledovaniyu 

r. Volkhova (Materials on the Investigation of the River Volkhov), 

no. 17.

The procedures are given for computing unsteady flow by the method 

of finite differences, as applied to the problems of daily discharge fluctuations on the Volkhov River.

194. Koch. A., and Carstanjen, M., 1926, Von der Bewegung des Wassers und 

dabei auftretenden Kraeften [Movement of waters and the associated 

forces]: Berlin Springer-Verlag, sec. 7, p. 132-150.

Discussion is given for waves subject to momentum control that is neglecting energy losses. The celerity of steep positive and negative waves 

is developed as approximation C= ± (Vi+3^2/4+Vff#i where ( ) is for 

a wave of small height traveling upstream, and (+) is for a wave traveling 

downstream. Friction is neglected. Other aspects of wave celerities are 

analyzed.

195. Krey, H., 1926, Die Flutwelle in Flussmuendungen und Meeresbuchten 

[The tidal wave in estuaries and sea bays]: Mitt, der Versuchsanstalt 

fuer Wasserbau u. Schiffbau, [Berlin], no. 3, Selbstverlag.

196. Makkaveev, V. M., 1926, Gidromekhanicheskiy anali Nevskikh navodneniy [Hydromechanical analysis of Neva inundations]: Izdatel'stov 

Gidrologischeskogo Inst. [U.S.S.R.], v. 1.

197. Makkaveev, V. M., 1926, Rol'treniya i neprizmatichosti rusla pri dlinnykh 

volnakh [The role of friction resistance and nonprismatic shape of 

channel for the long waves]: Izdatel'stov Nauchno-Meliorativnogo 

Inst. [U.S.S.R.], no. 11-13.

198. Moellner, M., 1926, Die Wellen, Schwingungen und die Naturkraefte 

[Waves, oscillations and natural forces]: Braunschweig [Germany].

199. Struik, D. J., 1926, Determination rigoureuse des ondes irrotationnelles 

pe"riodiques dans un canal a profondeur finie [Accurate determination 

of irrotational periodic waves in a canal with finite depth]: Math* 

Annalen [Germany], v. 95, p. 595-634.

The periodic irrotational waves in a straight canal having a horizontal 

bottom and vertical walls, and in which the liquid moves along the canal, 

are studied analytically, primarily on the basis of mathematical approach
74 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

introduced by Levi-CivitiL It is shown that the waves are symmetrical. 

Other characteristics of waves are treated, such as the equation of free 

water surface (wave shape), the mean depth in canal, the maximum depth, 

the maximum depression, the water velocity, the energy, and so forth.

200. Thorade, H., 1926, Fortschreitende Wellen bei Veraenderlicher Wassertiefe 

[Progressive waves in the changing water depth]: Mitt. Math. Gesell. 

[Hamburg], v. 6, no. 5, p. 203-225; see also Ann. Hydr., v. 54, p. 217- 

222, 1926.

Green's law is discussed and found not to be applicable for long waves. 

Stationary waves as a basic form are discussed. The solutions of wave 

equations are given in the case of changing depth for both convex and 

concave conditions. It is concluded that a very long progressive wave, by 

traveling along a sloping bottom between two points having constant depths 

of HI and Hz, is transformed to a progressive wave of nonuniform celerity, 

having an amplitude that oscillates between its original amplitude 

and an amplitude of ^HJHz times the original.

201. Weinstein, A., 1926, Sur la vitesse de propagation de 1'onde solitaire [On

the celerity of the solitary wave]: Reale Accad. Lincei [Rome] Rend.,

Classe sci. fis.-matematichee naturali, sec. 6, v. 3, p. 463-468.

Using the method of Levi-Civita (1925), the celerity of the symmetrical

stationary solitary wave is derived as CP^gHoil + h/Ho (21/20) -{-(h/H0)*]

in a uniform rectangular canal with the initial depth HQ and wave height h

middle point of wave of an infinite length). Since C2 , as estimated by the

first two terms, alone is greater than the true C2 , the correction (third

term on the right side) is negative, and, for h/H0 <l/5, this term cannot

ordinarily be greater than 4 percent of the first two terms.

202. Winkel, R., 1926, Das Verhalten von Hebungs-und Senkungswellen bei 

verschiedener Fliessbewegung [The behavior of rising and falling waves 

in the different water flows]: Deutsche Wasserwirtschaft [Germany], 

year 21, no. 1, Berlin, p. 4-6.

Types of positive and negative waves in a canal are discussed, with the 

application of basic formula for celerity C= V± -^/gA/B.

203. Winkel, R., 1926, Besondere Wellenerscheinungen in Schiffahrts Kanaelen 

infolge von Schleusungen [Special wave phenomena in the navigation 

canals due to lock operations]: Die Bautechnik [Germany], year 4, 

no. 9, p. 110-111.

A full negative wave, created in upstream direction by filling the locks, 

is discussed. The wave transformation by progressing, its influence on a 

boat in the canal (oscillations of a boat due to wave movement), and the 

wave reflection are analyzed.

204. Boess, Paul, 1927, Berechnung der Wasserspiegellage [Computation of 

water-surface elevations]: 2. Teil, Zeitlich veraenderliche Wasserbewegungen in off en en Gerinnen [Unsteady water movement in open 

channels]: Forschungsarbeiten aus dem Gebiete des Ingenieur-Wesens, 

V.D.I. Verlag, G.m.b.H., no. 284, p. 63-96.

Restatement is made of types of surges in channels, with formulas given 

for known celerity and surge heights. The experiments to verify the surges 

are described. The computed and observed heights of positive surges 

 created by sudden closure (total or partial) of a gate in a canal are compared 

both for the subcritical and supercritical flow. The minimum height
BIBLIOGRAPHY 75

(H+h) is given, in the case of total closure with constant Q but changing 

H, when H is equal to critical depth Hc> (H+K) minimum is equal to 

(2.25 Hc), but the height of energy line is only (1.5 H). Agreement between computed and experimental results is good. In the ease of partial 

closure, the height h for a AQ (change of Q) computed by formula agrees 

with the experimental results. For negative surges, starting from water at 

rest in channel, the computation gives Fmax=0.430 ^fgH; h=QA77 H, in 

the ease of total opening; Qmai 0.225 B V0 Hm and the coefficients for an 

approximation are 0.50, 0.50, and 0.250, respectively, for rectangular channels. The experiments gave somewhat smaller results for slope of channel 

bottom other than zero. The same is given for sudden partial openings. 

For celerity, a new_formula, C=4gH/(F'+4 Vff-fiO, is given, based on experiments, with ~*igH as a good approximation. Relation of h, Q, and V 

is discussed. The height of hydraulic jump is developed from the above 

analysis. The experiments with surges in the headrace canal of the Itter 

powerplant are described and results given.

205. Dahl, 1927, Om svallninger i kanaler oeh tuber [On waves in canals and 

pipes]: Handlinger [Stockholm], no. 63.

206. Hill, R. A., 1928, Graphics of temporary flood storage: Eng. News-Rec.,

v. 100, p. 657-659.

This paper describes a method, which employs the use of mass curves 

and a polar diagram of outflow rate, for calculating the balance between 

reservoir inflow and outflow duiing floods. The example given is for a 

complex outflow system. The step-by-step integration by means of mass 

curves uses the polar diagram to relate slope, or rate of outflow discharge, 

to storage volume.

207. Puls, L. G., 1928, Construction of floodflow routing curves: U.S. 70th 

Cong., 1st sess., House Doe. 185, p. 46-52 and U.S. 71st Cong., 2d 

sess., House Doe. 328, p. 190-191.

The storage equation is used for flood routing along the river. The 

liver is divided into reaches, and for each reach the storage-height curve 

for the lower end of the reach is determined, as well as the rating curve of 

outflow discharge, and the two factors, PF+ Q/2 and W Q/2. The routing 

procedures are explained.

208. Shiroki, S. P., 1929, Sravnitel'naya otsenka metodov opredeleniya kolebaniy uravnya pri regulirovanii raskhoda [Comparative appraisal of 

the methods for determination of stage fluctuations in runoff regulation]: M. I. I. T. [U.S.S.R.] Trudy (Proc.), no. 11.

209. Forchheimer, Philipp, 1930, Hydraulik [Hydraulics]: 1st ed., Leipzig,

Teubner, p. 246-303, 411-432, 465-484.

The basic partial differential equation for unsteady flow and wave celerity 

are rederived and discussed. The wave energy and wave foi m are analy/ed. 

Dam breaches and created waves are treated using Ritter's original data. 

Tidal wave in estuaries is also treated. Flood waves in riveis are analyzed, 

especially with reference to celerity, maxima values (S, V, Q, PT), influence 

of cross-sectional shape on wave celerity (as the ratio C/Vin the function 

of cross-sectional shape), form changes, and wave attenuations showing 

AQ=(B2QAx)/[2S(<)Q/<)H) 3 r], where r is curvature radius of the curve 

Q f(%) at the peak. The wave-routing method is described in detail by 

use of the storage equation and mass curves. Othei procedures based on 

inflow and outflow hydrographs are also described. A good historical
76 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

review of flood-routing procedures for reservoirs is given. The chapter on? 

wave movement treats sea waves. Very extensive bibliographical references 

are given in footnotes.

210. Makkaveev, V. M., 1930, Teoreticheskoe izuchenie volnoobraznogo 

dviheniya vodnykh mass [Theoretical study of wave movement of 

water masses]: Gosudarstveniy Gidrologicheskiy Inst. [U.S.S.R.l; 

Trudy.

211. Besson, F. S., 1931, Reservoir effect upon river floods: The Mil. Engineer,,

p. 577-580.

The storage-discharge relationship is determined by the use of weighted 

inflow (3Pi + P2)/8 and weighted outflow (3Qi + &)/8, as a function of 

[(3Pi + P2)/8+TFi], where Wi is the storage at the beginning of the timeincrement. This weighting of inflows and outflows violates the continuity 

equation, according to Thomas (see Thomas' paper of 1934).

212. Egiazarov, I. V., 1931, K sutochnomu regulirovaniyu gidroelektricheskikh, 

stantsiy. Eksperimenal'noe issledovanie otritsatel'noy volny izliva, 

v dlinnom lotke [On the daily regulation of hydroelectric stations. 

Experimental investigations of the movement of negative wave along 

a canal]: Nauchnomeliorativniy Inst. [Leningrad], Izv., no. 3, (with* 

English summary).

213. Fenchel, W., 1931, Sulle onde di canale di tipo permanente [Canal wavesof permanent type]: Reale Accad. Lincei Atti [Rome], v. 13, May 17, 

p. 740-743.

With reference to Levi-Civita's papers (1907, 1924), it is shown that a< 

wave motion will be of permanent type if the propagation celeiity is constant and if the surface profile appears unchanged by wave progression., 

Levi-Civita's requirement of a constant relative discharge is not necessary.

214. Goodrich, R. D., 1931, Rapid calculation of reservoir discharge: Civil Eng.,.

v. 1, p. 417-418.

The trial-and-error method for flood routing by the use of storage equation (P Q=dW/d£), for the selected time interval is given, using the outflow-storage factor equal to (Pi + Pa + TFi Qi= W^ + Qz, so that Wz and Qi 

are obtained by successive operations in tabulating the above values.

215. Scobey, F. C., 1931, Unusual flow phenomena: Civil Eng. v. 1, p. 1101-1103. 

A brief illustrated discussion of hydraulic jump, stationary waves, overrunning waves, curves, and air en trainmen t is presented.

216. Shulits, Samuel, 1931, Graphical integration in hydraulic problems: Soc.,

Civil Engineers [Boston] Jour., v. 18, no. 8, p. 287-300. 

In graphical integration use is made of the so-called summation curve, 

the curve of the integral of the function being investigated. The construction and properties of the summation curve are presented here, and itsuse is illustrated by two examples. [From author's synopsis.]

217. Thorade, H., 1931, Probleme der Wasserwellen [Problems of water waves]: 

Hamburg [Germany], Verlag Henri Grand, p. 1-219; also see Problemeder Kosmischen Physik, Heft 13-14, and chap. 32 Flusstiden, 148-152,. 

Hamburg.

This is a clearly written book relating mainly to ocean waves but covering 

also flood waves and the effect of channel friction on waves. The second 

part of this work, p. 72-194, treats principally the translation waves. The-
BIBLIOGRAPHY 77

work of many investigators is reviewed and their equations for wave-celerity 

are given and explained. The book is comparatively free from involved 

mathematical discussion. Sixteen pages of references to literature (up 

to 1930) on sea and canal waves are given at the end of the book. 

[Abstract by R. E. Horton.]

218. Velikanov, M. A., 1931, Dvizhenie volny pri proryve plotiny [The wave 

movement created by dam breach]: Gidrotekhnicheskoe Stoitel'stvo 

[U.S.S.R.], no. 2, p. 20-21.

The author claims that the crest of a wave due to dam breach moves 

with celerity C=i/gH (1+3A/4H). He derives the approximate relationship between wave height and distance traveled by the wave crest.

219. Bakhmetev, B. A., 1932, Hydraulics of Open Channels: New York,

McGraw-Hill, Book Co., p. 1-329.

Though this book treats only steady flow, it is useful for the study of those 

characteristics of channel flows which are common to both steady and 

unsteady flow.

220. Barrows, H. K., 1932, Velocity of flow in natural streams: Am. Geophys.

Union Trans., v. 13, p. 339-349.

Studies of a 230-mile reach of the Delaware River between Trenton, 

New Jersey, and Downsville, New York, included comparisons of observed 

and computed times of transit of water. Port Jervis, New York (drainagearea 3070 square miles), approximately midway on the River between 

Trenton (drainage area 6,800 square miles) and Downsville (drainage area 

380 square miles), was taken as the index station and seven assumed index 

flows from 0.15 to 5.00 sec-feet per square mile were used as the basis for 

analysis. Many cross sections of the River were taken in the 230-mile reach 

and stage-discharge curves were made for each profile station. Concordant 

profiles based on different rates of flow at the index station were constructed 

to harmonize observations of discharge and profile. Times of transit were 

computed and corrected on the basis of float measurements which were 

made simultaneously in five sections of the River averaging about 50 miles 

each in September 1929, and March 1930. The former float observations 

were made at an average stage representing 0.45 sec-feet per square mile at 

Port Jervis, and the latter at a stage representing 3.6 sec-feet per square 

mile. Computed times of transit were materially longer than those based 

upon the float measurements, and the computed times were corrected to 

make them consistent with the float runs in September. Analyses of 

flood-wave velocities by L. K. Sherman showed that the flood-wave velocity 

was approximately double the velocity of ordinary flow at the rising stage 

of the River. Observations of wave velocities in the River below the 

Hawley Plant of the Pennsylvania Power and Light Company on Wallenpaupack River checked the computed time of flow by Sherman formula 

fairly well. 

[Abstract by F. T. Mavis.]

221. Bednarski, E. J., 1932, Graphical analysis of spillway capacity: Western 

Construction News, Feb. 25, 113-115 (also erratum, loc. cit., Mar. 25, 

p. 168).

The graphical solution of the flood-spillway problem is carried out by 

using the storage equation, the mass curves of inflow and outflow, and the 

spillway rating curve, expressed in power form.

728-245 64 6
78 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

222. Brown, E. I., 1932, The flow of water in tidal canals: Am. Soc. Civil 

Engineers Trans., v. 96, p. 749-834.

The author gives a very complete summary of the theories that have 

been presented at various times and indicates their limitations. He 

proposes a new theory which assumes the reflection of the canal wave from 

the basin or reservoir which is fed by the canal, and he states that the degree 

of reflection will be governed not merely by the canal itself, but also by 

the basin.

Based on the studies of Bourdelles (1898, 1899, 1900) and Bonnet (1922- 

1923), the derivations are given for: effect of friction in reducing height of 

wave in an infinitely long canal; time required for crest of wave to reach 

a given point; energy of wave; influence of change of width; interference 

of two opposite waves; propagation of the tide in a canal connected to 

infinitely large reservoir; propagation of tide in a canal feeding a reservoir 

of limited capacity; reflection of the canal wave from a tidal reservoir of 

limited capacity; and propagation of the tide in a canal connecting two 

bodies of water, each having an independent tide of its own. The theory 

is applied to the Suez Canal showing comparisons of the computed and 

observed data.

223. Calame, J., 1932, Calcul de 1'onde de translation dans les canaux d'usine 

[The computation of translation wave in powerplant canals]: Ed. 

La Concorde, Lausanne [Switzerland].

224. Cherkasov, A. A., 1932, O skorosti dvizheniya volny popuska [On the

celerity of release wave]: Vsesoyuzniy Nauchno-issledovatelskiy Inst.

Gidrotekhniki i Melioratsii [Moscow] Trudy, v. 7, p. 109-128.

The experiments of wave movement in a dry channel are made in a.

canal 15.45 meters long, 0.2 meter wide and 0.3 meter deep, with roughness (wood) n 0.0093. Three cases of wave movement are studied:

H=Q,V=Q, S = Q; H0 =0, S = Q; and H=Q, S>0. A fourth case is studied,

which is the same as the third case but with a partly opened gate. The

velocity of the wavehead is given from results as V=3.l-\lh(2H-\-K)/

^h+0.0025H/h, and for H=0, and h>Q, V=3.l^h^

The celerity of wave is given as C f(n, R, V, x, A). An empirical 

formula, based on experimental results, is developed, and the velocities 

are given for the four above cases.

225. East, R. L., 1932, Spillway-discharge calculations: Civil Eng., v. 2, no. 1, 

p. 34-36.

The semigraphical flood-routing method discussed was developed by 

W. A. Robertson. It is similar to the Goodrich method (1931) and is 

based on the method outlined by R. E. Horton (1918) that employs the 

simple storage equation (P Q dW/dt). Two curves are used: 

^1= JTi-QiAZ/2 and F2 = JT2 + Q2Af/2 (with A£=</12 in this study), F2 =P+F, 

where P=(Pi + P2)AZ/2. Values are obtained by a cumulative process 

of tabulation.

226. Eisner, F., 1932, Stroemungslehre der Rohre und offenen Gerinnen, 

II Teil (Eisner) Offene Gerinnen [Fluid dynamics of pipe and open 

channel, part 2 (Eisner) open channel]: Akad. Verlagsgeselschaft 

m.b.H. [Leipzig]. 

Paragraphs 6 and 7 of chapter 3 treat translation waves.
BIBLIOGRAPHY 79

227. Lamb, Sir Horace, 1932, Hydrodynamics: 6th ed., New York, Dover

Pubs., p. 250-561.

A complete discussion of the hydrodynamics of waves, with many 

references.

228. Levi-CivitS,, Tullio, 1932, Caracte"ristiques des systemes diffeYentielles 

et propagation des ondes [Characteristics of differential systems and 

propagation of waves]: Paris.

229. Meyer-Peter, E., and Favre, H., 1932, Ueber die Eigenschaften von Schwaellen and dei Berechnung von Unterwasserstollen [On the characteristics 

of surges and the computations for tailwater galleries]: Mitt, der 

Versuchsanstalt fuer Wasserbau, E.T.H., Schweiz. Bauzietung, v. 

100, no. 4, p. 43-50, and no. 5, p. 61-66.

Surges in the tailrace canals of water powerplants are studied for three 

phases of surge action: (1) when the surge does not touch the ceiling of 

the gallery; (2) when the surge fills the gallery; and (3) surge action at 

outlet in the canal. The analytical studies made employ the continuity 

and momentum equations. The experimental results obtained at the 

Wettingen water powerplant are added, and comparison is made between 

computed and experimental results.

230.^ Sherman, L. K., 1932, Streamflow from rainfall by unitgraph method:

Eng. News-Rec., April 7, p. 501-505.

The concept of the unitgraph is presented with an analysis of unitgraph 

determination and application. The paper discusses the unitgraphs for 

varying rates of rainfall. It discusses base flow, gives an example of 

unitgraph (Big Muddy River), and shows the computation of the hydrograph. Percentage of runoff and accuracy of the method are discussed.

231. Takaya, S., 1932, Some problems on the motion of water waves: Geophys.

Mag. [Japan], v. 6, p. 347.

A generalization of Green's laws of wave propagation in a canal of 

gradually varying section having transverse dimensions that vary slightly 

within the limits of a wavelength.

232. Arakawa, H., 1933, Generalized Green's law on wave motion in a canal of

variable section: Geophys. Mag. [Japan], v. 7, p. 319-325. 

A generalization of Green's law, demonstrating that: the greater the 

slope of the canal bottom, the greater becomes the amplitude of the sea 

wave traveling up the canal; the smaller the variation of canal width, the 

greater is the amplitude of this wave; the increase of amplitude is greater 

for the short waves than for the long waves; for small wave oscillations, 

the amplitude of waves on a sloping canal depends on the bottom slope 

and on the width of canal.

233. Barrows, H. K., 1933, Reservoir-storage above the spillway-level: Civil

Eng., v. 3, p. 233.

A graphical procedure is given for the determination of reservoir storage 

above the spillway level that reduces the outflow from the reservoir. A 

simple short method, employing the mass curves of inflow and spillway 

discharge, is used. (This method was devised by R. S. Holmgren in the 

1928-30 investigations of floods in Vermont.) As the starting point, 

storage W represents the increment of difference in level AH. Using the 

slope of the outflow mass curve for AH, a new point on the inflow mass
80 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

curve is obtained for the new level JET+AH, and the computation process 

is repeated.

234. Bernadsky, N. M., 1933, Rechnaya gidravlika, ee teoriya i metodologiya 

[River hydraulics, its theory and methodology]: Moscow-Leningrad. 

Translations from this book m Am. Soc. Civil Engineering, Manuals 

of Engineering Practice, no. 35, p. 53, 1957.

For the computation of elements of the free surface profiles in unsteady 

flow, the basic curves are developed and followed by the computation 

of instantaneous regimes. This is a method of plotting the instantaneous 

water levels in given times, with their control a posteriori, and is an approximate integration of differential equations of unsteady flow in finite differences form, with constant time intervals. The graphs of discharge and 

level distributions along the channel are obtained in the time moments 

which are selected in advance. The classical wave celerity formulas are 

discussed; these, according to the author, are valid only for small wave 

amplitudes, when flow resistance can be neglected. For wave amplitude 

greater than the original water depth, the author recommends a new 

celerity formula derived from continuity equation with the friction resistance taken into account, C V±{i^/g(HQ +h). The value of coefficient 

M is given in function of_the ratio H0/(H0+K). For the wave in a dry 

bed, M=2, so that C=

235. Drisko, J. B., 1933, Report on wave studies: On file with Mass. Inst.

Tech., Civil Eng. Dept.

This report contains a summary discussion of various types of waves, 

for example, solitary, tidal, oscillatory, which is followed by a description 

of experiments made in a small channel. The experimental work covered 

solitary waves, waves propagated into a channel by tides, power- (increment-) waves, and drop- (decrement-) waves. Simultaneous stagetime 

graphs at four points, recorded on a chronograph, accompany the report. 

[Author's abstract.]

236. Drisko, J. B., 1933, Wave motion in a channel: Am. Geophys. Union 

Trans., n. 14, p. 516.

This paper discusses briefly the nature of powerwaves (abrupt increase 

of flow in a channel) , mentions their relation to solitary waves of translation, 

and describes experimental work on various wave-types done in a small 

channel. The experimental work provided substantiation of J. Scott 

Russell's work with solitary waves, showed something of the nature of tidal 

waves propagated into a canal, and of the behavior of power waves. 

[Author's abstract.]

237. Egiazarov, I. V., 1933, Opisanie laboratoriy gosudarstvennogo nauchnoissledovatel'skogo instituta gidrotekhniki, Leningradskogo

gidrotekhnicheskogo Instituta (VTUZ-a) [The description of the

laboratory of government scientific institute for hydraulic engineering,

or of the Leningrad hydraulic institute (VTUZ)]: Izdatel'stov

Nauchno-issledovatel'skiy Inst. Gidrotekhniki [Leningrad], p. 1-192.

The experiments for unsteady flow are described, as are the measuring

and recording devices used, and all laboratory activities involved.

238. Gibson, A. H., 1933, Construction and operation of tidal model of the 

Severn Estuary: Great Britain, H. M. Stationery Office, sec. 9.
BIBLIOGRAPHY 81

The height and rate of travel of the bore is studied in the model of the 

Severn River. Despite the small dimensions of the model, close agreement 

is shown between phenomena observed in the river model and in the natural 

river. The speed of the bore depends on factors as indicated by

-v,

whereas the actual celerity is approximately 95 percent of that given by 

this formula.

239. Khristianovich, S. A., 1933, Otrazhenie dlinnoy volny konechnoy amplitudy 

[Reflection of long wave of finite amplitude]: Gosudarstveniy Gidrologicheskiy Inst. [U.S.S.R.] Zapishi, v. 9, p. 27-66.

The paper deals with the reflection of a very long wave of finite amplitude 

which propagates in a prismatic channel. The method of characteristics 

is used, and the equations are linearized equations. The reflection caused 

by a sudden change of cross section is studied. The depths and velocities of 

the reflected wave correspond to those of the primitive wave. When the 

reflection is created by a wall, the corresponding depths are related by the 

equation

where H=primitive water depth in canal, HI = depth of wave with fixed 

direction, and H0 =the corresponding depth in the section where reflection 

takes place. The solution is valid until the reflected wave is formed. 

[From author's abstract, given in French at the end of the paper.]

240. King, W. H., 1933, Translatory waves in open channels: Civil Eng., v. 3,

p. 319-321.

In general when the discharge in a channel is suddenly changed a positive 

or negative accelerating wave is formed. If the change acts to increase 

the depth, the wave has an abrupt face, and, if it acts to decrease the depth, 

the wave has a sloping face. A number of formulas for practical application 

are given. Channel friction has a modifying effect on the height and 

velocity of a wave and the profile of the water surface behind it. In 

deep water, for waves of moderate height, the velocity of wave varies 

approximately as the square root of the maximum depth of flow and can 

often be determined closely enough by neglecting channel friction. The 

principles involved in computing the water profile back of a wave by analysis of short reaches are comparatively simple, it is stated, but the computations may be long and tedious. 

[Abstract by F. T. Mavis.]

241. Makkaveev, V. M., 1933, Gidromekhanicheskaya zadacha v probleme 

nagonnykh navodneniy [Hydromechanical task in the problem of 

surge inundations]: Sbornik rabot po zashchite Leningrada ot navodneniy. (Collection of works treating the protection of Leningrad 

from inundation) [Leningrad].

242. Puppini, U., 1933, Influenza sui regimi fluviali dell'esercizio non uniforme

di impianti idraulici [The influence of unsteady powerplant operations

on river regimes]: Elettrotecnica [Italy], v. 20, p. 505-510.

Effects of a periodic change of discharge are propagated along a channel.

The paper stresses that the attenuation of a wave thus created is slow and

that unsteady operation of a water powerplant will be felt many kilometers
82 FLOUD-ROUTING METHODS AND UNSTEADY FLOW

downstream. Integration of partial differential equations is performed 

analytically and discussed. It is concluded that the velocity, depth, and 

discharge of waves propagate not only with attenuation, but with a change 

of form, which tends toward a pure sinusoidal form. An example is given 

of a flood of Adige River in Trento and Mori.

243. Sherman, L. K., 1933, Formula for wave velocity in open channels verified:

Civil Eng., p. 473.

Discussing King's paper (1933), the author gives Graeffs (1875) or 

Seddon's formula (1899) as C=(Q2 Qi)/(A2 Ai), with its properties 

emphasized.

244. Thijsse, J. Th., 1933, Influence of the closing of the Zuyder Zee on the tidal 

regimen along Dutch coasts: Permanent Internat. Assoc. Navigation 

Cong. Bull., 8th year, no. 15, p. 59-82.

An explanation of the Zuyder Zee problem is given, followed by a study 

of unsteady flow. The study is confined to determination of the nature 

of a tidal wave that passes through a channel, setting up frictional resistance 

to its movement. Using Lorentz's linearization approach, the celerity of 

the wave is given by a new formula C=-JgH cos e/cos %e with tge= (8gVm)f 

(37rnc2 //), where n = 0 for uniform movement and e=ir/2, and where 

resistance is nil, e=0.

245. Trifonov, E. K., 1933, Materialy po eksperimentalnomu issledovaniyu 

dvizhenia polozhitelnoy volny po sukhomu dnu [Experimental investigation of propagation of positive wave along dry bottom]: Nauchnoissledovatelskiy Inst. Gidrotekhniky Izv., (Sci. Research Inst, 

Hydrotechnics Trans.) [Leningrad], v. 10, p. 169-188 (English summary 

p. 182-188).

The experiments of wave movement on a dry channel were made in a 

canal 0.4 meters wide and 30 meters long. The initial wave heights were 

30 and 40 centimeters, with canal slope S=0.004. Changes of bed materials 

and of roughness were made. The number and purpose of experiments 

were limited and only general conclusions were made. Wave profiles were 

given but the formation of waves was not studied, and the shape of waveheads was determined by rather simple means. The conditions under 

which the experiments were made are described in detail.

246. Bachet, 1934, Note sur la propagation et 1'annonce des crues [Note on the 

propagation and forecast of floods]: Fonts et Chauss6es Annales [Paris], 

year 104, no. 34, p. 409-465.

The historical development of flood prediction on the Loire by the 

Federal Service is described. The early predictions were by the doublecolumn tabular method of giving the effects of flood waves from two 

separate sources from which the maximum could be obtained. On the basis 

of an extended mathematical analysis, a method of graphical determination 

of flood heights by the use of transparent scales was developed. The 

effects of attenuation, overbank storage eddies at confluence, and multiple 

tributaries are analyzed separately and reduced to a straight-line diagram. 

In doing this for a large system of tributaries considerable judgment is 

necessary. An illustration is given of the application of this method to 

floods on the lower Loire River at Montjean, and the results are compared 

with actual floods. The positive and negative errors were found to be
BIBLIOGRAPHY 83

evenly divided, in the neighborhood of 0.1 meter, and did not exceed 0.25

meter.

[Abstract by C. A. Wright.]

The discharge of unsteady flow, q, is classified as discharge of steady 

flow r and complementary discharge p (positive or negative). Using the 

equation q = k^S0 +f)H/()x it is found approximately that p = (r/2*So)c)^r/c)a;. 

Any change of unsteady flow is considered to be accompanied by a vertical 

and a horizontal change. Passing from discharges to water depths, similar 

relations are obtained. The confluence of rivers is analyzed, and the 

relations for three stations are developed. In flood routing for flood 

forecasts, special templates are constructed,with the methods of construction 

given, and the example of the Loire River shown.

247. Bonvicini, D., 1934, Sulla propagazione delle perturbazioni di regime nei 

corsi a pelo libero [Propagation of perturbations in a stream with free 

surface]: Elettrotecnica [Italy], v. 21, p. 652-656; see Sci. Abs. B., v. 

37, p. 692, 1934.

The problem of small fluctuations in the mean velocity and head of water 

in a free-flowing stream is studied. The loss of energy through friction is 

estimated according to the Manning formula and account is taken of any 

variation in the mean radius. It is concluded that waves propagate from 

upstream to downstream with small alternation. The integration is done 

by expressing hydraulic magnitudes, composed of values pertinent to steady 

flow, with additional components due, to unsteady flow. The solution of 

obtained partial differential equations is given in general form with real 

and complex numbers, then discussed, and examples are given.

248. Chary, 1934, Influence du calibrage partial d'un corns d'eau sur la propagation des crues [Influence of improvements in a water course on the 

propagation of flood waves]: Fonts et Chaussees Annales [Paris], 

year 104, no. 32, p. 394-399.

The object of this study was to determine what effect improvement of 

a river channel for the reduction of flood stages at one point had upon 

corresponding river stages at stations downstream. A graphical analysis 

is given, based upon two successive stations along a river, at which the 

stage-discharge relation and the inundation volumes are known. Discharge-time curves for a flood are drawn for the upstream station in relation to the downstream station, both before and after the improvements 

are made. The volume of inundation is determined in each case for a 

flat flood wave and for a sharp peaked flood wave. An example was 

given at Beuvron where the river valley inundation was reduced half by 

the improvements, without increasing the flood stage downstream from 

Beuvron by an appreciable amount. 

[Abstract by C. A. Wright.]

249. Cornish, V., 1934, Ocean waves and kindred geophysical phenomena:

Cambridge, Cambridge Univ. Press, p. 1-159.

The third chapter deals with bores and progressive waves in rivers. The 

creation of a progressive wave observed in the Rapids of Niagara is described. The excess velocity of roll waves over that of the current is in 

accordance with the formula given by J. S. Russell for the velocity of 

solitary waves. The friction of the bottom is regarded as the origin of 

the roll waves.
84 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

250. Crengg, H., 1934, Schwallversuche in Oberwassergraben des Murkraftwerkes Mixnitz-Frohnleiten [The experiments with surges in head

canal of the hydroelectric power plant on the river Mur, MixnitzFrohnleiten]: Die Wasserwirsschaft [Germany], no. 22, p. 213.

The experiments are made for the movement of surges (positive and

negative waves) in a canal with bottom width 6.20 meters; depth 4.70

meters; side slope 1:1.5; length, 6,950 meters; bottom slope 0.00025; and

maximum discharge Q=110 m3 per sec. The experiments made involved

nonuniform steady flow, with 0.69 meter and 2.62 meters of backwater

at the powerplant, and with surface line slopes of 0.00184 and 0.0000475.

The first experiment is made with the discharge decreasing from 94 m3 per

sec to zero, in 2 sec (positive surge), and the second experiment with discharge decreasing from 60 m3 per sec to zero, in 2 sec. The resultSrOf wave

movement are given and discussed.

251. Egia?arov, I. V., 1934, Sutochno regulirovanie Volkhovskoy gidroelektricheskoy stantsii [Analysis of pondage conditions at the Volkhov 

hydroelectric plant]: Nauchno-issledovatel'sky Inst. Gidrotekhniki 

Izv. (Sci. Research Inst. Hydrotechnics Trans.) [Leningrad], v. 12, 

p. 5-23 (brief abstract in English, p. 21-23).

The celerity of disturbances in headrace and tailrace canals is measured. 

The height of the wave created by discharge change in a powerplant was 

computed by author's formula (1924) as

where k is a shape coefficient of wave form, and AQ is the change of discharge 

in time Ai, but for small waves

where h is decrease of level in headrace canal, and W= volume of water 

passed in trailrace canal for time t. The analysis of level, fluctuations in 

both canals is made on the basis of observed results. The observed wave 

heights check closely with those computed by the above formulas. Wave 

propagation in canals covered with ice is investigated.

252. Egiazarov, I. V., 1934, Gidroelektricheskie silovye ustanovki [Hydroelectric power plants, elements of water power development]: 3d ed., 

Moscow, Leningrad, v. 1, p. 137-152.

The book treats unsteady flow in the canals of water powerplants and 

presents a summarized exposition on wave movement and methods of 

computation.

253. Hegly, 1934, Sur la propagation d'une onde solitaire dans un canal re"duit, 

a section trapezoidale [Propagation of a solitary wave in a reduced 

channel of trapezoidal section]: Acad. sci. [Paris] Comptes rendus, 

v. 199, p. 826-828.

Experiments using a model of a trapezoidal canal verify the basic formula 

for wave celerity C=^g(Ho+K), and show small differences between computed and observed celerities.

254. Mastitskiy, N. V., 1934, Graficheskiy method postroeniya mgnovennykh 

prodol'nykh profiley svobodnoy poverkhnosti pri neustanovivshemsya 

dvizhenii vody v rechnom rusle [Graphical method of construction of
BIBLIOGRAPHY 85

instantaneous longitudinal profiles of free surface during unsteady flow 

in the river channel]: Gidrotekhnicheskoe Stroitel'stvo [U.S.S.R.], 

no. 7, p. 7-14.

This paper presents a graphical flood-routing method, based on a continuity equation, in finite differences form, for selected elementary time 

intervals and reach lengths that uses the rating cuives and the relation of 

channel storage to the water depth. This method is feasible only for the 

determination of discharges; for the computation of water depths, additional 

graphical plottings are necessary. The slope of water surface in the reacb 

is taken into account by deriving the curve

and by considering the wave celerity in the reach for different depths or 

discharges. Ten steps are given for the step-by-step computation.

255. Proskuryakov, B. V., 1934, Primer primeneniya metoda proektirovaniya 

neustanovivshegosiya dvizheniya potoka, predlozhennoga inzh. Bernadskim N. M. K raschetu katastroficheskogo povodka v sluchaye 

mgnovennog razrusheniya plotiny [An example of application of the 

method of computation of unsteady flow, proposed by engineer N. M. 

Bernadsky in paper entitled "To the computation of catastrophic 

flood in the case of instantaneous breach of a dam"]: Nauchnoissledovatel'skiy Inst. Gidrotekhniky [Leningrad] Izv. (Proc.), v. 11,. 

p. 75-95.

This article contains an example of the method devised by N. M. Bernadsky for unsteady flow computations as applied to a particular case of flow 

resulting from a break in a dam. In the first part of the article a general 

description of the method is given. The method consists in simultaneously 

solving, graphically and by gradual approximations, the equations of continuity and equilibrium. Most terms in the dynamic differential equation for 

unsteady movement are neglected. The equations have the following shape r

AS= ( V*Bm)/c2 + (A/g) d V/dt

where dfl/d/= change of elevation in a unit of time ;dQ/d# = change of discharge along a unit of length of the river; Bm = average width of the channel; 

c=coefficient in Chezy's formula; g= acceleration of gravity; dV/dt=derivative of the velocity. At the same time, marginal conditions for the waves 

of upper and lower pools must be satisfied. These conditions are as follows:

(a) equation of the celerity of propagation of the upper-pool wave, C=VgH;

(b) equation of the celerity of propagation of the lower-pool wave, C= 2i/gH;

(c) condition of continuity, which requires that the discharge of water contained in the propagating wave should be equal to that passed through the 

terminal part. The condition of constant equality of volumes of water 

existing in the lower pool and issuing from the upper pool, must be fulfilled 

when plotting each instantaneous surface level. The procedure consists of 

plotting the instantaneous surface with subsequent checks and requires 

satisfaction of all above-mentioned conditions. In the second part of the 

article a practical example is given of an application of the method to a case 

of dam breach. In the third part a comparison is made between instantaneous calculated levels and those measured in laboratory. 

[From the author's abstract.]
86 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

256. Rzhanitsyn, N. A., 1934, Rechnaya gidravlika [River hydraulics]: Moscow

Leningrad, pt. 1, p. 96-209.

Comparison of the natural observations of unsteady flow with the results 

of flood routing by Mastitskiy's method is given in this book. The agreement found was considered to be adequate, in view of likely inaccuracies in 

observations and in the selection of a reasonable time interval for flood 

routing. This book is an extensive treatment of unsteady flow, with a 

general discussion of flood-wave movement, of Mastitskiy's method, 

Bernadsky's method, and other methods of flood routing. In deriving the 

outflow hydrograph, the mathematical integration is shown for a given 

inflow hydrograph and for given storage-discharge relations. The author 

described his method of graphical integration of the simple equation. The 

wave created by dam breach is treated in chapter 4, with a summary of the 

results of previous studies of this subject.

257. Schultze, E., 1934, Die Bestimmung der Abflussverhaeltnisse in Tidegebiet 

[The determination of flow conditions in the tidal region]: Die Bautechnik [Germany], year 12, no. 34, p. 438-443, and no. 38, p. 493-497. 

Results of measurements of velocity of tide movement (by floating object) 

at five places on a tidal river are given; these results show values obtained 

for discharge, mean velocity, and cioss-sectional area. The current-meter 

measurements are also given for some tidal waves. The analysis of wave 

motion is made by two partial differential equations (dynamic and continuity), and harmonic analysis is introduced for tidal waves. The application is discussed, with the solution by numerical approximation, using harmonic analysis (taking into account wave reflection and attenuation), and 

using the equation for surge movement. Comparison is made of the results 

obtained by three methods of computation, and the need for model studies 

emphasized.

258. Thomas, H. A., 1934, The hydraulics of flood movements in rivers: Carnegie 

Inst. Tech. [Pittsburgh] Eng. Bull., p. 1-70 (1934, 1937). German 

abstract in V. D. I. Zeitschr., v. 79, no. 18, p. 559, 1935. 

This report presents a systematic analysis of unsteady flow in rivers and 

of the approximate flood-routing methods that have been developed. The 

following are discussed: review of laws of steady and unsteady flows; 

dynamic propagation of stable wave forms; difficulties of integration by 

exact methods and boundary conditions; use of hydraulic models for unsteady flow (which is recommended for accurate flood routing); approximate 

methods of flood routing in uniform channels and in actual rivers. Three 

approximate methods are analyzed: first approximation, with simple 

storage equation for reach Aa; and time interval Af, based on the relationship of storage and outflow discharge (method found to be lacking in 

accuracy); second approximation, in which the slope of the reach is considered to be a straight line, with or without corrections for velocity head 

and acceleration term; and third approximation employing two differential 

equations for development of the finite differences method. The applicability of methods is discussed. Model tests are made to check the above 

methods.

259. Vreedenburgh, C. G. J., 1934, Een eenvondige afleiding van de formule van 

Airy voor de voort-plantingssnelheid van golven in water met eindige 

deipte [Simple derivation of Airy equation for celerity of water waves 

of finite depth]: Ingenieur [Netherlands], v. 49, no. 22, p. 212-213.
BIBLIOGRAPHY 87

A theoretical derivation of celerity formula which is given as C= 

V (gX/27r) tanh (27rfl/X), with a relationship between 5 = fl/X and the celerity. 

For C=V0# or C=T/g(H0+h) (J. S. Russell formula), for 

C is the above formula with elliptic trajectories (Airy), and for 

C -\g\l2ir with circular trajectories (Gerstner).

260. Egiazarov, I. V., 1934-35, Regulation of the water level in the reaches of 

canalized rivers and regulation of the flow below the last lock-dam 

according to whether the water power is used or not: Permanent 

Internat. Assoc. Navigation Cong., 16th, Brussels, 1934, Sec. 1, 

Question 2, and Nauchni-issledovatel'skiy Inst. Gidrotekhniki [Leningrad] Izv. (in Russian), v. 17.

261. Bernard, M. M., 1935, An approach to determinate stream flow: Am. Soc.

Civil Engineers Trans., v. 100, p. 347-395.

A continuation of Sherman's study of the unit hydrograph, which introduces the distribution graph as a way of converting rainfall to stream flow. 

The distribution graph is shown to be a function of watershed characteristics, and the method of developing the distribution graph without recourse to streamflow records is demonstrated by use of the physiographic 

characteristics of six individual watersheds. Limitations of this method 

are discussed.

262. Biot, M. A., 1935, Quadratic wave equation-flood waves in a channel with

quadratic friction: [U.S.] Natl. Acad. Sci. Proc., v. 21, p. 436-443. 

An exact solution of the equation of wave propagation with quadratic 

damping is demonstrated. It shows that high amplitude waves are more 

quickly damped and that this damping effect depends on both the volume 

of wave and the friction coefficient of the channel. The solution may be 

interpreted physically as representing certain types of flood waves.

263. Brown, E. I., 1935, A study of the effect upon navigation and upon the

upkeep of the banks and bed of canals and canalized rivers: a) of

undulatory movements and of currents produced, in the reaches

adjacent to locks with a high lift, particularly by the rapid filling and

emptying of these works; b) of the rise or fall of the water surface

caused by changes in the supply, whether natural or artificial, or by

the action of the prevailing winds on long reaches: Permanent Internat.

Assoc. Navigation Cong., 16th, Brussels, 1935, trans. (discussions).

The results of wave movement due to lock operations in the Panama

Canal are given, together with observations of wave movement due to

operations of a lock elsewhere in the United States. For the most part,

the canal cross sections are large in comparison with the discharges, so that

wave heights are relatively small.

264. Clemens, G. R., 1935, The reservoir as a flood-control structure: Am. Soc.

Civil Engineers Trans., v. 100, p. 879-927.

A graphical method based on the storage equation, called "reach reservoir 

method," for flood routing in reservoirs and in valley storage is given. 

The curves Fl = Wl+Q1 /2 and F2=W2 -Q2!2 are used, as well as Q=f(H) 

and W f(H), with known inflows. Curves of different scales can be 

plotted to increase the accuracy of the method. The time travel of the 

wave, the local inflows, and the valley-storage effect are discussed.
88 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

265. Dachler, Robert, 1935, Betrachtung ueber nicht-stationaeres Fliessen, im

besonderem ueber die Hochwasserwelle [Deliberations on unsteady

flow, especially the flood wave]: Wasserkraft u. Wasserwirtschaft

[Germany], v. 30, no. 8, p. 88-91, and no. 9, p. 103-108.

The instantaneous slope in a cross section is S=Sf+Sb+St, in which

S/=friction resistance slope, S6=(l/20)d( F2)dz, and 8t=(lfg)&V/M. The

relation of maxima for S, V, Q and H is shown, and the rating curve loop

is discussed. The order of magnitude of slopes Sb and St is given in an

example, and it is shown that they can be neglected when considering

friction slope. An approximate graphical procedure is developed for the

computation of wave attenuation. The celerity formula C (!/B)dQ/dH

is redeveloped and discussed.

266. Dalle, Valle, and Visentini, M., 1935, Regulation of the water level in the 

reaches of canalized rivers and regulation of the flow below the last 

lock dam according to whether the water power is or is not used: 

Permanent Internat. Assoc. Navigation Gong., 16th, Brussels, 1935, 

Trans. (discussions).

The summary of experiments with wave movement on the Adige River 

(Italy) is given. The length of the reach for experiments was 100 km. 

Waves that are artificially created by sudden water release from the reservoir of Cardano powerplant are measured along the river, and the celerity 

and wave attenuation are studied.

267. Deymie, Ph., 1935, Propagation d'une intumescence allongee [Propagation 

of a long wave]: Rev. Gen. 1'Hydraulique [France], v. 1, no. 3, p. 138- 

142.

Results of large scale experiments made on the Seine are discussed. It 

is rarely possible to operate on this scale, especially when it is necessary to 

use water carrying a heavy load of traffic. Experimental results agreed 

well the theoretical calculations. They were also compared with results 

obtained on the Aar River in Switzerland under similar conditions but with 

a steeper slope and a bed containing larger sized material. (Another 

series of observations and calculations for the Seine will follow in a later 

article.) The celerity computed by formula C= V+-\/gH as 6.14 m per see 

was confirmed by the experiments on the Seine. A function is introduced 

in the partial differential equation, so that 5^/5i=Q, and 5^/5a;= H. 

The partial differential equations of second order, which are linear in relation 

to second derivatives, are given. The author passes then to the charaoteristics which are given by three conditions with dx/dt=V±-JgH. He 

concludes that the celerity of wave front is independent of the form of wave 

which follows the front, and of the flow resistance (with the assumption that 

first derivatives in (H, Q)-space stay continuous on the wave front). The 

discontinuities in h and Q propagate with celerity V±-JgH. The computed h is found to be in agreement with the observed. Supported by analytical analysis, the wave is considered in three parts: front, with attenuation 

by exponential law h=ae~bf; backfront, with attenuation as e~kt/Pf*, and 

central zone of the wave. The theoretical background is the RiemanHadamard method of integration.

268. Deymig, Ph., 1935, Note sur la propagation des intumescences allongees 

[An account of the propagation of elongated intumescences]: Permanent Internat. Assoc. Navigation Gong. Bull., 20 »>. 64-70.
BIBLIOGRAPHY 89

The differential equations of unsteady flow are reduced to an AmpereMonge equation. It is deduced from the characteristics of this later 

equation that if there is initially a uniform regimen characterized by a 

velocity V0, a depth H0, and a quantity Q0, the celerity C of the front of an 

intumescence is C= Vo-\-^gH0. This relation is independent of the shape 

and the height of the intumescence and of the extent of the friction. The 

propagation of discontinuities in Q and H obey the same law. The law of 

decrease in the frontal height of a discontinuous intumescence is not analyzed. The law of attenuation of the frontal height h for an intumescence 

for which the flow superposed on the initial regimen is constant and equal 

to Q0 is h=(Q'o/V0)e~ax where a is a function of bed-slope S0, of V0 and HQ, 

the velocities and the depths of the primitive regimen, and x is the distance 

traversed by the front. There can be no steep back face to an intumescence 

produced by a suddenly applied constant excess flow of short duration. 

[Abstract by G. H. Keulegan.]

269. Egiazarov, I. V., 1935, Teoreticheskoe i eksperimentalnoe issledovanie 

otritsatel'noy volny [Theoretical and experimental investigations of 

negative wave]t Gidroelektricheskaya Laboratory a (Hydroelectric 

Laboratory) [Leningrad] Bull. 5, paper 6.

This is an experimental study of the influence of friction and slope upon 

the propagation and shape of negative wave. The first approximations of 

these two influences are given by a number of formulas for wave shape and 

velocity distribution.

270. Evangelisti, Giuseppe, 1935, Lo studio dei fenomeni di colpo d'ariete per 

mezzo deo calcolo simbolico [The study of phenomena of water hammer 

treated by symbolic calculation]: Elettrotecnica [Italy], v. 22, p. 134- 

141.

A method of investigation is explained based on the employment of 

functional operators, with which can be treated the phenomena of propagation of hydrodynamic perturbations reducible to the equations of vibrating 

cords. Examples of practical interest are given which can be dealt with 

by this method in a relatively simple way compared with the usual means 

employed. 

[Abstract by Joseph H. Sorenson.]

271. Favre, H., 1935, Etude the"orique et exp&lmentale des ondes de translation 

dans les canaux decourverts [The theoretical and experimental study 

of translation waves in open channels]: Paris, Dunod, p. 1-209. See 

also a shorter version in Rev. Gen. PHydraulique, v. 1, no. 3, p. 157 

The works of De Saint-Venant (1871) and Boussinesq (1877) are summarized. Gradually varied waves and surges, as well as their combinations, 

are analyzed, with some contributions in the discussions by the author. 

Of principal importance are the reported results of experiments, carried 

out in a canal, measuring 0.42, 0.40, and 73.58 meters (width, height, and 

length, respectively). Details of the measurements and experimental 

work are discussed. The given formulas of second approximation for 

positive surges are not valid as soon as fe>JFf/4; The undular shape of 

nearly equal waves of wavefront is studied; for H given the ratio of small 

wavelength to mean height of wave increases by a decrease of mean wave 

height. For the negative surges the formulas are not applicable to experimental results as soon as h^>H/5, with the shape of wave front changing
90 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

by its progression. The experiments are well in agreement with the 

theory in the case of reflection of surges. For small h, the shapes of surges 

can be approximated by a straight line. An example for the computation 

of wave translation in a canal of water powerplant with trapezoidal cross 

section is given.

272. Heduy, M., 1935, A study of the effect upon navigation and upon the 

upkeep of the banks and bed of canals and canalized rivers a symposium [see no. 263]: Permanent Internat. Assoc. Navigation Cong., 

16th, Brussels, 1935, Trans. (discussions).

Starting from the elementary theory of wave movement in canals (based 

on the Lagrange celerity formula and the wave height created by discharge 

change AQ), the results of laboratory model studies (Sauley, Metz) and of 

observations in nature regarding wave movement are given. The study 

is made for wave heights that are small in comparison with the water 

depth, in which case it is found that the wave height is independent of 

time of creating AQ difference.

273. Koval'skiy, M. I., 1935, Issledovanie transformatsii rechnogo povodka 

graficheskim metodom N. V. Mastitskogo [Investigation of flood 

routing by Mastitskiy's graphical method]: Gidrotekhnicheskoe 

Stoitel'stvo [U.S.S.R.], no. 5, p. 7-12.

Natural observations of unsteady flow are compared with the results of 

flood routing by Mastitskiy's method. Agreement was found to be good 

within the limits of error incurred in observations and time-interval selection 

for routing. The Oka River reach from Kaluga to Aleksin is used, and 

flood waves of 1908 and 1932 are shown as examples. The basis for routing 

is the storage-discharge relationship, W=/(Q), for the reach of river studied.

274. Kurihara, K., 1935, On the transmissibility of long waves along a canal 

when there are abrupt changes in depth: Japan Acad. [Japan] Proc., 

v. 11, no. 8, p. 316-318.

The problem of a partly reflected and partly transmitted water wave 

occurring at a point where there is an abrupt change in the cross section 

of the canal is expanded to the problem involving two such points, between 

which lies a reach of the canal that is either shallower or deeper than the 

adjoining reaches. It is concluded, by analysis, that the transmissibility 

is independent of the depth of the intervening reach. The maximum 

transmissibility occurs when the length of reach is an even multiple of 

one-fourth of the wavelength in that reach, and the minimum transmissibility occurs when the depth is an odd multiple of one-fourth of the 

wavelength.

275. Kuskov, L. S., 1935, K voprosu o dvizhenie polozhitel'noy volny po sukhomu dnu [On the problem of movement of positive wave along a dry

bed]: Gidrotekhnicheskoe Stroitel'stvo [U.S.S.R.], no. 5.

The methods of Bernadsky (1933) and Cherkasov (1932) are described

and analyzed. The author points out that the theoretical coefficient /*

in Bernadsky's formula, given as ju=2 for the positive wave moving in a

dry bed, coincides with experimental results, if the roughness coefficient

n=0.010-0.011. For waves moving in a dry bed, the author employs an

empirical relation of ju in formula C=fj,-)/gh, and the roughness coefficient

n, with mean values for n=0.010, M = 2.0; for n=0.0l7, ju=1.01; and for

ra=0.0225, M=0.67.
BIBLIOGRAPHY 91

276. Lacaze, B., 1935, Une crue artificielle sur 1'Aar [An artificial flood on River

Aar]: Rev. Industrielle [France], v. 65, no. 2309, p. 161-171. 

Report on artificial flood-wave produced on the Aar River, Switzerland, 

on February 6, 1920, based on report by K. Kobelt, published by Service 

Federal des Eaux, Switzerland; observations on progress on wave; bearing 

of observations on effect of pumped storage released into the river. 

[Abstract by Joseph H. Sorenson.]

277. Lamoen, J., 1935, La propagation de la mare"e dans les fleuves maritimes 

et dans les bras de mer [Propagation of the tide in tidal rivers and in 

areas of the sea]: Liege Univ., Cours d'hydraulique Applique"e, Nat. 

Sci. Conf., 2d, Brussels, June 19-23, Contr., p. 15-22. 

This article is a mathematical treatment of the tidal phenomena present 

in tidal rivers; special attention given to the Maritime Scheldt and its tributaries and to the Zuider Zee. The author gives a short summary of the 

theory of the oscillation of a heavy liquid in an open channel, discusses 

potential and kinetic energy of waves, and treats the calculation for compound channel cross sections. A short discussion is made of tidal conditions in the Zuider Zee and the Scheldt. Waves of translation are briefly 

described, and the author closes his paper with a paragraph on model 

reproduction of tidal rivers.

278. Masse", P. R., 1935, Sur une equation aux derives partielles de la the"orie 

des intumescences [On an equation with partial derivatives of the theory 

of intumescences]: Acad. sci. [Paris] Comptes rendus, v. 200, p. 109- 

110.

By integration of basic equations of unsteady flow, the following conclusions are drawn: the values of discharge, Q are attenuated at the front side 

of the wave according to exponential law; a flood wave having the initial 

propagation celerity is transformed by the effect of friction to a wave which 

travels at the assymptotic celerity 3F/2, where F=the mean velocity of 

uniform flow; wave-level increase is governed by the same law as discharge 

increase, Q/h being equal to the celerity of the front of wave, which changes 

to 3F/2 at the middle of intumescence.

279. Masse", P. R., 1935, Sur divers problemes aux limites de la the"orie des intumescences [On different problems at the limits of the theory of intumescences]: Acad. sci. [Paris] Comptes rendus, v. 200, p. 376-377. 

Referring to an earlier paper (1935, Comptes rendus) it is possible to 

relate the upstream and downstream wave propagation with the homologous points, Q'' Qe~^x , where Q' = discharge in upstream and Q in downstream direction, a = mean velocity of uniform flow, and x distance along 

the canal. This relation is used for the theory of wave reflections. The 

addition of discharge, occurring in the middle of the canal, and of wave 

propagation in the canal, when the movement is steady but nonuniform, 

is studied analytically.

280. Masse", P. R., 1935, These sur 1'amortissement des intumescences qui se 

produisent dans les eaux courantes [Thesis on the attenuation of intumescences produced in flowing water]: Paris, Herman et Cie, Thesis 

series A, no. 1564, order 2430, p. 1-79.

The fundamental partial differential equations for unsteady flow are 

rederived, and the basic equation of characteristics is shown. By assuming 

small differences for depth, velocity, and discharge between steady and
92 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

unsteady flow, a general partial differential equation of these departures is 

derived. By assuming a linear-type representation of that equation, the 

integration is made, and the problem of wave propagation in the downstream 

direction is solved by using complex variables. Special attention is given 

to study of the wavefront. The asymptotic intumescence corresponding 

to an initial limited disturbance is analyzed. These solutions are compared 

with those of Boussinesq (1877), and the differences are explained. The 

wave propagates so as to approach celerity to the limit 3F/2, which makes 

(HSo-cU2) nearly zero, or so approaching the law of floods; C is resistance 

factor in Chezy's formula. Other problems studied are: Cauchy's problem 

of injection of water in the middle of channel with resultant creation of two 

waves; wave reflections; and wave propagation on nonuniform flow. Applications are given, and numerical solutions are shown, in the application of 

the theory. Experiments made on the Seine River, (those of Chatou), as 

well as those made on the Aar River, are discribed and compared with the 

theory.

281. Mass6, P. R., 1935, L'amortissement des intumescences [The damping of 

translatory waves]: Rev. G6n. 1'Hydraulique [France], v. 1, no. 6, 

p. 300-308.

Surges on rivers are studied analytically. The base is the partial differential equation of second order, linear, and having constant coefficients. 

It is called "equation of the intumescences" by the author. It is expressed 

in reduced values of time t' = (pV£)/(gS), and length

as

where

&=1.027o, and a= , ' YM , a and 0 are

coefficients dependent on velocity distributions and are 1.096 and 1.040, 

respectively. The characteristics are C\ and C2, with C=&+Vl + &2. In 

flood, C= (3/2) V. The author uses the integration of the above equation 

based on the Poincare telegraph-equation method. The boundary conditions are determined by a convenient method of integration in the complex 

domain. The discussion of differences (of front celerity) between computed 

and observed values, as well as other aspects of movement of surges is given 

at the end of the paper.

282. Masse1 , P. R., 1935, Hydrodynamique fluviale, regimes variables [River 

hydrodynamics, variable regimes]: Paris, Hermann.

283. Norman, A., 1935, A study of the effect upon navigation and upon the 

upkeep of the banks and bed of canals and canalized rivers a symposium [see No. 263]: Permanent Internat. Assoc. Navigation Cong., 

16th Brussels, 1935, Trans. (discussions).

The results of experiments with waves produced in the natural canals 

(Trolhattan, Sweden) by lock operation are given.

284. Posey, C. J., 1935, Slide-rule for routing floods through storage reservoirs 

or lakes: Eng. News-Rec., v. 114, p. 580-581.
BIBLIOGRAPHY 93

This method involves short, uniform time-increments and requires a 

separate set of scales for each reservoir having a different volume-depth or 

outflow-depth relation. Lengths along slide and stock represent total 

volumes as "day-second-feet," the former bearing a simple scale with 

graduations defined by Fz= Wz+Qdf2, and the latter bearing two opposed 

scales with a common origin defined by I=Pt, and Fi=W\ Qtt/2, in 

which P*= average inflow rate, i=length of step (time increment), Wi and 

W} are storage at beginning and end of step, respectively, and Qi and Q2= 

outflow rates at beginning and end of step, respectively. The outflow 

rate is assumed to be a known function of the total storage, and the storage 

equation is of the form (F*i+F2). Given the inflow rate and the outflow 

rate at the beginning of a step, the outflow rate at the end of the step can 

be obtained directly by means of the slide-rule or by means of a nomograph 

illustrated. 

{Abstract by P. T. Mavis.]

285. Rappert, C., 1935, Nicht-fitationaere Abflussvorgaenge in Fluessen [Unsteady flow in rivers]: Ver. Deutsch Ingenieurfe [Germany], v. 79, no. 

18, p. 559.

Thomas' study of the hydraulics of flood movements in rivers (1934) is 

discussed briefly.

286. Schaffernak, Friedrich, 1935, Hydrographie [Hydrology]: Vienna, Julius

Springer, p. 1-438.

Uses of mass curves of inflow or outflow with continuity equation applied 

to flood routing are discussed (p. 243-253). The stage relationship along 

a river is shown (p. 287-288 and p. 375-381). Routing of flood waves 

through reservoirs, based on the continuity equation, is shown for different 

cases, when some relationships are known and two remaining relationships 

are to be determined. Two graphical methods are used: (1) inflow and 

outflow hydrographs and (2) the mass curves of these hydrographs (p. 

393-426). Many graphical short cuts are given.

287. Simonov, W. P., 1935, O rasprostranenii dlinnoy volny malogo podema 

odnogo napravleniya [On the propagation of long waves of small 

amplitude in one direction]: Zapiski Gosudarsvenogo Gidrologicheskogo Inst. [U.S.S.R.], v. 14, p. 125-199.

The author treats the influence of resistance on long wave propagation 

in one direction in a prismatic channel. The flow resistance is assumed to 

be proportional to the mean velocity, and the amplitude of the wave is 

assumed to be small. The equations of characteristics are developed and 

used. The method of successive approximation is applied, but the series 

obtained by this method are inconvenient for the computation. The reconstruction of the series by which they are made convergent and more 

convenient for the computation, is given in this paper. A particular 

problem, solved numerically, is given as an example.

288. Sum, T., 1935, A study of the effect upon navigation and upon the upkeep 

of the banks and bed of canals and canalized rivers a symposium 

[see no. 263]: Permanent Internat. Assoc. Navigation Cong., 16th, 

Brussels, 1935, Trans. (discussions).

The experiments with waves caused by lock operations in the Horin 

lock in Czechoslovakia are given.

728-245 64 7
94 FLOOD-ROUTING METHODS AND UNSTEADY PLOW

289. Thijsse, J. Th., 1935, A study of the effect upon navigation and upon the 

upkeep of the banks and bed of canals and canalized rivers a symposium [see no. 263]: Permanent Internat. Assoc. Navigation Cong., 

16th, Brussels, 1935, Trans. (discussions).

The paper deals with the influence of wave height on the wave celerity 

and with the attenuation of wave height by flow resistance. For wave 

heights at the distance x from the wave origin or when wave heights are 

known, the formula is given (without derivation) as h=h0/(l + gh&x/^c2RH), 

where h0 =maximum wave height, ft=any wave height, c=Chezy's coefficient, $= 1.5 (or 3.0), and the influence of flow resistance on celerity is given 

as

C=V0#Vcos 0/cos (|0), where tan e 3gVm&i/8irnc2H,

and where n=angular velocity of wave for the formulas of periods T=2»/n. 

The formulas were derived in 1924 and 1933. The results agree well with 

the observations in Zuider Zee. Details are given in the description of the 

wave reflections caused by narrowing or broadening of a channel, with the 

use of coefficient ft=B2CztBiCi. The comparison with observations is 

given.

290. Voegerl, Franz, 1935, Beitrag und Beispiel zur Schwallberechnung [Contribution and example for wave computation]: Wasserkraft und Wasserwirtschaft [Munich], v. 30, no. 22, p. 257-262.

The celerities, heights, and velocities of surges produced in canals of 

trapezoidal cross section are derived as a function of cross-sectional characteristics and as a function of created surges. The celerities are given in a

general form, with C= V± -\/gA/B as a special case for small surge heights. 

The surges in reservoirs and rivers that move along backwater curves are 

studied, and formulas for wave heights and celerities are derived. An 

example is given, and the results of wave experiments in the MaxnitzForhnleiten Canal are discussed.

291. Weber, C., Kanalwellen mit geringer Wellenhoehe [Canal waves with small 

wave height]: Zeitschr. Angew. Math. Mech. [Berlin], v. 15, no. 6, 

p. 380-381.

The potential movement of ideal liquid in a canal is studied analytically, 

with assumption that the movements are small, so that the equation for 

motion can be taken without the quadratic term. Three cases are studied: 

(1) when the gate at the end of canal does not move; (2) when the gate 

moves periodically; and (3) when the gate is motionless prior to the moment t, then is in motion from t to t\.

292. Weidner, M. E., 1935, A study of the effect upon navigation and upon the

upkeep of the banks and bed of canals and canalized rivers a

symposium [see no. 263]: Permanent Internat. Assoc. Navigation

Cong., 16th, Brussels, 1935, Trans., p. 127-136 (discussions).

The results of experiments with wave movements in canals in the natural

state are given for the canal from Wesel to Datteln; the canalized part of

Neckar; the canalized Main River; and the Danube at the Kochletlock

at Passau. No generalization is made, except by comparing the results

of observations with the maximum wave height, as computed by formula

h=QI(B^~gT[-BV), and with the celerity C=^fgH V (the two values

were close because the wave heights were small).
BIBLIOGRAPHY 95

293. Angelini, A. M., 1936, Sulla propazione delle perturbazioni nei corsi 

d'acqua [Propagation of perturbations in water courses]: Ellettrotecnica [Italy], v. 23, p. 486-494.

Starting with the simplified equations of disturbed flow in the channels, 

obtained by Puppini (1933) with the small intumescences, the propagation 

law for waves of small variations in discharge and depth is determined for 

any wave shape (periodic or nonperiodic). A graphical procedure is given 

for the practical solution of problems in order to avoid laborious computations. The graphical procedure is also applied to a more complex and more 

general case for any variation of form and amplitude. A numerical 

example is given and some general considerations are derived from the 

computation. The computation is made by using the method of operators 

(developed by Heaviside, Giorgi, and others), which is feasible for these 

problems. Some properties of operational calculus are shown and 

interpreted.

294. Bolotov, V. V., 1936, Prakticheskiy metod rascheta kolebaniy v byefakh' 

gidrostantsiy pri sutochnom regulirovanii [Practical method of calculating water variations in pools of hydroelectric plant under daily 

regulation]: IndustriaFniy Inst. Trudy [Leningrad], no. 11, Razdel 

gidrotekhnike, Vypusk, 1, p. 3-32.

The author analyzes the records of the Volkhow and Svir hydroelectric 

plants. He concludes that the elevations of the water level in the upper 

and lower pools may be represented as a straight line function depending 

on two factors: water capacity of the plant and additional water volume 

received from its upper pool. The author suggests a practical method 

of adopting, for his formulas, numerical coefficients developed from the 

operational records. As a result, he recommends this relationship as a 

method for calculating water-level variations in pools of hydroelectric 

plants, when their rates of water-capacity variations have been adopted. 

[From author's summary.]

295. Brown, E. I., 1936, Flow in tidal rivers: U.S. Engineer School, Fort Belvoir, 

Va. Engineering Procedure as Applied to Tidal Canals and Estuaries, 

Pt. 2.

Propagation of the tide in rivers. Theory of river tides. Consideration 

of termination of estuary in a dead end by a fall or a dam, which will be 

the case of a fully reflected wave in the indefinite canal. Consideration of 

case in which estuary merges without interruption into bed of a river free 

of tidal effects. Consideration of case where maritime portion is limited 

by dam or other obstruction reflecting primary wave, but with addition 

of fluvial flow.

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

February 1954.]

296. Chertousov, M. D., 1936, Opredelenie maksimal'noy otmetki urovnya 

vody v derivatsionnykh kanalakh GES pri sbrose nagruzki stantsii 

[Determination of maximum elevation of water surface in headracecanals of hydroelectric power plants during shutdowns]: Nauchnoissledovatel'skiy Inst. Gidrotekhniki [Leningrad] Izv. (Sci. Research, 

Inst. Hydrotechnics Trans.), v. 19, p. 82-104 (English summary, p. 

103-104).
% FLOOD-ROUTING METHODS AND UNSTEADY FLOW

The author contends that the maximum level created in the headrace 

will be at the upstream end of the canal in the case of the sudden closing 

of gates of a powerplant. Using Crengg's experiments (1934) and the experiment in the Niva no. 2 canal, it is shown that in actual conditions the rise 

of the water level in the forebay, near the plant, continues during a sufficiently long period of time after the moment when the positive wave 

reaches the upstream end of the canal. The maximum elevation in the 

forebay is higher than that at the upstream end. The author proposes to 

determine the maximum elevation at the forebays at the time that a 

negative wave, originating at the upstream end of the canal, reaches the 

forebay: Hm»x=H«+(Ha Hi)Tz/t, where Hmax =maximum elevation at 

forebay, # =elevation at forebay when the positive wave reaches the upstream end, HZ and HI, the elevations at two intermediate points of canals 

when the positive wave passes them, f=travel time of the positive wave 

between sections 1 and 2, and T2 =travel time of the negative wave along 

the entire canal.

297. Egiazarov, I. V., 1936, Neustanovishegosya volnovoga dvizheniya v 

dlinnykh byefakh [Unsteady wave movements in long pools): Nauchnoissledovatel'skiy Inst. Gidrotekniki [Leningrad] Izv. (Sci. Research 

Inst. Hydrotechnics Trans.), v. 16 p. 105-138 (English summary, p. 

135-138).

The first part is a presentation of the papers read at the 16th Congress 

of the Permanent International Association for Navigation Congresses. 

The second part deals with the results of investigation of wave movement, studied theoretically and in nature by the Hydroelectric Laboratory 

in Leningrad. Intumescences are evidences of degeneration occurring in 

positive and negative waves. The celerities of these waves are analyzed, 

with some simplification of earlier formulas, with C=-\/gH(l-{-h'/H), 

where h' is the mean height of wave (h 3h'/2). The reflection of waves is 

discussed. The comparison of theoretical results with results of observation of waves in canals is made with good agreement. The study is made 

of the shape of the head of a positive wave, the reflection of waves from 

great water surfaces, the maximum increase of elevation, and the influence 

of a broadened part of a canal on waves.

298. Favre, H., 1936, Le probleme des vagues [The wave problem]: Schweiz.

Bauzeitung [Switzerland], v. 108, no. 1, p. 1-4, and no. 2, p. 18-20. 

The propagation of waves in deep and shallow water is studied. The 

celerity C=V(,0L)/2w)f for deep water with length L, is compared with 

more exact formulas and with results of observations. The reflection of 

waves is discussed.

299. Fison, David, 1936, Manner of flow of river in flood: Inst. of Engineers

[Australia] Jour., v. 5, no. 3, p. 73, and v. 8, no. 5* p. 161-169. 

Some views of the manner of flow of a river in flood are advanced in the 

first part of this paper. In the second part these views are treated somewhat differently, modified, and brought nearer to practical application. It 

is shown: (1) that the speed of increments of the rate of discharge of a 

river, through any cross section, is equal to the differential coefficient with 

regard to height of the rating curve for that cross section, divided by the 

surface width at the cross section; (2) that there is strong evidence that 

the rating curves, that is, height against discharge, at average cross sections are curves of sectional areas multiplied by a constant for the section;
BIBLIOGRAPHY 97

(3) that variable flow in a stream can be measured without recourse to 

current meters or any current measurements whatsoever; and (4) that 

temporary storage, as separated, by definition, from less frequent classes 

of valley storage, does not reduce the maximum rate of discharge of a river. 

A method of delineating flood flow is illustrated that permits the study 

of the behavior of flood water in very accurate detail.

300. Horton, R. E., 1936, Natural stream channel-storage: Am. Geophys. 

Union Trans., pt. II, July, p. 406-418, (corrections 1937, p. 440-441). 

The relationship, Q=KWM, between the river discharge Q at the recession part of hydrograph (with ground-water flows deducted) and the total 

upstream channel storage, W, is analyzed for different streams. The 

parameters K and M are determined, and their variations are discussed.

301. International Boundary Commission, U.S. and Mexico, 1936, Flow of 

the Rio Grande and tributary contributions: U.S. Dept. State, Water 

Bull., no. 6, p. 82-94.

Data is given that shows the approximate travel time and flattening 

of flood crests on the Rio Grande.

302. Khristianovich, S. A., 1936, Sur le caleul dea mouvements non permanents 

dans les canaux et les rivieres [On the computation of unsteady flow 

in canals and rivers]: Russian edition: O raschete neustanovivshikhsya 

dvizheniy v kanalakh i rekakh: Hydrological Conference of Baltic 

Countries, 5th, Finland, 1936, report no. 16, D. p. 1-40; Russian ed.: 

Gosudarstveniy Gidrologieheskiy Inst. Trudy [U.S.S.R.], v. 5, p. 

25-53 1937.

The equations of characteristics are developed from partial differential 

equations of unsteady flow and are discussed, first with the friction included, then neglected. Four theorems are given for the solution of equations of characteristics, with the analysis of the positive downstream and 

upstream waves, for mixed waves, and for waves not subject to destruction. 

The surges as shock waves are studied. The following cases of long waves 

are discussed: propagation of the waves in one direction, the meeting of 

two opposite waves, the decomposition of nonpermanent movement in 

both positive and negative waves, and the reflection of a long wave. 

The approximate method of computation of a long wave by method of 

characteristics is given for the case when f rietion resistance can be neglected; 

a numerical example is shown.

303. Khristianovich, S. A., 1936, Razlozhenie neustanovivshegosya dvizheniya 

v kanale na pryamuyu i obratnuyu volnu [Decomposition of unsteady 

movement in canals in a direct and a reverse wave]: Gosudarstveniy 

Gidrologieheskiy Inst. [U.S.S.R.], Zapiski, v. 15.

304. Khristianovich, S. A., 1936, O volnakh, voznikayusichik pri razrushenii 

plotiny [On the waves created by dam breaches}: Gosudarstveniy 

Gidrologicheskiy Inst. [U.S.S.R.] Zapiski, v. 15.

30!). Lamoen, J., 1936, Sur 1'hydraulique des fleuves & marges [On the hydraulics 

of rivers with tidal effects]: Rev. Gen. Hydraulique [France], nos. 10, 

11, 12, p. 533-545, 595-600, and 643-654.

The forecast of tidal movements in estuaries after the rivers are corrected 

(Zuider Zee closure) is the subject of the paper. The general properties 

of tides in rivers are discussed. The method of determining the instan-
98 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

taneous discharges in a tidal river is given. Tidal currents and shapes of 

beds of tidal rivers are analyzed. Tide graphs are discussed and the theory 

of water oscillations in a canal is given, as are two partial differential 

equations for unsteady flow and the expression for potential energy of waves. 

The law of similitude is given for the studied phenomena. Propagation 

of tidal waves along rivers is theoretically studied, and the formulas of 

wave status are developed. After introduction of the flow resistance, the 

partial differential equation is reduced to that of the telegraph. The case 

of inclined channel is analyzed. The cases of a horizontal canal with flow 

resistance taken into consideration; a canal without upstream inflow, the 

tides of Zuider Zee, and a canal where the upstream inflows may not be 

neglected are developed and discussed. The results depend mainly on 

the accuracy of determined flow resistance coefficients.

306. Lane, E. W., and Baldwin, O. T., 1936 Flush wave velocities in sewers:

Eng. News-Rec., June 11, p. 848-849.

Describes the results of experiments on waves produced by flush tanks 

in sewers with very flat gradients, including diagrams showing the formation of the wave front and the velocity of movement. Shows why the wave 

front moves faster than the velocity of the water forming the wave. 

[Abstract by E. W. Lane.]

307. Mass6, P. R., 1936, Le probleme du mascaret [The problem of tidal bore]:

Rev. Gen. Hydraulique [France], v. 2, no. 9, p. 489-498. 

The author continues his investigation of this tidal bore as a follow up 

of his paper of 1935. The assumptions for the equation of intumescences 

are discussed (V= V0 -}-v, and H=H0 -}-h, in which v and h are small values, 

as differences among the unsteady and steady flows). Lorentz's introduction of linearization of resistance losses: kV=(gbV2)/H, in which k 

= (8gfbymax)/(3wfl), is discussed; its introduction reduces the equation to the 

form of the telegraph equation. The history of studies of mascaret (tidal 

bore) is given. The theory of tidal bore is developed, based on the theory 

of wave shock in movement of gases (according to Hugeniot). The celerity 

of the wave front is assumed to be V-{-\/gH and that of the body of the 

wave to be 3V/2. The method of characteristics is used. The integrated 

equation is given, then applied to floods and to tidal movements in rivers. 

The point x is determined where the singularity in tidal movement appears 

as a bore. The equation for different cases is discussed, and the applications are made using observed data on tidal bores of the Seine and Dordogne 

Rivers, which show good agreement when compared with the results 

computed by this method.

308. Meinzer, O. E., Cady, R. C., Leggette, R. M., and Fishel, V. C., 1936, 

The channel-storage method of determining effiuent seepage: Am. 

Geophys. Union Trans., v. 17, p. 415-418.

For a selected small river, the relation of the gage height to the total 

upstream channel storage is derived as a straight line, showing departures 

up to 7 percent.

309. Pettis, C. R., 1936, Relation of rainfall to flood runoff: Mil. Engineer,

v. 28, p. 94-98.

One diagram of a composite flood-wave shows wave velocities in terms 

of the average velocity of the entire cross-section at crest. This diagram 

shows that the average ratio of wave-velocity to average discharge-velocity
BIBLIOGRAPHY 99

at crest is 1.35. Much of the discussion is relative to the author's width 

formula for floods, originated in 1927 and published in Engineering NewsRecord, June 21, 1934. 

[Abstract by F. T. Mavis.]

310. Shilov, V. T., 1936, Issledovanie neustanovivshegosya dvizheniya v 

derivatsionnom kanale gidrostantsii Niva II [Study of unsteady flow 

in the canals of water powerplant Niva II]: Gidrotekhnicheskoe 

Stoitel'stvo [U.S.S.R.], no. 12, p. 26-30.

The experimental results of wave movement in the Niva II powerplant 

canal are given and then compared with the theoretical formulas for celerity 

and wave heights. It is found that celerity is in all cases smaller than 

that computed by the Lagrange formula, and the author supposes that 

this is due to the neglect of friction resistance in the theoretical studies. 

The wave heights computed by Johnson's and Egiazarov's formulas were 

close to the observed values.

311. U.S. Army Corps of Engineers, 1936, Method of flood routing: U.S. Army 

Corps of Engineers, Zanesville Dist., App. 5 of Survey Kept, on Muskingum River (report not published; mimeo. copies distributed with 

Corps of Engineers Circular letter R and H, no. 47.)

3115. U.S. Army Corps of Engineers, 1936, Flood routing method of computing 

reservoir effects: U.S. Army Corps of Engineers, "308" report on Ohio 

River below Wheeling, W. Va., U.S. 74th Cong., 1st sess., House 

Doc. 306.

Appendix A, Details in connection with flood control, contains the 

computation of storage curves, routing tables, natural inflow tables, and 

modified flows.

313. U.S. Army Corps of Engineers, 1936, Method of flood routing: U.S. Army

Corps of Engineers, Providence Dist., sec. 1, v. 1 (App. G to Rept.

on Survey for Flood Control, Conn. River Valley, 1936), p. 1-13.

The influence of changing slope in a reach for flood routing is herein

considered. Though theoretically not correct, the assumption is advanced

that the ratio of valley storage to a weighted flow determined from both

inflow and outflow is constant for any reach and is dependent upon the

physical shape of the valley within the reach. This assumption is shown

to be practically acceptable because it permits a good approximation.

The principle given as a formula is

in which K= ratio of storage increment in reach in day-second-feet to corresponding flow increment in second-feet; T=time unit of computation in 

days or fractions of a day; X= fractions of weighted flow increment that is 

derived from inflow increment. The general procedure of flood routing 

is described and application of method to Dover-Newcomerstown reach 

is given.

Bergeron, Louis, 1937, Me"thode graphique ge"ne"rale de calcul des propagations d'ondes planes [General graphical method of computation of 

the propagation of plane waves]: Soc. Ing&iieurs Civils [France] Me"m., 

year 90, no. 4, p. 407-497.
FLOOD-ROUTING METHODS AND UNSTEADY FLOW

This report presents the graphical method of study of wave propagation 

and is of general interest. In pages 476-480 the positive and negative 

surges in channels (bore and depression) are studied by this method, using 

the celerity formula C=^JgH. An example is given of the movement of a 

positive wave in a canal, Q, discharge increasing from 0 to Q, and from the 

time 0 to time, 8 time-units 1/^gHo later, where 1 is a selective unit reach 

length of the canal, and .Ho = the initial depth.

315. Cagniard, L., 1937, Hydrodynamique fluviale; Regimes variables [Fluvial 

hydrodynamics; variable regimes]: Rev. Gen. Hydraulique [France], 

v. 3, no. 15, p. 128-136.

The author continues the discussion of Masses approach to the problem 

of intumescences, given in his paper of 1935. Using the same basic differential equation and reduced values as Masse" used, the author starts with 

the Carson method of integration and arrives at an expression for elevations 

h. He first discusses the wave movement in the downstream direction, 

with wavefront and discontinuities, and the pseudowave called the "flood 

wave." Then the movement of the wave in the upstream direction isanalyzed. The corresponding formulas and characteristic curves are 

developed for both cases for h and H.

316. Callet, 1937, Note sur la propagation des crues [Note on the propagation)

of floods}: Fonts et ChaussSes Annales [Paris], year 107, no. 3, p. 37-46.

The author starts from Bachet's work (1934) and his formula for dis-.

charge of steady flow Ar=? (5jB/5a;)Ax, in which p== complementary discharge..

Using a reach of the river as an example, he finds that p is small, and

dH/dx is only 5 percent of the river slope. By the analysis of other factors,

he finds that

Ar= - (

and with A #=? difference of levels between two stations, and T= travel 

time between them, then for Qm*i and <2ro i n

Ar = - (1/2A ff) d[T*rd/?/d*]/d^ - ( T*/2Afl) rd2#/d*2 ,

with the approximate expression Ar=( T2ry)/(2Afl), where 7= curvature 

of the stage hydrograph at the peak. The application is given for the 

Rhine River.

317. Drioli, Carlo, 1937, Esperienze sul moto perturbato nei canali industrial! 

[Experiments on surges in industrial canals]: Energia Elettrica [Italy } r 

v. 14, no. 4, pt. 1, p. 285-305, and app., p. 306-311; and no. 5, pt. 2, 

p. 382-402.

318. Egiazarov, I. V., 1937, Teoreticheskiy raschet neustanovivshegosya volnovogodvizheniya v dlinnykh byefakh i ego sopostavlenie s eksperimentom i 

naturoy [Theoretical computation of unsteady wave movement in 

long canals and its comparison with experiments and nature]: Gidrotekhnicheskoe Stroitel'stvo [U.S.S.R.], no. 6, p. 36-42, and no. 7, p, 

19-25.

The movement of steep positive and negative waves along the pools and 

channels is studied. The author gives his equation relating H0, h, and 

AQ> aad the equation of traveled length a? by the wave in function of all 

involved parameters. The experimental data is compared with the computed values, with good agreement of two. The reflection of surge waves 

is studied, as is the fluctuation created by wave reflection. The computa-
BIBLIOGRAPHY 101

 tion method for wave movement is given in the case in which a reflection 

exists. The condition of friction resistance by wave reflection is analyzed. 

A comparison is given of theoretically determined values Tdth the values 

from nature, by considering slope, resistance, and reflection. There 

is sufficient agreement between them. A simplified computational method, 

using the Maxnitz Canal experiments as an example, is given.

319. Egiazarov, I. V., 1937, Neustanovivsheesya dvizhenie v dlinnykh byefakh 

[Unsteady wave motion in long pools]: lav. Nauchno-issledovatel'skogo 

Inst. Gidrotekhniki [U.S.S.R.], v. 21, p. 57-129.

The first part of the paper is a historical analysis of studies in movement 

of surge waves that includes a physical picture of surge-wave movement 

in long pools, and the names for differenct parts of waves are given. There 

follows a summary of some other of the author's works and of his formulas. 

The third part deals with experiments of positive and negative waves, 

which are compared with theoretical results based on the author's formula. 

Good agreement is shown. The fourth part deals with the total or partial 

reflection of waves and with the effect of flow resistance on the propagation 

and fluctuation of waves, created by reflections. The fifth part contains the 

comparison of results of experiments in large natural channels with theoretical 

results and results of laboratory experiments. The sixth part gives the 

simplified computations and shows their practical significance. The last 

part treats the simularity laws for model studies of unsteady flow and discusses the program of future investigations. Extensive conclusions and 

summaries are included; a summary in English (p. 129-132) and 131 

bibliographical entries (p. 125-129).

320. Goodridge, R. S. , 1937 A graphic method of routing floods through

reservoirs: Am. Geophys. Union Trans., v. 18, p. 433-440. 

A semigraphical method for integrating the storage equation is presented. 

For given inflow hydrograph, storage-elevation function and outflow discharge-elevation function (rating curve), the outflow hydrograph and 

storage-time function are determined. The shortcut method described 

uses a process of direct integration, without employing mass curves. It is 

based on the use of a selected time unit Af (which is variable) for the use 

equation W= QA< and uses T as a constant (time required -to -fill a given 

volume by a given discharge). T depends on the selected increments for 

storage and discharge.

321. Horton, R. E., 1937, Natural stream channel-storage (second paper): Am. 

Geophys. Union Trans., v. 18, p. 440-456.

Corrections of the statements in the author's paper of 1936 are given* 

It is shown that the observed values of M in the storage-outflow discharge 

function Q=KWM are in good agreement with the values to be expected 

from hydraulic considerations.

Data given, derived from analyses of stream rises, show the volume of 

channel storage during recession when the channel outflow rate is 1 cu ft per 

sec per mile. It is shown that volume of channel storage is related both to the 

size of the drainage basin and the stream density. It is also shown that the 

rating curve of the channel storage-outflow for a complete rise, including both 

rising and receding stages, usually has the form of a hysteresis loop. It is 

shown that channel storage was concentrated in a single detention reservoir 

having the same volume and water-surface area as that of the total channel 

storage and the same outflow capacity. This relation makes it possible to
102 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

treat analytically the subject of outflow graphs as modulated by channel 

storage. The effect of channel storage in reducing flood crests and regulating outflow is complicated, in most cases, by the presence of groundwater inflow and outflow. Expressions are derived for channel outflow for 

both the rising and receding sides of the channel outflow hydrograph, 

and examples are given showing the effect of a constant inflow of ground 

water.

322. Kertselli, S. A., 1937, Tablitsy dlya vychisleniya skorostey rasprostraneniya 

frontov voln po potprtomu byefu dlya pryamougol'nogo i parabolicheskogo rusel [Tables for calculations of unsteady flow in open channels of rectangular and parabolic cross section]: Gosudarstveniy 

Gidrologicheskiy Inst. [U.S.S.R.] Trudy, v. 5, p. 54-61. 

This paper contains a table of the integrals

= f-T^-t dl>> and CX (T,)= f-^ : dr,, JIT i J IT *

for a=3 z=

which are required for the computation of unsteady flow in prismatic open 

channels, with friction being taken into account. The case x = 3 pertains 

to the rectangular cross section, and a; =4 to the parabolic prismatic cross 

section. The integrals correspond to the Bakhmetev function formulas.

323. Khristianovich, S. A., 1937, Razrushenie dlinnoy volny odnogo napravleniya

[Direction reversal of a long wave with a unique direction]: Gosudarstveniy Gidrologicheskiy Inst. [U.S.S.R.] Trudy, v. 5, p. 62-70.

The paper is a study of unsteady flow in a prismatic channel for the case

when a long wave progressing in one direction is reversed. This motion is

a "wave in one direction" and is the same as that produced in a channel

by a dam breach, or as that of a permanent "shock wave," but the composition of the reversed long wave is changed. The depth of the resultant

"shock wave" is not constant, but is variable with time; this variability

depends essentially on the wave form that has created the reversal. With

time, the "shock wave" depth may approximate zero.

324. Lane, E. W., 1937, Predicting stages for the lower Mississippi; Civil Eng.,

v. 7, p. 122-125.

This article gives a method of predicting stages on the lower Mississippi, 

based on the stage-relation method. It describes the method used for 

reaches between which there are no important tributaries entering the 

stream and also where important tributaries join. It shows the result of 

the application of the methods to previous floods and the procedure by 

which the lowering due to crevasses in past floods can be computed. Data 

are also given on the time of travel for flood peaks of various heights. 

[Author's abstract.]

325. Masse", P. R., 1937, Des intumescences dans les torrents [Translatory waves

on torrents]: Rev. Ge"n. Hydraulique [France], v. 3, no. 18, p. 305-306.

Deymie"'s (1935) exponential law of wave-front absorption with coefficient (gS/VC)(l V /2-^fgh) is stressed. It is shown that after travelling

some distance the wave front becomes imperceptible. The wave then

propagates with C=3F/2, in which the attenuation is expressed as the

inverse of the square root of the length. In upstream direction, attenuation

always follows the exponential law. Wave propagation along torrents is
BIBLIOGRAPHY 103

similar to wave propagation on rivers. The author concluded that if 

\ =i/gH, the wave cannot propagate a torrent upstream. He introduced 

the other limit, C 2 ifgh., above which the torrent is rapid and below 

which the torrent is moderate. Below the limit the wave attenuates as 

in a river, but much more slowly. Above the limit the coefficient of absorption changes sign, the asymptotic celerity 31/2 is greater than I -}-^fgh't 

and the wave front increases exponentially, becoming a bore. He suggests 

the experiments to verify this theory.

3215. Masse1 , P. R., 1937, The problem of river-bores : Permanent Internat. Assoc.

Navigation Gong. Bull., no. 23, p. 59-68.

General theory of variable regimens in fluvial hydrodynamics and its application to problem of river-bores. Principal aspects of phenomenon and 

explanation of same.

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

February 1954.]

327. Schoklitsch, Armin, 1937, Hydraulic Structures (translated from the German by Sanuel Shulits) : Am. Soc. of Mech. Engineers [New York], 

v. 1, p. 65-67.

The mass-curve method of graphical integration of the storage-differential equation is presented.

328. Sidenko, A. I., 1937, K analizu neustanovivshegosya dvizheniya po issled 

ovaniyam v nature [On the analysis of unsteady flow by field measurements]: Izv. Nauchnoissledovatel'skogo Inst. Gidrotekhniki [U.S.S.R.J, 

v. 20, p. 112-131 (English summary, p. 131-133).

Theoretical values of wave motion computed by Egiazarov method (1937) 

are compared with actual observations at the White Sea Canal. The 

experiments are made in the canal 1,400 meters long and 42 meters wide, 

and results for negative wave are compared with those^obtained by formula

given by Egiazarov, and x=(t ti)C\ with

given by De Saint- Venant (1871). The formulas are given for positive 

waves employing similar comparisons. A formula is developed for Sy, the 

height at wavefront. Nine experiments are compared with computed 

values, and departures are discussed. It is concluded that the two groups 

of results are in satisfactory agreement, and Egiazarov equations are considered to be a good approximation. The complex wave, considered to be 

composed of negative and positive waves, can be computed according to 

Egiazarov method, and for the case of small change of discharge, the friction 

losses can be neglected for several kilometers. The reflections of waves are 

studied experimentally, and the prediction of resulting waves, composed of 

negative and positive parts, and reflections, is discussed.

329. Alien, J., 1938, Experiments on water waves of translation in small channels:

Philos. Mag. [London], v. 25, sec. 7, p. 754-768.

In an extremely tortuous channel (exemplified, here, by the Mersey River 

channel studied at scale model) the wave celerity is dependent upon the 

depth of the deepest axis of channel (thalweg) and independent of the 

velocity of the stream in which the wave travels. It is observed that the
104 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

-'Celerity is greater than that corresponding to the mean depth and l«ss than 

that corresponding to the greatest depth. In a straight, triangular channel 

the celerity is that calculated for the average depth, but in the trapezoidal 

cross section, the effective depth He is somewhat greater than the average 

depth, and He= (bH+%SH*)(b+SH), in which6=bottom width, #=depth, 

S side-slope. Gibson's celerity formula (1923) is herein confirmed for 

straight or nearly straight and uniform channels, with depth measured along 

the thalweg. Celerities of tidal bores computed for the natural river and 

for its model agree well.

330. Beckman, H. C., 1938, Rate of travel of a change in discharge, with W. B. 

Langbein's and J. H. Morgan's discussions: U.S. Geol. Survey, Water 

Resources Bull., p. 61-66, p. 131-132, and p. 196-198. 

The use of the celerity formula C= (!/B)dQ/dH is discussed.

331. Belokon', P. N., 1938, O stepeni tochnosti graficheskogo metoda inzh.

N. V. Mastitskogo po postroeniyu prodol'nykh profiley svobodnykh

poverkhnostey pri neustanovivshemsya dvizhenii vody y rechnom rusle

[On the degree of accuracy of the Mastitskiy's graphical method for the

construction of water-surface profiles of unsteady flow in the river

channel]: Gidrotekhnicheskoe Stroitel'stvo [U.S.S.R.], no. 3, p. 13-14.

Comparison is made of natural observations in unsteady flow with the

results of flood routing by Mastitskiy's method. The comparison shows

that this method gives results close to those observed. The departures

between the two results are within the limits of error of observations. The

decrease of time intervals decreases the departures.

332. Brigham, R. H., 1938, Movement of wave crests and troughs in rivers:

U.S. Geol. Survey, Water Resources Bull., p. 38-39. 

The movement of wave crests created by powerplant operations is compared with the movement of the flood wave in rivers. Differences of 

celerity and of attenuation of waves are pointed out.

333. Prank, J., and Schueller, J., 193&, Schwingungen in der Zuleitungs und 

Ableitungskanaelen von Wasserkraftanlagen [Oscillations in headrace 

and tailrace canals of water powerplants]: Berlin, Springer, p. 1-200. 

The celerity formulas for surges are developed for different cross sections, 

and families of curves are used to facilitate the computation. Experiments 

in natural canals are given as support of the formulas. The phenomenon 

of secondary waves in a positive surge is discussed. The influence of friction on celerity is considered, using Forchheimer's and Favre's methods. 

The change of waves, reflection, and further progression in narrowing parts 

of channels is discussed, principally with use of the method of characteristics 

in simple form. An extensive application of the theory to the powerplant 

canals is given, as are many examples.

334. Henry, Marc, 1938, Propagation des intumescences dans un canal 

rectangulaire [Propagation of translatory waves in a rectangular 

channel]: Rev. G6n. Hydraulique [France], v. 4, no. 19, p. 17-24, and 

no. 20, p. 65-71.

Starting from Deymi6's equation (1935) and his developments, two 

characteristics are introduced: y=V±C, and z=f(V±2C). The role of 

characteristics in propagation is special, namely, when the function z of V 

and Cis demanded, so that its derivative along the direction y and the value
BIBLIOGRAPHY 105

y itself are simultaneously the function of V and C, the above relations for y 

and 2 are obtained. On that property the author bases its method of 

characteristics. The case where S (\Vz)J(gR=o is analyzed,, where 

F± 2C= constant on a characteristic (and can, change from one to-another). 

The conditions for the discontinuity in the general case are studied, and 

then the general solution without discontinuity is analyzed. The last 

part of the report is the study of propagation of discontinuity, a>nd then 

the results are compared with Bergeron's study (1937) on graphical method 

of computation of wave propagation.

335. Horton, R. E., 1938, Definitions and classification of flood waves-, Permanent

Internat. Assoc. Navigation Cong. Bull., no. 25, p. 55-70. 

Waves are classified as being either longitudinal or transverse. They 

are further classified as waves that are subject wholly or mainly to 

gravitational or momentum control and waves that are largely subject to 

friction control. These classifications are subdivided with reference to 

orbital or nontranslatory waves and translatory waves. Wave classification 

is also made according to occurrence as deep-water or shallow-water 

waves. Other waves discussed are: seiches, impulse waves (abrupt 

change of inflow or outflow), solitary waves, power waves (in waterpower canals), breaker waves, bores, mud-flow waves, surges, flood waves 

(normal flood waves, cloudburst flood waves), and alternating flood 

waves (reflection waves). Definitions are given for single, multiple, 

and train waves (positive and negative). Waves are then classified 

according to rate of change of regimen, according to direction of motion 

of the wave crest relative to the direction of stream1 flow, and so forth. 

Classification in eight types of single-faced (monoclinal) waves and eight, 

types of two-faced (or normal) flood waves is. discussed.

336. Horton, R. E., 1938, Seddon's and Forchheimer's formulas for crestvelocity of flood-waves subject to channel-friction control: Am. 

Geophys. Union Trans., v. 19, p. 374-382.

Analysis is made of Seddon's formula for wave-crest celerity C (lf 

B)dQ/dH, where B is the width of canal; and of Forchheimer's formula 

C=(1/-B)[(Q3 Qi)/(H% HI)], where Qj and H2 are upstream constant 

discharge and depth, and Qi and HI are downstream constant discharge 

and depth. Both are derived from continuity equations. Using the 

actual data for comparison and control it is shown that Forchheimer's* 

formula fits the data better.

337. Horton, R. E., 1938, Channel waves subject chiefly to momentum control:

U.S. Dept. Agr., Soil Conserv. Service T. P. 16, p. 1-51, and

Permanent Internat. Assoc. Navigation Comg. Bull., no. 27, 193&.

This paper describes a series of experiments made at the author's

laboratory, using an experimental slope channel 5% inches wide and 120

feet in length, with recording gages at 25 foot intervals. Water was

supplied from a constant-level orifice box, with both screw and level

movements, so that the rate of flow could be changed either abruptly or

gradually. Altogether about 190 experiments were carried out, the

results of which are given in the paper. Cases covered include, for

different slopes and depths, the following: instantaneous increase in

discharge; instantaneous decrease in discharge; gradual increase in discharge;

gradual decrease in discharge; waves with triangular profile; waves with

rectangular profile; disturbance waves in still water; and alternating
106 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

waves, such as occur when an outlet is opened or closed in a pond. No 

attempt is made to develop a new formula for wave velocity, but in conjunction with the observed velocities and other data there is given for 

each experiment the velocity of the wave for the same conditions as 

computed by two formulas of Bazin-Darcy, by the Leach-King formula, 

and the Koch and Carstanjen formula, and it is shown that all of these 

formulas give reasonably good agreement with the experimental data for 

simple increment waves. The phenomena of alternating waves which 

travel successively upstream and downstream through a mill-pond are 

particularly interesting, as it is shown that the water-level in the pond 

changes abruptly by definite quanta each time the wave changes direction 

at one end of the channel.

In conclusion, the author points out that in these experiments the effect 

of channel-friction was apparently relatively small, and waves of the 

character of those covered by the experiments are subject chiefly to momentum control. In the case of triangular and rectangular waves, the 

wave-length did not exceed 1000 to 2000 times the wave-height. The 

author points out that in the case of natural river-floods, the wave-length 

is often-times of the order of 25,000 to 100,000 times the wave-height, and 

suggests that there is a gradual transition for waves of the same type as 

between short waves (that is, wave-length less than 1,000 times waveheight) and long waves, such as occur in river-channels, with an increasing 

degree of channel friction control and a decreasing effect of momentum 

as the ratio of wave-length to wave-height increases. 

[Author's abstract.]

338. Hung-Chi, Lay, 1938, A digest of the English literature of flood waves:

Iowa Univ., M.S. thesis, p. 1-158.

Approximately 70 papers in the English literature of flood waves are 

reviewed and correlated. Bibliography contains 69 entries.

339. Khristianovich, S. A., 1938, Neustanovivsheesia dvizhenie V kanalakh i 

rekakh [Unsteady motion in channels and rivers] : Sbornik "Nekatorye 

novie voprosy mekhaniki sploshnoi sredy" (Several new questions on 

the mechanics of continuous media) . Acad. Sci. Math. Inst. [U.S.S.R.], 

p. 12-154.

340. Knappen, T. T., 1938, Discussion of the paper "National aspects for flood 

control" (A symposium): Am. Soc. Civil Engineers Trans., v. 103, p. 

667-678.

The value of the experimental approach by use of the model is doubted, 

and the Muskingum method is emphasized. The ratio of the storage 

increment ATT to the weighted flow increment AQ is introduced by T. S. 

Burns, F. B. Harkness and G. T. McCarthy. (See No. 342.) 4 fit is shown 

that

ATT_Af[0.5(P2 + Pi) -0.5(Q»-Qi)]

where X= fraction of the weighted flow increment that is derived from inflow 

increment; At is the time unit; and

A<?= (&-QO +X[(P2 -Pi) -(<?2-Qi)] 

is prism storage plus wedge-storage increment.
BIBLIOGRAPHY 107

341. Langbein, W. B., 1938, Some channel-storage studies and their application 

to the determination of infiltration: Am. Geophys. Union Trans., 

v. 19, p. 435-437.

The relation between channel storage and discharge is studied, and 

the characteristics of recession curves are analyzed on the basis of relationships revealed by the consecutive daily discharge. The relation of storage 

W, discharge q (as the yield in cfs per sq mi), and the river drainage area 

is studied, showing that M, in the formula g=DTFM, decreases in the downstream direction. A classification of recession curves is introduced, 

based on the exponent M, which is representative of the geomorphological 

characteristics of channels. A new method is derived for the computation 

of infiltration during rainfall by the determination of the river-inflow 

hydrograph.

342. McCarthy, G. T., 1938, The unit hydrograph and flood routing: U.S. 

Engineer School, Fort Belvoir, Va. [1940].

A monograph entitled "The unit-hydrograph and flood-routing" was 

prepared by Gerald T. McCarthy for presentation at the Conference of 

the North Atlantic Division of the United States Corps of Engineers at 

New London, Connecticut, June 24, 1938, this being published in limited 

edition in the United States Engineer Office at Providence, R.I. This 

describes the "coefficient" method of flood routing, which is reported 

to have been used with marked success in recent flood-routing studies, 

especially those on the Muskingum River System in Ohio and the Connecticut River System in New England. This method is considered to 

be an improvement over previously used methods, chiefly in making a 

definite allowance for "Wedge-storage" effects and in more systematic 

utilization of the available natural flood-flow data. Its original features 

are said to be especially due to the work of Mr. McCarthy. 

[Abstract by H. A. Thomas.]

343. Moots, E. E., 1938, A study in the theory of flood-waves: Am. Geophys. 

Union Trans., v. 19, p. 383-386.

The hydromechanic equations of a simple flood wave in a wide rectangular 

channel are developed and discussed. The paper includes: (1) a restatement of the fundamental equations governing a simple flood wave in a wide 

rectangular channel; (2) the development and solution of the equations of 

the roll wave as a special case of the flood wave; (3) a discussion of the order 

of occurrence of the three maxima of mean velocity, discharge, and stage; 

(4) the determination of the celerity of a flood wave as the rate of movement 

of a constant flow; (5) the determination of the celerity of a flood wave 

having small amplitude; (6) the development and solution of the equations 

of the rising flood wave.

The conclusions are:

1. The profile of the roll wave after equilibrium has been established may 

be obtained in terms of elementary functions from the general 

hydrodynamic equations of a simple flood wave.

2. The profile of the monoclinal, or rising flood wave, after equilibrium 

has been established, may be obtained in terms of elementary functions from the general hydrodynamic equations of a simple flood wave.

3. The celerity of the monoclinal, or rising flood wave in equilibrium, may 

be obtained from Kleitz's equation for the velocity of a constant flow.
108 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

4, The celerity of a flood wave of small height is found to be greater than 

the mean velocity of the channel in normal flow.

344. Moots, E. E., 1938, A study in flood waves: Iowa Univ. Eng. Studies Bull.

14, p. 1-24,

Two basic differential equations for unsteady flow are redeveloped. The 

equation for the roll wave (Wave of a dam breach) is rederived as »=(!/ 

S) [H+ HZ log (1 H/H2)], where x.== the abscissa of wave shape, H = ordinate, 

£F2 ^ starting depth, and <S^=river-bed slope. The sequence of attaining 

maxima is: velocity (V), discharge (Q) and stage (H), in that order, according to Kleitz (1877). For constant Q traveling (dQ=o), it is shown that

C=

There is a discussion of the changes of C that occur throughout the reach, 

particularly at the time of occurrence of maximum V, Q, and H. Qfna»- 

celerities are given for the wave with constant celerity, the wave with small 

amplitude, the roll way& and the solitary wave. Equations are derived 

for mean velocity and shape of the solitary wave. The laboratory tests 

made for this study, employing a rising wave (from Q\ to Qz), show that thecelerity of the toe is very close to Russell's and Bazin's celerities for solitary 

waves.

345. Rouse, H., 1938, Fluid mechanics for hydraulic engineers: New Yorkr

McGraw-HiU Book Co., p. 353-402.

Chapter 15 deals with general characteristics of wave phenomena and 

with properties of gravity waves: celerity formula, velocity due to acceleration by wave movement, characteristic dimensions of surface waves, etc. 

Chapter 16 deals with channel solitary waves (celerity and shape, generation), with surges (generation, celerity, undular profile, progress of wave 

with time), roll waves, hydraulic jump as a standing surge, and wave effect 

on boundaries.

346. Snyder, Franklin, 1938, Synthetic Unit Graphs: Am. Geophys. Union

Trans., v. 19, p. 447-454.

The lag of time from center of mass of rainfall to peak of runoff is used 

as the principal drainage-basin characteristic in the derivation of syntheticunit graphs. The peak rate of the unit graph is expressed as a function of 

the lag. A distribution graph is detei mined by means of the lag. By use 

of the distribution graph and the known peak of runoff, a unit graph can 

be constructed for the area in question. Use and limitations of the synthetic unit graph in flood forecasting are discussed.

347. Steinberg, I. H., 1938, A method of flood routing: Civil Eng., v. 8, p.

476-477.

This report presents a method for routing floods in the upper Mississippi 

River. Many outflow-storage and storage-factor (storage -f outflow) 

curves are used, as developed from the general function W=f(Q, Qt, M)f 

where Q=outflow, Qt=tributary flow entering river downstream of the 

reach, the M = ihe storage factor. Taking constant Q* or M, two families 

of curves W=f(Q) are developed. The M parameter family, for Q t 

= constant, consists of parallel sloping straight lines, where the slope depends 

upon the time interval used, and the distance between lines is dependent 

upon the unit of length of the storage scale (M = Q/2+W") or ( Q/2
BIBLIOGRAPHY

+ M=W)'. The intersect of M and Qt of the two families, gives Q and W. 

The routing is made by tabulation, using developed families of curves.

348. Taylor, E. H., 1938,. Analysis of the positi-ve surge in a rectangular open

channel: Civil Eng., N. 8, p. 685-686.

The movement of steep positive wave up the channel, with wave height 

h, depths HI and H% and velocities V\ and V2, is given by equation 

(S3 -252)/(S + l) = 4fc2X, with F2 =(1-A;) Vl} S=(#2 -#i)/#i, andX= V^/2gH. 

For the hydraulic jump V\H\ V<tHi, the equation becomes 82 + 38 + 2=4X.

3>49» Bachet and Beau1 , 1939, Rapport sur la prevision des crues [Report, on the 

flood forecast]: Internat. Assoc. Sci. Hydrology Trans., General Mtg., 

Washington, D.G., 1939, v. 1, ques. 2, rept. 4, p. 1-19. 

The forecast of floods is based on the stages; the actual discharge is 

divided as: (1) the discharge of steady flow and (2) a complementary discharge (positive or negative) resulting from unsteady flow. A template is 

presented for flood routing along the river. According to the author, flood 

forecasting by the given method (when the method is applicable) amounts 

to the simple integration of partial differential equations.

35Q. Deymi6, Ph., 1939^ Propagation d'une intumescenee allongee (Problem aval) 

[Propagation of an elongated- intumescence (tailwater problem)]: New 

York, John Wiley and. Sons, p* 537-544 (laternat. Coag. AppL Mech., 

5th Mtg., New York, 1939, Proc.).

The surges created by powerplant releases are studied analytically, and 

the results are verified by experiments on the Seine River in Paris (1933). 

Two partial differential equations are integrated using some simplifications. 

The surge is studied according to its three parts: the wave front, which is 

alternated by progressing according to exponential law; the residiial wave 

(back part of wave), which is damping asymptotically and which cannot 

have a "black front"; and the central zone of wave, which is composed of 

the residual waves of the two other parts. The surge approximates the 

celerity 3 V/2, when the canal length is infinite. Due to friction, the surge 

tends to assume the characteristics of the permanent wave. Experimental 

results at the River Seine confirmed the theory.

351. Egiazarov, I. V., 1939, Proekt tekhnieheskikh uslovij i norm rascheta 

svobodnoy poverkhnosti potoka pri neustanovivshemsya dvizhenii v 

otkrytykh dlinnykh byefakh [The project of engineering conditions and 

norms for the computation of the free surface movement of unsteady 

flow in the long open channels]: Sbornikt Proekty Tekhnieheskikh 

Usloviy i Norm Gidrotekhnicheskogo Proektivovaniya (Collection: 

Projects of Technical Conditions and Norms for Designing in Hydraulic 

Engineering), [Moscow-Leningrad].

352. Mass6, P. R., 1939, Recherches sur la theerie dea eaux eourantes [Research 

on the theory of flowing waters]: New York, John Wiley and Sons, 

p. 545-549, (Internat. Cong. AppL Mechs., 5th Mtg. New York, 

1939, Proe.).

Two general partial differential equations for unsteady flow are integrated 

in the complex plane [Masses thesis of 1935]. These lead to the conclusion 

that: (1) intumescence tends to propagate asymptotically with the celerity 

728-245 64 8
110 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

3F/2 (known as "law of floods"); (2) for the tidal wave, C, M and N must 

be expressed as:

and

N=(gS/CV)(l + V/2

and the condition necessary for conversion of the tidal wave to the bore is

353. Pillsbury, G. B., 1939, Tidal Hydraulics: U.S. Army Corps Engineers, 

Prof. Paper 34 [1940], Rev. Ed. 1956.

354. Rutter, E. J., Graves, Q. B., and Snyder, F. F., 1939, Flood routing (with 

discussions): Am. Soc. Civil Engineers Trans., v. 104, p. 275-313.

This paper describes the procedure developed by the engineers of the 

Tennessee Valley Authority for routing flood through the Tennessee 

River in order to predetermine the effects of various dams in modifying 

flood-stages. While in their broad aspects, the flood-routing methods 

described in this paper are not essentially different from ones which have 

been used in other previous investigations, nevertheless a description of 

the application and extension of these methods to meet the special problems encountered on the complex river-system of the Tennessee constitutes 

an important contribution to the literature on flood-waves and reflects 

current progress in the development of practical methods to analyze 

flood-wave phenomena on actual river-systems. 

[Abstract by H. A. Thomas.]

Routing was accomplished by dividing the river length into reaches and 

applying the storage equation to each reach. Total inflow into each reach 

was determined day by day from the routed outflow of the adjacent upstream reach, published discharges of metered streams flowing into the 

reach, and inflow estimated from rainfall on the unmeasured areas draining 

into the reach. Total flood volume at each dam site was assumed to be 

known. The storage in the reach for various flows are determined from 

topographic maps and cross sections of the river valley in conjunction with 

flow profiles determined from backwater curves. The outflow was then 

derived from the relation of storage and discharge for the reach. 

[From author's conclusions.]

In the discussion (p. 197), R. D. Goodrich proposed the use of first 

(A) and second (A2) differences of inflow and outflow, when floods rise and 

fall rapidly; C. O. Wisler proposed the plotting of curves (W+QM) and 

(W-QAt), R. W. Powell suggested the use of W=f(OAP+Q+W), in 

which 0.4 is the weight for inflow.

355. Supino, Giulio, 1939, Sur la propagation des ondes dans les canaux [On 

the wave propagation in canals]: Rev. G6n. Hydraulique [France], 

v. 5, no. 29, p. 260-262.

The integration of two partial differential equations is made by linearization of equations for small wave heights (H=H0 +h, V=V0+v), with 

So=kVo2. By neglecting the small, second-order terms of the second 

member, the general form of solution is given and discussed. The solution of the complete equation is obtained by means of successive approximations, and two solutions of linearized and of complete equations are 

compared. The author concludes that the attenuation of the wave as
BIBLIOGRAPHY 111

obtained by linearized equation is smaller than the attenuation which 

results from the integration of the complete equation.

356. Zoch, R. T., 1939, The mathematical synthesis of the flood hydrograph: 

Am. Geophys. Union Trans., v. 20, p. 207-218. See also the author's 

earlier papers: Monthly Weather Review, Sept. 1934, Apr. 1936, Apr. 

1937.

In studying the function of rainfall infiltration, the flood hydrograph is 

analyzed for the synthesized evidences of this function. Parts of the 

hydrograph are shown to represent the concentration, saturation, transition 

and depletion phases of flood runoff. General aspects of the mathematical 

approach to the rainfall-runoff relationship are discussed.

357. Alien, J., and Matheson, J. L., 1940, An experimental investigation of the 

propagation of tides in parallel and in convergent channels: Inst. 

Civil Engineers [England] Jour., p. 107-118.

The propagation of tidal waves in parallel and convergent channels is 

studied experimentally and compared with the theoretical formulas. 

The results demonstrate that, within certain limits, close agreement 

exists between theory and experiments, and they confirm that the amplification is greater in a parallel inlet than in a convergent inlet of equal 

channel length and mean depth. No significant difference was observed 

in the behavior of two convergent channels, one of 6° and the other of 

of 13° apex angle.

358. Dantscher, Kasper, 1940, Wanderwellen in Schiffahrtskanaelen [Traveling 

waves in navigation canals]: Wasserkraft u. Wasserwirtschaft [Munich] 

v. 35, no. 7, p. 145-147.

This is a discussion, with examples, of the traveling of surges in navigation 

channels, and especially the influence of the velocities produced by wave 

progression on the resistance to ship movements.

359. Dantscher, Kasper, 1940, Die Wanderwelle in Schiffahrtskanal [The traveling wave in a navigation canal]: Wasserkraft u. Wasserwirtschaf t 

[Munich], v. 35, no. 10, p. 226-229.

The study of surge propagation in a navigation canal, where the surge 

must travel through transitions from a large to a narrow or narrow to 

large section of canal, is given. The celerities for two sections are developed 

and related to section characteristics. The change of wave heights due 

to the change of cross sections is determined. The velocities in the canal, 

produced by movement of the changing waves, are determined and their 

influence on navigation is discussed.

360. Keulegan, G. H., and Patterson, G. W., 1940, Mathematical theory of 

irrotational translation waves: U.S. Nat. Bur. Standards Jour. Research, v. 27, p. 47-101.

This paper is the first of a series dealing with the motion of flood waves 

and other waves of translation in open channels. The case treated is that 

of waves for which the forces of fluid friction are negligible with respect to 

the inertia and gravitational forces. The irrotational motion of a perfect 

liquid in a horizontal rectangular canal when the original surface is disturbed 

is investigated on the assumption that the horizontal velocity in a cross 

section is approximately uniform. The results are also applicable to motion 

in a canal of uniform slope containing water originally moving with a
112 FLOOD-ROUTING METHODS AN» UNSTEADY FLOW

uniform velocity. Special emphasis is laid on disturbances, which are 

propagated without change of form, and in these cases formulas are derived 

for the wave profile and velocity of propagation. Formulas are also derived 

which give the deformation, energy, motion of the center of gravity, and 

moment of instability of an arbitrary intumescence. Consideration is 

given to the maximum height of a wave of a permanent form. Formulas 

have been compared with the available experimental data. Of special 

interest is the comparison of the shape of the undulations composing the 

head of aa initial surge with the characteristics of the cnoidal wave. 

[Authors' abstract.]

361. Keulegan, G. H., and Patterson, G. W., 1940, A criterion for instability 

of flow in steep channels: Am. Geophys. Union Trans., v. 21, p. 

594-596.

In the case of a symmetrical wave traveling along a wide channel having 

bottom slope i, depth H0, and mean velocity V0, before the wave arrives, 

the criterion developed for instability of wave (steepening of wave with 

time and distance travelled) is

 >(9/8A0, or Vo2>(9/4)!7ffo, 

where ffH0i= (X0/2) F02.

This formula is based on Manning's formula. H. Jeffreys (1925) and H. A. 

Thomas (1939) developed the criterion *>2X, based on Chezy's formula*

362. Laden, N. R., Reilly, T. L., and Minnotte, J. S., 1940, Synthetic unit 

hydrograph, distribution graphs and flood routing in the Upper Ohio 

River Basin: Am. Geophys. Union Trans., v. 21, p. 649-659. 

The coefficient method or Muskingum method for flood routing is discussed. The authors present an outline of the method used in determining 

the unrated tributary inflows. The unit-graph and synthetic-graph 

methods are discussed. The adopted method uses McCarthy's method 

and Snyder's synthetic distribution graph for determination of inflow 

hydrographs.

363. Langbein, W. B., 1940, Channel storage and unit hydrograph studies:

Am. Geophys. Union Trans., v. 21, p, 620-627.

The paper discusses the channel phase in rainfall-runoff relation. The 

storage of the reach, given by McCarthy as W=K[xP-3r (1 x)Q], is studied 

through the analysis of coefficients K and x, and the equation is modified as.

S i

where St is storage of a reach in time t; P is the inflow at time preceding t 

by 2Kx, and Q is outflow following t by 2Kx. An approximate form 

St=KQ(i+2K:f) is suggested. The use of the lag interval (2Kx) is discussed, 

and examples are given.

364. Meleshchenko, N. T., 1940, Primenenie teorii dlinnykh voln maloy amplitudy k voprosam sutochnogo regulirovaniya [The application of the 

theory of long waves of small amplitude to the problems of daily 

regulations]: Vsesoyuzniy Nauchno-issledovatel'skiy Inst. Gidrotekkniki i Melioratsii, Izv. [U.S.S.R.], v. 28.

365. Posey, C. J., and Fu Te, I., 1940, Functional design of flood control 

reservoirs: Am. Soc. Civil Engineers Trans., v. 105, p. 1638-1674.
BIBLIOGRAPHY 113

The functional design of a flood control reservoir has as its objective the 

determination of the relation between the storage space that must be 

provided and the corresponding reduction of the flood peak. The method 

of incorporating this relation into the study of the economic balance of a 

flood control system has been explained in detail by Sherman M. Woodward, M. Am. Soc. C.E. Mr. Woodward's "five-sixths rule," which heretofore has provided the only direct method for the functional design computations, is limited in its application to reservoirs of such proportions that 

the maximum outflow is small compared with the average inflow during 

the flood, and to reservoirs with orifice-type outlets and a certain type of 

depth-capacity relationship. In this paper the writers have generalized 

Mr. Woodward's method and have extended it to apply to reservoirs with 

either orifice- or weir-type outlets, in valleys of a wide range of morphological configuration. Although the relationships derived can be used in the 

design of multiple-purpose reservoirs, the present discussion is restricted 

to reservoirs for flood control only. 

[Authors' synopsis.]

366. Ramponi, Francesco, 1940, Resultati sperimentali sulla propagazione delle 

perturbazioni di regime nei canali [The experimental results of the 

propagation of regimen changes in channels]: Energia Elettrica [Italy], 

v. 17, no. 11, p. 643-653.

The author first gives a summary of his note on theoretical treatment 

of wave propagation in open channel, as presented to Second Congress of 

V.M.I, in Bologna, Italy, 1940. Starting with a particular solution of 

partial differential equation for unsteady flow in channel, he derives the 

parameters for that particular solution. As the final result, he gives the 

celerity and wave attenuation as function of period variation. He describes 

his experiments with flood waves on the Isarco and Adige Rivers and gives 

his results for celerities and attenuations, with the parameter R in his 

solution being computed. The relation of B to period T=27r/w is given 

for the experimental data. The second group of experiments, performed 

in the laboratory canal, is a study of short-period disturbances. The experiments and devices used are described. The results are given in terms 

of the relation of celerity or attenuation to the period T, showing an increase of attenuation by an increase of amplitude, and a decrease of celerity 

by an increase of period T.

367. Tarpley, J. F., 1940, A new integrating machine: Mil. Engineer, v. 32,

no. 181, p. 39-43.

This paper describes the use of a mechanical integrator to be used in 

routing floods through reservoirs. Based on the simple storage equation, 

the device employs drums, gears, and a differential motor. The operator 

is required to follow three curves in order to translate the recordings on 

three drums and obtain the simultaneous values of: P=f(i)-rnflow hydrograph, Q=f(H)-outflpw rating curve, and W=f(H)-storage function. At 

the|same time, the integrator is giving values of Q=f(<) and H=J(J) on 

two other drums.

368. Thomas, H. A., 1940, Graphical integration of flood wave equations: Am.

Geophys. Union Trans., v. 21, p. 596-602.

The two partial differential equations for unsteady flow are rederived. 

A trial-and-error process of determining local inflow from given stage pro-
114 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

files for different time instants is given by using the stage-surface profile 

for a constant discharge as the reference base, from which the depth h ia 

measured. By the repeating process, finite relations of Q=/(0 for given 

stations and A=f(x) for given instants are determined. Having Ai and 

Az for both families of curves, SAQ and AW are obtained, where SAQ ia 

the difference between outflow and inflow and AW is the storage of a 

reach Az during AJ. Local inflow SP=AW SAQ. Solution of the trialand-error process in the case of known local inflow is also given. The 

author recognizes that the method is laborious, but considers it justified.

369. Thomas, H. A., 1940, The propagation of waves in steep prismatic conduits: Iowa Univ., Hydraulic Conf. Proc. Eng. Studies Bull. 20, p. 

214-229.

Roll waves on a steep channel are analyzed by application of the movingbelt analogy. The hydraulic theory of flow on the surface of a movingbelt conveyor is developed. The formulas are given for different computed 

quantities. Stationary wave trains on a moving belt are analyzed, and 

relationships are given for the various characteristics of wave train and 

channel.

370. Volkov, I. M., 1940, Resul'taty naturnykh nablyudeniy popuskovykh voln 

napolneniya na rr. Topse i Tsareve [The results of observations in 

nature of filling release waves on the rivers Topsa and Tsareva]: 

Sbornik Arhangel'skogo Lesotekhnicheskogo Inst. [U.S.S.R.].

371. Arkhangel'skiy, V. A., 1941, K raschetam neustanovivshegosya dvizheniya 

and v kanalakh i rekakh [On the computation of unsteady flow in 

canals and rivers]: Akad. Nauk [U.S.S.R.], Inst. Mekhaniki, Inzhenernyy Sbornik, v. 1, no. 1, p. 129-136.

A general grapho-analytical procedure used as a finite-differences method 

for approximate integration of two partial differential equations for unsteady flow is given, when velocity and acceleration heads are neglected 

in the momentum equation. The channel is divided in reaches Ax, and 

time interval AJ for integration. For each reach the relations Hi"=f(Qi") 

and jyr2'=/(Q2/) are developed with index 2 referring to the end of timeinterval AZ, and Q', H' for the beginning and Q" and H" for the end of 

the reach Az. Special families of curves [H=f(Q) for given AH or H"} 

are to be developed for each reach in order to facilitate the integration. 

A combined integration graph is given for many reaches. The method ia 

discussed as applied to the system of several channels and to surges. The 

author's integration shows that the results agree well with observations.

372. Arkangel'skiy, V. A., 1941, Obzor sposobov proektirovaniya rezhima 

otkrytykh vodotokov v sluchae neustanovivshegosya dvizheniya 

[Review of methods of computing unsteady flow of water in channels]: 

Akad. Nauk [U.S.S.R.], Inst. Mekhaniki, Inzhenernyy Sbornik, v. 2, 

p. 321-326.

The methods of integrating two partial differential equations for unsteady 

flow, either by the method of characteristics or by the method of finite 

differences as applied to the two partial differential equations are discussed. 

The author discusses his method of finite differences (1941), which neglects 

the inertia term, obtaining
BIBLIOGRAPHY 115

with c=Chezy's coefficient. If the inertia term cannot be neglected, it is 

used as a correction

_ _____

where AdH is to be added to 5H. The initial and boundary conditions are 

discussed, and the results are compared with the observed flood waves.

373. Arredi, Filippo, 1941, Sulle onde di traslazione nei canali [On the translatory 

waves in channels]: Acqua [Italy], no. 5.

374. Coulson, C. A., 1941, Waves: London, Oliver and Boyd, p. 60-86.

This book is a mathematical account of the common types of wave 

motion, including waves in liquids. Following a summary of hydrodynamie 

formulas, application to tidal waves and surface waves is shown. The 

equations for tidal waves in a canal are given, defining celerity, shape 

development, and particle paths of such waves.

375. Dietz, D. N., 1941, A new method for calculating the conduct of translation waves in prismatic canals: Physica [Netherlands], v. 8, no. 2, 

p. 177-195.

The author improves the wave-celerity formula by including two resistance terms, so that

where c=Eytelwein's resistance coefficient, C= celerity of a given height, 

H=(Z H)IH with Z momentary average depth, and with

The author concludes that without the influence of resistance each positive 

wave would become steeper in front and would be flattened in back. 

The resistance effect upon the wave is shown mainly by the term

indicating that the wave will be retarded at the front and accelerated at 

the back, and that effect increases as the slope S of the water surface 

decreases.

376. Gilcrest, B. R., and Marsh, L. E., 1941, Channel storage and discharge 

relations in the lower Ohio River Valley: Am. Geophys. Union 

Trans., v. 22, p. 637-649.

The procedure for development of storage discharge relation is treated 

with basic equation Q2 Qi=(t/K)(Pi Qi), where f=time in days and 

K=raiio between corresponding increments of storage W and discharge 

Q for steady flow (K=dW/dQ). Examples of routings based on that 

equation are given. Coefficients C0, Ci, and <72, related to t, K, P, and AT 

(a ratio of wedge storage to corresponding difference P Q, with dischargeassumed to vary with distance), are determined and analyzed.

377. Horton, R. E., 1941, Virtual channel-inflow graphs: Am. Geophys. Union

Trans., p. 811-820.

The virtual channel-inflow graph is conceived as a synthetic inflow" 

graph, which meets four required conditions and, by the routing procedure,.
116 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

is'produced as a likeness of the outflow graph. The four conditions required 

are: (1) inflow stops at second inflection point of outflow graph; (2) inflow 

graph intersects outflow graph at its maximum; (3) both graphs start at 

the same time; (4) their volumes are equal. Simple forms of synthetic 

inflow graphs are discussed and channel storage is analyzed by the use 

of virtual-inflow graphs. The conditions under which these graphs are 

valid and should be used are discussed; examples are given.

378. Horton, R. E., 1941, Flood crest reduction by channel storage: Am.

Geophys. Union Trans., p. 820-836.

Using his virtual-inflow graph (1941) the author gives the ratio of inflow 

and outflow crests as Qm/Pm =S(l Wm/W), where TTm =maximum channel 

storage, W= volume of flood wave, and S=l (1 c)t0fb, where 2o=time 

of occurrence of channel-outflow crest, &=time base of channel virtual 

inflow graph, and c=ratio of total outflow W\ at crest time to the area of 

the triangle [c= Wif(Qmt0f2')]. The above equation is applied to many 

examples. For maximum advantage in the use of this equation many more 

determinations of Wm/W, c, and t^/b are needed to contribute to its application. The relationship of S and c is given as determined from above equation and from observed data; good agreement is shown.

379. Messina, U., 1941, II grado di approssimazione di una formula semplice 

per la determinazione delle altezze delle onde di traslazione nei canali 

[Degree of approximation of a simple formula for determination of 

translation wave heights in channels]: Acqua [Italy], no. 10.

380. Meyer, O. H., 1941, Flood-routing on the Sacramento River: Am. Geophys.

Union Trans., p. 118-124.

In considering the judgments to be made in preparation for a flood-routing 

system, the author discusses such criteria for selection of river reaches as 

location of major inflows, critical points (stream junctions, flow divisions, 

and weirs), and check-points (actual gage records). The method of estimating local inflows by synthetic unit hydrographs and by discharge 

correlation is discussed. The channel storage-outflow discharge relationship is determined both by the flow-line method and by empirical curves 

(which were preferred to the flow-line method). A method is given for 

using the rating curve of the upstream station to develop discharge-rating 

curves and storage-outflow curves for locations downstream where only 

gage-height records are available. Three methods of routing are used: 

(1) the lag method, using six-hour time intervals; (2) the Steinberg method; 

(3) the flood-routing machine, for long reaches (employing a family of 

storage-outflow curves having inflow as the parameters). Equal preference was given to the first and third methods; the Steinberg method was 

replaced by the labor-saving machine.

381. Meyer, O. H., 1941, Simplified flood routing: Civil Eng., v. 11, no. 5, p.

306-307.

The routing procedure is made in tabular form by using the simple 

storage equation, and storage-discharge relationship, based on the method 

of derivation given by Rutter, Graves, andSnyder (1939).

382. Rakhmanov, A. N., 1941, O neustanovivshemsya dvizhenii v nizhnikh i 

verkhnikh byef akh rechnykh gidrouzlov {On unsteady flow in headrace 

and jtailraoe canals of hydraulic structures]: Vsesoyuzniy Nauchno-
BIBLIOGRAPHY 117

issledovatel'skiy Inst. Gidrotekhinki i Meliorated, Izv. [U.S.S.R.], 

v. 30.

The method developed by the author is often used for flood routing in 

natural water courses. Two functions H=fi(x,f) and Q=f2(x,t) are graphically represented. These functions having been derived from initial and 

boundary conditions that are more or less arbitrarily used (derived from 

previously solved parts of the flood-routing solution). Control is obtained 

by using two differential equations for unsteady flow. Basically, the 

storage equation is applied, thus giving to the method characteristics of 

the volumetric approach. For the long channel with gradually changing 

wave, results by this approach show good agreement with the observations. 

This method is used by the author for flood routing on the Volga, Kama, 

and other rivers with a flat slope. Regardless of the somewhat arbitrary 

nature of the approach, the results have proved to be sufficiently accurate 

for many flood-routing purposes.

383. Schultze, Edgar, 1941, Die Berechnung der Gezeiten in Flussmuendungen 

[Calculation of tides in estuaries]: Die Bautechnik, v. 19, no. 12/13,, 

p. 135-150, Mar. 21. [U.S. Army, Corps Engineers, Waterways 

Expt. Sta., Translation no. 43-17, Dec. 1942, by H. B. Edwards]. 

Theoretical mathematical discussion of methods of computing tide levels 

in river estuaries.

[Annotations in Bibliography on Tidal Hydraulics, Corps of Engineers,. 

February 1954.]

884. Tatum, F. E., 1941, Flood routing by method of successive averages: U.S. 

Army Corps Engineers, Rock Island Dist. [Illinois], p. 1-10.

The equation fora hydrograph recession curve is QilQ2=K(k~^v, where 

ti and fe represent the times measured from the second inflection point of 

hydrograph, and Qi and Qa are discharges at h and fe, respectively. All 

recession curves having the same K will be parallel on semilog paper. K 

is generally constant for a particular river station, except that uneven rainfall distribution changes the K-value somewhat. Because K remains 

constant for a short pattern, the storage-discharge functions are identical 

for all reaches of equal time of travel, and for two recession curves (P and 

Q) with equal K, Q2 =(Pi-f P2)/2. The first step is to determine the length 

of the time period. The time of travel should not be too long, because the 

obtained ordinates would not determine the true shape of the hydrograph, 

especially at critical points. (A long reach can be broken into small subreaches.) The second step is to determine the number of routing steps. 

Two general guides for procedure are: to determine the number of time 

units (periods) between the peaks, then to double this number; to route a 

few critical points on the recession curve and by trial to determine which 

number of routing reaches causes the recession curve to coincide with the 

recession curve of the downstream hydrograph (when there is considerable 

inflow). This method becomes a single procedure of obtaining the average 

of the inflow at the beginning and the end of the period. Application of the 

method is shown.

385. Turner, H. M., and Burdoin, A., 1941, The flood hydrograph: Soc. Civil 

Engineers [Boston] Jour., v. 28, p. 232-281.

386. Wilson, W. T., 1941, A graphical flood-routing method: Am. Geophys.. 

Union Trans., v. 21, p. 893-898.
118 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

To evaluate the influence of storage in a reach, the author suggests that 

the attenuation be determined by routing a "storage component" (as 

though a reach were a reservoir) and a "translation component." The 

actual graph would be between the two, but no method is given for determining the exact amount of attenuation between two limits. The timeconversion factor is analyzed and, as Langbein gives in discussion, is equal 

to the time distance between centers of gravity of inflow and outflow 

hydrographs.

387. Woodward, S. M., and Posey, C. J., 1941, Hydraulics of steady flow in

open channels: New York, John Wiley & Sons, p. 133-145. 

This paper describes a method of flood routing that uses the storage 

equation and two auxiliary functions: (W+y2QAf) and (W y2QAf). The 

computation is performed with Posey's slide rule. A slide-rule computation of pool-level routing that employs a variable stage-discharge relationship is described. The use of slide rule is discussed.

388. Arredi, Filippo, 1942, Metodo del calcalo della propagazione delle perturbazioni graduali nelle correnti aperte [Method of computation for 

propagation of gradual perturbation in open channels]: Acqua [Italy], 

no. 2.

389. Barrows, H. K., 1942, A study of valley storage and its effect upon the

flood hydrograph: Am. Geophys. Union Trans., v. 23, p. 483-488. 

The influence of valley storage upon flood hydrographs is discussed 

together with analysis of observed data. The recession curve is used for 

analysis of valley storage and permits computation of the inflow hydrograph from the actual flood hydrograph.

:390. Evangelisti, Giuseppe, 1942, Sulla propagazione delle piccole onde nei 

canali a sezione variable [On the propagation of small waves in canals 

with variable cross section]: Annali di Matematica Pura ed Applicata 

[Italy], Jan. 25.

391. Ippen, T. A., 1942, Gas-wave analogies in open-channel flow: Iowa Univ.,

2d Hydraulic Conf. Proc., Eng. Studies, Bull. 27, p. 248-265. 

The paper summarizes the general results of the theoretical and experimental investigations of shooting or rapid flow in hydraulic structures, 

bringing these results to the attention of hydraulic engineers as useful tools. 

Analogy is drawn between supersonic flow of gages and supercritical flow 

of water. Special attention is given to standing waves. Wavefronts of 

small height and assumed constant energy are analyzed. The method of 

characteristics is applied to supercritical flow with constant energy, and the 

method of using shock polars for supercritical flow with high wavefronts 

and energy dissipation is presented. The application of theory is discussed.

-392. Langbein, W. B., 1942, Storage in relation to flood waves, Part XI-H of 

Hydrology (O. L. Meinzer, ed.): Physics of the Earth, IX, New 

York, McGraw-Hill Book Co., p. 561-572.

The storage of a river reach and its influence on flood waves are discussed. 

The general relation of storage and discharge is

W=- [xP> 

a
BIBLIOGRAPHY 119

in which constants a and n express the stage-discharge characteristics of 

the control sections at the respective ends of the reach, and 6 and m evaluate 

the mean storage-volume characteristics, with Q = aHn and W=bHm. 

The coefficients K and x in Muskingum formula are analyzed, with x 

defining also the attenuation of flood wave (known as attenuation factor). 

Other relationships of storage in river reaches are discussed.

393. Levin, Leon, 1942, M4thode graphique de calcul du mouvement nonpermanent dans les canaux en ecoulement libre [Graphical method for 

computation of unsteady flow in open channels]: Ge'nie Civil [France], 

v. 119, no. 11-12, p. 109-113.

The Bergeron method (an alternative of the method of characteristics) 

is used for travel of surges along a channel. The effect of resistance along 

a reach is replaced by the concentrated head loss (KQ2 ) at the end of the 

reach. The experiments in a canal have shown that this method gives 

results that are in close agreement with experimental results only for cases 

involving very small wave heights.

The author introduces the effect of friction resistance by means of a 

parameter depending oh depth H. When the resistance along the channel 

cannot be considered to be negligible, the singular concentrated resistance 

at the end of each reach replaces the effect of channel resistance. Experiments made in the hydraulic laboratory of Hydro-technical Society of 

France, in Grenoble, have confirmed the author's approach.

394. Linsley, R. K., 1942, River forecasting methods: U.S. Weather Bur. Man., 

Nov., p. 1-100 [reprinted 1945].

In the author's discussion entitled "translation of hydrograph downstream," the gage relationships are given with and without backwater 

effect. Peak-discharge relations are given and discussed, and the lagging 

of the hydrograph is explained. The streamflow-routing method under 

consideration is based on the storage equation and relates average storage 

to levels of outflow and inflow. The method of storage factors (W Qi/2) 

and (TF+Q2/2) is used.

395. Posey, C. J., 1942, Flood wave characteristics as related to flood routing: Iowa 

Univ., 2d Hydraulic Conf. Proc., Eng. Studies, Bull. 27, p. 224-233.

The Jones and Grimm formulas for the relation of discharges in steady 

and unsteady flows are compared analytically, and the Jones formula is 

proven to be less accurate than the Grimm formula. A flood-routing 

method is described, based on the storage equation, that uses a double 

family of curves with the storage factor (TF+J4QAO=/(AH), where A.ff= 

fall in the reach. For one family of curves, depth (H) at the end of the 

reach is a parameter. The other family of curves uses discharge (Q) at 

the end of reach as a parameter. The routing table is given.

396. R&ne'nieras, Gaston, 1942, Adaptation de la methode Bergeron & I'e'tude 

de la propagation des intumescences [Adaptation of Bergeron method 

to the study of wave propagation]: Unpublished manuscript to be 

found in Archives of "La Soc. Hydrotech. France," [Paris].

The Bergeron method is adapted for unsteady flow in channels. The 

author discusses the use of -\/H instead of H in the equation of characteristics, noting the advantage of the straight line of graph -*/H=f(V) which has 

a slope of 45°. This concept was later developed in detail by Cray a.
120 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

397. Shepley, J. M., and Walton, C. B., 1942, Solving reservoir problems with

circular point-by-point computer: Civil Eng., v. 12, p. 154-155. 

The use of a device patterned after the circular slide rule is described for 

flood routing based on the storage equation. The storage factors FI = 

W/At+Q/2 and Fn=W/M Q/2 are used. Reservoir elevations are represented by concentric circles. The average inflows are measured as the 

angular distance between two index lines. A limited amount of numerical 

integration is required.

398. U.S. Beach Erosion Board, 1942, A summary of the theory of oscillatory

waves: U.S. Beach Erosion Board Tech. Kept. 2, p. 1-43. 

This summary of the theories of oscillatory wave motion on a free water 

surface includes studies made of wave characteristics, small amplitude 

waves, waves of finite amplitude, wave groups and transmission of energy, 

waves on a sloping bottom, wave refraction, and damping of oscillatory 

waves. Much of the theory discussed requires experimental verification, 

and mention is made of the extent of confirmation obtained.

399. Wisler, C. 0., and Brater, E. F., 1942, Am. Soc. Civil Engineers Trans., 

v. 107, p. 1519-1562.

A method of flood routing is described, the successful use of which depends only upon the availability of dependable streamflow records during 

a typical flood at various points on the main stream or on the tributaries 

whose flow is to be routed downstream. No cross sections of stream channel 

or velocities of flow are required. Nor are discharge records on all of the 

tributaries needed. A hydrograph of inflow from the unmeasured area is 

directly computed. This flow and that at each of the upstream stations is 

then routed downstream.

These routed flows show the extent to which each of the upper tributaries 

contribute to the flood peak at each downstream point. A check on the 

accuracy of the results is provided by adding the routed flows and comparing 

the resulting hydrograph with the actual records.

The entire procedure is based upon the storage equation and upon the 

principle that, for all high stages, there is a straight line relationship between the volume of storage contained in any reach of river channel and 

the sum of the inflow rate at the upper end and the outflow rate at the lower 

end of that reach. Except perhaps for unusual channel conditions, this 

relationship holds true. 

[Authors' synopsis.}

400. Arkhangel'skiy, V. A., 1943, Effektivnyy priem raschetov neustanoivshegosya dvizheniya v otkrytykh vodotokakh pri spokoynykh izmeneniyakh rezhima [An effective method of calculating unsteady flows in 

open channels]: Akad. Nauk [U.S.S.R.], Inst. Mekhaniki, Inzhenyernyy 

Sbornik, v. 3, no. 1, p. 124-127.

This paper presents improvements that shorten the author's method 

(1941) for integrating partial differential equations of unsteady flow by a 

method of finite differences. Both computation methods neglect the inertia 

term (l/g)dVfdt.

401. Keulegan, G. H., and Patterson, G. W., 1943, Effect of turbulence and 

channel slope on translation waves: U.S. Natl. Bur. Standards, Jour. 

Research Paper 1544, v. 30, p. 461-512.
BIBLIOGRAPHY 121

This paper is the second of a series dealing with the motion of flood waves 

and other waves of translation in open channels. The first paper considered 

waves controlled solely by inertia forces; the present one is an analysis of 

the combined effects of turbulent friction and inertia. The basic equation 

of motion for gradually varied unsteady flow in prismatic channels is derived 

from fundamental principles. The effect of the velocity distribution in 

the original undisturbed current on the motion of short waves is investigated 

and the effects of wave height, curvature of profile, and fluid friction on the 

celerity of a wave-volume element is analyzed in detail. The deformation 

of a straight sloping front and the change of height of an abrupt wave front 

is treated. Special emphasis is laid on disturbances of negligible curvature 

and practical methods of handling engineering problems arising in connection with the operation of locks or hydroelectirc canals are given. 

[Authors' abstract.]

402. Turner, H. M., 1943, The flood hydrograph and valley storage: Am. Geophys

Union Trans., v. 24, p. 609-618.

The difference shown between the rising limb of the actual hydrograph 

and of the computed hydrograph (using storage computed from recession 

curves) is greater than the river stage at which the rise begins. This 

difference is also greater for flood hydrographs having the longest time between the rising and falling limbs. In such cases, the greatest difference 

between actual and computed hydrographs occurs at the lowest stages.

403. Herbert, D. J., and Lowe, F. C., 1944, Progress report on model studies of 

the Sacramento-San Joaquin Delta, Central Valley Project, California: 

U.S. Bur. Reclamation, Hydraulic Lab. Kept. no. Hyd-142, Apr. 10.

404. Kohler, M. A., 1944, The use of crest-stage relations in forecasting the 

rise and fall of the flood hydrograph: U.S. Weather Bur., Hydrol. 

Director Office (mimeo.), Aug., p. 1-19.

In certain areas, where the stage-discharge relation changes frequently 

with changes in channel regime, all forecasting procedures must necessarily 

be expressed in terms of discharge. If these changes are gradual, however, 

the procedure may be developed in terms of stage from the most recent 

data, and revised as subsequent floods provide additional data.

The procedures described in the following paragraphs are, in fact, solutions of the storage equation; that is outflow is equal to inflow plus or 

minus changes in storage.

Within a fixed period of time the change in storage is determined by the 

reach characteristics, which are constant, and the four variables PI, Pz, 

Qi, and Q2. The procedures are based on a statistical determination of 

factors inherent in discharge and stage records. The constants of the 

regression equations themselves reflect physical characteristics which 

determine the relative storage in any reach.

The reader is encouraged to demonstrate to himself the physical basis 

for the procedure in its application to the solution of the storage equation 

in a particular reach for which the storage volume is actually known. 

[From author's synopsis.]

405. Kohler, M. A., 1944, A forecasting technique for routing and combining

flow in terms of stage: Am. Geophys. Union Trans., p. 1030-1035. 

A method of flood forecasting for downstream points on large rivers is 

given, using records of river stages and particularly those showing change
122 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

in stage during the period equal to the wave-propagation time between 

two mainstream stations. During this time of wave propagation between 

the two stations, stage departures that occur in straight-line relationship 

represent change in stage on major tributaries. As an example of the 

method, the Mississippi River is used with the Arkansas River and White 

River as tributaries.

406. Linsley, R. K., 1944, The development of flow and stage relations for 

river forecasting: U.S. Weather Bur., Feb. (mimeo.), p. 1-14. 

This paper discusses choice of the gage relation to be used as the forecast 

tool, primarily for crest-stage predictions, describing the simple gage 

relationships that may be developed (stage relations, flow relations, and 

flow-stage relations). It further discusses selection of data, cautioning 

that the upstream stages used to relate to stages at a downstream point 

must be those that actually influence the crest at the downstream point. 

It is noted that in order to obtain a true stage relationship, the two stages 

used for establishing a particular point on the curve of relationship must 

be separated by a time interval equal to the time of travel.

407. Linsley, R. K., 1944, Use of nomographs in solving stream-flow routing

problems: Civil Eng., v. 14, no. 5, p. 209-210.

The nomograms are designed to speed the solutions of storage differential 

equation in the form

Three vertical straight lines are used with scales shown, left and right of 

the line, for variables: line I, with variables (W\ %Qi+At) and Qi; line II, 

with variables Q2 and (W2+%Q2At); and line III, with relationship to 

runoff during the time interval At, inches over the basin, and total runoff 

during the time interval. The variables can be set up along the three lines 

in different ways. Each river needs individual treatment in planning the 

setup of scales.

408. Clark, C. O., 1945, Flood-storage accounting: Am. Geophys. Union Trans.,

p. 1016-1030.

Flood-storage accounting is the flood-routing procedure applied in 

reverse,, and it uses known hydrographs to calculate the runoff on the 

ground. It is based on the fundamental storage equation, inflow minus 

outflow = the change of storage. The Muskingum flood-routing method is 

used for the computation of inflow hydrographs in major and minor watercourses. The effective rainfall which produces the inflow hydrograph is 

derived by the use of the unit hydrograph.

409. Clark, C. O., 1945, Storage and the unit hydrograph: Am. Soc. Civil

Engineers Trans., v. 110, p. 1419-1488.

The purpose of this paper is to clarify the inherent relationship between 

fundamental tools (storage and unit hydrograph) and to show how this relationship may be used to derive accurate unit hydrographs for very short 

periods of initial runoff which accurately reflect the influence of shape of 

drainage area upon the shape of the hydrograph, allow the segregation of 

elements of the hydrograph attributable to particular components of the 

drainage area, and permit the definition of the calculations on intangible 

factors of personal judgment.
BIBLIOGRAPHY 123

After showing that the constant time units characteristics of unit hydrographs are the characteristics induced by storage capacity of the streams 

and that storage capacity and discharge capacity are each a limiting factor 

on the other, the paper illustrates the incorporation of these characteristics 

in hydrograph calculations. 

[From author's synopsis.]

410. Cooperrider, C. K., Cassidy, H. O., and Niederhof, C. H., 1945, Forecasting 

stream flow of the Salt River, Arizona: Am. Geophys. Union Trans., 

v. 26, p. 275-281.

The forecast is made for a small tributary stream on the basis of that 

stream's flow correlation with Salt River. An equation is given which 

defines the standard error of forecast. The estimated or forecasted flow of 

test cases came within 2.2 percent of the actual flow.

411. De Marchi, Giulio, 1945, Onde di depiessione provocate da apertura di 

paratoia in un canale indefinite [Depression waves created by opening 

of gates in canal of infinite length]: Energia Elettrica [Italy], v. 22, no. 

1-2, p. 1-13.

The experiments by Egiazarov have shown the possibility of applying 

the Saint-Venant theory to the movement of depression waves created by 

sudden opening of a gate at the extremity of a canal of indefinite length 

with rectangular cross section. The outflow discharge is smaller than the 

discharge which corresponds to the maximum energy level in the canal. 

The Saint-Venant theory is applied, and the results are discussed for the case 

of the sudden total opening of a gate, as wide and deep as the canal. The 

equation is obtained: H=(l/9g)(2T/gH0 x/t)*, between depth H and the 

initial depth H0, with z=distance from gate, and i=time. The profiles of 

waves are given, and the results of experiments by Trifonov are compared 

with computed data, and discussed. The same negative waves are studied 

in the cases of a bottom gate, or a surface reversible gate, with sudden 

openings, as well as the case with a gradual opening of gates.

412. Neyrpic, 1945, Report on Special Rhine River flood model experiments: 

Ateliers Neyret-Beylier and Piccard-Pitet (actually Neyrpic), Grenoble 

[France].

Report describes a model study of dam breaches and sudden water release 

on the Rhine River. Experiments are described, and results are given.

413. Vedernikov, V. V., 1945, Conditions at the front of a translation wave disturbing the steady motion of a real fluid [Usloviya na fronte volny 

popuska, narushayushchey ustanovivsheesya dvizhenie real'noy 

zhidkosti]: Akad. Nauk [U.S.S.R.] Doklady, v. 48, no. 4, p. 239-242 

(version in English).

The trajectory of the front of a translation wave disturbing a steady flow 

is a characteristic by means of which the partial derivative cannot generally 

be computed. If the condition establishing a relation between the independent variables and the first derivatives is fulfilled, then one of the partial 

derivatives remains arbitrary, and it is possible to compute the derivatives 

by means of the characteristics. It is shown that this condition is fulfilled 

at the front of a wave disturbing a steady flow. A convenient way of computing the derivatives is indicated and the formulas are derived for computing the derivatives of the first and second order in a partial differential 

equation for unsteady flow in a prismatic channel for any distance from the 

initial section.
124 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

414. Vedernikov, V. V., 1945, V raschetu neustanovivshegosya dvizheniya 

zhidkosti v otkrytom rusle [On the computation of unsteady motion 

of liquid in an open channel]: Akad. Nauk [U.S.S.R.], Izv., Otdetenie Tekhnicheskikh Nauk, no. 4, p. 499-503.

The Khristianovich approach to the Massau method of characteristics is 

discussed. In the formulas developed by Khristianovich (1938), the term 

K=g(So V2{c*R) must be determined in the region of computational 

differences of characteristics. This author's study makes it possible to obtain the formula with a change of K in the first approximation. The basis 

for this solution is the Boussinesq assumption of a linear-resistance law for 

waves of small amplitude. In differing from the Boussinesq approach, the 

author treats the wave of finite amplitude, with the linearization of resistance in a more general form, in the region of computational differences of 

characteristics.

415. Wilkinson, J. H., 1945, Translatory waves in natural channels: Am. Soc. 

Civil Engineers Trans., v. 110, p. 1203-1236.

The phenomena of translatory wave travel in natural channels have been 

studied for many years without entirely satisfactory results. Published 

studies have been largely theoretical, supplemented by laboratory tests in 

some instances, but with few observed data on waves in natural channels. 

Stream flow regulation by storage reservoirs in the Tennessee River Basin 

during the past few years has developed numerous translatory waves of 

various magnitudes. Under such conditions, waves can be selected for 

study which travel considerable distances without being appreciably 

affected by tributary inflows.

This paper presents data on natural channel translatory waves on the 

Clinch and the lower Tennessee rivers, and on Wheeler Reservoir in Alabama for a considerable range of flow in each type of channel. Theoretical 

velocities are computed for these waves by formulas and are compared with 

observed velocities. Results indicate that wave velocities on the Clinch 

River can be computed with reasonable accuracy, and that velocities for 

the rising face of a wave on the lower Tennessee River can be approximated. 

Computed velocities for waves in Wheeler Reservoir are in poor agreement 

with observed velocities. 

[Author's synopsis.]

416. ArkhangeFskiy, V. A., 1946, Dvizhenie v forme preryvnoy volny v otkrytykh vodotokakh [Flow in the form of a breaker]: Akad. Nauk 

[U.S.S.R.J, Inst. Mekhaniki, Inzhenernyy Sbornik, v. 2, no. 2, p. 

119-126.

The problem of the steep wave (surge) moving on a long wave is studied 

by combining the method of characteristics, given for Z (level) and Q 

(discharge) in the x-, f-plane, with application of the author's graphical 

method of finite differences. The trial- and error-procedure of obtaining 

the celerity function C j(f) in x-, <-plane is given for steep wave movement 

on the unsteady flow of a long wave. The case of steady flow is introduced. 

The time when the steep wavefront will arrive at a given place, x, is computed by the use of developed procedures, and some corrections are 

introduced.

417. Cheng, H. M., 1946, A graphical solution for flood routing problems: Civil 

Eng. [London], v. 16, Mar., p. 126-128; Corrigenda, p. 404, Sept.
BIBLIOGRAPHY 125

The storage equation P Q=dW/dt, for finite At, is integrated graphically 

using the form

Three graphical methods of integrating are given: (1) using factors W 

YzQM, and W+%QM; (2) using only factor W+%QM; and (3) using only 

W=f(Q) and three parallel lines for PA*, +YzQAt and J4QA*.

418. Craya, A., 1946, Calcul graphique des regimes variables dans les canaux 

[Graphical computation of variable regimes of flows in channels]: 

Houille Blanche [France], new ser., no. 1, Nov. 1945-Jan. 1946; 

p. 19-38, and no. 2, Mar. 1946, p. 117-130 (Partly translated into 

English by A. Duncan, TVA) .

Graphical methods based on characteristics are studied. For the train 

of elementary waves it is found that V 2-\fgH= constant for the downstream 

wave and V-\- 2rfgH= constant for a wave in upstream direction. These 

equations give the characteristics (lines) in the coordinate system (V, ^}gH), 

with slopes 0.5 and 0.5. For the observer traveling upstream with the 

wave train and with the celerity_V C, the velocity and wave height vary 

as V 2C= constant, with C="\/gH, For the observer traveling downstream 

with celerity V+C, the relation is V-\-2C= constant. Waves resulting 

from the merging of two waves approaching from opposite directions 

are studied in terms of their characteristics. The graphical representation is given with straight-line characteristics. Problems are solved 

or cases involving: the jump in bottom; the bottom having continuous 

slope; friction losses due to flow resistance; both sloping bottom and flow 

resistance; changing cross section; wave translation; prismatic channel of 

any cross section; the boundary conditions; propagation of elementary 

wave (estimate of wavefront in the case of canal which is initially without 

water movement, where flow is uniform or variable) ; and shock waves.

419. Oraya, A., 1946,, La propagation des ondes dans les e"coulements & surface 

libre graduellement varie"e [Wave propagation in the flow with free, 

gradually varied surface]: Centre de Perfectionnement Tech., Maison 

de la Chimie [France], Jan., no. 1549.

420. Pupping Umberto, 1946, Sulla forma dell'equazione del regime vario nelle 

correnti liquide [On the form of equation for unsteady flow]: Energia 

Elettrica [Italy], no. 7, p. 261-265.

The equation of unsteady flow is presented in two forms, the first being 

obtained by the terms of the equation relating to the filament of velocity 

flux in the field of discharge, and the second being obtained by means 

of the field "cross section of flow" determination. It is demonstrated that 

the first form is preferable, either because of its immediate significance 

of energy, or because it may be adapted to a simplified form which leads, 

in an approximate way, to an equation of the usual practical form.

421. Re", R., 1946, Etude du lacher instantane" d'une retenue d'eau dans un 

canal par la me'thode graphique [A study of sudden water release from 

a body of water to a canal by the graphical method]: Houille Blanche 

[France], no. 3, p. 181-187.

Craya's development of method of characteristics (1945-46) is applied 

to Rhine River for the purpose of wave-height forecast in case of dam 

breach. The problem is presented in a schematic form. The principles 

of integration by characteristics are given in summary form with special

728-245 64 9
126 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

attention given to shock waves (bores). The graphical procedure of 

integrating is described, and precautionary advice is given. The results 

are considered and conclusions drawn at the end. Charts giving practical 

illustrations of the method accompany the paper.

422. U.S. Engineer School, 1946, Engineering construction, flood control, 

(Chap. 5, Flood routing, and Chap. 6, Reservoirs): U.S. Engineer 

School, Fort Belvoir, Va., p. 127-224.

A definitive discussion of flood routing is given, including generalized 

theories of flood wave and channel. Pauls method for flood routing (with 

factors W+Q/2, W Q/2) is described; Steinberg's and McCarthy's (Muskingum) methods are given. All three methods are based on the storage 

equation. Special attention is given to the effects of levees, floodways, 

and channel improvement. With reference to reservoirs, the storage 

equation is solved by trial-and-error, Goodridge, graphical, and masscurve methods. Integrating machines and various aspects of flood routing 

through reservoirs are discussed in detail.

423. Zamarin, E. A., 1946, Dvizhenie povodka pri proryve plotiny [Movement 

of flood wave after the dam breach]: Nauchnye Zapiski (scientific 

proceedings) Moskovskogo Gidromeliorativnogo Inst. [U.S.S.R.].

424. Anonymous, 1947, Electrical computer aids flood studies: Civil Eng., v.

17, no. 3, p. 18.

This is a brief note describing an electrical analog device employing an 

oscilloscope to observe the change in shape of flood waves.

425. Arkhangel'skiy, V. A., 1947, Raschety neustanovivshegosiya dvizheniya 

v otkrytykh vodotokakh [The computations of unsteady flow in open 

channels]: Akad. Nauk [U.S.S.R.], p. 1-134.

This is a complete treatment of the author's contributions to the study 

of unsteady flow. There is a general discussion of unsteady flow and of 

the differential equations for unsteady flow in prismatic channels. Khristianovich's approach to the integration of these equations by the method 

of characteristics is given in detail. The author's method of expressing 

finite-difference equations, or the method of instantaneous regimes, is 

described. By this method the inertia terms in the Saint-Venant equations 

are ignored or replaced by a correction coefficient. The variables are 

assumed to change linearly within the time A£, and reach &x. The boundary conditions are discussed. The procedure of integration is explained. 

Both the method of characteristics and the method of finite differences are 

then applied to unsteady flow in form of surges, with an analysis of wave 

reflection given. The sources of data and the representations of the data 

are discussed, and examples of computation by both methods are given. 

Also examples of flood-wave transformation are shown for the case of 

dam breach and of powerplant operation, with unsteady channel flow. 

All seven chapters have summaries in English.

426. Bruman, J. R., 1947, Application of the water channel compressible gas

analogy: North Am. Aviation, Inc., Rept. NA-47-87. 

This report contains a brief discussion of the analogy between flow of a 

free-surface liquid (that is, under action of gravity) in a shallow channel 

and the two-dimensional flow of a compressible gas.

Results of reasearch to determine the optimum conditions of water depth 

and model size are presented. The basic equipment used is a shallow basin
BIBLIOGRAPHY 127

about four feet wide by twenty feet long, traversed by a light carriage on 

which models are mounted. Findings of this work are embodied in a set 

of design recommendations for a complete water channel set up. 

[From the summary in the paper.]

427. Dean, W. R., 1947, Note on waves on the surface of running water: Philos,

Soc. [Cambridge] Proc., v. 43, p. 96-99.

A solution is given for local disturbance induced on the surface of a steady 

uniform flow by application of a constant pressure. Treatment of the 

problem is purely analytical.

428. Dronkers, J. J., 1947, Methoden van getijberekening [Methods of tidal 

calculation]: De Ingenieur [Netherland], no. 49, Bouw. -en WaterBouwkunde 13, p. 129-188.

Two methods of tidal calculations are given, the choice of method being 

dependent upon the desired accuracy of results and on the nature of the 

rivers or estuaries in question. They are called the linear and the exact 

methods. The exact method takes into account the irregularities of rivers 

or estuaries, but when rivers are considered regular, the linear method 

gives more rapid but less accurate solutions. The exact method uses 

finite differences, whereas, the linear method uses the Fourier-series for 

tides, with integration along the regular channel.

429. Haines, N. S., 1947, Varying reservoir levels and discharges forecast by

new method: Eng. News Rec., v. 138, June 26, p. 69-71. 

A method of routing in reservoirs that uses rate of rise of the water surface 

as a function of inflow rate and elevation of pool.

430. Holsters, H., 1947, Le calcul du mouvement non-permanent dans les 

rivieres par le methode dite des lignes d'influence [The computation 

of unsteady flow in rivers by the so-called "influence-lines" method]: 

Rev. Ge"n. 1'Hydraulique [France], v. 13, no. 37, p. 36-39; no. 38, p. 

93-94; no. 39, p. 121-130; no. 40, p. 202-206; and no. 41, p. 237-245. 

The method of characteristics for integration of unsteady flow equations, 

developed by Massau (1899-1905), is adapted for practical purposes. The 

author calls the characteristics "the influence lines" in order to distinguish 

his construction of lines from the rigorously exact lines (characteristics). 

The base of the computation method is a net of characteristics that correspond to zero velocity of stream and to a depth invariable in time (that is, 

the mean depth for a tidal river). The theory of Massau is presented. 

Using two partial differential equations and two total differentials and 

determinants, the following two characteristics are obtained: dx= -\--\JgH 

dt, and (level) dh=±dQ/B^gH±Q\Q\/c2B2H^ Jf (d/i/dpo along a propagated discontinuity is greater than V(V + 2-\lgH)(^gH V)l^H, there 

will be a finite wave discontinuity. The method of influence lines is given. 

Wind influence, Chezy coefficient, and many applications are studied. 

These include modification of permanent regime, tidal movement in rivers, 

the analysis of influence on floods on a main river by a dam and reservoir 

on a tributary river, problems relating to the Escant River, and polders. 

Model verifications are analyzed.

431. Lin, Pin-Nam, 1947, Unsteady flow problems from Massau's line of attack:

Iowa Univ. M.S. thesis.

In this paper is attempted the introduction of a workable scheme to 

adapt Massau's theoretical treatment to practical problems. The paper
128 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

consists of three parts, the first part being devoted to a brief account and 

discussion of Massau's treatment, the second part to the introduction of a 

graphical solution, and the third part to two numerical examples. 

[From author's abstract.]

432. Mastitskiy, N. V., 1947, Priblizhennyy graficheskiy metod rascheta 

neustanovivshegosya dvizheniya potoka [Approximate graphical 

method of computation of unsteady flow]: Tran. Neustanovivsheesya 

Dvizhenie Vodnogo Potoka v Otkrytom Rusle, Akad. Nauk [U.S.S.R.]. 

The method developed by the author (1934) is improved, and the computation of basic hydraulic elements of movement is made by unique construction.

433. Nosek, T. M., and Dice, R. I., 1947, A theoretical study of flood waves 

resulting from sudden dam destruction: Mass. Inst. Technology, M.S. 

thesis, p. 1-28.

For the study of dam breach waves the Craya method (1946) was used 

in the solution of two simulated dam failures for which very accurate model 

results were available. The modified Craya method was used for routing 

of the wave resulting from the failure of St. Francis Dam, for which accurate records exist. The Craya method was found to be the best theoretical 

method, in the absence of model studies, for predicting flood-wave phenomena resulting from sudden dam destruction. This method gave results 

within 5 percent of accurate model tests for relatively uniform channel 

cross section, slope, and friction factor. The modified Craya method gave 

results within 10 percent of the recorded maximum water elevations resulting from the failure of St. Francis Dam.

434. Potapov, M. V., 1947, Priblizhennyy gidravlicheskiy method rascheta 

dvizheniya volny popuska [Approximate hydraulic method of computation of the movement of release waves]: Sbornik Neustanovivshesya 

Dvizhenie Vodnogo Potoka v Otkrytom Rusle, Akad. Nauk [U.S.S.R.]. 

Reprinted in Sochineniya [Works], v. 3, p. 449-471, 1951. 

For simple schematic flood waves (triangular, trapezoidal), flood-wave 

attenuation, which occurs as a function of channel length, is shown as the 

change of the peak discharge and of maximum stages. A correction coefficient is introduced in simple formulas, and a discussion of the method of 

determination of the cofficient is given. The application is presented for 

any channel cross-sectional shape and for the parabolic shape. The time 

of arrival of the maxima is computed, and the manner of determining the 

hydrographs and the graphs of instantaneous regimes is shown. The channel 

with changing hydraulic elements is discussed.

435. Schultz, E. R., 1947, Preliminary flood-routing studies for the Yangtze 

River Gorge Dam: Col. Univ. M.S. thesis.

436. Steinberg, I. H., 1947, A flood routing device: Am. Geophys. Union Trans., 

v. 28, p. 247-254.

A flood-routing device, based on the simple storage equation and the coefficient method for the storage-discharge relation of a river reach, is 

developed by the use of two slides to which are attached arms with adjustable slopes. One of the slides has a linear scale graduated in inches and 

tenths of an inch. The equation for routing is given and the application 

is discussed. Two examples are given.
BIBLIOGRAPHY 129

437. Stroband, H. J., 1947, Een bijdrage tot de kennis van de getijberebing op 

benedenrivieren en zeearmen [A contribution to the knowledge of tidal 

calculations for river and sea estuaries]: De Ingenieur, no. 36, Bouw-en 

Water-Bouwkunde 10, p. 89-95.

An approximation is added to the method of linearizing the quadratic 

resistance term, in order to account for the effects of partial tides as well as 

the main tide.

438. Ursell, F., 1947, The effect of a fixed vertical barrier on surface waves in

deep water: Philos. Soc. [Cambridge] Proc., v. 43, no. 3, p. 374-382. 

The two-dimensional reflection of surface waves from a vertical barrier 

in deep water is studied theoretically.

439. U.S. Bureau of Reclamation, 1947, Flood routing: U.S. Bur. Reclamation 

Manual, chap. 6.10, pt. 6, Flood Hydrology, IV, Water Studies, 

Dec. 30.

Different simple methods of flood routing along channels are described: 

Puls method (and example), modified Puls method (and example), 

Stern berg method (using storage factor, and example), Rutter, Groves, 

and Snyder method, and Muskingum method. Flood routing through 

reservoirs is briefly discussed with reference to the modified Puls method 

and the trial-and-error method.

440. U.S. Geological Survey, 1947, Channel storage and flood routing: U.S.

Geol. Survey, Water Resources Div., Handb. Hydrologists, p. 1-16. 

The general relation between upstream and downstream hydrographs is 

discussed. Included are: stage-storage and discharge-storage methods 

of flood routing, and computation of storage volume in the coefficient 

(Muskingum) method, with evaluation of coefficient K and of x. Applications are shown.

441. Vedernikov, V. V., 1947, Volni popuskov real'noy zhidkosti [Release waves 

of the real liquid]: Sbornik Neustanovivsheesia dvizhenie vodnogo 

potoka v otkritom rusle [Transactions, Unsteady movement of the 

water current in an open channel]. Akad. Nauk [U.S.S.R.]. 

Special attention is given to the ways that wave movement of real 

liquids differs from wave movement of ideal liquids. The author gives 

the criteria for the stability of initial flow, of attenuation, and for breaking 

of translation waves. By theoretical analysis of unstable initial flow 

and the formation of unattenuated waves, the author comes to the conclusion that the classification of flow as subcritical and supercritical is 

not sufficient. He adds also "oversupercritical flow." When applied 

to large rectangular channels, the criteria are expressed as:

(1) subcritical V0<Ve =^gHm ; (2) supercritical Ve <V0<Vc "=B^mf2', 

and (3) oversupercritical V0 ^>VC ",

with Hni=A/B. For the first two classifications, waves attenuate when 

they propagate, the flow is stable, and the wave breaks only if its creation 

is sufficiently sudden. In the case of oversupercritical flow, every disturbed 

surface is not attenuated, but is increased resulting in formation of new 

shorter waves having greater heights, so that the wave breaks by progressing. 

Oversupercritical flow is not stable, and the waves created contract. 

It is characteristic of such flow that the negative wave does not become 

less steep, but may, on the contrary, become steeper by wave translation.
130 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

442. Veen Van, Johan, 1947, The calculation of tides in new channels: Am.

Geophys. Union Trans., v. 28, p. 861-866.

The numerical solution of the large number of simultaneous partial 

differential equations used in the theoretical design of the Netherlands 

waterways has previously been done manually at enormous expense of 

time and money. This paper shows how the same ends may be accomplished 

by an electrical network analogy.

[From Bibliography on Tidal Hydraulics, Corps of Engineers, February 

1954.]

443. Bachet, 1948, Methodes graphiques d'annonce des crues [Graphical methods

of flood forecast]: Houille Blanche [France], no. B, p. 720-728. 

The same concept is used as in the author's earlier paper (1934), namely, 

that a discharge (q) is the sum of the steady flow (r) and the changing 

discharge with g=r+p. It is developed that dx=(dr/dH)(dt/B), and 

dp = (dp/dx)dx, so that the wave is propagated by a translation 

(l/B)dr/dH and a vertical change dr= (dp/doOcfc, called "attenuation." 

The fundamental equation is BdH/dt-{-()pl()x=o, where dH is the height 

variation due to attenuation. If two cross sections of the propagating 

wave are under consideration, the lost volume due to attenuation is 

equal to the sum of complimentary discharges lost from those sections 

during the time period considered. The complementary discharge is 

found to be p=(l/2S)(d#ydr)(rdr/dz). The flood routing is based on 

translation and attenuation factors. The functions r=fi(x C£) and 

p fifr Cf) are introduced. The procedure for practical flood routing 

is developed, based on the formula &H=6T(&H/dP'), where 6 and r are 

two constants to be computed. The templates are given for the graphical 

flood routing.

444. Barrows, H. K., 1948, Floods: Their Hydrology and Control (Flood routing 

and effect of reservoirs upon flood peaks, p. 73-79): New York, McGraw-Hill Book Co., p. 1-432.

The Muskingum flood-routing method and procedure of routing is given 

and the effect of reservoirs on flood-peak attenuation is briefly discussed.

445. Bourguignon, M. P., 1948, Releves d'intumescences dans les ovarages 

d'amane'e et de restitution de 1'usine de Kembs [Surge records in the 

head-race and tail-race canals of the Kembs water powerplant]: 

Houille Blanche [France], no. B, p. 699-719.

The records which have been obtained give an approximate confirmation 

of the results of theoretical and experimental studies made up to 1947. 

It is found that the propagation celerities of waves approximate Boussinesq's formula

C= V±T/gH[l + (1 -2H/35)3A/4HL

where C=propagation celerity, V=velocity in the canal, ff=mean depth 

of water, h=height of wavefront, B=width of water surface. Observations for secondary waves agree with Favre's results. Wavefronts have 

the tendency to attenuate and to approach the solitary wave, in this respect.

446. Courant, R., and Friedrichs, K. O., 1948, Supersonic flow and shock waves:

New York Inter-science Publishers Inc.

The shallow water theory is briefly discussed on pages 32-35. The 

treatment of supersonic flow and shock waves is useful whenever an analogy 

can be drawn to compare waves in compressible fluids with the water waves.
BIBLIOGRAPHY 131

447. Gulp, M. M., 1948, The effect of spillway storage on the design of upstream

reservoirs: Agr. Eng., v. 28, p. 344-346.

Starting from the simple triangular shape of the inflow hydrograph and 

the spillway storage as a function of reservoir characteristics, a formula 

for the ratio P=Qm/Pm (peak of outflow to peak of inflow), is given as

where A =area of river basin in acres; D average depth of flood runoff 

in inches, a = area of reservoir surface at crest-level of spillway in acres, 

d=stage in feet above the spillway crest; S=average slope of banks of 

reservoir through range of stage d, in percent.

448. Friedrichs, K. O., 1948, On the derivation of the shallow water theory 

(An appendix to the article: The formation of breakers and bores, by 

J. J. Stoker): New York Univ. Commun. Pure and Appl. Math., v. 1, 

no. 1, p. 81-85.

The shallow-water theory is derived by means of a formal perturbation 

procedure that is essentially a development with respect to a parameter 

<r = Kh, in which K is the maximum initial curvature of the free surface 

and h is the depth. When <r is small, the water is said to be shallow. It 

is shown that in the derivation of higher order approximations for flows 

in shallow open channels, it is not necessary to make as many special 

physical assumptions as were made by Boussinesq (1877).

449. Gil'denblat, Ya. D., Makulov, V. V., and Semikolenov, A. S., 1948, Neustanovivshiysya rezhim nizhnego byefa GES [Unsteady flow regime 

in the tail-race canal of hydroelectric power stations]: Sbornik Problemy 

regulirovanya rechnogo stoka [Collection: Problems of river flow 

regulation]: Akad. Nauk [U.S.S.R.], no. 2, p. 43-142. 

The first part is the investigation of organization, methodology, and 

results of observations of the hydraulic regime of tailrace levels of hydroelectric power stations during daily regulations. The second part is the 

comparison of the theoretical results with the data of observations in the 

nature. The method of finite differences (instantaneous regimes) applied 

to two partial differential equations and the method of characteristics are 

used; the results of both methods are compared with the observations. 

The departures are relatively small.

450. Keller, J. B., 1948, The solitary wave and periodic waves in shallow water:

New York Univ. Commun. Appl. Math, v. 1, no. 4, p. 323-342. 

The author contends that the existence of the solitary wave has not 

yet been proved mathematically and that the method of proceeding to 

higher approximations is also obscure. He further contends that when 

the actual solutions for stationary waves are found, progressive waves 

may be obtained from them by the addition of constant fluid velocity. 

This paper discusses waves of permanent type in shallow water. The 

method is that of expending systematically the solution of the exact hydrodynamic problem in powers of a dimensionless parameter a=(<ah), where 

h is the depth of undisturbed fluid, and « is the curvature at some point 

on the surface. The solution of permanent form is given by the first and 

second approximations.

451. Keulegan, G. H., 1948, Gradual damping of solitary waves: U.S. Natl. 

Bur. Standards Jour. Research, v. 40, p. 487-498.
132 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

The problem of damping due to the viscous action of translation waves 

is treated. A short explanation of Boussinesq's boundary layer theory 

for wave motion is given and expressions for the damping of rectangular 

and solitary waves are derived. Russell's experimental results for solitary 

waves are compared with the theory, and satisfactory agreement is found 

to exist. This verification permits the use of the resultant formulas in 

model tests on harbors, and so forth, to correct for the dissipative effects 

of damping of shallow-water waves.

452. Lemoine, R., 1948, Sur les ondes positives de translation dans les canaux 

et sur le ressaut ondu!6 de f aible amplitude [On the positive translation 

waves in the canals and on the undular hydraulic jump with sma1 ! 

amplitude]: Houille Blanche [France], no. 2, p. 183-185. 

The yielding of waves in a powerplant's headrace canal, as a consequence 

of a sudden switch-off of the units, is analyzed for the Kembs water powerplant. According to the surge theories (and methods by Mass6 and Craya), 

an increase in the front's steepness could be anticipated. Experimental 

results show that no such phenomenon occurs, and undular waves of the 

jump are observed. Based on Favre's work (1935), the undular jump 

traveling upstream is analyzed and explained. The difference is emphasized between a positive wave propagated upstream in water at rest and 

a positive wave (more stable) traveling on flowing water.

453. Linsley, R. K., Foskett, L. W., and Kohler, M. A., 1948, Use of electrical 

analogy in flood wave analysis: Internat. Assoc. Sci. Hydrology, 

Oslo, Aug. 1948, Comptes rendus et rapports, no. 19, v. 1, p. 221-227. 

Electrical analogy is used to solve the flood-routing problem of the 

storage equation, using the Muskingum method. It is shown that 

W=K[XP+(l X)Q], where #=storage factor with dimension of time, 

and Z=weighting factor for inflow. The electric current flowing into a 

condenser is equal to the time rate of change of charge of W on the condenser (dWjdt), and the total charge on the condensers C\ and C2, at any 

time, depends on the potential drop across them. It is proved that the 

electrical storage in the devised circuit is analogous to the channel storage 

assumed in the Muskingum method, when C\=Cz and when the two resistances are equal, or Rt =R3. The method is described, and examples of 

flood routing for different K and X are given.

454. Linsley, R. K., Foskett, L. W., and Kohler, M, A., 1948, Electronic device

speeds flood routing: Eng. News-Rec., v. 141, no. 26, p. 64-65. 

This is a description of an electronic device for speeding flood-routing 

computations based on the Muskingum method of expressing the storageflow relationship. The device includes condensers, resistors, self-balancing 

potentiometers, and other equipment.

455. Meleshchenko, N. T., and Yakubov, M. S., 1948, Metodika raschetov 

neustanovishegosya dvizheniya v otkrytykh ruslakh po methodu 

S. A. Khristianovicha [The methodology of computation of uusteady 

water movement in open channels by the method of S. A. Khristianovich]: Vsesoyuzniy Nauchno-issledovatel'niy Inst. Gidrotekhniki 

[U.S.S.R.] Izv. (Trans.), v. 38, p. 29-70.

This paper results from studies made during the period 1938-40. The 

basic two partial differential equations and elementary derivation of 

equations of characteristics for prismatic channels are given. The types
BIBLIOGRAPHY 133

of equations of characteristics for prismatic channels (simple equations), and 

nonprismatic channels, as well as for natural channels (complex equations), 

are developed. The finite differences for numerical computation by method 

of characteristics are introduced, employing characteristics in the plane 

(x,f) either as straight lines or as curves. The computation applicable 

in the case of great resistance forces is analyzed with discussion of initial 

conditions and boundary conditions, for plane (x,t) considered for both 

steady and unsteady flow. The cases of lateral inflows, and channel 

junction are discussed. Computation of unsteady flow in prismatic and 

natural channels by means of the method of characteristics is shown in 

two examples. A simplified method for computation of daily water regulation is given and a nomographic procedure for computing unsteady 

flow in a prismatic channel is developed, and an example of use for daily 

regulation in powerplant canals is given. The method of characteristics 

is discussed, noting the mechanical character of the computation when 

employed for standard operations, and noting further that the method is 

practical for use in those cases when forces of inertia prevail over resistance forces.

456. Meleshchenko, N. T., and Yankubov, M. S., 1948, Methodika rascheta 

preryvnoy volny v prizmaticheskom rusle [The methodology of computation of positive surges in the prismatic channel]: Izv. Vsesoyuznogo 

Nauchno-issledovatel'nogo Inst. Gidrotekhniki [U.S.S.R.], v. 38, p. 

71-94.

This work was done in 1940. It is the continuation of work done in 

1938-40 and published in 1948 on the method of characteristics. The 

movement of a positive surge (traveling bore) is studied and computed by 

method of characteristics. First, the equations for steep positive wave 

traveling downstream are derived in a simplified form, in which C= Vm + 

aCm, where a is a function of HI and Hz (depths in front of and behind the 

surge), Vm =i(Vi+Vj)f and Cm=|(d+C2), where C^Vs^i and C2= 

-JgH2. The procedure of wave computation is given, as is discussion of 

initial and boundary conditions and an example of computation. An 

approximate and simplified computation of wavefront deformation is 

derived, inasmuch as the more exact procedure is laborious. Also, a 

simplified computation of the change of free water surface at the place of 

created surge is developed. An example of simplified computation of 

positive surge waves is given for powerplant operations.

457. Putman, J. H., 1948, Unsteady flow in open channels: Am. Geophys. Union

Trans., v. 29, p. 227-232.

Massau's method (1900) of characteristics in integrating two basic 

partial-differential equations is given as a short summary.

458. Stoker, J. J., 1948, The formation of breakers and bores; the theory of

nonlinear wave propagation in shallow water and open channels:

New York Univ., Commun. Appl. Math., v. 1, no. 1, p. 1-87.

As this subject is also treated extensively in the author's book "Water

Waves" (1957), only the content of the paper is here reported. The

author's contribution is mainly in the mathematical treatment of breaker

and bore formation, particularly in his use of gas-wave versus water-wave

analogy. Basic differential equations are given in the introduction. The

paper discusses: nonlinear shallow-water theory; integration of differential
134 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

equations by the method of characteristics; the motion of a single wave; 

propagation of pulses into still water of constant depth; propagation of 

depression waves into still water of constant depth; shock conditions; 

constant shocks as bores, hydraulic jump, reflection from a rigid wall; the 

breaking of a dam; flood waves in rivers; breaking of waves in shallow water 

of uniform depth; and breaking of waves on a uniformly sloping beach.

459. Veen Van, Johan, 1948, Estimates of tidal currents: Panama Canal Spec.

Eng. Div., I.C.S. Memo 299, Feb. 18.

Results of computation of velocities to be expected in an open Panama 

Sea-Level Canal. Studies made by Dr. Van Veen, Chief Engineer at the 

Rijkswaterstaat, The Hague, Netherlands, from data furnished by the 

Special Engineering Division, Panama Canal. Comparison between 

velocities computed by Dr. Van Veen, and those of the Special Engineering 

Division using Gen. Pillsbury's method. Includes article "Analogy between 

Tides and A.C. Electricity" by Dr. Van Veen.

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

February 1954.]

460. Viti, M., 1948, Le equazioni differenziali del moto lineare non permanente 

per i liquidi incomprimibili [The differential equations of the linear 

unsteady motion for noncompressible liquids]: Acqua [Italy], Mar., 

no. 1-3.

461. Barber, N. F., 1949, Behaviour of waves on tidal streams: Royal Soc.

[London], ser. A, v. 198, no. 1052, p. 81-93.

Discussion of the manner in which waves change characteristics when 

passing through regions where water has streaming motion; application to 

tidal streams where velocity depends on both time and position; and 

experimental data.

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

February 1954.]

462. Citrini, Dulio, 1949, Sull'attenuazione di un'onda positiva ad opera di uno 

sforatore laterale [On the attenuation of a positive wave affected by 

a side spillway]: Energia Elettrica [Italy], v. 26, no. 10, p. 589-599. 

Reprinted as Memorie e Studi, Istituto di Idraulica e Costruzione 

Idrauliche, Milano, no. 76, 1949.

The passage of a positive (downstream or upstream directed) wave along 

an open channel which has a lateral spillway is studied by the use of characteristics. Although the quantitative results are considered to be only 

approximate, the study presents clear evidence of interrelationships of 

wave movement (particularly wave attenuation) and the geometric and 

hydraulic elements which characterize the stream current and the spillway. 

The method of characteristic lines is given and applied. The conclusion 

is that the spillway does not influence the wave celerity. Relative-value 

relationships, giving attenuation of depth or discharge, are developed as a 

function of the ratio of spillway length to canal width.

463. Clark, C. O., 1949, Application of flood storage accounting methods: Am.

Geophys. Union Trans., v. 30, p. 528-532.

Employing his flood storage accounting method, discussed in an earlier 

paper (1945), the author demonstrates its use in the day-by-day application 

to forecast flow of the Roanoke River.
BIBLIOGRAPHY 135

464. Crossley, H. E., Jr., 1949, The analogy between surface shock waves in a 

liquid and shocks in compressible gases. Experimental study of 

hydraulic jump interactions: Calif. Inst. Technology Lab. Rept. N-54.

465. Dmitriyev, G. T., 1949, Vychisleniye karakteristik ustanovivshegosya 

plavno izmenyayushchegosya dvizheniya v prizmaticheskikh ruslakh 

[Computation of characteristics of a steady but gradually varying 

movement in prismatic channels]: Akad. Nauk [U.S.S.R.], Doklady, 

v. 68, no. 5, p. 825-827.

Using Agroskin's method, as developed from the theory of steady but 

gradually varying flow, characteristics of the differential equations for 

unsteady flow are analyzed for two waves of opposite flow direction. The 

analysis is given for three different slopes of canal bottom, S: S=O; /S>O; 

and S<O. The wavefront shape is derived for the waves.

466. Dressier, R. F., 1949, Mathematical solution of the problem of roll-waves 

in inclined open channels: New York, Interscience Publ. Inc, NewYork Univ. Inst. Math. Sci., Commun. Pure Appl. Math., v. 2, no. 

2, 3, p. 149-194.

The purpose of this paper is to obtain solutions which are periodic with 

respect to distance, describing the phenomenon called "roll-waves." The 

discontinuous periodic solutions can be constructed by joining together 

sections of a continuous solution through shocks (or "bores")- It is shown 

first that no continuous solutions can be periodic and that only one special 

continuous solution can be used as the basis for constructing discontinuous 

periodic solutions. The analysis is based upon the non-linear partial differential equations of the "shallow water theory," augmented by the 

Chezy formula to allow for turbulent resistance. The Bresse profile equation is obtained in a form applicable for progressing wave flows. Shock 

conditions are derived for the case of an arbitrary continuous channel bed 

and for a flow subject to a resisting force. It is proved that roll-waves 

cannot occur if the resistance is zero or if the resistance exceeds a certain 

critical value. As the resistance decreases, the size of the waves decreases 

also; and if the resistance becomes too large, the profiles reverse their direction and can no longer be joined by shocks. This critical value is reached 

when the (dimensionless) resistance coefficient equals one-fourth the 

value of the channel slope. The resistance must also act in such a manner 

that it decreases as the water depth increases. The analysis proves that 

the ratio of wave height to wave length of roll-waves is always independent 

of the speed of the waves. Explicit expressions for water height and shock 

height as functions of wave length are derived. The investigation studies 

the static discharge rate as a function of the wave speed, and asymptotic 

formulas for the wave speed in terms of the average discharge rate are 

derived. Twelve sets of curves are presented. For prescribed values of 

slope, resistance, and wave speed, there is a one-parameter family of rollwave solutions. If the wave length is also prescribed, the solution will 

then be unique. 

[From author's summary.]

467. Dronkers, J. J., 1949, Aperc.u des methodes pour la determination du movement de marge dans les embouchures et les fleuves & marge neerlandais 

[Review of methods for the determination of tidal movements in the 

entrances and in tidal rivers of Netherlands]: Internat. Cong. Navigation, Lisbon, rap 17c, sec. II., quest. 1, p. 159-178.
136 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

468. Johnstone, D., and Cross, W. P., 1949, Elements of applied hydrology, 

Chap. 7, Flood routing: New York, The Ronald Press Co., p. 160-191. 

The chapter includes discussion of the simple storage equation; an example of flood routing through reservoirs and retarding basins with storage 

as a function of discharge alone; an example of flood routing by use of a 

mass diagram and a storage-factor (2W/M+Q) curve; discussion of flood 

routing in a stream, where storage is used as a function of inflow and 

outflow, derivation is given of the storage relationship for a reach, and an 

example of the flood routing is shown. Special problems are discussed, 

such as the accounting for intermediate inflow and backwater effects. 

Effects of flood-control projects are shown by computations. Mechanical 

aids and short cuts are discussed.

469. Keller, J. B., 1949, The solitary wave and periodic waves in shallow water:

Acad. Sci. [New York] Annals, v. 51, act. 3, p. 345-350. 

This paper gives a brief summary of studies of the permanent waves, 

with reference to the solitary permanent wave. The author concludes 

that the existence of the solitary permanent wave has not yet been proved 

mathematically. He claims that solutions are found for stationary waves 

and that progressive waves are obtained from them by the addition of a 

constant fluid velocity. The shape of the wave is developed and discussed. 

The author theorizes that, if the water at infinity were stationary, the 

water under the crest of a solitary wave would be moving. According 

to the solutions of Korteweg and De Vries (1895), and Rayleigh (1911) 

the water under the crest of a solitary wave would then be stationary.

470. Kreisel, G., 1949, Surface waves: Appl. Math. Quart., v. 7, p. 21-44. 

This paper is a mathematical treatment of the asymptotic behavior of 

wave motions and the definition of the reflection coefficient, the objective 

being to determine the reflection from obstacles in arbitrary depth. Such 

reflected and transmitted wave trains are determined as coming up against 

obstacles, which lie in the bottom of a canal or are fixed in the surface. 

The simple harmonic oscillations are considered.

471. Lamoen, J., 1949, Tides and current velocities in a sea-level canal: Eng.

[London], v. 168, no. 4357, p. 97-99.

The method of characteristics is herein used for solving the tidal-current 

problem, based on Massau's (1900) and Holster's (1947) studies of this 

method, but the application given differs slightly from that in Holster's 

original text. More emphasis is placed on the method of treatment than on 

the theory. The example of an open-sea Panama Canal is shown, as is the 

good agreement between the computed results and results of the model test 

(Meyers-Shultz, 1949), within the framework of assumptions made. The 

process of computation is useful whenever the method of characteristics is 

used for flood routing, using the same basic assumptions. The linear law 

of friction resistance is a basic assumption of this computation process.

472. Linsley, R. K, Kohler, M. A., and Paulhus, J. L. H., 1949, Applied Hydrology: 1st ed., New York, McGraw-Hill Book Co., p. 465-541. 

Waves of translation and of oscillation are considered here. Translation 

waves treated include monoclinal rising waves, abrupt translatory waves, 

surges, flood waves due to dam failure and seiches. Oscillatory waves and 

tides are briefly discussed. Natural flood waves and channel storage are 

discussed, and it is shown how the channel storage function can be derived
BIBLIOGRAPHY 137

from hydrographs. With reference to routing based on storage equations, 

the following procedures are given: Muskingum, semigraphical, graphical, 

and by routing machines. Nonstorage routing procedures discussed 

(based on correlations and on successive averages) include methods of adjustment for local inflow, variable stage-discharge relations, and gage 

relations. Stage routing is shown, as are the typical curves used. Travel 

time is analyzed.

473. Lowell, S. C., 1949, The propagation of waves in shallow waters: New 

York Univ, Ph. D. thesis.

474. Meyers, J. S., and Schultz, E. R., 1949, Tidal currents: Am. Soc. Civil

Engineers Trans., v. 114, p. 665-684.

The tidal currents in an open-sea Panama Canal are treated by means of 

a small-scale model. The study is chiefly concerned with velocities of 

currents.

475. Munk, W. H., 1949, The solitary wave theory and its application to surf 

problems: Acad. Sci. [New York] Annals, v. 51, act. 3, p. 376-424.

Testing of the solitary wave theory, as applied to surf problems, was performed by field observations, by laboratory experiments, and by theoretical 

approaches.

The results of observations made to test the solitary wave theory exhibit 

so much scatter that they cannot be said to constitute a confirmation of 

theory comparable to an inductive laboratory confirmation of physical law. 

However, verifying evidence is strong enough to indicate that the solitary 

wave theory provides a useful approach to the study of various surf phenomena.

The difficulties of obtaining reliable observations of some of the features 

described are very great. In the field, extreme variability in the wave 

train itself is likely to obscure meaningful relationships. In the laboratory, 

it is difficult to avoid interference by reflected waves and surging. Because 

of these inherent uncertainties in measurements, a theoretical framework 

is almost essential for establishing useful relationships.

476. Ramponi, Francesco, 1949, Modello di un'onda di piena fluviale [Model

of a river flood wave]: Energia Elettrica [Italy], no. 11, 12, p. 699-700.

A discussion of the selection of satisfactory scales to use for deformed

river models, having a fixed bed, as exemplified in the case of a reach of the

Adige River for the flood regime. The relation of scales for different

hydraulic elements is given as the study result.

477. Ruggiero, C., 1949, Ondes graduelles dans les cours d'eau [Gradual waves 

in channels]: Intern. Assoc. Hydraulic Research, Convention, Grenoble [France], 1949, Rept.

478. Scimemi, Ettore, 1949, Risultati sperimentali sulla propagazione delle 

perturbazioni di regime nei canali [Experimental results of the propagation of regimen perturbations in channels]: Energia Elettrica [Italy], 

no. 11, 12, p. 691-696.

The paper treats the results of a study of wave movement that involves 

experiments in nature (canal of powerplant Cismon-Moline, Italy). Model 

experiments are described, and the computations of wave characteristics 

given. The canal of the powerplant and of the experimental laboratory 

(canal model 1:25) is described, as are the experiments. Forchheimer's
138 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

formula for wave celerity is used. The results of two types of experiments 

are compared with the results by computation. The maximum wave 

height, as computed by the Forchheimer formula for initial positive wave, 

may increase by a combination of reflected waves, but can be substantially 

decreased if the spillways are well designed. Wave profiles for different 

types of spillways are given.

479. Sokolovskiy, D. L., 1949, O metodakh ucheta transformatsii maksimumov 

v prudakh pri proektivovanii iskusstvennykh sooruzheniy [On the 

methods of computation of transformation of flood peaks in the pools 

for the design of artificial structures]: Stroitel'stvo Dorog [Highway 

Construction] [U.S.S.R.], no. 4.

480. Sorenson, K. E., 1949, Curves solve reservoir flood-routing equations: Civil

Eng., v. 19, no. 11, p. 56-57.

The flood routing is based on the storage equation, arranged in the form 

(W! + %(QiM) + [%(Pi + Pt)-Qi\ A0 = Tr»+H(Q,A0. The following curves 

are used: W+y2 (QM)=f(H), W=f(H), Q=f(H), and P=y2 (Pl + Pj=f(t), 

Q=f[W+%(QA£)]. The functions H=f(t), and Q=f(t) are computed. A 

graphical method is used, and a combined graphical and nomographic 

solution is shown.

481. Stoker, J. J., 1949, The breaking of waves in shallow water: Acad. Sci.

[New York] Annals, v. 51, act. 3, p. 360-375.

The principal objective of the paper is to interpret certain well known 

phenomena in gas dynamics in terms of water-wave phenomena, considering, particularly, the analogy between the shock wave in compressible gas 

and a breaker of water wave. The method of characteristics is 

used for solutions. The author explains why waves break in shallow 

water and applies his theory to problems found in studying the down-river 

progress of flood waves and surges. He concludes that waves break in 

different ways, depending upon the particular circumstances, among which 

an important factor is the comparison between the depth of water and the 

wavelength.

482. U.S. Army Corps of Engineers, 1949, Hydraulic design; surges in canals: 

U.S. Army Coips Engineers, Civil Works Construction, Eng. Manual, 

pt. 106, chap. 6, p. 1-13.

The chapter is confined to the problem of surges that occur in navigation 

canals as a result of lock operations. The generation of surges is discussed. 

The analysis of reflection, stability, and longevity of surges is given. The 

surge control and model test are presented briefly. A table at the end of 

chapter gives equations of the physical characteristics for four kinds of surge: 

positive upstieam and downstream surges, and negative upstream and 

downstream surges.

483. Volker, F., and Schoenfeld, J. C., 1949, Le regime d'une riviere sous Peffet 

de debits variables [The regime of a river under the variable flows]: 

Internat. Cong. Navigation, Lisbon, 1949, rap. 17e, sec. 1, quest. 3, 

p. 59-78.

484. Adler, G. F. W., 1950, Model tests on Clachan underground power station:

English Elec. Jour. [Great Britain], v. 11, no. 4, p. 119-127. 

Experiments on wave movement in tailrace underground channels having 

free water surface are described and analyzed. Calculation of wave
BIBLIOGRAPHY 139

heights and celerities by the momentum method gives reliable results in 

the case of leasonably uniform velocity distribution and a low rate-of-change 

of discharge. The real values are somewhat lower than the computed 

because of friction and wave reflection.

485. Arthur, R. S., 1950, Refraction of shallow water waves: The combined 

effect of currents and underwater topography: Am. Geophys. Union 

Trans., v. 31, p. 549-552.

A solution is given for determining the refraction effect according to 

Fermat's Principle for the case of shallow water waves moving in any 

given distribution of current and depth. Application is made to an analytic 

model of an intense rip current, and the results are compared to actual rips. 

[Author's abstract.]

486. Bergeron, Louis, 1950, Du coup de belier en hydraulique au coup de foudre 

en electricity. M6thode graphique generate [From the water hammer 

in hydraulics to the voltage surge along electrical power lines]: Paris, 

Dunod, p. 284-289.

The same graphical construction that is used to follow wave progression 

in the case of a water hammer in a conduit is applied to surge movement 

in an open surface canal. This is an application of the method of characteristics, used in a simplified form.

487. Biesel, F., 1950, Etude thebrique de la houle en eau courante [Theoretical 

study of wave in running water]: Houille Blanche [France], no. A, 

p. 279-285.

The propagation of a wave in channels with running water is studied 

analytically employing approximations of the first order. This is a twodimensional problem considering an ideal fluid in a large channel. The 

author's conclusions are as follows:

1. For very short waves, the difference between wave propagation celerity 

and the velocity of flowing water at the surface is equivalent to the 

propagation celerity of very short waves in still water.

2. The difference between propagation celerity of long waves and the mean 

velocity of flowing water is generally greater than the propagation 

celerity of long waves in water at rest.

3. This difference is greater by the amount of difference in surface and 

bottom velocities.

4. In general, the mean critical velocity is slightly greater than ifgH.

488. Citrini, Dulio, 1950, Sull'efficacia di uno sfioratore laterale nelle manovre 

di arresto complete [On the effectiveness of a side weir in the unsteady 

motion following full rejection of load]: Energia Elettrica [Italy], v. 27, 

no. 2, 1950 p. 77-80. Reprinted as Memorie e Studi, Inst. di Idraulica 

e Costruzione Idrauliche, Milano, no. 78, 1950.

The validity of a formula for computing the attenuation of a positive 

wave in a canal, the result of the operation of a lateral spillway, is proved 

in the limits defined in the paper, and on the basis of the recent experiments. 

This formula was developed in an earlier study (Citrini, 1949). A graph 

is developed from the same formula, which is of immediate practical value 

for special operations producing complete stoppage of flow in the canal. 

[Author's summary translated.]
140 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

489. Finzi, Bruno, 1950, Caratteristiche dei sistemi differenziali e propagazione 

ondosa [The characteristics of differential systems and wave propagation]: Energia Elettrica [Italy], v. 22, no. 4, p. 189-195.

This theoretical study contains : an analysis of wavefronts that considers 

the function of wavefront movement and develops celerity

a discussion of the method of characteristics; a discussion of the kinematic 

and dynamic conditions relating to discontinuities; analysis of the propagation of sound in liquids; and analysis of the propagation of tidal waves, 

in which is developed the wave celerity formula C=-\/g(H+h).

490. Gentiljni, Bruno, 1950, L'azione di uno sfioratore laterale sull'onda positive 

assendente in un canale (risultati di nuove experienze) [The effect of 

a side weir on a positive ascending wave in a canal (the results of the 

new experiments)]: Energia Elettrica [Italy], v. 27, no. 1, p. 1-10. 

Reprinted in Memorie e Studi, Istituto di Idraulica e Contruzione 

Idrauliche, Milano, no. 79, 1950.

The effect of a lateral longitudinal spillway on a translation wave (positive 

upstream) is studied experimentally, in the case when the wave is created 

by sudden stoppage of flow in a canal which has this lateral spillway. 

Conditions of the experiments are described and results given for the 

spillway effects on wave attenuation. The lateral spillway produces an 

attenuation which corresponds to that computed by the theoretical approach 

(Drioli, 1937). No difference in wave attenuation is noted, whether the 

spillway is on one side of the canal or whether there are two spillways, one 

on each side, having a total length equal to that of the single lateral spillway.

491. Gilcrest, B. R., 1950, Flood routing: Engineering Hydraulics (H. 

Rouse, ed.), New York, John Wiley and Sons, Chap. 10, p. 635-710.

The chapter is divided in four parts:

1. Introduction, which treats the general flow-routing problems and their 

classification;

2. Mathematics of flood routing, which treats the differential equations for 

unsteady flow, rate of travel of flood waves, wave attenuation, discharge in unsteady flow, two approximation methods based on neglect 

of the momentum equation, and the third approximation method, 

based on two differential equations;

3. Routing of floods through reservoirs, which treats the reservoir-storage 

characteristics, the use of storage factors (or working values W+ %Q&t 

or W/&t-\-Q/2); the use of slide rules for routing, the various factors 

affecting reservoir storage, the mechanisms for routing; and

4. Routing of floods through open channels which treats the stage-discharge-storage relations, attenuation of flood waves, application of 

coefficients in Muskingum method, the analysis of wedge storage, 

routing through junctions, complete methods of solution, solution by 

the method of characteristics, and the mechanical and electrical 

devices for flood routing.

492. Guerrero, E. C., 1950, Analisis del paso de avenidas por vasos reguladores 

[The passage of floods through control reservoirs] : Ingeniera Hidraulica 

en Mexico, v. 4, no. 4, p. 21-28.
BIBLIOGRAPHY 141

Graphical and numerical flood-routing methods based on the storage 

equation and currently in use are described and applied. Particular 

attention is given to the Goodridge method.

493. Keulegan, G. H., 1950, Wave motion: Engineering Hydraulics (H. Rouse,

ed.), New York, John Wiley and Sons, Chap. 11, p. 711-768. 

The chapter contains 7 parts: (1) preliminary considerations, which 

treats the classification of waves and method of analysis; (2) shallow-water 

waves, which treats translation waves, solitary waves, negative waves, and 

boundary resistance; (3) deep-water waves, which treats waves of finite 

amplitudes, configuration of waves at maximum height, and effect of internal 

resistance; (4) oscillatory waves in shallow water; (5) transformation of 

waves, which discusses shallow-water waves in channels of variable cross 

section, deep-water waves over sloping beds, and wave pressure on vertical 

walls; (6) open-channel surges, which shows surge characteristics, propagation of a discontinuous surge front, intermittent surges on steep slopes, 

and propagation of surges on a dry bed; and (7) internal waves due to the 

variation of density, the propagation of interfacial waves and their stability, 

and density currents.

494. Kritskiy, S. N., and Menkel', M. F., 1950 [Russian], Gidrologicheskie 

osnovy rechnoy gidrotekhniki [Hydrologic basis of river hydraulics 

engineering]: Akad. Nauk [U.S.S.R.].

495. Seemann, D., 1950, Die Kriegsbeschaedigungen der Edertalsperrmauer, 

die Wiederherstellungsarbeiten und die angestellten Untersuchungen 

ueber die Standfestigkeit der Mauer [War damage to the Eder Dam; 

repair work and testing the stability of the dam]: Die Wasserwirtschaft 

[Germany], year 41, p. 1-7 and p. 49-55.

Stage hydrographs of the flood wave, created by the destruction of the 

dam, are given for the dam site and for six other sites 51 to 362 kilometers 

downstream.

496. Sretenskiy, L. N., 1950, Volny [Waves]: In "Mekhanika v SSSR za tridtsat 

let (1917-47)", Gosudarstvennoe Izdatel'stvo Tekhniko-teoreticheskoy 

Literatury [Moscow, Leningrad], p. 279-299 (Hydrodynamics).

The contributions by the scientists of U.S.S.R. in the field of wave theory 

are reviewed from the standpoint of the mathematical approaches employed. 

Included are discussions of the results of studies of waves of finite amplitude, 

infinitely small waves (two dimensional problems, and three dimensional 

problems), and the theory of tidal waves. The references included cover a 

total of 80 Russian theoretical studies in the field of wave theory.

497. Supino, G., 1950, Sur 1'amortissement des intumescences dans les canaux 

[On the attenuation of steep waves in channels]: Rev. Ge"n. 1'Hydraulique [Paris], no. 57, p. 144-147.

It is shown by analysis that it is possible for a wave progressing downstream in rapid flow, V2jgH>2.25, to increase in height. This theory, 

also held by Masse", could be confirmed only by experience. By use of 

Lamoen's method of linearization, only attenuation of the wave could be 

shown. This study recommends the author's first method of linearization.

498. Supino, Giulio, 1950, La propagazione delle onde nei canali [Wave propagation

in channels]: Energia Elettrica [Italy], v. 22, no. 4, p. 196-205. 

728-245 64 10
142 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

This paper studies the wave propagation in channels with free surface 

flow using the two hydraulic equations (continuity and dynamic). It is 

shown: (1) that the different forms of dynamic equation are essentially 

equivalent for the study of linearized waves; (2) that it is possible to integrate the nonlinear equation in some cases; (3) that the inconvenience 

stressed by Bonvicini (1934) can be eliminated; this is for the wave which 

stops attenuating for the slopes 2.6 times the critical slope; (4) that some 

results can be obtained very rapidly by the theory of characteristics, and 

some limitations are put forward for the use of this theory. 

[Author's summary translated.]

499. Wicker, C. F., and Rosenzweig, O., 1950, Theories on tidal hydraulics; 

Comm. Tidal Hydraulics, Rep. 1, p. 101-125.

500. Chow, Ven Te, 1951, A practical procedure of flood routing: Civil Engineering and Public Works Rev. [Great Britain], v. 46, no. 542, p. 586- 

588. Reprinted as: A practical procedure for flood routing, III. 111. 

Univ. Civil Eng. Studies, Hydr. Eng. Ser., no. 1, 1951. 

The basic storage equation is used in the form

2 TTi/A T- Ql + Pj + P, = 2 W,!& T+ Q2.

The historical data of floods in a reach are used for the construction of 

characteristics curves. The AT unit is used to that 2/AT=l, and curves 

W+Q and W Q are used. An example is given. Instead of using loop 

curves for storage of river reaches (for rising and falling stages), a mean 

W-\-Q f(W) curve is used, and a graphical procedure is given for computing the mean curve. Five related factors or conditions for use of the 

above procedure are discussed: selection of reaches, routing period, storage 

function, flow less than critical, and local inflows.

501. Daily, J. W., and Stephan, S. C., 1951, Characteristics of the solitary wave:

Am. Soc. Civil Engineers Trans. v. 118, p. 575-587. 

The solitary wave is analyzed with reference to its wave classification and 

discussed in terms of its three principal characteristics of celerity, form, and 

velocity of liquid particles. The energy loss due to friction is also discussed. 

Results of earlier studies of these properties are reviewed. Experiments 

made at the Massachusetts Institute of Technology concerning these 

properties are discussed in comparison with the theoretical studies, showing 

that the experiments produced a value for celerity that is somewhat smaller 

than that obtained by theoretical formulas and produced an expression for wave 

form that is very close to the Boussinesq sech-function. The study further 

concludes that theories for motion and displacement of liquid particles have 

been confirmed only qualitatively, and that 90 percent of wave energy is 

contained within the limits y/h >0.46.

502. Drouhin, Mallet, and Pacquant, 1951, Contribution a I'e'tude des debits 

de crue et du dimensionnement des eVacuateurs [Contribution of 

the study of flood discharge and the design of spillways]: Internat. 

Comm. Large Dams, 4th Cong., New Delhi, 1951, Trans. v, 2, p. 461- 

578.

For the determination of spillway characteristics, a parameter r = Pmai/ 

Qmax is introduced in the formula. The storage function is used, and a 

method of computing spillway length is given. If, however, Qmai is imposed, the maximum water stage at the dam can be determined by the formula developed. Graphical methods for flood routing are given, using the
BIBLIOGRAPHY 143

storage factors TF+^QAi and W %ki in different combinations, based on 

the storage equation.

503. Dupouy, M., 1951, Ondes de crues et reglettes Bachet [Flood waves and

Bachet templates]: Houille Blanche [France], no. A, p. 230-231. 

Comparison is made of the transitory regimes of telegraphic signals with 

unsteady flow, which linearizes hydraulic regimes by taking small increments dQ of discharge as discontinuous changes and derives the relation 

between a hydrograph Q=f(t) and the so-called "percussional admittance" 

of signal transmission. The Bachet templates (1934) for flood forecasting 

can theoretically be based on this relationship. Rating-curve relation used 

for plotting curves of templates is introduced, and use of the templates is 

discussed. The author makes a theoretical approach from other angles to 

derive templates for flood routing.

504. Eckart, C., 1951, Surface waves on water of variable depth: Scripps Ins.

Oceanography, Lecture Notes, Fall Semester 1950-51, Ref. 51.12,

Wave Rept. 100.

Starting from the linearized equations of hydrodynamics, it is possible 

to give a logically consistent treatment of the propagation of waves in 

water of variable depth. This theory does not, in all respects, conform 

to observation; it was the intention to exhibit the assumptions that underlie 

the theory, so that later investigators might alter them with a view to 

improving the theory. The later parts of Section 7 show in detail how 

the theory fails when the water depth becomes very small. 

[From author's preface.]

505. Frank, J., 1951, Betrachtungen ueber den Ausfluss beim Bruch von 

Stauwaenden [Considerations on the outflow from dam breaches]: 

Schweizer. Bauzeitung [Zurich], no. 29, p. 401-406.

Outflow hydrographs resulting from dam breaches are studied by a 

simple analytical approach. Six assumptions are made: friction along 

the reservoir and singular resistances are negligible; the breach is rectangular; the crest is horizontal and perpendicular to flow; the reservoir has 

a constant mean depth, and horizontal bottom; width of the water surface 

is either very great or infinite; the water in the reservoir is at rest before 

the breach occurs and there is no tailwater effect. Starting from De 

Saint-Venant and Ritter equations, new equations are developed for the 

maximum discharge. The Eder Dam breach is used to evaluate the 

maximum discharge by means of the developed formulas.

506. Freeman, J. C., Jr., 1951, The solution of nonlinear meteorological 

problems by the method of characteristics: Am. Meteorol. Soc. 

Compendium Meteorol. p. 421-433.

507. Harkness, F. B., 1951, Harkness flood router; Device for graphically 

representing flood conditions; specification of construction and 

operation: U.S. Patent Office, Patent File no. 2, 550, 692. 

A mechanical flood-routing device is described that is based on the 

continuity (storage) equation. Starting from the inflow discharge hydrograph, and considering the storage-out-flow conditions, the flood-routing 

device traces directly the outflow discharge hydrograph.

508. Hayami, Sho., 1951, On the propagation of flood waves: Kyoto Univ. 

[Japan], Disaster Prevention Research Inst., Bull. 1, p. 1-16.
144 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Introducing the effect of longitudinal diffusion, caused by the mixing 

into the equation of continuity and assuming the mean flow, taken over 

a suitable range, to be steady and uniform, the differential equation of 

flood waves was derived. It is an equation of diffusion containing a term 

of advection. As the equation is nonlinear, an approximate method of 

solution was discussed and solutions were obtained under several conditions. 

They explain well the properties of flood waves. The approximate 

equation of flood waves is linear, and a flood of any form is, therefore, 

supposed to be composed of many elementary flood waves of simple 

character: unit graph, or unit flood. A method of computing the unit 

graph was described and some numerical examples were shown. In the 

last, some of the results of observations made on an artificial unit flood in 

the Yedo River were compared with the theoretical computations. Their 

agreement is excellent. 

[From author's synopsis.]

509. Hayashi, Taizo, 1951, Mathematical Theory of Flood Waves: Japanese

Natl. Gong. Appl. Mech., 1st, Proc., p. 431-436.

This paper deals with the mathematical theory of flood waves in open 

channels of uniform rectangular cross section. The basic equations used 

are the equation of motion for gradually varied unsteady flow and the 

equation of continuity. Solution is carried out by a method of successive 

approximations with respect to a small parameter <r, which is defined by

where the numerator is the vertical acceleration of the water level of the 

peak of a flood wave at a cross section, say at the origin, g=the intensity 

of gravity and /So = slope of channel. By the second approximation, the 

law of diminution of flood waves is given, where the factor of a diminution 

is expressed as a function of a Froude number, the width of the channel, 

and the small parameter a. The law is compared with available data 

ofjfield experiments. 

[Author's abstract.]

510. Hayashi, Taizo, 1951, Mathematical study of the motion of intumescences 

in open channels of uniform slope: Japan Soc. Civil Engineers Trans. 

[Tokyo], no. 11, p. 1-11.

This paper is the first of a series dealing with the motion of translation 

waves in open channels. The case treated is that of an arbitrary intumescence with a small height in a uniform channel with a rectangular crqss 

section containing water originally moving with a uniform velocity. Approximate formulas are derived which give the deformation and motion 

of the intumescence. The case of the intumescence of the form of a rectangular type is also dealt with and for that the form of the rigorous 

solution is shown. Special emphasis is laid on the dispersive property of 

intumescences. A method of illustration of the dispersion is presented 

and both the propagation speed and the dispersion of intumescences of 

any arbitrary shape are illustrated. The results are all derived with the 

method of operational calculus on the plane of the complex variables. 

[Author's English synopsis.]

511. Iwagaki, Yuichi, 1951, Theory of flow on road surface: Kyoto Univ. 

[Japan] Eng. Fac. Mem., v. 13, no. 3, p. 139-147.
BIBLIOGRAPHY 145

By solving the momentum equation of a thin sheet flow on road surface 

numerically with the condition of continuity obtained under the condition 

that rain falls on road uniformly, water depth and mean velocity of thin 

sheet flow and also frictional velocity related with soil erosion of road 

surface are computed, and then the effects of camber shape and longitudinal 

slope of road surface on its drainage and stabilization are discussed. 

[Author's abstract.]

512. Levin, L6on, 1951, Etude experimental du r6gime transitoire engendre1 par 

la rupture d'un barrage [The experimental study of the transient 

regime created by a dam breach]: Acad. sci. [Paris] Comptes rendus, 

v. 233, p. 646-648.

Shape of the wave travelling downstream immediately subsequent to 

breach of the dam is studied experimentally in a small canal 0.115 meter 

wide by 2.0 meters long. The study is particularly concerned with effects 

of canal shape, original depth of water below dam, and friction. It is 

considered that the first two factors are the most important and that the 

influence of friction can be neglected. The wave shapes are classified in 

four categories.

513. Levin, L6on, 1951, Eksperimentalna hidraulika [Experimental hydraulics], 

Part 4, Unsteady flow in canals and rivers: Belgrade [Yugoslavia], 

p. 161-216.

Analysis is made of waves in channels that develop by progressing, and 

classical theories of surges and gradually varied waves are discussed. The 

methods of finite differences and characteristics are given. One chapter 

is devoted to a discussion of the author's graphical method, outlined in 

detail in his later papers (1952). Special attention is given to the similitude 

between models and prototypes for the experimental studies of unsteady 

flow. Various model techniques are discussed in terms of the experiment 

used by the author and others.

514. Nekrassov, A. I., 1951, Tochnaya teoriya voln ustanovivshegosya vida na

poverkhnosti tyazheloy zhidkosti [The exact theory of steady waves

on the surface of a heavy fluid]: Izdat. Akad. Nauk [Moscow], p. 1-94.

The first chapter deals with surface waves in water of infinite depth.

The second chapter treats mathematically the solitary wave of shallow

water, starting from the theory and formulas for deep-water surface waves.

The paper concludes with treatment of a class of nonlinear integral

equations.

515. Potapov, M. V., 1951, Sochineniya [Works]; Book 3 on runoff regulation: 

Sel'khozgiz, Moscow, v. 3, p. 1-471.

In discussing various treatments of unsteady flow, the author states that 

the use of finite-differences methods, taking into account the inertia terms, 

is a very tedious procedure. This method requires detailed topographic 

data of the channel, and 8 to 10 man-days of work, under the supervision 

of a highly specialized engineer, for the computation and plotting of the 

graphs depicting the travel of one wave.

Use of mass-curves of hydrographs is discussed (p. 81-99). The integration of storage equation for spillway discharges is given in approximate 

form for tabular procedure, as well as for more accurate tabular procedure. 

The author's graphical method, using curves W=f(Q/2), is given as in a 

simple graphical method for equal At (also in nomographical form). Ap-
146 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

proximate analytical methods are discussed and compared (p. 179-205)f 

with reference to the principles of unsteady flow in channels (p. 205-215).. 

The effect of lateral inflow and backcurves is analyzed (p. 215-218).

516. Ransford, G. D., 1951, A contribution to the first-order theory of translation waves: Houille Blanche [France], no. 5, p. 758-763. 

The Masse1 equation for the dynamic partial differential equation of 

unsteady flow is used ip the form where Q, H and V are replaced by 

Qo+q, H0 -{-h, and Vo+t>, where g, h, and v are quantities that differ very 

little for unsteady flow, and where reduced variables are used in dimensionless form. The purpose of this equation was to obtain a solution of 

the type fp (x)et>', by integration. The author considers that this type of 

solution is incomplete, because singular solutions also exist in the form 

/pCoOte"'. This is proved in the paper. By comparing results of an example, the theory of waves of the first order needs to be modified along, 

the lines indicated above, and according to the Boussinesq formula, the 

change in shape of the front of wave is due to the term: tevt.

517. Rich, R. G., 1951, Hydraulic transients, Chapter 8, Surges in power canals

and tidal harmonics: New York, McGraw-Hill Book Co., p. 214-242. 

Johnson's treatment (1922) of the rejection and demand surges in power 

canals is given, with formulas for changes in levels and for celerity. The 

effects on surges of channel-bottom slope and channel friction in the power 

canal is discussed. The numerical computation of the surge in power 

canal by the trial-and-error method is given. The movement of tidal 

waves along the channel is presented in general and for the special case 

of the channel having a lock at one end. The method of Brown (1932) 

for tidal wave movement in the channel and an example of computation 

is given. The infinite-series solutions of tidal harmonics is given at the 

end of the chapter.

518. Schoenfeld, J. C., 1951, Propagation of tides and similar waves: 'S-Gravenhage, Ministerie Van Verkeer en Waterstaad Staatsdrukkerij-en 

Uitgeverijbedrijf [Netherlands], p. 1-232.

This Ph. D. thesis is one of most extensive works on the unsteady tidal 

and water movement in channels. Theoretical and practical aspects of 

integration problem are given, especially by method of characteristics. 

Chapter 1, Introduction, gives or derives the basic theories and differential 

equations and discusses them. Chapter 2 gives the mathematical and 

physical approach and describes the theory and procedure of the method 

of characteristics for integration of the partial differential equation. 

Chapter 3 analyzes the use of characteristics in wave propagation and 

studies various methods for wave development in propagation as well a» 

reflection of waves. Chapter 4 deals with harmonic analysis of wave 

motions. Chapter 5 considers the relation of characteristics and harmonic conceptions for linear effects of wave propagation. Chapter 6 

treats the same relation, but for nonlinear effects. This theoretical treatment constitutes Part I, which deals mostly with mathematical and 

physical aspects of waves. In Part II the empirical approach is emphasized. Chapter 7 deals with laboratory experiments of tidal motion in 

channels. Chapter 8 discusses computations of wave propagation by the 

characteristics method. Chapter 9 deals with harmonic computationmethods, and Chapter 10 with continuity-dynamic computation methods 

(iterative and difference methods); Chapter 11 is a comparative study and
BIBLIOGRAPHY 147

discussion of computation methods. The appendices contain: Chapter 12, 

some practical integration problems and their solution by characteristics; 

Chapter 13, some considerations on the accuracy of the experiments and 

the computations; and Chapter 14, some auxiliary analyses in connection 

with the computation.

519. Schoenfeld, J. C., 1951, Distortion of long waves; equilibrium and stability: 

Internat. Assoc. Sci. Hydrology, U.G.G.I., v. 4, p. 140-157.

Two causes of distortion of long waves are considered. The differences 

in celerities (dispersion) tend to steepen the wavefront and to flatten the 

waveback; the friction resistance tends to flatten both the wavefront and 

waveback (attenuation), or to steepen them (amplification). The wavefront of the long wave either develops into a surge, or a decline (tidal 

waves, flood waves, waves in power and navigation canals). The conditions of wave equilibrium are studied. If V<^C at the toe of a wave, 

C=2-\fgH/[l + (2h/K) (dk/dH)], in which k = coefficient in Chezy formula, 

and the wave front is stable. When F>C, the waveback is stable. The 

waves having constant profile are determined analytically: waves traveling 

down a river, waves traveling up a river, and waves in a canal having a 

horizontal bottom. The stability of different waves is studied, and criteria 

for stability are developed.

520. Swain, F. E., 1951, Determination of flows in interconrected estuarine 

channels produced by the combined effects of tidal fluctuations and 

gravity flows: Am. Geophys. Union Trans., v. 32, p. 653-672.

A method is presented for determining the variable flows produced in 

an estuarine channel system by gravity flows from the uplands and the 

effects of tidal heights and phases throughout the system. The procedure 

makes use of a theory developed to establish the magnitude of "surge" or 

"bore" waves produced in canals by a disturbance of flow. The tidal 

height variations are represented as a series of incremental changes in 

water surface height which in turn produce small bore waves. The effect 

of channel friction, length, and cross section are taken into account. A 

suggested procedure for solving a given problem is presented together 

with an illustrative example which gives the results of a study made for 

the delta region of the Sacramerto and San Joaquin Rivers in central 

California. 

[Author's abstract.]

521. Tison, L. J., 1951, Le propagation des ondes de crue dans une region 

lacustre [Flood-wave propagation in a lacustrine region]: Centre 

d'e"tudes de recherches et d'essais scientifiques des constructions du 

ge*nie civil et d'hydraulique fluviale Bull. (Centr. Rech. Const. Bull.), 

[Liege, Belgium], v. 5, p. 371-386.

Starting from De Marchi's study (1948) and Fantoli's basic approach to 

the integration of storage equation, by using a sine function for the inflow 

hydrograph in the reservoir, the formulas for the peak discharge attenuation 

and time lag between the peaks of inflow and outflow discharge hydrographs 

are applied to the lakes of Sotkamo River Basin.

522. Baines, W. D., 1952, Water surface elevations and tidal discharges in the 

Fraser River Estuary, Jan. 23, 24, 1952: Canada, Natl. Research 

Council rept. MH-32, Apr. 8.
148 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Report on a survey made during the low-water period of the Fraser 

River and presentation of results obtained for water surface elevations, 

local low and high water, and total discharge. Method of computing tidal 

discharge is included.

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

June 1955.]

523. Biesel, F., 1952, Study of wave propagation in water of gradually varying 

depth: U.S. Natl. Bur. Standards, fire. 521, p. 243-253.

The present paper is a theoretical study of periodic waves progressing 

in water of variable depth. Although the theory can be extended to threedimensional motion, it will only consider the two-dimensional case which 

can be briefly described as that of waves advancing in a straight channel 

with rectangular cross section and variable depth. This problem has 

already been studied by various authors, but the solutions obtained have 

always been restricted to flat sea bottoms of constant slope and can be 

worked out in practice only for a limited number of singular values of the 

bottom slope (cf. the works of Miche, Stoker, Loewy, etc.) or for small 

bottom slopes and very small relative depth (Lowell, Miche). The solution obtained by the present theory is subject to the restriction that the 

slope of the bottom is so small that its square is negligible. Bottom curvature and higher derivatives of depth with respect to distance along the wave 

course are also neglected. Provided these assumptions are satisfied, the 

bottom may have any shape whatsoever.

In the first part of this paper, investigations are limited to first order 

theories, that is, the squares of the velocities due to the presence of waves are 

neglected (this is the case for all previous theories of waves on sloping bottoms). In the second part, a second-order correction is introduced, and 

it is shown by numerical examples that this correction may be important 

in some respects. 

[Author's abstract.]

524. Blackmore, W. E., 1952, Line diagrams for problems of storage: Water 

Power [London], Aug.-Sept., p. 299-303, and p. 355-357.

A graphical method based on the storage equation and called z-line 

(zig-zag line) is described. The procedure is based on a constant reservoir 

area A (small spillway fluctuations). The time intervals (T) are constant, 

and a slope line dH/dQ=T/2A is used. The graphical construction uses 

Qi from the rating curve Q=f(H), then P=(Pi + P2)/2 from inflow hydrograph P=f(f) and the slope line dHfdQ= T/2A to obtain the point Q2, and 

so forth. A similar procedure, starting from an estuary tidal curve, is used 

to determine levels in tidal basins. Other hydraulic problems are solved by 

this method.

525. Clausnitzer, R., 1952, Der Schwellbetrieb in Flusskraftwerken, seine Moeglichkeiten und Vorteile [The possibilities and advantages in the use of 

canals as pools with wave movement in the operation of water power 

plants]: Die Wasserwirtschaft [Germany], year 42, no. 11, p. 342-347.

By considering canal storage to be effectively the san e as pool storage 

in the case of powerplant operation that creates wave movement along the 

canals, experimental data are here given that describe wave movement in 

the Iller powerplant canal, particularly the change of water levels during 

the operation of the plant
BIBLIOGRAPHY 149

526. Craya, A., 1952, The criterion for the possibility of roll-wave formation: 

U.S. Natl. Bur. Standards Circ. 521, p. 141-151.

The theoretical interpretation of roll waves, proceeding from the equations 

of De Saint-Venant, can follow two primary lines of approach: The first 

considers the stability of flow in a channel, i.e., the damping or amplification 

of a perturbance of small initial amplitude; the second proceeds from a 

systematic analysis of the quasisteady regime introduced by H. A. Thomas.

The object of the study is to analyze, compare, and clarify these two 

points of view for a prismatic channel of arbitary cross section and for 

resistance laws of general form. Both lead to the same criterion for the 

condition of roll-wave formation, which coincides for particular conditions 

with the formula obtained by Vedernikov from a somewhat different 

stability derivation. 

[Author's abstract.]

527. Cuenod, Michel, and Gardel, Andre, 1952, fitude des ondes de translation 

de faible amplitude dans les cas des canaux d'amene'e des usines hydroelectriques [Study of translation waves of small amplitude in the canals 

of water powerplants]: Bull. Tech. de la Suisse Eomande [Switzerland], 

year 78, no. 7, p. 93-102.

Using operational calculus for study of the movement of waves of small 

amplitude in channels, the authors have developed a general analytical 

solution that taken into account the resistance factor. The boundary 

conditions at upstream and downstream ends of powerplant canals are 

given, and the particular solution in operational form is developed. For a 

sudden change occurring at the end of a canal, the inverse transformation of 

Laplace is used for functions obtained. The wave elevation in a canal is 

thus obtained as the function of position and time. Canals of both infinite 

and finite length are treated, reflection waves are taken into account in the 

last case and friction resistance is or is not considered. The study is related 

to the stability of powerplant regulation.

528. Daily, J. W., and Stephan, S. C., Jr., 1952, The solitary wave; its celerity, 

profile, internal velocities and amplitude attenuation: Mass. Inst. 

Technology Hydrodynamics Lab. Tech. Rept. 8, p. 1-56. 

History, definition, and previous investigations of the solitary wave are 

first considered. A table summarizes results of previous investigations in 

terms of the celerity, profile, fluid element velocity, and attenuation, as 

found by each investigation. Experimental setups are described and 

discussed. Included is a summarizing report and discussion of experimental 

data with respect to the following topics: (1) Profile the deviations from 

the Boussinesq theoretical profiles lead to the conclusion that the amplitude 

and depths experimentally examined vary, plotting around the theoretical 

profile. (2) Celerity the equations of celerity presented by Rayleigh 

and Boussinesq are close to experimental results for all values of h/H0) in 

which h is the amplitude. (3) Attenuation due to friction both the 

amplitude and wave volume decrease (as is concluded by thorough study 

performed both analytically and experimentally. (4) Particle path and 

velocity the theoretical values of fluid-element velocity are greater for 

velocities at the center of the wave than those values found by experiments, 

and the difference increases with an increase of h/H0 . The maximum 

amplitude value is A/H0 =0.6, approximately. The friction effects on wave 

attenuation are studied experimentally for two values of roughness, but 

the results are not generalized.
150 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

529. Davies, T. V., 1952, Symmetrical, finite amplitude gravity waves: U.S.

Natl. Bur. Standards Circ. 521, p. 55-60.

The problem of the steady propagation of gravity waves of finite amplitude in a tank with a horizontal base is a nonlinear problem of a special 

type. The stream function satisfies a linear partial differential equation 

and the nonlinearity enters through the free surface boundary condition. 

The classical theory converts the nonlinear boundary condition into a 

linear condition using the perturbation method and is therefore restricted 

to infinitesimal amplitude waves. It is possible, however, to deal with the 

finite amplitude problems by replacing the exact nonlinear condition by a 

new nonlinear condition that is a close approximation and for which the 

exact solution may be determined. This enables one to discuss all waves 

up to and including the limiting case when breaking occurs at the crest. 

The Levi-Civita approach is used for this purpose, and the application of 

the method to the solution of the solitary wave problem is demonstrated. 

[From author's abstract.]

530. Davis, C. V., 1952, Handbook of Applied Hydraulics: 2d ed., New York,

McGraw-Hill Book Co.

The Muskingum flood-routing method along channels (by P. Z. Kirpich, 

p. 1197-1198), and reservoir flood routing as graphical procedure by use of 

storage equation ( by K. E. Sorensen, p. 1235-1236) are given.

531. Dressier, R. F., 1952, Hydraulic resistance effect upon the dam-break 

functions: U.S. Nat. Bur. Standards Jour. Research v. 49, no. 3, 

p. 217-225.

The dam-break solution, a known centered simple-wave when resistance is neglected, is studied with the Chezy resistance formula added to 

the nonlinear shallow-water equations. Resistance transforms the wavefront from a characteristic curve into an envelope of characteristics. The 

flow near the tip differs from the other parts, due to a distinct boundarylayer type of region adjacent to the wavefront envelope. Then a perturbation leads to a system of partial differential equations with variable 

coefficients. Initial conditions are derived for the singularity at the origin. 

By studying its characteristic equations, this system is solved explicitly 

for the correction functions. Except at the tip, resistance raises the water 

surface and lowers velocities. These functions, no longer simple waves, 

possess concurrent straight characteristics lines that map into another 

set of the same type. The critical flow locus moves downstream, faster 

for more resistance and discharge rates are reduced. The method fails 

in the tip layer because the asymptotic expansions for the first derivatives 

lose validity there. Estimates are made indirectly for the wavefront velocity by observing where the boundary-layer becomes predominant. 

[Author's abstract.]

532. Dressier, R. F., 1952, Stability of uniform flow and roll-wave formation:

U.S. Natl. Bur. Standards Cir. 521, p. 237-241.

A uniform open channel flow down an inclined aqueduct sometimes 

changes into a complicated periodic motion with a progressing-wave 

profile, the profile moving downstream faster than the water particles. 

This phenomenon called "roll-waves" is investigated by three completely 

different methods:

(1) The construction of actual roll-wave solutions is effected by joining 

certain special continuous solutions together by means of moving dis-
BIBLIOGRAPHY 151

continuities or bores. Of the three shock conditions of mass, momentum, 

and energy, the energy condition requires the use of a resistance term 

whose magnitude varies inversely with the hydraulic radius, as well as 

directly with the square of the velocity; otherwise, the equations are not 

sufficiently non-linear to produce roll-wave solutions. These waves cannot exist if resistance is absent, or if it exceeds a certain critical value. 

A two-parameter family of solutions is obtained; in all cases, the flow is 

Bubcritical at the peaks and supercritical in the valleys.

(2) By a perturbation procedure using maximum curvature of the 

profile as expansion parameter, continuous periodic progressing waves 

are found in terms of elliptic functions. These waves possess greater 

profile curvature than the discontinuous solutions of the non-linear shallow 

water theory. The perturbation also yields a two-parameter family of 

waves; these also are supercritical at peaks and subcritical in the valleys.

(3) Using a stability analysis which observes the location of roots in 

the right or left half-plane, the same critical condition on resistance causing 

instability is obtained as in Section 2. This method likewise shows that 

a resistance term varying only with velocity will produce uniform flows 

which are always stable, hence no roll-waves. A more general analysis 

indicates that instability may occur whenever resistance varies directly 

with any power of velocity, and inversely with any nonzero power of the 

hydraulic radius. 

[Author's abstract.]

533. Eckart, C., 1952, The propagation of gravity waves from deep to shallow

water: U.S. Natl. Bur. Standards Circ. 521, p. 165-173. 

An approximate wave equation has been derived for the propagation 

of gravity waves on water of any depth. Although the precise conditions 

for the validity of the approximations have not been established, they 

appear to be justifiable in a wide variety of cases. 

[From author's abstract.]

534. Hensen, W., 1952, Ueber die Fortschrittsgeschwindigkeit der Tidewelle 

in einem Flusse [On the celerity of tidal wave in a river]: Hann. 

Versuchsanstalt fuer Grundbau u. Wasserbau Mitt. [Franzius-Inst. 

der Hoch-Schule], Hanover [Germany], v. 2, p. 89-106.

535. Hunt, J. N., 1952, Viscous damping of waves over an inclined bed in a 

channel of finite width: Houille Blanche [France], no. 6, p. 836-842. 

[French, English],

The calculation of energy dissipation in the layers of water near the bed 

and near the walls is made for wave propagation in a canal of finite and 

uniform width, when the wavelength is large in comparison with the water 

depth. An expression for the damping of wave amplitude is deduced, 

which confirms results already obtained for the case in which the canal 

width is infinite. The example given is for the case in which the canal bed 

has a slight slope.

536. Kalinin, G. P., 1952, Osnovy methodiki kratkosrochnykh prognozov 

vodnogo rezhima [The basis of the methods for short-term forecast 

of runoff regime]: Tsentralniy Inst. Prognozov Trudy (Central Inst. 

for Forecase Proc.) [U.S.S.R.], no. 28 (55).

537. King, R. E., 1952, Stage predictions for flood control operations: Am. Soc. 

Civil Engineers Trans., v. 117, p. 690-698.
152 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Extensive study of the problem of flood-stage forecasting for the lower 

Mississippi River has led to the following conclusions:

1. Reliable forecasts for the lower Mississippi River cannot be made by 

using any rigid formula or procedure.

2. Procedures which do not evaluate the effects of current channel conditions, tributary or diversion flows, storage, and precipitation will not 

give reliable forecasts.

3. Procedures dependent on the use of volumetric principles will give better 

results than those which are based entirely on stage heights.

4. Stages on the lower Mississippi River can be forecast with sufficient 

accuracy to warrant the use of these forecasts as the basis for planning 

and executing flood-fighting activities.

[Author's conclusions.]

538. Kotik, J., 1952, Existence and uniqueness theorems for linear water waves: 

Mass. Inst. Technology.

539. Kritskiy, S. N., and Menkel', M. P., 1952, [Russian], Vodokhozyaystvennye 

raschety Regulirovanie polovodiy i povodkov [Water resources 

computations; Regulation of floods and floodwaves]: Gidrometeoizdat 

[Leningrad], p. 225-261.

The schematic or constructed discharge hydrographs are discussed for 

cases in which the return period for both peak and volume of flood are 

known. In other cases a binomial curve for discharge hydrograph is 

assumed (using the Euler integral of second order). The continuity and 

dynamic equations of unsteady flow are discussed, especially with reference 

to the hydraulic characteristics and relations for a given channel reach, 

the initial and the boundary conditions, and various methods of integrating the two partial differential equations. Particular attention is given 

to the approximate methods of flood routing. Wave celerity is studied 

as the sum of water velocity and the expression A(dV/dA), and the celerity 

formula is developed as

For three types of channel cross section it is shown that: rectangular, 

C=1.67 V; triangular, (7=1.33 V; and for the section composed of two 

quadratic parabolas (with the sharp point at the junction), C= 1.22V. For 

simple schematic hydrographs the decrease of peak by storage is related 

to the flood volume and storage volume.

540. Lacombe, H., 1952, Quelques aspects du probleme des marges fluviales et 

de la formation du mascaret [Some aspects of the problem of tides in 

rivers and of the bore formation]: C.O.E.C. Bull., p. 228-251.

541. Levin, Le"on, 1952, Elements de depart du mouvement nonpermanent a 

la suite d'une rupture de barrage [Basic principles of the start of unsteady flow following the breach of a dam]: Electrotech., Hydraulique, 

Radio [France], no. 172.

This paper describes the experimental results already given in author's 

other papers and in his book.

542. Levin, L&m, 1952, Razvoy valova od rushenya visokikh brana [Evolution 

of waves created by the breach of large dams]: Yugoslav Natl. Comm. 

Large Dams, 2d Mtg., 1952, Trans., p. 104-118. (Translated into
BIBLIOGRAPHY 153

English by Military Hydrology and Hydraulics Division, Army Map

Service, Washington, D.C., Jan. 1955.)

This is an experimental study dealing with the evaluation and shape of 

waves created by dam breach. A graphical method of flood routing is 

used, based on a modified method of characteristics which employs the 

form coefficient e=AfHB, making it possible to use straight-line characteristics for any cross section form. The comparison of wave-shape detail, 

permitted by the graphical method developed by the author, is given special 

attention with respect to use in the model study of dam breach involving 

Medyuvrshye dam and reservoir.

543. Levin, Le"on, 1952, Mouvement nonpermanent sur les cours d'eau & la 

suite de rupture de barrage [Unsteady flow in rivers created by dam 

breach]: Rev. Ge"n. 1'Hydraulique [Paris], no. 72, p. 297-315. 

The method for computing unsteady flow by De Saint-Venant's equation 

should not be used where the wave heights are great in comparison to the 

initial river depths (Favre limits the application to the ratio h/H l/S). 

Wave profiles after dam breach are given for different tailwater depths, 

resulting from both experimental and analytical study. A new graphical 

method based on the characteristics is developed which is suitable for 

large ratios of h/H (12 15, of which are experimentally verified). The 

method is applicable to any cross section characterized by the form coefficient e=A/BH. A graph is given for computation of height coefficient 

and celerity coefficient for a given value of e. Integration is by straight 

lines only. An example of computation is shown for the dam and reservoir 

Medjuvrshye (Yugoslavia) and the experimental results of the model 

study are compared with results by the graphical method; good agreement 

is obtained.

544. Lin, Pin-Nam, 1952, Numerical analysis of continuous unsteady flow in 

open channels: Am. Geophys. Union Trans., v. 33, p. 226-234 (discussion by J. C. Schoenfeld and Pin-Nam Lin, v. 34, p. 792-795, 

1953).

In this paper is presented a simplified method of solving Massau's 

equations of characteristics for unsteady flow having a continuous surface 

profile. This method eliminates much of the trial-and-error process 

formerly required.

Although Massau's rigorous analysis of unsteady flow in open channels 

was published some 50 years ago, its practical application has not been 

very extensive; for, among other reasons, the application of Massau's 

method is handicapped by the tiring labor involved in the trial method of 

solving his equations of characteristics. By developing a scheme of using a 

constant time interval for each step of computation, it is shown that the 

trial-and-error process can be avoided, except in the laying out of characteristics in the x-t plane. The latter operation being guided either by the 

characteristics laid out in a previous step or by the initial condition, is 

generally easy to perform. Since the method has the distinct feature of 

using a constant value of At throughout a step of computation, it may be 

called the method of constant A£. 

[From author's abstract and introduction.]

545. Nougaro, J., 1952, Recherches exp6rimentales sur les intumescences dans 

les canaux de"couverts [Experimental research on steep waves in open 

channels]: Acad. Sci. [Paris] Comptes rendus, v. 235, no. 15, p. 788-791.
154 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

The celerities of steep waves in canals are first determined by the graphical 

method of characteristics, and are then verified both by experiments in 

nature (as exemplified by operations of the Camon and Valentine powerplants on the Garonne River) and by experiments employing a model canal. 

Results agree well.

546. Ogiyevskiy, A. V., 1952, Gidrologiya Sushi Sel'khozgiz [U.S.S.R.].

Methods are given for computing the maximum reservoir outflow to be 

accounted for in emptying reservoir storage preparatory to storing flood 

inflows. The inflows are represented by schematic hydrographs, either 

triangular or trapezoidal in shape. Employing the schematic form of 

inflow hydrograph, the routing equation is shown to be:

where Qw=peak of outflow, Pw=peak of inflow, TP,=storage between the 

level of spillway crest and the maximum attained level, TTr=preflood 

reduced storage, Wf= flood- wave volume, a=t/T (t= duration of flood 

maximum of inflow hydrograph having trapezoidal shape, and T=the 

duration of base of inflow hydrograph; l=Q and a=0 for triangular shape).

547. Paynter, H. M., 1952, Methods and results from M.I.T. studies in unsteady flow: Soc. Civil Engineers [Boston] Jour., v. 39, no. 2, p. 

120-165.

Part 2 treats the graphical solutions of transient problems by slopeline 

method, in which integration of function y=f(x) by differences ^y=Yz(fi 

+/2) Az, where /i and /2 are the values of / for the beginning and the end of 

Az. It is shown that

Ay=*yi+Ay2= (Az/2)/i+ (Ax/2)/,.

The values A^ and Ay2 are determined from slopes ± Yz&x and values /t 

and /2 . The simple graphical integration is shown. The graphical integration of function dy/dx=f(x) g(y) is shown in a similar way. This 

technique is applied to flood routing through reservoirs using the equation 

dW/dt= P(f) Q(t), or using the inflow hydrograph P(t) and the storagedischarge relationship Q(W}. The integration of Af gives Q(f) directly. 

The second-order differential equation is also solved graphically and applied 

to surge tanks. Part 3 treats flood routing by admittance methods using 

the electrical network theory, where impedance and admittance functions 

characterize the behavior of transients and oscillations in a linear system. 

By using Duhamel's superposition theorem, the admittance has analogy 

in river channel storage and travel time, and the input in the inflow and 

the output in the outflow being analagous. The relation is given for coefficient routing method (Muskingum), where the admittance is analogous to 

the function of time. The routing procedure is outlined in the case of a 

given admittance. The admittance for floods is given as the sum of time 

delay and storage effect.

548. Pohle, F. V., 1952, Motion of water due to breaking of a dam and related 

problems: U.S. Nat. Bur. Standards Circ. 521, p. 47-53.

The two-dimensional hydro dynamical equations of motion, expressed in 

Lagrangian representation, are used to investigate the motion of an ideal 

fluid. This representation has the far-reaching advantage in problems 

with time-dependent free boundaries that the independent space variables
BIBLIOGRAPHY 155

(a, 6) are the initial coordinates of the particles: the region occupied by 

(a, 6) is thus a fixed region independent of t.

The displacements X(a, 6; t), Y(a, 6; f) and the pressure P(a, b; t) are 

expanded in powers of the time t. Equating to zero the coefficients of 

powers of t leads to a systematic procedure for the determination of the 

successive terms in the three expansions. Each term is a solution of the 

Poisson equation, in which the nonhomogeneous terms are known functions of previously determined quantities. In all cases considered, higher 

approximations require determination of a Green's function, which is the 

same for all approximations.

The method is applied to the initial stages of the breaking of a dam. 

[Author's abstract.]

549. Roseau, M., 1952, Reflexion des ondes dans un canal de profondeur variable 

[Wave reflection in a canal of variable depth]: Acad. sci. [Paris] Comptes rendus, v. 234, no. 3, p. 297-299.

An analytical study of wave reflection that computes the reflection 

coefficient of waves and studies the relationship of amplitude of stationary 

waves at the extremes of a canal of indefinite length to constant and 

infinite depth.

550. Russell, R. C. H., and Macmillan, D. H., 1952, Waves and tides: Hutchison's Sci. Tech. Pubs. [London], p. 1-348.

Contents of Part I, Waves, by R. C. H. Russell: Characteristics of 

ocean waves. Ideal waves. The generation of waves by wind. Waves 

near the shore. Reflection, diffraction, refraction and wave-induced currents. Movement of material by the sea. The effect of wave action on 

structures. Wave measuring. Part II, Tides, By D. H. Macmillan. 

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

February 1954.]

551. Sokolovskiy, D. L., 1952, Rechnoy stok, metody issledovanniy i raschetov 

[River runoff, methods of investigation and computation]: Gidromet, 

Izd. [Leningrad], chap. 7, 8, p. 226-348.

Among other aspects of flood investigation, the book deals with the 

approximation of hydrographs by analytical expressions, travel time of 

flood waves, and flood forecasting.

552. Tsinger, V. N., 1952, Uchet srabotki vodokhranilishch pri raschetakh 

snizheniya maksimal'nykh raskhodov [Taking into account the 

lowering of reservoir levels in the computation of the decrease of 

maximum outflow discharge]: Gidrotekhnika i Melioratsiya [U.S.S.R.], 

no. 11.

Formulas are developed for computation of maximum discharges over 

spillways for the case involving lowering of the reservoir levels prior to 

flood inflow, aild assuming inflow hydrographs of simple triangular and 

trapezoidal shapes.

653. Uchida, Shigeo, 1952, On the analysis of flood wave in a reservoir by the 

method of characteristics: Japanese Natl. Cong. for Appl. Mech., 

2d, Proc., p. 271-276, (published by Science Council of Japan [Tokyo], 

May 1953).
156 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

A generalized form of characteristic equations for unsteady flow in open 

channels having varying cross sections is introduced for the purpose of analyzing the reservoir effects. The method uses a simplified form for cross sections 

that employs the hydrograph plane for construction of overflow conditions. 

An example is given of the calculation of flood-wave movement through an 

idealized reservoir. Computed hydrographs of uniform-channel routing 

are compared with observed hydrographs; fair agreement is shown. 

Deformation of flood waves in reservoirs is discussed and examples are given.

554. Ursell, F., 1952, Discrete and continuous spectra in the theory of gravity

waves: U.S. Natl. Bur. Standards Circ. 521, p. 1-5. 

It is shown how systems with discrete and continuous spectra differ in 

their physical behavior. It has generally been assumed that waves in an 

infinite canal have a continuous spectrum; examples are here given to 

show that the spectrum is mixed on a sloping beach and when there are 

internal boundaries. The corresponding oscillations are three-dimensional. 

The relation of the theory of spectra to the problem of uniqueness is 

discussed. 

[Author's abstract.]

555. Zheleznyak, I. A., 1952, Opredelyeniye raschetnikh raskhodov vodosbrosnykh sooruzheniy s uchetom akkumulatsii [The determination of 

discharges over spillways when the storage is taken into consideration]: 

Gidrotekhnika i Melioratsiya [U.S.S.R.], no. 1.

The method of computing maximum outflow discharge is given for the 

case involving preflood reduction of reservoir storage and a spillway crest 

that is equipped with gates.

556. Apte", A., 1953, Sur une solution approche"e du proble"me de 1'onde solitaire 

[On an approximate solution of the problem of solitary wave]: Acad. 

sci. [Paris] Comptes rendus, v. 236, no. 26, p. 2477-2479. 

This paper presents an analytical approach to the problem of solitary 

wave having crest at angular point.

557. Baines, W. D., 1953, Survey of tidal effects of flow in the Fraser River 

Estuary, June 10 and 11, 1952: Canada Natl. Research Council rept. 

MH-40, Apr. 10.

Description of survey made on two typical days during 1952 freshet. 

Presentation of measured water surface elevations and computed discharges 

at 25 stations in the estuary. Curves of local low and high tides are 

included and are compared to results obtained in a similar survey on 

January 23 and 24, 1952. Description of the dynamics of the flow as 

indicated by results so far obtained.

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

June 1955.]

558. Becker, E., 1953, Analogic zwischen Wasserschwall and Verdichtungsstoss

[Analogy between surge and shock wave of compressible fluid]:

Ingenieur Archiv [Germany], v. 21, no. 1, p. 42-54.

When applied for gas, dynamic approximations, the analogy between

surge and shock waves does not hold because the thermodynamic conditions

applicable to the two fluids are different. The fact that water surge

is not an exact discontinuity accounts for this difference. The conditions

required for analogy are discussed. Surge formulas are given for small

and great surges occurring in very shallow water as well as in deep water.
BIBLIOGRAPHY 157

559. Bergeron, Louis, 1953, Me'thode graphique pour le calcul des ondes de 

translation [Graphical method for the computation of translatory 

waves]: Soc. Franchise des Mecaniciens [France] Bull. 7.

560. Burns, J. C., 1953, Long waves in running water: Philos. Soc. [Cambridge] 

Proc., v. 49, p. 695-706.

Classical shallow-water theory for the propagation of long waves in 

running water is modified by the inclusion of the effects of the vorticity 

present in the main stream as the result of the action of viscosity. When 

this vorticity is assumed constant, a non-linear theory can be used, but for 

more general velocity distributions in the main stream it is necessary to 

linearize the problem.

In the linearized theory, a general equation is obtained connecting the 

wave velocity with the velocity in the undisturbed stream and this is solved 

in several special cases. It is shown generally that the wave velocity 

relative to the mean flow is always greater than the value given by the 

classical theory. The wave velocity relative to the bottom of the stream 

has two values, one less than the minimum stream velocity and the other 

greater than the maximum stream velocity. 

[Author's abstract.]

561. DeMarchi, Giulio, 1953, Action of side weirs and tilting gates on translation 

waves in canals: Minn. Internat. Hydraulics Conf. (Joint Mtg. 

Internat. Assoc. Hydraulic Research and Hydraulics Div. of Am. Soc. 

Civil Engineers), Minneapolis, 1953, Proc., p. 537-545. 

This is an experimental study of the surge-reduction effect of a side weir, 

placed along a canal, and of frontal tilting gages in the forebay of a powerplant. The wave heights reduced by side spillways are computed by the 

Citrini method (1949, 1950) and compared with the experimental results, 

thus verifying the consistency of the Citrini theory. The tilting gate can 

be effective if operated either before or at the same time as the turbine 

operations.

562. DeMarchi, Giulio, 1953, Azione de uno sfioratore a ventola sull'onda positiva provocata dall'arresto delle macchine nel canale adduttore di un

impianto idroelettrico [On the action of a side weir, controlled by a

tilting gate, upon the positive translation wave generated in the canal

of a powerplant by the sudden closing of the turbines]: Energia

Elettrica [Milano], v. 30, no. 12, p. 12-20. Reprinted as Istituto di

Idraulica e Costurzioni Idrauliche [Milano] Memorie e studi, no. 110,

1953: [On model tests for the Tornavento Power Plant].

A report is given concerning a first series of experiments carried out at the

Hydraulic Laboratory of the Politenico of Milano on a model of the canal

forebay and power-station of the Tornavento hydroelectric plant. The aim

of the experiments was to determine the characteristics of the ascending

positive wave generated in the canal by the closing of the turbines and to

study the action exerted upon such wave by the sudden opening of the

tilting-gates of the spillway in the forebay. The experimental layout and

the instrumentation are described and the results obtained with a special

water-level recorder placed in the canal are presented and discussed. The

conclusion is that, in the case studied, the effect of the gates is substantially

one of stabilizing the reduction of the wave height due to the widening of

the canal at the location of the forebay.

[Author's abstract.]

728-245 64 11
158 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

563. Dmitriyev, A. A., 1953, Prokhozhdeniye dlinnykh voln cherez prepyatstviya 

pri chastichnom otrazhenii [Passage of long waves over obstacles in 

the case of a partial reflection with the given reflection coefficients]: 

Akad. Nauk [U.S.S.R.] Doklady, v. 15, no. 4, p. 509-512. 

This is an analytical study of wave reflection and wave transmission in 

the case of one concentrated resistance and of two resistances, located L 

and 2L distances from the coordinate origin. It is concluded that, under 

actual conditions, the dissipation of energy must decrease with the coefficient of wave transmission.

564. Dressier, R. F., and Pohle, F. V., 1953, Resistance effects on hydraulic 

instability: New York, Interscience Publ. Inc., New York Univ., 

Inst. Math. Sci., Commun. Pure Appl. Math., v. 6, no. 1, p. 93-96. 

The Vedernikov and Craya results of condition for instability are obtained by other methods and assumptions. The developed criterion for 

Chezy's formula

is for Manning resistance formula F2 = 9/4 gH cos 0. Any flow governed 

by a resistance varying with any power of velocity, but independent of 

depth, must remain stable. The next result is that both effects of depth 

and velocity dependence for resistance, operating simultaneously, are 

needed to produce instability.

565. Frank, Josef, 1953, Die Hydraulik der Durchlaufspeicherung [The hydraulics of using pools in chain for flow regulations]: Der Bauingenieur 

[Germany], year 28, no. 2, p. 39-44 and 87-91.

Analysis is made of the movement of positive and negative surge waves 

in reservoirs used by powerplants as regulating pools for their daily operations. The celerity formulas, in approximately exact form, serve for 

routing of surges. Wave crossings, new waves occurring in opposite directions, and wave reflection at the end of pools are studied; new water depths 

are determined. Secondary phenomena are discussed. A graphical procedure is given, as is a detailed numerical example.

566. Glover, R. E., Herbert, D. J., and Daum, C. R., 1953, Electrical analogies 

and electronic computers (a symposium); Application to a hydraulic 

problem: Am. Soc. Civil Engineers Trans., v. 118, p. 1010-1027. 

This paper discusses the general conditions of the problem of flow distribution in a network of estuarine channels to which an analog computer 

model was applied. After developing the analog requirements, the model 

is described, with emphasis on the electronic circuit that provides the 

required square-law resistance. The equations correlating electrical and 

hydraulic quantities are developed from the basic electrical and hydraulic 

relationships. Finally, the methods by which the required boundary conditions were duplicated are discussed. 

[Author's abstract.]

567. Hayashi, Taizo, 1953, Mathematical theory and experiment of flood waves: 

Japan Soc. Civil Engineers Trans. [Tokyo], no. 18, p. 13-26.

This paper describes mathematical theory and laboratory experiment of 

the flood wave in uniform open channel. The theory is developed by the 

method of successive approximations with respect to a small parameter
BIBLIOGRAPHY 159

ff which is defined by o-=V F(Q)/g/So, where #(0) denotes the vertical 

accelerative rate of rising of the stage of the flood wave at the moment of 

passage of its crest at a cross section of the channel, g the acceleration of 

gravity and So slope of the channel. The laboratory experiment was 

conducted primarily for the verification of the theory. Flood wave with 

any desired value was generated in flowing water with any desired value 

of Froude number and specifically, the rate of attenuation and the speed 

of propagation, both of which were related to the crest of the wave, were 

measured and are compared with those given by the theory. 

[Author's abstract.]

568. Holsters, H., 1953, Le calcul du movement non-permanent dans les rivieres 

par la m6thode dite des "lignes d'influence" [The computation of 

unsteady motion in river channels by the method called "influence 

lines"]: Houille Blanche [France], no. 4, p. 495-509. 

This paper on nonpermanent flow in rivers contains: fundamental 

equations, ignoring secondary terms of inertia and their solution by methods 

of finite differences (exact methods); a simplified method called "influence 

lines," (as characteristics in simplified form are called); repercussions 

of the arbitrary choice of mean depth and of velocity on the calculated 

evolution, first, with reference to an elementary wave profile, and second 

with reference to tidal curves (choice of too large a value for depth regularizes the profile without modifying the general form, while choice of a too small a 

value for depth leads to rapidly diverging oscillations). Examples of calculation are given. Influence of secondary terms of inertia is studied. 

The paper discusses the errors introduced by simplifications in the form 

of influence lines (introduced first by author's paper in 1947).

569. Hopkins, C. D., Jr., 1953, The celerity of very large flood waves in the 

Arkansas Basin, with discussion: Am. Geophys. Union Trans., v. 34, 

p. 594-596 (discussions: v. 35, p. 513-518).

Numerous streams in the Arkansas River basin provide the data shown 

as crest discharge-travel time and crest stage-travel time relationships, 

supporting the theory that large flood waves travel much more rapidly 

than flood waves of moderate size.

570. Ichiye, Takashi, 1953, On the abnormal high waters in rivers: Oceanographical Mag., v. 5, no. 1, p. 45-60.

The rivers and estuaries in the city of Osaka often overflow, chiefly due 

to the meteorological high waters caused by typhoons, and cause serious 

damage to structures and other facilities. The properties of flood waves 

of meteorological tides progressing up the rivers and some discussion about 

them are presented. Data collected during three typhoons: Muroto on 

Sept. 21, 1934; Jane on Sept. 3, 1950; and Ruth on Oct. 15, 1951, were 

were used as a basis for the theoretical computations.

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

May 1959.]

571. Keulegan, G. H., 1953, Characteristics of internal solitary waves: U.S.

Natl. Bur. Standards Jour. Research, v. 51, no. 3, p. 133-140. 

This is an application of the method of approximations initiated by 

Boussinesq to the disturbances of the interface points for waves of permanent form and the interval solitary wave. The system considered 

is a layer of liquid on another layer of greater density, the liquids of the
160 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

layers being initially at rest and of constant total depth. The form of 

the wave is established. The dependence of wave velocity on wave 

height, on density differences, and on layer thickness is determined. 

[Author's abstract.]

572. Kohler, M. A., 1953, Electrical analogies and electronic computers (a 

symposium); Application to stream-flow routing: Am. Soc. Civil Engineers Trans., v. 118, p. 1028-1045.

Late in 1948, the Weather Bureau, United States Department of 

Commerce, developed an electronic device for stream-flow routing that 

has proved to be highly effective in the preparation of river stage forecasts. 

Although originally designed for routing flows from point to point along a 

stream, subsequent studies indicate that the equipment is equally applicable to the direct routing of effective rainfall (runoff) over relatively 

large basins. This application of the flow analog and the conditions under 

which the original circuit fails to provide a satisfactory reproduction of 

the outflow hydrograph are discussed in this paper. The basis for the 

circuit employed in the analog and the method of operating the equipment 

are also discussed briefly. 

[Author's synopsis.]

573. Kozeny, Josef, 1953, Hydraulik, F/III, Die nichtstationaere Bewegung: 

[Hydraulics, Chapter F/III, The unsteady flow]: Vienna, SpringerVerlag, p. 251-267.

Starting from two partial differential equations for unsteady flow, the 

wave heights are expressed as

y=A sin (WL)[z-(F0 +V<7Ho")*]+exp. (-^0V0t/2H0),

in which L = wave length, ^d = resistance coefficient, and where V 0 and HO 

are values for a steady flow. The case is analyzed when the curvature of 

surface profile is taken into consideration, and especially its influence 

on the wave celerity. The energy of wave is given. Surges (bore and 

depression) are treated in a way similar to that of De Saint-Venant (1871) 

and of Boess; examples are given for the computation of height, celerity, 

and velocity of waves. The changes in positive and negative surges as 

they progress is studied, and formulas for height, celerity, and time of the 

wave break are given. The change of cross section and its effect on surges 

is analyzed, and examples of lock operations are given.

574. Labaye, G., and Duranton, R., 1953, Utilisation des galeries souterraines 

en reservoirs d'eclusees [Use of tunnels as storage reservoirs]: Houille 

Blanche [France], no. B, p. 735-746.

The use of tunnels as storage reservoirs is studied through the determination of initial surface (after the tunnel ceases to be under pressure). Study 

is made of unsteady transitory movement until critical regime is attained. 

Results are generalized.

575. Mason, M., 1953, Surface water wave theories: Am. Soc. Civil Engineers

Trans., v. 118, p. 546-574.

This paper discusses the characteristics of oscillatory surface waves and

summarizes the development of pertinent theory. The more important

equations that characterize wave formation and movement are presented.

The method by which waves are believed to be generated is described,

and a theory of the growth of waves is formulated. Several charts
BIBLIOGRAPHY 161

provide convenient means of determining wave characteristics and wave 

effects from a knowledge of the limiting factors. Other subjects, including 

wave refraction, diffration, and reflection, are also briefly treated. 

[From author's abstract.]

576. Messerle, H. K., 1953, Electronic high speed simulation of hydraulic problems: Inst. Engineers [Australia] Jour., v. 25, no. 3, p. 35-41.

An electronic high speed simulator has been developed at the University 

of Melbourne for the solution of the pondage equation relating the waterflow and the water level in a reservoir. The instrument has been used for 

the investigation of general flood-routing problems and is described in this 

paper.

The solution of more complex hydraulic problems is investigated also, 

and it is shown that the principles used for the simulation of the pondage 

equation can be extended quite readily to the analysis of general hydraulic 

systems.

The experimental results obtained on the pondage simulator have been 

sufficiently accurate for practical purposes and the setting up and solution 

of any particular pondage problem for a number of boundary conditions 

can be carried out within about 10 minutes. The effects of varying the 

circuit parameters can be investigated rapidly once a problem has been 

set up. It is also important to one reservoir, one inflow and one outflow 

function, which can be solved with fixed dimensional computed ranges 

simply by applying the ideal of analogue models. Several independent 

inflows arising from various tributaries can be allowed for by providing 

additional inflow generators. 

[From author's conclusions.]

577. Milyukov, P. I., 1953, O sootnoshenii mezhdu skorostyu dobeganiya 

ravnogo raskhoda i skorostyu techeniya [On the relation between the 

celerity of the uniform discharge and the velocity of flow]: Meteorologiya i Gidrologiya [U.S.S.R.], no. 8.

578. Nougaro, Jean, 1953, Theoretical and experimental studies of the propagation of translation waves in open channels: Minn. Internat. Hydraulics Convention (Joint Mg. Internat. Assoc. Hydraulic Research 

and Hydraulics Div. of Am. Soc. Civil Engineers), 1953, Proc., p. 

555-559.

A graphical method for predicting the rise in the water level upon passage 

of a translation wave is presented.

The classical characteristics lines of Bergeron are replaced by characteristic curves which eliminate most of the hypothesis postulated by Bergeron.

A system of curves on tracing paper moving on the proper coordinates 

particularly facilitates the graphical construction. The author points out 

how to take into account the head losses, the slope of the channel bed, and 

special conditions resulting from the section change of a bifurcation, a 

large basin, etc.

The author points out that many experiments justify the application of 

the formula C=V0# ± V and indicates some application of his graphical 

method.

The results obtained are in accordance with those experimentally 

obtained. 

[Author's abstract.]
162 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

579. Nougaro, Jean, 1953, Recherches expe'rimentales sur les intumescences 

dans les canaux decouverts [Experimental research on the intumescences in open channels]: Soc. Francaise des Mecaniciens Bull. [France], 

no. 9, 3d trimestre, p. 23-35.

580. Nougaro, Jean, 1953, Etude theorique et experimental de la propagation 

des intumescences dans les canaux decouverts [Theoretical and experimental study of the propagation of translation waves in open channels]: Pubs. Sci. et Techs, du Ministere de 1'Air [France], no. 284, p. 

1-155.

581. Pleshkov, Ya. F., 1953, Vliyanie predpavodochnoy srabotki vodokhranilishcha na raschetnyy maksimal'nyy raskhod [The influence of preflood decrease of storage in reservoirs on the maximum outflow 

discharge to be computed]: Gidrotekhnicheskoe Stroitel'stvo 

[U.S.S.R.], no. 9.

The influence of the preflood decrease of storage on maximum outflow 

discharge over a free spillway is studied; it is concluded that in some cases 

the peak outflow may not be reduced, or may even be increased.

582. Riggs, H. C., 1953, A method of forecasting low flow streams: Am. GeophysUnion Trans., v. 34, p. 427-434.

The normal base-flow depletion curve is used, and examples are shown, to 

provide a method of forecasting low flow.

583. Schoenfeld, J. C., 1953, Inleiding tot de methode der Karakteristieken 

[Introduction to the method of characteristics]: Rijkswaterstaat commun. [Den Haag, Netherlands], Nota C.S.D 53-2.

584. Serre, F., 1953, Contribution a 1'etude des 6coulements permanents et variables dans les canaux [Contribution to the study of permanent and 

nonpermanent flows in channels]: Houille Blanche [France], p. 374-388 

and p. 830-872 (unsteady flow, p. 857-872).

Study of nonpermanent flow showing derivation of partial differential 

equations of flow. The advantages which can be gained in the study of 

swells, waves, and breaking jumps under variable flow are emphasized. 

The assumption of uniform distribution of velocities is made. The differential equations for the waves having constant celerity are derived, and 

are similar to differential equations derived for steady flow. In a canal 

having initial constant velocity V and depth H, there can be an infinity of 

waves of finite amplitude (h), each having a celerity CVand the length 2L. 

The solitary wave is that where celerity, Cr ±^/gH = ±^!g(H-\-h), is 

maximum.

585. Shuleykin, V. V., 1953, Razrushenie voln pod deystviem melkovodya 

[Breaking of waves under the effect of shallow water]: Akad. Nauk 

[U.S.S.R.] Doklady, v. 43, no. 3, p. 463-466.

The conditions are studied under which a wave would break on a sloping 

breach of shallow water. The profile of a wave before breaking is analytically determined and found to compare well with the wave profile 

photographed in the model basin. The stability and instability of wave 

profiles are discussed at the end of paper.

586. Sorenson, K. E., 1953, Graphical solution of hydraulic problems: Am. Soc. 

Civil Engineers Trans., v. 118, p. 61-77.
BIBLIOGRAPHY 163

Graphical solution of first-order differential equations and application of 

that procedure to various hydraulic problems, including flood routing 

through reservoirs, is shown.

587. Stoker, J. J. 1953, Numerical solution of flood prediction and river regulalation problems; Derivation of basic theory and formulation of numerical methods of attack: New York Univ., Inst., Math. Sci. Rep. IMM200.

The report treats rederivation of two partial differential equations for 

unsteady flow and discusses the numerical methods to be used in integrating 

these equations for solutions to practical problems. Thomas's approach 

(1937), using the finite-differences method, which is laborious to apply by 

standard computation methods, is improved by the use of digital computers 

and by development of appropriate numerical procedures. The experience 

and calculating equipment of gas dynamics (with the same equations applicable for unsteady flow of water in channels) assist in problem solutions. 

The numerical procedures of solving equations by the method of characteristics is discarded, and a method of finite differences of integration in 

(x, f) -plane, covered by a rectangular net of points, is derived, wherein the 

derivatives in differential equations are replaced by difference quotients. 

The relation of Ax and At is determined by the use of characteristics, so that 

the value At is limited as soon as Ax is selected.

588. Stoker, J. J., 1953, Unsteady waves on a running stream: New York Univ.,

Comm. Pure and Appl. Math., v. 6, p. 471-481.

The problem is investigated mathematically in terms of the classical 

hydrodynamic theory of irrotational flow in an incompressible perfect fluid. 

The dimensionless parameter gH/V2 is used for the discussion of the problem. The solution of the unsteady wave problem is developed, and the 

state of steadiness is discussed as the limit of unsteady motion for infinite 

time.

589. Supino, Giulio, 1953, La propagazione delle onde di translazione [The 

propagation of translation waves]: Energia Elettrica [Italy], v. 30, 

no. 4, p. 201-210.

The solution of partial differential equations, already known for rectangular channels, is extended to apply to any prismatic channel. Assuming that the changes of discharge and area are small, in the comparison of 

unsteady flow with steady flow, and assuming neglect of friction, the 

Boussinesq celerity formula is obtained and extended for any prismatic 

channel. Translation waves are studied with respect to nonuniform 

steady flow. The linearized solution for the wave of the regime is obtained 

(being the solution that gives the same celerity for the water depth and 

for the discharge). The wave attenuation is determined, and an example 

of its use is given. The waves having different celerities for depths and 

discharges are also studied. The initial conditions are discussed.

590. Supino, Giulio, 1953, Propagazione di piccole onde su un moto-base permanente [The propagation of small waves on a basic nonuniform

unsteady flow]: Energia Elettrica [Italy], v. 30, no. 6, p. 333-340.

This paper is a continuation of a previous paper by the author (Energia

Elettrica, 1953). The movement of small waves on a nonuniform steady

flow is studied, especially in the case of supercritical flow. Starting from

the general partial differential equation for discharge and area changes due
164 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

to unsteady flow and for short travel time of wave, the method of characteristics is used for derivation of the equation along a "band of characteristic." From this band of characteristic the celerity of wavefront is 

determined for changing velocity, area, and width for a nonuniform steady 

flow. This method is not appropriately applied to long waves, for which 

the earlier paper (1953) gives the best solution. In the case of small 

waves, the solution for coefficients in partial differential equation is developed for known stage and discharge hydrographs at the initial point 

(as is the case for supercritical flow).

591. Supino, Giulio, 1953, La riduzione delle piene del Reno da Cento alia 

Bastia [The attenuation of River Reno flood from Cento to Bastia]: 

Giornale del Genio Civile [Italy], year 91, no. 1-2, p. 34-40. 

The continuity equation (P Q = dW/df) is integrated by use of a sine 

function for P. The results developed for Q are applied to the Reno River 

in a reach of 57.4 kilometers that is practically without tributaries. The 

results are compared with the field data. It is concluded that the flood 

of 1951 has the nature of a rare flood, not in terms of volume, but in terms 

of the type of flood wave.

592. Tsinger, V. N., 1953, Raschet snizheniya maksimal'nykh raskhodov 

vodokhranilisch s uchetom predpavodochnoy srabotki [The computation of the decrease of maximum outflow discharge from reservoirs 

when reservoir level is decreased prior to the flood]: Gidrotekhnicheskoe 

Stroitel'stvo [U.S.S.R.], no. 4.

Maximum discharge at the free spillway, when the inflow hydrograph is 

approximated by a triangle, is developed as Qm =(Wf WT W3)Pmj 

(Wf T/KWfWr), where $m =maximum outflow, Pm = maximum inflow 

discharge, WT emptied storage before the flood inflows and the spill of 

water begins; Ws=volume of stored water between the level of spillway 

and the maximum attained level; Wf flood volume; and K ti/T, ratio 

of the duration of flood peak to total duration of flood.

593. Ursell, F., 1953, The long-wave paradox in the theory of gravity waves:

Philos. Soc. [Cambridge] Proc., v. 49, p. 685-694.

The theory of long waves in shallow water under gravity employs two 

different approaches, which have given rise to a well known paradox remarked by Stokes, but hitherto not fully resolved. On the one hand it 

was shown by Airy (1845) that, if the pressure at any point in the fluid 

is equal to the hydrostatic head due to the column of water above it, then 

no wave form can be propagated without change in shallow water of uniform 

depth; on the other hand, it was shown by Rayleigh's theory (1876) of the 

solitary wave that this conclusion may be incorrect. In the present paper 

an attempt is made to elucidate the paradox. Waves of small amplitude 

h and large horizontal wavelength X (compared with the depth H of the 

water) are studied, and is shown that Airy's conclusions is valid if h\2/H3 

is large, whereas the solitary wave has h\2/H3 of order unity. Equations 

of motion are derived corresponding to large, moderate and small values 

of h\2/H3 ; these can be summarized in a single equation for the profile y(x,t):

^7/2 ($11 ^^ ^3 1$ fffi O^V 

 . . Oii TT (jf± I ~~ -4 

due to Boussinesq (3). It is also shown that the linearized theory of surface
BIBLIOGRAPHY 165

waves is valid only if h/\ and h\z/H3 are both small. Some remarks are 

made on the generation of the solitary wave, and on the breaking of waves 

on a shelving beach.

[Author's abstract.]

594. U.S. Army Corps of Engineers, Waterways Experiment Station, 1953, 

Study of hydrographs in triangular flume: U.S. Army Corps of Engineers, Waterway Expt. Sta., Interim rept. 1. folder 1, July 1953, p. 

4-6; folder 2, figs. 5-411; Interim rept. 2, folder 1, p. 1-6; folder 2, 

figs. 14-405.

Steep-front triangular waves are studied experimentally in a triangular 

flume, for the purpose of obtaining data to assist in the determination of 

coefficients and curves for routing floods with steep fronts. This information is needed for study of the effects on wave attenuation of channel 

slope, roughness, and discharge. The experiments are described, and percentages of discharge reduction along the flume are given. The experimental results are given in figures and tables.

595. U.S. Army Corps of Engineers, 1953, Routing of floods through river chan" 

nels: U.S. Army Corps of Engineers Manual, Sept. 1953, pt. 114i 

Chap. 8, p. 1-33, pis. 1-14.

Two basic types of approximate flood-routing methods are demonstrated: (1) "storage routing" methods, in which energy factors are largely 

neglected and only the effects of storage in the intervening reach and local 

inflows are considered in estimating changes in the flood wave as it passes 

through a reach of channel; and (2) methods of approximating flood-wave 

shape at the lower end of a reach by time displacement of values of average 

inflow into the reach. Of the storage-routing methods, the coefficient (or 

Muskingum) method is given principal attention; other methods are explained or compared in terms of the factors determining these routing 

coefficients.

596. Wells, L. W., 1953, Some remarks on shallow water theory: New York

Univ. Inst. Math, and Mech. Rept. IMM-NYU-198, p. 1-18. 

The report is a simplified analytical treatment of the approximate theory 

derived from the exact hydrodynamic theory of gravity waves of small 

surface amplitude, which makes the assumption that the length of wave is 

sufficiently large when compared with the depth of water. Results obtained are of general interest and application to shallow-water theory in 

treating the reflection of waves from an obstacle of finite length.

597. Zheleznyak, I. A., 1953, Opredelennie maksimarnykh raskhodov s uchetom 

reguliruyushchey emkosti vodokhranilishcha [The determination of 

maximum discharges by taking into account the regulating storage]: 

Gidrotekhnicheskoe Stroitel'stvo [U.S.S.R.], no 9, p. 21-25. 

In the case of spillways with gates on the crest, preflood reduction of 

reservoir storage does not reduce the peak of outflow discharge, when 

Wr<^Wn, where Wr=preflood reduction of storage and TFn the volume 

of inflow hydrograph from the beginning of inflow to the moment when the 

outflow is equal to the discharge of normal level (with all gates open).

598. Appleby, F. V., 1954, Runoff dynamics; A heat conduction analogue of 

storage flow in channel networks: Internat. Assoc. of Sci. Hydrology, 

Rome, 1954, Proc., no. 38, v. 3, p. 338-348.
166 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

The normal differential equation of storage flow is developed into a form 

analogous to that for linear heat flow, where storage corresponds with temperature. Inherent in the resulting equations are two parameters, the 

outflow velocity V and the storage transit factor K, which is a response 

time and mean time of concentration of the network. The product of these 

two factors is a characteristic length e, the product of this length and the 

surface width of flow at a control section measures the total water surface 

area in the system; or, if combined with area of flow at the outfall, the 

volume of storage. V2K is directly analogous to diff usivity in heat flow and 

a measure of the power of the system to eliminate its accumulated flood 

waters. Time factors in general are discussed, with formulae for the lag of 

peak flow and the regulation of the peak under unit runoff conditions. 

Examples are given using runoff data from large catchment areas and 

urban drainage networks. The development of a heat conduction analogue 

as a working instrument is described, as is its application to studies in 

general of the hydrograph of flow, and in particular the study of brief 

inter-periodic correlation between runoff and rainfall. 

[Author's abstract.]

599. Benjamin, T. B., and Lighthill, M. J., 1954, On cnoidal waves and bores:

Royal Soc. [London] Proc., v. 224, no. 1159, p. 448-460. 

From the theory of stationary wave trains a new presentation of cnoidal 

wave theory is given. The theory permits study of the development of the 

wave train behind a bore. Such a wave train, which is present in all but the 

very strong bores, is shown to be capable of absorbing almost all the energy 

that, according to classical theory, is liberated at the bore, although some 

minute residual dissipation of energy still appears to be necessary.

600. Chitale, S. V., 1954, Bores in tidal rivers with special reference to Hooghly:

Irrig. and Power [India], v. 11, no. 1, p. 110-120.

Includes discussions and presents equations pertaining to formation 

and propagation of the bore, variations in the bore height, and speed of 

the bore.

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

June 1955.]

601. Cocchi, G., 1954, Sulla propagazione delle onde in alveo a forte pendenza 

[On wave propagation in channels of steep slope]: Accad. Sci. dell'1st. 

Bologna [Italy].

602. Dressier, R. F., 1954, Comparison of theories and experiments for the 

hydraulic dam-break wave: Internat. Assoc. Sci. Pubs. no. 38, v. 3, 

p. 319-328.

Experimental data obtained at the U.S. Bureau of Standards are presented for trajectories of the forward wave front, the backward wave 

front, and for flow profiles, over long time intervals for three different 

channel bottoms which range from smooth to very rough Chezy coefficients. 

The roughness has been calibrated over a wide range of discharge rates 

in each case. The experiments are compared with previous data of Schoklitsch and Egiazarov; they are then compared with mathematical results 

obtained by Dressier and Whitham, based upon two different approaches. 

In one theory, the tin region is neglected; in the other, the tip is analyzed 

and the agreement with each other. The experimental data deviate in a 

consistent way from these theoretical results, indicating more generally 

that the Chezy resistance function, normally measured for steady flows,
BIBLIOGRAPHY 167

may be inadequate to describe a highly unsteady flow or the nature of 

turbulent resistence in a tip region. The experiments show a division 

of the flow into distinct regimes, depending initially upon laminar viscous 

effects only, then upon inviscid hydrostatic pressure effects only, and 

finally upon turbulent resistive action. These transitions occur over a 

very short time interval; quantitative data for this was obtained from 

high speed motion picture photography. This furnishes estimates on 

the time necessary for the formation of fully developed turbulence arising 

from the state of rest. Wave profiles immediately after gate opening are 

also compared with theoretical results of Pohle, based upon an analysis of 

velocity-potential flow in Lagrangian coordinates. 

[Author's abstract.]

603. Friedrichs, K. O., and Hyers, D. H., 1954, The existence of solitary waves:

New York Univ., Commun. on Pure and Appl. Math., v. 7, p. 517-550.

The authors prove the existance of stable solitary waves in a direct

mathematical manner, assuming that, aside from pressure forces, gravity

is the only force acting on the fluid.

604. Gherardelli, Luigi, 1954, Sul moto vario in canali prismatici attorno al 

regime critico [On the unsteady movement in prismatic canals around 

the critical regime]: Energia Elettrica [Italy], v. 31, no. 1, p. 34-38. 

Using the equations of characteristics and their properties, the propagation of waves is studied for the case when the water depth is close to the 

critical depth Hc. It is shown that the wave has oscillations as soon as 

the discharge and depth approach those of critical flow. With respect to 

rivers, critical flow is a zone rather than a determined depth or discharge. 

In this region the given function Fi(x Ct) is indeterminate, a casual 

function, having anomalies that may be of sizable amplitude. The physical 

aspects of phenomena taking place in rising and falling limbs of a hydrograph, when the depth passes the critical depth, are discussed, and the 

patterns of secondary oscillations are outlined.

605. Haws, E. T., 1954, Surges and waves in open channels: Water Power

[London], v. 6, no. 11, p. 419-422.

The surges occurring in powerplant channels are studied analytically. 

The rise in water level resulting from a positive upstream surge traveling on 

the downstream-flowing current is derived; particular attention is given 

to the level rise that occurs after the surge has passed. A similar analysis 

made for a variable channel section. The celerity for very low waves is 

derived as approximately that of the classical form C= F+V0#». 

Slow changes in flow and superelevation of water surface occurring at 

bends in the channel are discussed.

606. Hoffman, G. R., 1954, Tidal calculations applied to the Estuary of the 

River Great Ouse: Inst. Civil Engineers Proc., v. 3, no. 3, p. 809-829.

The paper describes calculations made to reproduce tidal levels observed 

during a flood of the River Great Ouse on 16 March 1937. It outlines 

briefly the method of calculation and gives an account of the various 

alternatives tried before agreement between observed and calculated levels 

was finally obtained. In the lower part of the estuary, where the tidal 

range is large compared with the mean depth at half tide, the losses at 

bridge piers, bends, and obstructions had to be separated from the boundary 

roughness loss which was represented by a Manning type equation. In
168 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

addition, it was found necessary to average the surveyed cross sections 

in a reach by a rather more elaborate method than that generally used. 

It is concluded that tidal calculations in relatively deep estuaries give 

reasonably accurate results and that in certain cases the method may 

compare favorably, both in time and cost, with model experiments. In 

shallow estuaries, however, the accuracy of predicted levels is reduced. 

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

June 1955.]

607. Isaacson, E., Stoker, J. J., and Troesch, B. A., 1954, Numerical solution of 

flood prediction and river regulation problems (Ohio-Mississippi 

floods), report II: New York Univ., Inst. Math. Sci. Kept. IMMNYU-205, p. 1-46.

Problems of flood prediction and regulation on the Ohio and Mississippi 

rivers are studied by means of the finite-differences method (Stoker, 1953) 

on the digital computer, UNIVAC. Flood problems are posed, and techniques for solution are formulated employing a computational model of 

the Ohio River. Description is given of the finite differences employed 

and of the mesh system used for Af and Az. Special attention is given 

to problems resulting from junction of the Ohio and Mississippi rivers. 

Expansion of solutions approximately of the first characteristics, and 

theoretical analyses of steady flows and of steady progressing waves are 

shown in two appendices.

608. Ishihara, T., Hayami, Sho., and Hayami, Shi., 1954, On the electronic 

analog computer for flood routing: Japan Acad. Sci. Proc., v. 30, 

no. 9, p. 891-895.

Starting from Hayami's theory (1952) of flood waves, in which the 

basic difference equation is approximately

where D = a constant determined by the hydraulic resistance and bottom 

slope of channel, and p= numerical constant defined by the various 

irregularities of river reach, the electronic analog computer is developed, 

in order to avoid the defect of principle that is inherent in the Muskingum 

method. The computer that has been developed to solve the above 

equation is described, and examples of its application to river flood routing 

are given. The estimates of D and p are discussed.

609. Iwagaki, Y., and Sueishi, T., 1954, Approximate method for calculation

of unsteady flow in open channels with lateral flow: Japanese Natl.

Cong. for Appl. Mech., 4th, Proc., rept. II/3, p. 235-240 (published

by Science Council of Japan [Tokyo], March 1955).

The authors propose an approximate method for calculation of unsteady

flow in open channels with lateral inflow, using the characteristic curves.

Hydrographs resulting from abrupt increase and decrease of rate of lateral

inflow are obtained by this method, and the calculated hydrographs are

compared with experimental results.

[Abstract in the report.]

610. Iwagaki, Y., and Sueishi, T., 1954, On the unsteady flow in open channels 

with uniform lateral inflow: Japan Soc. Civil Engineers Proc. [Tokyo], 

v. 39, no. 11, p. 575-583.
BIBLIOGRAPHY 169

611. Moklyak, V. I., 1954, Rozrakhunok neodnochasovogo dobichannya vodi pri

nerivnomirnomu raspodili shvidskosti v pototsi [Computation of

unequal travel times of water with nonuniform distribution of velocities

in channel]: Akad. Nauk [Ukrainian S.S.R.] Doklady, no. 4.

Because the velocity distribution in a cross section is not uniform, the

water passing a given cross section at a given moment will arrive at the

downstream section at different times. Special curves are derived for a

rectangular section for the purpose of computing travel time for different

parts of initial discharge having nearly the same velocity in a cross section.

612. Morikawa, G. K., 1954, On the theory of flood waves in rivers: New York

Univ., Inst. Math. Sci. Kept. IMM-NYU-210, p. 1-24. 

Using a coordinate system that moves downstream with C, the celerity 

of steady progressing wave, it is found that for the monoclinal wave 

(Hi, H2 ; Vi, F2) there is

Using the stretching transformation for the combined space and time 

variables, a natural way of obtaining appropriate asymptotic representations is derived. The best stretching transformation is given for flood 

routing.

613. Nougaro, Jean, 1954, Pribilizhno odredyivanye visine nestatsionarnikh 

talasa u otvorenom kanalu [An approximate computation of surge 

heights in an open canal]: First Mtg. Hydraulics, Belgrade, 1954, 

Proc., p. 73-78.

It is useful sometimes for a preliminary study to show the approximate 

value of the wave height in an open canal. For this purpose, equations 

are set up and then expressed in terms of graphs. The author is considering cases occurring most frequently in the operation of hydroelectric 

power plants, i.e.: (1) Sudden stopping of turbines, for the wave in headwater canal; (2) Sudden starting of the turbines, for the wave in headwater 

canal; (3) Linear starting of the turbines, for the tailwater canal, taking 

or not taking into account the influence of the mean velocity of the water 

flow on the celerity of wave propagation. Numerous experimental data 

show the feasibility of use of graphs, given in ftiis paper. 

[From author's abstract.]

614. Nougaro, Jean, 1954, Influeneia de una singularidad de un canal sobre la 

propagacion de intumescencias [Influence of a singularity of a canal 

on the propagation of intumescenses]: Facultad de Ingenieria [Montevideo] Bull., v. 5, no. 14, p. 349-390.

When the resistance to water movement is taken into account, then an 

approximate celerity formula for rectangular channel is derived as

±F

where F=mean velocity for depth H, H original depth in canal, ft=wave 

height (surge), n.=Strickler's roughness coefficient, R=hydraulic radius, 

and L=length of canal. For a singularity having the head loss h', the 

propagation celerity through the irregularity is shown by approximation 

to be

C=T/g[H-h'(H/h-h/H+2)] ± V.
170 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

The approximation is verified experimentally. It is also verified experimentally that a jump in bottom of AZ gives a change in h (wave height) 

by t\h= V2/\Z/( V2 -i-gH). Equations for a trapezoidal canal are also 

given.

615. Penati, Savio, 1954, Azione di uno sfioratore a ventola sull'onda positiva 

provocata dall'arresto delle macchine nel canale adduttore di un 

impiante idroelettrico [On the action of a side weir, controlled by a 

tilting gate, upon the positive translation wave generated in the canal 

of a power plant by the sudden closing of the turbines]: Energia 

Elettrica [Italy], v. 31, no. 10, p. 733-741. Reprinted as Istituto 

di Idraulica e Costruzioni Idrauliche [Milano] Memorie e studi, no. 

115, 1954 [On new model tests for the Tornavento Power Plant]. 

This study gives the second series of experiments made using a model 

of Tornavento powerplant. Description is given of the instruments and 

the procedure used in registering water levels when surges occur due to 

sudden stoppage of powerplant operation, for the case when a tilting gate 

with spillway in the channel forebay is used. These results, employing 

more accurate techniques, confirm the previous results. They show that 

the surge reduction is appreciable, but that the secondary oscillations 

depend on the time duration of closure of powerplant, and they propagate 

farther upstream as this closure time increases. The result of measurement of surge clearly confirm the values obtained by formula derived 

theoretically.

616. Schoenfeld, J. C., 1954, Analogy of hydraulic, mechanical, acoustic, and 

electric systems: App. Sci. Research., Sec. B., p. 417-450.

617. Supino, Giulio, 1954, Le oscillazioni del risalto idraulico durante una

propagazione ondosa [The oscillations of hydraulic jump during wave

propagation]: Energia Elettrica [Italy], v. 31, no. 2, p. 90-94.

This paper is a study of the movement of the hydraulic jump due to

wave propagation and its passage through the hydraulic jump. The

study is limited to the rectangular and very wide channel. The position

of the jump as it occurs in nonuniform steady flow is first determined.

Analysis is then made of the effect, upon position of the jump, of a wave

that is generated upstream from the jump and moves downstream. The

analytical approximation is obtained. If the jump is to remain at the

same place during wave movements, the downstream change of level must

be regulated to produce this condition.

618. Tifonov, E. K., 1954, Neustanovivsheesya dvizhenie vody v otkrytykh 

ruslakh pri nulevoy nachal'noy glubnne, -volna posukhu [Unsteady 

flow in open channels with zero initial depth, for wave in dry bed]: 

V.N.I.I.G. im. Vedeneeva (manuscript), [Leningrad]. 

The celerity of positive downstream surges, in dry bed, is studied experimentally in a canal 75 meters long. Results show that the celerity of wavefront decreases with an increase of distance travelled by the surge, and 

that it approaches asymptotically the velocity V of steady flow corresponding to the constant discharge which is introduced at the beginning of the 

canal. The mean velocity at the beginning of the canal approaches also 

the mean velocity of steady flow, but the celerity of surge is somewhat 

greater than the velocity at the beginning of the canal. However, the 

difference is small, decreasing with an increase of distance travelled by the
BIBLIOGRAPHY 171

surge. (This summary and tables of results are given in the paper of 

Kovalenko, 1957.

619. Ambraseys, N. W., 1955, Application of the continuity equation for sewer 

system computations: Internat. Assoc. Hydraulic Research, 6th Gen. 

Mtg., The Hague, 1955, Proc., v. 4, p. D3-1 to D3-10.

A method for the determination of sewage-pipe diameters is developed, 

based on the use of the storage in the system and employing the balance 

equation for inflow minus outflow equals the stored water in a given time 

unit.

620. Bata, G., 1955, Utilisation des valeurs r^duites dans le domaine des regimes 

transitories des canaux ouverts et leur application dans le cas des 

gal cries utilises en reservoirs d'eclusees [Use of relative values for 

unsteady flow in open channels and their application in the case of 

tunnels used as storage reservoirs]: Internat. Assoc. Hydraulic 

Research, 6th Gen. Mtg., The Hague, 1955, Proc., v. 4, p. D8-1 to 

D8-10.

Two forms of relative values are studied for unsteady flpw. In the first 

group the relative (or reduced) values are obtained in De Saint-Venant's 

equations by using Q and H of the initial steady uniform flow. In the 

second case the dimensionless terms are those resulting from Froude's 

similitude, the reference length being the height H of the steady flow. The 

regions are precisely defined for suitability of groups of values to be used, 

and applications are made to the problem of emptying a tunnel being used 

as a storage reservoir. Use of relative values for unsteady flow in channels 

was suggested by use of relative values for unsteady flow in conduits under 

pressure.

621. Binnie, A. M., Orkney, J. C., 1955, Experiments on the flow of water from a 

reservoir through an open horizontal channel; II, The formation of hydraulic jump; Royal Soc. [London] Proc., ser. A, v. 230, no. 1181, p. 

237-246.

By experimental study of the hydraulic jump it is revealed that there is 

little difference between the height of the leading wave in the smooth undular jump and the theoretical height of a solitary wave formed at the 

same Froude number. Rayleigh's classical theory (1914) gave a fair 

approximation of mean depth in the undular train of waves.

622. Bose, N. K., and Sinha, G., 1955, A study of discharge variation in tidal 

channels: Internat. Assoc. Hydraulic Research, 6th Gen. Mtg., The 

Hague, 1955, Proc., v. 1, Tidal Hydraulics and Tidal Models, Paper A2-

Hydraulic observations taken in 1954 in a tidal creek near Calcutta are 

analyzed with the purpose of finding a relation between the wetted perimeter 

P and the discharge Q, similar to Lacey's formula for uniform flow in stable

alluvial channels: P equals cVQ, where c is a constant. For the ebb 

flow such a fixed relation was found at the beginning of this stage of the 

flow. For the flood flow, however, this relation between P and Q was found 

during the greater part of the flow. Further analyses are being continued 

and the relations between P and Q will be examined more closely.

[Abstract from Bibliography on tidal hydraulics, Corps of Engineers, May 

1959.]
172 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

623. Dahl, N. J., 1955, On non-permanent flow in open canals: Internat. Assoc.

Hydraulic Research, 6th Gen. Mtg., The Hague, 1955, Proc., v. 4,

p. D19-1 to D19-16.

Cases are treated for oscillations without the loss of energy and oscillations 

without distortion. The general case is stated, and a particular solution is 

shown; examples are given. The theoretical results and previous full-scale 

test results are compared and discussed. The following propagation 

celerities are given: (1) wave in deep water, C^g-\/l/2ir, where I-wavelength; 

(2) wave in shallow water without distortion, C V -\-\gH; (3) wave in 

shallow water with distortion (very long waves), C«3F/2, where V=original mean velocity of water in the channel.

624. Dronkers, J. J., and Schoenfeld, J. C., 1955, Tidal computations in shallow 

waters: Am. Soc. Civil Engineers, Proc., v. 81, Separate 714 (discussions, Paper 841, p. 41, and HY 5, Paper 1082-3). Also Rijkswaterstaat commun. no. 1, [Den Haag, Netherlands] 1959. 

The computation of tidal elevations and currents in shallow coastal 

waters may serve various practical purposes. Mathematically the problem 

is so involved that no simple procedure to be generally accepted exists. 

There are mainly three ways of approach which are illustrated in this 

paper by expounding the computation methods developed in Europe, in 

particular in the Netherlands. It depends on the sort of problem to be 

solved which method should be adopted as the most appropriate. Contents: Introduction (purposes of computation; nature of the problem; 

historical survey of tidal computation; list of basic symbols). The Basis 

of Tidal Computations (schematization of an estuary; the differential 

equations; particularizing conditions). Integration by harmonic components (confined to the periodic tide; equations are linearized making it 

possible to consider the tide as sinusoidal; nonlinear terms are treated and 

the interaction of harmonic components is investigated; are developed 

more in detail). Direct integration (finite difference methods are discussed, 

both in the original quad-scheme and in the more recent cross-scheme; 

principles of the more refined methods of power series and iteration are 

discussed, followed by various applications of these methods). Integration 

along characteristics (principle of the characteristics and their bearing on 

the phenomenon of propagation discussed in connection with linear equations without resistance; amendments to be made in order to deal with the 

resistance are discussed; the variability of the velocities of propagation is 

dealt with; shock wave conditions are considered).

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

May 1957.]

625. Dziatlik, H., 1955, Model elektryczny rzeki [Electrical model for a river]: 

Przeglad Elektrotechniczny [Poland], 21 July 1955, p. 447-445. 

Unsteady flow in open channels is considered to be an important and 

difficult problem of modern hydroengineering. Elucidation of this problem 

depends on the solution of De Saint-Venant differential equations, but the 

various graphical and analytical integration methods hitherto practiced 

are, as a rule, troublesome or inaccurate. The new method is based on the 

contention that the equations for a long-distance electric transmission line 

having variable parameters can be made to be analogous in all respects, 

so that even the most complex cases of unsteady flow in rivers can be studied 

according to the transient phenomena in long-distance lines. The analogy
BIBLIOGRAPHY 173

of electric and hydraulic variables is given by comparing basic differential 

equations.

626. Einstein, H. A., and Fuchs, R. A., 1955, Computation of tides and tidal 

currents U.S. Practice: Am. Soc. Civil Engineers Proc., v. 81, Separate 715.

The authors have recently made a survey of past and present calculation methods used for the prediction of tidal stages and flows in canals and 

estuaries. The results of the study conducted under contract for the 

Committee on Tidal Hydraulics, Corps of Engineers, U.S. Army, are in 

part, offered in this paper. The report is limited to practice in the United 

States as developed and applied before the Second World War. Describes 

and discusses prominent older methods and indicates where the methods 

may be modified for added accuracy or simplicity. A resume of the 

complicating aspects of the tidal flow problem in general and the various 

effects and causes which must go into a mathematical description of such 

flows is given. Discusses the differential equations of tidal flow and presents the equations. The various solutions of the equations, obtained by the 

introduction of simplifying assumptions, which are presented and discussed 

are: Parson's harmonic theory (Parsons, W. B., The Cape Cod Canal); 

Brown's reflected wave theory (Brown, E. I., Flow of Water in Tidal 

Canals); and Pillsbury's theory (Pillsbury, G, B., Tidal Hydraulics). 

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

May 1957.]

627. Evangelisti, Giuseppe, 1955, On the tidal waves in a canal with variable 

cross-section: Internat. Assoc. Hydraulic Research, 6th Gen. Mtg., 

The Hague, 1955, Proc. v. 1, p. A10-1 to A10-2.

The theoretical approach to wave movement in a canal having variable 

cross-sections, showing that the wave-phase velocity is independent of 

wave frequency, is given by ^gH, as if the canal were uniform. The waves 

undergo appreciable distortion, and their amplitude varies along the canal 

in such a way that a quantity (proportional to the energy of the wave) remains constant.

628. Hunt, J. N., 1955, On the solitary wave of finite amplitude [A proposed 

1'onde solitaire d'amplitude finie]: Houille Blanche [France], no. 2, 

p. 197-203.

It is shown that the equation for the profile of the solitary wave and the 

velocity of propagation can be obtained by a convenient process of successive 

approximations without determining the velocity potential throughout the 

plane of flow. The method employs an operational form, of the free-surface 

condition given by Levi-Civit& (1911). The wave profile and velocity are 

found to the third order in the ratio h/H, where h=the wave amplitude and 

H=the undisturbed depth. For small values of this ratio the solution reduces to that of Boussinesq. The velocity equation is in good agreement 

with measurements reported by Daily and Stephan (1953) :

(7i/.ff) ma* =0.92; comparison with C/^gH gives the difference 0.6 percent for 

h/H =0.5 and 2 percent for h/H=O.Q2.

728-245 6
174 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

629. Ippen, A. T., Kulin, Gershon, and Raza, M. A., 1955, Damping characteristics of the solitary wave: Mass, Inst. Technology Hydrodynamics Lab. 

Tech. Kept. 16, p. 1-40.

This report presents the results of an investigation of the damping of 

solitary waves in water of constant depth. The damping was studied 

principally by direct observation of the decrease in amplitude of waves as 

they proceeded up and down a 32-foot Lucite tank. Also reported is a 

preliminary investigation of the damping process by direct measurement of 

boundary layer velocities with a specially designed differential gage.

Damping was observed over bottoms roughened with uniform sands of 

diameter ranging from 0.0056 feet to 0.0198 feet, and the results are compared with previously published smooth-bottom data. Still-water depths 

ranged from 0.17 feet to 0.40 feet. Approximate expressions for wave 

Reynolds number and boundary layer thickness are derived and the latter 

is compared with available experimental results.

It was found that in a tank of the size used in this investigation, Reynolds 

number sufficiently high to permit the existence of true roughwall friction 

in the roughened bottom runs were not developed.

Results of near-bottom velocity observations with a differential gage verified the existence of a boundary layer increasing in thickness from front to 

rear of the wave. 

[From author's abstract.]

630. Ishihara, Tojiro, and Ishihara, Yasuo, 1955, On an electronic analog computer for flood routing: Japan Soc. Civil Engineers [Tokyo] Trans., no. 24, 

p. 44-57.

The flood is a very complex phenomenon. According to Dr. Hayami's 

theory of flood waves, however, which takes into account the complex 

features of river, the fundamental equation becomes, approximately

dtf 

+ da: 5x2

where H: water depth, t: time, x: distance, and A, /*: numerical constants. 

In this paper, a method to initiate this equation by electric circuits and to 

construct them was mentioned, that is, an electronic analog computer was 

designed and constructed. Using this computer, the authors obtained 

some solutions of flood waves, and discussed the characteristics of flood and 

the significance of constants A and of the equation mentioned above. As 

an example, the results of actual flood routing for the Kiso River were explained and it was found that this computer produced with good accuracy. 

[Author's English synopsis.]

631. Iwasa, Y., 1955, Analytical considerations on cnoidal and solitary waves:

Kyoto Univ. [Japan] Eng. Fac. Mem., v. 17, no. 4, p. 264-276. 

The mathematical theory on progressive translation waves of the permanent type is developed. The analysis of the second approximation 

based upon the momentum and energy approaches, considering the effects 

of the vertical acceleration, shows that the surface profile of waves is 

expressed in terms of an elliptic function which is designated as the cnoidalfunction, the name being derived therefrom as analogous to sinusoidal 

waves, and the solitary wave is included as a special case of the cnoidal 

waves.
BIBLIOGRAPHY 175

It should be noticed that the behavior of cnoidal waves may be reduced 

to that of classical sinusoidal waves as a special limiting approximation. 

[Author's abstract.]

632. Kazarnovskiy, Yu. E., 1955, Uchet transformatsii pavodka v vodokhranilishchakh s sifonnymi vodosbrosami [Flood- wave transformation in 

reservoirs with syphon spillways]: Gidrotekhnika i Melioratsiya 

[U.S.S.R.], no. 8, p. 55-57.

In flood routing based on Kocherin's formula for the triangular-shaped 

hydrograph, the author develops the following formula for damping of 

flood peak (routed through a reservoir having a free spillway automatically 

operated) :

_Qm _. Wl -*-pm -1 w

where Pm=peak of inflow, Qm peak of outflow hydrograph, W= volume 

of flood wave, W\ = volume between the maximum and starting (normal) 

reservoir levels. For routing through a reservoir having syphon spillways, 

the formula is: [(1 + K)/2K](1-^W\/W, where K=Q0/Qm, Q0 being the 

outflow for normal water level in reservoir. Because, for the most part, 

#>0.9, [(1 + X)/2X]«1, so that y=l-^fW\|W.

633. Kovacs, Gyorgy, 1955, Az arhullamok levonulasdra jellemzo" hidro!6giai 

mennyise'gek meghatarozdsa [Determination of the hydraulic magnitudes characterizing the propagation of flood waves]: Hidrol6giai 

Kozlony [Hungary], v. 35, no. 11-12, p. 394-423. (English summary 

p. 422-423.)

This study uses the Bernoulli equation for unsteady flow and the continuity equation. The basis of the study is the analysis of rating curves, 

especially of loop curves pertaining to flood waves. The parts of the loop 

corresponding to a rising limb (monoclinal) and falling limb (also monoclinal) are specially treated and fitted by power function, and then applied to 

one-peak floods. The actual data of the Tisza and Danube Rivers support 

all derivations for rating curves and flood waves. The author considers 

that this procedure enables the determination of time and local variations 

of discharge for flood waves.

634. Ligthill, M. J., and Whitham, G. B., 1955, On kinematic waves I, Flood 

movements in long rivers: Royal Soc. [London] Proc., A 229, p. 

281-316.

The kinematic waves are first defined as destinct from the dynamic 

waves. The flood waves in long rivers are treated mostly as kinematic 

waves, although dynamic waves also appear. From the full equation of 

motion for an idealized problem, it is shown that at Froude numbers 

appropriate to flood waves, the dynamic waves are rapidly attenuated, 

and the main disturbance is carried downstream by the kinematic waves. 

The kinematic shock wave (monoclinal flood wave) is investigated. The 

final section treats the application of the theory of kinematic waves to the 

determination of flood movement.

635. Moklyak, V. I., 1955, K raschetu transformatsii povodkov v estastvenny 

ruslakh [On the flood routing in natural channels]: Ukrainskiy 

N.I.G.M.I. [U.S.S.R.], Trudy (Proc.), Vypusk (issue) no. 3.
176 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

636. Nash, J. E., and Farrell, J. P., 1955, Graphical solution of the flood routing 

equation for linear storage-discharge relation: Am. Geophys. Union 

Trans., v. 36, p. 319-320.

Starting from the equation W=KQ and P=Q + KdQ/dt, the experiment 

relation of Q and P is developed analytically as P Q=(P Q<>) e tjK, 

where Q0 is the outflow for t=Q. A template is developed for Q = Q0t tjK, 

and the routing for constant P is given as an example of the template's use.

637. Nougaro, Jean, 1955, M6thode graphique pour le calcul de la propagation 

des intumescences dans les canaux decouverts [Graphical method for 

the computation of propagation of tranlatory waves in open channels]: 

Internat. Assoc. Hydraulic Research, 6th Gen. Mtg., The Hague, 

1955, Proc., v. 4, p. D5-1 to D5-15.

The paper deals with the principle of a new graphical method for the 

study of intumescences. It is based on and deduced from Bergeron's 

graphical method, by using linear characteristics (l/B~)(l/[V±C]) in coordinate system (Q, PT) . These straight lines can be replaced in a rectangular channel by the characteristic curves

Q + Q'xQo log H± (2/3)BH-<i/gH.

The applications of this method are made for both laboratory and powerplant canals, and the results justify the application, according to the author. 

This method allows an approximate relation

HI =

where H0 is the initial water depth in rectangular channel, and HI is the 

depth after the passage of wave, especially for the sudden changes in water 

powerplant operations and for surges created in both headwater and tailwater canals. 

[From author's abstract.]

638. Proudman, J., 1955, The propagation of tide and surge in an estuary:

Royal Soc. [London] Proc., A, v. 231, p. 8-24.

Theoretical investigation of the distribution, along an estuary, of a 

combination of tide and surge which have been generated in the open sea. 

The following results relate to the same sequence of meteorological conditions over the sea. For a single progressive wave, the height of a surge 

whose maximum occurs near to the time of tidal low water, and decreases 

as the range of tide increases. To the order of approximation followed in 

the paper, these differences are due to friction and increases with distance 

from the sea. For a standing oscillation, the following results relate to the 

head of the estuary. When the primary surge rises to its maximum more 

rapidly than it falls from it, and when this maximum occurs near to the time 

of tidal high water, the effect of shallow water is to make the surge increase 

as the range of tide increases, and the effect of friction is to make the surge 

decrease as the range of tide increases. On applying the formulae of the 

paper relating to progressive waves to the Thames Estuary, it appears that, 

owing to the small depth of water at mean level, the details only hold for 

a few miles from the sea. But the tendencies enumerated should hold all 

the way to London.

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers; 

February 1954.]
BIBLIOGRAPHY 177

639. Proudman, J., 1955, The effect of friction on a progressive wave of tide and 

surge in an estuary: Royal Soc. [London] Proc., A, v. 233, p. 407-418.

This is a second paper (for the first see: The propagation of tide and 

surge in an estuary, by J. Proudman) on the theoretical distribution, along 

an estuary, of a combination of tide and surge which have been generated 

in the open sea. It differs from the first paper in that, in the basic equations, 

the nonlinear inertia term is neglected, and only one phase of the wave is 

considered at a time. Also, the formulae are now valid for any height of 

tide or surge at the mouth of the estuary and for any distance from the sea. 

For the same sequence of meteorological conditions over the sea and when 

the height of surge at the mouth of the estuary is greater than a certain 

fraction of the range of tide there, the surge up the estuary is less at the 

time of high water than it is at the time of low water, and it decreases as 

the range of tide increases. This is a wide extension of the range of validity 

of one of the results of the first paper.

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

February 1954.]

640. Richards, B. D., 1955, Flood estimation and control; Ch. XI, Flood Control: 

London, Chapman and Hall, Ltd., p. 134-153.

This paper descusses the flood absorption effect of regulating reservoirs 

having various kinds of outlets, constant or changing inflows, and floodstorage basins.

641. Sato, S., Kikkawa, H., and Kishi, T., 1955, On the hydraulic characteristics 

of the tidal reach of the Tone River: Internat. Assoc. Hydraulic 

Research, 6th Gen. Mtg., The Hague, 1955, Proc., v. 1, Paper A6.

This paper reports the results of field observations made in the tidal 

reach of the Tone River in 1950 and 1954. It discusses: (1) the characteristic of the equation of motion, velocity profile, and the propagation velocity 

of river tide; (2) attenuation of tidal amplitude along the river course; (3) 

the characteristics of the suspended load transportation and the shearing 

force acting on river bed; and (4) the turbulent mixing between salt and 

fresh water, and the intrusion velocity of the salt water. 

[Abstract from Bibliography on Tidal Hydraulics Corps of Engineers? 

May 1959.]

642. Schoenfeld, J. C., 1955, Theoretical considerations on an experimental 

bore: Intnat. Assoc. Hydraulic Research, 6th Gen. Mtg., The 

Hague, 1955, Proc., v. 1, p. A15-1 to A15-12.

Both the propagation and the shape of the bore are considered thoretically, and these considerations are checked by experimental evidence 

obtained on a tide with a bore in a laboratory canal.

The propagation is treated by considering the bore as a mobile hydraulic 

jump. The theory is checked twice, first by combining the data on the 

bore when it passed the observation places, and secondly by computing 

the whole tide with the bore by the method of characteristics.

The influence of the vertical accelerations on the shape of the bore is 

treated by considerating elementary short wave solutions of sinusoidal 

and exponential form. The bore is compared to the solitary wave. The 

influence of capillarity is discussed. The theory is checked by means of 

length profiles of the observed bore. Finally an energy budget is re-
178 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

constructed, which yields the amount of deceleration losses. The breaking 

of the bore is discussed. 

[Author's abstract.]

643. Schoenfeld, J. C., 1955, Discussion of the paper "Graphical method for 

calculating the propagation of translatory waves in open channels" 

by Nougaro, J: Intnat. Assoc. Hydraulic Research, 6th Gen. Mtg.r 

The Hague, 1955, Proc., v. 4, p. HI-3 to HI-5.

Use of the method of characteristics for integration of two De SaintVenant equations is discussed. Because of the symmetry of differential 

equations and contresubcharacteristics (characteristics projected on surfaces of dependent variables), it is shown that it is best to use H and V 

(depth and velocity) or He and Q (total energy head and discharge) as 

dependent variables, rather than to use other combinations. It is advantageous to use the combination He and Q when the cross-sectional area 

varies with the length of channel.

644. Schoenfeld, J. C., 1955, Discussions on the papers "Nonpermanent flow 

in open canals" by N. J. Dahl and "On tidal waves in a canal withvariable cross-section" by G. Evangelisti: Intnat. Assoc. Hydraulic 

Research, 6th Gen. Mtg., The Hague, 1955, Proc., v. 1, p. HI-3 

to HI-6.

Long waves in channels undergo two distinct forms of distortion, as 

caused by : (1) friction, which is expressed by the zero-order terms, 

linear or nonlinear, in the differential equations; and (2) variability of 

cross-sectional area with height of water, which is expressed by the nonlinear character of the first-order terms. In this view, the author's studies 

and results are compared with the results of Evangelisti and of Dahl,

645. Schoenfeld, J. C., 1955, Tides in funnel-shaped channels: Rijkswaterstaat 

Commun. [Den Haag, Netherlands], Nota CSD 55-16.

646. Schoenfeld, J. C., 1955, Getijberekening door Integratie langs Karakteristieken [Tidal computation by integration along characteristics]: 

Rijkswaterstaat Commun. [Den Haag, Netherlands], Nota CSD 55-14, 

p. 13-18.

647. Sueishi, Tomitaro, 1955, On the runoff analysis by the method of characteristics [Hydraulic studies on the runoff phenomena of rain water, 

2d report]: Japan Soc. Civil Engineers [Tokyo] Trans., no. 29, p. 74-87, 

In this paper, the approximate method calculating the unsteady flow in 

open channels with uniform lateral inflow using the characteristic curves, 

described in the author's previous paper, is practically applied to the 

analysis of runoff phenomena in natural rivers, and a new method estimating the runoff from rainfall is established.

Application of this method to runoff analysis in the Daido River in 

Japan shows that the method of characteristics is very available for the 

estimation of the runoff in rivers with relatively steep slopes. 

[Author's abstract in English].

648. Yuichi, Iwagaki, 1955, Fundamental studies of the runoff analysis by 

characteristics: Kyoto Disaster Prevention Research Inst., Kyoto 

Univ. [Japan], Bull. 10, p. 1-25.

In treating the hydraulic analysis of surface runoff phenomena, the 

author's basic idea is expressed as follows: the surface runoff mechanism of 

rain water in a mountainous district consists of the combination of overland
BIBLIOGRAPHY 179

flow and flow in open channels with lateral inflow. Consequently, the unsteady flows with lateral inflow must be solved for various conditions, 

individually.

In this paper, an approximate method for calculation of unsteady flow 

in open channels with lateral inflow, using the characteristics, is presented. 

Hydrographs resulting from the abrupt increase and decrease of rate of 

lateral inflow are obtained by this approach, and moreover the calculated 

hydrograph are compared with the experimental results. 

[Author's synopsis.]

649. Whitham, G. B., 1955, The effects of hydraulic resistance in the dambreak problem: Royal Soc. [London] Proc., v. 227. no. 1170, p. 399-407.

When resistance is neglected, the solution of the simple dam-break problem is readily obtained on the basis of shallow-water theory, and the results 

are well known. However, near the head of the wave, where the water 

surface meets the ground, resistance effects cannot be neglected; there is, 

in fact, a type of boundary layer near the wave-front. In this paper, the 

Pohlhausen method (which is used in conventional boundary-layer problems) is applied to study of the effect of this "boundary layer." In particular, the retardation of the wavefront behind the position predicted by 

the simple theory is found. 

[Author's abstract.]

650. Zheleznyak, I. A., 1955, Raschet transormatsii maksimal'nykh raskhodov 

s uchetom predpavodkovogo oporozhnyeniya vodokhranilishcha [Computation of maximum discharge modification taking into account the 

emptying of a reservoir before the flood]: Akad. Nauk [Ukrainian] 

S.S.R. Research and Computations in Hydrology, v. 13 (XX), p. 

112-121.

The study takes into account the effect on reduction of the maximum 

outflow discharge of prior emptying of a part of storage. Tsinger's formula 

(1953) is discussed, and its misconceptions are stressed. Pleshkov's (1953) 

position is criticized, and an explanation is given for those instances when 

reduction in the peak will not take place. Assuming preflood reduction of 

storage, approximate formulas for maximum outflow are developed for 

free spillway operation and for operation of gages on the crest. The author 

employs Kocherin's assumption of the linear function of the outflow rating 

curves and states that the formulas permit an error of several percentage 

points. The limitations of this procedure are discussed, and it is shown 

that the procedure is valid only if Wr (reduced storage) is smaller than Wn 

(see Zheleznyak, 1953).

651. Abbott, M. R., 1956, A Theory of the propagation of bores in channels and 

rivers: Philos. Soc. [Cambridge] Proc., v. 52, no. 2, p. 344-362.

A theory is presented of the nonlinear propagation of waves and bores 

in channels of varying cross section with a basic steady flow governed by 

frictional resistance; this corresponds to the flow in tidal rivers. The 

theory provides a condition on the tidal range required to produce a bore, 

in terms of the geometry and friction parameters of the river, and the 

propagal ion of such a bore is then described. The theory is applied to the 

Severn River and the results agree satisfactorily with observation. Results 

for the special case of waves moving into still water in a channel of varying 

section are also noted in detail.
180 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, May 

1959.]

652. Adachi, Shohei, 1956, On the propagation of flood wave in the transient 

region between the river channel and the storage region of reservoir: 

Japanese Natl. Gong. Appl. Mech. Proc., rept. 11/27, p. 367-371 

(published by Science Council of Japan [Tokyo], March 1957). 

A flood wave has the character of a long wave in the storage region 

where the slope of water surface is nearly always zero. The propagation 

velocity of a flood wave is usually larger in this region than in the river 

channel. In the actual reservoir, however, the storage effect acts to delay 

the propagation of a flood wave. Therefore, between the river channel 

and the storage region of a reservoir, there must be the transient region 

where the propagation of a flood wave may be retarded by the storage 

effect. Authors have confirmed the existence of this region by the model 

experiment of Maruyama Reservoir in the Kiso River. 

[Abstract in the report.]

653. Alekseev, G. A., 1956, Priblizhennye methody rascheta transformatsii 

pavodka vodokhranilishchem na osnove skhematizatsii gidrografov 

pritoka i sbrosa [Approximate methods of flood routing through reservoirs by using the schematic hydrographs of inflow and outflow]: 

Gosudarstveniy Gidrologicheskiy Inst. [U.S.S.R.] Trudy, No. 52 

(106).

654. Arsenishvili, K. I., 1956, Kriterii volnoobrazovaniya v kanalakh s bol'shimy

uklonami [Criteria of wave formation in canals with great slopes]:

Gidrotekhnicheskoe Stroitel'stvo [U.S.S.R.], no. 3, p. 41-44.

Results of experimental studies of roll-wave formations in steep channels

are reported. The experimental check of the existing criteria (Vedernikov,

Mostkov, Keuligan-Patterson, Thomas, Dressier, and so forth) of roll-wave

formation shows that these criteria, in some cases, neither describe the

true character of wave movement nor take into consideration the shape of

the cross section. The author gives the following empirical criteria:

Criterion A^: 0.02>S0>0.30; Criterion A2 : Hv/P>0.10, where 80=bottom

slope, H0 = depth for full canal with steady flow, and P= wetted parameters

for steady flow. Criterion A2 determines the regime, without roll waves.

for slopes of waves according to Criterion A\.

655. Blind, Hans, 1956, Nichtstationaere Stroemungen in Unterwasserstollen 

[Unsteady flow in tailrace outlet tunnels]: In Veroeffentlichungen zur 

Erforschung der Druckstossprobleme [F. Toelke, ed.], Berlin, Springer, 

p. 67-108.

This theoretical and experimental study of water regimen in the tailrace 

tunnels of underground water powerplants also treats the movement of 

surges (bores and depressions) along the circular tunnels, when water is 

flowing as free surface flow. The celerity of waves, which is the criterion 

for change of surges by progression, is analyzed both by the theoretical and 

experimental method.

656. Chow, Ven Te, 1956, Hydrologic studies of floods in the United States 

(Flood-routing): Internat. Assoc. Sci. Hydrology, U.G.G.I., 1956, 

v. 2, pub. 41, p. 143-147.

This is a short review of flood-routing methods in the United States, 

including: (1) analytical, graphical, semigraphical, nomographical, slide-
BIBLIOGRAPHY 181

rule, circular computers, and Muskingum methods; (2) mechanical and 

electronic instrumental routing procedures; (3) model routing; and (4) 

stage routing.

657. Cuenod, M., 1956, Contribution & I'e'tude des crues-Determination de la 

relation dynamique entre les precipitations et le debit des cours d'eau 

au moyen du calcul a 1'aide des suites [Contribution to the study of 

floods; calculation, by using a series method, of the dynamic relation 

between rainfall and river flow]: Houille Blanche [France], no. 3, 

p. 391-402.

The device of the index (or characteristics) hydrograph is introduced to 

represent the rising limb of a hydrograph resulting from a given value of 

precipitation that starts suddenly and persists indefinitely. The hydrographs are derived by using a series of units for the precipitation and the 

corresponding series of resulting discharges. This method is a variation of 

the classical unit hydrograph theory. An example of its use for the Krummbach River is given and discussed.

658. Dooge, J. C. I., 1956, Synthetic unit hydrographs based on triangular 

inflow: Iowa Univ., Dept. Mech. and Hydraulics, M.S. thesis, p. 1-103. 

This study is concerned with the problem of synthetic unit hydrographs, 

that is, the prediction of surface runoff on the basis of only such information 

as is available from a topographic map of the catchment area. A theory of 

the unit hydrograph is presented which links the unit hydrograph principle 

with the presence of linear storage in the catchment area. The nature of 

the runoff process and the morphology of natural catchments are discussed 

in order to provide a basis for a synthetic method. It is argued that the 

instantaneous unit hydrograph can be produced by routing a triangular 

inflow through a single element of linear reservoir storage, and hydrographs 

derived in this way are shown to give excellent agreement with both the 

dimensions and shape of unit hydrographs derived by other means. In the 

final chapter a practical procedure is outlined and two examples are worked.

659. Egiazarin, B. O., 1956, Priblizhennyy raschet neustanovivshegosya dvizheniya vody v nizhnem b'efe GES pri sutochnom regulirovanii [An approximate method of computation of unsteady flow in tailwater 

channels, made by employing use of powerplant records]: Gidrotekhnicheskoe Stroitel'stvo [U.S.S.R.], no. 7, p. 48-51. 

This approximate method is derived from the actual data obtained on 

powerplant canals and laboratory channels. The formulas for the parameter a (a degree of wave deformation) are derived for different cases. A 

practical procedure is given for use of formulas in computing wave characteristics.

660. Escande, L., and Nougaro, J., 1956, Regime variable dans un canal d'amenee 

associe" & une galerie en charge [Unsteady flow in a headrace canal 

leading to a pressure tunnel]: Houille Blanche [France], no. 2, p. 

240-249.

This report is a study of two problems relating to waves in a headrace 

canal that is connected to a pressure tunnel having a mass-oscillations 

setup. The first problem deals with a tunnel-surge-tank system connected 

to an open canal. The second problem deals with a syphon set in a headrace canal. The use of theoretical, analytical, and graphical methods is 

demonstrated. The results thus obtained show satisfactory agreement 

with the experimental results.
182 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

661. Gil'denblat, Ya. D., 1956, Nekatorye voprosy, svyazannye s raschetami 

neustanovivshegosya dvizheniya i otsenkoy raspolagaemykh po naporu 

pikovykh moshchnostey gidroelektrostantsii [Some questions related 

to the computation of unsteady flow and the estimation of peak power 

loads of water powerplants, as determined by amount of head available]: Akad. Nauk [U.S.S.R.], Problemy regulirovaya rechnogo stoka 

[Problems of river runoff regulations], v. 6, p. 263-277. 

Unsteady movement in the tailwater of powerplants, as represented by 

the unsteady-load diagram, is studied in order to analyze power loads. 

The boundary conditions at the powerplant and at the beginning of tailwater channel are discussed. A study is made of the position of positive 

surges in each moment, which information is necessary for selecting At and 

Aa; for flood routing. Rating curves at selected stations are used for 

analysis. A formula for the optimum length of reach is given, and the 

selection of reaches and of time interval is discussed. A method of accounting for winter conditions is shown. An approximate (straight-line) relationship between the decrease of powerplant head (by tailwater-level increase) 

and the difference between peak load and mean load is developed, as are 

other significant relationships.

662. Isaacson, E., Stoker, J. J., and Troesch, B. A., 1956, Numerical solution

of flood prediction and river regulation problems, Report III: New

York Univ., Inst. Math. Sci. Rept. 6 IMM-NYU-235, p. 1-70.

The flood-routing procedure employing the finite-differences method,

and using the digital computer UNIVAC (Stoker 1953), is applied as a

numerical solution for prediction of the 1945 and 1948 floods in the Ohio

River, for prediction of the 1947 flood through the junction of the Ohio and

Mississippi Rivers, and of the floods of 1950 and 1948 through the Kentucky

Reservoir. The factors affecting accuracy are discussed, and suggestions

are given for improvement of the numerical methods. The model studies

of river flood movements are compared with numerical studies using

digital computers.

663. Ishihara, T., Hayami, Sho., and Hayami, Shi., 1956, Electronic analog 

computer for flood flows: Regional Tech. Conf. Water Resources 

Devel. in Asia and Far East, Proc., ECAFE, Folod Control Ser. no. 9, 

Bangkok, p. 170-174.

Based on Hayami's equations (1954) for flood flows, the electronic 

analog computer is developed and described. Its suitability for accounting 

for the nonlinear resistance losses along the river is notable.

664. Ishihara, Tojiro, and Ishihara, Yasuo, 1956, Electronic analog computer 

for flood flows in the Yodo River: Japan Soc. Civil Engineers Proc. 

[Tokyo], v. 41, no. 8, p. 21-24.

665. Iwasa, Y., 1956, Hydraulic characteristics of solitary waves: Japan Soc. 

Civil Engineers Ann. Convention [Tokyo], May 1956.

666. Jaeger, Charles, 1956, Engineering Fluid Mechanics: [English ed. translated

by P. O. Wolf] London Blackie and Son Limited, chap. 8, p. 359-392.

German edition, Basle, Switzerland, Verlag Birkhaeuser, 1949.

The chapter on unsteady flows discusses propagation of small waves on

flowing water and surges in open channels. It contains analysis of surges

if change of discharge is known and if change of depth at inlet is known.

A graphical method of analysis is given for translatory waves in open
BIBLIOGRAPHY 183

channels (see Craya, 1945, 1946), and for underground powerstations 

(tailrace canals).

667. Kinosita, T., 1956, Hydrodynamic study of the flood flow: Internat. 

Assoc. Sci. Hydrology, U.G.G.I., Symposia Darcy, Dijon [France], 

1956, v. 2, pub. 41, p. 173-183.

The paper deals with numerical integration of two partial differential 

equations for unsteady flow. By neglecting the inertia term (dF/d<+ FdF/ 

dz = 0), the boundary condition is determined by water depth alone, and 

Q is determined from a momentum equation. A mesh point scheme of 

Ax and Af is given for the finite differences method of integrating two equations, in which emphasis is on the use of mesh points for boundary and for 

initial conditions. An expression is given for the accumulation of errors 

due to replacement of differentials by differences. The restrictions for 

A< are introduced (for any cross section and for rectangular and triangular 

channels). The examples for river and model channel are given. The 

peak velocity is studied as a function of wave steepness.

668. Kohler, M. A., 1956, River and water-supply forecasting: Acad. Sci.

[New York] Trans., ser. 2, v. 18, no. 8, p. 732-745. 

This is a general discussion of flood forecasting which treats the subject 

of streamflow routing. The basic storage equation is used, and the storage 

of a reach is expressed as S=K[xP+(l x)Q], in which K and x are experimental constants.

669. Kozdk, Miklos, N6hdny a nempermanens szabadfelszinu vizmozgas 

szdmitdsdra szolgalo eljdrds ismertetese [Some methods for the 

computation of unsteady flow with free water surface]: Hidrologiai 

Kozlony [Hungary], v. 36, no. 1, p. 17-32.

The present paper discusses methods of analysis of unsteady, gradually 

varying, free surface flow. The following conclusions can be drawn:

(1) Because of its mathematical exactness the method of characteristics 

is advantageous, provided that all the relevant hydraulic coefficients can 

be determined exactly according to actual conditions. Greatest care 

should therefore be exercised in applying this method to watercourses 

having greatly varying cross sections. Irregularities of the bed and particularly wave reflections occurring at river bends in natural watercourses 

made the reliability of the establishment of wave front velocities highly 

questionable. Values obtained by computation and by actual measurements, however, show a fair agreement up to lengths of 50 kilometers.

(2) The method of finite differences is applicable to all cases. A disadvantage of this method is its reduced exactness, which, however, is sufficient 

for practical purposes. Actual computation work is at the same time 

facilitated. 

[From author's summary, given in English.]

670. Kritskiy, S. N., and Menkel', M. F., 1956, O rasplastyvanii volny

vysokikh vod pri prodvizhenii po prizmaticheskomu ruslu [Attenuation

of the flood wave in the movement along the prismatic channel]:

Akad. Nauk [U.S.S.R.], Problemy regulirovaniya rechnogo stoka

[Problems of river runoff regulations], v. 6, p. 248-262.

A schematic approach is used to study wave attenuation along a river

channel. The maximum discharge attenuation is analysed in terms of the

path x, maximum discharge QQ at starting station, roughness coefficient n,
184 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

volume of flood wave W, and river-bed slope So. A simple starting wave 

is assumed, having Hm&* = H and triangular shape of the length s on each 

side of H. An approximate formula is derived which gives the peak at the 

section x as :

Q= Qo

where area A aHm+l, H depth, W 2aHm+ls/(m + 2'). It is concluded that 

the rate of maximum discharge attenuation along river increases with 

sharpness of hydrograph (characterized by Qo/W) and with increase of 

channel roughness, but decreases with an increase of river slope S0 . The 

routing of a simple triangular wave by instantaneous wave profiles (method 

of finite differences) is used to derive attenuation. Results show a close 

check of the developed formula.

671. Long, R.R., 1956, Long waves in a two-fluid system: Jour. Meteorology* 

v. 13, no. 1, p. 70-74.

The differential equations governing the unsteady motion of a system 

of two superimposed liquids are integrated by use of the method of characteristics. The solution differs significantly from that obtained by assuming 

a very thin lower layer.

If the basic shear across the interface is small, a wave breaks forward 

(as an ordinary water wave) only if the amplitude is fairly small and if the 

lower layer is thinner than the upper layer. If the upper fluid is thinner, 

the wave breaks backward. The speed of the various points of the wave 

does not vary monotonically with amplitude.

If a shear exists, higher velocities in the upper layer in the direction of 

wave. The opposite shear may cause a wave to break backward. 

[Author's abstract.]

672. Long, R. R., 1956, Solitary waves in one- and two-fluid systems: Tellus, 

v. 8, no. 4, p. 460-471.

A theoretical discussion is given of the solitary wave. Part I is concerned 

with the wave on a water surface; Part II considers the solitary wave at the 

interface of two superimposed liquids of different density, bounded above 

and below by rigid surfaces.

In Part I higher approximations are obtained for the speed of propagation, 

and limits to the wave speed are derived from the momentum theorem. 

The results include a correction to the second approximation of Weinstein.

In Part II the first two approximations to the wave speed are derived. 

The equations of the second approximation yield limits on the wave 

amplitude. 

[Author's abstract.]

673. Messerle, H. K., 1956, Differential analyser solution of hydraulic problems 

in hydroelectric systems: Houille Blanche [France], no. 6, p. 813-836.

The use of automatic computers in hydraulic analysis is discussed in 

this paper, and special reference is made to the mechanical differential 

analyser, which has been found particularly suitable for problems arising 

in hydraulic systems. The programming of a mechanical differential 

analyser for the solution of general hydraulic problems is described, and 

it is shown how the results obtained may be used to determine the optimum 

design fea ures.

One of the problems analyzed on the mechanical differential analyser of 

the C.S.I.R.O., Mathematical Instruments Section, University of Sydney,
BIBLIOGRAPHY 185

is the diversion of the Tooma River, a project of the Snowy Mountains 

Hydro-Electric Authority in Australia. This investigation is discussed in 

detail as a typical example.

The paper is useful for the study of advantages and disadvantages, as 

well as for study of the assumptions made in the use of mechanical differential analyser.

674. Nougaro, Jean, 1956, Reflexion et transmission d'une intumescence a un 

changement de section dans un canal decouvert [Reflection and transmission of a wave at a change of cross section in an open channel]: 

Acad. sci. [Paris] Comptes rendus, v. 243, no. 15, p. 1016-1019. 

The method is given for determination of the waves reflected and transmitted as a result of sudden change of canal cross section, in a transitory 

movement. Formulas for the new depth are developed and are checked 

by experiments; adequate agreement is shown.

675. Nougaro, Jean, 1956, Sur l'amortissement de la hauteur d'une intumescence

positive dans un canal decouvert a fond horizontal [On the attenuation

of the height of a positive wave in an open canal with horizontal bottom]: Acad. sci. [Paris] Comptes rendus, v. 242, no. flQ, p. 1263-1265.

An analytical method using characteristics (straight lines) is developed

for computing the damping of a steep wave as it progresses both in upstream

and downstream directions. The characteristics resulting from friction

loss is introduced as a first approximation. Expressions for change in

A/i are developed, and it is concluded that the damping of a positive surge

in downstream direction is smaller than that of a positive wave in upstream

direction. The formulas can be used for any cross section if the equivalent

depth in rectangular channel is used.

676. Nougaro, Jean, 1956, Recherches experimentales sur l'amortissement des 

intumescences dans les canaux decouverts [Experimental research of 

the wave absorbtion in the open channels]: Acad. sci. [Paris] Comptes 

rendus, v. 242, no. 16, p. 1953-1956.

It is shown that if a discharge is changed from Q0 to Qi (from V0 to V\) in 

a channel from

where Aft is the damping of created wave of height h, and V is the velocity 

at the point where the damping is Ah. Graphical solutions are given for 

three cases and compared with the experiments in a model 3 percent 

size of the Palaminy powerplant canal. Comparison is also made with 

the experiments envolving natural canal (Pebernat), and good agreement 

is shown.

677. Nougaro, Jean, 1956, Influencia de una singularidad de un canal, sobre la

propagacion de intumescencias [The influence on wave propagation of

a singularity in a canal]: Bol. de la Facultad de Ingeneria y Agrimensura de Montevideo [Uraguay], v. 5, no. 14, p. 349-390.

The wave propagation in a canal becomes a complex study when it

involves a singularity in resistance to flow. The paper deals with the

influence of this singularity on wave celerity and with the variation of

elevation created by tli'e disturbance. For a large resistance or construction,

the approximate celerity is C= V±-Jg(H+h)/T/l + ghB/Hn*RV3.

The formula is checked by experimental results. A similar expression is
186 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

given for rapid change in bottom level. The problem of unsteady flow irt 

a canal attached to the tunnel of a powerplant is studied, as is the influence 

on unsteady flow of a syphon in the middle of a canal.

678. Pan-Zhu, E., 1956, Raschet neustanovivshegosya dvizhenya v otkrytykh 

vodotokakh po metodu karakteristik [Computation of unsteady movement in surface channels by method of characteristics]: Akad. Nauk 

[U.S.S.R.] Trans., Div. Tech. Sci. (Izv.), no. 4, p. 42-57. 

A simplified approach to the method of characteristics for integrating the 

partial differential equation of unsteady flow is given. From he equations 

of characteristics in differential form, the form of finite differences for all 

four variables (x, t, V, H) is introduced. An auxiliary function F is introduced, as function of A#, Af and Ay. The depth Hx is expressed as a function of depths Ha and H6 (at points a and 6) and of F, and nomograms for 

solving the relation, and for determining F, are given. The nomographical 

solution for many relationships is a feature of this approach. The wavecomputation procedure is given by steps, with computation examples. The 

method was applied to regular channel, however, the possible application 

of the method for natural channels is discussed.

679. Ransford, G. D., 1956, The propagation and the reflection of abrupt translation waves in still water of steadily decreasing depth: Houille 

Blanche [France], no. 3, p. 406-414.

By analysis and employing the method of characteristics (according to 

Massau), the author concludes that the movement of a positive wave, 

having discontinuity or finite wave slope at the wave tip, at the point of 

departure, will always conclude by breaking in a channel having either 

horizontal or sloping bottom in the direction of movement. The negative 

wave tip with the same departure condition will flatten out indefinitely in 

the horizontal channel, but in the sloping channel the situation depends on 

the slope relationship of the wavefront and the bottom of the channel. 

For bZ/bS (wavefront slope) <t/2, the wave steepens: for dZ/dS=i/2 the 

wave stays unchanged, and for b.Z/c)*S>i/2 the wave flattens out, where 

i= bottom slope.

680. Rockwood, D. M., and Hildebrand, C. E., 1956, An electronic analog for 

multiple-stage reservoir type storage routing: U.S. Army Corps 

Engineers, Civil Works Inv., Proj. CW 171, Tech. Bull. 18, 27 Mar. 

p. 1-12.

A working model of an electronic analog for multiple-stage reservoir type 

storage routing has been built and successfully tested. It may be used for 

routing rainfall or snowmelt runoff or streamflow through basin, channel, 

or reservoir storage. The advantages of this analog over other types currently in use are: (1) Flexibility of operation a variable number of stages 

can be coupled, the time of storage of any of the stages can be varied during 

the routing: and (2) Simplicity and low cost the parts for the working 

model described herein are all standard radio components and cost less 

than $100 exclusive of the recording potentiometer. 

[From author's summary.]

681. Rudinger, George, 1956, Boundary conditions in nonsteady flow: Internat. Convention Appl. Mech, 9th, Brussells, Proc., v. 3, p. 152-164 

[1957].

Problems involving quasi-one-dimensional nonsteady flow in a duct 

require the knowledge of the conditions that govern the reflection of pres-
BIBLIOGRAPHY 187

sure waves from various flow boundaries. Some of these boundary conditions are accurately known while others must be treated by means of suitable assumptions. A brief review of the present knowledge in this field is 

presented. For wave reflections from an open end of a duct, it has been 

customary to assume that the boundary conditions in nonsteady flow are 

the same as in steady flow. Actually, the steady-flow boundary conditions 

require some time to re-establish themselves after having been disturbed 

by an incident wave. 

[Part of author's abstract.]

682. Schoenfeld, J. C., 1956, Getijberekening met de ARMAC [Tidal computation by the ARMAC]: Rijkswaterstaat Commun. [Den Haag, Netherlands], Nota CSD 56-9.

683. Semenido, V. I., 1956, Proverka raschetov neustanovivshegosiya dvizheniya 

v otkrytom rusle [Verification of computation of unsteady movement 

in open channel]: Gidrotekhnika i Melioratsiya [U.S.S.R.], no. 7, p. 

51-52.

The celerity of surges is measured in a canal 7,000 meters long, about 

16 feet deep, and about 60 to 70 square meters in cross section. The 

discharge is increased from 165 to 210 cubic meters per second, and then, 

after 16 minutes, is decreased to 165. Results obtained are compared with 

those of a grapho-analytical computation by V. A. Arkhangelskiy and with 

the celerity computed by the formula C= V+ijgA/B. Good agreement 

is shown between measured and computed celerities.

684. Supino, G., 1956, Onde di ampiezza crescente su moto-base permanente 

[The waves with increasing height on the nonuniform steady baseflow]: 

Energia Elettrica [Italy], v. 33, no. 11, p. 1-7.

This paper is the continuation of earlier papers published in Energia 

Elettrica (1953, 1954) and treats analytically the case of the wave having 

increasing height, which is compared with results of experiments made in 

a laboratory canal. The wave movement of small waves is generally 

studied in comparison to basic nonuniform steady flow, when the heights of 

discharges propagate with the same difference of phase from infinity 

upstream to infinity downstream, or the other way around. This study 

eliminates first and third assumptions and continues the treatment given 

in 1953, it gives a new theoretical solution. Among other results that are 

verified experimentally is the conclusion that in the propagation of wave, 

a wave exists with an important characteristic, that of increasing amplitude, 

starting from value zero.

685. Szesztay, K., 1956, Some methodical problems of flood forecasting in drainage basins of great extension: Internat. Assoc. Sci. Hydrology, 

U.G.G.I., Symposia Darcy, Dijon [France], 1956, v. 2, pub. 14, p. 

198-216.

This is a description of a method of forecasting stages that is based on 

water-level relationships. The selection of parameters is analyzed, and by a 

coaxial method of graphical correlation, the forecast graphs are developed.

686. Tsinger, V. N., 1956, Transformatsiya maksimalnykh raskhodov vodokhranilishchami [Transformation of maximum flood discharges by 

reservoirs]: Gosudarstveniy Gidrologicheskiy Inst. [U.S.S.R.] Trudy, 

no. 52 (106).
188 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

687. U.S. Soil Conservation Service, 1956, Flood routing through reservoirs: 

U.S. Soil Conserv. Service, Eng. Handb. Hydraulics, sec. 5, pt. 5.8, 

p. 1-14.

Two flood-routing methods based on the storage equation are formulated: 

(1) a graphical method, using storage-discharge curves for reservoirs and 

mass curves for inflow and outflow; (2) a semigraphical method, using the 

storage-discharge relation for reservoir inflow hydrograph, and the conversion-time interval, T, as time required for a flow as measured by one unit 

(say, 1 inch) of ordinate on the flow scale to accumulate to the same unit 

(1 inch) of storage on the storage scale.

688. Afanasev, A. I., 1957, Nekatorye voprosy ucheta transformatsii povodochnoy volny pri gidrologicheskikh prognozakh po zapasam vody v 

ruslovoy seti na primere r. Dunaya [Some problems of computation 

in flood routing for the hydrological forecasts by using the water 

storage in river channels (River Danube used as an example)]: 

Tsentralniy Inst. Prognozov [U.S.S.R.] Trudy, No. 59.

689. Befani, A. N., 1957, Voprosy teorii i rascheta stoka [Problems of theory 

and computation of runoff]: Meteorologiya i Gidrologiya [U.S.S.R.], 

no. 12, p. 6-15.

This paper commences with a general discussion of the water-balance 

equation, the genesis of water concentration, and rainfall-runoff relationships 

(given by formulas) . Surface runoff is analyzed by means of the differential 

equation of flow on a sloping surface. The continuity equation of river 

flow, considering lateral inflow per unit of channel length, is given as

Wave celerity is derived as a function of water velocity together with 

functions of coefficients representing form of longitudinal profiles, relative 

damping, and relative nonuniform side inflow. The formulas for flood 

peaks qma,x, (yield) or Qma x are given for different practical cases, taking 

into account the coefficient of transformation by river channel storage. 

The formulas for spring runoff, underground discharge, seasonal and annual 

flows are also given.

690. Blackmore, W. E., 1957, The slope-template method for some hydraulic

problems: Water Power [London], March, p. 83-88. 

The solution starts with two functions, X fi(t) and Y=f2 (f), related as 

dY/(?t=f3(X Y). Given the curve representative of the variation X, 

the slope-template method allows the immediate plotting of the curve of 

Y. The template (a device consisting of a rectangular piece of transparent 

celluloid) is described. The above equation is used for mass curves of 

inflows and outflows, and for routing floods on the basis that the outflow 

discharge (slope on the mass curve of outflow) is related to the difference of 

accumulated volume of inflow and outflow. The template has a sloping 

line for Q, and a curve to determine the relation of Q and W=P&t Q&t. 

If the hydrographs are used instead of mass curves, a curve of template is 

drawn giving the rate of increase of outflow against (Q P). Generally 

a family of template-drawn curves, one curve for each value of outflow, 

is used. The method of constructing the curves is given.

691. Bonnet, L., 1957, Contribution & 1'etude des fleuves & mare'e. Probleme 

des discontinuites [Contribution to the study of tidal rivers. Problem 

of discontinuities]: Travaux Publics de Belgique Annales, Mem. 

[Belgium], no. 5, p. 5-18.
BIBLIOGRAPHY 189

This study treats the effects on river tidal movement of marked changes 

occurring in river cross sections, due to widening or narrowing of the section 

or deepening of the channel. Simple formulas are developed for the effect 

of these discontinuities on tidal-river waves, on wave energy and amplitude. 

The results are verified, in a tidal-river model, for abrupt widening, narrowing, heightening, or deepening of channel. Good agreement is shown 

between results by formulas and results of experiments in a model.

692. Butler, S. S., 1957, Engineering Hydrology: Englewood Cliffs, N.J.,

Prentice-Hall, p. 203-212.

A brief treatment of mass curves and flood routing, which gives emphasis 

to the modified Puls method of flood routing.

693. Chertousov, M. D., 1957, Spetsialniy kurs [Hydraulics, special course, 

chapter 13 on unsteady water movement in open channels]: Gosenergoizdat [Moscow], p. 391-453,

This is an exposition on waves in channel; special emphasis is on the work 

done by Russian hydraulicians. The fundamental partial differential 

equations are derived and discussed; special attention is given to the integration of the equations in special cases. The celerity and discharge of 

waves are given. Waves of small amplitude, particularly those created 

by pondage in powerplant operation, are treated, and cases are discussed. 

Emphasis is on practical application.

694 Doodson, A. T., 1957, The analysis and prediction of tides in shallow water:

Internat. Hydrographic Rev., May.

The considerations that have led to the development of practicable 

methods of overcoming the difficulties in predicting shallow water tides 

are explained. The harmonic method of analysis and prediction of tides 

used for deep water cannot be used for shallow water with any degree 

of accuracy. The mode of generation of the shallow water tides and the 

interrelations of the harmonic constituents are discussed. The problem 

of predicting the times and heights of high and low water is then discussed 

with only a minimum of mathematical explanation. Instructions for 

analysis and prediction are given which can be followed by computers who 

are little concerned with theory. These methods, which are evolved at 

the Liverpool Observatory and Tidal Institute, have been in use since 1926. 

[Abstract from Bibliography on tidal hydraulics, Corps of Engineers, 

May 1959.]

695. Druml, F. U., and Lawler, E. A., 1957, Application of differential analyser 

and digital computers to hydraulic problems: Am. Soc. Civil Engineers 

Mtg., Jackson, Miss., 1957, p. 1-9.

The economic feasibility of the use of computers is briefly described. 

The use of UNIVAC I for solution of two partial differential equations 

for unsteady flow is discussed. Use of the Goodyear electronic differential 

analyser is described using Muskingum routing equations applied to 

determine movement of surges in sewers, and using the storage equation 

to determine pool elevations. The Burroughs E-101 computer is discussed with respect to use for unit hydrograph computations and in statistical studies.

696. Escoffier, F. F., 1957, Determining the coefficients for the Muskingum 

method directly from the physical characteristics of a uniform channel 

(and linearizing the friction term, and the similarity of the flood 

728-245 64 13
190 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

wave and canonical heat equations): U.S. Army Corps of Engineers, 

Mil. Hydrology Bull. 10, Apps. B and C, p. 37-71.

The quantities K, X, C2 for use in the conventional Muskingum floodrouting method are derived directly from the physical characteristics 

of a uniform channel. Certain sources of error in the conventional method 

are indicated and a modified method is then developed to eliminate them. 

It is shown that for a given stage there exists an ideal length of reach for use 

in the modified Muskingum method and it is suggested that this same ideal 

length will prove useful in the conventional method as well. The first portion of Appendix C shows how the friction term of the flood wave equation 

used in Appendix B can be converted to a linear form, thereby facilitating 

the process by which 1he flood wave equation can be converted to a form 

similar to the canonical heat equation. The second portion of Appendix 

C demonstrates how, by a simple change of variables, the flood wave equation can be converted to the classic canonical heat equation. 

[Author's abstract.]

697. Frank, Joseph, 1957, Nichtstationaere Vorgaenge in den Zuleitungs-und 

Ableitungskanaelen von Wasserkraftwerken [Unsteady phenomena 

in the tailrace and headrace canals of water powerplantsj: 2d. ed., 

Berlin, Springer, p. 1-333.

This is an extention of the work done collaboratively by this author and 

J. Schueller (first edition, 1938). This study elaborates on the method of 

characteristics used for the transformation of waves in narrow parts of 

channel and gives a more detailed analysis than the first study.

698. Haindl. K., 1957, Translachni vlny na nadkitickem proudeni [Translation 

waves in supercritical flow]: Vodni Hospodarstvi [Czechoslovakia], 

no. 5, p. 127-128.

This paper discusses the occurrence of secondary waves that propagate 

in the direction of slope of the channel, creating complications in a canal. 

The theory of these waves, results of observations made by specialists 

in the U.S.S.R., and measures taken to avoid occurrence of such waves are 

discussed.

699. Ippen, A. T., and Kulin, Gershon, 1957, The effect of boundary resistance 

on solitary waves: Houille Blanche [France], no. 3, p. 390-407.

This paper reports on the results of an experimental investigation of the 

attenuation of solitary wave amplitude conducted in the Hydrodynamics 

Laboratory of the Massachusetts Institute of Technology. The tests were 

in a lucite channel 32 ft. long and W% in. wide in which initial water depths 

ranged from approximately 0.2 ft to 0.4 ft. Some runs were made with the 

tank bottom artificially roughened with gravel of uniform diameter. Attenuation results are compared with available theory.

In another phase of the study, it was attempted to approach the damping 

problem by direct measurement of the transient boundary layer near the 

bottom. A special differential gage for measuring the low transient velocities was designed, and the preliminary results obtained with it are described.

700. Ippen, A. T., and Mitchell, M. M., 1957, The damping of the solitary wave 

from boundary shear measurements: Mass. Inst. Technology, Hydrodynamics Lab. Tech. Kept. 23, p. 1-50.
BIBLIOGRAPHY 191

Due to the unsteady nature of the flow under the solitary wave, the 

instantaneous shear stresses cannot adequately be described by steadystate relations except at the point of zero acceleration under the wave crest.

In considering the entire wave, the effects of the unsteady motion cancel 

out so that an average resistance coefficient can be related to a wave Reynolds number. Correlation to the Blasius theory for steady laminar flow 

over a flat plat was obtained.

As for the attentuation runs, much higher damping developed from the 

rough bottom than from the smooth bottom. The rate of damping could 

be related systematically to the absolute roughness and the wave properties.

The unsteady nature of the flow causes inertial as well as viscous effects 

in the shear measurements, which are explained by virtual mass effects on 

the roughness particles. These local inertial effects cannot be separated 

from the local viscous effects, but cancel out when the entire wave i* 

considered.

The average resistance coefficients obtained from the various waves fof 

the two roughnesses fall mostly into the transition range of Reynolds numbers beyond which the resistance coefficients seem less dependent ofl 

viscosity. 

[From authors' abstract.]

701. Ishihara, Yasuo, 1957, On the application of an electronic analog computer 

for flood routing to actual rivers: Japan Soc. Civil Engineers Trana. 

[Tokyo], no. 43, p. 43-47.

The principle of an electronic analog computer for flood routing, based 

upon Dr. Hayami's excellent theory, and its characters were already explained in the author's previous papers. In this paper, the methods of 

application of this computer to actual rivers with various figures, inflow 

from tributaries, outflow to distributaries, retardation pool for flood protection, etc., are discussed and it is found to be applicable to actual rivers, changing 

the constants contained in the computer. As some examples of application, 

the results of flood routing in the Kiso and Yodo Rivers are shown. 

[Synopsis in English at the end of transactions.]

702. Iwasa, Y., 1957, Attenuation of solitary waves on a smooth bed: Am. Soc. 

Civil Engineers Trans., v. 124, p. 193-206.

The hydraulic characteristics of solitary waves have been considered in a 

preceding paper (1955). In this paper, further development of the study 

of solitary waves is treated. Especially considered is the mathematical 

analysis of the attentuation process of solitary waves. The theoretical results are compared to experimental data of Russell and Ippen.

It should also be noticed that the total kinetic energy minus the potential 

energy is twice the amount of the kinetic energy of the vertical motion, as 

Starr (1947) has already verified. 

[Author's summary.]

703. Kalinin, G. P., 1957, Runoff calculations and forecast according to water 

storage in the river network and to its water inflow: Internat. Assoc.. 

Sci. Hydrology, U.G.G.I., Mtg., Toronto, 1957, v. 3, pub. 45, p. 78-88.

For flood-forecast purposes, water stored m the channel network of a 

river is studied with respect to its effect upon runoff and particularly upon 

flood peaks. According to the author, the approximations made by use of 

simple formulas, are justified by the disadvantage of exact routing methods.
192 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

704. Kalinin, G. P., and Milyukov, P. I., 1957, O raschete neustanovivshegosya 

dvizheniya vody v otkrytykh ruslakh [On the computation of unsteady 

flow in open channels]: Meteorologiya i Gidrologiya [U.S.S.R.], no. 

10, p. 10-18.

An approximate treatment of wave movement in river channels that 

neglects the inertia terms and assumes that the discharge Qt, at a point 

in unsteady flow, is Qi=Qo(l + J4A/S/S0), where Qo^steady flow for the 

surface slope (/So) and A/S=additional slope (<Si = *S0 +A*S). When 

A/S=0.1/S0, the error is 1 percent, and when A/S=+2Si, the error is 6 

percent. When the discharge is the same as in steady flow, the length L 

is found to be L=Qo/(2SodQ0 ldH*). This length is called the "characteristic 

reach of river channel" and can be taken as constant, regardless of change 

of levels, that involves only small errors. Also, the storage-discharge 

relationship is approximately constant for the length L. The example of 

water powerplant at Ivankovo is analyzed with characteristic reaches. 

By assuming that storage-discharge relation is linear, the coefficient of 

proportionality is analyzed, and it is shown that the outflow must be an 

exponential function for constant inflow. Assuming constant inflow during 

time intervals Ai, it is possible to determine the approximate outflows. 

The author considers the accuracy of this approximation to be close to 

that of exact methods.

705. Kishi, Tsutomu, 1957, On the highest progressive wave in shallow water: 

Public Works Research Inst. Jour. Research [Tokyo], March, v. 2, 

Research paper 4, p. 93-97.

The highest progressive wave in shallow water has been one of the most 

important problems not only in the dynamical point of view but also in 

the engineering purpose.

The author successfully applies the Rayleigh method to calculate the 

momentum which is contained in a wave, and consequently to give the 

expression of the highest wave.

The analysis gives good agreement with Messrs. F. Suquet and A. 

Wallet's experiment.

Also some characteristics of the highest wave are discussed in this paper. 

[Author's summary.]

706. Kivisild, H. R., 1957, Hydraulic studies in estuaries: Conf. Coastal Engineering, 6th, Gainesville, Palm Beach, and Miami Beach, Fla., 

Procs., p. 562-572.

There is a region in estuaries where water velocities are far below critical 

and where sea-level variations greatly affect the hydraulic conditions, but 

where still-distinct channels exist. In this region the water levels are 

usually as much influenced by tides and meteorological conditions as by 

river discharges. Floods may arise from high discharges as well as from 

storm surges. In this study relations are presented where hydrological 

and meteorological factors are included. The affected area is treated as a 

system of channels with more or less unidirectional flow in each. Frequently the flow conditions vary considerably over the length or width of 

a sea or river arm. The determination of hydraulic parameters is, therefore, quite difficult. In this paper, methods for a rational estimate of 

parameters have been shown. Using these parameters, the influences of 

channel topography, river flow, and meteorology are considered in a system 

of equations. These equations are transposed to an applicable form for
BIBLIOGRAPHY 193

integration by finite differences, which in dynamic cases could be carried 

out along characteristics.

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

May 1959.]

707. Kovalenko, E. P., 1957, Nekatorye sluchai rascheta pryamoy polozhitel'noy 

volny popuska [Some cases of computation of positive release waves]: 

Akad. Nauk [Minsk, U.S.S.R.], Inst. Energetiki Trudy [Inst. Energetics 

Trans.], v. 2, p. 165-183.

The author states that the methods of characteristics and of finite 

differences are difficult to apply, and that there is need for more simple 

computing procedures to apply to the practical problems of movement 

of surges along canals. Accordingly, he proposes an approximate method 

for the computation of a positive surge traveling along a dry natural 

channel of the regular form and having constant bottom slope. The 

discharge at the upstream end is increased from zero to discharge, Q. The 

celerity of the surge created will decrease and, by progressing, will approximate the velocity of steady movement for the discharge, Q. According 

to the author, this assumption is supported by experiments of Tifonov 

(1954). The length of the steep wavefront will increase with progression, 

and after some distance the form will be stabilized. The same results are 

obtained by the experiments of Volkov (1953), in a canal of 24.57 meters. 

Moots' work (1938) is criticized, and particularly his formula for the 

progress of wave. The author derives the relationship Q^=Qo+C^Ax> 

where C= celerity, Qo=initial discharge, AJd x=change of cross section by 

the increase of discharge, and Qx =the discharge at the moment considered. 

The celerity C=(Qi Qo)/(Ai A), where 4i=cross section for steady 

movement with Qi and A = cross section at the considered moment. Using 

two partial differential equations (neglecting the inertia term), the following formula is obtained:

, c*RdA dx

in which c and R are from Chezy's formula, and S0=bottom slope. This 

formula is applied to different channel forms, and is compared with the 

experiments; good agreement is obtained. Examples of computations 

are shown.

708. Levi, I. I., 1957, Dinamika ruslovikh potokov [Dynamics of channel 

flow; on hydraulic regimes of channel flows]: Gos. Energeticheskoe 

Izdatel'stvo [Moscow, Leningrad], p. 15-34.

The author presents the fundamental features of unsteady flow and 

characteristics of channels. The method of characteristics, as elaborated 

by Khristianovich, is given.

709. Mayer, P. G., 1957, A study of roll waves and slug flows in inclined open 

channels: Cornell Univ., Ph. D. thesis.

710. Morikawa, G. K., 1957, Non-linear diffusion of flood waves in rivers: New

York Univ., Commun. Pure and Appl. Math., v. 10, no. 2, p. 291-303.

This report presents a consideration of surface waves in which resistance

plays an important but not completely dominant role. By use of the

stretching transformations the asymptotic representation and the conse-
194 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

quent asymptotic solutions of the two basic partial differential equations 

are obtained.

711. Nougaro, Jean, and Barbe, A., 1957, Amortissement, reflexion et transmission des intumescences dans les canaux de"couverts [Absorption, 

reflection and transmission of steep waves in open canals]: Internat. 

Assoc. Hydraulic Research, 7th Convention, Lisbon, 1957, sec. D, 

p. D36, 1-21.

This paper first treats the influence of the roughness of the canal on loss 

of wave energy, determining the decrease of wave celerity in the two 

following cases: (1) high linear head losses, (2) local head loss. The decrease 

of the amplitude of the wave is defined by means of very simple relations, 

deduced from a graphical method improved by the author and applicable 

in the case of a canal having a horizontal apron, and also in the case of a 

canal having a certain slope. Tests have been made to confirm the theoretical studies for the different cases considered. The paper next treats the 

reflection and transmission of the waves in a canal where a change in 

section has occurred. Relations permitting determination of the fractions 

of the waves which are reflected and transmitted have been established; 

they are based on the fundamental theorems of hydraulics and on graphical 

methods. Tests made in the Toulouse Laboratory canals confirmed the 

solutions proposed. 

[Adapted from authors' abstract.]

712. Ray, W. E., and Mondschein, H. F., 1957, A method of forecasting stages 

on flat rivers: Am. Geophys. Union Trans., v. 38, p. 698-707. 

In the course of daily forecasting of river stages at selected points along 

the Illinois and Mississippi Rivers, it has been noticed that conventional 

mainstem forecasting procedures proved somewhat inadequate in cases 

where rivers were generally flat. Such methods usually involve multiple 

graphical correlation of two inflow with two outflow stage values, or 

Muskingum routing methods. In the study reported here, an entirely 

new approach to the problem on the Illinois River was developed. It 

involved the use of a downstream stage parameter as an index of slope 

and channel storage, and when used in conjunction with modifications of 

standard procedures for predicting discharge inflows, could more adequately 

reproduce the observed stage hydrograph at the upstream point. Results 

achieved using this new approach are demonstrated in two actual flood 

rises.

[Authors' abstract.]

713. Sandover, J. A., and Zienkiewicz, O. C., 1957, Experiments on surge waves 1

Water Power [London], v. 9, no. 11, p. 418-424.

A description of experiments made to ascertain whether the profiles of 

various types of surge waves, recorded in a model channel, correspond with 

those deduced analytically. The undular wavefront is studied. The 

experiments are described in detail. The dimensionless parameters are 

compared, taking into account the original depth, the maximum and 

minimum depth, length of the first undular wave, and surge celerity. 

Favre's and Lemoine's theoretical results for solitary wave and steep-front 

wave are compared with experimental results. Suggestions are given for 

the computation of surge celerity (by simple hydraulic jump formula) and 

for computation of the maximum height of undular wave (by simple 

solitary-wave expression).
BIBLIOGRAPHY 195

714. Schoenfeld, J. C., 1957 Analogue methods for storm surge problems: 

Rijkswaterstaat Commun. [Den Haag, Netherlands], Nota CSD 57-5.

715. Stoker, J. J., 1957, Water Waves: New York, Inter-science Publishers, 

p. 1-567.

The book is a comprehensive account of the mathematical theory of 

wave motion in liquids having a free surface and subjected to gravitational 

and other forces. Applications to a wide variety of physical problems are 

given. As a thorough discussion of wave theory, the book treats types of 

physical and mathematical problems involved as well as methods used. 

Its four parts consider: (1) basic hydrodynamics theory, (2) approximate 

wave theory based on small wave amplitude, (3) waves in shallow water 

approximated by nonlinear theory, with amplitudes not necessarily small, 

and (4) exact theory, using the exact nonlinear free-surface conditions.

Special attention is given to the methods of finite differences and characteristics of flood waves. Dam-breach waves are analyzed mathematically in full extent.

716. Syrov, Yu. P., and Kovalenko, E. P., 1957, Neustanovivshesya rezhimy v 

byefakh malykh gidroelekrostantsiy [Unsteady regimes in the headwater and tailwater canals of small water powerplants]: Akad. Nauk 

[Minsk, U.S.S.R.], Inst. Energetiki Trudy [Inst. Energetics Trans.], 

v. 2, p. 148-164.

The propagation of surges along the headwater and tailwater canals of 

small water powerplants is registered at selected stations. It is concluded 

that Lagrange's formula, C=V-\-T/gHm, for small depths gives values 

greater than those measured. A correction coefficient a is introducted so 

that C= V-\-a-\jgHm. #m=mean depth before the wave arrives, and V= 

mean water velocity, a ranging in the cases studied from approximately 

0.38 to 0.50 for the positive surges, and 0.24 to 0.42 for the negative surges.

717. U.S. Army Corps of Engineers, 1957, Flow through a breached dam: U.S.

Army Corps Engineers, Wash. Dist., Mil. Hydrology Bull. 9, June.

This bulletin describes the computation of outflow resulting from dam

breach under the following conditions of breaching: (1) relatively small

breaches of various shapes; and (2) Relatively large breaches of rectangular

shape, in which frictional resistance of flow through the reservoir becomes

an important factor.

718. U.S. Army Corps of Engineers, 1957, Artificial flood waves: U.S. Army

Corps Engineers, Wash. Dist., Mil. Hydrology Bull. 10, June. 

The bulletin provides a simplified and practical method for predicting 

the magnitude, progress, and other hydraulic characteristics of an artificial 

flood wave produced by dam destruction. This method attempts to improve upon the Muskingum method of flood routing and to produce results 

that are sufficiently accurate for military purposes.

719. U.S. Army Corps of Engineers, 1957, Regulation of stream flow for military 

purposes: U.S. Army Corps Engineers, Wash. Dist., Mil. Hydrology 

Bull. 11, June.

This bulletin presents computation methods of operation of outlet dams 

for controlled variation in river depth, width, and velocity at downstream 

points. Also included are methods for determining the magnitude, duration, and timing of reservoir releases to obtain the desired hydraulic effects 

at downstream points.
196 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

720. U.S. Soil Conserv. Service, 1957, Flood routing: U.S. Soil Conserv. Service,

Eng. Handb., Hydrology, sec. 4, suppl. A, pt. 3, 17, p. 1-28. 

In the subchapter on flood routing, three methods, based on the storage 

equation, are given: (1) storage-indication method, which uses two storage 

factors (W/At Q/2) and (TF/Ai+Q/2), and tabular integration; (2) Goodrich-Wisler method, which differs from the above method only in the plotting 

and the use of the working curves (storage factors); and (3) Wilson method, 

a graphical method that gives results equal to the above methods, when the 

storage factors, plotted on log paper, approximate a straight line, with 

slope as unity. Other methods for approximate routing are given.

721. Viti, Mario, 1957, Contribute allo stuido del moto vario nei canali [Contribution to the sudy of unsteady movement in channels]: Ingegnere 

[Italy], v. 31, no. 10, p. 903-914.

This paper treats the differential equations of unsteady flow by using 

two characteristics. Characteristics lines are given and discussed, and 

their advantages are stressed. The simplification introduced is discussed, 

and the integration of equations for characteristics is given. The numerical 

applications follow the theoretical analysis. Special attention is given 

in the paper to the treatment of varied flow by use of characteristics.

722. Yamada, H., 1957, On the highest solitary wave: Research Inst. Appl. 

Mech. Rept. [Japan], v. 5, no. 18, p. 53-67.

723. Zienkiewicz, O. C., and Sandover, J, A., 1957, The undular surge wave: 

Internat. Assoc. Hydraulic Research, 7th Gen. Mtg., Lisbon, 1957, 

Proc., v. 2, sec. D.

When sudden changes of flow occur in a channel, the well-known phenomenon of surge waves results. For a certain ratio of the average depths 

on either side of such a surge the profile resembles a breaking hydraulic 

jump. Similarity as in the case of such undular, positive waves moving 

in still water of constant depth is studied. Accurate measurements of 

profiles are taken throughout the undular range, enlarging and supplementing the original results of H. Favre. It is found that the height of 

the first wave crest in all tests closely approximates to the height of a solitary wave moving with the same velocity. The wavelength and the heights 

of the succeeding troughs and crests are considerably influenced by channel 

friction. A development of the variable flow equations originated by F. 

Serre allows a step-by-step computation to be made for the whole of the 

wave profiles. With a suitable choice of friction coefficient remarkable 

agreement between computed and measured profiles has been obtained. 

[Authors' abstract.]

724. Dressier, R. F., 1958, Unsteady non-linear waves in sloping channels:

U.S. Natl. Bur. Standards Rept. 5768, p. 1-14.

It is shown in general that the exact solution to every non-degenerate 

unsteady water wave problem in a straight channel inclined at arbitrary 

slope, governed by the non-linear hydraulic equations, can be obtained 

in terms of the complete elliptic integral of the second kin, E. By means 

of a non-Newtonian reference frame, every such wave problem for a sloping 

channel can be replaced by an associated problem for a horizontal channel. 

For the latter, the partial differential equations become reducible and thus 

permit hydrograph inversion. The Riemann integration method for the 

resulting Euler-Poisson equation yields an auxiliary function for these 

hydraulic problems which is transformable into a Legendre function and
BIBLIOGRAPHY 197

then into the elliptic integral. In particular, the procedure is applied to 

obtain the exact solution for the water wave in a sloping channel produced 

by sudden release of the triangular wedge of water (the reservoir) initially 

at rest behind a vertical wall. The behavior of the solution is exhibited 

for convenience in two level-line charts, and representative wave profiles 

and velocity distributions are presented. 

[Author's abstract.]

725. Felkel, K., 1958, Die Berechnung der nicht-stationaeren Fliessbewegung 

des Wassers in offenen Gerinnen [Computation of unsteady flow 

in open channels]: Die Bautechnik [Germany], year 35, no. 6, p. 

216-222.

An approximation of the computation of unsteady flow in a river reach 

that is made of the assumption that the discharge distribution along the 

reach can be represented as Q x ~Q BAHAL/At, where Q* is discharge, 

AL is distance from the outlet end, and Q is the discharge at the end. For 

AL = L (length of reach), Qf.=P. The method of computation based on 

this assumption is outlined. The movement of surges is analyzed, taking 

into account the effect of friction resistance.

726. Gregor, L., 1958, Modifikace Boussinesqovy rovnice nestacionarniho 

proudeni [The modification of Boussinesq's equation for unsteady 

flow]: Vodni Hospodarstvi [Czechoslovakia], v. 81, no. 3, p. 74-78.

Analysis of the equation for unsteady flow in open channels is given; 

the significance of critical flow is emphasized. Losses due to friction are 

considered to be nearly the same for unsteady flow as for steady flow. 

The Froude number is used as the criterion for the degree of wave deformation. It is shown that Forchheimer's solution for wave deformation 

can be derived from the continuity equation. Kleitz' solutions are discussed, and Boussinesq's equation for unsteady flow is extended.

727. Guerrini, Pietro, 1958, Sopra il calcolo delle onde di translazione nei 

canali prismatici [On the computation of translation waves in prismatic 

channels]: Energia Elettrica [Italy], v. 35, no. 2, p. 125-131.

This paper presents simplifications of and additions to the Craya method 

of using characteristics lines for computation of unsteady flow in channels. 

It is shown that characteristic curve lines may be replaced by straight lines. 

If the mean depth is Hm =A/B, and depth is H, then for every cross section 

(rectangular, parabolic, triangular) having constant dH/dHm, characteristic 

curve lines can be reduced to straight lines. Polygonal cross sections, 

however, would require tedious integration, as is shown by the tabulation of 

results given.

728. Halek, V., 1958, Zkushenosti z resheni hydraulickych problemu methodou 

elektroanalogie [Experience with the solutions of hydraulic problems 

employing the method of electrical analogy]: Vodni Hospodarstvi 

[Czechoslovakia], no. 10, p. 297-305.

The author describes the electric analogy apparatus which has been 

designed at the Hydraulic Research Institute in Brno for purposes of 

research. The paper sums up the experience gained in measuring and 

preparing electrical models and indicates new ways of simplifying experimental techniques. Solutions of problems involving unsteady flows are 

also discussed.
198 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

729. Isaacon, E., Stoker, J. J., and Troesch, A., 1958, Numerical solution of 

flow problems in rivers: Am. Soc. Civil Engineers Proc., v. 84, no. 

HY 5, Paper 1810, p. 1-18.

The numerical method of integrating two basic partial differential 

equations by the finite-differences procedure is described. The computation is done by a digital computer, UNIVAC. Examples show that the 

numerical computation is flexible. By this procedure the method of 

characteristics is used to determine the time-difference range for At, when 

Aa; is selected, and there is good analysis of the mesh of points (At, Ax) 

in the plane (t, x). Also, the method permits analysis of flow at river 

junctions (where backwater effects may pertain) and is therefore favored 

over flood routing methods based on the continuity equation alone. 

Comparison of flood waves computed on UNIVAC and observed flood 

waves gives good agreement in all cases when background data is sufficient 

and accurate.

730. Kalinin, G. P., and Milyukov, P. I., 1958, O raschete neustanovivshegosya 

dvizheniya vody po ruslam pri pomoshchi krivykh dobeganiya [On the 

computation of unsteady water flow along the channels by the use of 

reach-travel curves]: Meteorologiya i Gidrologiya [U.S.S.R.], no. 7, 

p. 18-25.

The flood-routing starts from the characteristic reach introduced by the 

authors (1957) with L=(Q0/S0')dH/dQo, computed from steady flow. 

Storage W is related to Q as Q=W/t, where t is constant, and is the same 

for all characteristic reaches. Using these simplifications and assumptions 

the storage equation is integrated, starting from the inflow discharge and 

developing the outflow discharge at the end of the reach. The reachtravel curve is called F(t) Q/P, where P is inflow and Q is outflow of a 

group of reaches. The formula for Q for n-characteristic reaches is developed 

and F(f) comptued, where the peak of F(f) increases with an increase of 

number of reaches. The formulas for F(f) are given, Poisson's function 

being used as a standard curve. This method is applied with good agreement to the data for unsteady flow in the Irankovo powerplant canal.

731. Kawabata, S., 1958, Study of tidal river. Graphical solution of unsteady 

flow and its application: Public Work Research Inst. Rept., Ministry 

of Construction [Japan], v. 99, March, p. 1-11.

732. Kohler, M. A., 1958, Mechanical analogs aid graphical flood routing: Am. 

Soc. Civil Engineers Proc., v. 84, no. HY 2, Paper 1585, p. 1-14.

A new graphical procedure of flood routing that is aided by the use of 

mechanic analogs is described.

Graphical techniques for routing directly on the plotted hydrograph 

charts have been discussed for (1) the case of simple linear storage-discharge 

function, (2) the case of the Muskingum equation, (3) the case of linear 

storage function in conjunction with constant lag (translation), and (4) the 

case of both lag and the storage factor varying as complex functions of 

flow in the reach. The fourth category represents what is believed to be 

one of the most flexible methods of routing yet devised. Amazingly 

reliable results have been obtained for reaches which are incompatible 

with storage functions of the Muskingum and other recognized methods 

of routing. The time required for procedure development compared 

favorably with any other known technique. Moreover, the graphical 

solutions illustrated are operationally rapid and reliable. The basic
BIBLIOGRAPHY 199

features of mechanical analogs comparable to the graphical solutions 

described have also been discussed. An analog which will facilitate 

variable K and lag has now been constructed and is undergoing tests. 

[Author's summary and conclusions.]

733. Lawler, E. A., and Druml, F. U., 1958, Hydraulic problem solution on 

electronic computers: Am. Soc. Civil Engineers Proc., v. 84, no. 

WW I, paper 1515, p. 1-38.

This paper discusses the application of three types of electronic computers and demonstrates the wide range of problems readily solved by these 

modern tools. Tedious, cumbersome, conventional methods of solution 

are replaced by quicker and frequently more accurate methods. The use 

of computer Univac I for flood routing by finite difference method (New 

York Univ. studies); the use of Goodyear Electronic Differential Analyzer 

(1954) for flood routing by Muskingum method for surges in sewers and for 

fluctuations in pool elevations and others; and the use of computer Burroughs E-101 for unit-hydrograph computations and for statistical computations, are discussed. The conditions under which the computers are 

useful are analyzed at the beginning of paper.

734.^'Linsley, R. K., Kohler, M. A., and Paulhus, J. L. H., 1958, Hydrology for

Engineers: New York, McGraw-Hill Book Co., p. 216-244. 

This is a condensed exposition of methods of flood routing, and the major 

emphasis is on flood routing procedures which are based either on the storage 

equation (for both reservoirs and channels), or on stage relations. The 

routing aids are included.

735. Moklyak, V. I., 1958, K osnovam rascheta transformatsii pavodkov [On 

the basis of flood routing]: Meteorologiya i Gidrologiya [U.S.S.R.], 

no. 1, p. 37-40.

According to the author, one of the main causes of the attenuation of 

waves is the difference in travel required for the component parts of the 

initial discharge occurring in the starting cross section. This difference 13 

due to the varying velocities at which the water travels in different filaments. The travel velocities from one cross section to another are introduced, and special travel curves are developed. This approach is applied 

to the results of experiments (GiFdenblat and others, 1948), and good 

agreement is shown with experimental observed values of attenuation. 

This study emphasizes the nonuniformity of velocity distributions in a 

cross section.

730. Nougaro, Jean, and Duffour, Pierre, 1958, Methode graphique pour le 

calcul de la propagation des intumescences dans un canal de section 

quelconque [The graphical method for computation of wave propagation in a channel of any cross-sectional shape]: Acad. Sci. [Paris] 

Comptes rendus, v. 247, no. 12, p. 860-861.

The graphical method of computing wave propagation, as developed by 

Nougaro (1955, Comptes Rendus, v. 240, p. 1689) for rectangular channels, 

and applied with some modifications and approximations to trapezoidal 

channels, is here extended to channels of any cross-sectional shape, in which 

B (width of channel) assumes the function of area (or depth).

737. Nougaro, Jean, and Duffour, Pierre, 1958, Recherches experimentales sur 

les intumescences dans les canaux de section quelconque. Application de la methode graphique S(Q) pour le calcul de la propagation des
200 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

intumescences [Experimental research on the intumescences in channels 

with any cross section. Application of graphical method S(Q) for the 

computation of propagation of intumescences]: Acad. sci. [Paris] 

Comptes rendus, v. 247, no. 25, p. 2292-2295.

The experiments of movement of surges in laboratory channels that are 

models of powerplant canals and galleries, are described. The sudden or 

linear changes of powerplant operations are studied, and the observed waves 

are compared with the waves computed by graphical constructions, the 

H(Q) and S(Q) methods as proposed by Nougaro (1955). The comparison 

shows good agreement of the two sets of results, thus justifying use of the 

proposed graphical method.

738. Pezzoli, G., 1958, Osservazioni sulle perturbazioni nei canali a pelo libero

e spora alcuni aspetti particolari dei moti ondosi [Observations on the

disturbances in canals with free water surface, and on some particular

aspects of undular motions]: Energia Elettrica [Italy], no. 1, p. 1-8.

The "small wave" concept is considered unclear in earlier works and is

defined more precisely by using three parameters: height of disturbance

or small wave amplitude (K), water depth (H), and wavelength (L). The

direction of propagation of small waves (being either in the downstream

direction or in both directions) is shown to be no criterion for determination

of the type of flow (whether subcritical or supercritical). For $/=bottom

slope, and Se=slope, in the case of critical velocity, it is derived that small

waves will not propagate upstream in moving water when ($//Sc)>(L/

2itH), which is valid for (L/H) <2.7. For L/H in limits 2.7-25, and for

(L/fZ)>25, the other conditions are given. The problem of genesis of

uniform velocity of the transporting water, associated with any wave

motion, is studied. The author contends that this generally neglected

aspect of wave motion is of interest in revealing how periodic disturbances

tend to propagate on particular profiles of steady flow, including uniform

flow.

739. Proudman, J., 1958, On the series that represent tides and surges in an

estuary: Jour. Fluid Mech. [Great Britain], v. 3, pt. 4, p. 411-417. 

Paper is concerned with a progressive wave of general form in an infinitely 

long estuary of uniform cross section when there is a permanent current 

independent of the wave. The only approximation is the neglect of friction. Explicit formulas in the form of infinite series are found for the 

surface elevation and for the current. In the special case in which there 

is no permanent current and when the oscillation at the mouth of the estuary 

reduces to a single harmonic constitutent, the first five harmonic shallow 

water constituents at any place up the estuary are evaluated. 

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

May 1959.]

740. Rockwood, D. M., 1958, Columbia Basin stream flow routing computer:

Am. Soc. Civil Engineers Trans., v. 126, pt. 4, Paper 3119, p. 32-56.

A method for using digital computers for stream-flow routing in Columbia

Basin is described. Water excesses of rainfall and snowmelt are routed

by subbasins to synthesize streamflow, which in turn is routed through

lakes and channels. A new routing technique is introduced by use of the

digital computer. The storage equation is used and linear relation of

storage and outflow discharge is applied, Ta=W]Q = dW!dH)f(dQldH).

Basic codes and routines, input data, derivation of basin storage coefficients
BIBLIOGRAPHY 201

(Ts=KQt~°-2 for Columbia), derivation of channel storage coefficients 

and lake-routing coefficients are treated. The method provides the framework for completely automatic determination of the components of streamflow in the Columbia River Basin, and the method is feasible only with 

an electronic computer. (Discussion by Willard M. Snyder, D. M. Rockwood, and Author's closure.)

741. Scholer, H. A., 1958, Tides in rivers and coastal inlets: Inst. Engineers 

[Australia] Jour., v. 30, no. 4-5, p. 125-136.

The object of the paper is (a) to describe features of tides in rivers and 

coastal inlets that might be of interest to the engineer and the hydrographic 

surveyor and (b) to present formulae and methods of computation for 

predicting tidal behavior. In Section I, after the introduction, some general features of ocean tides are described. A method of determining representative ocean tides for the purpose of estimating tidal behavior in rivers 

and coastal inlets is then presented. A general mathematical discussion follows, and the equations of tidal flow, on which the paper is based, 

are derived. The basic equations are applied to three special cases of 

tidal flow, which serve to bring out salient features of tidal behavior in 

rivers and coastal inlets. General methods of evaluating friction coefficients 

from analyses of tidal observations are applied to the Clarence River. 

In Section 2 a general method of numerical computation of tides in rivers 

and coastal inlets is presented, and an outline given of its application to 

tides in a single channel, a channel with tributaries, and a branched channel. 

[Abstract from Bibliography on Tidal Hydraulics, Corps of Engineers, 

May 1959.]

742. Sinha, G., 1958, Mathematical computation of tidal propagation in estuarine rivers: Irrig. and Power [India], v. 15, no. 1, p. 85-99.

The question of improvement of tidal rivers is usually investigated with 

the help of hydraulic scale models, since direct mathematical treatment is 

not possible owing to the complicated structure of the mathematics of 

tidal mechanics. However, there are certain inherent limitations of tidal 

models which make their operation more difficult than models of nontidal 

rivers. This naturally affects the results obtained from the model experiments to some extent. There is, however, no other way of verifying these 

results. This has encouraged attempts at mathematical treatments 

with the help of simplified approximation of the channel contour, and 

methods of mathematical computation of the tidal propagation have been 

evolved in different countries, especially in the Netherlands where the 

sea and tidal action are of profound importance. Of the different methods 

of computation the iteration has been found to be particularly useful and 

has been applied successfully for computations of many rivers. The paper 

gives, in brief, an idea of the simplifications introduced, the different 

methods of solution and their applications. In particular, the solution 

by the iteration method and its practical application have been discussed. 

[Abstract from Bibliography on Tidal hydraulics, Corps of Engineers, 

May 1959.]

743. Volkov, I. M., 1958, Obzor rezultatov issledovaniy dvizheniya volny 

popuska po sukhomu ruslu [Survey of the results from the investigation 

of release-wave movement along a dry river bed]: Akad. Nauk [Kazakh. 

S.S.R.], Inst. Energetiki Trudy, v. 1, p. 3-18
202 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

This is a short review of most of the important works on unsteady flow 

in U.S.S.R., from about 1930-48. It is concluded that a small amount of 

attenuation of waves occurs in movement along the dry channel. The 

characteristics of progressing wave heads in dry channel (infiltration, 

eddies, air-water mixture, friction losses, excessive erosion, and sedimenttransport capacity) are outlined and discussed; particular attention is 

directed toward torrents. Bernadski's (1933) formula for celerity of waves 

in dry channels is derived (C=2-\/gh) and analyzed. Some of Bernadski's 

assumptions are criticized. The use of the formula C= 2-Jgh is considered 

to be good, but due to the attenuation of wave head it is difficult to determine the height h. The results of Tikhonov's paper (1933) and Cherkasov's 

 (1932) paper are discussed, and many objections made to the assumptions, 

results, and conclusions. The results of many papers dealing with unsteady 

ilow in dry channels are analyzed, the objective being the solution of the 

problems of rivers used for floating logs. The status of the problem of 

wave movement in dry channels is given and is clearly differentiated from 

the problem of wave movement in full flowing channel. The need for more 

extensive systematic experiments is emphasized.

744. Yamada, Hikoji, 1958, Permanent gravity waves on water of uniform 

depth: Research Inst. Appl. Mech., Rept. [Japan], v. 6, no. 23, 

p. 127-139.

Levi-Civit&'s formulation (as an eigenvalue problem) of permanent 

water-wave problem, which deals with a field function, holomorphic in the 

unit circle and satisfying a certain boundary condition on the circumference, 

are extended to the periodic waves in a canal of uniform depth. Deepwater waves and solitary waves are two special cases. The application of 

an iterative procedure on this formulation supplied sufficiently accurate 

results. An example is given. Some discussions on mass transport »nd 

eigenvalue are also included. 

[Author's summary.]

745. Yano, Katsumasa, 1958, On the characteristics of the flood stream: Kyoto 

Univ. [Japan], Disaster Prevention Research Inst., Ann. no. 2.

In this paper, flood flow in river channels are theoretically solved under 

the assumptions that the acceleration term in the equation of motion is 

negligible and the Froude number is approximately constant in the small 

reach of river channel. The theoretical solution is too complex to be applied 

to real rivers, so the method of the practical numerical calculation using the 

table of Gauss' error function is proposed.

In conclusion, the flattening of the flood becomes remarkable with the 

increase in slope and resistance of river bed, but it is almost independent of 

initial water depth. The velocity of propagation of flood peak becomes 

large with the increase in slope of river bed and initial water depth.

746. Yuan-Po, Kou, 1958, Investigation of storage effects of reservoirs subjected

to superfloods: Iowa Univ. M. S. thesis.

A theoretical study to find out what effect reservoirs of various characteristics will have in decreasing the peak of increasingly greater superfloods. 

A quick, approximate method has been devised which takes the principal 

variables into account.

747. Hensen, W., 1958-59, Die Berechnung von Tidewellen in Tidefluessen 

[Computation of tidal waves in tidal rivers]: Die Kueste [Germany], 

no. 7, p. 1-19.
BIBLIOGRAPHY 203

748. Brabant, C. E., 1959, Electronic computers in flood control studies: Mil.

Engineer, v. 51, no. 342, p. 293-296.

The following computer topics are discussed: Programs, results obtained, 

improvements in application, preparation of a program, and time and ost 

advantages.

749. Charrueau, A., 1959, Sur les ondes de choc [On the shock waves]: Fonts 

et Chauss6es [France], Annales no. 5, p. 497-549.

750. Chow, Ven Te, 1959, Open-Channel Hydraulics, Part V. Unsteady Flow:

New York, McGraw-Hill Book Co., p. 523-621.

Part V of the book contains three chapters concerning the unsteady 

flow, each accompanied by long reference list (total 163 references). Chapter 18 treats gradually varied unsteady flow, considering continuity of 

unsteady flow, dynamic equation, monoclinal rising wave, dynamic equation 

for uniformly progressive flow (showing the wave profile of this flow), wave 

propagation, solution of the unsteady surface flow. Chapter 19 treats 

rapidly varied unsteady flow, considering uniformly progressive flow, the 

waving hydraulic jump, positive surges, negative surges, surges in powerplant and navigation canals, surges through channel transitions and channel 

junctions, and pulsating flow. Chapter 20 treats flood routing, considering 

the method of characteristics, the method of diffusion analogy, including 

principles and methods of hydrologic routing, and a simple hydrologic 

method of routing.

751. Darman, Z. I., 1959, O sposobakh predskazaniya srokov nastupleniya pika 

polovodya na ravninnykh rekakh [On the procedures of forecasting 

the times of peak discharge of floods on the lowland rivers]: Tsentralniy Inst. Prognozov Trudy [Central Inst. for Forecast, Trans.] 

[U.S.S.R.], fasc. 82, p. 64-82.

For the purpose of forecasting the time of occurrence of flood peaks on 

small rivers due to snowmelt, a study is made of the travel time of peaks 

of small rivers, where time of concentration of flood flow generally varies 

with the magnitude of the flood. For some rivers, the duration of peak 

period is found to be related to the magnitude of the peak, a decrease in 

period length being shown for an increase of peak discharge.

752. Dooge, J. C. I., 1959, A general theory of the unit hydrograph: Am.

Geophys. Union Trans., v. 64, p. 241-256.

On the basic assumption that the reservoir (storage) action in a catchment can be separated from translation of waves, the general equation of 

the unit hydrograph is given in function of the main parameters. The 

derivation is analytical, and existing solutions are discussed for different 

shapes of catchments. The general solution and the errors of approximation are given.

753. Dronkers, J. K., and Schoenfeld, J. C., 1959, Tidal computations in shallow 

water: Rijkswaterstaat Commun. [Den Haag, Netherlands], no. 1, 

p. 4-60.

754. Eagleson, P. S., 1959, The damping of oscillatory waves by laminar boundary layers: U.S. Beach Erosion Board Tech. Memo. 117, p. 1-38. 

The report presents the results of an analytical and experimental investigation of the shearing stresses exerted on a smooth bottom by the passage 

of oscillatory water waves. Average resistance coefficients and damping
204 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

coefficients were derived in terms of the pertinent physical properties of 

the waves, using existing small amplitude wave theory and assuming nonseparating flow in a laminar boundary layer. The measured bottom 

shearing stresses greatly exceeded those predicted by theory for the range 

of waves tested.

755. Einstein, H. A., and Harder, J. A., 1959, Electric analog model of a tidal 

estuary: Am. Soc. Civil Engineers Trans., v. 126, pt. 4, Paper 3277, 

p. 855-869.

Five hundred miles of channel in the Delta Region, California, which 

are subject to sea water encroachment, may be protected by salinity 

barriers and a master levee system. An electric analog model developed 

at the University of California, Berkeley, Calif., predicts tidal amplitude 

and flows resulting from such modifications to the hydraulic system. 

[Authors' synopsis.]

756. Felkel, K., 1959, Beitrag zur berechnung von schwall und sunk [Contribution to the computation of the positive and negative bore]: Die 

Wasserwirtschaft [Germany], no. 9, p. 242-246.

757. Guyot, M. Th., Nougaro, Jean, and Thirriot, Claude, 1959, Methode de 

calcul num6rique de la variation de niveau consecutif au passage 

d'une intumescence [Method for numerical computation of consecutive variation of elevation during the passage of an intumescence]: 

Acad. sci. [Paris] Comptes rendus, v. 248, no. 21, p. 2950-2952. 

A method of numerical computation for the elevation or depression of

consecutive water levels is presented for the passage of an intumescence.

This method is a numerical transposition of the graphical method proposed

by one of the authors (J. Nougaro, 1955, the Hague).

[Translated synopsis from the paper.]

758. Guyot, M. Th., Nougaro, Jean, and Thirriot, Claude, 1959, Calcul nume'rique des regimes transistoires dans les canaux decouverts [Numerical computation of transient regimes in open channels]: Acad. sci. 

[Paris] Comptes rendus, v. 249, p. 1858-1860.

The method of numerical computation already described (Comptes 

rendus, no. 248, p. 2950) has been applied to diverse laboratory experiments, and two examples are given: (1) the sudden stop of flow at the 

downstream end of a horizontal section having rectangular cross section; 

(2) the sudden stop of flow at the downstream end of a horizontal canal 

having trapezoidal cross section.

759. Hayashi, T., and Nougaro, Jean, 1959, Sur la similitude des regimes non 

permanents dans les canaux decouverts [On the similitude of unsteady 

regimes in open channels]: Acad. sci. [Paris] Comptes rendus, v. 249, 

no. 12, p. 1028-1030.

Different scales have been used for the similitude of nonpermanent 

regimes in open channels. The method proposed is being justified by the 

comparison of theoretical and experimental results, these last being derived 

by the similitude of laboratory experiments.

760. Hill, P. G., and Stenning, A. H., 1959, Laminar boundary layers in oscillatory flow: Am. Soc. Mech, Engineers, Paper 59-A-265, p. 1-17.

761. Holsters, H., 1959, Stabilite et convergence dans les calculs nume'riques 

du mouvement non permanent dans les rivieres par des "methodes
BIBLIOGRAPHY 205

pas-a-pas" [Stability and convergence in the numerical computations 

of unsteady movement in rivers by the step method]: C.E.R.E.S. 

[Belgium], v. 10, p. 389-450.

762. Ishihara, Yasuo, 1959, On an electronic analog computer for runoff: 

Japan Soc. Civil Engineers Trans., no. 60, January.

The fundamental equation is derived in diffusion type including convection by introducing the longitudinal mixing process of flows as in 

flood routing presented by Hayami, and by considering the areal fluctuations of rainfalls. The equation is two dimensional and nonlinear.

Integrating the above equation along contour lines of catchment basin 

and after some approximations, the modified equation, which is onedimensional but nonlinear, is derived. The electric circuit of the direct 

analog computer, therefore, is virtually equivalent to the one for flood 

routing.

Some examples of runoff estimations in the Yura River Valley, which 

is the model drainage basin having about 350 sq km are conclusively described. The results by this computer method indicate a good coincidence 

with the actual hydrographs in this basin.

763. Kalinin, G. P., and Milyukov, P. I., 1959, Prognoz i raschet uravney vody 

nizhe GES na osnove vodomernikh nablyudeniy [The forecast and 

computation of water levels downstream from powerplants, on the 

basis of the stage observations]: Meteorologiya i Gidrologiya 

[U.S.S.R.], no. 8, p. 13-19.

A brief discussion of forecasting methods based on stage observations is 

given. An analytical method is presented, based on a procedure employing a short time interval At, during which it can be assumed that 

travel time T=dW/dQ is constant (W= storage of the reach, Q = outflow 

discharge of the reach) and that there is a linear change of inflow (q) 

along the reach. For Ai<^ r the continuity equation gives AQ = (qm Q)K1} 

in which Ki 1 exp ( Ai/r), gm =mean q. KI is to be determined: 

(1) by W=f(Q), which is determined from channel morphology; (2) from 

the change of discharge in the upstream and downstream gage stations 

as Ki=AQ/(qm Q) (3) by the travel time of individual flood wave phases; 

(4) by an empirical approach. These various ways are discussed, and 

practical application is made.

764. Kunstatsky, J., 1959, Pohyb translacni vlny v prazdnem koryte [Movement of a translation wave in dry bed]: Vodohospodarsky Casopis 

[Czechoslovakia], v. 8, no. 4, p. 287-299.

765. Laboratoire National d'Hydraulique, 1959, Mouvement nonpermanent dans 

les canaux a section prismatique et variable quelconque [Unsteady 

movement in channels with prismatic or any variable cross section]: 

Lab. Natl. Hydraulique, Chatou [France], Rept. T218B.

766. Laboratoire National d'Hydraulique, 1959, Intumescence consecutive a 

une variation brusque de d6bit [Intumescence following a sudden 

discharge change]: Lab. Natl. Hydraulique, Chatou [France], Rept. 1, 

T220B.

767. Laurenson, E. M., 1959, Storage analysis and flood routing in long river

reaches: Jour. Geophys. Research, v. 64, no. 12, p. 2423-2433. 

Numerous flood-routing studies have indicated that the maximum length 

of reach through which a flood can be routed in a single step is such that 

728-245 64 14
206 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

time of travel through the reach does not exceed about half of the period of 

rise of the inflow flood. This paper presents a method of analyzing the 

storage characteristics of reaches considerably longer than this by arbitrarily inserting a number of hydrographs between the inflow and outflow 

hydrographs in such a way as to produce single-valued storage-discharge 

relations for the shorter reaches so formed. These storage-discharge 

relations are then used in conjunction with a graphical flood-routing procedure to route the inflow successively through the several short reaches 

and so reproduce the outflow hydrograph. 

[Abstract from Journal of Geophysical Research, December 1959.]

768. Liggett, J. A., 1959, Unsteady open channel flow with lateral inflow: Stanford Univ. Tech. Kept. 2, p. 1-73.

769. Ludewig, Dietrich, 1959, Rechenschieber zur Berechnung des Seerueckhaltes [Slide rule for the computation of storage in reservoirs]: Wasserwirtschaft-Wassertechnik [Germany], no. 3, p. 100-103.

Based on the storage equation and the use of time intervals At, the slide

rule is designed to compute the storage in the movement of a flood wave

through the reservoir. Scales on the fixed part of the slide rule are used

thus: the lower scale for P (inflow) and Q (outflow), and the upper scale for

HI and HZ (the heights at the beginning and the end of At); scales on the

moving part are used thus: the lower scale for W/At and the upper scale for

HI and H2 . In this first form the slide rule is used for two procedures:

when different openings of an outlet under pressure are operated and when

both the spillway and bottom outlet are operated. In a second form the

slide rule is used for fixed outlet shapes (without moving gates or valves),

and the At is constant.

770. Mayer, P. G., 1959, Roll waves and slug flows in inclined open channels:

Am. Soc. Civil Engineers Trans., Paper 3158, p. 505-564 [1961]. 

Roll waves and slug flows are established as two distinctly different wave 

patterns and are phenomenologically studied in an inclined open channel. 

The basic characteristics of flow are expressed in terms of pertinent physical 

properties. A theory is presented and reference is made to similar phenomena in allied fields in which mathematical analyses exist. It is hoped 

that this study will extend the present knowledge (1961) of unsteady 

phenomena in open channel flow. (Discussions by F. F. Escoffier, R. H. 

Taylor, J. F. Kennedy; Tojiro Ishihara, Yuichi Iwagaki, and Yoshiaki 

Iwasa; and author's closure.) 

[Author's abstract.]

771. Mkhitaryan, A. M., 1959, O valnakh na bystrotoke [On the waves in a 

steep canal]: Akad. Nauk [U.S.S.R.], Izv., Odelenie Tekhnicheskikh 

Nauk, Energetika i Avtomatika, no. 1, p. 90-99.

Waves created in steep channels are analyzed. The fundamental equations are given, as are the conditions for the creation of wavetrain, the 

solution of fundamental equations, and analysis of results. The celerity of 

waves is derived, examples are given, and conclusions are drawn.

772. Nash, J. E., 1959, A note on the Muskingum flood-routing method: Jour.

Geophys. Research, v. 64, no. 8, p. 1053-1056.

An exact method of solution of the flood-routing equation, when the 

storage is a linear function of weighted inflow and outflow, is developed. 

This operation is shown to be equivalent to routing a multiple of the inflow
BIBLIOGRAPHY 207

through reservoir storage and subtracting the excess inflow. Modified 

coefficients for the Muskingum equation are developed which do not 

depend on the routing interval being small relative to K. 

[Author's abstract.]

773. Nechaeva, N. S., 1959, Naibolee rasprostranennye v S. SH. A. methody 

rascheta transformatsii voln povodkov pri peremeshchenii po rechnomu 

rusle [The methods most currently used in the U.S.A. for the flood 

routing along rivers]: Tsentralniy Inst. Prognozov [U.S.S.R.], Trudy 

(Proc.), no. 94, p. 79-91.

774. Pavlov, G. G., 1959, Opredelenie vysoty podpornoy volny v derivatsii 

GES pri nalichii sliva [Determination of the height of a surge wave in 

the canal of a water powerplant, equipped with spillway]: Gidrotekhnicheskoe Stroitel'stvo [U.S.S.R.], no. 4, p. 43-45.

A graphical method is geven for the computation of wave height in the 

case of a canal having a spillway, where a decrease in powerplant discharge 

creates the spill of water and a small wave traveling upstream. From 

spillway rating and the channel-wave characteristics y=}(Q), where y=wave 

height, and Q=change of discharge passing under the wave, the intersection of curves gives the wave height and Q.

775. Poggi, B., 1959, Correnti stratificate. Onde di traslazione in alvei prismatice [The stratified flows. The translation waves in prismatic channels]: 

Energia Elettrica [Italy], no. 8, p. 685-691.

770. Rubbert, F. K., 1959, Die Tiberechnung als Problem der numerischen 

Analysis [Tidal computation as a problem of numerical analysis, a 

report of the principles and methods]: Mitteilungsblatt der Bundes 

Anstalt fuer Wasserbau [Karlsruhe, Germany], v. 12, p. 29-51.

777. Sapozhnikov, V. I., 1959, Ob ispol'zovanii krivykh dobeganiya vody dlya 

prognoza stoka [The use of curves of water travel time for the flow 

forecast]: Tsentralniy Inst. Prognozov [U.S.S.R.] Trudy (Proc.), no. 

84, p. 54-64.

The curves of water-concentration, or water-travel, time from a part 

of a river basin to the gaging station are used as a basis for flow forecast. 

The study is analytical, emphasizing the problems usually associated with 

the synthetic hydrographs and synthetic unit hydrographs.

778. Schnoor, E., 1959, Anwendung des "Differenzenverfahrens" bei der Tidewellenberechnung in den von den gezeiten beeinflukten Fluessen 

[Application of "finite-differences method" in tidal wave computation 

in the tidal rivers]: Der Bauingenieur (Germany), v. 34, no. 6, p. 

231-240. Discussion of this paper by D. Rose, Bauingenieur [Germany], v. 35, no. 8, p. 327.

The finite-differences method as applied to the two De Saint-Venant 

partial differential equations is given and discussed. Once Ax is selected, 

the selection of At is limited by giving the limit value for At/Ax. The 

procedure and the comparison with tidal model experiments are discussed.

779. Schoenfeld, J. C., and Verhagen, C. M., 1959, Development of the tidal 

analogue technique in Holland: Internat. Days for Analog Computatations, Strasburg [France], Sept. 1-6, 1958, p. 376-380.
208 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

780. Scholer, H. A., and Germanis, E., 1959, Unsteady flow in rivers and artificial canals: Inst. Engineers, Civil Engineering Trans. [Australia], 

v. CE I, no. 1, p. 27-37.

This paper is concerned with unsteady flows in rivers and artificial 

canals produced by disturbances of the order of magnitude of floods and 

tides. With such disturbances the vertical accelerations of the water 

particles are negligible compared with the horizontal accelerations.

The object of this paper is to demonstrate a new approach to problems 

of unsteady flow in rivers and artificial canals. Cases of unsteady flow 

are described. General formulas, based on the method of characteristic 

lines, are developed from the point of view of programming for an electronic digital computer. A specific problem is selected and the general 

scheme of programming this problem for the computer is described. 

[From authors' summary and introduction.]

781. Sekerz-Zen'kovich, T. Ja., 1959, Propagation of a free tidal wave in a 

canal of variable depth: Akad. Nauk [U.S.S.R.], Morskoy Gidrofisicheskiy Inst. Trudy, v. 18, p. 85-93.

782. Thirriot, Claude, 1959, Sur les Phe"nomenes de propagation d'une onde de 

choc dans une galerie de fuite d'usine hydroelectrique souterraine. 

[On the phenomena of shock-wave propagation in a tailrace gallery 

of an underground water power station]: Acad. sci. [Paris] Comptes 

rendus, v. 249, p. 2716-2718.

The case of a positive intumescence moving in a gallery, when the bore 

partly touches the ceiling of the gallery, is analyzed analytically by neglecting the effect of the air on the water mass.

783. Tonini, Dino, 1959, Su di una generalizzazione della equazione di continuita nel campo idrologico [A generalization of the continuity equation in hydrology]: Italian Natl. Hydraulics Conf. Rept., p. 1-17 

(manuscript presented to conference).

The continuity equation

is analyzed, and some characteristics of specific volumes and times are 

introduced and discussed. Integration of the above storage or continuity 

equation is given for specific inflow hydrographs, and the resulting outflow 

hydrographs are developed, taking into account the type of storagedischarge relationship involved.

784. U.S. Army Corps of Engineers, 1959, Design memorandum no. 8, Columbia 

Lock and Dam, Control and regulation of Walter F. George powerhouse inflows analyzed by means of Univac I electronic computer; 

Appendix, Univac I computer program for routing flows through 

Columbia Reservoirs; Part I, Program setup, code and operating instructions ;Part II, Flow Charts: U.S. Army Corps Engineers, Mobile 

Dist., December.

This is a study outlining the basic regulation plan for the Columbia pool 

under normal operating conditions. The study includes an analysis of 

the effects of these inflows on stages and mean velocities at regular intervals 

along the entire length of the pool. The analysis is based on the results 

of wave-routing computations performed by means of a UNIVAC I elec-
BIBLIOGRAPHY 209

tronic computer. Memorandum No. 8 presents the UNIVAC program 

in detail, including the basic equations, flow charts, annotated code, and 

operating instructions.

785. Uspenskiy, P. N., 1959, Wave propagation in a channel subject to rotational 

motion: Akad. Nauk [U.S.S.R.], Morskoy Gidrofisicheskiy Inst. Trudy, 

v. 15, p. 17-33.

786. Vapnyar, D. U., 1959, Vliyanie treniya na dvizhenie svobodnoy prilivnoy

volny v kanale postoyannogo secheniya [Effect of resistance on the

movement of tidal wave in a channel of constant cross section]:

Meteorologiya i Gidrologiya [U.S.S.R.], no. 7, p. 21-25.

The form of a wave of type y=yo cos at that is propagated in a narrow

channel is given as an equation that is the product of exponential and

cosine functions. The celerity is given as C=-\/2gH/(-\/l-}-fi+l), where

n=k/a, in which fc= coefficient of friction. The friction has negligible

influence on C for £T<OOO m. The friction has greater influence on the

short wave than on the long wave.

787. Whittington, R. B., 1959, Reservoir flood influx problems: Inst. Water 

Engineers Jour. [Great Britain], v. 13, no. 7, p. 628-632.

788. Yevdjevich, V. M., 1959, Computation of the outflow from a breached dam:

U.S. Natl. Bur. Standards Rept. 6473, p. 1-27.

The openings of breached dams are classified as small, medium and 

large openings. For the first two, the procedures are given for computation 

of the outflow hydrographs after the breach, using Savage River Reservoir, 

Maryland, as an example.

789. Yevdje\ich, V. M., 1959, Analytical integration of the differential equation 

for water storage: U.S. Natl. Bur. of Standards Jour. Research, B. 

Math, and Math. Phys., 63 B, no. 1, p. 43-52.

The integration of the storage differential equation at present is usually 

done mostly by graphical or numerical procedures. An approach to the 

analytical integration of that equation is the subject of this paper. A 

new method of fitting the given background curves by mathematically 

tractable expressions is introduced. The storage-outflow discharge relation 

is expressed in the form of a power function. A general differential equation 

for water storage y' + cPy2 cyk = o is derived, with c and k constant for 

the given reservoir, outflow shape and type of flow, and P being the inflow 

hydrograph. The analytical solutions of this equation for P oP constant, 

and certain P=f(f) are given for the integrable cases [tables 1 to 3, eqs. 

(12) to (29)]. The application of the results obtained is discussed at the 

end of the paper. 

[Author's abstract.]

790. Yevdjevich, V. M., 1959, Effect of sudden water release on the reservoir

free-outflow hydrograph: U.S. Natl. Bur. Standards, Jour. Research,

B. Math, and Math. Phys., 63 B, no. 2, p. 117-129.

The free-outflow hydrograph is studied in the case of a sudden water

release from a reservoir. The outflow hydrograph is called free when it is

not affected by the tailwater levels. Both the effect of the steep negative

wave, created by sudden water release, and the effect of the flow resistance

are analyzed through the use of a fictitious inflow hydrograph. The

water accelerated by the steep negative wave movement, whether or not

the flow resistance is taken into account, gives this fictitious hydrograph.
210 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

The procedures are given for the computation of the wave celerities and 

the new water velocities along the reservoir, for the computation of wavefront heights and the maximum outflow discharge, and for the determination of the fictitious and total inflow hydrograph. The examples are given 

for the pyramidal reservoirs of rectangular and parabolic cross sections. 

A general procedure for the determination of the reservoir free-outflow 

hydrograph in an approximate form is derived. The example of the 

Savage River Reservoir (Maryland) is given. 

[Author's abstract.]

791. Yuy, Vi-Chzhun', 1959, Krivaya ruslovogo dobeganiya i vliyanie na nee 

izmeneniya chisla kharakternykh uchastkov [The curve of channel 

travel time, and the influence on it by the change of number of characteristic reaches]: Meteorlogiya i Gidrologiya [U.S.S.R.], no. 11, p. 

27-34.

The author starts from the Kalinin and Milyukov paper (1958), which 

introduces the curve of travel time and the length of characteristic reach. 

This length is small for small rivers. After discussing the properties of the 

approach by Kalinin and Malyukov, the author proves that for the interval 

of time At it is permissible to use average discharges in computations. 

At is used in functional relationship to discharge change in the given reach. 

The analysis of influence of the number of characteristic reaches on the 

curve of travel time is given in tables, thus representing (as a parameter) 

the number of computational time-units required. Results of the study 

have been applied in some sections of the Amur River.

792. Zheleznyak, I. A., 1959, Transformation of floods by a system of reservoirs: 

Hydrologic Convention, 3d, [U.S.S.R.], v. 4: Section for the water 

resources development, Leningrad, p. 107-114.

793. Honda, M., 1959-60, A theoretical investigation of the interaction between 

shock waves and boundary layers: Tohoku Univ. [Japan], Inst. High 

Speed Mech. Repts., v. 11, p. 23-53.

794. Annemueller, H., 1960, Der Einfluss des Turbinenschnellschlusses auf die 

Schiffahrt in Kraftwasserstrassen. [The effect of rapid shutoffs of 

turbines on the navigation along canals of water power stations]: Der 

Bauingenieur [Germany], no. 9, p. 338-342.

For a sudden shutoff of turbines, the height of the surge is

and the celerity of the surge is

795. Ball, F. K., 1960, Finite tidal waves propagated without change of shape:

Jour. Fluid Mech. [Great Britain], v. 9, pt. 4, p. 506-512. 

Coriolis terms are introduced into the equations governing the motion 

of a finite tidal wave. Various types of solution are found for waves which 

travel without change of shape, and which are periodic with sharp crests 

and broad troughs. The classical result that such waves cannot be propagated without change of shape is therefore untrue in these circumstances. 

[Author's abstract.]
BIBLIOGRAPHY 211

796. Bezanov, K. A., 1960, K teorii difraktsii Udarnykh voln. [Contribution 

to the theory of diffraction of shock waves]: Prikladnaya Matematika 

i Mekhanika [U.S.S.R.J, v. 24, no. 4, p. 718-722.

This paper investigates the diffraction of impacting waves at the straight 

bank, which has a small angle in comparison with the direction of movement of the front of impact wave.

797. Carter, R. W., and Godfrey, R. G., 1960, Storage and Flood-Routing: 

U.S. Geol. Survey Water-Supply Paper 1543-B, 104 p.

The basic equations used in flood routing are developed from the law 

of continuity. In each method the assumptions are discussed to enable 

the user to select an appropriate technique.

In the stage-storage method the storage is related to the mean gage 

height in the reach under consideration. In the discharge-storage method 

storage is determined from weighted values of inflow and outflow discharge. 

In the reservoir-storage method storage is considered as a function of 

outflow discharge alone.

A detailed example is given for each method to illustrate that particular 

technique. 

[Authors' abstract.]

798. Gessner, Walter, 1960, Schwall, Sunk and Wassersprung. Berechnung 

auf der Grundlage dimensionloser Kenngroessen. [The positive and 

negative bores, and hydraulic jump. Computation with dimensionless values]: Der Bauingenieur [Germany], no. 8, p. 286-291. 

The formulas for surges are derived for a prismatic channel and a horizontal 

bottom, and for the conditions when: the front of surge does not change 

by progressing; the friction resistance is replaced; the discharge is changed 

suddenly; and the velocities are uniformly distributed in a cross section. 

The formulas, tables, and graphs for the computation of characteristic 

surges are expressed in dimensionless values (£=AA/A) and in Froude 

number.

779. Goroshnikov, E. A., 1960, Transformatsiya makjimal nykh urovnejr 

vesennego polovodya v ustevom uchastke Severnoy Dviny [Transformation of maximum level of spring flood in the estuarine portion 

of the North Dvina]: Meteorologiya i Gidrologiya [U.S.S.R.], no. 

8, p. 38-41.

The attenuation of peak levels of floods is studied along 140 km long 

mouth reach of North Dvina River. An exponential function is fitted 

to the relation of peak level and position x of cross sections along this 

reach. From the error of a sine type function of x, the other fitting is 

a sum of exponential and sine functions, and the third order fitting adds 

also a third linear function.

800. Guyot, M. Th., Nougaro, Jean, and Thirriot, Claude, 1960, Me*thode

de calcul numerique des phe"nomenes transitoires dans un canal

ouvert [Method of numerical computation of transient phenomena

in an open channel]: Acad. sci. [Paris] Comptes rendus, v. 250, p. 55-57.

This paper presents the principle and the realization of a more rigorous

method for numerical computation of level variation due to the passage

of an intumescence than was presented in two preceding papers by the

same authors (Comptes rendus, v. 248, p. 2950 and v. 249, p. 1858; see

items 757 and 758). The flow chart for the digital computer is given.
212 FLOOD-EOUTING METHODS AND UNSTEADY FLOW

801. Guyot, M. Th., Nougaro, Jean, and Thirriot, Claude, 1960, Etude 

numerique des regimes transitoires dans les canaux [Numerical study 

of transient regimes in channels]: Houille Blanche [France], no. B., 

p. 814-832.

Numerical calculation on an IBM 650 ordinator of the propagation of 

solitary waves in free flow canals, considering transient conditions. The 

three assumptions used are:

(1) Breakdown of the movement into successive steps;

(2) Negligible vertical accelerations;

(3) Small wave heights above the initial water level. 

Starting from Saint Venant's equations continuity and dynamic equations 

and with the above assumptions, the two following methods are developed:

(a) Numerical transposition of a graphical method put forward by 

M. Nougaro at the 6th General Meeting of the IAHR (Paper D-5, the 

Hague, 1955);

(b) A method directly based on the theory of characteristics which though 

more complex, is also more rigorous. 

[Abstract from Houille Blanche.]

802. Hampel, R., 1960, Bruchversuch an einer Bogensperre der Wildbachverbauung [Dam breach experiment on the arch dam for sediment 

control]: Oesterreichische Wasserwirtschaft [Austria], nos. 8 and 9, 

p. 187-193.

803. Kalinin, G. P., and Levin, A. G., 1960, Ispolzovanie electronnoy modeliruyushchey ustanovki dlya prognoza dozhdevikh povodkov [The 

use of electronic model device for the forecast of rainfall floods]: 

Meteorologiya i Gidrologiya [U.S.S.R.], no. 12.

804. Kalinin, G. P., and Milyukov, P. I., 1960, Raschet uravney vody v nizhnikh byefakh GES [Computation of Water levels in tail-race canals 

of water power plants]: Meteorologiya i Gidrologiya [U.S.S.R.], 

no. 2, p. 19-23.

The computation of water levels below a hydroelectric powerplant 

is carried out by using the cross section of one half of characteristic river 

reach, or

l=(Q/2S)(dH/dQ),

and the flow rating curve Q=f(H). Good agreement between the computed and observed level fluctuation at that cross section is shown.

805. Kalinin, G. P., Milyukov, P. I., and Nechayeva, N. S., 1960, Prostaya 

elektronnaya modeliruyushchaya ustanovka dlya prognoza pavodkov 

[The simple electronic model equipment for the flood forecast]: Meteorologiya i Gidrologiya [U.S.S.R.], no. 8, p. 20-28.

The basic formulas and principles underlining the electric analog 

integrating device are given. The device is described and the electric 

schematics are given. Discussion of application of device ends the paper.

806. Klemens, Vit, 1960, Zjednodusene reseni transformace povodnove vlyn 

[Simplified solution of flood wave routings in reservoirs]: Vodohospodarsky Casopis [Czeckoslovakia], v. 8, no. 4, p. 317-326.

807. Laitone, E. V., 1960, The second approximation to cnoidal and solitary 

waves: Jour. Fluid Mech. [Great Britain], v. 9, pt. 3, p. 430-444.
BIBLIOGRAPHY 213

The expansion method introduced by Friedrichs (1948) for the systematic 

development of shallow-water theory for water waves of large wave length 

was used by Keller (1948) to obtain the first approximation for the finiteamplitude solitary wave of Boussinesq (1872) and Rayleigh (1876), as well 

as for periodic waves of permanent type, corresponding to the cnoidal 

waves of Korteweg and de Vries (1895).

The present investigation extends Friedrich's method as so to include 

terms up to the fourth order from shallow-water theory for a flat horizontal 

bottom, and thereby obtains the complete second approximations to both 

cnoidal and solitary waves. These second approximations show that unlike the first approximation, the vertical motions cannot be considered 

as negligible, and that the pressure variation is no longer hydrostatic. 

[Author's abstract.]

808. LeMehaute, Bernard, 1960, Periodical gravity wave on a discontinuity: 

Soc. Civil Engineers Am. Proc., v. 86, no. HY 9, Paper 2646, p. 11-41; 

Am. Soc. Civil Engineers Trans., v. 126, Paper 3212, p. 1006-1036. 

A general theory is presented to study the transmission and reflection 

of gravity waves on a discontinuity to the first order of approximation. 

Particular attention is given to the cases of an obstruction, a change 

of width, and a change of depth. In the latter case only, the gravity wave 

arriving at an angle is studied. 

[Author's synopsis.]

809. Nash, J. E., 1960, A unit hydrograph study with particular reference to 

British catchments: Inst. Civil Engineers, Proc., v. 17, p. 249-282. 

The moments of the instantaneous unit hydrograph are correlated with 

the topographical characteristics for a large number of British catchments, 

and a general equation for the instantaneous unit hydrograph chosen. 

The use of the correlation to predict the hydrograph for catchments where 

sufficient data on rainfall and streamflow are not available is explained, 

and examples given. 

[Author's synopsis.]

810. Nechaeva, N. S., 1960, Opredelenie kriteriya rasplastyvaniya pavodochnykh 

voln i voln popuskov [Determination of the attenuation criterion of 

flood waves and of release waves]: Tsentralniy Inst. Prognozov 

[U.S.S.R.] Trudy (Proc.), no. 105, p. 53-66.

As an analogy to reservoir effects, the attenuation of schematic regular 

flood hydrographs along a river channel is studied. For a river reach, the 

ratio k=QdlQu (Qi=downstream peak discharge, Qu =upstream peak 

discharge) is expressed in different simplified ways: kl=l W/12Q, W= 

storage in river reach, SQ=hydrograph volume; or fe (1 TF/2SQ) 

(1 aW/22Q), where a = correction coefficient; or fc3=(l TF/2SQ) (1 

a!TT/2ZQ) (1 a2TF/2SQ); and so on, in which final kn=(l W/N2Q)irt 

where N= 1-5. The next formula is given as k=f(N, t/r), where r dW/dQ, 

and t=duration of unit flood release.

811. Nougaro, Jean, Thirriot, Claude, and Guyot, M. T., 1960, Etude numerique 

des regimes transitoires dans les canaux d6conuverts [Numerical study 

of transient regimens in the open channels]: Conf. on Hydraulics, 

Budapest [Hungary], 1960, rept. 6, p. 1-10.

This paper summarizes some of the works already published by the three 

authors. The method of characteristics is applied for the intumescences
214 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

traveling along a canal. The integration is carried out by finite differences, 

programmed for a digital computer. The principles and examples of 

computation flow charts are described, and examples of computations are 

given.

812. Partenscky, H. W., 1960, Waves in navigation canals due to lock filling: 

Am. Soc. Civil Engineers Proc., v. 86, no. WW 1, p. 85-122. Discussion by Escoffier, F. F., v. 86, no. WW 3, p. 163-171. 

Wave celerities and wave deformations along canals, created by navigation lock operations, are studied. The special emphasis is on the wave 

height, wave celerity, wave deformations caused by changes in cross section, 

theoretical investigations of the fillings phenomena, formation of front of 

negative waves, profiles of waves due to lock filling, deformation of trailing 

part of waves, and so forth. Model tests are used to verify the wave theory, 

the analysis of the surge being measured in the prototype.

813. Rose, D., 1960, Ueber die quantitative Ermittlung der Gezeiten und 

Gezeitenstroemie in Flachwassergebieten mit dem Differenzverfahren 

[On the quantitative determination of tides and tidal flows in shallow 

water regions by finite-differences method]: Francius-Inst. Mitt. 

[Hanover, Germany], no. 18, p. 1-159.

814. Sinha, G., 1960, Methods of mathematical computation for tidal rivers and 

their application as practiced in the Netherlands: Inst. Engineers 

Jour. [India], v. 40n., no. 9, pt. 1, p. 556-570.

815. Supino, Giulio, 1960, Sopra le onde di traslazione nei canali, La equazione 

linearizzata {On the translation waves in channels, a linearized equation]: Accad. Nazionale dei Lincei [Rome] Rend. (Class of Physical, 

Math., and Natural Sciences), ser. 8, v. 29, fasc. 5 and 6, Note I, 

Nov., p. 239-243 and Note II, Dec., p. 472-476.

In the first note a particularly simple procedure is developed for the 

limitations and approximations related to the attenuation of waves and to 

their propagation celerity under linearized conditions. The solution of the 

two De Saint-Venant linearized partial differential equations has the form

A. f <p=-z 0>* sin v ( t 

which does not have a general character. The solutions represent only the 

waves propagating with constant celerity (equal for discharges and levels), 

called the "regime waves." In the second note, some limitations and 

approximations for t\ (coefficient of attenuation) and a (wave celerity) 

are analyzed.

816. Ter-Krikorov, A. M., 1960, Sushchestvovanie periodicheskikh voln 

vyrozhdayushskikhsya v uedinennuyu [The existence of periodic 

waves degenerating into a solitary wave]: Prikladnaya Matematika 

i Mekhanika [U.S.S.R.], v. 24, no. 4, p. 622-636.

The long wave on the water surface degenates into a solitary wave, when 

the wavelength tends to be infinite (cnoidal waves). The paper gives 

proof of the existence of cnoidal waves, which is valid for the total range of 

cnoidal waves.

817. Urban, Jaroslav, 1960, Transformace povodonve vlny v nadrzi za soucasne 

manipulace se stavidly [Transformation of flood wave in reservoir
BIBLIOGRAPHY 215

with simultaneous maneuver of control gates]: Vodohospodarsky 

Casopis [Czeckoslovakia], v. 8, no. 4, p. 299-316.

818. Ursell, F., 1960, Steady wave patterns on a nonuniform steady fluid flow:

Jour. Fluid Mech. [Great Britain], v. 9, pt. 3, p. 333-346. 

A steady slightly nonuniform flow with a free surface is subject to a 

concentrated surface pressure which gives rise to a pattern of surface waves. 

(For gravity waves on deep water this is the well-known Kelvin ship-wave 

pattern.) The motion is assumed inviscid, and the waves are assumed 

small. A theory is developed for the wave pattern, based on the following 

assumptions:

(1) The stream velocity component normal to a wave crest is equal 

to the phase velocity based on the local wavelength.

(2) The separation between consecutive crests is equal to the local 

wavelength.

These assumptions are expressed in mathematical form, and the existence 

of a set of characteristic curves (associated with the group velocity) is 

deduced from them. These characteristics are not identical with the 

crests. Let the additional assumption be made that:

(3) The characteristics all pass through the point disturbance; the 

characteristics are then completely defined and may be constructed by a 

step-by-step process starting at the point disturbance. The same construction gives the direction of the wave crests at all points. The wave 

crests can then be deduced.

Assumptions of the same type as (1) and (2) have long been familiar 

in various applications of ray tracing. For uniform flows the present 

theory gives the same pattern as the method of stationary phase. 

[Author's abstract.]

819. Wiegel, R. L., 1960, A presentation of cnoidal wave theory for practical

application: Jour. Fluid Mech. [Great Britain], v. 7, pt. 2, p. 273-286.

Cnoidal wave theory is appropriate to periodic waves progressing in

water whose depth is less than about one-tenth the wavelength. The

leading results of existing theories are modified and given in a more practical

form, and the graphs necessary to their use by engineers are presented.

As well as results for the wave celerity and shape, expressions and graphs

for the water particle velocity and local acceleration fields are given. A

few comparisons between theory and laboratory measurements are included.

[Author's abstract.]

820. Zhidkov, A. P., 1960, Raschet neustanovivshegosiya dvizheniya vody po 

metodu G. P. Kalinina i P. I. Milyukova dlya nizhnego byefe Rybinskoy GES pri sutochnom regulirovanii [Computation of unsteady 

water flow by the method of G. P. Kalinin and P. I. Mi yukov for the 

tailrace level of Rybinsk water powerplant during daily regulations]: 

Tsentralniy Inst. Prognozov [U.S.S.R.] Trudy (Pioc.), no. 96, p. 

85-126.

In this report the concept of characteristic river reach is used. The 

reach is determined under the condition that the discharge of that reach is 

uniquely determined, and that the storage effect is the same as that of a 

reservoir storage. Storage is L=(Q/S) (dH/dQ), in which Q, S and H are 

discharge, slope, and level of water surface, respectively. The simple 

storage equation is applied for flood routing. The example of the Ribinsk 

powerplant is used to compare this method, the method of finite differences
216 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

applied to two partial differential equations, and the observed fluctuations 

in flood routing. The precision of anticipated fluctuations was the same for 

both computational methods. The use of Kalinin-Milyukov method is 

recommended, and suggestions for further research are given.

821. Benney, D. J., 1961, A non-linear theory for oscillations in a parallel flow:

Jour. Fluid Mech. (U.S.), v. 10, pt. 2, p. 209-236.

Three-dimensional periodic oscillations in the shear flow region between 

two parallel streams are considered up to that second order of the oscillation amplitude. It is shown that, as an integral part of the oscillation, there 

is a mean secondary flow in the nature of a longitudinal vortex. Despite 

the dissimilarity of the profile of the basic flows, several of the principal 

features of the calculated results can be compared with those observed for 

the Blasius flow by Schubauer and Klebanoff and Tidstrom at the National 

Bureau of Standards. 

[Author's abstract.]

822. Escande, L., Nougaro, Jean, Castex, L., and Barthet, H., 1961, Influence 

de quelques parametres sur une onde de crue subute a 1'aval d'un 

barrage [The influence of certain parameters on a sudden flood of wave 

downstream from a dam]: Houille Blanche [France], no. 5, p. 565-575.

Initial experimental results obtained with a 1:300 scale model of the 

Truyere valley below the Sarrans dam.

Wave celerity, form and amplitude, as governed by various parameters 

such as impounded water level, initial rate of flow downstream from the 

dam, and river bed roughness. 

[Abstract from Houille Blanche.]

823. Faure, J., and Nahas, N., 1961, Etude num6rique et expeYimentale d'intumescences & forte courbure du front [A numerical and experimental 

study of steep-fronted solitary waves]: Houille Blanche [France], no. 

5, p. 576-587.

The use of an electronic ordinator to resolve the propagation of a steepfronted wave on a dry or wet bed, which is governed both by De SaintVenant's equations and the shock wave (or wavefront) equations.

Consideration of the most general case of a rectangular valley of constant 

width (dry or wet downstream bed, gradient and friction), and of the case 

of any prismatic or rectangular valley with a wet downstream bed.

Comparison between theoretical and experimental results. 

[Abstract from Houille Blanche.]

824. Gerritt, Abraham, 1961, Some aspects of surface water wave scale effects: 

Am. Soc. Civil Engineers Proc., v. 87, no. HY 1, Paper 2708, p. 41-56. 

A study of the scale effects due to surface tension for model studies of 

water gravity waves generated by moving pressure disturbance is presented. 

The region of the wave system, generated by the moving pressure disturbance in which scale effects occur, has been determined by explaining the 

physical meaning of the asymptotes occurring in the fish-line problem. 

[Author's synopsis.]

825. Gherardelli, Luigi, 1961, Contribute alia teoria del mato vario delle correnti 

a pelo libero, Applicazione del methodo di integrazione di Riemann 

[Contribution to the theory of variable motion for open water-courses. 

Application of Riemann integration method]: Energia Elettrica 

[Italy], v. 38, no. 1, 14 p.
BIBLIOGRAPHY 217

Reference is made herein to the linear equation of variable (unsteady) 

motion for linear flows. This equation may be easily solved by the Riemann integration method after a suitable change of variables. The 

process suits the numerical calculations. 

[Author's synopsis.]

826. Guelton, M., Weingaertner, P., and Sevin, Ph., 1961, Fonctionnement en 

e'cluse'es du canal industriel de Basse-Durance [Lock-controlled operation of the industrial diversion works on the Lower Durance]: Houille 

Blanche [France], no. 5, p. 597-612.

Description of the diversion works, which include five successive canal 

sections and discharges into the "Etang de Berre" at sea level. Other 

features are: total length 80 km; total gross head 256 m; flow capacity 250 

cu. M per sec; output 500,000 kw.

Description of the "by the lockful" operation of the system, which can 

take on the total load or be shut down very quickly. Load variations can 

be distributed simultaneously over all five power stations, so that they act 

as a single 500,000 kw unit.

Description of a study of the solitary waves resulting from varying 

conditions due to the upstream and downstream power stations in each 

canal reach.

Description of the provisions made in the successive canal reaches to 

absorb such waves and the means provided in the individual power station 

relief works to ensure the safety of the various canal sections. 

[Abstract from Houille Blanche.]

827. Kalinin, G. P., 1961, Vizallasok 6s vizhozamok e!6rejelzese a nempermanens 

vizniozgds alapegyenleteinek KozelitS megoladasa alapjan [Forecasting stages and discharges by the approximate solution of the fundamental equations of unsteady flow]: Vizugyi Kozlemenyek [Hungary], 

no. 4, p. 420-436.

Using the storage differential equations for flood routing, the author 

stresses that the basic problem in routing is the selection of the reach Arc. 

The "characteristic bed reach" is introduced, as in author's earlier papers 

with hydraulic interpretation, which offers a theoretical explanation for the 

trend of the numerical constants of the Muskingum routing methods. A 

comparison of the hydraulic basis of the method with the unit hydrograph 

method yielded the same results as those obtained by I. E. Nash, using a 

different approach in 1960. 

[English summary, p. 51.]

828. Laboratoire National d'Hydraulique, 1961, Intumescence consecutive a 

une variation brusque de de"bit [Intumescence following a sudden discharge change]: Lab. Natl. Hydraulique, Chatou [France], Kept. no. 

2, T.359B.

829. Laboratoire National d'Hydraulique, 1961, Calcul sur ordinateur 

e"lectronique: mouvement non permanent dans les canaux, onde de 

crue [Computation on the electronic computer; unsteady movement 

in channels, flood wave]: Lab. Natl. Hydraulique, Chatou, [France], 

Kept. T.400B.

830. Levin, A. G., and Zhidkov, A. P., 1961, Predvychislenie khoda uravney 

reki Volgi nizhe Stalingradskoy GES metodom elektromodelirovaniya 

[Computation of level hydrograph of Volga River below Stalingrad
218 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Hydroelectric Powerplant by the method of electric models]: Meteorologiya i Gidrologiya [U.S.S.R.], no. 8, p. 38-41.

The application of an electric analog device for flood routing, developed 

by Kalinin, Milyuko\, and Nechaeva, is applied to the Volga River below 

the Stalingrad Hydroelectric Power Station.

831. Montuori, G., 1961, La formazione spontanea dei treni d'onde su canali a 

pendenza molto forte [The spontaneous formation of wave trains in 

the canals with very steep slope]: Energia Elettrica [Italy], no. 2, p. 

127-141.

832. Northrop, W. L., and Timberman, C. W., 1961, Use of computers for 

Kansas River flood studies: Am. Soc. Civil Engineers Proc., v. 87- no. 

HY 4, Paper 2866, p. 113-150.

A general digital computer program is described that produces flow 

hydrographs for an extensive network of basin subareas using unit hydrographs and routing coefficients. This concept is expanded to determination 

of releases for a system of reservoirs. Development of an electronic analog 

for reservoir operation is also described. 

[Authors' synopsis.]

833. Preissmann, A., and Gunge, J. A., 1961, Calcul du mascaret sur machine 

e'lectronique [Tidal bore calculation on an electronic computer]: 

Houille Blanche [France], no. 5, p. 588-596.

Fundamental difficulties encountered in the numerical resolution of 

De Saint-Venant's equations by the finite-differences method, in the case 

of discontinuous flow conditions. Introduction of a "psuedo viscosity" 

as an approximate representation of the loss of head due to the travelling 

surge wave and hence the development of the bore. Preliminary systematic calculations on simplified examples showing up the various factors 

liable to affect the formation and propagation of the bores. 

[Abstract from Houille Blanche.]

834. Preissmann, A., and Werner, G., 1961, Application du calcul des intumescences sur machine e'lectronique & divers cas pratiques [Application of 

translation wave calculations on an electronic computer to practical 

cases] Houille Blanche [France], no. 5, p. 613-621. 

The three following possible applications of electronic computer to 

solitary wave calculations are discussed:

1. Systematic trial flood wave calculations, in order to assess the validity 

of certain semi-empirical rules governing flood propagation;

2. Study of the stability of control afforded by automatic gates;

3. Calculation of flows in a tunnel alternating between pressure flow and

free surface conditions. 

[Abstract from Houille Blanche.]

835. Rantz, S. E., 1961, Surges in natural channels: U.S. Geol. Survey WaterSupply Paper 1369-C, 90 p.

This report presents the result? of an investigation of the travel of surges 

in a natural channel. It is demonstrated that the initial element of a flat 

wave front travels with varying velocity in accordance with Seddon's 

principle. Progressive flattening of the wave form is shown to be related 

to channel storage. The investigation was made on a 12.7-mile reach of 

Mokelumne River below Pardee Reservoir in California. Data for 10 

surges are tabulated and analyzed. 

[Author's abstract.]
BIBLIOGRAPHY 219

836. Sorensen, Torben, and Larsen, lan, 1961, Hydraulic calculations of storm

flood levels in connected shallow basins with special reference to the

Limfjord, Denmark: Internat. Assoc. Hydraulic Research, 9th

Convention, Dubrovnik (Yugoslavia), 1961, Proc., Paper 3, 41, p. 1-8.

In this paper the calculation of storm flood levels in connected shallow

basins is treated as a quasistationary case taking into account the wind

set-up in the basins and the combination of wind set-up and flow in the

connecting channels.

The calculations were made on a digital computer, the Danish DASK 

machine being used for this purpose.

The water levels in the Limfjord were calculated for 55 different storms 

between 1931 and 1959 to establish frequency distributions of the highest 

high waters at 12 different points. 

[Authors' abstract.]

837. Supino, Giulio, 1961, Sopra le onde di traslazione nei candli, B) II caso 

non-lineare [On the translation waves in channels. B) the nonlinear case]: Accad. Nazionale dei Lincei (Rome), Rend. (Class of 

Physical, Math., and Natural Sciences), ser. 8, v. 30, fasc. 2, p. 140-148. 

The integration of the two De Saint-Venant partial nonlinear differential 

equations is treated, after they have been reduced to second-order partial 

differential equations. Different cases for solutions are analyzed and 

discussed.

838. Thirriot, Claude, 1961, fitude de la convergence d'un precede" numerique 

de calcul des intumescences [Study of the convergence of a numerical 

procedure for computation of intumescences]: Acad. sci. [Paris] 

Comptes rendus, v. 252, p. 1421-1423.

Following the previous notes (Comptes rendus v. 248, p. 2950, 1959; 

v. 249, p. 1858, 1959, and v. 250, p. 55, 1960) of applying the method of 

characteristics and IBM 650 digital computer to computation of intumescences, the comparison of formulas of first and second order permits the 

selection of finite difference AQ (discharge). The departure between 

results given by the two formulas are analyzed, and an approximation 

of errors involved is developed.

839. Thirriot, Claude, 1961, fitude adimensionnelle des intumescences [Dimensionless study of intumescences]: Acad. sci. [Paris] Comptes rendus, 

v. 252, p. 3395-3397.

The relative (dimensionless) values are introduced for length, velocity 

area, and time, and the two De Saint-Venant equations are correspondingly 

changed. The use of the equations in demensionless form, and their 

discussion are given.

840. Thirriot, Claude, 1961, filude des phenomenes pneumatiques provoque"s 

par la propagation d'intumescence dans une galerie de fuite d'usine 

hydroclectrique souterraine [Study of compressed air phenomena 

created by a propagation of intumescence (bore) in a tailrace gallery 

of an underground hydroelectric power station]: Acad. sci. [Paris] 

Comptes rendus, v. 252, p. 4108-4110.

Following the note (Comptes Rendus v. 249, 1959, p. 2718), the shock 

wave downstream from the bore in a gallery is represented schematically 

by two partial differential equations. In this note the equations are 

reduced to dimensionless values. The Mach number is used. The equa-
220 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

tions are then linearized, and solutions are obtained by using the operational computation. The computation is also carried out on IBM 650 

digital computer, and the velocities of air at the end of gallery are given 

in function of time.

841. Wolska-Boschenek, J., 1961, Probleme aux limites pour un domaine non 

borne" dans la the'orie du mouvement non stationnaire d'un liquide 

visqueux [Boundary problem for a non limited domain in the theory 

of unsteady movement of a viscous fluid]: Archive for Rational 

Mechanics and Analysis [Germany], v. 7, no. 3, p. 196-211.

842. Yano, Katsumasa, 1961, Theoretical research on the surging phenomena 

of the high tide by the typhoon into the rivers and canals: Kyoto 

Univ. [Japan], Disaster Prevention Research Inst. Pub., p. 194-197. 

Basing on the data of the Ide-Bay Typhoon, when the records of the 

travelling-up phenomena of the high tide into the Kiso-River and others 

were clearly recognized, the author tried to solve their phenomena theoretically and found some interesting characteristics, by solving the differential equations by means of Rieman integral method. 

[Author's English synopsis.]

843. Yevdjevich, V. M., 1961, Unsteady free surface flow in a storm drain 

(general and analytical study): Colo. State Univ. Eng. Research, Rpt. 

CER 61 VMY 38, p. 1-76.

As a preliminary general study the broad scope of this report is the 

outline of problems, the selection of mathematical tools and procedures, 

and the elaboration of a general approach for further studies by hydraulic 

model and by digital computer investigations in order to develop a set of 

routing methods for storm drain floods.

The initial and boundary conditions, applications, and the general approach selected are briefly enumerated in the introduction. The integration of differential equations by method of finite differences is treated in 

detail, special attention is given to boundary problems.

844. Harder, J. A., 1962, Analog models for flood control systems: Am. Soc. 

Civil Engineers Proc., v. 88, no. HY 2, Paper 3074, p. 63-74.

The success of applying analog model techniques to tidal flow channels 

has led to the extension of the technique to the simulation of flood control 

systems. The similarity relationships for long period flood waves are 

derived.

Methods for the simulation of the channel sections are described, with 

particular attention paid to the nonlinear properties of the stage-discharge 

relation and the stage-storage relation. The introduction of boundary 

conditions in the form of streamflows and rainfall excesses is programmed 

for each six hours in the prototype, for a total period of ten days, by special 

pinboards. Each simulated flood requires five milliseconds, after which 

initial conditions of flow, at the initial instant of the ten-day period, are 

reestablished in preparation for the next flood period. Time history of 

stage or discharge at any two of fifty stations can be displayed simultaneously on an oscilloscope screen. Provision is made for the insertion of 

up to six reservoirs into the analog, each of which is provided with a programmable output simulating the action of reservoir release gates. It is 

expected that the analog will be useful both in project planning and during 

flood emergencies. 

[Author's synopsis.]
INDEX BY AUTHORS

[This index by authors refers to the numbers of the references in the Chronological Bibliography. The index 

is arranged alphabetically according to author's last name. The Russian names are transcribed. For 

each author's name the number of each reference and the year of publication are shown inside the parentheses. When there are two or more authors, each reference is indexed under the names of all coauthors. 

Two or more authors are indicated by an asterisk (*) before the identifying number of such a reference: 

one asterisk if there are two authors, two asterisks if there are three authors, three asterisks if there are 

four authors, and so forth]

Abbott, M. B., 651 (1956).

Adachi, Shohei, 652 (1956).

Adler, G. F. W., 484 (1950).

Alfanasev, A. I., 688 (1957).

Airy, Sir George Biddle, 16 (1845).

Alekseev, G. A., 653 (1956).

Alibrandi, M. P., 133 (1917).

Alien, J., 329 (1938); *357 (1940).

Ambraseys, N. W., 619 (1955).

Angelini, A. M., 293 (1936).

Annemueller, H., 794 (1960).

Anonymous, 424 (1947).

Appleby, F. V., 598 (1954).

Apte,A.,556(1953).

Arakawa, H., 232 (1933).

Arkhangel'skiy, V. A., 371 (1941); 372 (1941); 400

(1943); 416 (1946); 425 (1947). 

Arredi, Filippo, 373 (1941); 388 (1942). 

Arsenishvili, K. I., 654 (1956). 

Arthur, R. S., 485 (1950). 

Auria, L. d'., 67 (1890).

Bachet, 246 (1934); *349 (1939); 443 0948).

Baines, W. D., 522 (1952); 557 (1953).

Bakhmetev, B. A., 108 (1912); 128 (1915); 219 (1932).

Baldwin, O. T., *306 (1936).

Ball, F. K., 795 (1960).

Barbe, A., *711 (1957).

Barber, N. F., 461 (1949).

Barrows, H. K., 220 (1932); 233 (1933); 389 (1942);

444 (1948).

Barthet, H., ***822 (1961). 

Basset, A. B., 61 (1888). 

Bata, G., 620 (1955). 

Baumgarten, 20 (1848). 

Bazin, H., 27 (1862); 30 (1865); 55 (1885). 

Beau, *349 (1939). 

Becker, E., 558 (1953). 

Beckman, H. C., 330 (1938). 

Bednarski, E. J., 221 (1932). 

Befani,A. N.,689 (1957). 

Belokon', P. N., 331 (1938). 

Benjamin, T. B., *599 (1954). 

Benny, D. J., 821 (1961). 

Bergeron, Louis, 314 (1937); 486 (1950); 559 (1953).

728-245 64 15

Bernadsky, N. M., 193 (1926); 234 (1933).

Bernard, M. M., 261 (1935).

Besson, F. S., 211 (1931).

Bezanov, K. A., 796 (1960).

Bidone, George, 8 (1824).

Biesel, F., 487 (1950); 523 (1952),

Binnie, A. M., *621 (1955).

Biot, M. A. 262 (1935).

Blackmore, W. E., 524 (1952); 690 (1957).

Blind, Hans, 655(1956).

Bobylev, D.,79 (1898).

Boess, Paul, 204 (1927).

Bolotov, V. V., 294 (1936).

Bonneau, L., 162 (1922); 177 (1924).

Bonnet, L., 168 (1922-23); 69! J957).

Bonvicini, D., 247 (1934).

Bose, N. K., *622 (1955).

Bouasse, H., 178 (1924).

Bourdelles, 80 (1898); 84 (1899); 86 (1900).

Bourguignon, M. P., 445 (1948).

Boussinesq, Joseph, 33 (1871); 36 (1872); 38 (1873);

45 (1877); 46 (1877); 77 (1897). 

Brabant, C. E., 748 (1959). 

Brater, E. F., *399 (1942). 

Brigham, R. H., 332 (1938). 

Brown, E. I., 222 (1932); 263 (1935); 295 (1936). 

Bruman, J. R., 426 (1947). 

Burdoin, A., *385 (1941). 

Burns, J. C., 560 (1953). 

Bulter, S. S., 692 (1957).

Cady, R. C., ***308 (1936).

Cagniard, L.,315 (1937).

Calame, J., 223 (1932).

Caligny, Anatole de, 21 (1848); 23 (1861); 31 (1866);

37 (1872).

Calland, N., 29 (1864). 

Callet, 316 (1937). 

Carstanjen, M., *194 (1926). 

Carter, R. W., *797 (1960). 

Cassidy, H. O., **410 (1945). 

Castex, L., ***822 (1961). 

Cauchy, A. L., 10 (1827). 

Charrueau, A., 749 (1959). 

Chary, 248 (1934).

221
222 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Cheng, H. M., 417 (1946).

Cherkasov, A. A., 224 (1932).

Chertousov, M. D.. 296 (1936); 693 (1957).

Chitale, 8. V., 600 (1954).

Chow, Yen Te, 500 (1951); 666 (1956); 750 (1959).

Chrystal, 96 (1904).

Clsotti, Umberto, 106 (1910-11); 109 (1912); 110 

(1912); 111 (1912); 116 (1913); 117 (1913); 127 

(1914-15); 138 (1918); 139 (1918); 144 (1919); 149 

(1920).

Citrini, Dulio, 462 (1949); 488 (1950).

Clark, C. O., 408 (1945); 409 (1945); 463 (1949).

Clausnitzer, R., 525 (1952).

Clemens, G. R., 264 (1935).

Cocchi, G.,601 (1954).

Comoy, M., 50 (1881).

Cooperrider, C. K., **410 (1945).

Cornish, Vaughan, 99 (1907); 105 (1910); 249 (1934)

Coulson, C. A., 374 (1941).

Couraut, R., *446 (1948)

Craya, A., 418 (1946); 419 (1946); 526 (1952).

Crengg, H., 250(1934).

Cross, W. P., *468 (1949).

Crossley, H. E., Jr., 464 (1949).

Cuenod, Michel, *527 (1952); 657 (1956).

Gulp, M. M., 447 (1948).

Cunge, J. A., *833 (1961).

Dachler, Robert, 265 (1935).

Dahl, 205 (1927); 623 (1955).

Daily, J. W., *501 (1951); *528 (1952).

Dalle, Valle, *266 (1935).

Dantscher, Kasper, 358 (1940); 359 (1940).

Darman, Z. I., 751 (1959).

Davies, T. V., 529 (1952).

Davis, Calvin Victor, 530 (1952).

Damn, C. R., "566 (1953).

Dean, W. R., 427 (1947).

De Marchi, Giulio, 411 (1945); 561 (1953); 562 (1953).

De Saint-Venant, Barre, 32 (1870);34 (1871); 56 (1885).

De Vries, G., *74 (1895).

Deymie, Ph., 267 (1935); 268 (1935); 350 (1939).

Dice, R. I., *433 (1947).

Dietz, D. N., 375 (1941).

Dmitriyev, A. A., 563 (1953).

Dmitriyev, G. T., 465 (1949).

Doodson, A. T., 694 (1957).

Dooge, J. C. I., 658 (1956); 752 (1959).

Drioli, Carlo, 317 (1937).

Drisko, John B., 235 (1933); 236 (1933).

Dressier, R. F., 466 (1949); 531 (1952); 6E2 (1952); *564

(1953); 602 (1954); 724 (1958). 

Dronkers, J. J., 428 (1947); 467 (1949); *624 (1955);

*753 (1959).

Drouhin, **502 (1951). 

Druml, F. U., *695 (1957); *733 (1958). 

Duffour, Pierre, *736 (1958); *737 (1958). 

Dupouy, M., 503(1951). 

Duranton, R., *574 (1953). 

Dziatlik, H., 625 (1955).

Eagleson, P. 8., 754 (1959). 

Earnshaw, S., 18 (1847). 

East, R. L., 225 (1932). 

Eckart, C., 504 (1951); 533 (1952). 

Egiazarin, B. O., 659 (1956).

Egiazarov, I. V., 179 (1924); 212 (1931); 237 (1933); 251 

(1934); 252 (1934); 260 (1934-35); 269 (1935); 297 

(1936); 318 (1937); 319 (1937); 351 (1939).

Einstein, H. A., *626 (1955); *755 (1959).

Eisner, F., 192 (1925-26); 226 (1932).

Ekdahl, O. Z., 62 (1888); 112 (1912).

Escande, L., *660 (1956); *»* 822 (1961).

Escoffler, F. F., 696 (1957).

Evangelist!, Giuseppe, 270 (1935); 390 (1942); 627 

(1955).

Fantoli, G., 78 (1897); 187 (1925).

Farrell, J. P., *636 (1955).

Faure, J., *823 (1961).

Favre, H., *229 (1932); 271 (1935); 298 (1936).

Feifel, Eugen, 140 (1918).

Felkel, K., 725 (1958); 756 (1959).

Fenchel, W., 213 (1931).

Finzi, Bruno, 489 (1950).

Fishel, V. C., ***308 (1936.)

Flson, David, 299 (1936).

Flamant, A., 63 (1889); 87 (1900); 103 (1909).

Forchheimer, Ph., 94 (1903); 97 (1904); *150 (1920);

169 (1923); 180 (1924); 209 (1930). 

Foskett, L. W., **453 (1948); 454 (1948). 

Francis, J.B., 64(1889). 

Frank, Joseph, *333 (1938); 505 (1951); 565 (1953); 697

(1957).

Freeman, J. C., 3r., 506 (1951). 

Friedrichs, K. O., 448 (1948); '448(1948); '603(1954). 

Fry, S. A., 131 (1916). 

Fuchs, R. A., *626 (1955). 

Fuller, W. E., 119 (1914). 

Fu Te, I., '365 (1940).

Gardel, Andre, '527 (1952).

Gentilini, Bruno, 490 (1950).

Germanis, E., '780 (1959).

Gerritt, Abraham, 824 (1961).

Gerstner, Franz von, 5 (1802).

Gessner, Walter, 798 (1960.

Gibson, A. H., 170 (1923); 238 (1933).

Gilcrest, B. R., *376 (1941); 491 (1950).

Gil'denblat, Ya. D., '449 (1948); 661 (1956).

Gherardelli, Luigi, 604 (1954); 825 (1961).

Glover, R. E., "'566 (1953).

Godfrey, R. G., '797 (1960).

Goodrich, R. D., 214 (1931).

Goodridge, R. S., 320 (1937).

Goroshnikov, E. A., 799 (1960).

Graeff, 39 (1875); 40 (1875); 41 (1875); 51 (1883).

Graves, Q. B., "354 (1939).

Green, G., 11 (1837); 13 (1839).

Gregor, L., 726 (1958).

Grover, N. C., 163 (1922).

Grunsky, O. E., 101 (1908).

Guelton, M., **826 (1961).

Guerrero, E. C., 492 (1950).

Guerrini, Pietro, 727 (1958).

Guyot, M. Th., **757 (1959); **758 (1959); '*800

L960); "801 (1960); "811 (1960). 

Gwyther, R. F., 88 (1900); 89 (1900).

Haerens, E., 85 (1899). 

Hagen, G., 24 (1861). 

Haindl, K., 698 (1957). 

Haines, N. S., 429 (1947).
INDEX 223

Halek, V., 728 (1958).

Hampel, R., 802 (1960).

Harder, J. A., *755 (1959); 844 (1962).

Harkness, Frank B., 507 (1951).

Havestadt, A., 53 (1884).

Hayami, Shi, **608 (1954); **663 (1956).

Hayami, Sho, 508 (1951); **608 (1954); **663 (1956).

Hayashi, Taizo, 509 (1951); 510 (1951); 567 (1953); 759

(1959).

Haws, E. T., 605 (1954). 

Heduy, M., 272 (1935). 

Hegly, 253 (1934). 

Henry, Marc, 334 (1938). 

Hensen, W., 534 (1952); 747 (1958-59). 

Herbert, D. J., *403 (1944); **566 (1953). 

Herschel, Clemens, 42 (1875). 

Hildebrand, C. E., *680 (1956). 

Hill, P. G., *760 (1959). 

Hill, R. A., 206 (1928). 

Hoffman, G. R., 606 (1954). 

Holsters, H., 430 (1947); 568 (1953); 76l](1959). 

Honda, M., 793 (1959-60). 

Hopkins, C. D., Jr., 569 (1953). 

Horton, R. E., 141 (1918); 300 (1936); 321 (1937); 335

(1938); 336 (1938); 337 (1938); 377 (1941); 378 (1941). 

Hunt, J. N., 535 (1952); 628 (1955). 

Hyers, D. H., *603 (1954).

Ichiye, Takashi, 570 (1953).

Intnat. Boundary Comm., U.S. and Mexico, 301

(1936). 

Ippen, A. T., 391 (1942); **629 (1955); *699 (1957);

*700 (1957).

Isaacson, E., **607 (1954); **662 (1956); **729 (1958). 

Ishihara, Tojior, **608 (1954); *630 (1955); **663 (1956);

*664 (1956). 

Ishihara, Yasuo, *630 (1955); *664 (1956); 701 (1957);

762 (1959).

Iwagaki, Y., 511 (1951); *609 (1954); *610 (1954). 

Iwasa, Y., 665 (1956); 631 (1955); 702 (1957).

Jacob, C. C., 113 (1912).

Jaeger, Charles, 666 (1956).

Jeffreys, Harold, 188 (1925).

Jones, B. E., 132 (1916).

Johnson, R. D., 135 (1917); 164 (1922).

Johnstone, D., *468 (1949).

Kabelac, Karl, 104 (1909).

Kalinin, G. P., 536 (1952); 703 (1957); *704 (1957); *730

(1958); *763 (1959); *803 (1960); *804 (1960); **805

(1960); 827 (1961). 

Karlson, 120 (1914). 

Karteweg, D. J., *74 (1895). 

Kawabata, S., 731 (1958). 

Kazarnovskiy, Yu E., 632 (1955). 

Keeland, P., 14 (1839). 

Keller, J. B., 450 (1948); 469 (1949). 

Kelvin, Lord (Thomson, William), 57 (1886); 59

(1887); 60 (1887). 

Kertselli, S. A., 322 (1937). 

Keulegan, G. H., *360 (1940); *361 (1940); *401 (1943);

451 (1948); 493 (1950); 571 (1953). 

Khristianovich, S. A., 239 (1933); 302 (1936); 303

(1936); 304 (1936); 323 (1937); 339 (1938). 

Kikkawa, H., **641 (1955).

King, R. E., 537 (1952).

King, W. H., 240 (1933).

Kinosita, T., 667 (1956).

Kishi, Tsutomu, **641 (1955); 705 (1957).

Kivisild, H. R. 706 (1957).

Kleitz, Oh., 47 (1877).

Klemens, Vit, 806 (1960).

Klunzinger, P., 58 (1886); 76 (1896).

Knappen, T. T., 340 (1938).

Kobelt, Karl, 153 (1921).

Koch, A., *194 (1926).

Kohler, M. A., 404 (1944); 405 (1944); **453 (1948);

454 (1948); **472 (1949); 572 (1953); 668 (1956); 7?2

(1958); **734 (1958). 

Kotik, J., 538 (1952). 

Kovacs, Gyorgy, 633 (1955). 

Kovalenko, E. P., 707 (1957); 716 (1957). 

Koval'skiy, M. I., 273 (1935). 

Kozak, Miklos, 669 (1956). 

Kozeny, Josef, 121 (1914); 573 (1953). 

Kreisel, G., 470 (1949). 

Krey, H., 154 (1921); 171 (1923); 195 (1926). 

Kritskiy, S. N., *494 (1950); *539 (1952); *670 (1956). 

Kulin, Gershon, **629 (1955); *699 (1957). 

Kunstatsky, J., 764 (1959). 

Kurihara, K., 274 (1935). 

Kuskov. L. S., 275 (1935).

Labaye, G., *574 (1953).

Laboratoire National d'Hydraulique, Chatou (Paris,

France), 765 (1959); 766 (1959); 828 (1961); 829 (1961). 

Lacaze, B., 276 (1935). 

Lacombe, H., 540 (1952). 

laden, N. R., **362 (1940). 

Lagrange, I. L., 3 (1781); 4 (1788). 

Laitone, E. V., 807 (1960). 

Lamb, Sir Horace, 227 (1932). 

Lamoen, J., 277 (1935); 305 (1936); 471 (1949). 

Lane, E. W., *306 (1936); 324 (1937). 

Langbein, W. B., 341 (1938); 363 (1940); 392 (1942). 

Laplace, P. S., 2 (1775-76). 

Larsen, lan, *836 (1961). 

Laurenson, E. M., 767 (1959). 

Lawler, E. A., *695 (1957); *733 (1958). 

Lay, Hung-Chi, 338 (1938). 

Lechalas, M. C., 54 (1884). 

Leggette, R. M., ***308 (1936). 

LeMehaute, Bernard, 808 (1960). 

Lemoine, R., 452 (1948). 

Levi-Civita, Tullio, 100 (1907); 114 (1912); 155 (1921);

165 (1922); 181 (1924); 182 (1924); 189 (1925); 228

(1932).

Levi, I. L, 708 (1957). 

Levin, A. G., *803 (1960); *830 (1961). 

Levin, Leon, 393 (1942); 512 (1951); 513 (1951); 541

(1952); 542 (1952); 543 (1952). 

Levy, Maurice, 81 (1898). 

Liggett, J. A., 768 (1959). 

Lighthill, M. J., *599 (1954); *634 (1955). 

Lin, Pin-Nam, 431 (1947); 544 (1952). 

Lin, Ping-Yi, 190 (1925). 

Linsley, R. K., 394 (1942); 406 (1944); 407 (1944); **453

(1948); **454 (1948); **472 (1949); **734 (1958). 

Long, R. R., 671 (1956); 672 (1956). 

Lowe, F. C., *403 (1944). 

Lowell, S. C., 473 (1949).
224 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Ludewig, Dietrich, 769 (1959).

Macmillan, D. H., *550 (1952).

Macy, F. H., 172 (1923).

Maier, E., *151 (1920).

Maillet, Edmond, 92 (1901); 95 (1903); 102 (1908);

142 (1918); 145 (1919); 166 (1922). 

Makkaveev, V. M., 196 (1926); 197 (1926); 210 (1930);

241 (1933).

Mallet, ** 502 (1951). 

Marsh, L. E., *376 (1941). 

Mason, M., 575 (1953). 

Massau, Junius, 65 (1889); 66 (1889); 90 (1900); 98

(1905); 122 (1914). 

Masse, Pierre, Rene, 278 (1935); 279 (1935); 280 (1935);

281 (1935); 282 (1935); 307 (1936); 325 (1937); 236 (1937);

352 (1939).

Mastitskiy, N. V., 254 (1934); 432 (1947). 

Matheson, J. L., *357 (1940). 

Mayer, P. G., 709 (1957); 770 (1959). 

McCarthy, G. T., 342 (1938). 

McCowan, J., 69 (1891); 73 (1894). 

Meinzer, O. E., ***308 (1936). 

Meleshchenko, N. T., 364 (1940); *455 (1948); *456

(1948).

Menkel', M. F., 494 (1950); *539 (1952); *670 (1956). 

Messerle,H. K., 576 (1953); 673 (1956). 

Messina, U., 379 (1941). 

Meyer, O. H., 380 (1941); 381 (1941). 

Meyer-Peter, E., *229 (1932). 

Meyers, J. S., *474 (1949).

Miami Valley Flood-Protection Work, 135 (1917). 

Milyukov, P. I., 577 (1953); *704 (1957); *730 (1958);

*763 (1959); 804 (1960); **805 (1960). 

Minnotte, J. S., **362 (1940). 

Mises, von R., 123 (1914). 

Mitchell, M. M., *700 (1957). 

Mkhitaryan, A. M., 771 (1959). 

Moellner, M. 198 (1926).

Moklyak, V. I., 611 (1954); 635 (1955); 735 (1958). 

Mondschein, H. F., *712 (1957). 

Montuori, G., 831 (1961). 

Moots, E. E., 343 (1938); 344 (1938). 

Morikawa, G. K., 612 (1954); 710 (1957). 

Mueller, R., 118 (1913). 

Mullins, J., 68(1890). 

Munk, W. H., 475 (1949).

Nahas, N., *823 (1961).

Nash, J. E., *636 (1955); 772 (1959); 809 (1960).

Nechaeva, N. S., 773 (1959); **805 (1960); 810 (1960).

Nekrasova, A., 146 (1919); 156 (1921).

Nekrassov, A. I., 514 (1951).

Newton, Sir, J., 1 (1685-86).

Neyrpic, 412 (1945).

Niederhof, C. H., **410 (1945).

Norman, A., 283(1935).

Northrop, W. L., *832 (1961).

Nosek, T. M., *433 (1947).

Nougaro, Jean, 545 (1952); 578 (1953); 579 (1953); 580 

(1953); 613 (1954); 614 (1954); 637 (1955); *660 (1956); 

674 (1956); 675 (1956); 676 (1956); 677 (1956); *711 

(1957); *736 (1958); *737 (1958); **757 (1959); **758 

(1959); **800 (1960); **801 (1960); **811 (1960); ***822 

(1961).

Oeltjen, J., 147 (1919). 

Ogiyevskiy, A. V., 546 (1952). 

Orkney, J. C., *621 (1955).

Pacak, A., **173 (1923).

Pacquant, **502 (1951).

Palatini, Attilio, 129 (1915); 130 (1915).

Palmer, W. S., 115 (1912).

Pan-Zhu, E., 678(1956).

Pardoe, W. S., 157 (1921).

Parsons, W. B., 143 (1918).

Partenscky, H. W., 812 (1960).

Partiot, H. L., 22 (1858); 25 (1861); 26 (1861); 35 (1871);

70 (1892); 93 (1901).

Patterson, G. W., 360 (1940); *361 (1940); *401 (1943). 

Paulhus, J. L. H., **472 (1949); **734 (1958). 

Pavlov, G. G., 774 (1959). 

Paynter, H. M., 547 (1952). 

Penati, Savio, 615 (1954). 

Pettis, C. R., 309 (1936). 

Pezzoli, G., 738 (1958). 

Pigeaud, 148 (1919). 

Pillsbury, G. B., 353 (1956). 

Pleshkov, Ya. F., 581 (1953). 

Poggi B., 775 (1959). 

Pohle, F. V., 548 (1952); *564 (1C63). 

Poisson, 7 (1816). 

Polak, Ignatz, 75 (1895). 

Posey, C. J., 284 (1935); *365 (1940); *387 (1941); 395

(1942).

Potapov, M. V., 434 (1947); 515 (1951). 

Preissmann, A., *833 (1961); *834 (1961). 

Proskuryakov, B. V., 255 (1934). 

Proudman, J., 191 (1925); 638 (1955); 639 (1955); 739

(1958).

Puls, Louis G., 207 (1928). 

Puppini, U., 242 (1933); 420 (1946). 

Putman, J. H., 457 (1948).

Rakhmanov, A. N., 382 (1941).

Ramponi, Francesco, 366 (1940); 476 (1949).

Rankine, W. J. M., 28 (1863).

Ransford, G. D., 516 (1951); 679 (1956).

Rantz, S. E., 835 (1961).

Rappert, C., 285 (1935).

Ray, W. E., 712 (1957).

Rayleigh, Lord, 43 (1876); 107 (1911); 124 (1914).

Raza, M. A., **629 (1955).

Re, R., 421 (1946).

Reilly, T. L., **362 (1940).

Reineke, H., 158 (1921).

Remenieras, Gaston, 396 (1942).

Reynolds, Osborne, 48 (1877).

Rich, R. George, 517 (1951 >

Richards, B. D., 640 (1955).

Riggs, H. C., 582 (1953).

Rippl, W., 52 (1883).

Ritter, A., 71(1892).

Rockwood, D. M., *680 (1956); 740 (1958).

Rose, D., 813 (1960).

Roseau, M., 549 (1952).

Rosenzweig, O., *499 (1950).

Roudzki, 82 (1898).

Rouse, H., 345 (1938).

Rubbert, F. K., 776 (1959).
INDEX 225

Eudinger, George, 681 (1956).

Euemelin, Th., *150 (1920).

Kuggiero, C., 477 (1949).

Bnnning, T. E., 125 (1914).

Eussell, J. S. (Scott), 12 (1837); 15 (1842-43); 44 (1876).

Eussell, R. C. H., *550 (1952).

Eutter, E. J., **354(1939).

Ezhanitsyn, N. A., 256 (1934).

Samsioe, 183 (1924).

Sandover, J. A., *713 (1957); *723 (1957).

Sapozhnikov, V. I., 777 (1959).

Sato, S., **641 (1955).

Schaffernak, Friedrich, 286 (1935).

Schnoor, E., 778(1959).

Schoenfeld, J. C., 483 (1949); 518 (1951); 519 (1951);

583 (1953); 616 (1954); 624 (1955); 642 (1955); 643 

(1955); 644 (1955); 645 (1955); 646 (1955); 682 (1956); 

714 (1957); *753 (1959); *779 (1959). 

Schoklitsch, Armin, 136 (1917); 327 (1937). 

Scholer, H. A., 714 (1958); *780 (1959). 

Schueller, J., *333 (1938). 

Schultz, E. R., 435 (1947); *474 (1949). 

Schultze, Edgar, 257 (1934); 383 (1941). 

Sciememi, Ettore, 478 (1949). 

Scobey, C. F., 215 (1931). 

Seemann, D., 495 (1950). 

Seddon, James A., 83 (1898); 91 (1900). 

Sekerz-Zen'kovich, T. Ja., 781 (1959). 

Semenido, V. I., 683 (1956). 

Semikolenov. A. S., **449 (1948). 

Sen, N., 184 (1924). 

Serre, F., 584 (1953). 

Sevin, Ph., **826 (1961). 

Shepley, J. M., *397 (1942). 

Sherman, L. K., 174 (1923); 230 (1932); 243 (1933). 

Shilov, V. T., 310(1936). 

Shlroki, S. P., 208 (1929). 

Shuleykin, V. V., 585 (1953). 

Shulits, Samuel, 216 (1931). 

Sidenko, A. I., 328 (1937). 

Simonov, W. P., 287 (1935). 

Sinha, G., *622 (1955); 742 (1958); 814 (1960). 

Smetana, J., **173 (1923). 

Snyder, F. F., 346 (1938); **354 (1939). 

Sokolovskiy, D. L., 479 (1949); 551 (1952). 

Sorenson, K. E., 480 (1949); 586 (1953). 

Sorensen, Torben, *836 (1961). 

Spaeth, K., *151 (1920). 

Sretenskiy, L. N., 496 (1950). 

Steinberg, I. H., 347 (1938); 436 (1947). 

Stennlng, A. H., *760 (1959). 

Stephan, S. C., Jr., *501 (1951); *528 (1952). 

Stevens, J. C., 159 (1921). 

Stoker, J. J., 458 (1948); 481 (1949); 587 (1953); 588

(1953); **607 (1954); **662 (1956); 715 (1957); **729

(1958).

Stokes, G. G., 17 (1846); 19 (1847); 49 (1880). 

Stroband, H. J., 437 (1947). 

Struik, D. J., 199 (1926).

Sueishi, Tomitaro, *609 (1954); *610 (1954); 647 (1955). 

Sum, T., 288 (1935). 

Supino, Qiulio, 355 (1939); 497 (1950); 498 (1950); 689

(1953); 590 (1953); 591 (1953); 617 (1954); 684 (1956);

815 (1960); 837 (1961). 

Swain, F. E., 520 (1951).

Syrov, Yu. P., *716 (1957). 

Szesztay, K., 685 (1956).

Takaya, S., 231 (1932).

Tarpley, J. F., Jr., 367 (1940).

Tatum, F. E., 384(1941).

Taylor, E. H., 348(1938).

Taylor, G. I., 161 (1921-22).

Ter-Krikorov, A. M., 816 (9960).

Thirriot, Claude, **757 (1959); **758 (1959); 782

(1959); **800 (1960); **801 (1960); **811 (1960);

838 (1961); 839 (1961); 840 (1961). 

Thomas, H. A., 137 (1917); 258 (1934); 368 (1940);

369 (1940).

Thorade, H., 200 (1926); 217 (1931). 

Thijsse, J. Th., 185 (1924); 244 (1933); 289 (1935). 

Tifonov, E. K., 618 (1954). 

Till, J., **173 (1923). 

Timberman, C. W., 832 (1961). 

Tison, L. J., 521 (1951). 

Tonini, Dino, 783 (1959). 

Trifonov, E. K., 245 (1933).

Troesch, B. A., 607 (1954); **662 (1956); **729 (1958). 

Tsinger, V. N., 552 (1952); 592 (1953); 686 (1956). 

Turner, H. M., *385 (1941); 402 (1943).

Uchida, Shigeo, 553 (1952).

Urban, Jaroslav, 817 (1960).

Ursell, F., 438 (1947); 554 (1952); 593 (1953); 818 (1960).

U.S. Beach Erosion Board, 398 (1942).

U.S. Bureau of Eeclamation, 160 (1921); 439 (1947).

U.S. Corps of Engineers, 311 (1936); 312 (1936); 313

(1936); 482 (1949); 594 (1953); 595 (1953); 717 (1957);

718 (1957); 719 (1957); 784 (1959). 

U.S. Engineer School, 422 (1946). 

U.S. Geological Survey, 440 (1947). 

U.S. Soil Conservation Service, 687 (1956); 720 (1957). 

Uspenskiy, P. N., 785 (1959).

Vapnyar, D. U., 786 (1959).

Varlet, 175 (1923).

Vedernikov, V. V., 413 (1945); 414 (1945); 441 (1947).

Veen Van, Johan, 442 (1947); 459 (1948).

Velikanov, M. A., 218 (1931).

Verhagen, C. M., *779 (1959).

Visentini, M., *266 (1935).

Viti, Mario, 460 (1948); 721 (1957).

Voegerl, Franz, 290 (1935).

Volker, F., *483 (1949).

Volkov, I. M., 370 (1940); 743 (1958).

Vreedenburgh, C. G. J., 259 (1934).

Walton, C. B., *397 (1942).

Weber, C., 291 (1935).

Weber, Ernest, *9 (1825).

Weber, William, *9 (1825).

Wedderburn, J., 126 (1914).

Weidner, M. E., 292 (1935).

Weingaertner, P., **826 (1961).

Weinstein, A., 201 (1926).

Wells, L.W., 596(1953).

Werner, G., 834 (1961).

Wheeler, W. H., 72 (1893).

Whitham, G. B., *634 (1955); 649 (1955).

Whittington, R. B., 787 (1959).

Wicker, C. F., *499 (1950).
226 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Wiegel, R. L., 819(1960).

Wilkinson, J. H., 415 (1945).

Wilson, W. T., 386 (1941).

Winkel, R., 167 (1922); 176 (1923); 186 (1924);

(1928); 203 (1926). 

Wisler, O. O., *399 (1942). 

Wolska-Boschenek, J., 841 (1961). 

Woodward, S. M., 152 (1920); *387 (1941).

Yakubov, M. S., *455 (1948); *456 (1948). 

Yamada, Hikoji, 722 (1957); 744 (1958). 

Yano Katsumasa, 745 (1958); 842 (1961).

202

Yevdjevich, V. M., 788 (1959); 789 (1959); 790 (1959);

843 (1961).

Young, Of. T., 6 (1813). 

Yuan-po, Kou, 746 (1958). 

Yuichi, Iwagaki, 648 (1955). 

Yuy Vi-Chzhun', 791 (1959).

Zamarin, E. A., 423 (1946).

Zheleznyak, I. A., 555 (1952); 597 (1953); 650 (1955);

792 (1959).

Zhidkov, A. P., 820 (1960); 830 (9161). 

Zienkiewicz, O. C., *713 (1957); *723 (1957). 

Zoch, R. T., 356 (1939).

[Some entries in "The Chronological Bibliography," refer in the abstracts to authors and their works connected with the unsteady flow in channels and flood routing. The following index of these authors is 

given with the author's name, the year of the published work (which is not abstracted or annotated in 

this bibliography), the reference in the chronological bibliography which mentions the author and his 

work, and the year in which the reference was published. Some authors appear in the previous list and 

hi this index. The reference in this index is made only to the works not abstracted or annotated in the 

previous index. This index can orient the reader in searching for the additional references which are not 

abstracted or annotated here]

Agroskin; 465 (1949).

Babinet;32(1870).

Basset, A. B.; 74 (1895).

Belgrand; 95 (1903), 166 (1922).

Bernadsky, N. M.; 255 (1934).

Burns, T.S.; 340 (1938).

Oialdi; 23 (1861).

Church, I. P.; 134 (1917).

Clapp, W. B.; 101 (1908).

Darcy; 30 (1865), 136 (1917).

DeMarchi; (1948), 521 (1951).

Frizell, T. P.; 64 (1889).

Goupil; 133 (1917).

Goodrich, R. D.; 354 (1939).

Grimm; 395 (1942).

Harkners, F. B.; 340 (1938).

Harlacher, A. R.; 75 (1895).

Holmgren, R. S.; 233 (1933).

King; 337 (1938).

Kirpich, P. Z.; 530 (1952).

Kocherin, D. I.; 632 (1955), 650 (1955).

Krey; (1912), 186 (1924).

Lamb; 74 (1895).

Langbein, W. B.; 330 (1938), 386 (1941).

Leach; 337 (1938).

Loewy; 523 (1952).

Lorentz; 185 (1924), 244 (1933), 307 (1936).

Miche; 523 (1952).

Morgan,!. H.; 330 (1938).

Mostkov; 654 (1956).

Pettis, C. R.; 309 (1936).

Pillsbury, G. B.; 459 (1948).

Powell, R. W.; 354 (1939).

Robertson, W. A.; 225 (1932).

Schoenfeld, J. C.; 544 (1952).

Sorensen, K. E.; 530 (1952).

Starr; (1947), 702 (1957).

Suiquet, F.; 705 (1957).

Virla;22(1858).

Volkov, I. M.; 707 (1957).

Wahlman, P.; 164 (1922).

Wallet, A.; 705 (1957).

Whitham; 602 (1954).

Wisler, C. O.; 354 (1939).

Zeitlinger; 136 (1917).

INDEX BY SUBJECTS

[This index is based on the abstracts in the chronological bibliography and not on the original text in the 

references. The purpose of this index is only to orient the reader toward the main references; there is no 

intention of showing full coverage of a subject by listing all references. It was the tendency in compiling 

this index to represent a subject by a listing no more than approximately 10 percent of the total number of references]

Acceleration of rate of rise of the stage (level)..

Account (comprehensive) of the progress, 

status, summary of literature, and references 

on waves in channels. See Appraisal.

Amplification of waves. See Attenuation and 

amplification of waves.

567 Analogy of gas-waves and water waves. See 

Gas-wave analogy with open-channel flows.

Analogy of sound and light waves versus liquid 

waves... 

Analogy of water surges and shock waves. 

See Surges compared to shock waves.
INDEX 227

Analytical general treatment of waves. See 

General theories and laws of wave motion; 

Differential equations for unsteady flow; Integration of partial differential equations 

for unsteady flow, general; Tidal waves in 

channels, general.

Analytical integration of continuity equation. 

See Continuity equation, analytical integration.

Appraisal or summary of the progress, or of the 

status of literature on unsteady flow and 

flood routing (wave physics) _..........

17,19,61,87,178,179,198,208,209, 217, 222, 227, 

266, 258, 271, 285,319,328, 472,496, 501, 528, 624, 

626,629, 656, 693, 715, 734, 743,750. 

Aprons in the channel bottom and the resultant waves. See Ridges on the bottom 

and the resultant waves. 

Area-depth (level) functions for storage reservoirs... .__ _-- . - 39,115,789 

Attenuation or amplification of waves as they 

progress along the channel-. __ .... 15,

17, 18, 45, 58, 69, 74, 81, 119, 123, 133, 142, 145, 

151,152,163,177,209,222,242,247, 262, 266, 267, 

268, 278,289,297,301,325,332, 350, 355, 360,361, 

366, 386,434,441, 443, 445,451, 456,482, 484,488, 

490, 493,497, 498, 509, 510, 518, 519, 521, 526, 535, 

539, 553, 567, 575,589, 594,627,632, 641,644,659, 

670, 675, 676, 679, 684,699, 700, 702, 711, 715, 726, 

735, 743, 745, 754, 799, 810.

Bore created by the progressing of waves-. _. _ 16, 

50, 64, 69, 72, 90,162, 215, 238, 249, 307, 325, 326, 

352, 353, 458, 518, 519, 642, 651. 

See also Breaking of waves; Dam breach 

waves; Sudden water release waves; 

Dry bed waves; Surges. 

Boundary or initial conditions in wave movement (different aspects).._________ 258, 

281,372, 418,425, 455, 456, 493, 539, 566, 607, 661, 

662, 667, 681, 841.

Breaking of waves________________ 69, 

73,99,162,441,458,481,584,585, 593,642 

See also Tidal waves; passing to bores.

Celerity and friction resistance (effect of resistance on celerity)_____ 47,185, 234, 265, 275, 

278, 287, 289, 318, 319, 333, 375, 401, 614, 711, 786 

Celerity and singularity resistance (effect of

singularity resistance of the celerity)-- 614,677,711 

Celerity of waves, along backwater curves__ 290 

along reservoirs____________. _ 290,790 

analytical determination for small disturbances-.________________ 3, 

4, 7, 11, 13, 14, 15,18, 19, 32, 34, 36, 44, 45, 

47, 57, 59, 63, 69, 77, 87, 90, 94,133,140,194, 

234, 253, 259, 267, 268, 272, 287, 290,307, 316, 

343,344,360, 366, 393, 401, 418, 419, 427, 456, 

487, 571, 589, 605, 623, 627, 693, 716, 738. 

approximate form for flood peaks (waves). 

See Celerity of flood peak, approximations.

as function of many parameters____ 224, 

290,318,359,689

as function of mean water depth. __ 44,45,329 

as function of wave form____ 87,209,307,366

Celerity of waves, along backwater curves 

Continued

center of gravity of wave_____ 36,45,63,360 

comparison of different procedures of determination. See Comparison of analytically determined, experimental and 

observed celerities.

definition-...________ 63,87 

determined by experiments in canals and 

in hydraulic models.._________ 8, 

12,15, 27, 31, 97,136,140,174, 204, 224, 235, 

236, 238, 245, 253, 271, 272, 329,337,343,344, 

366, 393, 501, 545, 567, 615, 618, 623, 652, 655, 

677, 683, 711.

determined by experiments or observations in nature.- . 35, 

100,153,186, 204, 220, 224, 238, 250, 251, 266, 

267, 272, 276, 280, 292, 301, 307, 310, 333, 545 

determined by method of characteristics 545 

discharge__..._.____-__ 45,47,278,330,344 

dry-bed waves _ 224,234,255,275,618,707,743 

flood peaks, approximations.... 148,170,190,220, 

290,301,324,325,415,434,472,745,751 

general.___ ___ ___ __ 101, 

148, 153, 170, 190, 209, 220, 264, 266, 267, 

280, 281, 290, 325, 332, 337, 366, 472, 491, 

551, 567, 569, 704, 720, 751, 791. 

front of a wave____________ _ 267, 

268,278,307,344,519,531,590,669 

general. _ _ . 1, 

3, 4, 7,14,16, 27, 30, 32, 33, 34, 36, 43, 44, 

45, 46, 47, 48, 55, 56, 59, 61, 63, 67, 69, 77, 

87, 90, 94,106,133,140,170,194, 200, 201, 

209, 259, 267, 268, 270, 290, 298, 316, 343, 

344, 345, 360, 418, 427, 456, 487, 519, 545, 

584, 589, 627, 655, 738. 

See also other references given under

topics for wave celerities, 

group of waves..______-___ ... 48 

in changing water depth or section____ 200, 

209,329,359,390,504,652

in channels covered with ice.-__ 251 

in deep water-___ 59,105,146,165,259,298,623 

in flowing water compared with celerity

in water at rest______________ 487 

in form d Q/dA, for long waves with friction 

resistance prevailing, and as function of 

water velocity of wave_ ___ ... 40, 

41, 45, 47, 91, 94, 97, 148, 174, 209, 243, 

265, 278, 280, 281, 299, 307, 309, 325, 329, 

330, 336, 343, 344, 350, 478, 539, 560, 569, 

577, 623, 689, 707. 

See also Celerity of long waves, 

in trapezoidal, triangular and other crosssectional shapes of channels____..... 13, 

253,290,333,539,543,589,689,736 

long waves..____________ _ 45, 

67, 87,133,170,177, 200, 222, 238, 264, 265, 267, 

278, 337, 388, 487, 491, 519, 623, 641, 786. 

monoclinal waves_____________ 97 

174,336,343,344,366,415,569,612,716 

negative waves (surges) .. 8,27, 

30,90,150,180,194, 202,204,229,235, 236, 269, 

290, 292 297, 314, 318, 333, 334, 345, 517, 545, 

565, 573, 683, 713, 716, 736, 790.
228 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Celerity oAvaves, along backwater curves 

Continued

positive waves (surges)___________ 27, 

30,35,42,71,90,136,140,153,180,194,202,224, 

229, 235, 236, 245, 255, 275, 290, 292, 297, 301, 

314, 318, 325, 332, 333, 334, 345, 489, 517, 545, 

565, 573, 600, 615, 618, 642, 677, 683, 713, 716, 

736.

roll waves _ ____________ 94,771 

small disturbances. See Celerity, analytical determination for small disturbances.

solitary wave. ___________. 14, 

31, 33, 43, 44, 55, 56, 61, 69, 87, 117, 201, 204, 

343,344,345,360,501,519,528,584,628. 

steep wave traveling on a long wave ..... 416

variation- ____..____________ 624 

waves with great curvature_____ 45,401,573 

waves with great height._________ 234,401 

Center of gravity of waves___________ 63 

Changing cross section of channels and the 

resultant wave propagation_________ 126, 

158,177,184,191,197,200,219,222,231,232,248, 

277,297,359,390,418,434,493,520, 523, 573, 605, 

614,627,644,651,652,691,707,736, 750. 

Changes in river channels and the effects on 

floods....__..........._...... 177,219,248,434

Channel slope effect on waves. See Slope of

channels and the effect on waves. 

Channel storage effect on flood reduction. See 

Reservoir and channel storage effects on 

floods.

Characteristics (lines), their principles, application and discussion in different aspects... 65, 

66, 90, 98, 122, 138, 149, 177, 239, 267, 268, 280, 

281, 287, 302, 307, 314, 333, 334, 339, 391, 396, 

413, 414, 418, 419, 430, 431, 433, 462, 471, 481, 

486, 489, 493, 498, 506, 518, 531, 542, 543, 544, 

545, 553, 555, 556, 557. 559, 568, 578, 581, 583, 

587, 590, 604, 607, 624, 637, 642, 643, 646, 647, 

648, 666, 669, 671, 675, 676, 678, 679, 697, 708, 

711,715,721,727,729,736.

Characteristic river reach in flood routing__ 714,

730,791 

See also Selection of channel reaches in

flood routing.

Classification of routing procedures_____ 491, 750 

Classification of waves______ 15, 202, 336, 493, 501 

Cnoidal waves............... 74,360, 599, 631, 807, 819

Coefficient of cross-sectional form__.__ 542,543 

Comparison of analytically determined, experimental and observed celerities.____ 140, 

148, 150, 153, 174, 186, 220, 224, 229, 238, 253, 

267, 271, 280, 281, 289, 290, 292, 297, 298, 310, 

318, 319, 329, 332, 333, 336, 337, 343, 344, 350, 

360, 393, 415, 433, 445, 475, 478, 484, 501, 528, 

543, 545, 567, 578, 580, 614, 623, 628, 637, 652, 

669, 677, 683, 707, 711, 713, 716, 737, 831, 836. 

Comparison of computed and observed 

attenuation of waves....__________ 528,

567,670,684,699, 702, 735,812 

Comparison of computed and observed dambreach waves.__________________ 602

Comparison of computed and observed heights

.......__ 140,

153, 186, 204, 229, 238, 251, 255, 280, 281, 290, 

292, 297, 310, 318, 319, 328, 332, 350, 360, 393, 

433, 449, 451, 452, 475, 478, 484, 509, 543, 561, 

613, 614, 637, 660, 675, 676, 705, 713, 723, 737, 

812, 823. 

Comparison of computed and observed local

inflow hydrographs ____________ 609,610 

Comparison of computed and observed reflections of waves._____..... 271,289,318,674,711

Comparison of computed and observed roll

waves.____________________ 654 

Comparison of computed and observed shape 

of waves, and especially of solitary waves_ 501, 

509, 516,528, 542,543,713,723 

Comparison of computed and observed sudden

water release waves______ 411,452, 542,543, 707 

Comparison of computed and observed tidal

waves...___..._.__... 518,606,641,691, 742 

Comparison of routed and observed flood 

waves..._____.__246, 255, 256, 258, 331, 371, 

382, 399, 508, 553, 591, 648, 662,667, 729, 730 

Comparison of routed and observed flood 

waves, analysis of eirors.__________ 246, 

331,484,518,568, 704 

Composition of waves (front, central zone,

backfront)_______ 267,268, 280, 319, 337, 345,348 

Compound channel cross section and waves. _ 277 

Comprehensive account on waves. See Appraisal and summary. 

Computation of waves, surges, floods, and

so forth. See Examples of computation. 

Continuity equation applied to unsteady flow, 

general... _ 34,38, 

85, 159, 256, 491, 539, 591, 636, 689, 750, 783, 786 

See also Application of continuity equation 

in flood routing through reservoirs and 

channels.

Continuity equation in unsteady flow, analytical integration. .. ... . 159, 256, 591,636, 783, 786 

different types of integration. See Flood

routing.

Continuous wave treated as discontinuities in 

steps.-- 71,418,520

Correlation of stream flow for forecasting purposes 410 

Cross section form coefficient--_-.. 542,543 

Currents produced by waves..___.___ 23, 263 

See also Tidal waves maximum currents

produced by tidal waves. 

Curvature, stage hydrograph at peak and 

waves. . - 316,567

wave surface....___-___ 45,209,448,450

Dam breach waves__ 64, 

71,136, 209, 218, 255, 256,304,323,344, 412, 421, 

423, 425, 433, 458, 472, 495, 505, 512, 531, 541, 

542, 543, 548, 602, 649, 715, 717, 718, 724, 788, 

789, 790, 802, 723.

Damping of waves. See Attenuation and 

amplification of waves.

Decomposition of waves in a direct and a 

reverse wave ___._____.. 302,303,328,418

Deep water waves......_.__ 24,44,129,493,514

Definition of flood routing_ 152
INDEX 229

Deflnitioa of waves____________ 335,528,738 

Deformation of waves. See Attenuatioa aad

ampliflcatioa of waves.

Depth in the case of sudden water release__ 71,

136,218,224,255

Differences of variables for unsteady and 

steady flow, used in the analysis of waves 133, 

145, 246, 247, 280, 307, 315, 316, 349, 355, 443, 

516, 590.

Differential equations for wave movement and 

unsteady flow in channels._________ 7, 

10, 18, 34, 45, 47, 56, 77, 81, 85, 128, 129, 147, 

162, 177, 209, 262, 280, 282, 291, 293, 305, 326, 

339, 343, 344, 368, 401, 420, 425, 427, 448, 455, 

458, 460, 465, 487, 491, 498, 501, 504, 508, 518, 

529, 531, 533, 548, 568, 584, 587, 588, 593, 596, 

608, 623, 624, 627, 631, 641, 643, 684, 689, 693, 

715, 723, 724, 726, 744, 750. 

See also General laws and theories of wave motion; integration of differential equations for 

unsteady flow.

Differential equationsof wave with great curvature______________________ 45 

Diffusion and water waves________ 508,750 

Discharge along a flood wave_________ 20, 

182,279,305,491,725

Discharge in sudden water releases..____ 71, 

136,224,255,328,412,505,774,788,790 

Distribution (discharge) graph. See Unit

graph.

Dry bed waves__________________ 224, 

275,304,370,411,493,618,707,743,764,823 

D ynamic e quation of unsteady flow. See Momentum equation of unsteady flow.

Effect of channel slope on waves. See Slope

of channel and the waves. 

Effect of reservoir or channel storage on floods. 

See Reservoir and channel storage effect on 

flood.

Effect of spillways, gates, syphons, on surges 

in channels _ . _______ 192, 

462,478,488,490,561,615,660,677,774 

Elevation of waves and bodies of water. See

Depth, Height, or Stage.

Emptying time of reservoirs______ 39,136,320 

Energy of waves.. 15, 38, 45, 48, 63, 69, 87, 107, 140, 

199,209,222,277,305,360,501,573,642,702 

Errors when flood routing and observed floods

are compared_______ 246,331,484,518,568,704 

Examples of computation of attenuation of

waves.. . . 589,629,670,675,676,684,699 

Examples of computation of dam breach

waves,.__ ... 505,512,541,542,602,788,789,790 

Examples of computation of surges (positive 

and negative) in channels_._______ 180, 

192, 204, 255, 271, 280, 281, 287, 290, 314, 318, 

319, 328, 333, 358, 371, 393, 425, 431, 456, 484, 

517, 542, 565, 573, 637, 659, 675, 676, 707, 723 

Examples of computation of wave propagation 

along the channel. . 287,293,302,322,331,518,723 

See also Examples for surges, and for flood 

routing.

Examples of flood routing through channels.. 242, 

246, 247, 255, 256, 258, 273, 287, 293, 331, 347, 

354, 371, 376, 382, 399, 405, 421, 425, 430, 431, 

436, 453, 454, 455, 500, 502, 508, 515, 518, 543, 

553, 568, 591, 598, 607, 608, 630, 636, 648, 659, 

662, 664, 666, 667, 670, 673, 678, 701, 704, 711, 

729,730,791,797.

Examples of flood routing through reservoirs. 39, 

40,354,535,553,607,662,788,790.

Experiments and examples of computation of 

rool waves__________________ 654,771

Experiments with dam-breach waves____ 602

Experiments with observations of waves in 

natural or artificial channels_______.. 35, 

99,150,151,153,186, 204,220, 224,250, 266, 267, 

272,273,276,280, 283,288,290,292,294,306,217, 

328,331,333,350,366,415,433,445,475,478, 525, 

545,591,637,676,716.

Experiments with waves in canals and in 

hydraulic models__________ 8, 

12, 21, 27, 30, 31, 99,136,140,174, 204, 212, 224, 

235,236,237,245,253,255,258,269,271,272,319, 

329,337,393,403,412,421,430,433,474,475,476, 

478,485,488,490,502,512, 518,528, 541, 542,543, 

545, 561, 562,578,579,580, 594, 602,615,618,621, 

623,629,637,642, 652,655,674, 676,677,683,684, 

691,699,711,713,723,737.

Experiments with waves of great heights. 21, 

31,250,412,512,541,602,615,723

Experiments with waves, positive and negative waves (surges)____________.. 27, 

30, 42,150,151,186, 204, 212, 224, 235, 236, 250, 

255. 266,267,269,271,272,276,317,319,328,333, 

337,412 ,421,433,445,475,478,482,484,488,490, 

545,552,561, 562, 578, 579, 580,615,621,637,642, 

676,683,711,713,721,737,758.

Experiments with solitary waves. 21,31,501,528,699

Experiments with wave reflection -_ 674

Flood routing through channel net_ 442,472,598 

Flood routing through channels, general 51, 

108,112,147,148,158,166,185,193, 207, 208, 254, 

255, 256, 258, 273, 285, 299, 311, 340,347, 394,420, 

439, 472, 491, 494, 500, 539, 567, 568,595, 619,624, 

656,667, 693,715,720,730, 734, 740,750, 765, 773, 

780, 797, 841, 843.

Flood routing through channels by routing 

storage and translation components of 

waves___________ 386,443 

Flood routing through channels by using 

difference of variables for unsteady and 

for steady flow__________ 133, 

145, 246, 247, 280, 307, 315, 316, 349, 355, 443, 

516, 590.

digital computer______ _ 587, 

607, 656, 662, 682, 695, 729, 733, 740, 748, 780, 

784, 800, 801, 811, 823, 829, 832, 834, 836, 840. 

electrical analog devices, electronic analog 

computers.. . 424, 

453, 454, 472, 391, 547, 566, 572, 576, 608, 625, 

630, 656, 663, 664, 680, 695, 701, 728, 733, 755, 

762, 781, 803, 805, 830, 844. 

See also Flood routing through reservoirs by using electric analog devices.
230 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Flood routing through channels by using 

Continued

electrical network analogy 442,472,728,755 

See also Flood routing through reservoirs by using electrical analog devices.

fourier series for tidal flood waves___ 428,624 

functional operator's procedure of integration of two partial differential equations________________ 270,293,510,527 

mechanical devices and integrators..__ 491, 

507,656,673,732

See also Flood routing through reservoirs by using mechanical devices 

and integrators.

method of characteristics in different iorms. 65, 

66,90,98,239,372,393,396, 418,419,421,425, 

430, 431, 433, 449, 455, 456, 471, 486, 491, 513, 

518, 542, 543, 544, 553, 559, 565, 568, 578, 581, 

590, 624, 646, 647, 648, 666, 669, 678, 697, 708, 

715, 721, 727, 750, 757, 780, 800, 801, 811, 843. 

nomographs.. . 407,455,656,678 

See also Flood routing through reservoirs by using nomographs, 

slide rule _____436,656 

See also Flood routing through reservoirs by using the slide rule, 

stage relationship (forecasting procedure).. 92, 

95,102,148,166, 246, 286,324, 349,394,404, 405, 

406, 440, 472, 491, 536, 537, 551, 582, 656, 685, 

712, 763.

successive averages.__________ 384,472

templates and transparent scales_____ 246,

349,443,503,636,690

two partial differential equations and an 

approximation of the momentum equation_____________________ 177( 

242, 246, 255, 256, 258, 285, 293, 299, 539, 568, 

666, 696, 704, 745, 827.

two partial differential equations and linearization of friction-resistance losses . __ 185, 

244,471,518,696

See also Integration of partial differential equation by linearizing the 

friction-resistance losses, 

two partial differential equations and a 

numerical method of computation___ 587, 

607,662,666, 721, 729,733, 757 

Flood routing through channels by using continuity equation and 

graphical procedure of integration____ 256, 

382,432,472, 500,656,692,720,732 

See also Integration by the use of mass 

curve, storage-discharge relationship

mass curve integration procedure_____ 101,

286,690,692 

See also Mass-curve procedure in flood

routing through reservoirs, 

numerical tabular procedure of integration. 108, 

112, 207,254, 312,340, 347,381, 395,422, 491, 619, 

668,720,730,734,791.

propagation celerity or time of travel (or 

lag) in a reach, with or without rating 

curve: with At__________ 51,148,166,254, 

380,384,405,406,595,633,730,732,763,777,791

Flood routing through channels by using continuity equation and Continued 

storage-discharge relationship as coefficient 

(Muskingum) method.. 313, 

340, 342,344,362, 376,392, 408, 420, 436, 440, 444, 

453, 454,463, 472, 491, 530, 547, 595, 656,668, 688, 

695,696,718,732,733,734,740,750, 772. 

storage-discharge relationship in simple 

form for a reach. 258, 

273,285,312, 347,380,381,399, 417,432,440,453, 

454, 468,491, 537, 595, 656, 668,688, 704, 720,730, 

732, 734, 740, 763, 797,820. 

storage-stage relationship in a reach 179, 

207,254,371,394,404,472,537,656,797, 

Flood routing through channels in irregular

channels . 428 

Flood routing through channels to obtain the

unit graphs_ 658 

Flood routing through junctions of channels.. 455,

491,607,662,729

routing through reservoirs, general .. 39, 

52, 54, 62, 68, 83,108,112,113,125, 131, 135, 137, 

141,152,157,159,172,175,183,187, 208, 209, 264, 

284, 286,311, 320,327, 365,367, 422, 439, 468,472, 

480,491, 492, 502, 524, 547, 586, 687,740, 787, 789, 

790,797,806. 

Flood routing through reservoirs by using 

characteristics_..__ 553

circular slide rules.. 397,656

continuity (storage) equation 39,

40, 62, 68, 75, 83,108, 112,125,131,137,141, 152,

157,159,160,172,175, 206, 209, 214, 221, 225,233,

264, 284, 286, 320,327, 367,387, 422, 468, 479, 480,

491,492, 502, 515, 524, 547, 653, 686, 687, 740, 787,

797,806.

electric analog devices____ 424,547,576,608,

625,630,656,663,664,680,695,729,733,762,844

mechanical devices and integrators... 367,

380,422,468,472,491, 507,656,673,732

nomographs. __ 284,480,515,656

rate of rise of levels as function of inflow

discharge and reservoir level.. 42*

straight-line slide rules 284,

387,397,436,491,656,769

templates... _...... . 636,690

See also Flood routing through channels

by using templates.

Flood routing through reservoirs by using the 

continuity equation and 

constant AH, with a graphical procedure

of integration______ 125,152,233: 

constant time interval At and trial-anderror selection of Aff .... 68,135,137,141,

152,157,159,160,206,214,225,320,422,524,687 

graphical (short cut) procedures of integration_... 75,104,125,152, 157,160, 206, 220, 

233,264,286,320,468,480,515, 524,530,547,687 

mass-curves procedure of integration 52, 

113,118,137,152,160,172,175, 206, 209, 221, 233, 

286,327,422,515,687,690,692. 

polar diagram for the outflow rate 206- 

storage factors (TF±QA</2, or similar 

expressions)...__ 62,112,137,141,152, 214, 

225,264,284,354,387,422,468,480,491, 502,687
INDEX 231

Flood routing through reservoirs by using 

templates (see also Flood routing through 

channels by using templates):._____ 636, 690 

Filling time of reservoirs____________ 131 

Force of waves. See Surface shape of waves. 

Forces on the free surface of waves. See

Pressure on the free surface of waves. 

Friction in a channel reach treated as a concentrated singular loss of head at the end of 

a reach, instead of as continuous losses. 164,393,513 

Friction resistance and waves_________ 38, 

45, 69, 123, 163, 197, 217, 222, 244, 245, 262, 265, 

287, 305, 318, 319, 375, 401, 418, 430, 455, 501, 

517, 520, 527, 528, 531, 535, 578, 580, 602, 614, 

624, 629, 638, 639, 644, 649, 670, 675, 699, 700, 

711, 725, 754, 760, 786, 790, 793. 

Functional operators and wave propagation. . 270,

293,510,527

Gas-wave analogy with open-channel unsteady flow.__-____ 426,458,464,481,558,587

General laws and theories of wave motion__ 10, 

16, 37, 38, 45, 46, 69, 71, 74, 79, 81, 82, 86, 88, 

89, 96, 100, 103, 106, 107, 114, 124, 126, 127, 128, 

129, 130, 138, 144, 146,149,155,156,165 181,182, 

184, 185, 188, 199, 200, 201, 204, 210, 

213, 219, 227, 228, 231, 232, 258, 264, 280, 282, 

291, 293, 305, 326, 339, 343, 344, 368, 401, 420, 

425, 427, 448, 450, 458, 475, 493, 501, 504, 508, 

509, 510, 514, 518, 531, 548, 568, 571, 593, 596, 

624, 627, 631, 684, 705, 715, 724, 726, 744, 765, 

775, 785.

General review of the theory of waves. See 

Appraisal and summary.

Heavy liquid movement___________ 38,277 

Height of surges, determined by experiments. 140, 

186, 204, 251, 272, 292, 294, 296, 306, 310, 328, 

543, 614, 637, 659, 713, 723, 737. 

Determined by measurements in natural 

channels_________________ 140, 

186,251,272,292,294,306,310,328,637,659 

Height of surges or waves, analytical determination______________________ 8 

42, 69, 126, 140, 146, 156, 169, 171, 180, 186, 191, 

199, 200, 204, 251, 272, 290, 294, 296, 297, 318, 

328, 360, 379, 401, 517, 543, 565, 573, 600, 613, 

614, 637, 638, 659, 666, 675, 676, 713, 737, 757, 

774, 790.

Height of waves, as function of bottom ridges 

(aprons) and sills. _________ 109,418,614,677 

as function of changing cross section___ 126, 

171,184,297,333,359,614

as function of channel parameters____ 11, 

69, 106, 169, 171, 180, 190, 199, 251, 315, 360, 

379, 543, 573, 614, 675, 678, 757. 

Highest progressive wave.._.._______ 705 

Hydraulic jump and the passage of waves. 

See Position of the hydraulic jump during a 

wave passage.

Index graph. See Unit graph. 

Inflow hydrograph, fictitious_____ 788,790,798 

Inflow hydrograph and flood waves _ _ 40, 54, 75, 78, 

83, 312, 408, 463, 521, 539, 552, 783, 790

Influence of capilarity on waves..______ 642 

Initial conditions in wave movement. See 

Boundary and initial conditions in wave 

movement.

Instability (or stability) of waves.. 36,45,63,87,360, 

361, 441, 482, 519, 526, 532, 564, 585, 58S 

Integration of continuity equation analytically. 

See Continuity equation, analytical integration.

Integration of partial differential equation for 

unsteady flow, around the critical flow._ 604,726 

by approximations of equations______ 177, 

234, 242, 287, 291, 305, 348, 350, 355, 371, 372, 

425, 428, 491, 508, 509, 529, 539, 567, 568, 609, 

610, 639, 667, 704, 715, 827. 

by flnite-diflerences method.. . ... 85, 

147, 234, 371, 372, 400, 416, 425, 428, 449, 455, 

491, 513, 515, 518, 568, 587, 607, 624, 662, 667, 

669, 678, 706, 715, 729, 761, 813, 833. 

by graphical methods of integration.. 293, 547,

586, 690, 731 

by linearization (linear waves)......__185

280, 287, 307, 355, 414, 437, 471, 497, 503, 504, 

518, 529, 559, 589, 593, 624, 815. 

by method of characteristics___ 65, 

66, 90, 98,122,138,149, 239, 267, 268, 280, 181, 

287, 302, 307, 334, 339, 372, 391, 393, 413, 414, 

416, 418, 419, 425, 430, 431, 433, 449, 455, 456, 

457, 458, 471, 481, 486, 489, 491, 498, 506, 513, 

518, 531, 542, 543, 544, 553, 559, 568, 678, 580, 

583, 590, 604, 609, 610, 624, 638, 639, 642, 643, 

646, 647, 648, 666, 669, 675, 678, 708, 715, 721, 

727, 758, 780, 811, 843.

by numerical methods of integration ... 587, 

607, 662, 667, 715, 721, 729, 758, 761, 776, 800, 

801, 833, 834, 838. 

by particular solutions (complex plane,

Rieman integration, and so forth)...... 352,

366, 510, 516, 623, 724, 825, 842

for nonlinear relation__..__ 498,

514, 516, 518, 529, 715, 724, 821, 837

general_________ . 38,

65, 66, 106, 114, 129, 133, 185, 247, 257, 258,

262, 267, 268, 278, 280, 281, 282, 291, 293, 302,

305, 315, 322, 326, 339, 350, 352, 355, 360, 366,

414, 420, 425, 428, 430, 448, 450, 455, 458, 460,

487, 498, 504, 508, 509, 510, 414, 516, 518, 523,

527, 529, 531, 538, 539, 553, 559, 567, 568, 573,

580, 588, 589, 590, 593, 598, 604, 608, 624, 625,

627, 643, 667, 684, 693, 708, 710, 715, 723, 724,

726, 744, 750,780,815,825,837,841.

neglecting friction losses.. 302,

360,487,523,589,623,715.

using digital computers. 587, 

607, 662, 695, 729, 780, 800, 801, 811, 823, 829, 

832,834,836,840.

using electric analog. -See Flood routing 

through channels by using the electric 

analog devices.

using functional operators_270, 293,510, 527, 628 

using heat conduction analog_ 598,64* 

using iteration procedure___..624,628,742,744 

using method of telegraph-equations ... 281, 

305,307,503,547

using power series.. 624 

using stretching transformation_ .612,710
232 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Interference of opposite waves_222,302,328,418,565 

Intermittent surges on steep slopes. See Roll

waves. 

Internal waves due to the change of fluid

density................__................493,571

Jump in bottom and waves. See Ridges on

the bottom and waves.

Junctions of channels and waves_ _____ 455, 

491,578, 580,607, 729,750

Kinematic waves________________ 634

Lateral inflows and outflows. See Local inflows and outflows. 

Laws of wave motion. See General theories

and laws of wave motion.

Length of wave, and channel parameters. __ 11,

15,69,177

and wave celerity.._.______59,69,177,337 

Level hydrograph in reservoirs. See Stage

hydrograph in reservoirs. 

Linearization of friction losses.

See Integration of two partial differential 

equations, by linearization.

Local inflows and outflows and waves____ 264, 

368, 380, 399, 455, 468, 472, 500, 515, 609, 610, 

647,648, 689, 763, 768.

Lock manipulation and resultant waves. 

See Navigation lock operation and resultant 

waves.

Long waves...__________________ 74, 

87, 88, 89, 124, 178, 184, 197, 200, 302, 337, 364, 

388, 458, 477, 519, 560, 589, 590, 593, 644, 671, 

710, 750.

See also Solitary waves; Flood routing 

through channels.

Mass-curves as the procedure of integration 

In flood routing ________________ 52,216 

See also Flood routing through channels 

and through reservoirs by using masscurves.

Maximum discharge of a flood wave_____ 20, 

45,142,209,394, 788, 790

Maximum height of a flood wave_____45,209,360 

Maximum slope of a flood wave. _______ 209 

Maximum velocity of a flood wave ____ 20,45,209 

Merging of opposite waves. See Interference

of opposite waves.

Mesh system of points in the numerical integration. See Net points in finite differences 

method of integration.

Model similitude. See Similitude of models. 

Moment of instability, stability of waves.

See Instability, stability, of waves. 

Momentum partial differential equation for 

unsteady flow, approximations. ______ 145, 

162,371,491,539,568

comparison of different forms of equation.. 498

general... _._______....____34,56,

77, 85,162,164, 337,371,491, 511, 539, 568,705

order of magnitude of neglected factors by

approximation_.__..... _ 145,337, 371,568

Monoclinal waves._______________ 97, 

174, 337,348, 472,612,633, 750 

Moving coordinate system (or moving observer) with the wave. 44,100,139,181,182, 418,612

Navigation lock operation and resultant 

waves____....________.______ 31

151, 154, 167, 173, 176, 186, 203, 260, 263, 283, 

288,358,359,401,573, 750,812. 

Navigation lock operation waves, and waves 

related to operation of water powerplants 151, 

173, 260, 794, 826

Negative waves (surge waves of negative 

height)................____._.___._._-.-__ 8,

15, 27, 30, 69, 90, 103, 151,162, 180, 186, 202, 203, 

204, 212, 229, 235, 236, 240, 269, 271, 297, 314, 

318, 328, 333, 337, 411, 458, 482, 493, 565, 573, 

613, 666, 679, 697, 713, 737, 750, 790, 798. 

Net of points in numerical analysis in the 

finite differences method of integration of 

partial differential equation for unsteady 

flow...._________________.... 587,

607, 624,662,667, 729

Nonuniform base flow on which the unsteady 

flow takes place (waves on the nonuniform 

flow)-........._..._.-. ...-. _ .- 250,

279, 280, 468,515, 589, 590, 680,818 

Norms and conditions for designing channels 

for unsteady flow 351

Observations of waves in natural channels. 

See Experiments and observations of waves 

in natural channels.

Ondular shape of positive surges 271,333,345, 

360,445,452,604,621,698,713,723 

Operators. See Functional operators. 

Orbit of fluid particles in solitary wave movement in quiet channel water__ 13,48,501,528 

Orbit of fluid particles in wave motion 13,

31,48,528

Oscillatory (oscillating) progressive waves.-.. 15,

16,19,49,117,144,162,398,472, 493, 518,575

Outflow hydrograph in flood routing.. 39,40,54,

75, 78,788,789,790,798

Outflow rating curve and flood routing 39,83,

92,115,221

Partial differential equations for unsteady 

flow. See Differential equations for unsteady flow.

Periodic waves....______ 38,44, 59,107,127,165

189,242,291,450,518, 523,685,689,744,808,816

Permanent waves, celerity 100,139,570

definition...__....___.... 100

general___..__ 100,139,144,156,162,181,189,

213,280,350,360,450,469,519, 529,570,607,631

Position of hydraulic jump during the passage

of a wave 617 

Positive wave (surge waves) 15,27,30,42, 64, 

90, 103, 140, 162, 169, 180, 202, 204, 229, 235, 

236, 240, 250, 271, 275, 296, 297, 302, 314, 318, 

323, 333, 337, 348, 412, 452, 456, 458, 462, 482, 

488, 489, 490, 493, 561, 562, 565, 573, 605, 613, 

615, 642, 661, 666, 675, 676, 679, 697, 707, 713, 

723,737,750,798,823,835.
INDEX 233

Powerplant (or industrial) canals, waves in, 

general. __... 140,151,153,179,192,193,204,212, 

223, 229, 235, 236, 242, 250, 251, 252, 271, 294, 

296, 311, 317, 332, 333, 350, 364, 382, 401, 425, 

449, 452, 455, 456, 478, 484, 517, 527, 562, 565, J 

605, 613, 615, 637, 655, 659, 660, 661, 666, 673, 

693, 697, 704, 716, 730, 737, 750, 774, 794, 826, 

830.

powerplant load and waves_ 661 

powerplant stability of operation, and

water waves____.___ _ _. 527 

surface waves in tailrace galleries__ 229,449, 

484,655,666,804,820,834,840 

waves in channels due to powerplant

operations.__.. 242,276,294,350,794,820,826 

Pressure on the wave surface__.__.. 82,124,427 

Propagation of tidal bores in estuaries. See

Tidal waves, propagation in estuaries. 

Propagation of waves, in canals and channels, 

some general or particular aspects.... 38,87,96, 

105, 126, 177, 184, 189, 204, 205, 217, 223, 226, 

249, 271, 275, 277, 280, 282, 302, 360, 373, 401, 

415, 472, 473, 477, 481, 487, 493, 510, 513, 533, 

589,596, 705,750.

in channels covered with ice.. _..__ 251 

in channels subject to rotational motion. . 785

Quasipermanent regime of floods in large 

rivers.. . 45,526

Rate of rise of water stage and waves_..__ 567

Rating curves for unsteady flow____ 132,265,299,

321,380,395,406,472,633

Recession curve of hydrographs and storagedischarge function...__.__.__ 384,389, 402 

Reduced values for unsteady flow. See Relative dimensionless values used for unsteady 

flow, Differences of values for unsteady flow 

and steady flow

Reflection of waves, total or partial, or both, 

at the changes (sudden or otherwise) of cross 

section. . . __ 239,274,289,333,485,518,674,711,808 

by experimental studies..__..__ 328,337 

caused by the variable depth...__ 549,808 

combined effect of currents and underground topography. _.. 485 

friction taken into account . _ 318,319,563 

of long waves_____ __._____ 239, 

279, 280, 295, 298, 302, 323, 337, 438, 470, 518, 

549, 550, 563, 575, 596.

of negative waves..____________ 203, 

271, 297, 318, 319, 328, 337, 425, 458, 470, 482, 

485, 518, 527, 565, 796.

of positive waves...._ _.._____ 271, 

297, 318, 319, 328, 337, 425, 458, 470, 482, 485, 

518, 527, 565, 796.

of solitary waves. _ _ ___ 

of tidal waves_. _.__- 222,295,438,550 

Relative dimensionless values used for unsteady flow___________.___ 114,145, 

182, 213, 281, 315, 462, 516, 620, 713, 798, 839.

Relative wave discharge (discharge in relation 

to steady flow discharge) _..._.__ 114,

145,182,213,281,516,620. 

Reservoir and channel storage effects on floods, 

discharge, level, height___________ 41, 

54, 58, 76, 83, 115, 118, 119, 152, 183, 187, 211, 

233, 365, 378, 394, 444, 447, 468, 479, 502, 521, 

539, 546, 552, 553, 555, 581, 592, 597, 632, 640, 

650, 653, 670, 686, 689, 703, 719, 746, 787, 792, 

817.

on time shifting of the peak_______ 83,521 

Reservoir routing. See Flood routing through

reservoirs.

Reservoir stage hydrograph. See Stage hydrograph in reservoirs. 

Resistance and waves. See Friction resistance

and waves. 

Review of wave theory. See Appraisal and

summary.

Ridges on the bottom (cross section changes, 

jumps, sills) and the resultant waves, general..... . ... .. 57,59,109,110, 111, 116,418, 711

height of waves created__________ 109 

profile of waves created_________ 109,116 

River reaches. See Selection of river reaches. 

Roll waves in steep channels_________ 94, 

99, 162, 188, 189, 249, 343, 344, 345, 369, 441, 

466, 493, 526, 532, 601, 654, 709, 770, 771, 831. 

Routing of floods through reservoirs. See

Flood routing through reservoirs. 

Routing period. See Selection of time unit 

At for routing.

Schematic wave shapes for analytical treatment (wave profile, hydrograph)--.____ 58, 

76, 434, 447, 521, 539, 591, 592, 594, 632, 633, 653, 

658, 670, 730, 783. 

Secondary waves on other waves. See Ondular

waves on positive waves.

Seiches.________.__._________ 472

Selection of channel reaches (length) in fl00(j

routing. ...__ 380,

500, 661, 696, 704, 729, 730, 767, 791,820

Selection of time unit A£ for flood routing___ 500,

607,661,662,667, 729

Shape of waves. See Surface shape of waves. 

Shock-waves analogy. See Surges compared

with shock-waves.

Similitude of hydraulic models for wave movement, distorted models__________ 476,513 

general...________ 299,319,476,513,759,824 

Slope due to change of velocity head and acceleration......_...__.__....... 265,316,334

friction resistance in unsteady flow.._ 265,375 

Slope of channels and the effect on waves___ 45, 

158, 418,465,517,577, 580,594, 602, 670, 711 

Slopes in unsteady flow__.__.____ 265,334 

Small disturbance, celerity of waves. See 

Celerity, analytical determination for small 

disturbances; Celerity determined by experiments and observations. 

Small height, waves of... 45,287,390,590,596,715,738
234 FLOOD-ROUTING METHODS AND UNSTEADY FLOW

Solitary wave, compared with a bore. ___ 642,723 

general and special aspects._______ 13, 

15,16,18, 21, 31, 33, 36,43, 44, 48, 61, 63,69,73, 

74,87, 88,89,99,100,109,139,162,178,201,235, 

236, 280, 343, 344, 345, 445, 450, 451, 469, 475, 

493, 501, 514, 528, 529, 556, 571, 584, 593, 603, 

621, 628, 629, 642, 665, 671, 699, 700, 702, 713, 

722, 744,807,816.

Spectrum analysis applied to gravity waves _ 554 

Stability of waves. See Instability, stability,

of waves.

Stage hydrograph of reservoirs________ 83,125 

Standing waves. ._._. ._.-. 16,109,345, 391,638 

Stationary waves________________ 57, 

59, 74, 90, 105, 110, 111, 116, 124, 200, 201, 215, 

450,469,599. 

Storage-discharge (or stage) relationship by

statistical analysis_______________ 404 

Storage factors (TF±QA</5, or similar expressions), general____________ 62,347,354,387 

Storage function (storage-discharge or storagestage relationship for reservoirs and channels) 39, 

159, 179, 264, 273, 300, 308, 312, 313, 321, 341, 

354,363, 376, 377, 380, 384, 389, 392, 399, 402, 404, 

409,472, 491, 500, 502,668,696, 703, 704, 730, 740, 

763, 767, 783, 789. 

Storage in river valleys and the resultant

hydrograph._...._____________ 402,472 

Storage-outflow relationship__ ___.__ 159, 

160,206, 210,273,387, 740, 767,789 

Sudden change in bottom and the resultant 

waves. See Ridges on the bottom and the 

resultant waves.

Sudden water release, waves of 71, 

136,169, 209, 218, 224, 255, 275, 279, 280,304, 411, 

412, 434, 452, 505, 512, 531, 541, 542, 543, 548, 717, 

718, 719, 724, 766, 788, 790, 810, 828, 

Summary on waves. See Appraisal and

summary.

Summation curve of hydrograph. See Masscurves.

Surface area versus elevation for storage reservoirs. See Area-depth, level, height, functions for storage reservoirs.

Surface shape of waves, cnoidal waves... . 74,

360, 599, 631

dam breach waves.. . __.. 71,

136, 512, 541, 542, 543, 601, 724

general... _. 2,15,

19, 33, 35, 44, 45, 61, 74,129, 130, 140, 146, 162,

177, 199, 209, 262, 268, 269, 315, 360, 401, 411,

452, 465, 478, 510, 512, 516, 571, 572, 599, 628,

631, 642, 689.

negative waves. . 269,271,297,411

positive waves and of wavefront 271,

297, 360, 401, 452, 489, 510, 516, 642

solitary wave. 36,

43, 44,162, 344, 345, 360, 469, 501, 528, 571, 593,

628, 631.

symmetrical waves....__ __ -_ 33,199 

trochoidal waves....._ .. _.. 2, 28 

Surface waves In closed conduits (tunnels, 

pipes, galleries)-_.___ .... 574,619,620,655

Surges, compared with hydraulic jump 134,

204, 345, 348, 621, 642, 713, 723, 750

compared with shock waves.. 302,

323, 391, 418, 421, 446, 458, 464, 481, 558, 616,

624, 714, 749, 782, 793.

general and particular aspects of. . 8, 

15, 27, 30, 42, 103, 134, 140, 150, 151, 162, 163, 

178, 180, 186, 192, 202, 204, 212, 229, 235, 240, 

241, 249, 250, 269, 271, 275, 281, 290, 291, 292, 

296, 297, 302, 314, 317, 318, 319, 326, 333, 337, 

345, 348, 350, 358, 359, 393, 411, 412, 413, 416, 

418, 419, 421, 425, 445, 452, 456, 458, 472, 475, 

481, 482, 486, 489, 493, 510, 513, 517, 519, 520, 

558, 561, 562, 565, 573, 584, 599, 600, 605, 613, 

615, 624, 637, 638, 642, 655, 661, 666, 675, 676, 

679, 683, 697, 707, 713, 714, 716, 723, 725, 737, 

750, 756, 788, 789, 790, 798, 835, 840. 

See also Positive waves, Negative

waves, and Tidal bores.

moving on long waves__ _.. 416 

Synthetic inflow graph. See Virtual channelinflow graph. 

Synthetic graph. See Unit graph.

Theories of wave motion. See General

theories and laws of wave motion. 

Tidal waves moving along channels, channel 

cross section variation and tidal waves___ 627,

645,691

comparison of computed and observed (or

experimental) tidal waves._______ 161,

238,257,268, 307, 329, 357,570, 606,651,691

convergent channel and tidal wave.. _ 357.645

discharge variation in tidal wave.____ 622

electric analogy and tidal waves..____ 459,

566,755,779

estuarine channel system______ 520,566,741

examples of tidal wave computations. __ 67,

80, 81, 84, 143, 185, 196, 222, 238, 244, 277, 305,

307, 442, 459, 471, 517, 518, 520, 568, 570, 606,

646, 651, 755.

general . _. 6, 

16, 22, 25, 26, 29, 30, 50, 53, 67, 72, 80, 81, 84, 

86, 87, 90, 93, 107, 147, 158, 168, 178, 185, 191, 

196, 222, 241, 244, 257, 277, 305, 307, 352, 353, 

357, 374, 383, 428, 430, 437, 460, 467, 471, 499, 

517, 518, 519, 520, 540, 550, 557, 568, 624, 626, 

627, 638, 639, 691, 694, 731, 739, 741, 742, 745, 

753, 814, 836.

influence of friction resistance on tidal 

waves. _ _ . __ 81, 

143,168,185,244,520,638,639,651, 786 

maximum current...______ 81,459,471,472 

moving and celerity of tidal waves and 

bores_____________________ 22, 

25, 32, 34, 35, 50, 72, 81, 90, 143,158,161, 195, 

221, 222, 235, 236, 238, 244, 305, 307, 326, 329, 

353, 374, 428, 459, 471, 472, 489, 517, 518, 520, 

534, 600, 622, 624, 638, 639, 642, 651, 694, 706, 

731, 739, 741, 786, 833.

observations of tidal waves_ 25,67,72,161,238,

257,307,460,522,557,570,606,622, 626,641

passing to a bore _ . 16,

50, 64, 72, 90, 238, 307, 326, 352, 353, 458, 518,

519, 540, 642, 651, 833, 842.
INDEX 235

Tidal waves moving along channels, channel 

cross section variation and tidal waves 

Continued

propagation in estuaries and rivers____ 8, 

16, 22, 25, 34, 64, 67, 72, 81, 86, 87,161,168,195, 

196, 209, 238, 241, 277, 295, 305, 307, 352, 353, 

357, 383, 428, 430, 471, 474, 517, 518, 520, 522, 

534, 540, 550, 570, 606, 638, 639, 645, 646, 682, 

706, 739, 741, 742, 745, 753, 776, 778, 795, 813, 

814, 836, 842.

surface shape (profile) of tidal waves___ 34,

35,72,374

without change of shape__________ 795 

Time of emptying of reservoirs. See Emptying

time of reservoirs.

Torrent (river) bores_.______._ 64,325,326 

Trams of waves_____._ 94,99,369,599,771,831 

Trajectories of fluid particles. See Orbits of

fluid particles. 

Translatory waves. See Propagation of waves;

Solitary waves; Long waves. 

Transmissibility of waves at obstacles (Reflection of waves).._..__ 274,470,563,674,711,808 

Tributary inflow determined by flood routing. 362

Undular profile of waves. See ondular waves. 

Unit graph, synthetic graph, distribution 

graph, index graph, flood hydrograph _ 230,

261, 342, 346, 356, 362, 380, 385, 409, 508, 657,

658, 733, 752, 809.

Valley storage. See Storage function.

Variable cross section (depth) and waves. See 

Changing cross section and wave propagation and Changes in river channels and the 

effect on floods.

Velocity distribution and waves_______ 401,

584, 611,635,735

Velocity in sudden water releases_______ 71, 

169,224,290,306,618, 724,790

Velocity of fluid particles in wave movement. 5,

11,15,19,24,56,182, 269,501,528

Velocity of progressing flood wave ______ 20,

45,169,177,180,190,199,220,269,641,724

Velocity (mean) of solitary wave. _____ 344,628

Velocity of water accelerated by wave motion. 3,

4,33,34,36,45, 71,290, 345,359,573,790

Virtual channel-inflow graph ______ 377,378,658

Volume-depth (elevation, stage) function of

reservoirs. See Storage function. 

Volume of waves________________ 69

Wave celerity. See Celerity of waves.

Wave on dry bed. See Dry bed waves, Sudden water release waves, and Dam breach 

waves.

Waves in powerplant canals. See Powerplant 

canals.

Waves (surface) in tunnels, galleries, and 

pipes. See Surface waves in closed conduits.

Wave in transition zone between channel and 

reservoir________-__________ 652

Waves in supercritical flow....________ 590

Wedge storage____ _______ 491

Wind effect on waves in canals.._____ 263,430

o