TEXT FLY WITHIN
THE BOOK ONLY
u< OU
^ GO
DO
160917>m
STOKER
WATER WAVES
OSMANIA UNIVERSITY LIBRARY
Call No. ^VL-'Sj S£7 lO • Accession No.
Author S-Jfc fcfcT, T -
Title CO<X^L/L
This book should be returned on or before the date last marked below.
WATER WAVES
The Mathematical Theory with Applications
PURE AND APPLIED MATHEMATICS
A Series of Texts and Monographs
Edited by
R. COURANT • L. BERS . J. J. STOKER
VOLUME IV
Waves about a harbor
WATER WAVES
The Mathematical Theory
with Applications
J. J. STOKER
INSTITUTE OF MATHEMATICAL SCIENCES
NEW YORK UNIVERSITY, NEW YORK
19 f fill 57
INTERSCIENCE PUBLISHERS, INC., NEW YORK
INTERSCIENCE PUBLISHERS LTD., LONDON
All Rights Reserved
LIBRARY OF CONGRESS CATALOG CARD NUMBER
56-8228
INTERSCIENCE PUBLISHERS INC.
250 Fifth Avenue, New York 1, N.Y.
For Great Britain and Northern Ireland:
INTERSCIENCE PUBLISHERS LTD.
88/90 Chancery Lane, London W. C. 2
PRINTED IN THE NETHERLANDS
BY LATE HOITSEMA BROTHERS, GRONINGEN
To
NANCY
Introduction
1. Introduction
The purpose of this book is to present a connected account of the
mathematical theory of wave motion in liquids with a free surface
and subjected to gravitational and other forces, together with ap-
plications to a wide variety of concrete physical problems.
Surface wave problems have interested a considerable number of
mathematicians beginning apparently with Lagrange, and con-
tinuing with Cauchy and Poisson in France.* Later the British school
of mathematical physicists gave the problems a good deal of atten-
tion, and notable contributions were made by Airy, Stokes, Kelvin,
Rayleigh, and Lamb, to mention only some of the better known. In
the latter part of the nineteenth century the French once more took
up the subject vigorously, and the work done by St. Venant and
Boussinesq in this field has had a lasting effect: to this day the
French have remained active and successful in the field, and par-
ticularly in that part of it which might be called mathematical
hydraulics. Later, Poincar^ made outstanding contributions par-
ticularly with regard to figures of equilibrium of rotating and gravi-
tating liquids (a subject which will not be discussed in this book);
in this same field notable contributions were made even earlier
by Liapounoff. One of the most outstanding accomplishments in the
field from the purely mathematical point of view — the proof of the
existence of progressing waves of finite amplitude — was made by
Nckrassov [N.I], [N.lajf in 1921 and independently by a different
means by Levi-Civita [L.7] in 1925.
The literature concerning surface waves in water is very extensive.
In addition to a host of memoirs and papers in the scientific journals,
there are a number of books which deal with the subject at length.
First and foremost, of course, is the book of Lamb [L.3], almost
a third of which is concerned with gravity wave problems. There
are books by Bouasse [B.15], Thorade [T.4], and Sverdrup [S.39]
* This list would be considerably extended (to include Euler, the Bernoullis,
and others) if hydrostatics were to be regarded as an essential part of our subject.
t Numbers in square brackets refer to the bibliography at the end of the book.
X INTRODUCTION
devoted exclusively to the subject. The book by Thorade consists
almost entirely of relatively brief reviews of the literature up to
1931 — an indication of the extent and volume of the literature
on the subject. The book by Sverdrup was written with the special
needs of oceanographers in mind. One of the main purposes of the
present book is to treat some of the more recent additions to our
knowledge in the field of surface wave problems. In fact, a large part
of the book deals with problems the solutions of which have been
found during and since World War II; this material is not available
in the books just now mentioned.
The subject of surface gravity waves has great variety whether
regarded from the point of view of the types of physical problems
which occur, or from the point of view of the mathematical ideas
and methods needed to attack them. The physical problems range
from discussion of wave motion over sloping beaches to flood waves
in rivers, the motion of ships in a sea-way, free oscillations of enclosed
bodies of water such as lakes and harbors, and the propagation of
frontal discontinuities in the atmosphere, to mention just a few.
The mathematical tools employed comprise just about the whole of
the tools developed in the classical linear mathematical physics
concerned with partial differential equations, as well as a good part
of what has been learned about the nonlinear problems of mathe-
matical physics. Thus potential theory and the theory of the linear
wave equation, together with such tools as conformal mapping and
complex variable methods in general, the Laplace and Fourier
transform techniques, methods employing a Green's function, integral
equations, etc. are used. The nonlinear problems arc of both elliptic
and hyperbolic type.
In spite of the diversity of the material, the book is not a collection
of disconnected topics, written for specialists, and lacking unity and
coherence. Instead, considerable pains have been taken to supply
the fundamental background in hydrodynamics — and also in some
of the mathematics needed — and to plan the book in order that it
should be as much as possible a self-contained and readable whole.
Though the contents of the book are outlined in detail below, it has
some point to indicate briefly here its general plan. There arc four
main parts of the book:
Part I, comprising Chapters 1 and 2, presents the derivation of
the basic hydrodynamic theory for non-viscous incompressible fluids,
and also describes the two principal approximate theories which form
INTRODUCTION XI
the basis upon which most of the remainder of the book is built.
Part II, made up of Chapters 3 to 9 inclusive, is based on the ap-
proximate theory which results when the amplitude of the wave
motions considered is small. The result is a linear theory which from
the mathematical point of view is a highly interesting chapter in
potential theory. On the physical side the problems treated include
the propagation of waves from storms at sea, waves on sloping
beaches, diffraction of waves around a breakwater, waves on a
running stream, the motion of ships as floating rigid bodies in a sea-
way. Although this theory was known to Lagrange, it is often referred
to as the Cauchy-Poisson theory, perhaps because these two mathe-
maticians were the first to solve interesting problems by using it.
Part III, made up of Chapters 10 and 11, is concerned with problems
involving waves in shallow water. The approximate theory which
results from assuming the water to be shallow is not. a linear theory,
and wave motions with amplitudes which are not necessarily small
can be studied by its aid. The theory is often attributed to Stokes
and Airy, but was really known to Lagrange. If linearized by making
the additional assumption that the wave amplitudes are small, the
theory becomes the same as that employed as the mathematical
basis for the theory of the tides in the oceans. In the lowest order
of approximation the nonlinear shallow water theory results in a
system of hyperbolic partial differential equations, which in im-
portant special cases can be treated in a most illuminating way with
the aid of the method of characteristics. The mathematical methods
are treated in detail in Chapter 10. The physical problems treated in
Chapter 10 are quite varied; they include the propagation of unsteady
waves due to local disturbances into still water, the breaking of
waves, the solitary wave, floating breakwaters in shallow water. A
lengthy section on the motions of frontal discontinuities in the
atmosphere is included also in Chapter 10. In Chapter 11, entitled
Mathematical Hydraulics, the shallow water theory is employed to
study wave motions in rivers and other open channels which, unlike
the problems of the preceding chapter, are largely conditioned by
the necessity to consider resistances to the flow due to the rough
sides and bottom of the channel. Steady flows, and steady progressing
waves, including the problem of roll waves in steep channels, are
first studied. This is followed by a treatment of numerical methods
of solving problems concerning flood-waves in rivers, with the object
of making flood predictions through the use of modern high speed
XII INTRODUCTION
digital computers. That such methods can be used to furnish accurate
predictions has been verified for a flood in a 400-mile stretch of the
Ohio River, and for a flood coming down the Ohio River and passing
through its junction with the Mississippi River.
Part IV, consisting of Chapter 12, is concerned with problems
solved in terms of the exact theory, in particular, with the use of the
exact nonlinear free surface conditions. A proof of the existence of
periodic waves of finite amplitude, following Levi-Civita in a general
way, is included.
The amount of mathematical knowledge needed to read the book
varies in different parts. For considerable portions of Part II the
elements of the theory of functions of a complex variable are assumed
known, together with some of the standard facts in potential theory.
On the other hand Part III requires much less in the way of specific
knowledge, and, as was mentioned above, the basic theory of the
hyperbolic differential equations used there is developed in all detail
in the hope that this part would thus be made accessible to engineers,
for example, who have an interest in the mathematical treatment of
problems concerning flows and wave motions in open channels.
In general, the author has made considerable efforts to try to
achieve a reasonable balance between the mathematics and the
mechanics of the problems treated. Usually a discussion of the physical
factors and of the reasons for making simplified assumptions in each
new type of concrete problem precedes the precise formulation of the
mathematical problems. On the other hand, it is hoped that a clear
distinction between physical assumptions and mathematical deduc-
tions — so often shadowy and vague in the literature concerned
with the mechanics of continuous media — has always been main-
tained. Efforts also have been made to present important portions
of the book in such a way that they can be read to a large extent
independently of the rest of the book; this was done in some cases
at the expense of a certain amount of repetition, but it seemed to
the author more reasonable to save the time and efforts of the reader
than to save paper. Thus the portion of Chapter 10 concerned with
the dynamics of the motion of fronts in meteorology is largely
self-contained. The same is true of Chapter 11 on mathematical
hydraulics, and of Chapter 9 on the motion of ships.
Originally this book had been planned as a brief general introduc-
tion to the subject, but in the course of writing it many gaps and
inadequacies in the literature were noticed and some of them have
INTRODUCTION XIII
been filled in; thus a fair share of the material presented represents
the result of researches carried out quite recently. A few topics which
are even rather speculative have been dealt with at some length
(the theory of the motion of fronts in dynamic meteorology, given
in Chapter 10.12, for example); others (like the theory of waves on
sloping beaches) have been treated at some length as much because
the author had a special fondness for the material as for their intrinsic
mathematical interest. Thus the author has written a book which is
rather personal in character, and which contains a selection of
material chosen, very often, simply because it interested him, and
he has allowed his predilections and tastes free rein. In addition,
the book has a personal flavor from still another point of view since
a quite large proportion of the material presented is based on the work
of individual members of the Institute of Mathematical Sciences of
New York University, and on theses and reports written by students
attending the Institute. No attempt at completeness in citing the
literature, even the more recent literature, was made by the author;
on the other hand, a glance at the Bibliography (which includes
only works actually cited in the book) will indicate that the recent
literature has not by any means been neglected.
In early youth by good luck the author came upon the writings
of scientists of the British school of the latter half of the nineteenth
century. The works of Tyndall, Huxley, and Darwin, in particular,
made a lasting impression on him. This could happen, of course, only
because the books were written in an understandable way and also
in sucli a way as to create interest and enthusiasm: — but this was
one of the principal objects of this school of British scientists.
Naturally it is easier to write books on biological subjects for non-
specialists than it is to write them on subjects concerned with the
mathematical sciences — just because the time and effort needed to
acquire a knowledge of modern mathematical tools is very great.
That the task is not entirely hopeless, however, is indicated by John
TyndaU's book on sound, which should be regarded as a great classic
of scientific exposition. On the whole, the British school of popularizers
of science wrote for people presumed to have little or no foreknow-
ledge of the subjects treated. Now-a-days there exists a quite large
potential audience for books on subjects requiring some knowledge
of mathematics and physics, since a large number of specialists of
all kinds must have a basic training in these disciplines. The author
hopes that this book, which deals with so many phenomena of every
XIV INTRODUCTION
day occurrence in nature, might perhaps be found interesting, and
understandable in some parts at least, by readers who have some
mathematical training but lack specific knowledge of hydro-
dynamics.* For example, the introductory discussion of waves on
sloping beaches in Chapter 5, the purely geometrical discussion of
the wave patterns created by moving ships in Chapter 8, great parts
of Chapters 10 and 11 on waves in shallow water and flood waves in
rivers, as well as the general discussion in Chapter 10 concerning
the motion of fronts in the atmosphere, are in this category.
2. Outline of contents
It has already been stated that this book is planned as a coherent
and unified whole in spite of the variety and diversity of its contents
on both the mathematical and the physical sides. The possibility of
achieving such a purpose lies in the fortunate fact that the material
can be classified rather readily in terms of the types of mathematical
problems which occur, and this classification also leads to a reasonably
consistent ordering of the material with respect to the various types
of physical problems. The book is divided into four main parts.
Part I begins with a brief, but it is hoped adequate, development
of the hydrodynamics of perfect incompressible fluids in irrotational
flow without viscosity, with emphasis on those aspects of the subject
relevant to flows with a free surface. Unfortunately, the basic general
theory is unmanageable for the most part as a basis for the solution
of concrete problems because the nonlinear free surface conditions
make for insurmountable difficulties from the mathematical point
of view. It is therefore necessary to make restrictive assumptions
which have the effect of yielding more tractable mathematical
formulations. Fortunately there are at least two possibilities in this
respect which are not so restrictive as to limit too drastically the
physical interest, while at the same time they are such as to lead to
mathematical problems about which a great deal of knowledge is
available.
One of the two approximate theories results from the assumption
that the wave amplitudes are small, the other from the assumption
* The book by Rachel Carson [C.I 6] should be referred to here. This book is
entirely nonmathematical, but it is highly recommended for supplementary
reading. Parts of it are particularly relevant to some of the material in
Chapter 6 of the present book.
INTRODUCTION XV
that it is the depth of the liquid which is small — in both cases, of
course, the relevant quantities are supposed small in relation to some
other significant length, such as a wave length, for example. Both of
these approximate theories are derived as the lowest order terms
of formal developments with respect to an appropriate small dimen-
sionless parameter; by proceeding in this way, however, it can be
seen how the approximations could be carried out to include higher
order terms. The remainder of the book is largely devoted to the
working out of consequences of these two theories, based on concrete
physical problems: Part II is based on the small amplitude theory,
and Part III deals with applications of the shallow water theory.
In addition, there is a final chapter (Chapter 12) which makes up
Part IV, in which a few problems are solved in terms of the basic
general theory and the nonlinear boundary conditions are satisfied
exactly; this includes a proof along lines due to Levi-Civita, of the
existence, from the rigorous mathematical point of view, of progressing
waves of finite amplitude.
Part II, which is concerned with the first of the possibilities,
might be called the linearized exact theory, since it can be obtained
from the basic exact theory simply by linearizing the free surface
conditions on the assumption that the wave motions studied con-
stitute a small deviation from a constant flow with a horizontal free
surface. Since we deal only with irrotational flows, the result is a
theory based on the determination of a velocity potential in the space
variables (containing the time as a parameter, however) as a solution
of the Laplace equation satisfying certain linear boundary and initial
conditions. This linear theory thus belongs, generally speaking, to
potential theory.
There is such a variety of material to be treated in Part II, which
comprises Chapters 3 to 9, that a further division of it into sub-
divisions is useful, as follows: 1) subdivision A, dealing with wave
motions that arc simple harmonic oscillations in the time; 2) sub-
division B, dealing with unsteady, or transient, motions that arise
from initial disturbances starting from rest; and 3) subdivision C,
dealing with waves created in various ways on a running stream,
in contrast with subdivisions A and B in which all motions are
assumed to be small oscillations near the rest position of equilibrium
of the fluid.
Subdivision A is made up of Chapters 3, 4, and 5. In Chapter 3
the basically important standing and progressing waves in liquids
XVI INTRODUCTION
of uniform depth and infinite lateral extent are treated; the important
fact that these waves are subject to dispersion comes to light, and
the notion of group velocity thus arises. The problem of the uniqueness
of the solutions is considered — in fact, uniqueness questions are
intentionally stressed throughout Part II because they are interesting
mathematically and because they have been neglected for the most
part until rather recently. It might seem strange that there could be
any interesting unresolved uniqueness questions left in potential
theory at this late date; the reason for it is that the boundary con-
dition at a free surface is of the mixed type, i.e. it involves a linear
combination of the potential function and its normal derivative, and
this combination is such as to lead to the occurrence of non-trivial
solutions of the homogeneous problems in cases which would in the
more conventional problems of potential theory possess only iden-
tically constant solutions. In fact, it is this mixed boundary con-
dition at a free surface which makes Part II a highly interesting
chapter in potential theory — quite apart from the interest of the
problems on the physical side. Chapter 4 goes on to treat certain
simple harmonic forced oscillations, in contrast with the free oscil-
lations treated in Chapter 3. Chapter 5 is a long chapter which deals
with simple harmonic waves in cases in which the depth of the water
is not constant. A large part of the chapter concerns the propagation
of progressing waves over a uniformly sloping beach; various methods
of treating the problem are explained — in part with the object of
illustrating recently developed techniques useful for solving boundary
problems (both for harmonic functions and functions satisfying the
reduced wave equation) in which mixed boundary conditions occur.
Another problem treated (in Chapter 5.5) is the diffraction of waves
around a vertical wedge. This leads to a problem identical with the
classical diffraction problem first solved by Sommerfeld [S.I 2] for
the special case of a rigid half-plane barrier. Here again the uniqueness
question comes to the fore, and, as in many of the problems of Part II,
it involves consideration of so-called radiation conditions at infinity. A
uniqueness theorem is derived and also a new, and quite simple and
elementary, solution for Sommerfeld's diffraction problem is given.
It is a curious fact that these gravity wave problems, the solutions
of which are given in terms of functions satisfying the Laplace
equation, nevertheless require for the uniqueness of the solutions
that conditions at infinity of the radiation type, just as in the more
familiar problems based on the linear wave equation, be imposed;
I INTRODUCTION XVII
ordinarily in potential theory it is sufficient to require only boundedness
conditions at infinity to ensure uniqueness.
In subdivision B of Part II, comprised of Chapter 6, a variety of
problems involving transient motions is treated. Here initial con-
ditions at the time t — 0 are imposed. The technique of the Fourier
transform is explained and used to obtain solutions in the form of
integral representations. The important classical cases (treated first
by Cauchy and Poisson) of the circular waves due to disturbances at
a point of the free surface in an infinite ocean are studied in detail.
For this purpose it is very useful to discuss the integral representations
by using an asymptotic approximation due to Kelvin (and, indeed,
developed by him for the purpose of discussing the solutions of just
such surface wave problems) and called the principle, or method, of
stationary phase. These results then can be interpreted in a striking
way in terms of the notion of group velocity. Recently there have
been important applications of these results in oceanography: one
of them concerns the type of waves called tsunamis, which are
destructive waves in the ocean caused by earthquakes, another
concerns the location of storms at sea by analyzing wave records
on shore in the light of the theory at present under discussion. The
question of uniqueness of the transient solutions — again a problem
solved only recently — is treated in the final section of Chapter 6.
An opportunity is also afforded for a discussion of radiation con-
ditions (for simple harmonic waves) as limits as t -> oo in appropriate
problems concerning transients, in which boundedness conditions at
infinity suffice to ensure uniqueness.
The final subdivision of Part II, subdivision C, deals with small
disturbances created in a stream flowing initially with uniform
velocity and with a horizontal free surface. Chapter 7 treats waves in
streams having a uniform depth. Again, in the case of steady motions,
the question of appropriate conditions of the radiation type arises;
the matter is made especially interesting here because the circum-
stances with respect to radiation conditions depend radically on the
parameter U2/g/*, with U and h the velocity and depth at infinity, res-
pectively. Thus if U2/gh > 1, no radiation conditions need be im-
posed, if U2/gh < 1 they are needed, while if U2jgh = 1 something
quite exceptional occurs. These matters are studied, and their physical
interpretations are discussed in Chapter 7.3 and 7.4. In Chapter 8
Kelvin's theory of ship waves for the idealized case of a ship regarded
as a point disturbance moving over the surface of the water is treated
XVIII INTRODUCTION
in considerable detail. The principle of stationary phase leads to a
beautiful and elegant treatment of the nature of ship waves that is
purely geometrical in character. The cases of curved as well as
straight courses are considered, and photographs of ship waves taken
from airplanes are reproduced to indicate the good accord with
observations. Finally, in Chapter 9 a general theory (once more the
result of quite recent investigations) for the motion of ships, regarded
as floating rigid bodies, is presented. In this theory no restrictive
assumptions — regarding, for example, the coupling (or lack of
coupling, as in an old theory due to Krylov [K.20] between the
motion of the sea and the motion of the ship, or between the various
degrees of freedom of the ship — are made other than those needed to
linearize the problem. This means essentially that the ship must be
regarded as a thin disk so that it can slice its way through the water
(or glide over the surface, perhaps) with a finite velocity and still
create waves which do not have large amplitudes; in addition, it
is necessary to suppose that the motion of the ship is a small oscil-
lation relative to a motion of translation with uniform velocity. The
theory is obtained by making a formal development of all conditions
of the complete nonlinear boundary problem with respect to a para-
meter which is a thickness-length ratio of the ship. The resulting
theory contains the classical Michell-IIavelock theory for the wave
resistance of a ship in terms of the shape of its hull as the simplest
special case.
We turn next to Part III, which deals with applications of the
approximate theory which results from the assumption that it is the
depth of the liquid which is small, rather than the amplitude of the
surface waves as in Part II. The theory, called here the shallow
water theory, leads to a system of nonlinear partial differential
equations which are analogous to the differential equations for the
motion of compressible gases in certain cases. We proceed to outline
the contents of Part III, which is composed of two long chapters.
In Chapter 10 the mathematical methods based on the theory of
characteristics are developed in detail since they furnish the basis
for the discussion of practically all problems in Part III; it is hoped
that this preparatory discussion of the mathematical tools will make
Part III of the book accessible to engineers and others who have not
had advanced training in mathematical analysis and in the methods
of mathematical physics. In preparing this part of the book the
author's task was made relatively easy because of the existence of the
INTRODUCTION XIX
book by Courant and Friedrichs [C.9], which deals with gas dynamics;
the presentation of the basic theory given here is largely modeled
on the presentation given in that book. The concrete problems dealt
with in Chapter 10 are quite varied in character, including the
propagation of disturbances into still water, conditions for the
occurrence of a bore and a hydraulic jump (phenomena analogous to
the occurrence of shock waves in gas dynamics), the motion resulting
from the breaking of a dam, steady two dimensional motions at
supercritical velocity, and the breaking of waves in shallow water.
The famous problem of the solitary wave is discussed along the lines
used recently by Friedrichs and Hyers [F.13] to prove rigorously
the existence of the solitary wave from the mathematical point of
view; this problem requires carrying the perturbation series which
formulate the shallow water theory to terms of higher order. The
problem of the motion of frontal discontinuities in the atmosphere,
which lead to the development of cyclonic disturbances in middle
latitudes, is given a formulation — on the basis of hypotheses which
simplify the physical situation — which brings it within the scope
of a more general "shallow water theory". Admittedly (as has already
been noted earlier) this theory is somewhat speculative, but it is
nevertheless believed to have potentialities for clarifying some of
the mysteries concerning the dynamical causes for the development
and deepening of frontal disturbances in the atmosphere, especially
if modern high speed digital computing machines are used as an aid
in solving concrete problems numerically.
Chapter 10 concludes with the discussion of a few applications of
the linearized version of the shallow water theory. Such a linearization
results from assuming that the amplitude of the waves is small. The
most famous application of this theory is to the tides in the oceans
(and also in the atmosphere, for that matter); strange though it
seems at first sight, the oceans can be treated as shallow for this
phenomenon since the wave lengths of the motions are very long
because of the large periods of the disturbances caused by the moon
and the sun. This theory, as applied to the tides, is dealt with only
very summarily, since an extended treatment is given by Lamb
[L.3]. Instead, some problems connected with the design of floating
breakwaters in shallow water are discussed, together with brief
treatments of the oscillations in certain lakes (the lake at Geneva
in Switzerland, for example) called seiches, and oscillations in harbors.
Finally, Part III concludes with Chapter 11 on the subject of
XX INTRODUCTION
mathematical hydraulics, which is to be understood here as referring
to flows and wave motions in rivers and other open channels with
rough sides. The problems of this chapter are not essentially different,
as far as mathematical formulations go, from the problems treated
in the preceding Chapter 10. They differ, however, on the physical
side because of the inclusion of a force which is just as important as
gravity, namely a force of resistance caused by the rough sides and
bottom of the channels. This force is dealt with empirically by
adding a term to the equation expressing the law of conservation of
momentum that is proportional to the square of the velocity and
with a coefficient depending on the roughness and the so-called
hydraulic radius of the channel. The differential equations remain of
the same type as those dealt with in Chapter 10, and the same under-
lying theory based on the notion of the characteristics applies.
Steady motions in inclined channels are first dealt with. In par-
ticular, a method of solving the problem of the occurrence of roll
waves in steep channels is given; this is done by constructing a
progressing wave by piecing together continuous solutions through
bores spaced at periodic intervals, This is followed by the solution
of a problem of steady motion which is typical for the propagation
of a flood down a long river; in fact, data were chosen in such a way
as to approximate the case of a flood in the Ohio River. A treatment
is next given for a flood problem so formulated as to correspond
approximately to the case of a flood wave moving down the Ohio
to its junction with the Mississippi, and w'th the result that distur-
bances are propagated both upstream and downstream in the Missis-
sippi and a backwater effect is noticeable up the Ohio. In these
problems it is necessary to solve the differential equations numerically
(in contrast with most of the problems treated in Chapter 10, in
which interesting explicit solutions could be given), and methods of
doing so are explained in detail. In fact, a part of the elements of
numerical analysis as applied to solving hyperbolic partial differential
equations by the method of finite differences is developed. The results
of a numerical prediction of a flood over a stretch of 400 miles in
the Ohio River as it actually exists are given. The flood in question
was the 1945 flood — one of the largest on record — and the predic-
tions made (starting with the initial state of the river and using the
known flows into it from tributaries and local drainage) by numerical
integration on a high speed digital computer (the Univac) check
quite closely with the actually observed flood. Numerical predictions
INTRODUCTION XXI
were also made for the case of a flood (the 1947 flood in this case)
coming down the Ohio and passing through its junction with the
Mississippi; the accuracy of the prediction was good. This is a case
in which the simplified methods of the civil engineers do not work
well. These results, of course, have important implications for the
practical applications.
Finally Part IV, made up of Chapter 12, closes the book- with a
few solutions based on the exact nonlinear theory. One class of problems
is solved by assuming a solution in the form of power series in the
time, which implies that initial motions and motions for a short time
only can be determined in general. Nevertheless, some interesting
cases can be dealt with, even rather easily, by using the so-called
Lagrange representation, rather than the Euler representation which
is used otherwise throughout the book. The problem of the breaking
of a dam, and, more generally, problems of the collapse of columns
of a liquid resting on a rigid horizontal plane can be treated in this
way. The book ends with an exposition of the theory due to Levi-
Civita concerning the problem of the existence of progressing waves of
finite amplitude in water of infinite depth which satisfy exactly the
nonlinear free surface conditions.
Acknowledgments
Without the support of the Mathematics Branch and the Mechanics
Branch of the Office of Naval Research this book would not have been
written. The author takes pleasure in acknowledging the help and
encouragement given to him by the ONR in general, and by Dr. Joa-
chim Wcyl, Dr. Arthur Grad, and Dr. Philip Eiscnberg in particular.
Although she is no longer working in the ONR, it is neverthe-
less appropriate at this place to express special thanks to Dean
Mina llecs, who was head of the Mathematics Branch when this
book was begun.
Among those who collaborated with the author in the preparation
of the manuscript, Dr. Andreas Troesch should be singled out for
special thanks. His careful and critical reading of the manuscript re-
sulted in many improvements and the uncovering and correction of
errors and obscurities of all kinds. Another colleague, Professor E.
Isaacson, gave almost as freely of his time and attention, and also
aided materially in revising some of the more intricate portions of the
book. To these fellow workers the author feels deeply indebted.
Miss Helen Samoraj typed the entire manuscript in a most efficient
(and also good-humored) way, and uncovered many slips and in-
consistencies in the process.
The drawings for the book were made by Mrs. Beulah Marx and
Miss Lark in Joyner. The index was prepared by Dr. George Booth and
Dr. Walter Littman with the assistance of Mrs. Halina Montvila.
A considerable part of the material in the present book is the result
of researches carried out at the Institute of Mathematical Sciences of
New York University as part of its work under contracts with the
Office of Naval Research of the U.S. Department of Defense, and to a
lesser extent under a contract with the Ohio River Division of the
Corps of Engineers of the U.S. Army. The author wishes to express his
thanks generally to the Institute; the cooperative and friendly spirit
of its members, and the stimulating atmosphere it has provided have
resulted in the carrying out of quite a large number of researches in
the field of water waves. A good deal of these researches and new
results have come about through the efforts of Professors K. O. Fried-
xxiii
XXIV ACKNOWLEDGMENTS
richs, Fritz John, J. B. Keller, H. Lewy (of the University of Cali-
fornia), and A. S. Peters, together with their students or with visitors
at the Institute.
.1. J. STOKER
New York, N.Y.
January, 1957.
Contents
PART I
CHAPTKR PACK
Introduction ix
Acknowledgments xxiii
1. Basic Hydrodynamics 3
1.1 The laws of conservation of momentum and mass 3
1.2 Helmholty/s theorem 7
1.3 Potential flow and Bernoulli's law 9
1.4 Boundary conditions 10
1.5 Singularities of the velocity potential 12
1.0 Notions concerning energy and energy flux 13
1.7 Formulation of a surface wave problem 15
2. The Two Basic Approximate Theories 19
2.1 Theory of waves of small amplitude 19
2.2 Shallow water theory to lowest order. Tidal theory .... 22
2.3 Gas dynamics analogy 25
2.4 Systematic derivation of the shallow water theory 27
PART II
Subdivision A
Waves Simple Harmonic in the Time
3. Simple Harmonic Oscillations in Water of Constant Depth . . 37
3.1 Standing waves 37
3.2 Simple harmonic progressing waves 45
3.3 Energy transmission for simple harmonic waves of small ampli-
tude 47
3.4 Group velocity. Dispersion 51
4. Waves Maintained by Simple Harmonic Surface Pressure in
Water of Uniform Depth. Forced Oscillations 55
4.1 Introduction 55
4.2 The surface pressure is periodic for all values of a? 57
XXVI CONTENTS
CHAPTER PAGE
4.8 The variable surface pressure is confined to a segment of the
surface 58
4.4 Periodic progressing waves against a vertical cliff 07
5. Waves on Sloping Beaches and Past Obstacles 69
5.1 Introduction and summary 69
5.2 Two-dimensional waves over beaches sloping at angles co = n/2n 77
5.3 Three-dimensional waves against a vertical cliff 84
5.4 Waves on sloping beaches. General case 95
5.5 Diffraction of waves around a vertical wedge. Sommerfeld's
diffraction problem 109
5.6 Brief discussions of additional applications and of other methods
of solution 133
Subdivision B
Motions Starting from Rest. Transients
6. Unsteady Motions 149
6.1 General formulation of the problem of unsteady motions . . 149
6.2 Uniqueness of the unsteady motions in bounded domains . 150
6.3 Outline of the Fourier transform technique 153
6.4 Motions due to disturbances originating at the surface ... 156
6.5 Application of Kelvin's method of stationary phase .... 163
6.6 Discussion of the motion of the free surface due to disturbances
initiated when the water is at rest 167
6.7 Waves due to a periodic impulse applied to the water when
initially at rest. Derivation of the radiation condition for purely
periodic waves 174
6.8 Justification of the method of stationary phase 181
6.9 A time-dependent Green's function. Uniqueness of unsteady
motions in unbounded domains when obstacles are present . 187
Subdivision C
Waves on a Running Stream. Ship Waves
7. Two-dimensional Waves on a Running Stream in Water of
Uniform Depth 198
7.1 Steady motions in water of infinite depth with p = 0 on the
free surface 199
CONTENTS XXVII
CHAPTER PAGE
7.2 Steady motions in water of infinite depth with a disturbing pres-
sure on the free surface 201
7.3 Steady waves in water of constant finite depth 207
7.4 Unsteady waves created by a disturbance on the surface of a
running stream 210
8. Waves Caused by a Moving Pressure Point. Kelvin's Theory of
the Wave Pattern created by a Moving Ship 219
8.1 An idealized version of the ship wave problem. Treatment by
the method of stationary phase 219
8.2 The classical ship wave problem. Details of the solution . . 224
9. The Motion of a Ship, as a Floating Rigid Body, in a Seaway 245
9.1 Introduction and summary 245
9.2 General formulation of the problem 264
9.3 Linearization by a formal perturbation procedure 269
9.4 Method of solution of the problem of pitching and heaving of a
ship in a seaway having normal incidence 278
PART III
10. Long Waves in Shallow Water 291
10.1 Introductory remarks and recapitulation of the basic equations 291
10.2 Integration of the differential equations by the method of char-
acteristics 293
10.3 The notion of a simple wave 300
10.4 Propagation of disturbances into still water of constant depth 305
10.5 Propagation of depression waves into still water of constant
depth 308
10.6 Discontinuity, or shock, conditions 314
10.7 Constant shocks: bore, hydraulic jump, reflection from a rigid
wall 326
10.8 The breaking of a dam 333
10.9 The solitary wave 342
10.10 The breaking of waves in shallow water. Development of bores 351
10.11 Gravity waves in the atmosphere. Simplified version of the
problem of the motion of cold and warm fronts 374
10.12 Supercritical steady flows in two dimensions. Flow around
bends. Aerodynamic applications 405
10.13 Linear shallow water theory. Tides. Seiches. Oscillations in
harbors. Floating breakwaters 414
XXVIII CONTENTS
CHAPTER PAGE
11. Mathematical Hydraulics 451
11.1 Differential equations of flow in open channels 452
11.2 Steady flows. A junction problem 456
11.3 Progressing waves of fixed shape. Roll waves 461
11.4 Unsteady flows in open channels. The method of characteristics 409
11.5 Numerical methods for calculating solutions of the differential
equations for flow in open channels 474
11.6 Flood prediction in rivers. Floods in models of the Ohio River
and its junction with the Mississippi River 482
1 1 .7 Numerical prediction of an actual flood in the Ohio, and at its
junction with the Mississippi. Comparison of the predicted with
the observed floods 408
Appendix to Chapter 11. Expansion in the neighborhood of the first
characteristic 505
PART IV
12. Problems in which Free Surface Conditions are Satisfied Exactly.
The Breaking of a Dam. Levi-Civita's Theory 513
12.1 Motion of water due to breaking of a dam, and related problems 513
12.2 The existence of periodic waves of finite amplitude .... 522
12.2a Formulation of the problem 522
12.2b Outline of the procedure to be followed in proving the existence
of the function a)(%) 526
12.2c The solution of a class of linear problems 529
12.2d The solution of the nonlinear boundary value problem . . . 537
Bibliography 545
Author Index 561
Subject Index 563
PART I
CHAPTER 1
Basic Hydrodynamics
1.1. The laws of conservation of momentum and mass
As has been stated in the introduction, we deal exclusively in this
book with flows in water (and air) which are of such a nature as
to make it unnecessary to take into account the effects of viscosity
and compressibility. As a consequence of the neglect of internal
friction, or in other words of neglect of shear stresses, it is well
known that the stress system* in the liquid is a state of uniform
compression at each point. The intensity of the compressive stress
is called the pressure p.
The equation of motion of a fluid particle can then be obtained on
the basis of Newton's law of conservation of momentum, as follows.
A small rectangular element of the fluid is shown in Figure 1.1.1
Fig. 1.1.1. Pressure on a fluid element
with the pressure acting on the faces normal to the o?-axis. Newton's
law for the ^-direction is then
[ — (P
— Qa(*)
* We assume that the usual concepts of the general mechanics of continuous
media are known.
WATER WAVES
in which X is the external or body force component per unit mass
and a(x) is the acceleration component, both in the ^-direction, and
Q is the density. The quantities p9 X, and a(x) are in general functions
of x9 y, z, and t. Here, as always, we shall use letter subscripts to
denote differentiation, and this accounts for the symbol a(x} to denote
the component of a vector in the ^-direction. Upon passing to the
limit in allowing dx, dy, dz to approach zero we obtain the equation
of motion for the ^-direction in the form — px + qX = Qa(x}9 and
analogous expressions for the two other directions. Thus we have the
equations of motion
(1.1.1)
or, in vector form:
(1.1.2)
Px + % =
-Pv +Y =
— -P*
— grad p + F = a,
Q
with an obvious notation. The body force F plays a very important
role in our particular branch of hydrodynamics — in fact the main
results of the theory are entirely conditioned by the presence of the
gravitational force F = (0, — g, 0), in which g represents the acceler-
ation of gravity. It should be observed that we consider the positive
y-axis to be vertically upward, and the x, z-plane therefore to be horizontal
(usually it will be taken as the undisturbed water surface). This con-
vention regarding the disposition of the coordinate axes will be main-
tained, for the most part, throughout the book.
The differential equations (1.1.1) are in what is called the Lagrang-
ian form, in which one has in mind a direct description of the motion
of each individual fluid particle as a function of the time. It is more
useful for most purposes to work with the equations of motion in the
so-called Eulerian form. In this form of the equations one concen-
trates attention on the determination of the velocity distribution in
the region occupied by the fluid without trying to follow the motion of
the individual fluid particles, but rather observing the velocity
distribution at fixed points in space as a function of the time. In
BASIC HYDRODYNAMICS 5
other words, the velocity field, with components u9 v9 w9 is to be
determined as a function of the space variables and the time. After-
wards, if that is desired, the motion of the individual particles can
be obtained by integrating the system of ordinary differential equa-
tions x = u, y — v, z — w9 in which the dot over the quantities x9 y,
z means differentiation with respect to the time in following the
motion of an individual particle.-
In order to restate the equations of motion (1.1.1) in terms of the
Euler variables u, v, w9 and in order to carry out other important
operations as well, it is necessary to calculate time derivatives of
various functions associated with a given fluid particle in following
the motion of the particle. For example, we need to calculate the
time derivative of the velocity of a particle in order to obtain the
acceleration components occurring in (1.1.1), and quite a few other
quantities will occur later on for which such particle derivatives will
be needed. Suppose, then, that F(x9 y, z; t) is a function associated
with a particle which follows the path given by the vector
x = (x(t),y(t)9z(t));
it follows that
x = (x(t)9 y(t)9 z(t)) = (u, v9 w)
is the velocity vector associated with the particle. For this particle
the arguments x9 y9 z of the function F are of course the functions
of t which characterize the motion of the particle; as a consequence
we have
^ = Fxx + Fyy + Fzz + Ft
and hence the operation of taking the particle derivative d/dt is
defined as follows:
(1.1.3) 1( ) = u( )m+v( )y+w( ). + ( ),.
at
The distinction between dF/dt and dF/dt = Ft should be carefully
noted.
Since the acceleration a of a particle is given by a = (du/dt9 dv/dt9
dw/dt)9 in which (u9 v9 w) are the components of the velocity v of
6 WATER WAVES
the particle, it follows from (1.1.3) that the component a(x) — du/dt
is given by
du
— = uux + vuy + wuz + ut,
at
with similar expressions for the other components. The equations of
motion (1.1.1) are therefore given as follows in terms of the Euler
variables:
(1.1.4)
1
Ut + UUX + VUy + WUZ = — -- px ,
6
1
Vt + ^^ + TOV + WO* = — - Py — g,
e
i
I0f + U^ + U^y + WWZ = — - pz
Q
when we specify the external or body force to consist only of the
force of gravity.
Equations (1.1.4) form a set of three nonlinear partial differential
equations for the five quantities u9 v, w9 g, and p. Since the fluid is
assumed to be incompressible, the density Q can be taken as a known
constant. At the same time, the assumption of incompressibility leads
to a relatively simple differential equation expressing the law of
conservation of mass, and this equation constitutes the needed fourth
equation for the determination of the velocity components and the
pressure. Perhaps the simplest way to derive the mass conservation
law is to start from the relation
which states that the mass flux outward through any fixed closed
surface enclosing a region in which no liquid is created or destroyed
is zero. (By vn we mean the velocity component taken positive in the
direction of the outward normal to the surface.) An application of
Gauss's divergence theorem:
(1.1.5) JJei>n dS = JJJdiv (ev) dr
S R
to the above integral leads to the relation
fdiv (QV) dr = 0
R
BASIC HYDRODYNAMICS 7
for any arbitrary region R. It follows therefore that div (gv) = 0
everywhere, and since Q = constant, we have finally
(1.1.6) divv = ux +vv + wz = 0
as the expression of the law of conservation of mass. The equation
(1.1.6) is also frequently called the equation of continuity.
Equations (1.1.4) and (1.1.6) are sufficient, once appropriate
initial and boundary conditions (to be discussed shortly) are imposed,
to determine the velocity components u, v, w, and the pressure p
uniquely.
1.2. Helmholtz's theorem
Before discussing boundary conditions it is preferable to for-
mulate a few additional conservation laws which are consequences
of the assumptions made so far— in particular of the assumption that
internal fluid friction can be neglected.
The first of these laws to be discussed is the law of conservation
of circulation. The notion of circulation is defined as follows. Consider
a closed curve C which moves with the fluid (that is, C consists
always of the same particles of the fluid). The circulation F == F(t)
around C is defined by the line integral
(1.2.1) r(t) = j>udx+vdy+wdz
= <p vs ds
c
in which vs is the velocity component of the fluid tangent to C,
and ds is the element of arc length of C. The curve C is considered
as given by the vector x(a, t) with a a parameter on C such that
0 ^ a ^ 1 and x(0, t) = x(l, /). We are thus operating in terms of
the Lagrange system of variables rather than in terms of the Euler
system, and fixing a value of a has the effect of picking out a specific
particle on C.
i
We may write F(t) = \ v • xada in which v • xa is a scalar product
o
and xa, as usual, refers to differentiation with respect to a. For the
time derivative F we have therefore
8 WATER WAVES
1
f(t) = J(v • xa + v -
0
From the equation of motion (1.1.2) in the Lagrangian form with
a == v, F = (0, -g, 0) = — grad (gy), and from xa = va, the last
equation yields
i
(1.2.2) f(t) = n Xa ' grad p - gxa • grad y + v • v J da
o
i
= J - ~ Po - &Va + - (v • v)a Ida
0
= 0,
since the values of p, j/, and v coincide at a = 0 and or = 1, and p
and g are constants. The last equation evidently states that in a
nonviscous fluid the circulation around any closed curve consisting of
the same fluid particles is constant in time. This is the theorem of Helm-
holtz. The assumption of zero viscosity entered into our derivation
through the use of (1.1.2) as equation of motion.*
In this book we are interested in the special case in which the
circulation for all closed curves is zero. This case is very important
in the applications because it occurs whenever the fluid is assumed
to have been at rest or to have been moving with a constant velocity
at some particular time, so that v = const, holds at that time, and
hence F vanishes for all time. The cases in which the fluid motion
begins from such states are obviously very important.
The assumption that F vanishes for all closed curves has a number
of consequences which are basic for all that follows in this book.
The first conclusion from JT = 0 follows almost immediately from
Stokes's theorem:
(1.2.3) r=j>vsds = JJ(curl v)n dA,
c s
in which the surface integral is taken over any surface S spanning the
curve C. If 71 = 0 for all curves C, as we assume, it follows easily by a
well-known argument that the vector curl v vanishes everywhere:
* It should be added that the law of conservation of circulation holds under
much wider conditions than were assumed here (cf. [C.9], p. 19).
BASIC HYDRODYNAMICS 9
(1.2.4) curl v = (wy—vz, uz—wx, vx—uy) = 0,
and the flow is then said to be irrotational. In other words, a motion
in a nonviscous fluid which is irrotational at one instant always
remains irrotational. Throughout the rest of this book we shall assume
all flows to be irrotational.
1.3. Potential flow and Bernoulli's law
The assumption of irrotational flow results in a number of sim-
plifications in our theory which are of the greatest utility. In the
first place, the fact that curl v = 0 (cf. (1.2.4)) ensures, as is well
known, the existence of a single-valued velocity potential 0(x, y, 2; t)
in any simply connected region, from which the velocity field can be
derived by taking the gradient:
(1.3.1) v = grad 0 = (0X, 0y, 0Z),
or, in terms of the components of v:
(1.3.2) u = 0X9 v = 0y9 w = 0Z.
The velocity potential is, indeed, given by the line integral
<e,v,z
0(x9 y, 2; t) = I u dx + v dy + w dz.
The vanishing of curl v ensures that the expression to be integrated
is an exact differential. Once it is known that the velocity com-
ponents are determined by (1.3.2), it follows from the continuity
equation (1.1.6), i.e. div v = 0, that the velocity potential 0 is a
solution of the Laplace equation
(1.3.3) V2<Z> = 0XX + 0VV + 0ZZ = 0,
as one readily sees, and 0 is thus a harmonic function. This fact
represents a great simplification, since the velocity field is derivable
from a single function satisfying a linear differential equation which
has been very much studied and about which a great deal is known.
Still another important consequence of the irrotational character
of a flow can be obtained from the equations of motion (1.1.4). By
making use of (1.2.4), it is readily verified that the equations of
motion (1.1.4) can be written in the following vector form:
1 p
grad 0t + ~ grad (u2 + v2 + w2) = — grad - — grad (gy),
2 Q
use having been made of the fact that Q = constant. Integration
10 WATER WAVES
of this relation leads to the important equation expressing what is
called Bernoulli9 s law:
(1.3.4) 0t + I (u* + v* + te;2) + ? + gy = C(t),
2 Q
in which C(t) may depend on t, but not on the space variables. There
are two other forms of Bernoulli's law for the case of steady flows,
one of which applies along stream lines even though the flow is
not irrotational, but since we make no use of these laws in this book
we refrain from formulating them.
The potential equation (1.3.3) together with Bernoulli's law
(1.8.4) can be used to take the place of the equations of motion
(1.1.4) and the continuity equation (1.1.6) as a means of determining
the velocity components u, u, 10, and the pressure p: in effect, u, v9
and w are determined from the solution 0 of (1.3.3), after which the
pressure p can be obtained from (1.3.4). It is true that the pressure
appears to be determined only within a function which is the same
at each instant throughout the fluid. On physical grounds it is,
however, clear that a function of t alone added to the pressure p
has no effect on the motion of the fluid since no pressure gradients
result from such an addition to the pressure. In fact, if we set
= 0* -f fC(f ) df , then 0* is a harmonic function
with
grad 0 = grad 0* and the Bernoulli law with reference to it has a
vanishing right hand side. Thus we may take C(t) = 0 in (1.3.4)
without any essential loss of generality.
While it is true that the Laplace equation is a linear differential
equation, it does not follow that we shall be able to escape all of the
difficulties arising from the nonlinear character of the basic differen-
tial equations of motion (1.1.4). As we shall sec, the problems of
interest here remain essentially nonlinear because the Bernoulli law
(1.3.4), and another condition to be derived in the next section, give
rise to nonlinear boundary conditions at free surfaces. In the next
section we take up the important question of the boundary con-
ditions appropriate to various physical situations.
1.4. Boundary conditions
• We assume the fluid under consideration to have a boundary
surface S, fixed or moving, which separates it from some other
medium, and which has the property that any particle which is once
BASIC HYDRODYNAMICS 11
on the surface remains on it.* Examples of such boundary surfaces
of importance for us are those in which S is the surface of a fixed
rigid body in contact with the fluid— the bottom of the sea, for
example— or the free surface of the water in contact with the air.
If such a surface S were given, for example, by an equation
£(#, j/, z; t) — 0, it follows from (1.1.3) that the condition
(1.4.1) ^ = u£x + <, + «£. + C« = 0
dt
would hold on S. From (1.3.2) arid the fact that the vector (£X9 £y, £J
is a normal vector to S it follows that the condition (1.4.1) can be
written in the form
(,.4.2), <»<•=.. =„.,
a» v# + ti+ c;
in which d/dn denotes differentiation in the direction of the normal
to S and vn means the common velocity of fluid and boundary
surface in the direction normal to the surface.
In the important special case in which the boundary surface S
is fixed, i.e. it is independent of the time t, we have the condition
d0
(1.4.3) — = 0 on S.
on
This is the appropriate boundary condition at the bottom of the sea,
or at the walls of a tank containing water.
Another extremely important special case is that in which S is a
free surface of the liquid, i.e. a surface on which the pressure p is
prescribed but the form of the surface is not prescribed a priori.
We shall in general assume that such a free surface is given by the
equation
(1.4.4) y = ri(x,z\t).
On such a surface f = y — r)(a\ 3; t) — 0 for any particle, and hence
(1.4.1) yields the condition
(1.4.5) 0xr,x -&y + 0Z*)Z + 17, = 0 on S.
In addition, as remarked above, we assume that the pressure p is
given on S; as a consequence the Bernoulli law (1.3.4) yields the
condition:
* Actually, this property is a consequence of the basic assumption in con-
tinuum mechanics that the motion of the fluid can be described mathematically
as a topological deformation which depends continuously on the time t.
12 WATER WAVES
(1.4.6) ff, + 0t + I (01 + 02y + 01) + P- = 0 OR S.
2 Q
(As remarked earlier, we may take the quantity C(t) = 0 in (1.3.4).)
Thus the potential function 0 must satisfy the two nonlinear boundary
conditions (1.4.5) and (1.4.6) on a free surface. This is in sharp con-
trast to the single linear boundary condition (1.4.3) for a fixed
boundary surface, but it is not strange that two conditions should
be prescribed in the case of the free surface since an additional
unknown function rj(x, z; t)9 the vertical displacement of the free
surface, is involved in the latter case.
Later on we shall also be concerned with problems involving rigid
bodies floating in the water and S will be the portion of the rigid
body in contact with the water. In such cases the function r)(x, z\ t)
will be determined by the motion of the rigid body, which in turn
will be fixed (through the dynamical laws of rigid body mechanics)
by the pressure p between the body and the water in accordance
with (1.4.6). The detailed conditions for such cases will be worked
out later on at an appropriate place.
1.5. Singularities of the velocity potential
In our discussion up to now it has been tacitly assumed that all
quantities such as the pressure, velocity potential, velocity com-
ponents, etc. are regular functions of their arguments. It is, however,
often useful to permit singularities of one kind or another to occur
as an idealization of, or an approximation to, certain physical situations.
Perhaps the most useful such singularity is the point source or sink
which is given by the harmonic function
(1.5.1) 0
in three dimensions, and by
(1.5.2) 0 = — log r, r2 = x* + y*
2n
in two dimensions. Both of these functions yield flows which are
radially outward from the origin, and for which the flux per unit
time across a closed surface (for (1.5.1)) or a closed curve (for (1.5.2))
surrounding the origin has the value c, as one readily verifies since
d0/dn = d0fdr for r = constant. That these functions represent at
BASIC HYDRODYNAMICS 13
best idealizations of the physical situations implied in the words
source and sink is clear from the fact that they yield infinite velocities
at r = 0. Nevertheless, it is very useful here— as in other branches
of applied mathematics —to accept such infinities with the reservation
that the results obtained are not to be taken too literally in the
immediate vicinity of the singular point.
We shall have occasion to deal with other singularities than sources
or sinks, such as dipoles and multipoles, but these will be introduced
when needed.
1.6. Notions concerning energy and energy flux
In dealing with surface gravity waves in water it is important
and useful to analyze in some detail the flow of energy in the fluid
past a given surface S. Let R be the region occupied by water and
bounded by a "geometric" surface S which may, or may not, move
independently of the liquid. The energy E contained in R consists
of the kinetic energy of the water particles in R and their potential
energy due to gravity; hence E is given by
(1.6.1) E - Q R (0| + 01 + 01) + gy\ dx dy dz,
or, alternatively, by
(1.6.2) E = - JJJ(P + e^t)dx dy dz
R
upon applying Bernoulli's law (1.3.4) with C(t) = 0.
We wish to calculate dE/dt, having in mind that the region R
is not necessarily fixed, but may depend on the time t. Quite generally,
if E = f f f f(x, y, z; l)dx dy dz, it is well known that
R S
in which vn denotes the normal velocity of the boundary S of R
taken positive in the direction outward from R. In applying the
formula for dE/dt we make use of the definition of the function /
implied in (1.6.1) in the first term, but take / from (1.6.2) for the
second term. The result is
14 WATER WAVES
/ITT P P P
~dt = Q J J J (
s
The integrand in the first integral can be expressed in the form
&x(®t)x + ^y(^t)y + &z(®t)z = grad & • grad 0t
and hence the integral can be written as the following surface integral:
\ ^ dS,
s
in view of Green's formula and the fact that V20 = 0. Thus the
expression for dE/dt, the rate of change of the energy in R9 can be
put into the following form:
J Tjl /* /»
-f-JJ
(1.6.3) . = [Q0t(0n - vn) - pvn]dS.
s
We recall that vn means the normal velocity component of S, and
0n refers to the velocity component of the fluid taken in the direction
of the normal to S which points outward from R.
It happens frequently that the boundary surface S of R is made
up of a number of different pieces which have different properties
or for which various different conditions are prescribed. Suppose
first that a portion SP of S is a "physical" boundary containing
always the same fluid particles. Then 0n and vn are identical (cf.
(1.4.2)) and
(1.6.4.)
dE
dS.
dt
SP
If, in addition, the surface SP is fixed in space, i.e. vn = 0, the
contribution of SP to dE/dt evidently vanishes, as it should, since no
energy flows through a fixed boundary containing always the same
fluid particles. Similarly, the contribution to the energy flux also
vanishes in the important special case in which SF is a free surface
on which the pressure p vanishes; this result also accords with what
one expects on physical grounds.
Suppose now that SG is a "geometric" surface fixed in space, but
not necessarily consisting of the same particles of water. In this
case we have vn = 0 and the flow of energy through SG is given by
,,.6.5)
BASIC HYDRODYNAMICS 15
dE
-//<
An important special case for us is that in which 0 is the velocity
potential for a plane progressing wave given, for example, by
(1.6.6) 0(x, y, z; t) = <p(x-ct, y, z),
which represents a wave moving with constant velocity c in the
direction of the #-axis. The flux through a fixed plane surface S
orthogonal to the #-axis is easily seen from (1.6.5) to be given by
(iv
(1.6.7) -^ =
S
The negative sign results since our stipulations amount to saying
that the region R occupied by the fluid lies on the negative side
of S (i.e. on the side away from the positive normal, the #-axis);
and consequently the energy flux through S due to a progressing
wave moving in the positive direction of the normal (so that c is
positive) is such as to decrease the energy in jR, as one would expect.
It is to be noted that there is always a flow of energy through a
surface S orthogonal to the direction of a progressing wave if
0n E^= 0 — even though the motion of the individual particles of the
fluid should happen, for example, to be such that the particles move
in a direction opposite to that of the progressing wave.
1.7. Formulation of a surface wave problem
It is perhaps useful— although somewhat discouraging, it must be
admitted— to sum up the above discussion concerning the fun-
damental mathematical basis for our later developments by formula-
ting a rather general, but typical, problem in the hydrodynamics of
surface waves. The physical situation is indicated in Figure 1.7.1;
what is intended is a situation like that on any ocean beach. The
water is assumed to be initially at rest and to fill the space R defined by
— h((r, 2)^2/^0, — oo < z < oo,
and extending to + oo in the ^-direction. At the time t = 0, a given
disturbance is created on the surface of the water over a region D
(by the wind, perhaps), and one wishes to determine mathematically
the subsequent motion of the water; in particular, the form of the
16
WATER WAVES
Fig. 1.7.1. A very general surface wave problem
free surface y = r\(x, z; t) is to be determined. On the basis of these
assumptions the following conditions should be satisfied: First of
all, the differential equation to be satisfied by 0 is, of course, the
Laplace equation
(I.T.I) V*0 = 0XX + 0y
(xs(z9 t) ^ x < oo
^Q for - h(x9 z)^y^ rj(^ z; t)
[
00 < Z < 00
It is to be noted that xs(z; t)9 the abscissae of the water line on shore,
and rj(x9 z; t), the free surface elevation, are not known in advance
but are rather to be determined as an integral part of the solution.
As boundary condition to be satisfied at the bottom of the sea we
have
(1-7.2)
— - = 0 for y = — h(x9 z)9
on
while the free surface conditions are the kinematic condition (cf.
(1.4.5))
(1.T.3) 0xrjx -0y + &zr]z + rjt = 0 for y = rj(x9 z; t)9
and the dynamic condition
(1.T.4) gq + 0t + \(0l +01+ 0%) = F(x9 z; t) on y - rj(x9 z; t),
with F(x9 z; t) = 0 everywhere except over the region D where the
disturbance is created. At oo, i.e. for x -> oo and | z | -> oo, we
might prescribe that 0 and y remain bounded, or perhaps even that
they and certain of their derivatives tend to zero. Next we have the
initial conditions
(1.T.5) qfa z; t) = 0 for t = 0,
(1.T.6) 0X = 0y = 0Z s= 0 for t = 0,
BASIC HYDRODYNAMICS 17
appropriate to the condition of rest in an equilibrium position.
Finally we must prescribe conditions fixing the disturbance; this
could be done, for example, by giving the pressure p over the
disturbed region D of the surface, in other words by prescribing the
function F in (1.7.4) appropriately there.
One has only to write down the above formulation of our problem
to realize how difficult it is to solve it. In the first place the problem
is nonlinear, but what makes for perhaps even greater difficulties is
the fact that the free surface is not known a priori and hence the
domain in which the velocity potential is to be determined is not
known in advance— aside from the fact that its* boundary varies
with the time.
These are, however, not the only difficulties in the above problem.
If we assume that the function 0 is regular throughout the interior
of R and uniformly bounded (together with some of its derivatives,
perhaps) in R, the formulation of the problem given above would
seem to be reasonable from the point of view of mechanics. However,
the solution would probably not exist for all t > 0 for the following
reason: everyone who has visited an ocean beach is well aware that
the waves do not come in smoothly all the way to the shore (except
possibly in very calm weather), but, rather, they steepen in front,
curl over, and eventually break. In other words, any mathematical
formulation of the problem which would fit the commonly observed
facts even for a limited time would necessitate postulating the
existence of singularities of unknown location in both space and time.
Because of the difficulty of the general nonlinear theory very little
progress has been made in solving concrete problems which employ
it. An exception is the problem of proving the existence of two-
dimensional periodic progressing waves in water of uniform depth.
This was done first by Nckrassov [N.I], [N.I a] and by Levi-Civita
[L.7] for water of infinite depth, and later by Struik [S.29] for water of
constant finite depth. In Chapter 12 an account of Levi-Civita's theory
is given. In both cases the authors prove rigorously the existence of
waves having amplitudes near to zero by showing that perturbation
series in the amplitude converge. Another exception to the above
statement is the problem of the solitary wave, the existence of which,
from the mathematical point of view, has been proved recently by
Lavrentieff [L.4] and by Friedrichs and Hyers [F.13]; an account of
the work of the latter two authors is given in Chapter 10.9.
It seems likely that solutions of problems in the full nonlinear
18 WATER WAVES
version of the theory will, for a long time to come, continue to be
of the nature of existence theorems for motions of a rather special
nature.
In order to make progress with the theory of surface waves it is
in general necessary to simplify the theory by making special hypoth-
eses of one kind or another which suggest themselves on the basis
of the general physical circumstances contemplated in a given class
of problems. As we have already explained in the introduction, up
to now attention has been concentrated almost exclusively upon the
two approximate theories which result when either a) the amplitude
of the surface waves is considered small (with respect to wave length,
for example), or b) the depth of the water is considered small (again
with respect, say, to wave length). The first hypothesis leads to a
linear theory and to boundary value problems more or less of classical
type; while the second leads to a nonlinear theory for initial value
problems, which in lowest order is of the type employed in wave
propagation in compressible gases. If both hypotheses are made, the
result is a linear theory involving essentially the classical linear
wave equation; the present theory of the tides belongs in this class
of problems.
In the next chapter we derive the approximate theories arising
from the two hypotheses by starting from the general theory and
then developing formally with respect to an appropriate parameter—
essentially the surface wave amplitude in one case and the depth
of the water in the other— and in subsequent chapters we continue
by treating a variety of special problems in each of the two classes.
CHAPTER 2
The Two Basic Approximate Theories
2.1. Theory of waves of small amplitude
It has already been stated that the theory of waves of small
amplitude can be derived as an approximation to the general theory
presented in Chapter I on the basis of the assumption that the
velocity of the water particles, the free surface elevation y—r)(x, z; t),
and their derivatives, are all small quantities. We assume, in fact,
that the velocity potential 0 and the surface elevation rj possess the
following power series expansions with respect to a parameter e:
(2.1.1) 0 = e0(l) + e20(2) + e30(3) + . . ., and
It follows first of all that each of the functions 0<k)(x, y, z; t) is
a solution of the Laplace equation, i.e.
(2.1.3) V20><*> — 0.
We turn next to the discussion of the boundary conditions. At a
fixed physical boundary (cf. section 1.4) of the fluid we have clearly
the conditions
(2.1.4) -a-- = 0,
in which d/dn represents differentiation along the normal to the
boundary surface.
At a free surface S: y — r](x> z; t) on which the pressure is zero we
have two boundary conditions. One of them arises from the Bernoulli
law and has the form
grj + 0t + i(<Z>* + *I + 01) = 0 on S.
Upon insertion of (2.1.1) and (2.1.2) in this condition and developing
&t> &x> etc' systematically in powers of e (due regard being paid to
the fact that the functions 0^k)9 0(®, etc. are to be evaluated for
19
20 WATER WAVES
y = ri(x, z\ t) and that 77 in its turn is given in terms of e by
it) = 77<°> -)- erjM -\- . . . ) one finds readily the conditions
(2.1.5) ?7«» = 0,
(2.1.6) gqM + 0]11 = 0.
(2.1.7) W«> + <P« + i[(0»> )2 + (0<1} )2 + (#»> )•] + i7««*j« = 0
to be satisfied for y = T?(O), and since ?y(0) = 0 from (2.1.5) it follows
that the conditions (2.1.6), (2.1.7), etc. are all to be satisfied on the
originally undisturbed surface of the water y = 0. The other boundary
condition on S arises from the fact that the water particles stay
on S (cf. section 1.4); it is expressed in the form
#«ifc + #.% +*]t = ®y on S.
Insertion of the power series for 0 and 77 in this expression leads to
the conditions
(2.1.8) fjW - 0,
(2.1.9)
(2.1.10)
which are also to be satisfied for y = 0.
In view of the fact that iy(0) = 0, the free surface conditions can
be put in the form
(2.1.11) gqW+Qp^O,
(2.1.12) OTC» + 0W = _ J[(0W)« + (0^)2 + ((PW )•] « ,<i)0W,
(2.1.13) g^<w> + 0|w) = jP^-D,
in which the symbol pin-u refers to a certain combination of the
functions rj(k) and 0(k) with fc j£ n — 1, and all conditions are to be
satisfied for y = 0. Similarly, the other set of free surface conditions
becomes
TUP: TWO BASIC APPROXIMATE THEORIES 21
(2.1.14) fi
(2.1.15) ,» = *» - «>«,<« - *«,»> +
(2.1.16) T?<W)
in which G(n~l) depends only upon functions ?y(fc) and 0(k) with
k ^ n — I9 and once more all conditions are to be satisfied for
y _= o. This theory therefore is a development in the neighborhood
of the rest position of equilibrium of the water.
The relations (2.1.11) to (2.1.16) thus, in principle, furnish a means
of calculating successively the coefficients of the series (2.1.1) and
(2.1.2), assuming that such series exist: The conditions (2.1.11) and
(2.1.14) at the free surface together with appropriate conditions at
other boundaries, and initial conditions for / = 0, would in conjunc-
tion with V20(1) — 0 lead to unique solutions r/(l) and <2>(1). Once
rfl) and 0(l) are determined, they can be inserted in the conditions
(2.1.12) and (2.1.15) to yield two conditions for rj(2) and 0(2> which
with the subsidiary boundary and other conditions on 0(2) serve to
determine them, etc. One could interpret the work of Levi-Civita [L.7]
and Struik [S.29] referred to in section 1.7 as a method of proving
the existence of progressing waves which are periodic in x by showing
that the functions 0 and rj can indeed be represented as convergent
power series in e for e sufficiently small.
In what follows in Part II of this book we shall content ourselves
in the main with the degree of approximation implied in breaking
off the perturbation series after the terms e0(l} and erj(l) in the series
(2.1.1) and (2.1.2), i.e. we set 0 = e0(l) and rj = er](l). With this
stipulation the free surface conditions (2.1.11) and 2.1.14) yield
(2.1.17) gl?+0t = o)
I for y = 0.
(2.1.18) rit -0tf = oJ
By elimination of r\ between these two relations the single condition
on 0:
(2.1.19) 0tt + g0y = 0 for y = 0
is obtained; this condition is the one which will be used mainly in
Part II in order to determine 0 from V20 = 0, after which the free
surface elevation r\ can be determined from (2.1.17). The usual
method of obtaining the last three conditions is to reject all but
22 WATER WAVES
the linear terms in 77 and 0 and their derivatives in the kinematic
(cf. (1.4.5)) and dynamic (cf. (1.4.6)) free surface boundary con-
ditions. By proceeding in this way we can obtain a first approximation
to the pressure p (which was not considered in the above general
perturbation scheme) in the form:
(2.1.20) -= -gy-0t. *
Q
We can now see the great simplifications which result through
the linearization of the free surface conditions: not only does the
problem become linear, but also the domain in which its solution
is to be determined becomes fixed a priori and consequently the
surface wave problems in this formulation belong, from the mathe-
matical point of view, to the classical boundary problems of potential
theory.
2.2. Shallow water theory to lowest order. Tidal theory
A different kind of approximation from the foregoing linear theory
of waves of small amplitude results when it is assumed that the
depth of the water is sufficiently small compared with some other
significant length, such as, for example, the radius of curvature of
the water surface. In this theory it is not necessary to assume that
the displacement and slope of the water surface are small, and the
resulting theory is as a consequence not a linear theory. There are
many circumstances in nature under which such a theory leads to
a good approximation to the actual occurrences, as has already been
mentioned in the introduction. Among such occurrences are the tides
in the oceans, the "solitary wave" in sufficiently shallow water, and
the breaking of waves on shallow beaches. In addition, many pheno-
mena met with in hydraulics concerning flows in open channels such
as roll waves, flood waves in rivers, surges in channels due to sudden
influx of water, and other kindred phenomena, belong in the nonlinear
shallow water theory. Chapters 10 and 11 are devoted to the working
out of consequences of the shallow water theory.
The shallow water theory is, in its lowest approximation, the basic
theory used in hydraulics by engineers in dealing with flows in open
* In case the surface pressure pa(xt z; t) is not zero one finds readily that (2.1.17)
•is replaced by
(2.1.20)! gri + &t = - Po/Q,
while (2.1.18) remains unaltered.
THE TWO BASIC APPROXIMATE THEORIES 23
channels, and also the theory commonly referred to in the standard
treatises on hydrodynamics as the theory of long waves. We begin by
giving first a derivation of the theory for two-dimensional motion
along essentially the lines followed by Lamb [L.3], p. 254. As usual,
the undisturbed free surface of the water is taken as the o?-axis and
the t/-axis is taken vertically upwards. The bottom is given by
y = — h(x), so that h represents the variable depth of the undisturbed
water. The surface displacement is given by y = r](x, t). The velocity
components are denoted by u(x, y, t) and v(x, y, t).
The equation of continuity is
(2.2.1) Ux + Vy = 0.
The conditions to be satisfied at the free surface are the kinematical
condition:
(2.2.2) (r)t + ur,x - v) \y=ri = 0;
and the dynamical condition on the pressure:
(2.2.3) p !,_„ = 0.
At the bottom the condition is
(2.2.4) (uhx + v) |V=_A = 0.
Integration of (2.2.1) with respect to y yields
(2.2.5) f (Ux)dy +v\\ = Q.
J —h
Use of the condition (2.2.2) at y = 77 and (2.2.4) at y — — h yields
the relation
(2.2.6) r uxdy + r)t + u \ • ^ + u \_h • hx = 0.
J —h
We introduce the relation
g p(«) rn
(2.2.7) ^- u dy = u \ • rjx + u \y=_h • hx + \ ux dy.
v® J _fc(jB) J -h
and combine it with (2.2.6) to obtain
(2.2.8)
Up to this point no approximations have been introduced.
The shallow water theory is an approximate theory which results
from the assumption that the y- component of the acceleration of
24 WATER WAVES
the water particles has a negligible effect on the pressure p, or,
what amounts to the same thing, that the pressure p is given as in
hydrostatics * by
(2.2.9) p = gety - y).
The quantity Q is the density of the water. A number of consequences
of (2.2.9) are useful for our purposes. To begin with, we observe that
(2.2.10) px = ggjfc ,
so that px is independent of y. It follows that the ^-component of
the acceleration of the water particles is also independent of y\
and hence u, the ^-component of the velocity, is also independent
of y for all t if it was at any time, say at t = 0. We shall assume
this to be true in all cases —it is true for example in the important
special case in which the water was at rest at t = 0— so that u= u(x, t)
depends only on x and t from now on. As equation of motion in the
^-direction we may write, therefore, in view of (2.2.10):
(2.2.11) ut + uux = -gife.
This is simply the usual equation of motion in the Eulerian form,
use having been made of uy = 0. In addition, (2.2.8) may now be
written
(2.2.12) [u(ri + h)]x = -iji9
prj pr\
since u dy = u\ dy on account of the fact that u is independent
J — h J —h
of y. The two first order differential equations (2.2.11) and (2.2.12)
for the functions u(x, t) and r\(x, t) are the differential equations of
the nonlinear shallow water theory. Once the initial state of the fluid
is prescribed, i.e. once the values of u and 77 at the time t = 0 are
given, the equations (2.2.11) and (2.2.12) yield the subsequent
motion.
If in addition to the basic assumption of the shallow water theory
expressed by (2.2.9) we assume that u and 77, the particle velocity
and free surface elevation, and their derivatives are small quantities
whose squares and products can be neglected in comparison with
linear terms, it follows at once that equations (2.2.11) and (2.2.12)
simplify to
(2.2.13) ut = - gjfc ,
(2.2.14) (uh)x = - ifc ,
We have pv = — gg and (2.2.9) results through the use of p = 0 for y = r).
THE TWO BASIC APPROXIMATE THEORIES 25
from which 77 can be eliminated to yield for u the equation
(2.2.15) (uh)xx ~ — utt = 0.
S
If, in addition, the depth h is constant it follows readily that u
satisfies the linear wave equation
(2.2.16) uxx-—utt = 0.
In this case 77 satisfies the same equation. One observes therefore
the important result that the propagation speed of a disturbance is
given by Vgh. In principle, this linearized version of the shallow water
theory is the one which has always been used as the basis for the
theory of the tides. Of course, the tidal theory for the oceans requires
for its complete formulation the introduction of the external forces
acting on the water due to the gravitational attraction of the moon
and the sun, and also the Coriolis forces due to the rotation of the
earth, but nevertheless the basic fact about the tidal theory from
the standpoint of mathematics is that it belongs to the linear shallow
water theory. The actual oceans do not from most points of view
impress one as being shallow; in the present connection, however,
the depth is actually very small compared with the curvature of
the tidal wave surface so that the shallow water approximation is an
excellent one. That the tidal phenomena should be linear to a good
approximation would also seem rather obvious on account of the
small amplitudes of the tides compared with the dimensions of the
oceans. A few additional remarks about tidal theory and some other
applications of the linearized version of the shallow water theory to
concrete problems (seiches in lakes, and floating breakwaters, for
example) are given in Chapter 10.13.
2.3. Gas dynamics analogy
It is possible to introduce a different set of dependent variables
in such a way that the equations of the shallow water theory become
analogous to the fundamental differential equations of gas dynamics for
the case of a compressible flow involving only one space variable x.
(This seems to have been noticed first by Riabouchinsky [R.8].)
To this end we introduce the mass per unit area given by
(2.3.1) Q = Q(rj +h).
26 WATER WAVES
Since h depends only on x we have
(2.3.2) ft ==e^.
We next define the force p per unit width:
(2.3.3) p = \\pdy,
J — n
which, in view of (2.2.9) and (2.8.1), leads to
(2.3.4) P = ^(n+h}*=-j^Q*.
The relation between p and Q is thus of the form p = AQY with
y==2, that is, the "pressure" p and the "density" Q are connected
by an "adiabatic" relation with the fixed exponent 2.
Equation (2.2.11) may now be written
q(r) + h)(ut + uux) = - gQfa + h)qx
and this, in turn, may be expressed through use of (2.3.1) and
(2.3.4) as follows:
(2.3.5) Q(ut + uu9) = - px + gQhx ,
as one can readily verify.
The equation (2.2.12) may be written as
(2.3.6)
in view of (2.3.2) as well as (2.3.1). The differential equations (2.3.5)
and (2.3.6), together with the "adiabatic" lawp = gg*/2g given by
(2.3.4), are identical in form with the equations of compressible gas
dynamics for a one-dimensional flow except for the term gQhx on
the right hand side of (2.3.5), and this term vanishes if the original
undisturbed depth h of the water is constant. The "sound speed" c
corresponding to our equations (2.3.5) and (2.3.6) is, in analogy with
gas dynamics, given by c = VdpldQ, and this from (2.3.4) and (2.3.1 )
has the value
(2.8.7) C=— = Vg(rj + h).
It will be seen later that c(x, t) represents the local speed at which
a small disturbance advances relative to the water.
THE TWO BASIC APPROXIMATE THEORIES 27
2.4. Systematic derivation of the shallow water theory
It is of course a matter of importance to know under what cir-
cumstances the shallow water theory can be expected to furnish
sufficiently accurate results. The only assumption made above in
addition to the customary assumptions of hydrodynamics was that
the pressure is given as in hydrostatics by (2.2.9), but no assumption
was made regarding the magnitude of the surface elevation or the
velocity components. Consequently the shallow water theory may
be accurate for waves whose amplitude is not necessarily small,
provided that the hydrostatic pressure relation is not invalidated.
The above derivation of the shallow water theory is, however, open
to the objection that the role played by the undisturbed depth of the
water in determining the accuracy of the approximation is not put
in evidence. In fact, since we shall see later on that all motions die
out rather rapidly in the depth, it would at first sight seem reasonable
to expect that the hydrostatic law for the pressure would be, on the
whole, more accurate the deeper the water. That this is not the
case in general is well known, since the solutions for steady progressing
waves of small amplitude (i.e. for solutions obtained by the linearized
theory) in water of uniform but finite depth are approximated
accurately by the solutions of the shallow water theory (when it also
is linearized) only when the depth of the water is small compared
with the wave length (cf. Lamb [L.3], p. 368). It is possible to give
a quite different derivation of the shallow water theory in which the
equations (2.2.11) and (2.2.12) result from the exact hydrodynamical
equations as the approximation of lowest order in a perturbation
procedure involving a formal development of all quantities in powers
of the ratio of the original depth of the water to some other
characteristic length associated with the horizontal direction.* The
relation (2.2.9) is then found to be correct within quadratic terms
in this ratio. In this section we give such a systematic derivation
of the shallow water theory, following K. O. Friedrichs (see the
appendix to [S.19]), which, unlike the derivation given in section
* In this book the parameter of the shallow water theory is defined in two
different ways: in dealing with the breaking of waves in Chapter 10.10 it is
the ratio of the depth to a significant radius of curvature of the free surface;
in dealing with the solitary wave, however, it is essentially the ratio of the depth
to the quantity Uzjg, with U the propagation speed of the wave, and in this
case the development is carried out for Ua /gh near to one. In still other
problems it might well be defined differently in terms of parameters that are
characteristic for such problems.
28
WATER WAVES
2.2 above, is capable of yielding higher order approximations.
The disposition of the coordinate axes is taken in the usual manner,
with the x, 3-plane the undisturbed water surface and the i/-axis
positive upward. The free surface elevation is given by y — rj(x9 z, t)
and the bottom surface by y = — h(x, z). We recapitulate for the
sake of convenience the differential equations and boundary con-
ditions in terms of the Euler variables, that is, the equations of
continuity and motion, the vanishing of the rot at ion, and the boundary
conditions:
(2.4.1)
(2.4.2)
wz = 0,
wt + uwx
vu
vv
vw
px
wvz = --- py
wwz — ---- pz
(2.4.3)
(2.4.4)
(2.4.5)
(2.4.6)
= wx, vx = u
nt + urix + wr]z = v at y = 77,
p = 0 at y = 77,
uhx -\-v-\~ whz = 0 at y = — /i.
We now introduce dimensionless variables through the use of two
lengths d and k, with d intended to represent a typical depth and k
a typical length in the horizontal direction— it is characteristic of
the procedure followed here that the horizontal and vertical direc-
tions are not treated in the same way. The new independent variables
are as follows:
(2.4.7) x = x/k, y = y/d, z = z/k, r = t Vgd/k,
while the new dimensionless dependent variables are
u = ( Vgd)~lu, v = (kVgd/d)-1 v, w = ( Vgd)~l w
(2.4.8) p = — p,
^ = 7^/d, A = h/d.
In addition, we introduce the important parameter
(2.4.9) a =
THE TWO BASIC APPROXIMATE THEORIES 29
in terms of which all quantities will be developed; when this parameter
is small the water is considered to be shallow. This means, of course,
that d is small compared with fc, and hence that the x and z coor-
dinates (cf. (2.4.7)) are stretched differently from the y coordinate
and in a fashion which depends upon the development parameter.
Since it is the horizontal coordinate which is strongly stretched
relative to the depth coordinate, it seems reasonable to refer to the
resulting theory as a shallow water theory. The stretching process
combined with a development with respect to a is the characteristic
feature of what we call the shallow water theory throughout this
book. The dimensionless development parameter a has a physical
significance, of course, but its interpretation will vary depending
on the circumstances in individual cases, as has already been noted
above. For example, consider a problem in which the motion is to be
predicted starting from rest with initial elevation y — 770(#, j/, z)
prescribed; from (2.4.8) we have
y ^ r)0 =-- dfjQ(x9 y, z)
i- ix y
^'/I?0U' "<*'
from which we obtain
It is natural to assume that the dimensionless second derivative
tfficx will be at least bounded and consequently one sees that the
assumption that a is small might be interpreted in this case as
meaning that the product of the curvature of the free surface of the
water and a typical depth is a small quantity.
The object now is to consider a sequence of problems depending
on the small parameter a and then develop in powers of or. Introduc-
tion of the new variables in the equations (2.4.1) to (2.4.6) yields
(2.4.1)' aux + vy + awz = 0,
a[ut + uux + wuz + px] + vuy = 0,
(2.4.2)' • a[vt + uvx + wvz + py + 1] + TO, = 0,
. a[wt + uwx + wwz + pg] + vwy = 0,
(2.4.3)' wy = VZ9 uz = wx, vx = uy,
(2.4.4)' a[ijt + ur)x + wr]z] = v at y = 77,
80
WATER WAVES
(2.4.5)'
(2.4.6)'
a[uhx
p = 0 at y =
whz] -j-t; = Oa
=— h,
when bars over all quantities are dropped and r is replaced by t.
The next step is to assume power series developments for u, v,
w, rj, and p:
(2.4.10)
v =
w =
= p(0)
-f . . .,
4- . . .,
and insert them in the equations (2.4.1)' to (2.4.6)' to obtain, by
equating coefficients of like powers of a, equations for the successive
coefficients in the series, which are of course functions of x, y, z,
and t. The terms of zero order yield the equations
(2.4.1 )„
(2.4.2);
T.(0) n
y — '
/ a«%<°> = o,
7,(0)7,(°) _ o
\vvv — u»
(2.4.3);
(2-4.4);
(2-4.5);
(2.4.6);
M(0) _ ^(0) (0) _ (0)
Uy Wy. , VX Uy ,
«v ==
u(0> = 0 at y = T?(O),
p(o) = o at y = ^(0),
rj(o) = o at y = — A.
These equations yield the following:
(2.4.11) i><°> =0,
(2.4.12) w«» = w<°>(ff, z, <)»
(2.4.13) u<°) = u(0)(o;, 2, i),
(2.4.14) p<°>(0, i^W), a,*) = 0,
which contain the important results that the vertical velocity com-
ponent is zero and the horizontal velocity components are independent
of the vertical coordinate y in lowest order.
The first order terms arising from (2.4.1)' to (2.4.6)' in their turn
THE TWO BASIC APPROXIMATE THEORIES 81
yield the equations
(2.4.1); u«+»«=-*W,
t tt<0) + w(<%f + w(0)tt<0) + pf = 0,
(2.4.2); p<,°> +1=0,
. K>!0) + u^w(x°} + a;«»a><0) + p<°> - 0,
(2.4.4); rjf + u^rif + w™r)W = »<« at y = »?«»,
(2.4.6)i u^hx + ojtWA, + »<« = 0 at y = - h,
upon making use of (2.4.11), (2.4.12), and (2.4.13). Equation (2.4.1 )J
can be integrated at once since uw and w(0) are independent of y
to yield
(2.4.15) yd) - - (««> + o,<°> ) z, + F(<c, z, t),
with F an arbitrary function which can be determined by using
(2.4.6);; the result for u(1> is then
(2.4.16) »<« = - (M<°> + u,<0) ) y - [(«<»>/*), + (»«»)*).],._. .
To second order the vertical component of the velocity is thus linear
in the depth coordinate. In similar fashion the second of the equations
(2.4.2)j can be integrated and the additive arbitrary function of
x, z, t determined from (2.4.14); the result is
(2.4.17) p«»(0f y, z, t) - ijW(x9 z,t)-y
which is obviously the hydrostatic pressure relation (in dimension-
less form).
In the derivation of the shallow water theory given in the preceding
section this relation was taken as the starting point; here, it is
derived as the lowest order approximation in a formal perturbation
scheme. However, it is of course not true that we have proved that
(2.4.17) is in some sense an appropriate assumption: instead, it
should be admitted frankly that our dimensionless variables were
introduced in just such a way that (2.4.17) would result. If it could
be shown that our perturbation procedure really does yield a correct
asymptotic development (that the development converges seems
unlikely since the equations (2.4.1)' to (2.4.6)' degenerate in order
so greatly for a = 0) then the hydrostatic pressure assumption could
be considered as having been justified mathematically. A proof that
this is the case would be of great interest, since it would give a
mathematical justification for the shallow water theory; to do so in
82 WATER WAVES
a general way would seem to be a very difficult task, but Friedrichs
and Hyers [F.18] have shown that the development does yield
the existence of the solution in the important special case of the
solitary wave (cf. Chapter 10.9). (Keller [K.6] had shown earlier
that the formal procedure yields the solitary wave.) The problem
is of considerable mathematical interest also because of the following
intriguing circumstance: the approximation of lowest order to the
solution of a problem in potential theory is sought in the form of a
solution of a nonlinear wave equation, and this means that the
solution of a problem of elliptic type is approximated (at least in the
lowest order) by the solution of a problem of hyperbolic type.
The values of v(l) and p(0) given by (2.4.16) and (2.4.17) are now
inserted in the first and third equations of (2.4.2)J and in (2.4.4)!'
to yield finally
(2.4.18) uf + tt<°>t*W + w^uf + rif = 0,
(2.4.19) w><0) + i*<°> H>W + w^wf + i?<°> - 0,
(2.4.20) T?<°> + [ttWfoW + h)]x + [w<°>fo<o) + h)], = 0,
as definitive equations for t^0), w(Q\ and iy(0)— all of which, we repeat,
depend only upon #, 2, and /. If the superscript is dropped, w{Q) is
taken to be zero, and it is assumed that all quantities are independent
of z, one finds readily that these equations become identical with
equations (2.2.11) and (2.2.12) except for the factor g in (2.2.11)
which is missing here because of our introduction of a dimensionless
pressure.
It is clear that the above process can be continued to obtain the
higher order approximations. An example of such a calculation will
be given later in Chapter 10.9, where we shall see that the first non-
trivial term in the development which yields the solitary wave is of
second order.
PART II
Summary
In Part II we treat a variety of problems in terms of the theory
which arises through linearization of the free surface condition (cf.
the preceding chapter); thus the problems refer to waves of small
amplitude. To this theory the names of Cauchy and Poisson are
usually attached. The material falls into three different types, or
classes, of problems, as follows: A) Waves that are simple harmonic
in the time. These problems are treated in Chapters 3, 4, and 5 and
they include a study of the classical standing and progressing wave
solutions in water of uniform depth, and waves over sloping beaches
and past obstacles of one kind or another. The mathematical tools
employed here comprise, aside from classical methods in potential
theory, a thorough-going use of integrals in the complex domain.
B) Waves created by disturbances initiated at an instant when the water
is at rest. These problems, which are treated in Chapter 6, comprise
a variety of unsteady motions, including the propagation of waves
from a point impulse and from an oscillatory source. Uniqueness
theorems for the unsteady motions are derived. The principle mathe-
matical tool used in solving these problems is the Fourier transform.
The method of stationary phase is justified and used. C) Waves
arising from obstacles immersed in a running stream. This category
of problems differs from the first two in that the motion to be in-
vestigated is a small oscillation in the neighborhood of a uniform
flow, while the former cases concern small oscillations near the state
of rest. This difference is in one respect rather significant since the
problems of the first two types require no restriction on the shape of
immersed bodies, or obstacles, while the third type of problem
requires that the immersed bodies should be in the form of thin disks,
since otherwise the flow velocity would be changed by a finite amount,
and a linearization of the free surface condition would not then be
justified. In other words, the problems of this third type require
35
36 WATER WAVES
a linearization based on assuming a small thickness for any immersed
bodies, as well as a linearization with respect to the amplitude of the
surface waves. These problems are treated in Chapters 7, 8, and 9.
The classical case of the waves created by a small obstacle in a running
stream of uniform depth is first treated. This includes the classical
shipwave problem, discussed in Chapter 8, in which the "ship" is
treated as though it could be replaced by a point singularity. A
treatment is given in Chapter 9 of the problem of the waves created
by a ship moving through a sea of arbitrary waves, assuming the
ship to be a floating rigid body with six degrees of freedom and with
its motion determined by the propeller thrust and the pressure of the
water on its hull.
Finally, in an Appendix to Part II a brief summary of some of
the more recent literature concerned with the above types of problems
is given, since the cases selected for detailed treatment here do not
by any means exhaust the interesting problems which have been
solved.
SUBDIVISION A
WAVES SIMPLE HARMONIC IN THE TIME
CHAPTER 3
Simple Harmonic Oscillations in Water
of Constant Depth
3.1. Standing waves
In Chapter 2 we have derived the basic theory of irrotational
waves of small amplitude with the following results (in the lowest
order, that is). Assuming the #, 2-plane to coincide with the free
surface in its undisturbed position, with the t/-axis positive upward,
the velocity potential 0(x, y, z; t) satisfies the following conditions:
(8.1.1 ) V2<Z> = 0XX + 0
vy
in the region bounded above by the plane y = 0 and elsewhere by
any other given boundary surfaces. The free surface condition under
the assumption of zero pressure there is
(8.1.2) 0tt + g0y = 0 for y = 0.
The condition at fixed boundary surfaces is that d0/dn = 0; for
water of uniform depth h = const, we have therefore the condition
(8.1.3) 0y = 0 for y = - h.
Once the velocity potential 0 has been determined the elevation
Y)(x, z;t) of the free surface is given by
(3.1.4) 7? = --#,(*, 0, *; f).
%
Conditions at oo as well as appropriate initial conditions at t = 0
must also be prescribed.
In this section we are interested in those special types of standing
waves which are simple harmonic in the time; we therefore write
37
88 WATER WAVES
(3.1.5) 0(x, y, z; t) = eM<p(x, y, z) *
with <p a real function, and with the understanding that either the
real or the imaginary part of the right hand side is to be taken.
The problems to be treated here thus belong to the theory of small
oscillations of dynamical systems in the neighborhood of an equilib-
rium position.
The conditions on 0 given above translate into the following
conditions on <p:
(3.1.6) V2<p = 0, — h <y <Q9 — oo < #, * < oo,
(3.1.7) <p -—<p = Q, y = Q,
S
(3.1.8) <py = 0, y=-h.
As conditions at oo we assume that y and (py are uniformly bounded.**
Arbitrary initial conditions cannot now be prescribed, of course,
since we have assumed the behavior of our system to be simple
harmonic in the time. The free surface elevation is given by
tcs
(3.1.9) ri = -- eM . <p(x, 0, z).
S
We look first for standing wave motions which are two-dimensional,
so that (jp depends only upon x and y: <p — y(x, y), and also consider
first the case of water of infinite depth, i.e. h =- oo. One verifies
readily that the functions
(3.1.10)
\ (p = emv sin mx
are harmonic functions which satisfy the free surface condition
(3.1.7) provided that the constant m satisfies the relation
(3.1.11) m = o*/g.
In addition, the conditions at oo are satisfied. In particular, it is of
interest to observe that the oscillations die out exponentially in the
depth. The free surface elevation is then given by
* The most general standing wave would be given by 0 = f(t)(p(x, y, z). This
means, of course, that the shape of the wave in space is fixed within a multiplying
factor depending only on the time. Thus nodes, maxima and minima, etc. occur
at the same points independent of the time.
** This means that the vertical components of the displacement and velocity
are bounded at oo. One could prescribe more general conditions at oo without
impairing the uniqueness of the solutions of our boundary value problems, but
it does not seem worth while to do so in this case.
SIMPLE HARMONIC OSCILLATIONS 39
(3.1.12) ^_^
g \ sin mx
It should be pointed out specifically that our boundary problem,
though it is linear and homogeneous, has in addition to the solution
<p == 0 a two-parameter set of "non trivial" solutions obtained by
taking linear combinations of the two solutions given in (3.1.10).
The surface waves given by (3.1.10) are thus simple harmonic in
x as well as in t. The relation (3.1.11) furthermore states the very
important fact that the wave length A given by
(3.1.13) A == 2n/m = <2ng/o2
is not independent of the frequency of the oscillation, but varies
inversely as its square.
The above discussion yields standing wave solutions of physically
reasonable type, but one nevertheless wonders whether there might
not be others— for example, standing waves which are not simple
sine or cosine functions of x, but rather waves with amplitudes which,
for example, die out as x tends to infinity. Such waves do not occur
in two dimensions,* however, in the sense that all solutions for water
of infinite depth, except cp = 0, of the homogeneous boundary problem
formulated in (3.1.6) and (3.1.7) together with the condition that <p
and (py are uniformly bounded at oo are given by (3.1.10) with m
satisfying (3.1.11). This is a point worth pausing to prove, especially
since the method of proof foreshadows a mode of attack on our
problems which will be used in a more essential way later on. The
first step in the uniqueness proof is to introduce the function y(x, y)
defined by
(3.1.14) \p = (jpy — my, m > 0.
Since q> is a harmonic function, obviously yj is also a harmonic func-
tion. In addition, y) vanishes for y = 0 on account of its definition
and (3.1.7). Hence \p can be continued by reflection over the iT-axis
into a potential function which is regular and defined as a single-
valued function in the entire x, j/-plane. Since <p and q>y were assumed
to be uniformly bounded in the entire lower half plane it follows that
\p is bounded in the entire x, i/-plane since reflection in the #-axis
does not destroy boundedness properties. Thus ^ is a potential func-
tion which is regular and bounded in the entire x, t/-plane. By Liou-
* This statement is not valid in three dimensions as we shall see later on in
this section.
40 WATER WAVES
ville's theorem it is therefore a constant, and since y = 0 for y = 0,
the constant must be zero. Hence y vanishes identically. From (3.1.14)
it therefore follows that any solutions (p(x, y) of our boundary
value problem are also solutions of the differential equation
(3.1.15) <pv — nup = 0, — oo < y < 0.
The most general solution of this differential equation is given by
(3.1.16) <p = c(x)emv
with c(x) an arbitrary function of x alone. However, <p(x, y) is a
harmonic function and hence c(x) is a solution of
(8.1.17) — + m*c = 0
which, in turn, has as its general solution the linear combinations
of sin mx and cos mx. It follows, therefore, that the standing wave
solutions of our problem are indeed all of the form Aemv cos (w#+oc),*
with a and A arbitrary constants fixing the "phase" and the amplitude
of the wave, while m is a fixed constant which determines the wave
length A in terms of the given frequency a through (3.1.13).
In water of uniform finite depth h it is also quite easy to obtain
two-dimensional standing wave solutions of our boundary value
problem. One has, corresponding to the solutions (3.1.10) for water of
infinite depth, the harmonic functions
f (p = cosh m(y + h) cos mx,
(d.l.lS) 4 .
[<p = cosh m(y + h) sm mx,
as solutions which satisfy the boundary condition at the bottom,
while the free surface condition is satisfied provided that the con-
stant m satisfies the relation
(3.1.19) a2 = gm tanh mh
instead of the relation (3.1.11), as one readily sees. Since tanh mh-+I
as h -> oo it is clear that the relation (3.1.19) yields (3.1.11) as limit
relation for water of infinite depth. The uniqueness of the solutions
(3.1.18) for the two-dimensional case under the condition of boun-
dedness at oo was first proved by A. Weinstein [W.7] by a method
* It can now be seen that the negative sign in the free surface condition (3.1.7)
is decisive for our results: if this sign were reversed one would find that the
solution g? analogous to (8.1.16) would be bounded at oo only for c(x) = 0,
because (3.1.16) would now be replaced by c(x)e~mv, with m > 0.
SIMPLE HARMONIC OSCILLATIONS 41
different from the method used above for water of infinite depth
which can not be employed in this case (see [B. 12]).
It is of interest to calculate the motion of the individual water
particles. To this end let 6x and dy represent the displacements from
the mean position (x, y) of a given particle. Our basic assumptions
mean that dx9 dy and their derivatives are small quantities; it follows
therefore that we may write
— = u(xy y) = 0X — — m A cos at cosh m(y + h) sin mx
dt
ddy
—- = v(x, y) = 0y = mA cos erf sinh m(y + A) cos mx
within the accuracy of our basic approximation. The constant A is
an arbitrary factor fixing the amplitude of the wave. Hence we
have upon integration
dx = — sin at cosh m(y + h) sin mx,
(3.1.20)
7?? /-i
dy = sin at sinh m(y + /i) cos w#.
a
The motion of each particle takes place in a straight line the direction
of which varies from vertical under the wave crests (cos mx = 1) to
horizontal under the nodes (cos mx = 0). The motion also naturally
becomes purely horizontal on approaching the bottom y = — h.
These consequences of the theory are verified in practice, as indicated
in Fig. 3.1.1, taken from a paper by Ruellan ami Wallet (cf. [R.12]).
The photograph at the bottom makes the particle trajectories visible in
a standing wave; this is the final specimen in a series of photographs of
particle trajectories for a range of cases beginning with a pure pro-
gressing wave (ef. see. 3.2), and continuing with superpositions of pro-
gressing waves traveling in opposite directions and having the same
wave length but not the same amplitudes, finally ending with a
standing wave when the wave amplitudes of the two trains are equal.
We proceed next to study the special class of three-dimensional
standing waves that are simple harmonic in the time, arid which
depend only on the distance r from the t/-axis. In other words, we
seek standing waves having cylindrical symmetry. Again we seek
solutions of (3.1.6) which satisfy (3.1.7). Only the case of water of
infinite depth will be treated here, and hence (3.1.8) is replaced by
42
WATER WAVES
Fig. 3.1.1. Particle trajectories in progressing and standing waves
the condition that the solutions be bounded at oo in the negative
//-direction as well as in the x- and ^-directions. It is once more of
interest to derive all possible standing wave solutions which are
everywhere regular and bounded at oo because of the fact that the
solutions in the present case behave quite differently from those
obtained above for motions that are independent of the ^-coordinate.
SIMPLE HARMONIC OSCILLATIONS 43
In particular, we shall see that all bounded standing waves with
cylindrical symmetry die out at oo like the inverse square root of
the distance, while in two dimensions we have seen that the assump-
tion that the wave amplitude dies out at oo leads to waves of zero
amplitude everywhere.
It is natural to make use of cylindrical coordinates in deriving
our uniqueness theorem. Thus we write (3.1.6) in the form
(3.1.21) * _ (r -?^ + -? = 0, 0 ^ j/ > - oo, 0 ^ r < oo
r or \ dr J dy2
with r the distance from the j/-axis. The assumption that q> depends
only upon r and y and not on the angle 6 has already been used.
For our purposes it is useful to introduce a new independent variable
Q replacing r by means of the relation
(3.1.22) g = logr,
in terms of which (3.1.21) becomes
3V 2a>
(3.1.23) <?-* -Z + -Z = 0, y < 0, - oo < 0 < oo.
OQ* dy2
This equation holds, we observe, in the half-plane y < 0 of the
j/, g-plane. The boundary condition to be satisfied at y = 0 is (cf.
(3.1.7)):
(3.1.24) cpy — mq) = 0, m = a2/g.
We wish to find all regular solutions of (3.1.23) satisfying (3.1.24)
for which 99 and <py are bounded at oo. To this end we proceed along
much the same lines as above (cf. (3.1.14) and the reasoning imme-
diately following it) for the case of two dimensions, and introduce
the function y(g, y) by the identity
(3.1.25) y = <py — m<p, 2/<0, — oo < p < oo.
Since y; involves only a derivative of <p with respect to y and not
with respect to Q it follows at once that y is a solution of (3.1.23).
Since if) vanishes at y = 0 from (3.1.24) it follows easily that it can
be continued analytically into the upper half-plane y > 0 by setting
y(£> y) = — y(£> —y) an(l that the resulting function will be a
solution of (3.1.23) in the entire p, t/-plane. The function \p thus
obtained will be bounded in the entire plane, since it was bounded
in the lower half plane by virtue of the boundedness assump-
tions with respect to <p. A theorem of S. Bernstein now yields
44 WATER WAVES
the result that \p is everywhere constant* if it is a uniformly bounded
solution of (3.1.23) in the entire Q, t/-plane. Since ip vanishes on the
t/-axis it follows that y vanishes identically. Consequently we con-
clude from (3.1.25) that 9? satisfies the relation
(3.1.26) (py — m<p = 0, y < 0.
The most general function <p(g, y) satisfying this equation is
(3.1.27) (p = emvf($) = emvf(log r) = em^g(r)
with g(r) an arbitrary function. But (p(r, y) is also a solution of
(3.1.21) and hence g(r) is a solution of the ordinary differential
equation
(3.1.28) -:? I',!
r dr \ dr
or, in other words, g(r) is a Bessel function of ofrder zero:
(3.1.29) g(r) = AJ0(mr) + BY,(mr).
Since we restricted ourselves to bounded solutions only it follows
that all solutions <p(r, y) of our problem are given by
(3.1.30) g(r, y) = Ae^J^(mr), m = cr2/g,
with A an arbitrary constant. Upon reintroduction of the time
factor we have, therefore, as the only bounded velocity potentials
the functions
(3.1.31) 0(r, y; t) = AeiatemvJQ(mr).
As is well known, these functions behave for large values of r as
follows:
(3.1.32) 0(r, y; t) ~ Aeiaie™* • I/—- cos [mr — -\
' nmr \ 4/
and thus they die out like l/Vr> as stated above.
In two dimensions we were able to find bounded standing waves
of arbitrary phase (in the space variable) at oo. In the present case
of circular waves we have found bounded waves with only one phase
at oo. However, if we were to permit a logarithmic singularity at the
* What is needed is evidently a generalization of Liouville's theorem to the
elliptic equation (3.1.23) which has a variable coefficient. The theorem of Bern-
stein referred to is much more general than is required for this special purpose,
but it is also not entirely easy to prove (cf. E. Hopf [H.17] for a proof of it).
SIMPLE HARMONIC OSCILLATIONS 45
axis r = 0 and thus admit the singular Bessel function YQ(mr) as
a solution of (3.1.28), we would have as possible velocity potentials
the functions
(3.1.33) 0(r, y; t) = Beiatem^Y0(mr)
which behave for large r as follows:
(3.1.34) 0(r, y; t) ~ Beiate™v ]/— sin (mr - -\.
* nmr \ 4/
Admitting solutions with a logarithmic singularity on the i/-axis
thus leads to standing waves which behave at oo in the same way
as those which are everywhere bounded, except that they differ by
90° in phase at oo. Thus waves having an arbitrary phase at oo can
be constructed, but not without allowing a singularity. It has, however
not been shown that (3.1.31) and (3.1.33) yield all solutions with this
property.
3.2. Simple harmonic progressing waves
Since our boundary problem is linear and homogeneous we can
reintroduce the time factors cos at and sin at and take appropriate
linear combinations of the standing waves (3.1.5) to obtain simple
harmonic progressing wave solutions in water of uniform depth of
the form
(3.2.1) 0 = A cosh m(y + h) cos (mx ± at + a)
with m and a satisfying
(3.2.2) a2 = gm tanh mh,
as before.
The wave, or phase 9 speed c is of course given by
(3.2.3) c = a/m,
or, in terms of the wave length A — 2n/m by
(3.2.3), c = tanh *!*.
' %7l /
It is useful to write the relation (3.2.2) in terms of the wave length
X = 2n/m and then expand the function tanh mh in a power series
to obtain
46 WATER WAVES
We see therefore that
(3.2.5) a2 -> (y)2^ = m*&h as \ -*°*
and hence that
(3.2.6) c^=L Vgh if A/A is small.
This last relation embodies the important fact that the wave speed
becomes independent of the wave length when the depth is small compared
with the wave length, but varies as the square root of the depth. This
fact is in accord with what resulted in Chapter 2 upon linearizing
the shallow water theory (cf. (2.2.16)) and the sentence immediately
following), which led to the linear wave equation and to c = Vgh
as the propagation speed for disturbances. We can gain at least a
rough idea of the limits of accuracy of the linear shallow water
theory by comparing the values of c given by c2 = gh with those
given by the exact formula
(3.2.7) c* = tanh
2n A
for various values of the ratio A/A. One finds that c as given by
Vgh is in error by about 6 % if the wave length is ten times the
depth and by less than 2 % if the wave length is twenty times the
depth. The error of course increases or decreases with increase or
decrease in A/A.
In water of infinite depth, on the other hand, we have already
observed (cf. (3.2.2)) that
(3.2.8) c2 = gA/2jr.
Actually, the error in c as computed by the formula c2 = gA/2rc is
already less than 1/2 % if A/A > |. One might therefore feel justified
in concluding that variations in the bottom elevation will have but
slight effect on a progressing wave provided that they do not result
in depths which are less than half of the wave length, and observations
seem to bear this out. In other words, the wave would not "feel" the
bottom until the depth becomes less than about half a wave length.
It is of interest to determine the paths of the individual water
particles as the result of the passage of a progressing wave. As in the
preceding section we take dx and dy to represent the displacements
of a particle from its average position, and determine those dis-
placements from the equations
SIMPLE HARMONIC OSCILLATIONS 47
— = 0X = — Am cosh m(y + h) sin (mx -\-at-\- a),
dt
ddy
— -.-.- 0y = Am sinh m(y + ^) cos (mx + at + a),
(zr
since 0 is given by (3.2.1) in the present case. Integration of these
equations yields
(3.2.9)
dx = - — cosh m(y + /i) cos (rna: + at + a),
dy — --- sinh m(y + ^) sin (m# + (rf + a),
a
so that the path of a particle at depth y is an ellipse
dx2 ,fy2_l
a? ~¥ ~
with semi-axes a and 6 given by
.
a = - cosh m(y + h)
a
b = — sinh m(y + h).
a
On the bottom, y — — h, the ellipse degenerates into a horizontal
straight line, as one would expect. Both axes of the ellipse shorten
with increase in the depth. For experimental verification of these
results, the discussion with reference to Fig. 3.1.1 should be con-
sulted. In water of infinite depth the particle paths would be circles,
as one can readily verify. The fact that the displacement of the
particles dies out exponentially in the depth explains why a submarine
need only submerge a slight distance below the surface— a half wave
length, say — in order to remain practically unaffected even by severe
storms.
3.3. Energy transmission for simple harmonic waves of small
amplitude
In Chapter 1 the general formulas for the energy E stored in a
fluid and its flux or rate of transfer F across given surfaces were
derived for the most general types of motion. In this section
48 WATER WAVES
we apply these formulas to the special motions considered in the
present chapter, that is, under the assumption that the free surface
conditions are linearized. The formula for the energy E stored in a
region R is (cf. (1.6.1)):
(3.3.1) E = e [J(<^2 + 0; + <*>*2) + gy]dxdydz;
R
while the flux of energy F in a time T across a surface SG fixed in
space is given by (cf. (1.6.5)):
(8.3.2) F-
We suppose first that the motion considered is the superposition
of two standing waves which are simple harmonic in the time, as
follows:
(3.3.3) <P(x, y, z; t) = 9^(0?, y, z) cos at + 99, (#, j/, z) sin at.
Insertion of this in (3.3.2) with T = 2jc/a9 i.e. for a time interval
equal to the period of the oscillation, leads at once to the following
expression for the energy flux F through SG:
One observes that the energy flux over a period is zero if either (pl
or 9?2 vanishes, i.e. if the motion is a standing wave: a fact which
is not surprising since one expects an actual transport of energy
only if the motion has the character of a progressing wave. Still
another fact can be verified from (3.3.4) in our present cases, in which
(pl and (p2 are, as we know, harmonic functions: if SG is a fixed closed
surface in the fluid enclosing a region R Green's formula states that
provided that 9^ and <p2 have no singularities— sources or sinks for
example— in R. In this case the energy flux F clearly vanishes since
9?! and <p2 are harmonic. Also one sees by a similar reasoning that the
flux F over a period remains constant if SG is deformed without
passing over singularities. In particular, the energy flux through a
vertical plane passing from the bottom to the free surface of the water
SIMPLE HARMONIC OSCILLATIONS 49
in a two-dimensional motion would be the same (per unit width of
the plane) for all positions of the plane provided that no singularities
are passed over. This fact makes it possible, if one wishes, to con-
sider the energy in the fluid as though the energy itself were an
incompressible fluid, and to speak of its rate of flow.
In the literature dealing with waves in all sorts of media, but
particularly in dispersive media, it is indeed commonly the custom
to introduce the notion of the velocity of the flow of energy ac-
companying a progressing wave, and to bring this velocity in relation
to the kinematic notion of the group velocity (to be discussed in
the next section). The author has found it difficult to reconcile him-
self to these discussions, and feels that it would be better to discard
the difficult concept of the velocity of transmission of energy, since
this notion is not of primary importance, and nothing can be ac-
complished by its use which cannot be done just as well by using
the well-founded and clear-cut concept of the flux of energy through
a given surface. On the other hand, the notion is used in the literature
(and probably will continue to be used) and consequently a dis-
cussion of it is included here, following pretty much the derivation
given by Rayleigh in an appendix to the first volume of his Sound
[R.4], In the next section, where the notion of group velocity is
introduced, some further comments about the concept of the velocity
of transmission of energy will be made.
We consider the energy flux per unit breadth across a vertical
plane in the case of a simple harmonic progressing wave in water
of uniform depth (or, in view of the above remarks, across any surface
of unit breadth extending from the bottom to the free surface). The
velocity potential 0 is given by (cf. (3.2.1))
(3.3.5) 0 = A cosh m(y + h) cos (mx + at + a)
and (3.3.2) yields
(3.3.6) F = A*Q<jm J*+2*/<J ft* Cosh2 m(y + h)dy\ sin2 (mx + at) dt
for the flux across a strip of unit breadth in the time T = 2jt/a9
the period of the oscillation. Hence the average flux per unit time
is given by
(* q 7^ w F
(3.3.7) tav = -
since the average of sin2 6 over a period is 1/2. We have also taken
50 WATER WAVES
r) = 0 in the upper limit of the integral in (3.3.6) and thus neglected
a term of higher order in the amplitude. It is useful to rewrite the
formula (3.8.7) in the following form through use of the relations
or* = gm tanh mh and c = a/m:
(8.8.8) Fav = —si- cosh* mh • U,
O
with U a quantity having the dimensions of a velocity and given
by the relation
(3.3.9) U
Next we calculate the average energy stored in the water as a result
of the wave motion with respect to the length in the direction of
propagation of the wave. This is obtained from (3.3.1) by calculating
first the energy JEj over a wave length A = 2n/m at any arbitrary
fixed time. In the present case we have
EK — E0 = m2Q r JA [$A2 sinh2 m(y + h) cos2 (mx + at + a)
(3.3.10) + £ A2 cosh2 m(y + h) sin2 (mx + at + a)] dxdy
+
in which the constant EQ refers to the potential energy of the water
of depth h when at rest. On evaluating the integrals, and ignoring
certain terms of higher order, we obtain for the energy between two
planes a wave length apart arising from the passage of the wave the
expression
(3.3.11 ) EI - E0 = - A cosh2 nth,
2g
as one finds without difficulty. Thus the average energy Eav in the
fluid per unit length in the ^-direction which results from the motion
is given by
(3.3.12) Eav = - cosh2 mh.
2g
Upon comparison with equation (3.3.8) we observe that Eav is
exactly the coefficient of U in the formula (3.3.8) for the average
energy flux per unit time across a vertical plane. It therefore follows,
assuming that no energy is created or destroyed within the fluid
SIMPLE HARMONIC OSCILLATIONS 51
itself, that the energy is transmitted in the direction of propagation of
the wave on the average with the velocity U. As we see from (3.3.9)
the velocity U is not the same as the phase or propagation velocity c;
in fact, U is always less than c: for water of infinite depth it has the
value c/2 and it increases with decrease in depth, approaching the
phase velocity c as the depth approaches zero.
3.4. Group velocity. Dispersion
In any body of water the motion of the water in general consists
of a superposition of waves of various amplitudes and wave lengths.
For example, the motion of the water due to a disturbance over a
restricted area of the surface can be analyzed in terms of the super-
position of infinitely many simple harmonic wave trains of varying
amplitude and wave length; such an analysis will in fact be carried
out in Chapter 6. However, we know from our previous discussion
(cf. (3.2.7)) that the propagation speed of a train of waves is an
increasing function of the wave length— in other words, the wave
phenomena with which we are concerned arc subject to dispersion—
and thus one might expect that the waves would be sorted out as
time goes on into various groups of waves such that each group
would consist of waves having about the same wave length. We wish
to study the properties of such groups of waves having approximately
the same wave length.
Suppose, for example, that the motion can be described by the
superposition of two progressing waves given by
(341)
= A sin ([m + dm]x — [a + da]t)
with dm and da considered to be small quantities. The superposition
of the two wave trains yields
0 = 2A cos - (xdm — tda) sin I m + — \x — \a -\ -- \t\
(3.4.2) 2V \L 2 J L 2j /
= B sin (m'x — o't)
with m' = m + dm/2, a' = a + da/2. Since dm and da are small it
follows that the function B varies slowly in both x and t so that <P
is an amplitude-modulated sine curve at each instant of time, as
indicated schematically in Figure 3.4.1. In addition, the "groups"
of waves thus defined— in other words the configuration represented
52
WATER WAVES
by the dashed curves of Figure 8.4.1. —advance with the velocity
da/dm in the 0-direction. In our problem a will in general be a function
Fig. 3.4.1. Wave groups
of m so that the velocity U of the group is given approximately
by da/dm9 or, in terms of the wave length A = 27t/m and wave velocity
c = a/m, by
d(mc) . dc
dm dk
(3.4.3)
U =
The matter can also be approached in the following way (cf.
Sommerfeld [S.13]), which comes closer to the more usual cir-
cumstances. Instead of considering the superposition of only two
progressing waves, consider rather the superposition, by means of
an integral, of infinitely many waves with amplitudes and wave
lengths which vary over a small range:
(3.4.4) 0 = r*+*A(m) exp (i(mx - at)} dm.
J WQ— e
The quantity mx — at can be written in the form
(3.4.5) mx — at = m#c — aQt + (m — m0)x — (a — aQ)t.
From (3.4.4) one then finds
(3.4.6) 0 = C exp {i(m<p — a0t)}>
in which the amplitude factor C is given by
(8.4.7) C = r*+*A(m) exp{i[(m - mQ)x - (a - a0)*]} dm.
J WQ— «
We are interested here in seeking out those places and times (if any)
where the function C represents a wave progressing with little change
in form, since (3.4.6) will then furnish what we call a group of waves.
Since x and t occur only in the exponential term in (8.4.7), it follows
'that the values of interest are those for which this term must be
nearly constant, i.e. those for which (m — w0)# — (a — a0)t~ const.
SIMPLE HARMONIC OSCILLATIONS 58
It follows that the propagation speed of such a group is given by
(a — (70)/(ra — ra0), and if (m — w0) is small enough we obtain again
the formula (3.4.3).
Evidently, it is important for this discussion of the notion of group
velocity that the motion considered should consist of a superposition
of waves differing only slightly in frequency and amplitude. In
practice, the motions obtained in most cases —through use of the
Fourier integral technique, for example,— are the result of super-
position of waves whose frequencies vary from zero to infinity and
whose amplitudes also vary widely. However, as we shall see in
Chapter 6, it happens very frequently that the motion at certain
places and times is approximated with good accuracy by integrals
of the type given in (3.4.4) with e arbitrarily small. (This is, indeed,
the sense of the principle of stationary phase, to be treated in Chap-
ter 6.) In such cases, then, groups of waves do exist and the dis-
cussion above is pertinent.
In our problems the relation between wave speed and wave length
is given by (3.2.2) and consequently the velocity U of a group is
readily found, from (3.4.3), to be
2
We observe that the group velocity has the same value as was given
in the preceding section for the average rate of propagation of energy
in a uniform train of waves having the same wave length as those
of the group. In other words, the rate at which energy is propagated
is given by the group velocity and not the phase velocity. This is
often considered as the salient fact with respect to the notion of
group velocity. As indicated already in the preceding section, the
author does not share this view, but feels rather that the kinematic
concept of group velocity is of primary significance, while the notion
of velocity of propagation of energy might better be discarded. It
is true that the two velocities, in spite of the fact that one is derived
from dynamics while the other is of purely kinematic origin, turn
out to be the same— not only in this case, but in many others as
well*— but it is also true that they are not always the same— for
example, the two velocities are not the same if there is dissipation
of energy in the medium. In addition, we have seen in the preceding
section that the notion of velocity of energy can be derived when no
A general analysis of the reason for this has been given by Broer [B.I 8].
54 WATER WAVES
wave group exists at all —we in fact derived this velocity for the case
of a wave having but one harmonic component.
In Chapter 6 we shall have occasion to see how illuminating the
kinematic concept of a group and its velocity can be in interpreting
and understanding the complicated unsteady wave motions which
arise when local disturbances propagate into still water.
CHAPTER 4
Waves Maintained by Simple Harmonic Surface Pressure
in Water of Uniform Depth* Foreed Oscillations
4.1. Introduction
In our previous discussions we have considered always that the
pressure at the free surface was constant (usually zero) in both space
and time. In other words, only the free oscillations were treated and
the problems were, correspondingly, linear and homogeneous boun-
ary value problems. Here we wish to consider two problems in which
the surface pressure p0 is simple harmonic in the time and the resulting
motions are thus forced oscillations; the problems then also have a
nonhomogeneous boundary condition. In the first such problem we
assume that the motion is two-dimensional and that the surface pres-
sure is a periodic function of the space coordinate x over the entire
#-axis; in the second problem the surface pressure is assumed to be
zero except over a segment of finite length of the #-axis. In these
problems the depth of the water is assumed to be everywhere infinite,
but the corresponding problems in water of constant finite depth
can be, and have been, solved by much the same methods.
The formulation of the first two problems is as follows. A velocity
potential 0(x, y;t) is to be determined which is simple harmonic in
the time t and satisfies
(4.1.1) V20 = 0 for y < 0.
The surface pressure p(x; t) is given by
(4.1.2) p(x; t) = p(x) sin at,
and the boundary conditions at the free surface are the dynamical
condition (cf. (2.1.20)!)
(4.1.3) r]= -—&t
&
55
50 WATER WAVES
and the kinematic condition
(4.1.4) rit = ®y.
The last condition means that no kinematic constraint is imposed
on the surface— it can deform freely subject to the given pressure
distribution. In addition, we require that <Pt and 0y should be
uniformly bounded at oo. This means effectively that the vertical
displacement and vertical velocity components are bounded. In
section 4.3, the amplitude p(x) of the surface pressure p will have
discontinuities at two points and we shall impose appropriate con-
ditions on 0 at these points when we consider this case.
We seek the most general simple harmonic solutions of our problem;
they have the form
(4.1.5) 0 = <p(x, y) cos at + y(x, y) sin at.
The functions <p and y; are of course harmonic in the lower half
plane. The conditions (4.1.2), (4.1.3), and (4.1.4) arc easily seen to
yield for the function <p the boundary condition
a __
(4.1.6) (py — m<p = — — p(x) for y — 0
Ci O
with the constant m defined by
(4.1.7) m = a2/g;
while for y they yield the condition
(4.1.8) \py — my = 0 for y = 0.
The phase sin at assumed for p in (4.1.2) has the effect that ^ satisfies
the homogeneous free surface condition, as one sees.
We know from the first section of the preceding chapter that the
only bounded and regular harmonic functions y which satisfy the
condition (4.1.8) are given by
f cos mx
(4.1.9) V(%,y)
\«
1-
sin mx
In the next two sections we shall determine the function (p(x, y),
. i.e. that part of 0 which has the phase cos at, in accordance
with two different choices for the amplitude p(x) of the surface
pressure p.
SIMPLE HARMONIC SURFACE PRESSURE 57
4.2. The surface pressure is periodic for all values of x
We consider now the case in which the surface pressure is periodic
in x such that p(x) in (4.1.2) and (4.1.6) is given by
(4.2.1) p(x) = PsinAtf, — oo <x < oo.
One verifies at once that the following function (p(x, y):
aP e*v
(4.2.2) (p(x, y) = sin fa
Qg m — A
is a harmonic function which satisfies the free surface boundary
condition (4.1.0) imposed in the present case. Since the difference #
of two solutions q>l9 q>2 both satisfying all of our conditions would
satisfy the homogeneous boundary condition %v — m% — 0, it follows
that all solutions <p of our boundary value problem can be obtained by
adding to the special solution given by (4.2.2) any solution of the
homogeneous problem, and these latter solutions are the functions
y given by (4.1.9) since jj satisfies the same conditions as y. Therefore
the most general simple harmonic solutions of the type (4.1.5) are
given in the present case by
VaP eXv fcos0M?]l
(4.2.3) 0(<r, ?/; t) - sin fa + Ae™* \ . \\ cos at
[_Qg m — A [sin mx jj
{cos mx ]
\ sin at,
sin mx J
with A and B constants which are at our disposal. In other words,
the resulting motions are, as usual in linear vibrating systems, a
linear combination of the forced oscillation and the free oscillations.
These solutions— without the uniqueness proof— seem to have been
given first by Lamb [L.2].
We observe that the case A — m must be excluded, and that if A
is near to m large amplitudes of the surface waves arc to be expected.
This means physically, as one sees immediately, that waves of large
amplitude are created if the periodic surface pressure distribution
has nearly the wave length which belongs to a surface wave of the
same frequency for pressure zero at the surface— that is, the wave
length of the corresponding free oscillation.
If instead of (4.1.2) we take the surface pressure as a progressing
wave of the form
(4.2.4) p(x\ t) = H sin (at — fa)
58 WATER WAVES
it is readily found that progressing surface waves result which are
given by
Ha e^
(4.2.5) 0(x, y;t) = -- - cos (at - fa).
Qg m — /.
To this one may, of course, add any of the wave solutions which
occur under zero surface pressure. Again one observes an odd kind
of "resonance" phenomenon: large amplitudes are conditioned by
the wave length in space of the applied pressure once the frequency
has been fixed.
4.3. The variable surface pressure is confined to a segment of the
surface
In this section we consider the case in which the surface pressure p
I x \ < a
{P sin at.
with P a constant. Some of the motions which can arise under such
circumstances are discussed by Lamb [L.2] in the paper quoted above.
However, here as elsewhere, Lamb assumes fictitious damping
forces* in order to be rid of the free oscillations and thus achieve a
unique solution, and he also makes use of the Fourier integral tech-
nique which we prefer to replace by a different procedure. In fact,
the present problem is a key problem for this Part II and a peg upon
which a variety of observations important for other discussions in later
chapters will be hung. As we shall see, the present problem is also
decidedly interesting for its own sake, although Lamb strangely
enough made no attempt in his paper to point out the really striking
results.
In addition to prescribing the pressure p through (4.3.1) it is
necessary to add to the conditions imposed in section 4.1 appropriate
conditions at the points (it #, 0) where p has discontinuities. In
view of (4.1.3) it is clear that a finite discontinuity in 0t or r\ or
both must be admitted and it seems also likely that the derivatives
0X and 0y of 0 would be unbounded near these points. We shall make
* Lamb assumes resistances which are proportional to the velocity. In this
way the irrotational character of the flow is preserved, but it is difficult to see
how such resistances can be justified mechanically. It would seem preferable
to secure the uniqueness of the solution in unbounded domains by imposing
physically reasonable conditions on the behavior of the waves at infinity.
SIMPLE HARMONIC SURFACE PRESSURE 59
the following requirements
(4.3.2) 0t bounded; 0y = O(Q-I^£), e > 0
in a neighborhood of the points (^ a, 0) with Q the distance from
these points. This means, in particular, that the surface elevation is
bounded near these points and that the singularity admitted is not
as strong as that of a source or sink. We recall that 0t and 0y were
required to be uniformly bounded at oo.
The stipulations made so far do not ensure the uniqueness of the
solution 0 of our problem any more than similar conditions ensured
uniqueness of the solution of the problem treated in the preceding
section. However, we have in mind now a physical situation in which
we expect the solution to be unique: We imagine the motion resulting
from the applied surface pressure p given by (4.3.1) to be the limit
approached after a long time subsequent to the application of p to
the water when initially at rest. Under these circumstances one feels
instinctively that the motion of the water far away from the source
of the disturbance should have the character of a progressing wave
moving away from the source of the disturbance, since at no time
is there any reason why waves should initiate at infinity. (We shall
show (of. (6.7)) that the motion of the water arising from such initial
conditions actually does approach, as the time increases without
limit, the motion to be obtained here.) Consequently we add to our
conditions on 0 the condition— often called the Sommerfeld condition
in problems concerning electromagnetic wave propagation— that the
zvares should behave at oo like progressing waves moving away from
the source of the disturbance. As we shall see, this qualitative condition
leads to a unique solution of our problem.
In solving our problem there are some advantages to be gained by
not stipulating at the outset that the Sommerfeld condition should
be satisfied, but to obtain first all possible solutions of the form
(4.1.5), and only afterwards impose the condition. We have therefore
to find the harmonic functions (p which satisfy the condition (cf.
(4.1.6) and (4.3.1))
( c, | x \ ^ a
(4.3.3) <p - my = \ , y = 0
rv (0, | x | > a
with
Pa
(4.3.4) ra = a2/g, c = — —
68
60 WATER WAVES
on the free surface, and the boundedness conditions which follow
from those imposed on 0:
{cp and cpv bounded at oo,
i I V 7
<p bounded and <py = 0(e~"1+e), e > 0, at x = ± a.
The functions y in (4.1.5), i.e. those which yield the waves of phase
sin at in <Z>, satisfy the same conditions as in section 4.1 and are
therefore given by (4.1.9). We have therefore only to determine the
functions <p. To this end it is convenient to introduce new dimen-
sionless quantities
(4.3.6) xl = mx, yl = my, a± = ma
together with c^ = c/m so that the free surface condition (4.3.3)
takes the form
( cl , | xl \ ^ «!
(4.3.7) V _ v = , ft - 0.
In what follows we use the condition in this form but drop the sub-
scripts for the sake of convenience.
In most of the two-dimensional problems treated in the remainder
of Part II we make use of the fact that any harmonic function
(p(x, y) can be taken as the real part of an analytic function f(z) of
the complex variable z = x -\- iy and write
(4.3.8) f(z) = <p(x, y) + iy(x, y) = f(x + iy).
In our present problem f(z) is defined and analytic in the lower half
plane. To express the surface condition (4.3.7) in terms of f(z) we
write
id-~% - /) = die (if - f),
in which the symbol 8&e means that the real part of what follows is
to be taken. Consequently the free surface condition has the form:
(4.3.9) V.-V
SIMPLE HARMONIC SURFACE PRESSURE 61
We now introduce a new analytic function F(z) by the equation*
(4.3.10) F(z) = if'(z) - /(«)
and seek to determine F(z) uniquely through the conditions imposed
on 9? — Ste J(z). We observe to begin with that F(z) satisfies the
condition
f r \ T I <. n
c9 | x i ^ a
in view of (4.3.9). We show now that F(z) is uniquely determined
within an additive pure imaginary constant, as follows: Suppose
that G(z) - Fi(z) — F2(z) is the difference of two functions F(z)
satisfying the conditions resulting from (4.3.10) through those on
f(z). Then die G(z) would vanish on the entire real axis, except
possibly at x -- i a, as one sees from (4.3.11). Hence 3&eG(z) is a
potential function which can be continued analytically by reflection
over the real axis into the entire upper half plane; it will then be
defined and single-valued in the whole plane except for the points
(i fl, 0). At oo, 8&e G(z) is bounded in the lower half plane, while
£Jle G(z) = 0(g~~1+e), e > 0, at x = ± a in view of the regularity
conditions and the definition of G(z). These boundedness properties
are evidently preserved in the analytic continuation into the upper
half plane. Consequently Sfce G(z) has a removable singularity at the
points x — i a on the real axis since the singularity is weaker than
a pole of first order and the function is single-valued in the neigh-
borhood of these points. Thus 3te G(z) is a potential function which
is regular and bounded in the entire plane, and is zero on the real
axis; by Liouville's theorem it is therefore zero everywhere. Con-
sequently the analytic function G(z) is a pure imaginary constant,
and the result we want is obtained. On the other hand it is rather
easy to find a function F(z) which has the prescribed properties— for
example by first finding its real part from (4.3.11) through use of
the Poisson integral formula. We simply give it:
ic z — a
(4.3.12) F(z) = _logr:_;
one verifies readily that it has all of the required properties. We
take that branch of the logarithm which is real for (z —- a)/(z + a)
* This device has been used by Kotschin [K.I 4], and it was exploited by Lewy
[L.8] and the author [S.18] in studying waves on sloping beaches.
62
WATER WAVES
real and positive.
Ojice F(z) has been uniquely determined, the complex velocity
potential f(z) is restricted to the solutions of the first order ordinary
differential equation (4.3.10), which means that the solutions depend
only on the arbitrary constant which multiplies the non-vanishing
solution e~iz of the homogeneous equation if'(z) — f = 0. But
9te (A + iB)e~iz = ev(A cos x + B sin x) and these are the standing
wave solutions for the case of surface pressure p = 0. The most
general solution of (4.3.10), with F(z) given by (4.3.12), can be
written, as one can readily verify, in the form
(4.3.13)
c Cz t — a
/(*)=-*-« e«log— — dt,
n J zn t + a
with the initial point *0 and the path of integration any arbitrary
path in the slit plane. Changing #0 obviously would have the effect
of changing the additive solution of the homogeneous equation. It
is convenient to replace (4.3.13) by the following expression, obtained
through an integration by parts:
z - a C* /I 1
(4.3.14) /(*)
v y/v;
ci f
= — -
rc L
log
fo
/I
( --
\t-a
t+a
and at the same time to fix the path of integration as indicated in
t- plane
(a) (b)
Fig. 4.3.1a,b. Path of integration in f-plane
Figure 4.3.1. This path comes from oo along the positive imaginary
axis and encircles the origin, leaving it and the point (-—a, 0) to
SIMPLE HARMONIC SURFACE PRESSURE
63
the left. Use has been made of the fact that log (z— -a)/(*+a) ->0
when z -> oo; we observe also that the integrals converge on account
of the exponential factor.
That 95(3?, y) = &te f(z) as given through (4.3.14) satisfies the
boundary conditions imposed at the free surface and the regularity
condition at the points (i a, 0) is easy to verify. We proceed to
discuss the behavior of f(z) at oo (always for z in the lower half plane).
For this purpose it suffices to discuss the integrals
Pii
tOO
t± a
dt since the function log I —
behaves like l/z at oo (as one readily sees). To this end we integrate
once by parts to obtain
i Cz ei(t~z}
J(fy\ * I fit
2(z) — — "r— — * ,. , f,\2atm
£» i fl J £00 \* i" #)
We suppose that the curved part of the path of integration in Figure
4. 3. la is an arc of a circle. It follows at once that the complex number
t — z has a positive imaginary part on the path of integration as
long as the real part of z is negative, and hence we have
1
± #
f
J foo
dt
\t±a\
f
J 0
dt
\t±a\
Consequently I(z) behaves like l/z at infinity when the real part of
z is negative, and f(z) likewise. The situation is different, however,
if the real part of z is positive. To study this case, we add and subtract
circular arcs, as indicated in Figure 4.3.1b, in order to have an
integral over the entire circle enclosing the singularities at ± a
as well as over a path symmetrical to the path in Figure 4.3. la.
By the same argument as above, the contribution of the integral
over the latter path behaves like 1/2 at oo, and hence the non-
vanishing contribution arises as a sum of residues at the points ± #•
These contributions to I(z) are at once seen to have the values
2nie^ize^ia. Thus we may describe the behavior of f(z) as given
by (4.3.14) at oo as follows:
(4.3.15) /(*) =
'(7)
for Ste z < 0
— 4ci (sin a) e~iz + O I — I for die z > 0.
WATER WAVES
From (4.3.10) and (4.3.12) one sees that f(z) has the same behavior
at oo as f(z)9 except for a factor — i. Hence f(z)9 and with it
(jp(x, y) = 3te /(*), has the postulated behavior at oo. It is convenient
to write down explicitly the behavior of <p(x, y) at oo:
for x < 0,
(4.3.16) <p(x9 y) = 9te f(z) = } V r ' .
4c sin a ev sin x + O I — 1 for x > 0.
\rj
It follows that all simple harmonic solutions of our problem arc
given by linear combinations of
(4.3.17) 0(x9 y; t) = (die f(z) + Ae* sin x + Be* cos x) cos at
and
(4.3.18) <P(x9 y\ t) = (Cey sin x + Dev cos x) sin at
in which A, B, C, and 1) are arbitrary constants, and f(z) is given
by (4.3.14). In other words, the standing waves <p(x9 y) cos at just
found above, together with the standing waves which exist for
vanishing free surface pressure, constitute all possible standing waves.
We now impose the condition that the wave 0(x9 y\ t) we seek
behaves like an outgoing progressing wave at oo, i.e. that it behaves
like
S_: ev(H sin (x + at) + K cos (x + erf)) at x = — oo
and like
S+: ev(L sin (x — at) + M cos (x — at)) at x = + oo.
In view of the behavior of (p(x, y) = <%ef(z) at x = — oo (of. (4.3.16)),
i.e. the fact that it dies out there, it is clear that we may combine
the standing wave solutions (4.3.17) and (4.3.18) in such a way as
to obtain a progressing wave solution
(4.3.19) 0(x9 y; t) = e*(H sin (x + at) + K cos (x + at))
+ <p(x9 y) cos at
valid everywhere and which satisfies the condition £_, with the two
constants H and K still arbitrary. At x = + oo this wave has the
behavior
(4.3.20) &(x, y; t) = e*[(H sin (x + at) + K cos (x + at))
— 4c sin a sin x cos at] + O 1 — 1
SIMPLE HARMONIC SURFACE PRESSURE 65
in view of (4.3.16). In order that S+ should hold for this solution
for all t one sees readily that the constants // and K must satisfy
the linear equations
L — H — 4e sin a
(4.3.21 ) L = - H
\M = K, M = - K,
from which we conclude that
f L — — - 2c sin a, // = 2c sin a
<«•»> {*-*_..
Thus the solution is now uniquely determined through imposition
of the Sommerfcld condition, and can be expressed as follows:
(4.3.23) 0(x, y; t) ~ ---- sin maemv sin (mx ~ot) + O\—\, x > 0
/ 1 \
O\—\, x
\ r /
upon rcintroduction of the original variables and parameters (cf.
(4.3.6)), with 0(l/r) representing a function which dies out at
infinity like 1/r. The function 0 of course yields a wave with sym-
metrical properties with respect to the i/-axis. We observe that
the wave length A = 'Infm of these waves at oo is the same as that of
free oscillations of the same frequency, as one would expect.
The most striking thing about the solution is the fact that for
certain frequencies and certain lengths of the segment over which
the periodic pressure differs from zero, the amplitude of the progressing
wave is zero at oo; this occurs obviously for sin ma = 0, i.e. for
ma — kn, k — - 1, 2, 3, . . .. Since m — 2n/h with A the wave length
of a free oscillation of frequency a, it follows that the amplitude of
the progressing wave at oo vanishes when
(4.3.24) 2a - frA, k = 1, 2 ____ ,
i.e. when the length of the segment on which the pressure is applied
is an integral multiple of the wave length of the free oscillation having
the same frequency as the periodic pressure. This does not of course
mean that the entire disturbance vanishes, but only that the motion
in this case is a standing wave given by
(4.3.25) &(x, y; t) = y(x, y) cos at,
since the quantities // and K in (4.3.19) are now both zero. Since
<p now behaves like 1/r at both infinities, the amplitude of the standing
66 WATER WAVES
wave tends to zero at infinity. A wave generating device based on the
physical situation considered here would thus be ineffective at certain
frequencies. It is clear that no energy is carried off to infinity in
this case, and hence that the surface pressure p on the segment
— a 5* x ^ + a can do no net work on the water on the average.
Since r/t = 0V it follows that the rate at which work is done by the
pressure p (per unit width at right angles to the #, t/-plane) is
rpq>y cos atdx, and since p has the phase sin at it is indeed clear
•a
that the average rate of doing work is zero in this case.
There is a limit case of the present problem which has considerable
interest for us. It is the limit case in which the length of the segment
over which p is applied shrinks to zero while the amplitude P of p
increases without limit in such a way that the product 2aP approaches
a finite limit. In this way we obtain the solution for an oscillating
pressure point. One sees easily that the function f(z) given by (4.3.13),
which yields the forced oscillation in our problem, takes the following
form in the limit:
(4.3.26) i(z) = ^e-«^dt,
with C the real constant 2aPa/gg. At oo this function behaves as
follows
0 (— ) for die z < 0,
(4.3.27) /(«) = ' V*7
2Ci e~iz + 0 ( — I for S&e z > 0.
In this limit case of an oscillating pressure point we see that there are
no exceptional frequencies: application of the external force always
leads to transmission of energy through progressing waves at oo.
The singularity of f(z) at the origin is clearly a logarithmic singularity
since f(z) behaves near the origin like
C C*dt
(4.3.28) /(*) = -*-" _+....
n J t
We see that a logarithmic singularity is appropriate at a source or
sink of energy when the motion is periodic in the time.
SIMPLE HARMONIC SURFACE PRESSURE 67
4.4. Periodic progressing waves against a vertical clift
With the aid of the complex velocity potential defined by (4.3.13)
we can discuss a problem which is different from the one treated in
the preceding section. The problem in question is that of the deter-
mination of two-dimensional progressing waves moving toward a
vertical cliff, as indicated in Figure 4.4.1. The cliff is the vertical
Fig. 4.4.1. Waves against a vertical cliff
plane containing the //-axis. As in the preceding section, we assume
also that a periodic pressure (cf. (4.3.1)) is applied over the segment
0 <j| x ^ a at the free surface. To solve the problem we need only
combine the standing waves given by (4.3.17) and (4.3.18) in such
a way as to obtain progressing waves which move inward from the
two infinities, and this can be done in the same way as in section 4.3.
The result will be again a wave symmetrical with respect to the
t/-axis, and hence one for which 0X = 0 along the j/-axis; thus such
a wave satisfies the boundary condition appropriate to the vertical
cliff. We would find for the velocity potential 0 the expression, valid
for x > 0:
2Pa / 1 \
(4.4.1 ) 0(x, ij; t) = sin maem^[sm (nix + at)] + 0 I — I
to » '
with 0(l/r) a function behaving like 1/r at oo but with a singularity
at (a, 0). In order to obtain a system of waves which are not reflected
back to oo by the vertical cliff it was necessary to employ a mechanism
—the oscillating pressure over the segment 0 ^ x ^ a on the free
68 WATER WAVES
surface— which absorbs the energy brought toward shore by the in
coming wave. However, the particular mechanism chosen here, i.e.
one involving an oscillatory pressure having the same frequency as
the wave, will not always serve the purpose since the amplitude A
of the surface elevation of the progressing wave at oo is given, from
(4.4.1) and (4.1.4), by
2P
(4.4.2) A = — sin ma.
68
Thus the ratio of the pressure amplitude P applied on the water
surface near shore to the amplitude of the wave at oo would obviously
become oo when sin ma = 0. In other words, such a mechanism would
achieve its purpose for waves whose wave length A at oo satisfies
the relation a — k A/2, with k any integer, only if infinite pressure
fluctuations at the shore occur. Presumably this should be interpreted
as meaning that for these wave lengths the mechanism at shore is
not capable of absorbing all of the incoming energy, or in other words,
some reflection back to oo would occur. This remark has a certain
practical aspect: a device to obtain power from waves coming toward
a shore based on the mechanism considered here would function
differently at different wave lengths.
It is of interest in the present connection to consider the same limit
case as was discussed at the end of the preceding section, in which
the segment of length a shrinks to zero while Pa remains finite. In
this case no exceptional wave lengths or frequencies occur. However,
the limit complex potential now has a logarithmic singularity at the
shore line, as we noticed in the preceding section, and the amplitude
of the surface would therefore also be infinite at the shore line. What
would really happen, of course, is that the waves would break along
the shore line if no reflection of wave energy back to oo occurred,
and the infinite amplitude obtained with our theory represents the
best approximation to such an essentially nonlinear phenomenon
that the linear theory can furnish.
This limit case represents the simplest special case of the problem
of progressing waves over uniformly sloping beaches which will be
treated more generally in the next chapter. However, the present
case has furnished one important insight: a singularity of the complex
velocity potential is to be expected at the shore line if the condition
at oo forbids reflection of the waves back to oo, and the singularity
should be at least logarithmic in character.
CHAPTER 5
Waves on Sloping Beaches and Past Obstacles
5.1. Introduction and summary
Perhaps the most striking— and perhaps also the most fascinating-
single occurrence among all water wave phenomena encountered in
nature is the breaking of ocean waves on a gently sloping beach.
The purpose of the present chapter is to analyze mathematically the
behavior of progressing waves over a uniformly sloping beach insofar
as that is possible within the accuracy of the linearized theory for
waves of small amplitude; that is, within the accuracy of the theory
with which we are concerned in the present Part II. Later, in Chapter
10.10, we shall discuss the breaking of waves from the point of view
of the nonlinear shallow water theory.
To begin with, it is well to recall the main features of what is often
observed on almost any ocean beach in not too stormy weather.
Some distance out from the shore line a train of nearly uniform
progressing waves exists having wave lengths of the order of say
fifty to several hundred feet. These waves can be considered as simple
sine or cosine waves of small amplitude. As the waves move toward
shore, the line of the wave crests and troughs becomes more and
more nearly parallel to the shore line (no matter whether this was
the case in deep water or not), and the distance between successive
wave crests shortens slightly. At the same time the height of the
waves increases somewhat and their shape deviates more and more
from that given by a sine or cosine— in fact the water in the vicinity
of the crests tends to steepen and in the troughs to flatten out until
finally the front of the wave becomes nearly vertical and eventually
the water curls over at the crest and the wave breaks. These ob-
servations are all clearly borne out in Figures 5.1.1, 5.1.2, which
are photographs, given to the author by Walter Munk of the Scripps
Institution of Oceanography, of waves on actual beaches. At the
same time, it should be stated here that the breaking of waves also
occurs in a manner different from this— a fact which will be discussed
69
5.1,1. Waves on, a
Fig. 5.1.2. Breaking and diffraction of waves at an inlet
WAVES ON SLOPING BEACHKS AND PAST OBSTACLES 71
in Chapter 10.10 on the basis of other photographs of actual waves
and a nonlinear treatment of the problem.
It is clear that the linear theory we apply here can not in principle
yield large departures from the sine or cosine form of the waves in
deep water, and still less can it yield the actual breaking phenomena:
obviously these are nonlinear in character. On the other hand the
linear theory is to be applied and should yield a good approximation
for deep water and for the intermediate zone between deep water and
the actual surf region. However, the fact that breakers do in general
occur in nature cannot by any means be neglected even in formulating
the problems in terms of the linear theory, for the following reasons.
Suppose we consider a train of progressing waves coming from deep
water in toward shore. As we know from Chapter 3, such a train of
waves is accompanied by a flow of energy in the direction toward the
shore. If we assume that there is little or no reflection of the waves
from the shore— which observations show to be largely the case for a
gently sloping beach* —it follows that there must exist some mecha-
nism which absorbs the incoming energy; and that mechanism is of
course the breaking of the waves which converts the incoming wave
energy partially into heat through turbulence and partially into the
energy of a different type of flow, i.e. the undertow. In terms of the
linear theory about the only expedient which we have at our disposal
to take account of such an effect in a rough general way is to permit
that the wave amplitude may become very large at the shore line, or,
in mathematical terms, that the velocity potential should be per-
mitted to have an appropriate singularity at the shore line. As we
have already hinted at the end of the preceding chapter, the ap-
propriate singularity for a two-dimensional motion seems to be
logarithmic, and hence the wave amplitude would be logarithmically
infinite at the shore line. Indeed, it turns out that no progressing
wave solutions without reflection from the shore line exist at all
within the framework of the linear theory unless a singularity at
least as strong as a logarithmic singularity is admitted at the shore
line.
The actual procedure works out as follows: Once the frequency
of the wave motion has been fixed, two different types of standing
* This fact is also used in laboratory experiments with water waves: the
experimental tanks are often equipped with a sloping ''beach" at one or more
of the ends in order to absorb the energy of the incoming waves through breaking,
and thus prevent reflection from the ends of the tank. This makes it possible to
perform successive experiments without long waits for the motions to subside.
72 WATER WAVES
waves are obtained, one of which has finite amplitude, the other
infinite amplitude, at the shore line. These two different types of
standing waves behave at oo like the simple standing wave solu-
tions for water of infinite depth obtained in Chapter 3; i.e. one of
them behaves like emv sin (mx + a) while the other behaves like
emy cos {mx -}- a); hence the two may be combined with appropriate
time factors to yield arbitrary simple harmonic progressing waves
at oo. If the amplitude at oo is prescribed, and also the condition
(cf. the last two sections of the preceding chapter) requiring that the
wave at oo be a progressing wave moving toward shore, then the
solution is uniquely determined; in particular, the strength of the
logarithmic singularity at the shore line is uniquely fixed once the
amplitude of the incoming wave is prescribed at oo.
The fact that progressing waves over uniformly sloping beaches
can be uniquely characterized in the simple way just stated is not
a thing which has been known for a long time, but represents rather
an insight gained in relatively recent years (cf. the author's paper
[S.18] of 1947 and the other references given there). The method
employed in the author's paper makes essential use of an idea due
to H. Lewy to obtain the actual solutions for the case of two-dimen-
sional waves over beaches sloping at the angles Tt/Vn, with n an
integer; H. Lewy [L.8] extended his method also to solve the problem
for slope angles (p/2n)n9 with p an odd integer and n any integer
such that p < 2n. For the special slope angles 7t/2n the progressing
wave solutions were obtained first by Miche [M.8] (unknown to the
author at the time because of lack of communications during World
War II), and somewhat later by Bondi [B.14], but without uniqueness
statements. Actually, the special standing wave solutions for these
same slope angles which are finite at the shore line had already
been obtained by Hanson [H.3].
All of these solutions for the slope angles eo == jr/2n, become more
complicated and cumbersome as n becomes larger, that is, as the
beach slope becomes smaller. In fact, the solutions consist of finite
sums of complex exponentials and exponential integrals, and the
number of the terms in these sums increases with n. Actual ocean
beaches usually slope rather gently, so that many of the interesting
cases are just those in which the slope angle is small— of the order
of a few degrees, say. It is therefore important to give at least an
approximate representation of the solution of the problem valid for
small angles eo independent of the integer n. Such a representation
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 73
has been given by Friedrichs [F.I 4], To derive it the exact solution
is first obtained for integer n in the form of a single complex integral,
which can in turn be treated by the saddle point method to yield
asymptotic solutions valid for large n, that is, for beaches with small
slopes. The resulting asymptotic representation turns out to be very
accurate. A comparison with the exact numerical solution for co = 6°
shows the asymptotic solution to be practically identical with the
exact solution all the way from infinity to within a distance of less
than a wave length from the shore line. Eckart [E.2, 3] has devised
an approximate theory which gives good results in both deep and
shallow water.
For slope angles which are rational multiples of a right angle of
the special form co = pji/2n with p any odd integer smaller than 2n,
the problem of progressing waves has been treated by Lewy, as was
mentioned above. Thus the theory is available for cases in which co
is greater than jr/2, so that the ''beach" becomes an overhanging
cliff. The solution for a special case of this kind, i.e. for co = 135°
or p = 3, n = 2, has been carried out numerically by E. Isaacson
[1.2]. It turns out that there is at least one interesting contrast with
the solutions for waves over beaches in which co < n/2. In the latter
case it has been found that as a progressing wave moves in toward
shore the amplitude first decreases to a value below the value at oo,
before it increases and becomes very large at the shore line. (This
fact has also often been verified experimentally in wave tanks).
The same thing holds for standing waves: at a certain distance from
shore there exists always a crest which is lower than the crests at oo.
In the case of the overhanging cliff with co = 135°, however, the
reverse is found to be true: the first maximum going outward from
the shore line is about 1 % higher than the height of the crests at oo.
Still another fact regarding the behavior of the solutions near the
shore line is interesting. In all cases there exists just one standing
wave solution which has a finite amplitude at the shore line; Lewy
[L.8] has shown that the ratio of the amplitude there to the am-
plitude at oo is given in terms of the angle co by the formula (n/2co)11*.
Thus for angles co less than n/2 the amplitude of the standing wave
with finite amplitude is greater on shore than it is at infinity (becoming
very large as co becomes small) while for angles co greater than jr/2
the amplitude on shore is less than it is at oo. Since the observations
indicate that the standing wave of finite amplitude is likely to be the
wave which actually occurs in nature for angles co greater than
74 WATER WAVES
about 40°, the above results can be used to give a rational explanation
for what might be called the "wine glass" effect: wine is much more
apt to spill over the edge of a glass with an edge which is flared out-
ward than from a glass with an edge turned over slightly toward the
inside of the glass.
A limit case of the problem of the overhanging cliff has a special
interest, namely the case in which co approaches the value n and the
problem becomes what might be called the "dock problem": the
water surface is free up to a certain point but from there on it is
covered by a rigid horizontal plane. The solutions given by Lewy
are so complicated as p and n become large that it seems hopeless
to consider the limit of his solutions as co -> n. Friedrichs and Lewy
[F.I 2] have, however, attacked and solved the dock problem directly
for two-dimensional waves. For three-dimensional waves in water of
constant finite depth the problem has been solved by Hcins [H.I 3]
(also see [H.12]).
It would be somewhat unsatisfying to have solutions for the sloping
beach problem only for slope angles which are rational multiples
of n: it is clear that this limitation is imposed by the methods used
to solve the problem and not by any inherent characteristics of the
problem itself. The two-dimensional problem has, in fact, been solved
for all slope angles by Isaacson [I.I]. Isaacson obtained an integral
representation of Lewy's solutions for the angles pn/2n analogous
to the representation obtained by Friedrichs for the angles n/2n,
and then observed that his representation depended only upon the
ratio of p to n and not on these quantities separately. Thus the
solutions for all angles are given by this representation. Peters [P.5]
has solved the same problem by an entirely different method, which
makes no use of solutions for the special slope angles pyi/2n.
The problem of two-dimensional progressing waves over sloping
beaches thus has been completely solved as far as the theory of
waves of small amplitude is concerned. Only one solution for three-
dimensional motion has been mentioned so far, i.e. the solution by
Heins for three-dimensional motion in the case of the dock problem.
For certain slope angles co = n/2n the method used by the author
[S.18] can be extended in such a way as to solve the problem of
three-dimensional waves on sloping beaches; in the paper cited the
solution is carried out for the case co = n/29 i.e. for the case of waves
approaching at an angle and breaking on a vertical cliff. Roseau
[R.9] has used the same method for the case co = jr/4. Subsequently
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 75
the problem of three-dimensional waves on sloping beaches has been
solved by Peters [P.6] and Roseau [R.9], who make use of a certain
functional equation derived from a representation of the solution
by a Laplace integral. In section 5.4. we shall give an account of
this method of attack. Roseau [R.9] has solved the problem of waves
in an ocean having different constant depths at the two different
infinities in the .r-direction which are connected by a bottom of
variable depth.
Before outlining the actual contents of the present chapter, it
may be well to summarize the conclusions which have been obtained
from studying numerical solutions of the problems being considered
here, which have been carried out (cf. [S.18]) for two-dimensional
waves for slope angles co = 185°, 90°, 45°, and 6°, and for three-
dimensional waves for the case o> = 90°. The results for the case of an
overhanging cliff with a) -- 135° have already been discussed earlier.
In the other three cases the most striking and important result is
the following: The wave lengths and amplitudes change very little
from their values at oo until points about a wave length from shore
have been reached. Closer inshore the amplitude becomes large, as
it must in accord with our theory. It is a curious fact (already men-
tioned earlier) that the amplitude of a progressing wave becomes
less (for r/j - 6° about 10 % less) at a point near shore than its value
at oo, although it becomes infinite as the shore is approached. This
effect has often been observed experimentally. This statement holds
for the three-dimensional waves against a vertical cliff (with an
amplitude decrease of about 2 %), as well as for the two-dimensional
cases.
The exact numerical solution for the case of a beach sloping at
6° is useful for the purpose of a comparison with the results obtained
from the linear shallow water theory (treated in Chapter (10.13) of
Part III) and from the asymptotic approximation to the exact theory
obtained by Friedrichs [F.I 4], The linear shallow water theory, as
its name indicates, can in principle not furnish a good approximation
to the waves on sloping beaches in the deep water portion since it
yields waves whose amplitude tends to zero at oo. For a beach sloping
at 6°, for example, it is found that the shallow water theory furnishes
a good approximation to the exact solution for a distance of two or
three wave lengths outward from the shore line if the wave length
is, say, about eight times the maximum depth of the water in this
range; but the amplitudes furnished by the shallow water theory
76 WATER WAVES
would be 50 to 60 percent too small at about 15 wave lengths away
from the shore line. One of the asymptotic approximations to the
exact theory given by Friedrichs yields a good approximation over
practically the whole range from the shore line to infinity (it is in-
accurate only very close to shore); this approximation, which even
yields the decrease in amplitude under the value at oo mentioned above,
is almost identical with one obtained by Rankine (cf. Miche [M.8,
p. 287]) which is based upon an argument using energy flux con-
siderations in connection with the assumption that the speed of the
energy flux can be computed at each point in water of slowly varying
depth by using the formula (cf. (3.3.9)) which is appropriate in water
having everywhere the depth at the point in question. Friedrichs thus
gives a mathematical justification for such a procedure on beaches
of small slope.
It has already been made clear that the discussion in this chapter
cannot yield information about the breaking of waves, which is an
essentially nonlinear phenomenon. However, it is possible to analyze
the breaking phenomena in certain cases and within certain limitations
by making use of the nonlinear shallow water theory, as we shall see
in Part III. For this purpose, one needs to know in advance the
motion at some point in shallow water, and this presumably could
be done by using the methods of the present chapter, combined
possibly with the methods provided by the linear shallow water
theory.
The material in the subsequent sections of this chapter is ordered
as follows. In section 5.2. the problem of two-dimensional progressing
waves over beaches sloping at the angles 7t/2n, n an integer, is discussed
following the method of Lewy [L.8] and the author [S.18]. In section
5.3 the problem of three-dimensional waves against a vertical cliff
is treated, also using the author's method. The reasons for including
these treatments in spite of the fact that they yield results that are
included in the more general treatments of Peters [P.6] and Roseau
[R.9] is that they are interesting in themselves as an example of
method, and also they can be applied to other problems, such as the
problem of plane barriers inclined at the angles n/2n (cf. F. John
[J.4]), which have not been treated by other methods. In section 5.4,
the general problem of three-dimensional waves on beaches sloping
at any angle is treated following essentially the ideas of Peters.
In section 5.5 the problem of diffraction of waves around a rigid
vertical wedge is treated; in case the wedge reduces to a plane the
WAVES ON SLOPING BEACHES AND PAST OBSTACLES
77
problem becomes the classical diffraction problem of Sommerfeld
[S.I 2] for the case of diffraction of plane waves in two dimensions
around a half-plane barrier. A new uniqueness theorem and a new and
elementary solution for the problem are given. Methods of analyzing
the solution are also discussed; photographs of the waves in such cases
and comparisons of theory and experiment are made.
Finally, in section 5.6 a brief survey of a variety of solved and
unsolved problems which might have been included in this chapter,
with references to the literature, is given. Included are brief references
to researches in oceanography, seismology, and to a selection of papers
dealing with simple harmonic waves by using mathematical methods
different from those employed otherwise in this chapter. In parti-
cular, a number of papers employing integral equations as a basic
mathematical tool are mentioned and occasion is taken to explain
the Wiener-Hopf technique of solving certain singular integral
equations.
5.2. Two-dimensional waves over beaches sloping at angles a)=n/2n
We consider first the problem of two-dimensional progressing
waves over a beach sloping at the angle co = nf2n with n an integer
Fig. 5.2.1. Sloping beach problem
(cf. Figure 5.2.1), in spite of the fact that the problem can be solved,
as was mentioned in the preceding section, by a method which is not
78 WATER WAVES
restricted to special angles (cf. Peters [P.6], and Roseau [R.9]).
The problem is solved here by a method which makes essential use of
the fact that the slope angle has the special values indicated because
the method has some interest in itself, and it yields representations
which have been evaluated numerically in certain cases. In addition,
the relevant uniqueness theorems are obtained in a very natural way.
We assume that the velocity potential 0(x9 y; t) is taken in the
form 0 = eiat<p(x, y). Hence <p(x* y) is a harmonic function in the
sector of angle co — n/2n. The free surface boundary condition then
takes the form
a2
(5.2.1) <py — — y = 0, for y = 0, x > 0,
S
as we have often seen (cf. (3.1.7)), while the condition at the bottom is
(5.2.2) ?? = 0.
on
It is useful to introduce the same dimensionlcss independent variables
as were used in the preceding chapter:
(5.2.3) xl = mx, y± = my, m — a*/g.
The function q>(x, y) obviously remains harmonic in these variables,
and conditions (5.2.1) and (5.2.2) become
(5.2.1)' <py - <p = 0, y - 0, x > 0,
(5.2.2)' cpn = 0,
after dropping subscripts.
The simple harmonic standing waves in water of infinite depth
everywhere are given by
cos (x + a)
sin (x + a)'
we write these down because we expect that they will represent the
behavior of the standing waves in our case at large distances from
the origin, that is, far away from the shore line.
The solution of the problem is obtained in terms of the complex
potential f(z) defined by
(5.2.5) /(*) = f(x + iy) = <p(x, y) + iX(x, y).
The function f(z) should, like q>9 be regular and analytic in the
{oo
SH
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 79
entire sector (including the boundaries,* except for the origin).
The boundary conditions (5.2.1)' and (5.2.2)' are given in terms of
(5.2.6) <py - <p = & Ij- -
= 31 e (if - /) = 0 for z real and positive,
(5.2.7) Vn = 3le - (f(z)) = 0fc (- iexp (- in/2n) /')
on
= 0 for z — r exp {— w/2n}9 r > 0.
The second condition results from
•** ~ (/(*)) - #* (- - ~ (/(*))) = #* {-'>*/'(*)}.
3fl 1 r dO )
We introduce the two following linear differential operators:
(5.2.8) Li(D) = - i exp {- inftn) D,
(5.2.9) L2n(D) = i/) - 1
with Z> meaning d/dz. The basic idea of the method invented by
H. Lcwy is to find additional linear operators, L2, L3, . . ., £2n-i
such that the operation L^ • L2 • . . . • L2n applied on /(s) yields a
function F(z) whose real part vanishes on both boundaries of our sector.
Once this has been done, the function F(z) can be continued analyti-
cally over the boundaries of the sector by successive reflections to
yield a single-valued function defined in the entire complex plane
except possibly the origin. It can then be shown (see [S.18]), essen-
tially by using Liouville's theorem, that the function F(z) is uniquely
determined within a constant multiplying factor by boundedness
conditions on the complex potential f(z) at oo together with the order
of the singularity admitted at the origin. After F(z) has been thus
determined, the complex potential f(z) is obtained as a solution of
the ordinary differential equation L^L2 . . . L2nf(z) = F(z). Of course,
it is necessary in the end to determine the arbitrary constants in the
general solution of this differential equation in such a way as to
satisfy all conditions of the problem, and this can in fact be done
explicitly. It turns out that the resulting solution behaves at oo like
* Far less stringent conditions at the boundaries could be prescribed, since
analytic continuations over the boundaries can easily be obtained explicitly in
the present ease.
80 WATER WAVES
the known solutions for waves in water having infinite depth every-
where and that it is uniquely determined by prescribing the amplitude
of the wave at oo together with the assumption that it should be,
say, an incoming wave.
We proceed to carry out this program, without however giving
all of the details (which can be found in the author's paper [S.18]).
To begin with, the ordinary differential equation for f(z) and the
operators Lt are given by
(5.2.10) L(D)f = L! • L2 • L8 • . . . - L2nf
nD - I)/
with the complex constants v.k defined by
(5.2.11 ) a* = e~in (L + 1} , fc = 1, 2, . . ., 2n.
One observes that L±(D) and L2n(D) coincide with the definitions
given in (5.2.8) and (5.2.9). It is, in fact, not difficult to verify that
(5.2.12) &eF(z) = 0
on both boundaries of the sector, by making use of the properties
of the numbers a* and of the fact that Ste L^D) and Ste L2n(D)f
vanish on the bottom and the free surface, respectively, by virtue
of the boundary conditions (5.2.7) and (5.2.6).
So far we have not prescribed conditions on f(z) at oo and at the
origin, and we now proceed to do so. At the origin we assume, in
accordance with the remarks made in section 5.1 and the discussion
in the last section of the preceding chapter, that f(z) has at most a
logarithmic singularity; we interpret this to mean that | dkf(z)/dzk \ <
Mk/\ z \k in a neighborhood of the origin for k = 1, 2, . . ., 2/i, with
Mk certain constants. At oo we require that <p = ffle f(z) together
with | dkf(z)/dzk | for k = 1, 2, . . ., 2n be uniformly bounded when
z -> oo in the sector. (These conditions could be weakened con-
siderably, but they are convenient and are satisfied by the solutions
we obtain. ) In other words, although we expect the solutions of our
problem to behave at oo in accordance with (5.2.4) it is not necessary
to prescribe the behavior at oo so precisely since the boundedness
conditions yield solutions having this property automatically. Once
these conditions on f(z) have been prescribed we see that the function
F(z) defined by (5.2.10) has the following properties: 1) | F(z) \ is
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 81
uniformly bounded in the sector, and 2) | F(z) \ = 0(1 jz2n) in the
neighborhood of the origin.
We have already observed that 3te F(z) = 0 on both boundaries
of the sector and that F(z) can therefore be continued as a single-
valued function into the whole plane, except the origin, by the
reflection process. Here we make decisive use of the assumption
that co, the angle of the sector, is n/2n with n an integer. Since the
boundedness properties of F(z) at oo and the origin are preserved
in the reflection process, it is clear from well-known results concerning
analytic functions that F(z) is an analytic function over the whole
plane having a pole of order at most 2n at the origin. Since in ad-
dition the real part of F(z) vanishes on all rays z = r exp{ikn/2n},
k — 1, 2, . . ., 4tt, it follows that F(z) is given uniquely by
(5.2.13) F(z) =
Z2n
with A2n an arbitrary real constant which may in particular have
the value zero. Thus the complex potential f(z) we seek satisfies the
differential equation
(5.2.14) faDfoJ) - 1) ... (a2n.1D)(a2nZ) - I)/ - ^.
Our problem is reduced to finding a solution f(z) of this differential
equation which satisfies all of the conditions imposed on f(z). From
the discussion of section 5.1 we expect to find two solutions f^z)
and f2(z) of our problem which behave differently at the origin and
at oo; at the origin, in particular, we expect to find one solution,
say fi(z), to be bounded and the other, /2(s), to have a logarithmic
singularity.
The regular solution f^z) is the solution of (5.2.14) which one
obtains by taking for the real constant A2n the value zero, while
f2(z) results for A2n ^ 0. In other words the solution of the non-
homogeneous equation contains the desired singularity at the origin.
One finds for f^z) the solution
in which the constants ck and ftk are the following complex numbers:
82 WATER WAVES
(5.2.16)
. n + l k\\ n 2n (k-l
= exp {ml --- -- 1} cot — cot — . . .cot-
*\ \ 4 2/J 2n 2n 2n
-l)n
-- ,
2n
A: = 2, 3, ...,n
= cn.
The constants ck are obtained by adjusting the arbitrary constants
in the solution of (5.2.14) so that the boundary conditions on f(z)
at the free surface and the bottom are satisfied; that such a result
can be achieved by choosing a finite number of constants is at first
sight rather startling, but it must be possible if it is true that a func-
tion f(z) having the postulated properties exists since such a function
must satisfy the differential equation (5.2.14). The calculation of the
constants ck is straightforward, but not entirely trivial. The function
f^z) is uniquely given by (5.2.15) within a real multiplying factor.
As | z | -> oo in the sector, all terms clearly die out exponentially
except the term for k = n, which is cn exp {— iz}9 since all (iks
except f}n have negative real parts. Even the term for k — n dies
out exponentially except along lines parallel to the real axis. (The
value of cn, by the way, is exp {— - in(n — l)/4} since the cotangents
in (5.2.16) cancel each other for k = n.) This term thus yields the
asymptotic behavior of fi(z):
The solution /2(*) of the nonhomogeneous equation (5.2.14) which
satisfies the boundary conditions is as follows:
(5.2.18) f2(z)
n r r **Pk **' n
= £ a* ^k —dt- me****
*-l L J«00 t J
for the case in which the real constant A2n is set equal to one. The
constants {ik are defined in (5.2.16); and the constants ak are defined by
(5.2.19) ak = ck/{(n - I)\Vn}>
that is, they are a fixed multiple (for given n) of the constants ck
defined in (5.2.16). The constants ak9 like the ck, are uniquely deter-
mined within a real multiplying factor. The path of integration for
all integrals in (5.2.18) is indicated in Figure 5.2.2. That the points
• izftk lie in the lower half of the complex plane (as indicated in the
figure) can be seen from our definition of the constants (ik and the
fact that z is restricted to the sector — jr/2n ^ arg 2^0.
WAVES ON SLOPING BEACHES AND PAST OBSTACLES
83
Fig. 5.2.2. Path of integration in f-plane
The behavior of f2(z) at oo of course depends on the behavior
of the functions in (5.2.18). It is not hard to show— for example,
by the procedure used in arriving at the result given by (4.3.15)
in the preceding chapter— that these functions behave asymptotically
as follows:
(5.2.20)
fi*(*keit (()( } 9
>*i*k\ —dt~\ VI/
I J I . /1\
%/too * [2m — o[~] ,
0,
0, Jm (iz0k) ^ 0.
Once this fact is established it is clear from (5.2.19) and (5.2.18)
that f2(z) behaves asymptotically as follows:
(5.2.21)
n
,
(n —
since the term for k = n dominates all others (cf. (5.2.20)) and
Ste(izpk) > 0 in this case. Comparison of (5.2.21) with (5.2.17) shows
that the real parts of f^z) and f2(z) would be 90° out of phase at oo.
That the derivatives of /2(z) behave asymptotically in the same
fashion as f2(z) itself is easily seen, since the only terms in the deriva-
tives of (5.2.18) of a type different from those in (5.2.18) itself are
of the form bk/zk9 k an integer ^ 1. Finally, it is clear that f2(z)
has a logarithmic singularity at the origin. Hence f^z) and f2(z)
satisfy all requirements. Just as in the 90° case (cf. the last section
84 WATER WAVES
of the preceding chapter) it is now clear that f(z) = b^^z) + b%f2(z),
with bi and 62 any real constants, yields all standing wave solutions
of our problem.
The relations (5.2.17) and (5.2.21 ) yield for the asymptotic behavior
of the real potential functions (p^ and q>2 the relations:
(5.2.22) p^x, y) = 9keji~ ~t - ~\T~r e" cos \x + —7— n\
\ 4 /
t
(n —
7T / *W _ 1 \
(5.2.23) <p2(x, y) = 3tef2~ - - -.7-7- ** sin (a? + — - — n\
(n — - IJiyn \ 4 /
when it is observed that cn = exp {— CT(/& — l)/4). It is now possible
to construct either standing wave or progressing wave solutions which
behave at oo like the known solutions for steady progressing waves
in water which is everywhere infinite in depth. In particular we
observe that it makes sense to speak of the wave length at oo in our
cases and that the relation between wave length and frequency
satisfies asymptotically the relation which holds everywhere in water
of infinite depth. For this, it is only necessary to reintroduce the
original space variables by replacing x and y by mx and my, with
m = o2lg (cf. (5.2.3)), and to take note of (5.2.22) and (5.2.23).
Finally, we write down a solution 0(x, y\ t) which behaves at oo
like ey cos (x + t + a), i.e. like a steady progressing wave moving
toward shore:
(5.2.24) 0(x, y; t) = A[cpi(x, y) cos (t + a) — <p2(x, y) sin (t + a)].
As our discussion shows, this solution is uniquely determined as soon
as the amplitude is prescribed at oo (i.e. as soon as A is fixed) since
<pi(x, y) and (p2(x, y) yield the only standing wave solutions of our
problem and they are determined also within a real factor. As we
have already stated in the preceding section, the progressing wave
solutions (5.2.24) have been determined numerically (cf. [S.18]) for
slope angles to = 90°, 45°, and 6°, with results whose general features
were already discussed in that section.
5.3. Three-dimensional waves against a vertical cliff
It is possible to treat some three-dimensional problems of waves
pver sloping beaches by a method similar to the method used in the
preceding section for two-dimensional waves, in spite of the fact
that it is now no longer possible to make use of the theory of analytic
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 85
functions of a complex variable. In this section we illustrate the
method by treating the problem of progressing waves in an infinite
ocean bounded on one side by a vertical cliff when the wave crests
at oo may make any angle with the shore line (cf. [S.18]).
We seek solutions 0(x, y, z; t) of V2(x>1/^)0 = 0 in the region
x *^Q, y ^ 0, — oo < 2 < oo with the j/-axis taken normal to the
undisturbed free surface of the water and the z-axis* taken along
the "shore", i.e. at the water line on the vertical cliffs = 0. Progressing
waves moving toward shore are to be found such that the wave
crests (or other curves of constant phase) at large distances from
shore tend to a straight line which makes an arbitrary angle with
the shore line. For this purpose we seek solutions of the form
(5.3.1) 0(x, y, z; t) = exp {i(at + kz + ^)}(p(x9 y)
that is, solutions in which periodic factors in both z and t are split off.
As in the preceding section, we introduce new variables and para-
meters through the relations xl — MX, yl ~- my, zl = mz, ^ =- k/m,
m = o2jg and cbtain for <p the differential equation
(5.3.2) Vf,^ - Arty =- 0
and the free surface condition
(5.3.3) (py — (p = 0 for y = 0,
after dropping the subscript 1 on all quantities. The condition at
the cliff is, of course,
(5.3.4) --^ =0 for x = 0.
ox
At the origin x ~~ 0, y — 0 (i.e. at the shore line on the cliff) we
require, as in former cases, that (p should be of the form
(5.3.5) (p — Ip log r + <p, r <C 1,
for sufficiently small values of r = (x2 + f/2)1/2f. with <p and ^ certain
bounded functions with bounded first and second derivatives in a
neighborhood of the origin. The functions q> and !p should be considered
at present as certain given functions; later on, they will be chosen
specifically.
For large values of r we wish to have 0(x, y, z; t) behave like
* It has already been pointed out that functions of a complex variable are
not used in this section, so that the reintroduction of the letter 2 to represent a
space coordinate should cause no confusion with the use of the letter z as a complex
variable in earlier sections.
86 WATER WAVES
ev exp {i(at + kz + cue + /?)} with &2 + oc2 = 1 but k and a otherwise
arbitrary constants, so that progressing waves tending to an arbitrary
plane wave at oo can be obtained. This requires that <p(x, y) should
behave at oo like ev exp {i(onx +^2)} because of (5.3.1). However,
it is no more necessary here than it was in our former cases to require
that (p should behave in this specific way at oo; it suffices in fact
to require that
(5.8.6) \<P\ +\<Px\ + \<Pxy\ <M for r> R0,
i.e. that (p and the two derivatives of <p occurring in (5.3.6) should
be uniformly bounded at oo. As we shall see, this requirement leads
to solutions of the desired type.
We proceed to solve the boundary value problem formulated in
equations (5.3.2) to (5.3.6). The procedure we follow is analogous
to that used in the two-dimensional cases in every respect. To begin
with, we observe that
/} / /) \
(5.3.7) — ( — _ i\<p = o for both x = 0 and y = 0,
dx \dy J
because of the special form of the linear operator on the left hand
side together with the fact that (5.3.3) and (5.3.4) are to be satisfied.
A function yj(x, y) is introduced by the relation
(5-8-8) " - T* (TV -
The essential point of our method is that the function \p is determined
uniquely within an arbitrary factor if our function 9?, having the
properties postulated, exists. Furthermore, y> can then be given explic-
itly without difficulty. The properties of \p are as follows.
1. \p satisfies the same differential equation as y>, i.e. equation
(5.3.2), as one sees from the definition (5.3.8) of \p.
2. y) is regular in the quadrant x > 0, y < 0 and vanishes, in view
of (5.3.7), on x = 0, y < 0 and y = 0, x > 0. Hence y can be
continued over the boundaries by the reflection process to yield a
continuous and single-valued function having continuous second
derivatives yxx and y>yv (as one can readily see since V2y> — k2\p = 0,
and \p = 0 on the boundaries) in the entire x, t/-planc with the ex-
ception of the origin. (Here we use the fact that our domain is a sector
of angle rc/2.)
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 87
3. At the origin, \p has a possible singularity which is of the form
(jp(x, t/)//*2, with 99 regular, as one can see from (5.3.5) and (5.3.8).
This statement clearly holds for the function y when it has been
extended by reflection to a full neighborhood of the origin.
4. The condition (5.3.6) on q> clearly yields for \p the condition
that y is uniformly bounded at oo after \p has been extended to the
whole plane.
Thus \p is a solution of V2y> — k\ = 0 in the entire plane which
is uniformly bounded at oo. At the origin ip — q>/r2 + <p with 9? and
^ certain regular functions (q> — 0 not excluded). In addition, \p = 0
on the entire x and y axes. We shall show, following Weinstein
[W.5],* that the function
(5.3.9) \p(x, y) = AiH(£ (ikr) sin 20, r = Vx2 + y2, 0 ^ k ^ 1
is the unique solution for \p in polar coordinates (r, 0) with A an
arbitrary real constant, and H^ the Hankel function of order two
which tends to zero as r -> oo. The function ip has real values for r
real. (The notation given in Jahnke-Emde, Tables of Functions, is
used.)
The solution y is obtained by Weinstein in the following way.
In polar coordinates (r, 6) the differential equation for ^ is
_
3r2 ^ r dr r* *
For any fixed value of r the function \p can be developed in the
following sine series:
n=l
since y vanishes for 0 = 0, rc/2, n, 3n/2; and the coefficients cn(r)
are given by
cn(r) = Cn r/2y(r, O) sin 2nO dd, n = 1, 2, . . .,
Jo
with Cn a normalizing factor. From this formula one finds by differen-
tiations with respect to r and use of the differential equation for ^
that cn(r) satisfies the equation
* In the author's paper the solution \p was obtained, but with a less general
uniqueness statement.
88 WATER WAVES
The right hand side of this equation vanishes, as can be seen by
integrating the first term twice by parts and making use of the
boundary conditions \p = 0 for 0 = 0 and 6 = n/2. Thus the functions
cn(r) are Bessel functions, as follows:
cn(r) = Awi*^H<£(ikr) + BZaI2n(kr),
with A2n and B2n arbitrary real constants. The functions I2n are
unbounded at oo; the Hankel functions H^ behave like r~2n for
r -> 0 and tend to zero exponentially at oo. It follows therefore that
the Fourier series for ip in our case reduces to the single term given
by (5.3.9) because of the boundedness assumptions on \p.
For our purposes it is of advantage to write the solution \p in the
following form:
32
(5.3.10) ip = Ai -- H (1) (ikr), r = Vx2 + y2,
oxoy
in which A is any real constant and H(l) is the Hankel function of
order zero which is bounded as r -> oo. It is readily verified that this
solution differs from that given by (5.3.9) only by a constant multi-
plier: for example, by using the well-known identities involving the
derivatives of Bessel functions of different orders.
Once y> is determined we may write (5.3.8) in the form
(5.3.11) — ( -- I\w = Ai - HM (ikr), A arbitrary.
ox \oy / oxoy
This means that our function <p, if it exists, must satisfy (5.3.11) as
well as (5.3.2). By integration of (5.3.11) it turns out that we are
able to determine q> explicitly without great difficulty on account
of the simple form of the left hand side of (5.3.11). This we proceed
to do.
Integration of both sides of (5.3.11) with respect to x leads to
(5.3.12) --iv = Ai- H(» (ikr) + g(y),
in which g(y) is an arbitrary function. But g(y) must satisfy (5.3.2),
since all other terms in (5.3.12) satisfy it. Hence d2g/dy2 — k2g = 0.
In addition g(0) = 0, since the other terms in (5.3.12) vanish for
,y=0 because of (5.3.3) and the fact that dH™ l9y=(ik)'^(ylr)dH^ /dr.
Finally, g(y) is bounded as y -> — oo because of condition (5.3.6)
and the fact that dH(V /By tends to zero as r -+ oo. The function
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 89
g(y) is therefore readily seen to be identically zero. By integration
of (5.3.12) we obtain (after setting g(y) = 0):
(5.8.13) <p = Aiey T e~* ~ [H™ (ikVx2 + t2)]dt + B(x)c*.
J+oo ut
The function B(x) and the real constant A are arbitrary. The integral
converges, since d(H(V )/dt dies out exponentially as t -*• oo.
We shall see that two solutions ^ (x, y) and <p2(x, y) satisfying all
conditions of our problem can be obtained from (5.3.13) by taking
A = 0 in one case and A ^ 0 in the other case, and that these
solutions will be 90° "out of phase" at oo. (This is exactly analogous
to the behavior of the solutions in our previous two-dimensional cases. )
Consider first the case A = 0. The function 9? given by (5.3.13)
satisfies (5.3.2) only if
(5.3.14) - + (1 - k2)B(x) = 0.
It is important to recall that k2 < 1. The boundary condition
(px = o for x = 0 requires that Bx(0) = 0. The condition q>y — <p = 0
for y = 0 is automatically satisfied because of (5.3.12) and g(y) = 0.
Hence B(x) = Al cos Vl — k2x, with Al arbitrary, and the solution
(5.3.15) ^(cr, y) = A^v cos Vl — k*x.
This leads to solutions 0l in the form of standing waves,* as follows:
(5.3.15)' 0^, j/, 2; t) = Af^e* cos Vl~^~k*x • (C°S
(sin
for k2 < 1. If k = 1, the solution 0X given by (5.3.15)' continues to
be valid.
As we have already stated, we obtain solutions <p%(x,y) from
(5.3.13) for A ^ 0 which behave for large x like sin Vl — k2 x rather
than like cos Vl — k2 x9 and with these two types of solutions
progressing waves approaching an arbitrary plane wave at oo can
be constructed by superposition.
We begin by showing that (5.3.2) is satisfied for all x > 0, y < 0
by (p as given in (5.3.13) with A ^ 0, provided only that B(x)
* The standing wave solutions of this type (but not of the type with a singu-
larity) for beaches sloping at angles n/2n were obtained by Hanson [H.3] by a
quite different method.
90 WATER WAVES
satisfies (5.3.14). Since x > 0, it is permissible to differentiate under
the integral sign in (5.3.13), even though t takes on the value zero
(since the upper limit y is negative). By differentiating we obtain
(5.3.16) V2 <p- top = Ai {«* f V« I fj^ + (1 -
(v \
-
Since ffW is a solution of (5.3.2) the operator (92/9a?2 — A;2) oc-
curring under the integral sign can be replaced by — 92/9j/2 and hence
the integral can be written in the form
We introduce the following notation
and obtain through two integrations by parts the result
e*
in which we have made use of the fact that the boundary terms arc
zero at the lower limit + oo, since all derivatives of H^ (ikr) tend
to zero as r -> + oo. The integral of interest to us is given obviously
by II — /3 and this in turn is given by
U)
dt
_
3 a*/2 a?/
i
by use of the above relations for Im. Hence the quantity in the first
bracket in (5.3.16) is identically zero—in other words the term
containing the integral on the right hand side of (5.3.13) is a solution
of (5.3.2). Hence y is a solution of (5.3.2) in the case A ^ 0 if
B(x) satisfies (5.3.14). Since (5.3.12) holds and g(y) = 0 it follows
that the free surface condition (5.3.3) is satisfied by <p in view of
the fact that dH(V (ikr)/dy = 0 for y = 0.
We have still to show that a solution B(x) of (5.3.14) can be chosen
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 91
so that (px — 0 for x — 0, and that g? has the desired behavior for
large values of r. Actually, these two things go hand in hand. An
integration by parts in (5.3.13) yields the following for q>:
5.3.17)
(p=Aie* f V«#£l) (ikVx*+t
J QC
provided that x > 0. It should be recalled that the upper limit y
of the integral is negative; thus the integrand has a singularity for
x = 0 since t = 0 is included in the interval of integration and
11^ (ikr) is singular for r — 0. We shall show that lim dyjdx = 0
x->0
provided that #,(0) = — 2 A ^ 0. We have, for x > 0 and y < 0:
(jW
— -- Aiev
f V* |- [HM (ikVx* + t*)]dt
J 00 OX
dx
The second term on the right hand side is readily seen to approach
zero as x -> 0 since this term can be written as the product of x and
a factor which is bounded for y < 0. For the same reason it is clear
that the only contribution furnished by the integral in the limit
as x - > 0 arises from a neighborhood of t — 0 since the factor x may
be taken outside of the integral sign. We therefore consider the limit
lim f V< A [i//£
a->oJe OX
lim < [i//> (ikVx* +72)]d*, e > 0.
The function ill^ (ikr) has the following development valid near
r = 0:
HI^ (ikr) = — - [JQ(ikr) log r + p(r)~\
o v J n L ov / g -r FV ;j
in which p(r) represents a convergent power series containing only
even powers of r, and J0 is the regular Bessel function with the
following development
J0(ikr) = i +.-L- +....
It follows that
— [iHM (ikr)] = - - |- JQ(ikr) +J'o(ikr) - logr +xg(r) 1
ox n\r2 r J
1
0 2
92 WATER WAVES
in which g(r) = (l/r)dp/dr is bounded as o?->0 since j/<0. The con-
tribution of our integral in the limit is therefore easily seen to be
given by
2 f~* i * 2 f~fi a?
lim — - £-'— - - d* = lim — - — --- - dt.
By introducing u = </# as new integration variable and passing to
the limit we may write
2 f-e x , 2 f-00 dtt
lim - - - 0 dt = - - - - 2.
It therefore follows that lim d(p/dx = 0 provided that
(5.3.18) ^(0) = - 2A.
The function B(o?) which satisfies this condition and the differential
equation (5.3.14) is
o A __
(5.3.19) B(x) - -- -- sin Vl -
Vl -k2
Since H^ (ikr) dies out exponentially as r -> oo it follows that the
solution <p given by (5.3.17) with B(x) defined by (5.3.19) behaves
at oo like e* sin [(1 - k*)lf*x].
A solution 9?2 of our problem which is out of phase with q)l (cf.
(5.3.15)) is therefore given by
(5.3.20) <pi(x, y) = A2 ie* [*
n^y ___ __ "I
(ikVx* + i/2) - —.^.•= sin Vl - k2 x ,
Vl- k2 J
with ^f2 an arbitrary real constant. Standing wave solutions 02 are
then given by
(5.3.20)' <Z>2
fcos kz]
( V* ii\ • ) \.
\x> y) 1 . , f •
[sin kz\
By taking appropriate values of k progressing waves tending at oo
to any arbitrary plane wave solution for water of infinite depth can
be obtained by forming proper linear combinations of solutions of the
type (5.3.15)' and (5.3.20)'. For a progressing wave traveling toward
shore, for example, we may write
WAVES ON SLOPING BEACHES AND PAST OBSTACLES
93
(5.3.21 ) 0(x, y, z; t) = A
['
-A\p
A/1 — k2 "1
x, y) cos kz -\ -- q>2(x, y) sin kz cos at
Jfcs "1
992(#, y)cos kz sin at
in which Al and A2 in (5.3.15) and (5.3.20) are both taken equal
to ^. The solution (5.3.21) behaves at oo like Ae* cos (Vl —k2x+kz+at)
as one can readily verify by making use of the asymptotic behavior
of q>i(x, y) and q>2(x, y)* and it is the only such solution since <p^
and 9?2 are uniquely determined.
The special case k = I has a certain interest. It corresponds to
waves which at oo have their crests at right angles to the shore.
One readily sees from (5.3.15) and (5.3.20) that as k -> 1 the pro-
gressing wave solution (5.3.21) tends to
(5.3.22) 0(y, z; t) = Aev cos (z + at)
that is, the progressing wave solution for this case is independent
of a?, is free of a singularity at the origin, and the curves of constant
phase are straight lines at right angles to the shore line— all properties
that are to be expected.
The progressing wave solution (5.3.21) was studied numerically
-2
KO
K2.)J
Fig. 5.8.1. Standing wave solution for a vertical cliff (with crests at an angle
of 30° to shore)
* We remark once more that the original space and time variables can be
reintroduced simply by replacing «, y, z by ma?, my, mz and k by kjm.
94
WATER WAVES
for k = 1/2, i.e. for the case in which the wave crests tend at oo
to a straight line inclined at 30° to the shore line. The function
<p%(%9 0) is plotted in Figure 5.3.1. With the aid of these values the
contours for 0 were calculated and are given in Figure 5.3.2. These
are also essentially contour lines for the free surface elevation 77,
in accordance with the formula rj = <Pt \ v.0 . The water surface
g
is shown between a pair of successive "nodes" of <Z>, that is, curves
for which 0 = 0. These curves go into the 2-axis (the shore line)
under zero angle, as do all other contour lines. This is seen at once
from their equation (cf. (5.3.21) with at = n/2)
(5.3.23) (pi(x, 0) cos kz +
#, 0) sin kz = const.
Since <p2 -> oo as # -> 0 while 9^ remains bounded, it is clear that
sin kz must approach zero as x -> 0 on any such curve. That the
contours are all tangent to the s-axis at the points z = 27W, n an
integer, is also readily seen. It is interesting to observe that the
12
10
8
6
4
2
0
•2
-4
^
-091
.-09
l'
m \
- 0.0
0 I 2 3 4 x
Fig. 5.3.2. Level lines for a wave approaching a vertical cliff at an angle
height of the wave crest is lower at some points near to the cliff than
it is at oo. It may be that the wave crest is a ridge with a number
of saddle points.
WAVES ON SLOPING BEACHES AND PAST OBSTACLES
95
It should be pointed out that we are no more able to decide in the
present case than we were in the two-dimensional cases whether
the waves are reflected back to infinity from the shore, and if so to
what extent. Our numerical solution was obtained on the assumption
that no reflection takes place, which is probably not well justified
for the case of a vertical cliff, but would be for a beach of small slope.
5.4. Waves on sloping beaches. General case
We discuss here the most general case of periodic waves on sloping
beaches which behave at oo like an arbitrary progressing wave— in
particular, a wave with crests at an arbitrary angle to the shore line—
and for a beach sloping at any angle. As has been mentioned earlier,
this problem was first solved by Peters and Roseau (cf. the remarks
in section 5.1).
We seek a harmonic function 0(x, y, z; t) of the form exp {i(at+kz)}
' 9>(#9 y) in the region indicated in cross section in Figure 5.4.1. At
Fig. 5.4.1. Sloping beach of arbitrary angle
oo the function 0 should behave like exp {i(at+kz+vix)} • exp {a2y/g}
with k and a arbitrary. The function <p(x, y) is not a harmonic func-
tion, but satisfies, as one readily sees, the differential equation
(5.4.1) 9^* + p™ - *V = °>
the free surface condition
a2
(5.4.2) cpy — m<p = 0, y = 0, m = — ,
96 WATER WAVES
and the condition at the bottom*
(5.4.3) <pn = 0, y = — x tan co.
By introducing (as we have done before) the new dimensionless
quantities xl = mx, yl = my, ax = a/w, A?! = k/m the conditions of
the problem for <p(x, y) can be put in the form
(5.4.1)! <p9X + <pyy — k*<p = 0, 0 ^ fc ^ 1,
(5.4.2)! <pv - 9? = 0, y = 0,
(5.4.3)! y>n = 0, y = — x tan o>
after dropping subscripts. Since we require 99(0?, t/) to behave like
ei&x emy — exp {IK^XI + t/J at oo, it follows from (5.4.1) that
— a2 + m2 — &2 = 0 and hence that a? + A;* = 1. Thus fc in (5.4.1)!
(really it is A^) is, as indicated, restricted to the range 0 ^ k ^ 1,
and this fact is of importance in what follows.** Finally, we know
from past experience that a singularity must be permitted at the
origin. (In the problems treated earlier in this chapter we have
prescribed only boundedness conditions at oo in a way which led to
a statement concerning the uniqueness of the solution. In the present
case we do not obtain a similar uniqueness theorem— in fact, as has
been pointed out by Ursell [U.7, 8], Stokes showed that there exist
motions different from the state of rest and which die out at oo.
For these motions, however, the quantity k is larger than unity).
We seek functions (p(x, y) satisfying the above conditions as the
real or the imaginary part of a complex function f(z, z) which is
analytic in each of the variables z = x + iy and its conjugate
z = x — iy. In the two-dimensional cases, it was sufficient to consider
analytic functions f(z) of one complex variable, but in the present
case it is necessary to take more general functions since q>(x9 y) is
not a harmonic function. Note that we now use the variable z in a
different sense than above, where it is one of the space variables;
no confusion should result since the space variable z hardly occurs
again in the discussion to follow. It is useful to calculate some of the
derivatives of such functions with respect to x and y; we have,
clearly:
* Peters [P.6] solves the problem when the condition (5.4.3) is replaced by
the more general mixed boundary condition <pn + a<p — 0, a = const.
** Involved in this remark is the assumption that the derivatives of the solution
btehave asymptotically the same as the derivatives of its asymptotic development;
but this is indeed the case, as we could verify on the basis of our final represen-
tation of the solution.
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 97
fx =/**x +/!«« =/, +/z,
fy = i(h - /i),
ABB + fyy = 4/«Z •
Consequently our differential equation (5.4.1)! can be replaced by
the differential equation
since the real or the imaginary part of any solution of it is clearly
a solution of (5.4.1)!.
Among the solutions of the last equation are the following simple
special solutions (obtained, for example, by separating the variables
in writing / = f^z) • /,(«)):
kz 2
f(z, z) = C& + T c , f = const.,
which, when £ — — i for example, is of the form
Kkz\ \ [ ( k2\ \
I + — I y ! • exp | — i 1 1 — — ) x > , and this is a solution of
4/ ] { \ 4/ j
(5.4.1 )l which has the proper behavior at oo, at least. (Actually,
when combined with the factor eikz9 with z once more the space
variable, the result is a harmonic function yielding a plane wave in
water of infinite depth and satisfying the free surface condition).
One can obtain a great many more solutions by multiplying the
above special solution by an analytic function g(£) and integrating
along a path P in the complex £-plane:
(5.4.4) /(*, 5) - ±-. { ez^ + T f • g(C)d£.
'2m J P
By appropriate choices of the analytic function g(£) and the path P,
we might hope to satisfy the boundary conditions and the condition
at oo. This does, indeed, turn out to be the case.
Still another way to motivate taking (5.4.4) as the starting point
of our investigation is the following. It would seem reasonable to
look for solutions of (5.4.1) in the form of the exponential functions
<p = exp {mx + ly}. However, since we wish to work with analytic
functions of complex variables it would also seem reasonable to express
x and y in terms of z = x + iy and z = x — iy, and this would lead to
(/% -4- z\ /Z _ Z\ \
m j ___*! I — U [ - 1 1 . In order that this function
\ 2 / \ 2 /)
98
WATER WAVES
(which is clearly analytic in z and z separately) be a solution of
(5.4.1)! we must require that m* -f- Z2 — k2 = 0, and this leads at
{k2 z\
£z + 1 , with £ an ar-
4 £j
bitrary parameter, as one can readily verify. The method used by
Peters [P.6] to arrive at a representation of the form (5.4.4) is better
motivated though perhaps more complicated, since he operates with
(5.4.1) in polar coordinates, applies the Laplace transform with
respect to the radius vector, transforms the resulting equation to
the Laplace equation, and eventually arrives at (5.4.4).
One of the paths of integration used later on is indicated in Figure
5.4.2. The essential properties of this parth are: it is symmetrical
with respect to the real axis, goes to infinity in the negative direction
£- plane
Fig. 5.4.2. The path P in the f -plane
of the real axis, enters the origin tangentially to the real axis and
from the left, and contains in the region lying to the left of it a
number of poles of g(£). (The path is assumed to enter the origin
in the manner indicated so that the term z/£ in the exponential
factor will not make the integral diverge). Our discussion will take
the following course: We shall assume g(£) to be defined in the £-plane
slit along the negative real axis (and also on occasion on a Riemann
surface obtained by continuing analytically over the slit). The choice
of the symmetrical path P leads to a functional equation for g(£)
through use of the boundary conditions (5.4.2)! and (5.4.3)!, and vice
versa a solution g(£) of the functional equation leads to a function
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 99
9>(#> y) = ^te /(*»*) satisfying the boundary conditions. (By the
symbols Jm and Ste we mean, of course, that the imaginary, or real,
part of what follows is to be taken. ) We seek a solution of the functional
equation which is defined and regular in the slit £-plane, with at
most poles in the left half-plane (including certain first order poles
on the negative imaginary axis), and dying out at oo like 1/f. Once
such a function has been found, the prescribed conditions at oo will
be seen to follow by deforming the path P over the poles into a path
on the two edges of the slit along the negative real axis: the residues
at the poles on the negative imaginary axis clearly would yield con-
tributions of the type
{k2 z \
— irz H ---- ~» } , r > 0, which are easily seen to be
4 (-tr)J
of the desired type at oo, while the remaining poles and the integral
over the deformed path will be found to yield contributions that tend
to zero when 3te z -> + oo.
We begin this program by expressing the boundary conditions
(5.4.2 )j and (5.4.3 )t in terms of the function f(z, z). The first of
these conditions will be satisfied if the following condition holds:
(5.4.2)i ^M(fz — /5 + if) = 0, z real, positive,
as one readily sees. The condition (5.4.3)1 will be satisfied if
n • grad <p = 0, with n the unit normal at the bottom surface, i.e.
if Ste {n • grad /} = 0, and the latter is given by
&t {(/« + /•) sin <*> + {(fz - fz) cos «>} = 0,
or finally, in the form
(5.4.3); Sm {fze-i(0 - /-<?ift>} - 0, z = re~ta>9 r > 0.
Upon making use of (5.4.4) in (5.4.2)( the result is
(5.4.5) Sm ~ f c*t + * • [C - £ + tl g(C)C - 0,
2m Jp I 4£ J
z real, positive,
while (5.4.3 )[ yields
(5.4.6) Jm —. f e* + ? • [V** ~ ^ e^ g(C)dC = 0,
2m Jp I 4C J
z = re-*", r > 0.
To satisfy the boundary condition (5.4.5) it is sufficient to require
that g(f ) satisfies the condition
100 WATER WAVES
r k2 "i
(5.4.7) «//»£__+! g(£) = 0, f real, positive.
The proof is as follows: If (5.4.7) holds, then the integrand G(z, z, £)
in (5.4.5) is real for real z and real positive f . Hence G takes on values
G, G at conjugate points £, £ which are themselves conjugate, by the
Schwarz reflection principle. Since the path P is symmetrical, as
shown in Figure 5.4.2, it follows that d£ takes on values at £, £ that
are negative conjugates. Thus the integral (1 /2m) \ G d£ is real when
z is real and (5.4.7) holds. In considering next (5.4.6) we first introduce
a new variable s = £e~ia) to obtain for z = re~i<0 the condition.
replacing (5.4.6):
ir _L — r k2~\
(5.4.8) Jm — e ^ ** • \* -- g(seia>) eiu> ds = 0, r real.
2ni J p/ L 4*J
Here P' is the path obtained by rotating P (and the slit in the
£-plane as well, of course) clockwise about the origin through the angle
a). If g behaves properly at oo, and if the rotation of P' can be
accomplished without passing over any poles of the integrand, we
may deform P' back to P and obtain
(5.4.8)' Jm—{ *r* + ~T • \s - —1 g(seim) ei(0 ds = 0, r real.
p L 4*J
By the same argument as before we now see that the condition
(5.4.6) will be satisfied provided that g(£) satisfies the condition
(5.4.9) Jmg^e™}?* = 0, f real, positive.
Thus if the function g(£) satisfies the conditions (5.4.7) and
(5.4.9), the function f(z9 z) constructed by its aid will satisfy the
boundary conditions. As we have already remarked, g(£) must
satisfy still other conditions —at oo, for example. In addition, we
know from earlier discussions in this and the preceding chapter that
it is necessary to find two solutions (p(x, y) and <pi(x9 y)of our problem
which are "out of phase at oo", in order that a linear combination
of them with appropriate time factors will lead to a solution having
the form of an arbitrary progressing wave at oo. In this connection
we observe that if the path Pl of integration (as shown in Figure
5.4.3) is taken instead of the path P (it differs from P only in reversal
of direction of the portion in the upper half-plane), and if we define
<pi(x, y) as the imaginary part of /1(^, z) instead of its real part:
WAVES ON SLOPING BEACHES AND PAST OBSTACLES
101
x9 y) = Jm G(z9 2, £)d£ =
(5.4.10)
with G the same integrand as before, then <pi(x, y), by the same
argument as above, will satisfy the boundary conditions provided
that the function g(£) also in this case satisfies the conditions (5.4.7)
- plane
Fig. 5.4.3. The path Pl in the f -plane
and (5.4.9). It seems reasonable to expect that the integral over Pl
will behave the same as the integral over P when @te z is large and
positive (since the poles in the lower half-plane alone determine this
behavior and the paths P and Pl differ only in the upper half-
plane) except that a factor i will appear, and hence that y> and <pl
will differ in phase at + oo (in the variable #, that is) by 90°. This
docs indeed turn out to be the case.
Thus to satisfy the boundary conditions for both types of standing
wave solutions we have only to find a function g(£) satisfying the con-
ditions (5.4.7) and (5.4.9) which behaves properly at oo— the last
condition being needed in order that the path of integration can be
rotated in the manner specified in deriving (5.4.8)'. To this end we
derive a functional equation for g(£) by making use of these con-
ditions. From (5.4.7) we have, clearly:
(5.4.11) U ~ ~ + i\ g(0 =
while from (5.4.9) we have
~ ^ - *') «(?)•
real> Positive,
102 WATER WAVES
(5.4.12) g(C)*-ia) = g(C*2ia>''a>, £ ^al, positive,
both by virtue of the reflection principle. Eliminating g(£) from the
two equations we obtain
(5.4.13)
This functional equation was derived for £ real and positive, but
since g(£) is analytic it is clear that it holds throughout the domain
of regularity of g(£); it ig the basic functional equation for g(£), a
solution of which will yield the solution of our problem. Of course,
this equation is only a necessary condition that must be fulfilled if
the boundary conditions are satisfied; later on we shall show that the
solution of it we choose also satisfies the condition (5.4.11 ), and hence
the condition (5.4.12) will also be satisfied since (5.4.13) holds,
We proceed now to find a solution g(£) of (5.4.13) which has all of
the desired properties needed to identify (5.4.4) and (5.4.10) as
functions furnishing the solution of our problem, as has been done by
Peters in the paper cited above.
We therefore proceed to treat the functional equation (5.4.13),
which is easily put in the form:
<,teZi<»n £2 I ,/£ __ *!
(5.4.14) g( *<" 4
{ )
(C - «>i)(£ - »>
with r1>2 - -- - - .
The numbers rlt2 are real since we know that k lies between 0 and 1.
It is convenient to set
(5.4.15) ,(f, - *g>
(C + Wi)(t + ^^•2)
in which h(£), like g(£), is defined in the £-plane slit along the negative
real axis. The function &(£) will have poles in the left half-plane, but
only the poles at f = — irl and f = — ir2 of g(C) will be found to
contribute a non- vanishing residue of f(z) for die z -> + oo, and
"this in turn would guarantee that f(z) behaves at GO on the free
surface like Ae~**. For A(£) we have from (5.4.14) and (5.4.15) the
equation
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 103
,.41ftv 2 ,n
(5.4.16) - = - = m(L).
MO (f-friXf-fr,) k '
This equation is solved by introducing the function /(£) by
(5.4.17) log A(C) = l(£),
and one finds at once that /(£) satisfies the difference equation
(5.4.18) ( l(e**»t) - /(C) - log m(C) = w(f).
In solving this equation we shall begin by producing a solution
free of singularities in the sector — co ^ arg £ ^ co, after which the
function /&(£) — which is (cf. (5.4.17)) then also regular in the same
sector— can be continued analytically into the whole £-plane slit
along the negative real axis (or, if desired, into a Riemann surface
having the origin as its only branch point) by using (5.4.16). As an
aid in solving equation (5.4.18) we set
\co = OCTT, 0 < a ^ 1,
(5.4.19)
V 1C = Ta, *(*") - L(r), Mr") = W(r)9
and operate now in a r-plane. One observes that the sector — co
< arg f < co in the C-plane corresponds to the r-plane slit along
its negative real axis. For L(r) one then finds at once from (5.4.18)
the equation
(5.4.20) L(re2ni) - L(r) = W(r).
Our object in putting the functional equation into this form
(following Peters) is that a solution is now readily found by making
use of the Cauchy integral formula. Let us assume for this purpose
that L(r) is an analytic function in the closed r-plane slit along its
negative real axis* (which would imply that l(£) is regular in
the sector — o> ^ arg f ^ co, as we see from (5.4.19)); in such a
case L(r) can be represented by the Cauchy integral formula:
(5.4.21) L(r)= -L
with C the path in the {-plane indicated by Figure 5.4.4. If we
suppose in addition that L(£) dies out at least as rapidly as, say,
l/{ at oo, it is clear that we can let the radius R of the circular
part of C tend to infinity, draw the path of integration into the two
edges of the slit and, in the limit, find for L(r) the representation
We shall actually produce such a regular solution shortly.
104 WATER WAVES
(-plane
Fig. 5.4.4. Path C in the {-plane
(5-4.22) L(r) =
2m J 2m
— >
in readily understandable notation. On making use of (5.4.20), and
drawing the two integrals together, it is readily seen that L(r) is
given by
(5.4.23) L(r) = — . -- d£.
2jwJ_aof — r
The path of integration is the negative real {-axis, and W(£) is to
be evaluated for arg{ — — n. Since W(£) has no singularities (of.
(5.4.18)), it follows that L(r) as given by (5.4.23) is indeed regular
in the slit r-plane. L(r) also has no singularity on the slit except
at the origin, where it has a logarithmic singularity. Since the
numerator in the integrand behaves like l/{a, a > 0, at oo (cf.
(5.4.19), (5.4.18), (5.4.16)), it is clear that the function L(r) dies
out like I /r at oo in the r-plane. This function therefore has all of
the properties postulated in deriving (5.4.23) from (5.4.21), and
hence is a solution of the difference equation (5.4.20) in the slit
plane including the lower edge of the slit.
A solution of (5.4.18) can now be written down through use of
(5.4.19); the result is:
with w(£a) to be evaluated for arg { = — n. This solution is valid
so far only for f in the sector — a> ^ arg £ ^ co, where it is regular,
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 105
as we know from the discussion above. However, it is necessary to
define the function A(£) = el(^ (cf. (5.4.17)) in the entire slit £-
plane, and this can be done by analytic continuation with the aid
of the functional equation (5.4.16). In the process of analytic con-
tinuation, starting with the original sector in which /(£), and hence
A(£), is free of singularities, one sees that the only singularities which
could occur in continuing into the upper half-plane, say, would arise
from the function on the right hand side of the equation (5.4.16).
The only singularities of this function occur obviously at £ = irl 2.
Consequently no singularity of A(£) appears in the analytic con-
tinuation into the upper half-plane, through widening of the sector
in which &(£) is defined, until the points £ = irl and £ = ir2 have
been covered, and one sees readily from (5.4.16) that the first such
singularities of /i(£) —poles of first order— -appear at the points
ri,2 exP {*(2co + 7T/2)}, the next at r± 2 exp (i(4co + ?r/2)}, etc.,
though some, or all, of these poles may not appear on the first sheet
of the slit f -plane, depending on the value of the angle co. The con-
tinuation into the lower half-plane is accomplished by writing
(5.4.16) in the equivalent form
(5.4.16)'
Again we see that poles will occur in the lower half-plane in the
course of the analytic continuation, this time at rl%2 exp {-—i(2co +JT/2}
ri,2 CXP {—i(4a) +yt/2)}9 etc. The situation is indicated in Figure
5.4.5; /i(£) lacks the singularities of g(£) at the points — ?>! and
— ?V2 (cf. (5.4.15)). Thus the function A(£) is defined in the slit £-
plane. (It can also be continued analytically over the slit which
permits a rotation of the path of integration.) We see that /«(£)
may have poles in the open left* half-plane, on two circles of radii
r1 and r2, but the poles closest to the imaginary axis are at the
angular distance 2co from it. There is also a simple pole of A(£) at
the origin, but g(£) (cf. (5.4.15)) is regular there.
The behavior of /i(£) at oo in the slit plane is now easily discussed:
In the original sector we know from (5.4.24) that /(£) dies out at oo
like l/£1/a. Hence A(£) = el(^ is bounded in the sector, and since the
right hand side of (5.4.16) is clearly bounded at oo it follows that
A(£) is bounded at oo in the £-pUme.
The function g(£) = £*(£)/(£ + fViHf + trt) (cf. (5.4.15)) can now
be seen to have all of the properties needed to identify the functions
106
WATER WAVES
{-plane
Fig. 5.4.5. The singularities of /i(£) and
/(*) in (5.4.4) and f^z) in (5.4.10) as functions whose real part and
imaginary part, respectively, yield the desired standing wave solutions
of our problem. To this end we write down the integrals
(5.4.25)
-i*L'
§) .
(C
over the paths indicated in Figure 5.4.6, where the direction is in-
dicated only on the part of the path in the lower half-plane, since
the paths P9 Pl differ only in the direction in which the remainder
of the path is traversed.
Since /&(£) is bounded at oo, and 0te z > 0, the integrals clearly
converge. One sees also that the paths of integration can be rotated
through the angle co about the origin without passing over singularities
of the integrand, and also without changing the value of the in-
tegrals. (This was needed in deriving (5.4.8)'.) We prove next that
g(£) satisfies the boundary condition (5.4.11). To begin with, we shall
show that J(£) as defined by (5.4.24) is real when £ is real and positive.
Once this is admitted to be true, then h(£) as given by (5.4.17) would
have the same property, and the function g(£) defined by (5.4.15)
would easily be seen to satisfy the condition (5.4.11). We have, then,
only to show that /(£) is real for real £, and this can be seen as follows:
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 107
- plone
Fig. 5.4.0. The paths P, P^ in the C-plane
111 (5.4.21) log w(£a) is to be evaluated for arg £ = — jr. But in this
case one sees easily from the equation (5.4.16) defining w(fa) (with
a = ro/jr, ef. (5.4.19)) that m(^a) has its values on the unit eircle
when arg £ — — rr, and hence its logarithm is pure imaginary on
the path of integration; it follows at once from (5.4.24) that /(£)
is real for £ real and positive. Since g(C) was constructed in such a
way as to satisfy (5.4.13) we know that (5.4.12) is satisfied auto-
matically. Thus our standing wave solutions satisfy the boundary
conditions.
Finally, we observe that the behavior of / and fl for &e z -> oo
is what was prescribed. To this end we deform the path of integration
into a path running along the two banks of the slitted negative
real axis. The residues at £ = — ir^ 2 contribute terms already
discussed above which furnish the desired behavior for 3te z -> + oo.
We have, then, only to make sure that the residues at the remaining
poles and the integrals along the slit make contributions which die
out as 3te z -> + oo. As for the residues at the poles at the points
108 WATER WAVES
£w = r1>2 exp {± i(2nco + rc/2)}, n = 1, 2, . . ., we observe that these
contributions are of the form Aez^n9 but since — co ^ arg z ^ 0 it
is clear that these contributions die out exponentially when z tends
to infinity in the sector — CD ^ arg 2 fg 0. As for the integrals along
the slit, they are known to die out like 1/2, as we have seen in similar
cases before, or as one can verify by integration by parts. Thus all
of the conditions imposed on f(z) and f^z) are seen to be satisfied.
We observe, however, that the integrals in (5.4.25) over the paths
P and Pl converge only if 9&e z ^ 0, and hence this representation
of our solution is valid only if the bottom slopes down at an angle
^ jr/2. For an overhanging cliff, when co > n/29 the solution can be
obtained by first swinging the path of integration clockwise through
90° (and swinging the slit also, of course); the resulting integrals
would then be valid for all z such that ^m z ^ 0 and the solutions
would hold for 0 < co ^ n.
It is perhaps of interest to bring the final formulas together for
the simplest special case, i.e. the dock problem for two-dimensional
motion (first solved by Friedrichs and Lewy [F.12]), in which the
y
Fig. 5.4.7. The dock problem
angle co has the value n, as indicated in Figure 5.4.7.* In this case
the function /(£) is given by
(5.4.26)
and the integral defines it at once in the entire slit £-plane. The
standing wave solutions <p(x, y) — 3le f(z) and y1 (#, y) =Sm f^z)
are determined through
* As was mentioned in section 5.1, the dock problem in water of uniform
finite depth and for the three-dimensional case was first solved by Heins [H.I 3]
with the aid of the Wiener-Hopf technique.
WAVES ON SLOPING BEACHES AND PAST OBSTACLES
109
(5.4.27) f(z) = ~ | e^
i r .
(5.4.28) /t(z)
with A(C) defined by
(5.4.29)
As was remarked above, the integrals in (5.4.27) and (5.4.28) con-
verge only if &e z ^ 0. However, the analytic continuation into the
entire lower half-plane is achieved simply by swinging the paths
P and P! into the positive imaginary axis (which can be done since
A(C) is bounded at oo), while staying on the Riemann surface of A(£),
and these integrals are then valid for all z in the lower half-plane.
Finally, it is also of interest to remark that the functions f(z)
and f^z) do not behave in the same way at the origin: the first is
bounded there, and the second is not, and this behavior holds not
only for the spcciaj case of the dock problem, but also in all cases
under consideration here.
5.5. Diffraction of waves around a vertical wedge.
Sommerfeld's diffraction problem
In this section we are primarily concerned with the problem of
determining the effect of a barrier in the form of a vertical rigid
wedge, as indicated in Fig. 5.5.1, on a plane simple harmonic wave
y
i
Fig. 5.5.1. Diffraction of a plane wave by a vertical wedge
coming from infinity. In this case it is convenient to make use of
cylindrical coordinates (r, 0, y). We seek a harmonic function
110 WATER WAVES
&(r> 0> y\ t) in the region 0 < 0 <v, — h <y <Q, i.e. in the region
exterior to the wedge of angle 2n — v and in water of finite depth h
when at rest. The problem is reduced to one in the two independent
variables (r, 6) by setting
(5.5.1) <2>(r, 0, yi t) = /(r, 0) cosh m(y + h)eiot.
The boundary conditions 0e = 0 for 0 = 0, 6 = v corresponding
to the rigid walls of the wedge yield for /(r, 0) the boundary con-
ditions
(5.5.2) f0 = 0, 0-0, Q=v.
The free surface condition g0y + 0tt = 0 at y =- 0 yields the con-
dition
(5.5.3) m tanh mh — o2/g,
while the condition 0y = 0 at the bottom t/ = — A is satisfied
automatically. Once any real value for the frequency a is prescribed,
equation (5.5.3) is used to determine the real constant m — which
will turn out to be the wave number of the waves at oo — , and we
note that (5.5.3) has exactly one real solution of in except for sign;
if the water is infinitely deep we have in — <72/#, and the function
cosh m(y + h) in (5.5.1) is replaced by emv.
Thus the function /(r, 0) is to be determined as a solution of the
reduced wave equation
(5.5.4) V^0)/ + w2/ = 0, 0 < r < oo, 0 < 0 < v,
subject to the boundary conditions (5.5.2). Actually, we shall in the
end carry out the solution in detail only for the case of a reflecting
rigid plane strip (i.e. for the special case v — 2jr), but it will be seen
that the same method would furnish the result for any wedge. It is
convenient to introduce a new independent variable p, replacing r,
by the equation r = Q/m; in this variable equation (5.5.4) has the
form
(5.5.5) V^0)/+/ = 0, 0<e<oo, 0<0<*,
and we assume this equation as the basis for the discussion to follow.
So far we have not formulated conditions at oo, except for the
vague statement that we want to consider the effect of our wedge-
Shaped barrier on an incoming plane wave from infinity. Of course,
we then expect a reflected wave from the barrier and also diffraction
effects from the sharp corner at the origin. In conformity with our
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 111
general practice we wish to formulate these conditions at oo in such
a way that the solution of the problem will be uniquely determined.
It has some point to consider the question of reasonable conditions
at oo which determine unique solutions of the reduced wave equation
under more general circumstances than those considered in the
physical problem formulated above. For general domains it is not
known how to formulate these conditions at oo, and, in fact, it would
seem to be a very difficult task to do so since such a formulation
would almost certainly require consideration of many special cases.
In one special case, however, the appropriate condition to be im-
posed at infinity has been known for a long time. This is the case
in which any reflecting or refracting obstacles lie in a bounded domain
of the plane, or, stated otherwise, it is the case in which a full neigh-
borhood of the point at infinity is made up entirely of the homogeneous
medium in which the waves propagate. In this case, the condition
at oo which determines the "secondary" waves uniquely is Sommer-
feld's radiation condition, which states, roughly speaking, that these
waves behave like a cylindrical outgoing progressing wave at oo.
However, if the reflecting or refracting obstacles extend to infinity,
the Soinmorfeld condition may not be appropriate at all. Consider,
for example, the case in which the entire tT-axis is a reflecting barrier
(i.e. the case v — JT), and the primary wave is an incoming plane wave
from infinity. It is clear on physical grounds that the secondary wave
will be the reflected plane wave, which certainly does not behave
at oo like a cylindrical wave since, for example, its amplitude does
not even tend to zero at oo. Another case is that of Sommerfeld's
classical di (Traction problem in which an incoming plane wave is
reflected from a barrier consisting of the positive half of the ^-axis.
In this case, the secondary wave has both a reflected component
which has a non-zero amplitude at oo, and a diffracted part which
dies out at oo. A uniqueness theorem has been derived by Peters and
Stoker [P.10] which includes these special cases; we proceed to give
this proof both for its own sake and also because it points the way
to a straightforward and elementary solution of the special problem
formulated above. In Chapters 6 and 7 a different way of looking
at the problem of determining appropriate radiation conditions is
proposed; it involves considering simple harmonic waves (Chapter 6),
or steady waves (Chapter 7) as limits when t -> oo in appropriately
formulated initial value problems which correspond to unsteady
motions.
112
WATER WAVES
The uniqueness theorem, which is general enough to include the
problem above, is formulated in the following way: We assume that
/(#, y) is a complex- valued solution* of the equation
(5.5.6) V2/ + / = 0
in a domain D with boundary jT, part of which may extend to in-
finity. It is supposed that any circle C in the x, j/-plane cuts out of
D a domain in which the application of Green's formula is legitimate,
and, in addition, that the boundary curve F outside a sufficiently
Fig. 5.5.2. The domain D
large circle consists of a single half-ray R going to oo (cf. Figure
5.5.2).** On the boundary F the condition
(5.5.7) fn = 0
is imposed, i.e. the normal derivative of / vanishes, corresponding
to a reflecting barrier. (We could also replace this condition on part,
or all, of F by the condition / = 0. ) We now write the solutions of
f in D which satisfy (5.5.6) in the form
(5.5.8) f = g+h,
in order to formulate the conditions at oo in a convenient way.
What we have in mind is to separate the solution into a part h
which satisfies a radiation condition and a part g which contains,
* It is natural to consider such complex solutions, since, for example, a plane
'wave is obtained by taking f(q, 6) = exp { IQ cos (0 -f- a)}.
** Our theorem also holds if D is the more general domain in which the ray K
is replaced at oo by a sector, and the uniqueness proof given below holds with
insignificant modifications for this case also.
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 118
roughly speaking, the prescribed incoming wave together with any
secondary reflected or refracted waves which also do not satisfy a
radiation condition. More precisely, we require h to satisfy the
following radiation condition:
(5.5.9) lim
dh , •,.
— +^h
= 0.
Here C is taken to be a circle, with its center O (cf. Figure 5.5.2)
on the ray R going to infinity, and with radius Q so large that all
obstacle curves except a part of R lie in its interior. This condition
clearly follows from the well-known Sommerfeld radiation condition,
which requires that
(5.5.9), lim p* (V + ih\ -> 0
^oo \OQ J
uniformly in 0, and, incidentally, this is a condition independent of
the particular point from which Q is measured; we observe that if h
behaves at oo like e~~iQ/\/Q, i.e. like an outgoing cylindrical wave,
then condition (5.5.9)! is satisfied. We shall make use of the radiation
condition in the form (5.5.9) in much the same way as F. John [J.5]
who used it to obtain uniqueness theorems for (5.5.6) in cases other
than those treated here; his methods were in turn modeled on those
of Rcllich [R.7].
The behavior of the function g at infinity is prescribed as follows:
(5.5.10) g~g! +ga at oo,
with gl a function that is once for all prescribed,* while g2 is a function
satisfying the same radiation condition as /i, i.e. the condition
(5.5.9). (That the behavior of g at oo is fixed only within an additive
function satisfying the radiation condition is natural and inevitable. )
Finally, we prescribe regularity conditions at re-entrant points
(such as A, B, C in Figure 5.5.2) of the boundary of D; these con-
ditions are that
(5.5.11) f(Q,0)~cv fQ(Q*0)~^> k<l>
Qk
with (g, 6) polar coordinates centered at the particular singular point,
arid cl and c2 constants. (These conditions on / mean physically that
the radial velocity component may be infinite at a corner, but not
How the function gl should be chosen is a matter for later discussion.
114 WATER WAVES
as strongly as it would be for a source or sink.) At other boundary
points we require continuity of / and its normal derivative.
We can now state our theorem as follows:
Uniqueness theorem: A solution / of (5.5.6) in D is uniquely determined
if it 1) satisfies the boundary condition (5.5.9); 2) admits of a decom-
position of the form (5.5.8) with h a function satisfying (5.5.9), g a
function behaving as prescribed by (5.5.10) at oo; and 3) satisfies the
regularity conditions at the boundary of D.
The proof of this theorem will be given shortly, but we proceed
to discuss its implications here. The theorem is at first sight somewhat
unsatisfactory since it involves the assumption that every solution
considered can be decomposed according to (5.5.8), with g(p, 6) a
certain function the behavior of which at oo, in so far as the leading
term gl (cf. (5.5.10)) in its asymptotic development is concerned, is
not given a priori. However, it is not difficult in some instances at
least to guess, on the basis of physical arguments, how the function
g1(p, 6) should be defined. For example, suppose the domain D
consisted of the exterior of bounded obstacles only. In such a case it
seems clear that gi(g, 6) should be defined as the function describing
the incoming wave— either as a plane wave from infinity, say, or a
wave originating from an oscillatory source -—since bounded obstacles
give rise only to reflected and diffracted components which die out
at oo and which could be expected to satisfy the radiation condition.
Even if there is a ray in the boundary that goes to oo (as was postulated
above), it still would seem appropriate to take gt(p, 0) as the function
describing the incoming wave, provided that it arises from an oscil-
latory point source,* since such a source would hardly lead to reflcted
or refracted secondary waves that would violate the radiation con-
dition. However, if the incoming wave is a plane wave and an
obstacle extends to oo, one expects an outgoing reflected wave to
occur which would in general not satisfy the radiation condition;
in this case the function g1(g, 0) should be taken as the sum of the
incoming plane wave and an outgoing reflected wave. For example,
one might consider the case in which the entire #-axis is a reflecting
barrier, as in Figure 5.5.3. In this case one would in an altogether
natural way define gi((), 6) as the sum of the incoming and of the
reflected wave as follows:
* The same statement would doubtlessly hold if the disturbance originated
in a bounded region, since this case could be treated by making use of a distribu-
tion of oscillatory point sources.
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 115
(5.5.12) g^g, 0) == eiQ cos (*-a) + eiQ "
with a the angle of incidence of the incoming plane wave. If we were
then to set / = gl + h (i.e. we set g = gl everywhere) and prescribe
that h should satisfy the radiation condition, it is clear that we would
Fig. 5.5.3. Infinite straight line barrier
have a unique solution by taking h = 0. Our uniqueness theorem
does not apply directly here since there are two infinite reflecting
rays going to oo, but it could be easily modified so that it would
apply to this case. Thus we have — for the first time, it seems— a
/
/
/
Fig. 5.5.4. Sommerf eld's diffraction problem
uniqueness theorem for this particularly simple problem of the
reflection of a plane wave by a rigid plane. A less trivial example is
the classical Sommerfeld diffraction problem — in effect, a special
116 WATER WAVES
ease of the problem with which our present discussion began — in
which a plane wave coming from infinity at angle a to the #-axis is
reflected and diffracted by a rigid half-plane barrier along the
positive ir-axis, as indicated in Figure 5.5.4. In this case it seems
plausible to define the function g1(g, 6) as follows:
!eie cos (0-oe) + eig cos (0+<^ 0 < 0 < JT — OC
**cos(e-a), rc-a<0<7r+a
0, n + a < 0 < 2n.
This function is, of course, discontinuous, corresponding to the
division of the plane into the regions in which a) the incoming wave
and its reflection from the barrier coexist, b) the region in which
only the wave transmitted past the edge of the barrier exists, and
c) the region in the shadow created by the barrier. Again we would
be inclined to takeg = gl (cf. (5.5.8) and (5.5.10)) and set/ = gl + h,
with h satisfying the radiation condition. Of course, the function
h(Q9 6) in (5.5.8) representing the diffracted wave would then also
be discontinuous in that case since the sum gl + h is everywhere
continuous. It will be seen that the well-known solution given by
Sommerfeld can be decomposed in this way and that h then satisfies
the radiation condition. Our uniqueness theorem will thus be shown
to be applicable in at least the important special case of particular
interest in this section.
One might hazard a guess regarding the right way to determine
the function g in all cases involving unbounded domains: it seems
highly plausible that it would always be correctly given by the
methods of geometrical optics. By this we mean, from the mathemati-
cal point of view, that g would be the lowest order term in an asymp-
totic expansion of the solution / with respect to the frequency of the
motion that is valid for large frequencies; the methods of geometrical
optics would thus be available for determining g. However, to prove
a theorem of such generality would seem to be a very difficult task
since it would probably require some sort of representation for the
solution of wave propagation problems when more or less arbitrary
domains and boundary data are prescribed.
Once having proved that the solution of Sommerfeld's diffraction
problem could be decomposed in the way indicated above into the
.sum of two discontinuous functions, one of which satisfies the
radiation condition, it was observed that the latter fact opens the
way to a new solution of the diffraction problem which is entirely
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 117
elementary, straightforward, and which can be written down in a few
lines. In other words, once the reluctance to work with discontinuous
functions is overcome, the solution of the problem is reduced to
something quite elementary by comparison with other methods of
solution. The problem was solved long ago by Sommerfeld [S.12],
and afterwards by many others, including Macdonald [M.I], Bateman
[B.5], Copson [C.4], Schwinger [S.5], and Karp [K.3].
We shall first prove the uniqueness theorem. Afterwards, the simple
solution of Sommerf eld's problem just referred to will be derived;
this solution is in the form of a Fourier series. The Fourier series
solution is next transformed to furnish a variety of solutions given
by integral representations, including the familiar representation
given by Sommerfeld. The new representations are particularly
convenient for the purpose of discussing a number of properties of
the solution. In particular, two such representations can be used to
show that the function h in the decomposition / = gl + h (cf.
(5.5.13)) satisfies the radiation condition, and that our solution /
satisfies the regularity conditions at the origin; thus the solution is
shown, by virtue of our uniqueness theorem, to be the only one which
behaves at oo like gl plus a function satisfying the radiation condition.
The Stokes' phenomenon encountered in crossing the lines of discon-
tinuity of the functions gl and h is also discussed.
The uniqueness theorem formulated above is proved in the following
way. Suppose there were two solutions / and /* (cf. (5.5.8)) with
/* given by
(5.5.U) He, °) = «*(e» 0) + **(e» *)•
We introduce the difference ^(^, 0) of these solutions:
(5.5.15) *(e,0)=/(M)-/*(M)
= 8(Q, 0) - g*(Q, 0) + h(e, 0) - A*(e, 0)
and observe that %(Q, 0) satisfies the radiation condition (5.5.9), by
virtue of the Schwarz inequality, since h, A*, and the difference
g __ g* ail satisfy it by hypothesis; thus we have
2
(5.5.16) lim
im
->oo Jc
, •
+«*
= 0.
The complex-valued function # is decomposed into its real and
imaginary parts:
(5.5.17) x = Xi + *X»
and Green's formula
118
WATER WAVES
(5.5.18)
is applied to #x and #2 in the domain D* indicated in Figure 5.5.5.
The domain JD* is bounded by a circle C so large as to include all of
the obstacles in its interior except R, by curves which exclude the
Fig. 5.5.5. The domain D*
prolongation of R into the interior of C, and by curves excluding
the other bounded obstacles. By (5.5.15), # is a solution of (5.5.6)
which also clearly satisfies the boundary condition (5.5.7). Since
V2Xi = — Xi and V2#2 = — #2> it follows that the integrand of the
left-hand side of (5.5.18) vanishes. Because of the regularity conditions
at boundary points we are permitted to deform the boundary curve
J1* into the obstacle curves, and it then follows from the boundary
condition (5.5.7) that
(5.5.19)
since the contributions at the obstacles all vanish. We now make
use of the easily verified identity
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 119
(5.5.20)
to deduce from (5.5.19) the condition
(5.5.21) Jc(l*nl2 + \X\2)d* ~ jc\Xn + ix\*ds = 0,
from which we obtain, in view of (5.5.16) and %n = dx/dg on C:
(5.5.22) lim f \%\*ds = lim f
->oo * c ->oo ^
From the boundary condition (5.5.7), as applied on R, we see that
%(Q, 0) can be continued as a periodic function of period 4jr in 6
on C; hence # can be represented for all sufficiently large values of
Q by the Fourier series
00 nO
(5.5.23) * = I^n/2(0)cos-,
0 *
with Ani2(())9 the Fourier coefficient, a certain linear combination
of the Bessel functions Jw/2(g) and Fn/2(g), since % is a solution of
(5.5.6). The Fourier coefficients are given by
1 f2* nO
(5.5.24) '<„/,(<?) = - X(e,0)Cos-?.dO,
n Jo 2
and consequently we have
(5.5.25) le*^
dO
It follows at once from (5.5.22) that the Fourier coefficients behave
for large Q as follows:
(5.5.26) lim e* Anf2(e) = 0.
0— >00
Since the Bessel functions «/n/2(o) and Ynj2((>) all behave at oo like
1/V(?» i* follows that all of the coefficients ^4n/2(g) must vanish.
Consequently % vanishes identically outside a sufficiently large circle,
hence it vanishes* throughout its domain of definition, and the
* This could be proved in standard fashion since # is now seen to satisfy
homogeneous boundary conditions in the domain D* of Fig. 5.5.5.
120 WATER WAVES
uniqueness theorem is proved. As was stated above, this uniqueness
proof is much like that of Rellich [R.7].
The above proof can be modified easily in such a way as to apply
to a region with a sector, rather than a ray, cut out at oo. The only
difference is that the Fourier series for #(g, 0) would then not have
the period 4jr and that the Bessel functions involved would not be
of index n/2.
Once it has become clear that the decomposition of the solution
into the sum of the two discontinuous functions g(p, 6) and A(g, 6)
defined earlier is a procedure that is really natural and suitable for
this problem, one is then led to the idea that such a decomposition
might be explicitly used in such a way as to determine the solution
of the original problem ((cf. (5.5.8)) in a direct and straightforward
way. Our next purpose is to carry out such a procedure.
We set (cf. Figure (5.5.4) and equation (5.5.13)):
(5.5.27) /(0,
with g(p, 0) defined by
eiQ cos (0-oo _|_ £*<? cos (0+oc)^ o < 0 < tt — a
(5.5.28)
-*)9 7t-QL<0
0, n + oc < 0 < 2n.
In addition we have
(5.5.29) f0 = 0 for 0 = 0, 0 == 2rc
and we also require
(5.5.30) lim <\/Q (^- + <*) = 0 uniformly in 0,
0-^00 we /
since the validity of the radiation condition in this strong form can
be verified in the end.
The desired solution will be found by developing /(p, 0) into a
Fourier series in 0 for fixed p, and determining the coefficients of the
series through use of the radiation condition in the strong form
(5.5.30); afterwards, the series can easily be summed to yield a
convenient integral representation of the solution. That such a
* process will be successful can be seen very easily: The Fourier series
for /(g, 0) will, on account of the boundary condition (5.5.29) and the
fact that / is a solution of the reduced wave equation, be of the form
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 121
]£cnJn/2(0) cos n0/2; the Fourier coefficients for g(g, 6) as defined
by (5.5.28) are given in terms of integrals of the form
•2*1 „&
f2
ln=\
Jo
since this function also satisfies the condition ge = 0 for 0 = 0, 2n.
Since «/n/2((?)> and its derivatives as well, behave like l/\/(? f°r large
values of Q and the integrals In — by a straightforward application
of the method of stationary phase, for example, — also behave in
this way, it is clear that the limit relation (5.5.30) when used in
connection with (5.5.27) will serve to determine the coefficients cn.
We proceed to carry out this program. The finite Fourier transform
/ of / is introduced by the formula
— /* 2*i Aj/5
(5.5.31) J(Q, n) = f(o, Q) cos — dO. .
Jo 2
Since /9 — 0 for 6 — 0, 2jt we find for fm the transform
(5.5.32) J^-^-^J,
4
by using two integrations by parts. Since / is a solution of
(5.5.33) Q*fee + efQ + foe + Q2f = 0,
it follows that / is a solution of
(5.5.34) &QQ + QfQ + g2 ~ 1 = 0,
and solutions of this equation are given by
(5.5.35) J(e.n) = anJnl2(Q).
(The Bcsscl functions Ynj2(()) of the second kind are not introduced
because they are singular at the origin; the solution we want is in
any case obtained without their use.)
The transform of g(p, 0) is, of course, given by
/* 2*i A|/5
(5.5.36) g(0, n) = g(e, 6) cos — d0,
Jo 2
and we have, in view of (5.5.8), the relation:
C2n nQ f2* n6
(5.5.37) h(Q, 6) cos — dO = anJnl2(Q) - g(g9 6) cos — dO
Jo 2 Jo 2
or, also:
122 WATER WAVES
(5.5.88) h(e, n) = anJnl2(g) - g(e, n).
We must next apply the operation yp (9/^e +0 to both sides
of (5.5.87) and then make the passage to the limit, with the result*
*
(5.5.89) 0 = lim ^/Q (|- + i) \anJnl2(e) - ( *g(e, 6) cos ^ d6\.
*-»«> \OQ / L JO 2 J
Since the functions t/n/2(g) behave asymptotically as follows:
2
and since these asymptotic expansions can be differentiated, we have
(5.5.40) IL + i) Jn/2(Q) ~ I/A
as an easy calculation shows. The behavior of the integral over g
can be found easily by the well-known method of stationary phase,
which (cf. Ch. 6.8) states that
f
Ja
Q <P
in which a is a simple zero of the derivative <p'(Q) in the range
a < 6 < ft, and the ambiguous sign in the exponential is to be taken
the same as the sign of ^"(a). In the present case, in which g(p, 0)
is defined by (5.5.28) one sees at once that there are three points
of stationary phase, i.e. at 6 = a, 0 = n — a, and 0 — n + a. Of
the three contributions only the first, i.e. the contribution at 6 = a,**
furnishes a non-vanishing contribution for Q -> oo when the operator
Vp(9/9p + i) is applied to it; one finds, in fact:
(5.5.41) (I + A rg(e, 6) cos ^ dQ ~ 2 V^ cos ™ X«3.
we /Jo 2 f e 2
Use of (5.5.40) and (5.5.41) in (5.5.39) furnishes, finally, the coef-
ficients an:
(5.5.42) an = 2n cos — e?*T.
2
The Fourier series for /(g, 0) is
* It should be noted that the argument goes through if the radiation condition
is used in the weak form.
** This has physical significance, since it says that only the incoming wave is
effective in determining the Fourier coefficients of the solution.
WAVES ON SLOPING BEACHES AND PAST OBSTACLES
123
/(<?> 0) = /(g, o) + /(e, n) cos
aw TT ~i 2
or, from (5.5.35) and (5.5.42),
cos -.
It is not difficult to sum the series for /(p, 0). If we use the represen-
tation (for a derivation, see Courant-Hilbert [C.10, p. 413])
(5.5.43)
,(e) = J! f *-lH) c— i dc,
2^i Jp
where P is the path in the complex £-plane shown in Figure 5.5.6,
we find that /(p, 0) can be expressed as the integral of the sum of
£-plane
Fig. 5.5.6. The path P in the f-plane
a constant plus four geometric series. The summation of the geometric
series and a little algebra yields, finally, a solution in the form
(5.5.44)
1 f e~Wt)
ftp* 0) = - -- '
Sjti J p £
3w
"^
37i
We proceed to analyze the solution (5.5.44) of our problem with
respect to its behavior at oo and the origin, and we will show that
124 WATER WAVES
the conditions needed for the validity of the uniqueness theorem
proved above are satisfied. We will also transform it into the solution
given by Sommerfeld (cf. equation (5.5.47)). Not all of the details
of these calculations will be given: they can be found in the paper
by Peters and Stoker [P.19].
If we set
(5.5.45) /(g, 0) - J(e, 0+ot) + /(e, 0-<x)
and define /(g, «) by
(5.5.46) /(g,x) = — e 2
L
- 1 i 3;i
we see on comparison with (5.5.44) that (5.5.45) defines /(p, 0)
correctly as the solution we wish to investigate.
Let us first obtain the solution in the form given by Sommerfeld.
To this end, the denominators of the fractions in square brackets
in (5.5.46) are rationalized, and the fractions combined to yield
Qf l\ r 3 3* 1 .3n -,
1 re~2\C~C/ C2 + 2£2*14 cos-f +2t£S/4 cos-y + 1 LJ>
J(e>*) = 7-; — i — Fi~r^ ; —
4m Jp C L C2 + 2zC cos x — 1 J
One can then verify readily that / satisfies the differential equation
dl If --(c--)r -- t'~ x . -? i- * 21
^008- . .... . 3
2 f -^:--^ -
— ^ 2V c/ (f 2 + {,
2m Jp
If we use (5.5.43) and the well-known trigonometric formulas for
^i/2(?)> J-HZ(Q) we see ^^a* tta last equation is equivalent to
dl I/IT ~ M
— 2— + (2i cos K\! = — ^ — ^4 ^-^ cos - .
^ v ; r^e 2
A solution /# of the non-homogeneous equation which in general
vanishes as Q -> oo is readily found:
in
»oo/,-<A(l4-cosx)
= —
Vtoi
v /»oo/,
coijf f
2Je
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 125
Thus for / the appropriate solution of the differential equation
must be
1=6 + •*#•
Introduction of a new variable of integration z in the expression for
IN through the relation 2A cos2 x/2 = z2, and use of the formula
_—
e~iz2 dz = \/7t e 4
/— -00
leads with no difficulty to the expression
in
— /• / — x
6* f V 2c cos - a
(5.5.47) I(Q, x) = el* cos x 2 e~iz dz
and this leads, in conjunction with (5.5.45), to Sommerfeld's solution.
To derive the asymptotic behavior of 7(p, x) as Q -> oo we proceed
a little differently. The fractions in the square brackets in (5.5.46)
are combined, and some algebraic manipulation is applied, to yield
-ur
(Qt ' ~ 4^ i
4^ JP_1_ (C,/2
A new integration variable A is now introduced by the equation
•V/2 2v/2
with the result
-_
A — V2 ^4 cos -
2
The path P (cf. Fig. 5.5.6) is transformed into the path L shown in
Fig. 5.5.7, as one readily can see. The path L leaves the circle of
radius \/2 centered at the origin on its left. This representation of
the function 7(g, K) is obviously a good deal simpler than that
furnished by (5.5.46), and it is quite advantageous in studying the
properties of the solution: for one thing, the plane waves at oo
can be obtained as the residues at the poles
3*< Q i a
(5.5.49) A± = V2 c * cos -=-•
126
WATER WAVES
In fact, if there is a pole in the upper half of the A-plane (and there
may or may not be, depending on the values of both 6 and a) one
X-plane
Fig. 5.5.7. The path L in the A-plane
has, after deformation of the path L over it and into the real axis,
for I(Q, K) the result:
., (*\ e"iQ
*w- —
(5.5.50) I(o, 9c) = e
— V2 e 4 cos ~
v 2
piq cos x
"* A — A/2 e 4 cos -
2
If x = 7i — the only case in which there is a singularity on the real
z-plane
Fig. 5.5.8. The path C in the z-plane
axis, i.e. a pole at A = 0 — we assume that the path of integration
is deformed near the origin into the upper half-plane. It is convenient
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 127
to introduce the variable z = gA2 in the integral, with the result
(5.5.51) /($,*)= '-~- " ' C~*dZ
with Ax = \/2 e'3*'4 cos x/2, and C the path of integration shown
in Fig. 5.5.8 For large values of g, and assuming Ax ^ 0, the square
bracket in the integrand can be developed in powers of (Z/Q )1/2, and
we may write
e-*dz I C e~zdz
It is clear that we may allow e -> 0 (see Fig. 5.5.8) and hence the
path C can be deformed into the two banks of the slit along the
real axis; each of the terms in the square brackets then can be
evaluated in terms of the /^-function (cf., for example, MacRobert
[M.2], p. 143). It is thus clear that for Ax ^ 0, the leading term in
the asymptotic expansion of the integral in (5.5.51) behaves like
l/\/P» *n fact, we have for /(g, «):
(5.5.52)
Since /'(^) = V^ and ^x — V^ ^ 4 cos ~ wc have
A
(5.5.53) 7(p, x) ~ e** cosx - -- _—- 4-- .
zVzno cos -
^ 2
Of course, this holds only if H lies in the range 0 ^ ^ < n since a pole
occurs in the upper half of the A-plane only when cos */2 is positive
(cf. (5.5.50)). We must also exclude the value K — n, corresponding
to Ax = 0. Since K = 0 T a, we see that the values 0 = n ± a cor-
respond to the exceptional value K — n, and these values of 0, in
turn, are those which yield the lines in the physical plane across
which our solution / behaves discontinuously at oo. (Cf. Fig. 5.5.4).
128 WATER WAVES
The discussion of the last paragraph yields the result, in conjunction
with equation (5.5.45) which defines our solution in terms of /(p, H):
(5.5.54 ) /(0, 6 ) ~ e* cos
2V2no cos - - 2V2no cos -~
* 2 * 2
for large Q and for angles 6 such that 0 < 0 < n — a, and a in the
range 0 < a < n\ only in this case are there poles of both of the
integrals in (5.5.45) in the upper halfplane.
The discussion of the behavior of the solution in other sectors of
the physical plane and along the exceptional lines can be carried out
in the same way as above. For example, if A+ = \/2 £i371/4 cos ( 0 +oc/2 ) = 0,
and hence 0 = n — a, it follows that there is only one pole in the
upper halfplane and our solution /(g, n— a) is given by (cf. 5.5.48),
(5.5.49)):
e-iQ f e-Q&
?_ M L_
27riJL A
+
or also (cf. (5.5.51) and Fig. 5.5.8) by:
e~zdz
f(g9 n-*) =
e-iQ /• e-,
— dz.
Anijc z
The asymptotic behavior of / can now be determined in the same
way as above; the result is
_• _if!
(5.5.55) f(Q, n - a)
2 / - Jt — 2oc
2 V2jrp cos -
* 2
the second term resulting from the pole at the origin.
In this fashion the behavior of /(g, 6) for large values of Q is deter-
mined, and leads to
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 129
(5.5.56)
/(ft
eiQ cos (<>-a)
tg cos (0-a)
, ft — a < 0 <ft + a
0 , ft + a < 0 < 2ft.
This is, of course, a verification of one of the conditions imposed
at oo. In addition, the next terms in the asymptotic expansion, of
order 1/V0» are a'so determined, as follows:
(5.5.56)'
in
in
0-1a 2V^?cos9-±-a
2 ^2
, 0<0<ft— a
cos
-_ _ O«
in
2 V/2ftg i
__ a
, 0— ft — oc
~ , ft-a<0<ft+a
0 + a
2V2ftg
t ft + 2a
i
2
/ — 0-a / — 0+a
2v 2ft/) cos 2V2ft0 cos
2 2
We observe that these expansions do not hold uniformly in 0 because
of zeros in the denominators for 0 = n i a, i.e. at the lines of
discontinuity of the function g(p, 0).
With the aid of the function g(p, 0) defined in (5.5.56) we define
a function />(p, 0) by the equation
(5.5.57) /({), 0) = g({>, 0) + h(Q, 0).
Thus h is of necessity a discontinuous function since / is continuous
while g has jump discontinuities along the lines 0 = ft ± a. The
function h is given by (cf. (5.5.50)):
130 WATER WAVES
(5.5.58)
with the proviso that the integrals should be deformed into the
upper half-plane in the vicinity of the origin in case either A_ or k+
vanishes: i.e., in case 0 has one of its two critical values n± a.
That the sum g + h really is our solution / is rather clear in the
light of our discussion above; and that it has jump discontinuities
which just compensate those of g in order to make / continuous can
also be easily verified. We shall not carry out the calculation here.
The function A(g, 0), in view of (5.5.56) and (5.5.57) thus yields what
might be called the "scattered" part of the wave.
In order to show that our solution / satisfies the conditions of the
uniqueness theorem proved above, we proceed to show that h as
defined by (5.5.58) satisfies the radiation condition (5.5.9); afterwards
we will prove that / behaves at the origin as prescribed by (5.5.11).
Our solution / will thus be proved to be unique.
That the function A(p, 6) defined by (5.5.58) satisfies the radiation
condition is not at all obvious: one sees, for example (cf. (5.5.50)'),
that its behavior at oo is far from being uniform in the angle 0.
In fact, the transformation of h to be introduced below is motivated
by the desire to obtain an estimate for the quantity | dh/dQ + ih l>
which figures in the radiation condition, that is independent of 0;
and this in turn means an estimate independent of the quantities
A- and A+ defined by (5.5.49). The function A(g, 0) can first of all
be put in the form
fc/ m -*~*fj r e~Q*d* L3 r
Me* 0 = — — M- • -„— -.j- + V
m ( Jo A2 - A2_ Jo
as one readily verifies. We proceed as follows: First we write
, 6) = - ~ h • ("V*1 fV^'J'dfatt + A, • fV^2 f Y<^>'
M L Jo Jo Jo Jo
then carry out the integrations with respect to A to obtain
*~i(? r, r **-** , 5 r00 ^**
-7--U-- ---- i+V -- i
2Vm L Jo +0* Jo ((? +02
(p +0* Jo ((?
From this representation of h we obtain
3h tr* r, f00 /-«dt , . f00 /'A I
+ tA = __ A_- ---- 3+^' ------ , •
5p 4V7ri L Jo +t)i Jo +«)iJ
WAVES ON SLOPING BEACHES AND PAST OBSTACLES
131
It is important to observe that A2_ and Al have pure imaginary values,
as we know from (5.5.49). We also observe that the exceptional
lines 0 — n ± a, which correspond to A^ =0, simply have the effect
that one of the two terms in the brackets in the last equation vanishes.
From the Schwarz inequality we have
\dh
\d~Q
+ ih
/* GO A t IA 2 I 1 I 2 ' /* '
f e^-ldi * X, r f
I I i__±_!_ I
Jo (o +t)l &n Jo
Consider the first term on the right; we find:
2 ! r°° X-erf< i2 2 f00
Jo (g + <)z Jo (
(It
e«
<«dt '2
Since the same estimate holds for the second term, it follows that
dh .
and this estimate holds for all values of 0, since it holds for the two
exceptional values 6 = n ± a as well as for all other values in the
range 0 fg 0 rgj 2n. We have thus verified that the radiation condition
holds — in fact, we have shown that it holds in the strong form.
We proceed to show that /(p, 6) behaves properly at the origin.
To this end, we start with the solution in the form (cf. (5.5.48) and
(5.5.45)):
with L the path of Fig. (5.5.7). The transformation A = \/'z is then
made, so that the new path of integration D is like the path C in
Fig. 5.5.8 except that the circular part now has a radius large enough
to include the singularities of the integrands in its interior. We may
take the radius of the circular part of D to have the value 1/p, since
we care only for small values of Q in the present consideration.The
transformation QZ = u then leads to the following formula for /(g, Q):
132
WATER WAVES
with Dj a path of the same type as D except that the circular part
of Dl is now the circle of unit radius. For small values of Q the integrals
in the last expression can be expressed in the form
u
. . . \du
/* ft — W
+ e*^± — ^w
From this expansion we see clearly that
6— du + ....
as
0.
This completes the verification of the conditions needed for the
application of the uniqueness theorem to our solution /.
It has been shown by Putnam and Arthur [P.18] (see also Carr
and Stelzriede [C.I]) that the theory of diffraction of water waves
Fig. 5.5.9. Waves behind a breakwater
around a vertical barrier is in good accord with the physical facts,
the accuracy being particularly high in the shadow created by the
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 133
breakwater. Figure 5.5.9 is a photograph (given to the author by
J. H. Carr of the Hydrodynamics Laboratory at the California
Institute of Technology) of a model of a breakwater which gives some
indication of the wave pattern which results.
5.6. Brief discussions of additional applications and of other methods
of solution
The object of the present section is to point out a few further
problems and methods of dealing with problems concerned, for the
most part, with simple harmonic waves of small amplitude.
The first group of problems to be mentioned belongs, generally
speaking, to the field of oceanography. For general treatments of
this subject the book of Sverdrup, Johnson, and Fleming [S.32]
should be consulted. One type of problem of this category which
was investigated vigorously during World War II is the problem of
wave refraction along a coast, or, in other terms, the problem of
the modification in the shape of the wave crests and in the amplitude
of ocean waves as they move from deep water into shallow water.
We have seen in the preceding sections that it is not entirely easy
to give exact solutions in terms of the theory of waves of small
amplitude even in relatively simple cases, such, for example, as the
case of a uniformly sloping bottom. As a consequence, approximate
methods modeled after those of geometrical optics were devised,
beginning with the work of Sverdrup and Munk [S.35]. Basically,
these methods boil down to the assumption that the local propagation
speed of a wave of given length is known at any point from the for-
mulas derived in Chapter 4 for water of constant depth once the
depth of the water at that point is known; and that Iluygens' principle,
or variants of it, can be used to locate wave fronts or to construct
the rays orthogonal to them. The errors resulting from such an as-
sumption should not be very great in practice since the depth varia-
tions are usually rather gradual. Various schemes of a graphical
character have been devised to exploit this idea, for example by
Johnson, O'Brien, and Isaacs [J.7], Arthur [A.3], Munk and Traylor
[M.16], Suquet [S.30], and Pierson [P.8]. Figure 5.6.1 is a refraction
diagram for waves passing over a shoal in an otherwise level bottom
in the form of a flat circular hump, and Fig. 5.6.2 is a picture of the
actual waves. Both figures were taken from a paper by Pierson [P.8],
and they refer to waves in an experimental tank. As one sees, there
134
WATER WAVES
Fig. 5.6.1. Theoretical wave crest-orthogonal pattern for waves passing over a
clock glass. No phase shift
is fair general agreement in the wave patterns— even good agreement
in detail over a good part of the area. However, near the center of
the figures there are considerable discrepancies, since the theoretical
diagram shows, for instance, a sharp point in one of the wave crests
WAVES ON SLOPING BEACHES AND PAST OBSTACLES
135
which is lacking in the photograph. The fact is that there is a caustic
in the rays constructed by geometrical optics (i.e. the orthogonals
to the wave crests have an envelope), and in the vicinity of such a
region the approximation by geometrical optics is not good. One of
Fig. 5.6.2. Shadowgraph for waves of moderate length passing over a clock glass
the interesting features of Fig. 5.6.2 is that the shoal in the bottom
results in wave crests which cross each other on the lee side of the
shoal, although the oncoming waves form a single train of plane
waves. Figure 5.6.3 is an aerial photograph (again taken from the
130
WATER WAVES
paper by Pierson) showing the same effect in the ocean at a point
off the coast of New Jersey; the arrow points to a region where there
would appear to be three wave trains intersecting, but all of them
appear to arise from a single train coming in from deep water.
Fig. 5.6.3. Aerial photograph at Great Egg Inlet, New Jersey
In the case of sufficiently shallow water Lowell [L.I 6] has studied
the conditions under which the approximation by geometrical optics
is valid; his starting point is the linear shallow water theory (for
which see Ch. 10.13) in which the propagation speed of waves is
Vgh, with h the depth of the water, and it is thus independent of
the wave length. Eckart [E.2, 3] has devised an approximate theory
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 187
which makes it possible to deal with waves in both deep and shallow
water, as well as in the transition region between the two.
There is an interesting application of the theory of water waves to
a problem in seismology which will be explained here even though it
is necessary to go somewhat beyond the linear theory on which this
part of the book is based. We have seen in Chapter 3 above that the
displacements, velocities, and pressure variations in a simple harmonic
standing wave die out exponentially in water of infinite depth.
However, it was pointed out by Miche [M. 8] that this is true only of
the first order terms in the development of the basic nonlinear theory
with respect to the wave amplitude; if the development is carried out
formally to second order it turns out that the pressure fluctuates
with an amplitude that does not die out with the depth, but depends
on the square of the amplitude. (For progressing waves, this is not
true. ) In addition, the second order pressure variation has a frequency
which is double the frequency of the linear standing wave. (It is not
hard to see in a general way how this latter nonlinear effect arises
mathematically. In the Bernoulli law, the nonlinear term of the form
0y + 0% would lead, through an iteration process starting with
0 = Aemy cos mx cos at, to terms involving cos2 at and thus to
harmonics with the double frequency.) It happens that seismic waves
in the earth of very small amplitudes — called microseisms — and of
periods of from 3 to 10 seconds are observed by sensitive seismo-
graphs; these waves seem unlikely to be the result of earthquakes or
local causes; rather, a close connection between microseisms and dis-
turbed weather conditions over the ocean was noticed. However, since
it was thought that surface waves in the ocean lead to pressure
variations which die out so rapidly in the depth that they could not
be expected to generate observable waves in the earth, it was thought
unlikely that storms at sea could be a cause for microseisms. The result
of Miche stated above was invoked by Longuet-Higgins and Ursell
[L. 14] in 1948 to revive the idea that storms at sea can be the origin
of microseisms. (See also the paper of 1950 by Longuet-Higgins
[L.13].) In addition, Bernard [B.8] had collected evidence in 1941
indicating that the frequency of microseisms near Casablanca was
just double that of sea waves reaching the coast nearby; the same
ratio of frequencies was noticed by Deacon [D. 6] with respect to
microseisms recorded at Kew and waves recorded on the north coast
of Cornwall. Further confirmation of the correlation between sea
waves and microseisms is given in the paper of Darby shire [D. 4].
138 WATER WAVES
A reasonable explanation for the origin of microseisms thus seems to be
available. Of course, this explanation presupposes that standing
waves are generated, but Longuet-Higgins has shown that the needed
effects are present any time that two trains of progressing waves
moving in opposite directions are superimposed, and it is not hard to
imagine that such things would occur in a storm area — for example,
through the superposition of waves generated in different portions of
a given storm area. It might be added that Cooper and Longuet-
Higgins [C.3] have carried out experiments which confirm quantita-
tively the validity of the Miche theory of nonlinear standing waves. It
is perhaps also of interest to refer to a paper by Danel [D.2] in which
standing waves of large amplitude with sharp crests are discussed.
In Chapter 6 some references will be made to interesting studies
concerning the location of storms at sea as determined by observa-
tions on shore of the long waves which travel at relatively high speeds
outward from the storm area (cf. the paper by Deacon [D.6]).
The general problem of predicting the character of wave conditions
along a given shore is, of course, interesting for a variety of reasons,
including military reasons (see Bates [B.6], for example). Methods
for the forecasting of waves and swell, and of breakers and surf are
treated in two pamphlets [U.I, 2] issued by the U.S. Hydrographic
Office.
A necessary preliminary to forecasting studies, in general, is an
investigation of ways and means of recording, analyzing, and repre-
senting mathematically the surface of the ocean as it actually occurs
in nature. Among those who have studied such questions we mention
here Seiwell [S.9, 10] and Pierson [P.IO'J. The latter author concerns
himself particularly with the problem of obtaining mathematical re-
presentations of the sea surface which are on the one hand sufficiently
accurate, and on the other hand not so complicated as to be practically
unusable. The surface of the open sea is, in fact, usually extraordinar-
ily complicated. Figure 5.0.4 is a photograph of the sea (taken from
the paper by Pierson) which bears this out. Pierson first tries re-
presentations employing the Fourier integral and comes to the con-
clusion that such representations would be so awkward as to preclude
their use. (In Chapter 6 we shall have an opportunity to see that it is
indeed not easy to discuss the results of such representations even for
motions generated in the simplest conceivable fashion— by applying
an impulse at a point of the surface when the water is initially at rest,
for example.) Pierson then goes on to advocate a statistical approach
WAVES ON SLOPING BEACHES AND PAST OBSTACLES
139
to the problem in which various of the important parameters are
assumed to be distributed according to a Gaussian law. These deve-
lopments are far too extensive for inclusion in this book -—besides, the
Fig. 5.6.4. Surface waves on the open sea
author is, by temperament, more interested in deterministic theories
in mechanics than in those employing arguments from probability
and statistics, while knowing at the same time that such methods are
very often the best and most appropriate for dealing with the com-
plex problems which arise concretely in practice. It would, however,
seem to the author to be likely that any mathematical representations
of the surface of the sea— whether by the Fourier integral or any other
integrals— would of necessity be complex and cumbersome in propor-
tion to the complexity of that surface and the degree to which details
are desired.
Before leaving this subject, it is of interest to examine another pho-
tograph of waves given by Pierson [P.10], and shown in Fig. 5.6.5.
Near the right hand edge of the picture the wave crests of the pre-
dominant system are turned at about 45° to the coast line, and they
are broken rather than continuous; such wave systems are said to be
short-crested. About half-way toward shore it is seen that these
waves have arranged themselves more nearly parallel to the coast
(indicating, of course, that the water has become shallower) and at
the same time the crests are longer and less broken in appearance,
140
WATER WAVES
Fig. 5.6.5. Aerial photograph over Oracoke
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 141
though no single one of them can be identified for any great distance.
Near the shore, the wave crests are relatively long and nearly parallel
to it. On the photograph a second train of waves having a shorter
wave length and smaller amplitude can be detected; these waves are
traveling almost at right angles to the shore (they are probably caused
by a breeze blowing along the shore) and they are practically not
diffracted. Each of the two wave trains appears to move as though
the other were not present: the case of a linear superposition would
thus seem to be realized here. One observes also that there is a shoal,
as evidenced by the crossed wave trains and the white-water due to
breaking over the shoals.
We pass next to a brief discussion of a few problems in which our
emphasis is on the methods of solution, which are different from those
employed in the preceding chapters of Part II. The first such problem
to be discussed employs what is called the Wicner-Hopf method of
solving certain types of boundary problems by means of an ingenious,
though somewhat complicated, procedure which utilizes an integral
equation of a special form. This method has been used, as was men-
tioned in the introduction to this chapter, by Heins [11.12, 13] and by
Keller and Weitz [K.9] to solve the dock problem and other problems
having a similar character with respect to the geometry of the domains
in which the solution is sought. However, it is simpler to explain the
underlying ideas of the method by treating a different problem, i.e.
the problem of diffraction of waves around a vertical half-plane — in
other words, Sommerfeld's diffraction problem, which was treated by
a different method in the preceding section. We outline the method,
following the presentation of Karp [K.3]. The mathematical formula-
tion of the problem is as follows. A solution q>(a\ y) of the reduced
wave equation
(5.6.1) VV + &V = 0
is to be found subject to the boundary condition
(5.6.2) <py = 0 for y = 0, x > 0
and regular in the domain excluding this ray (cf. Fig. 5.6.6). In addi-
tion, a solution in the form
(5.6.3) <P=<PQ+Vi
with (jpQ defined by
(5.6.4) <p0 = elk<* cos e° + v sin V, 0 < 00 < 27r,
142
WATER WAVES
and with <p± prescribed to die out at oo is wanted. In other words, a
plane wave comes from infinity in a direction determined by the angle
00, and the scattered wave caused by the presence of the screen, and
Fig. 5.6.6. Diffraction around a screen
given by <pv is to be found. It is a peculiarity of the Wienor-IIopf
method— not only in the present problem but in other applications to
diffraction problems as well— that the constant k is assumed to be a
complex number (rather than a real number, as in the preceding
section ) given, say, by k = k± + ik^ with &2 small and positive. With
this stipulation it is possible to dispense with conditions on (p± of the
radiation type at oo, and to replace them by boundedness conditions.
We employ a Green's function in order to obtain a representation
of the solution in the form of an integral equation of the type to which
the Wiener-Hopf technique applies. In the present case the Green's
function G(x9 y\ #0, y0) is defined as that solution of (5.6.1) in the
whole plane which has a logarithmic singularity at the point (#0, yQ)
and dies out at oo (here the fact that k is complex plays a role). This
function is well-known; it is, in fact, the Hankel function
of the first kind:
(5.6.5) G(x, y;x0, y0) =
4
(k[(x - #0)2 + (y -
The next step is to apply Green's formula to the functions 9? and G in
the domain bounded by the circle C2 and the curves marked Cx in
Fig. 5.6.6. Because of the fact that G is symmetric, has a logarithmic
singularity, and that 99 and G both satisfy (5.6.1), it follows by argu-
ments that proceed exactly as in potential theory in similar cases that
?, y) can be represented in the form
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 143
(5.6.6) <p(x, y)
f r i dG
= fo>] 5-
Jo d?/o
+ exp {ik(x cos 00 + y sin 00)}
when the radius of the circle C2 is allowed to tend to oo, and the boun-
dary condition (5.6.2), the regularity conditions, and conditions at oo
are used. (A mild singularity at the edge x = 0, y = 0 of the screen
must also be permitted.) The symbol [9] under the integral sign re-
presents the jump in <p across the screen, which is of course not known
in advance. The object of the Wiener-Hopf technique is to determine
\y] by using the integral equation (5.6.6); once this is done (5.6.6)
yields the solution (p(x, y). The first step in this direction is to differen-
tiate both sides of (5.6.6) with respect to y, then set y --= 0 and
confine attention to positive values of x\ in view of the boundary
condition (5.6.2) we obtain in this way the integral equation
/•»
(5.6.7) 0 :- ik sin 0QeikxrQS°o + \(p(.rQ)]K(x — x^dx^ x > 0.
Jo
The kernel K(x — #0) of the integral equation is given by
d2G
(5.6.8) ^-*o)- v,-
oyoy0 „ „„,»<)
Equation (5.6.7) is a typical example of an integral equation solvable
by the Wiener-Hopf technique; its earmarks are that the kernel is a
function of (x x0) and the range of integration is the positive real
axis. ,
The starting point of the method is the observation that the integral
in (5.6.7) is strongly reminiscent of the convolution type of integral
in the theory of the Fourier transform. In fact, if the limits of integra-
tion in (5.6.7) were from — oo to + oo and the equation were valid
for all values of x, it could be solved at once by making use of the
convolution theorem. This theorem states that if
/(<*) = f °° fM exP {— ioucQ}dccQ and K(cn) = f °° K(x0) exp {—
J —00 J —00
— i.e. if / and AT are the Fourier transforms of / and k (cf. Chapter 6) —
then /(a)^(a) = f* i(xQ)K(x — x^dx^ in other words, the transform
J — -OC
of the integral on the right is the product of the Fourier transforms
of the function f(xQ) and K(x0) (cf. Sneddon [S.ll], p. 24). Conse-
quently if (5.6.7) held in the wider domain indicated, it could be
used to yield
144 WATER WAVES
0 - *(a) + [<p(a)]tf (a),
with /t(a) the transform of the nonhomogeneous term in the integral
equation. This relation in turn defines the transform [<p(a)j of
[(p(x0)] since /&(a) and K(<x.) are the transforms of known functions,
and hence [<P(MQ)] itself. We are, of course, not in a position to proceed
at once in this fashion; but the idea of the Wiener- Hopf method is to
extend the definitions of the functions involved in such a way that
one can do so. To this end the following definitions are made
g(x) = 0, x > 0; /(a?0) - \<p], XQ > 0
(5.6.9) - h(x) = 0, x < 0; f(x0) = 0, XQ < 0
h(x) = ik sin 6Qeikx cos \ x > 0.
Equation (5.6.7) can now be replaced by the equivalent equation
(5.6.10) g(x) = h(x) + f * f(x0)K(x — x0)dKQ, — oo < j < oo.
J— 00
Here g(x) is unknown for x < 0 and f(xQ)— the function we seek- is,
of course, unknown for x0 > 0; thus we have only one equation for
two unknown functions. Nevertheless, both functions can be
determined by making use of complex variable methods applied to
the Fourier transform of (5.6.10); we proceed to outline the method.
We have, to begin with, from (5.6.10):
(5.6.11) g(a)-Ma) + /(a)jT(a),
with A(a) and /?(a) known functions given by
(5.6.12) fe(oc) =
k sin 00
a — k cos 00'
(5.6.13) #(a) = ^-(k* -a2)*.
The equation (5.6.11) is next shown to be valid in a strip of the com-
plex a-plane which contains the real axis in its interior. We omit the
details of the discussion required to establish this fact; it follows in an
elementary way from the assumption that the constant k has a posi-
tive imaginary part, and from the conditions of regularity and boun-
dedness imposed on the solution 99 of the basic problem. K(<x.) is
factored* in the form (i/2)JSL(oO • JP+(a) with J?_(a) = (k - a)1/2,
K+(QL) = (A; + a)1/2 with K__ and K+ regular in lower and upper half-
* Such a manipulation occurs in general in using this technique; usually a
continued product expansion of the transform of the kernel is required.
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 145
planes, respectively. The equation (5.6.11) can then be expressed,
after some manipulation, in the form
(5.6.14) _
(k + a)1/2 a - k cos 00 [(* + a)1/2 (k + fccos i
~~ (A: + k cos 00)1/2(a - k cos 0^) ¥ ' ~~ a
where the symbols g+ and /_ refer to the fact that g(oc) and /(a) can be
shown to be regular in upper and lower half-planes of the complex
oc-plane, respectively, each of which overlaps the real axis. In fact, the
entire left side of (5.6.14) is regular in such an upper half-plane, and
similarly for the right hand side in a lower half-plane. Thus the two
sides of the equation define a function which is regular in the entire
plane, or, in other words, each side of the equation furnishes the
analytic continuation of the function defined by the other side.
Finally, it is rather easy to show, by studying the behavior of g(oc)
and /(a) at oo, that the entire function thus defined tends uniformly
to zero at oo; it is therefore identically zero. Thus (5.6.14) defines both
g(a) and /(a) since they can be obtained by equating both sides se-
parately to zero. Thus g(x) and /(#) arc determined, and the problem
is, in principle, solved.
The Wiener-Hopf method is, evidently, a most amusing and in-
genious procedure. However, it also has somewhat the air of a tour de
force which uses a good many tools from function theory (while
the problem itself can be solved very nicely without going into the
complex domain at all, as we have seen in the preceding section) and
it also employs the artificial device of assuming a positive imaginary
part for the wave number k. (This brings with it, we observe from
(5.6.4), that while the primary wave dies out as x -»• + oo, it be-
comes exponentially infinite as x -> — oo.) In addition, the problem
of diffraction by a wedge, rather than by a plane barrier, can not be
solved by the Wiener-Hopf method, but yields readily to solution by
the simple method presented in the preceding section. The author
hazards the opinion that problems solvable by the Wiener-Hopf
technique will in general prove to be solvable more easily by other
methods— for example, by more direct applications of complex in-
tegral representations, perhaps along the lines used to solve the
difficult mixed boundary problem treated in section 5.4 above.
We mention next two other papers in which integral equations are
146 WATER WAVES
employed to solve interesting water wave problems. The first of these
is the paper by Kreisel [K.19] in which two-dimensional simple
harmonic progressing waves in a channel of finite depth containing
rigid reflecting obstacles are treated. Integral equations are obtained
by using an appropriate Green's function; Kreisel then shows that
they can be solved by an iteration method provided that the domain
occupied by the water does not differ too much from an infinite strip
with parallel sides. (Roseau [R.9] has solved similar problems for
certain domains which are not restricted in this way. ) It is remarkable
that Kreisel is able to obtain in some important cases good and useablc
upper and lower bounds for the reflection and transmission coeffi-
cients. References have already been made to the papers by John
[J.5] on the motion of floating bodies. In the second of these papers
the problem of the creation of waves by a prescribed simple harmonic
motion of a floating body is formulated as an integral equation. This
integral equation does not fall immediately into the category of those
which can be treated by the Fredholm theory; in fact, its theory has
a number of interesting and unusual features since it turns out that
the homogeneous integral equation has non-trivial solutions which,
however, are of such a nature that the nonhomogeneous problem
nevertheless always possesses solutions.
Various problems concerning the effect of obstacles on waves, and
of the wave motions created by immersed oscillating bodies, have
been treated in a series of notable papers by Ursell [U.3, 4, 5 and
U.8, 9, 10]. Ursell usually employs the method of expansions in terms
of orthogonal functions, or representations by integrals of the Fourier
type, as tools for the solution of the problems.
Finally, it should be mentioned that the approximate variational
methods devised by Schwinger [S.5] to treat difficult problems in the
theory of electromagnetic waves can also be used to treat problems in
water waves (cf. Keller [K.7]). A notable feature of Schwinger's
method is that it is a technique which concentrates attention on the
quantities which are often of the greatest practical importance, i.e.
the reflection and transmission coefficients, and determines them,
moreover, without solving the entire problem. Rubin [R.13] has for-
mulated the problem of the finite dock— which has so far defied all
efforts to obtain an explicit integral representation for its solution —
as a variational problem of a somewhat unconventional type, and
proved the existence, on the basis of this formulation, of solutions be-
having at oo like progressing waves.
WAVES ON SLOPING BEACHES AND PAST OBSTACLES 147
An interesting type of problem which might well have been dis-
cussed at length in this book is the problem of internal waves. This
refers to the occurrence of gravity waves at an interface between two
liquids of different density. Such problems are discussed in Lamb
[L.3], p. 370. The case of internal waves in media with a continuous
variation in density has considerable importance also for tidal motions
in both the atmosphere (cf. Wilkes [W.2]) and the oceans (cf.
Fjeldstad [F.4]).
SUBDIVISION B
MOTIONS STARTING FROM REST. TRANSIENTS.
CHAPTER 6
Unsteady Motions
6.1. General formulation of the problem of unsteady motions
In the region occupied by the water we seek, as usual, a harmonic
function 0(x, y, z; t) which satisfies appropriate boundary conditions
and, in addition, appropriate conditions prescribed at the initial in-
stant / — 0. At the free surface we have the boundary conditions
(6.1.1) -#i+i?« = 0 \
^ 1 for y = 0, t > 0
(0.1.2) *t+gn=--v\
in terms of the vertical elevation rj(<x, z; t) of the free surface and the
pressure p(x, z; t) prescribed on the surface. As always in mechanics,
a specific motion is determined only when initial conditions at the
time t - 0 are given which furnish the position and velocity of all
particles in the system. This would mean prescribing appropriate
conditions on 0 throughout the fluid at the time t = 0, but since we
shall assume 0 to be a harmonic function at t = 0 as well as for t > 0
it is fairly clear that conditions prescribed at the boundaries of the
fluid only will suffice since 0 is then determined uniquely throughout
its domain of definition in terms of appropriate boundary conditions.*
As initial conditions at the free surface, for example, we might there-
fore take
(6.1.3) r,(x,z;0) = MX, z) \ = Q
(6.1.4) a
with /! and /2 arbitrary functions characterizing the initial elevation
and vertical velocity of the free surface.
In water wave problems it is of particular interest to consider cases
* We shall see later on (sections 6.2 and 6.9) that the solutions are indeed
uniquely determined when the initial conditions are prescribed only for the
particles at the boundary of the fluid.
149
150 WATER WAVES
in which the motion of the water is generated by applying an impul-
sive pressure to the surface when the water is initially at rest. To
obtain the condition appropriate for an initial impulse we start from
(6.1.2) and integrate it over the small time interval 0 <S t ^ r. The
result is
(6.1.5) pdt = — Q0(x, 0, *; r) — eg \
Jo Jo
r)dt,
since 0(x, y, z; 0) can be assumed to vanish. One now imagines that
r -> 0+ while p -> oo in such a way that the integral on the left tends
to a finite value— the impulse / per unit area. Since it is natural to
assume that 77 is finite it follows that the integral on the right vanishes
as r -> 0+, and we have the formula
(6.1.6) / = — Q0(x, 0, z; 0+)
for the initial impulse per unit area at the free surface in terms of the
value of 0 there. If/ is prescribed on the free surface (together with
appropriate conditions at other boundaries), it follows that
0(x, y, z; 0^ ) can be determined, or, in other words, the initial velocity
of all particles is known.
It is also useful to formulate the initial condition on 0 at the free
surface appropriate to the case in which the water is initially at rest
under zero pressure, but has an initial elevation rj(x9 z; 0). The condi-
tion is obtained at once from (6.1.2); it is
(6.1.7) 0t(x, 0, z; 0+) - - grj(x, z; 0+),
since p = 0 for t > 0. Prescribing the initial position and velocity of
the free surface is thus equivalent to prescribing the initial values of
0 and its first time derivative 0t. From now on the notation 0+ will
not be used in formulating initial conditions— instead we shall simply
write 0 instead of 0+.
6.2. Uniqueness of the unsteady motions in bounded domains
It is of some interest to consider the uniqueness of the unsteady
motions, for one thing because of the unusual feature pointed out
in the preceding section: it is sufficient to prescribe the initial position
and Velocity, not of all particles, but only of those on the boundary.
A uniqueness proof based on the law of conservation of energy will
be given.
To this end, consider the motion of a bounded volume of water con-
fined to a vessel with fixed sides but having a free surface (cf. Fig.
UNSTEADY MOTIONS
151
6.2.1). In Chapter 1 we have already discussed the notion of energy
and its time rate of change with the following results. For the energy
E itself we have, obviously:
h(x,z)
Fig. 6.2.1. Water contained in a vessel
(6.2.1) E(t) - Q JJJ
+
+
y] dx dy dz.
Here R refers to the volume occupied by the water at any instant.
The x, z-planc is, as usual, taken in the plane of equilibrium of the
free surface, and y = rj(x9 z; t) and y = h(x9 z) are assumed to be the
equations of the free surface and of the containing vessel, respectively.
The expression for E can now be written in the form
(6.2.2) E(t)
1 tf <«
+ 02Z) dx dy dz
By S is meant the projection on the #, s-plane of the free surface and
the containing vessel. In Chapter 1 the following expression for the
rate of change of the energy E was derived:
(6.2.3)
dE ff
— = JJ [Q&
- vn) - pvn] dS.
152 WATER WAVES
By R is meant the boundary surface of R, while vn means the normal
component of the velocity of R. It is essential for our uniqueness
proof to observe that in the special case in which p = 0 on the free
surface we have
dE
(6.2.4) — = 0, E = const.
This follows at once from the fact that vn = 0n = 0 on the fixed part
of the boundary, while vn = 0n and p = 0 on the free surface.
So far no use has been made of the fact that we consider only a
linear theory based on the assumption of small oscillations about the
equilibrium position. Suppose now that the initial position and velo-
city of the water particles has been prescribed, or, as we have seen in
the preceding section that rj(x9 z; 0) and 0(x9 y, z; 0) are given func-
tions:
( ^(*> *: °) = /i(*» *)
I 0(x, y, z; 0) = f2(x9 y, z).
We proceed next in the customary way that one uses to prove unique-
ness theorems in linear problems. Suppose that rjl9 019 and r\^ 02
are two solutions of the initial value problem. Then 0 = ^ — 02
and r\ = ijl — r]2 arc functions which satisfy all of the conditions
originally imposed on 0t and 17 f except that fl and /2 in (6.2.5) would
now both vanish, and the free surface pressure would also vanish
(cf. (6.1.2) and (6.1.7)). (Here the linearity of our problem is used in
an essential way. ) It follows therefore that dE/dt = 0, and E = const,
when applied to 0 and 77, as we have seen. But at the initial instant
rf = 0 and 0 = 0, so that
(0.2.6) E = - — htdxdz,
5
from (6.2.2) as applied to0 = <t>l — 02 and 17 = rjl — r]2. Consequent-
ly we have the result
(6.2.7) fff (0* + 01 + 01) dx dy dz + g ff ^ dx dz = 0,
j j j j j
R S
and this sum obviously vanishes only if grad 0 = 0 and rj = 0— in
other words it follows that 0l = 02 (except for an unessential addi-
tive constant), rjl = rj29 and the uniqueness of the solution of the
initial value problem is proved.
UNSTEADY MOTIONS 153
The proof given here applies only to a mass of water occupying a
bounded region. Nevertheless, it seems clear that the uniqueness of
the solution of the initial value problem is to be expected if the water
fills an unbounded region, provided that appropriate assumptions
concerning the behavior of the solution at oo are made. In the follow-
ing, a variety of such cases will be treated by making use of the tech-
nique of the Fourier transform and, although no explicit discussion
of the uniqueness question will be carried out, it is well-known that
uniqueness theorems (of a somewhat restricted character, it is true)
hold in such cases provided only that appropriate conditions at oo
are prescribed. Recently these uniqueness questions have been treated
by Kotik [K.I 7] and Finkelstein [F. 3]. The latter, for example, proves
the uniqueness of unsteady motions in unbounded domains in
which rigid obstacles occur, and both writers obtain their uniqueness
theorems by imposing relatively weak conditions at infinity. In sec. 9
of this chapter the theory devised by Finkelstein will be discussed.
6.3. Outline of the Fourier transform technique
As indicated above, the solutions of a series of problems of unsteady
motions in unbounded regions as determined through appropriate
initial conditions will be carried out by using the method of the
Fourier transform. The basis for the use of this method is the fact that
special solutions 0 of our free surface problems are given— in the
case of two-dimensional motion in water of infinite depth, for example
-b
(6.3.1) 0(a:9 y\ t) = e™* sin (at + a) cos m(x - r)
with
(6.3.2) a2 = gw
and for arbitrary values of a and r. From these solutions it is possible
to build up others by superposition, for example, in the form
(6.3.3) 0(x, y\ t) = h(a)e™v sin (at + <x.)dm
Joo
f(r) cos m(x — T) dr,
— 00
in which h(a) and f(r) are arbitrary functions. This in turn suggests
that the Fourier integral theorem could be used in order to satisfy
given initial conditions, since this theorem states that an arbitrary
154 WATER WAVES
function f(x) defined for — oo < x < oo can be represented in the
form
i r°° r°°
(6.3.4) f(x) = - da /(^) cos a(^ — x) dr\
n J o J ~oo
provided only that f(x) is sufficiently regular (for example, that J(x)
is piecewise continuous with a piecewise continous derivative is more
than sufficient) and that f(x) is absolutely integrable:
(6.3.5) r | /(a?) | do? < oo.
J —00
Indeed, we see that if we set h(a) = 1/n and a = yt/2 in (6.3.3) we
would have exactly the integral in (6.3.4) for t — 0 and y = 0, and
hence 0(x9 0; 0) would reduce to the arbitrarily given function f(x).
Thus a solution would be obtained for an arbitrarily prescribed initial
condition on 0.
It would be perfectly possible to solve the problems treated below
by a direct application of (6.3.4), and this is the course followed by
Lamb [L.3] in his Chapter IX. Actually the problems were solved
first by Cauchy and Poisson (in the early part of the nineteenth cen-
tury), who derived solutions given by integral representations before
the technique of the Fourier integral was known. It might be added
that these problems were considered so difficult that they formed the
subject of a prize problem of the Academic in Paris.
We prefer, in treating these problems, to make use of the technique
of the Fourier transform (following somewhat the presentation given
by Sneddon [S.ll ], Chapter 7) since the building up of the solution to
fulfill the prescribed conditions then takes place quite automatically.
However, the method is based entirely upon (6.3.4) and thus also
requires for its validity that the functions f(x) to which the technique
is applied should be represeritable by the Fourier integral. This is a
restriction of a non-trivial character: for example, the basically im-
portant solutions given by (6.3.1) are not representable by the Fou-
rier integral.
It is useful to express the Fourier integral in a form different from
(6.3.4). We write
f(
1 f°° f*
x) = - lim i(n)dri cos s(r] — x)ds.
n f->oo J -QO J 0
UNSTEADY MOTIONS 155
But since I cos s(rj — x)ds = 2 cos s(rj — - x)ds and
f * f ^
sin S(TI — x)ds = 0, we may write cos s(rj — x)ds =
J —5 JO
f *
% exp [is(x — rj)} ds, and hence
J — s
T /* GO /*00
(6.3.6) /(a?) = — e™*ds\ f(r))e^s dr\.
2^J~00 J-00
We now set
(6.3.7) t(s)
and call J(s) the Fourier transform of /(#). It follows at once from
(6.3.6) that the original function f(x) is obtained from its transform
l(s) by the inversion formula
In our differential equation problems it will be essential to express
the Fourier transform of the derivatives of a function in terms of the
transform of the function itself. Consider for this purpose the trans-
form of dnfldxn and integrate by parts (which requires that dnf/dxn
be continuous):
d»f
s - — e
i
•
J
If the (n — l)-st derivative is to possess a transform it must tend to
zero at ^ oo and hence we have
(6.3.9)
dxn dxn~l
that is, the transform of the n-th derivative is (is) times the transform
of the (n — l)-st derivative. By repeated application of this formula
we obtain the result
156 WATER WAVES
(6.8.10) - = (isrl
provided that f(x) and its first n derivatives are continuous and
that all of these functions possess transforms.
A rigorous justification of the transform technique used in the
following for solving problems involving partial differential equations
is not an entirely trivial affair (sec, for example, Courant-Hilbert
[C.10], vol. 2, p. 202 ff.). Such a justification could be given, but we
shall not carry it out here. Indeed, it would be reasonable to take the
attitude that one may proceed quite formally provided that one veri-
fies a posteriori that the solutions obtained in this way really satisfy
all conditions of the problem. This is usually not too difficult to do,
and, since the relevant uniqueness theorems are available, this course
is perfectly satisfactory.
6.4. Motions due to disturbances originating at the surface
We wish to determine first the motion in two dimensions due to the
application of an impulse over a segment of the surface —a ^x ^ a
at t = 0 when the water is at rest in the equilibrium position. We
suppose the depth h of the water to be constant and that it extends to
infinity in the horizontal direction. The velocity potential 0(cc, y; t)
must satisfy the following conditions. It must be a solution of the
Laplace equation:
(6.4.1) &xx+0yy = 0, -oo<o?<oo, -A^j/^0, *^0,
satisfying the boundary conditions
(6.4.2) 0tt + g$y = 0; y = 0, t >0
and
(6.4.3) 0y = 0, y = - h, t ^ 0.
The first of these conditions states that the pressure on the free
surface is zero for t > 0. As initial conditions we have, in view of
(6.1.6), (6.1.7), and the assumed physical situation:
(6.4.4) 0(x, 0; 0) = -- /(a?),
Q
(6.4.5) 0t(x, 0; 0) = 0,
with I(x) the impulse per unit area applied to the free surface. In
UNSTEADY MOTIONS 157
addition, we must impose conditions at oo. These are that 0 and
its first two derivatives with respect to x, y, and t should tend to zero
at oo in such a way that all of these functions possess Fourier trans-
forms with respect to x. This, in particular, requires that I(x) in
(6.4.4) should vanish at oo. Actually we consider only the special
case in which
T/ . ( P = const., | x | < a
(6.4.6) I(x) = '
I 0, | x | > a,
i.e. the case in which a uniform impulse is applied to the segment
| x | < a, the remainder of the surface being left undisturbed.
The solution 0(x, y; t) will now be determined by applying the
Fourier transform in x to the relations (6.4.1) — (6.4.5) with the object
(as always in such problems) of obtaining a simpler problem for the
transform 0(s, y; t) = (p(s, y; t). Once the transform 99 has been found
by solving the latter problem the inversion formula yields the solution
0. We begin by applying the transform to (6.4.1), i.e. by multiplying
by e~i8X and integrating over the interval — oo < x < oo; the result is
(6.4.7) - s*<p(89 y; t) + <pyy(s, y; t) - 0
in view of (6.3.10) and the assumed behavior of 0 at oo. (Clearly, it is
also necessary to suppose that the operation of differentiating 0
twice with respect to y can be interchanged with the operation of
integrating 0 over the infinite interval.) This step already achieves
one of the prime objects of the approach using a transform: the trans-
form cp satisfies an ordinary differential equation instead of the partial
differential equation satisfied by 0. The general solution of (6.4.7) is
(6.4.8) p(s, y, t) = A(s\ t)e \*\v + B(s; t)e~^v
in terms of the arbitrary "constants" A(s; t) and B(s; t). It is a simple
matter to find the appropriate special solution that also satisfies the
bottom condition (6.4.3), and from it to continue (just as is done in
what follows) in such a way as to find the solution for water of uniform
depth. However, we prefer to take the case of infinite depth and to
replace (6.4.3) by the condition that 0V -+ 0 when y -> — oo. The
transform <p then also must have this property so that we obtain for
<p($, y\ t) in this case the solutions
(6.4.9) <p(s, !/;*) = A(s; t)e\*\*.
158 WATER WAVES
The transform is next applied to the free surface condition (6.4.2) to
obtain
(6.4.10) cpit + g(jpy = 0, y = 0, t > 0
and upon insertion of <p($, 0; t) from (6.4.9) we find for A(s; t) the
differential equation
(6.4.11) Ati+g\8\A=0, t>0.
Finally, the initial conditions must be taken into account. The trans-
form of (6.4.5) leads, evidently, to the condition At(s; 0) = 0, and
the solution of (6.4.11) satisfying this condition is
(6.4.12) A(s\ t) = a(s) cos (Vg\s\t)
with a(s) still to be determined by using (6.4.4). From (6.4.4) we
have q?(s, 0; 0) = — (l/g)/(s) in which I(s) is, of course, the transform
of I(x) as given by (6.4.4); hence a(s) = — (I/Q)!(S) and we have for
3>(s, y; t) = (f(s, y; t) the result
(6.4.13) 0(8, y; *) - — 7(*)*W*cos (Vg|*| t).
The inversion formula (6.3.8) then leads immediately to the solution
I /»oo
(6.4.14) 0(x, y; t) = — — = I(s)e^veisx cos (Vg\s\ t) ds.
eV2nJ -a.
In our special case (cf. (6.4.6)) we have for
a 2 Pa sin sa
cos sx dx = •— :=- * — — !
- P Ca 2P Cc
T(*) = -7=\ r4~te = -—\
V2nJ -a V2n Jo
and hence finally for <Z>(#, t/; t) the solution
2Pa f00 sin sa qil y /— 4X ,
(6.4.15) 0(a?, y; t) = * v cos ^ cos ( Vg* 0*.
^rp Jo sa
as one can readily verify. For the free surface elevation we have
(from (6.1.2)):
cos
1 — 2 Pa f °° sin sa
(6.4.16) ri(x; t) = — - #e= - pz lim -- e*
S nQVg «->oJo ^
sin (Vgst)Vs ds.
UNSTEADY MOTIONS 159
One observes that the integrals converge well for all y < 0 because of
the exponential factor esy% i.e. everywhere except possibly on the free
surface. These formulas can now be used to obtain the solution for the
case of an impulse concentrated on the surface at x = 0; one need only
suppose that a -r 0 while P -> oo in such a way that the total im-
pulse 2Pa tends to a finite limit. For a unit total impulse we would
then obviously obtain for 0 and 77 the formulas:
1 /•» _
(6.4.17) 0(x, y; t) = — — es* cos sx cos (Vgs t) ds,
KQ Jo
If* _
(6.4.18) w(a?; 0 = — --- 7- lim esv cos sx sin ( Vgs t)\/s ds.
KQV8 v^o Jo
(We define r\(x\ t) as a limit for y -> 0 since the integral obviously
diverges for y = 0. This would, however, not be necessary in (6.4.16).)
By operating in the same way, one can easily obtain the solutions
corresponding to the case of an initial elevation of the free surface at
time t = 0, with no impulse applied. The only difference would be that
0 in (6.4.4) would be assumed to vanish while 0t in (6.4.5) would be
different from zero. We simply give the result of such a calculation,
but only for the limit case in which the initial elevation is concen-
trated at the origin. For 0 and r\ the formulas are:
VS f * /— ds
(6.4.19) 0(x, y; t) = — -^ e8* cos sx sin ( Vgs t) — >
n J0 ys
If00 _
(6.4.20) rj(x\ t) = - lim e*v cos sx cos (Vgs t) ds.
™ v-*0 Jo
There is no difficulty in treating problems having cylindrical sym-
metry that are exactly analogous to the above two-dimensional cases.
In these cases also one could begin with the solutions having symmetry
of this type that are simple harmonic in the time (cf. Chapter 3):
(6.4.21 ) 0(r, j/; t) — eM
with a2 — gm (for water of infinite depth). Here the quantity r is the
distance Vx* + z2 from the j/-axis, and J0(mr) is the Bessel function
of order zero that is regular at the origin. One could now build up more
complicated solutions by superposition of these solutions and satisfy
given initial conditions by using the Fourier-Bessel integral. This is
the method followed by Lamb [L.3], p. 429. Instead of this procedure,
160 WATER WAVES
one could make use of the Hankel transform in a fashion exactly anal-
ogous to the Fourier transform procedure used above (cf. Sneddon
[S.ll], p. 290, and Hinze [H.15]). We content ourselves here with
giving the result for the velocity potential 0(r, y; t) and the surface
elevation rj(r; t) due to the application of a concentrated unit impulse
at the origin at t = 0:
1 /•» _
(6.4.22) <P(r,y;t) = — -— esyJ<>(sr) cos (Vgst)s ds,
2nQJo
_ 1 /»oo __
(6.4.23) n(r; t) = - - T lim e» J0(sr) sin (Vgs t)s*12 ds.
ZnQVg *-»o Jo
*-»o
Naturally we want to discuss the character of the motions furnished
by the above relations, and in doing so we come upon a fact that holds
good in all problems of this type: it is a comparatively straightforward
matter to obtain an integral representation for the solution, but not
always an easy matter to carry out the details of the discussion of its
properties. The reason for this is not far to seek —it is due to the fact
that the solutions are given in terms of an integral over an integrand
which is oscillatory in character and which changes rather rapidly
over even small intervals of the integration variable for important
ranges in the values of the independent variables. Hence even a nu-
merical integration would not be easy to carry out. The fact is that the
motions are really of a complicated nature, as we shall see, and hence
a mathematical description of them can be expected to present some
difficulties. Indeed, the phenomena under consideration here arc
analogous to the refraction and diffraction phenomena of physical
optics and thus depend on intricate interference effects, which are
further complicated in the present instances by the fact that the wave
motions are subject to dispersion, as we have seen in Chapter 3.
Some insight into the nature of the solutions furnished by our for-
mulas can be obtained by expanding the integrands in power series
and integrating term by term (cf. Lamb. [L.3], p. 385).* The result
for r/(x; t) as given by (6.4.20), for example, is found to be (for x > 0):
It is clear that there is a singularity for x = 0, as one would expect.
* The subsequent discussion in this section follows closely the presentation
given by Lamb.
UNSTEADY MOTIONS 161
The series converges for all values of the dimensionless quantity
gt2l2x, but practically the series is useful only for small values of
gt2/2x, i.e. for small values of t, or large values of x. One observes also
that any particular "phase" of the disturbance— such as a zero of rj>
for example— must propagate with a constant acceleration, since any
such phase is clearly associated with a specific constant value of the
quantity gt2/2x.
It is in many respects more useful to find an asymptotic represen-
tation for the motion valid in the present case for large values of
the quantity gt2/2x, for which the power series are not very useful
because of their slow convergence. Indeed, the asymptotic represen-
tation yields all of the qualitative features contained in the exact
solution (6.4.24), and is also accurate even for rather small values of
the quantity gt2/2x (cf. Sneddon [S.ll], p. 287). For this purpose it
happens to be rather easy to work out an asymptotic development of
the solution that is valid for large values of gt2/2x9 and this we proceed
to do, following Lamb. We write (6.4.19) in the form
(6
-Iff00 °2y /a2* \
.4.25) 0(x,y;t) = --- e* sin I — + at] da
n Uo \ S I
r °*y . (a*x \ \
— eg sin I — — at] da }
Jo .\g I \
making use of a = Vgs, 2oda = gd#. New quantities | and o> are
introduced in (6.4.25) by the relations
from which
-co - g
The expression (6.4.25) is thus readily found to take the form
2gl/2 /»o>
(6.4.26) 0(x, 0; t) - -^ sin (|2 - o>2) d£
n% ' Jo
where f is introduced as new variable of integration and y is assumed
to vanish. The corresponding free surface elevation is given by
162
WATER WAVES
(6.4.27)
1
g'
as one readily verifies. In order to study the last expression we con-
sider the integral
(6.4.28) I c dc ==:: I c dc — I 6 dc*
i i I
JO */0 Jcu
It is well known that
(6.4.29)
while the second contribution can be treated as follows:
<-"2> dt
f
Jo
(6.4.30)
f * c«*-"» d!; = I ( °° -
Jo) 2Jco2 V*
i r °° i r°° a ~i
= _ H ^<'—*> + - < "i ^^-|2)^
^ L L» 2 J^ J
through introduction of t — £2 as new variable, and an integration
by parts. We show next that the final integral is of the order cw"1.
as follows:
| 3 [
\ t Z6 ut ^ I t
Jet)2 Jw2
r°°
Jo)2
dt =
Upon considering the real parts of (6.4.28), (6.4.29), (6.4.30), and
inserting in (6.4.27) one finds
,) = -^ (g)» [cos (g - J
(6.4.81)
in which the function O(co~l) refers, as one readily verifies, to a term
which behaves like (g/2/4#)~1/2. Consequently, if o>2 = g^2/4<r is suffi-
'ciently large, we may assume for the free surface elevation due to a
concentrated surface elevation at x = 0 and t = 0 the approximate
expression
UNSTEADY MOTIONS 163
1 /g*2U /g*2 n\
(6.4.32) ,<«.„-,_ (!-)*„(£--_).
By continuing the integration by parts, as in (6.4.30), it would be
possible to obtain approximations valid up to any order in the quanti-
ty co"1 = (gJ2/^)-*1'2, but such an expansion would not be convergent;
it is rather an asymptotic expansion correct within a certain order in
co"1 when an appropriate finite number of terms in the expansion is
taken. Expansions of this type are— as in other branches of mathe-
matical physics— very useful in many of our problems and we shall
have many other occasions to employ them.
The case of a concentrated point impulse applied at x — 0 at the
time t — 0 can be treated in exactly the same manner as the case
just considered: one has only to begin with the solution (6.4.17) in-
stead of (6.4.19), and proceed along similar lines. In particular, the
approximate solution valid (to the same order in a)~l) for large values
of g/2/4a? can be obtained; the result for the free surface elevation is
- 2 /g*2\3/2 . /gt2 x\
(6.4.33) ,j(x; t) ~ -- -— y- sin ^- - - .
' \AtxJ \±x 4/
The method used to derive these asymptotic formulas is rather
special: it cannot be very easily used to study the cylindrical waves
given by (6.4.23), for example. We turn, therefore, in the next section
to the derivation of asymptotic approximations in all of these cases
by the application of Kelvin's method of stationary phase. After-
wards, the motions themselves will be discussed in section 6.6 on the
basis of the approximate formulas.
6.5. Application of Kelvin's method of stationary phase.
The integrals of section 6.4 can all be put into the form
(6.5.1) I(k) = f \(£,k)eik*W d£
j (i
without much difficulty, and this is a form peculiarly suited to an
approximate treatment valid for large values of the real constant k.
In fact, Kelvin seems to have been led to the approximate method
known as the method of stationary phase through his interest in
problems concerning gravity waves, in particular the ship wave
problem. The general idea of the method of approximation is as fol-
164 WATER WAVES
lows. When k is large the function exp (ik<p(!;)} oscillates rapidly as £
changes, unless (p(£) is nearly constant, so that the positive and ne-
gative contributions to the value of I(k) largely cancel out, provided
that y>(f , k) is not a rapidly oscillating function of £ when k is large.
Hence one might expect the largest contributions to the integral to
arise from the neighborhoods of those points in the interval from a
to b at which <p(£ ), the phase of the oscillatory part of the integral,
varies most slowly, i.e., from neighborhoods of the points where
(p'(£ ) = o. This indeed turns out to be the case. In section 6.8 it will
be shown that
(6.5.2) /(*) = 2 V(«r. *)
By 0 ( 1 /&2/3) we mean a function which tends to zero like l/&2/3 as k -> oo.
In these expressions the sums are taken over all the zeros ocr of q/(£)
in the interior of the interval a ^ £ ^ b at which 9p"(ar) ^ 0 and over
the zeros as of q>'($) at which <p"(ocj = 0 but <p'"(ocj ^ 0. The sign
of the term ± jr/4 in the first sum should be taken to agree with the
sign of 9/'(ar). The relation (6.5.2) is valid if y(f, k) andg?(£) arc ana-
lytic functions of £ in a ?g { <£ 69 and if the only stationary points of
9?(|)are such that 9?"(£) and <p'"(l;) do not vanish simultaneously.*
We proceed to obtain the approximate solution (6.4.32) obtained
in the previous section once more by this method. The motion of the
water was to be determined for the case of an elevation of the water
surface concentrated at a point; the formula for the velocity potential
was put in the form (cf. (6.4.25)):
(6
1 f f00 *L /o*x \
.5.3) 0(x, y; t) = — - ] e o sin I — + at] do
n [Jo \ S J
f00 *1 /o*x \ ]
— e ^ sin 1 — — at\ da \
Jo \ g / J
* If a zero of <p'(£ ) of still higher order should occur, then terms of other types
would appear, and the error would die out less rapidly in A:. It should also be
noted that the coefficient function y of section 6.8 is assumed to be independent
of k, which is not true in some of the examples to follow. However, it is not
difficult to see that the proof of section 6.8 can be modified quite easily in such
a way as to include all of our cases.
UNSTEADY MOTIONS 165
This can in turn be put in the form
Iff00 f * \
(6.5.4) 0(x9 y; t) = -- em* ei(mx+°t} da - emy ei(mx~at} da
n Uo Jo )
with m — a2fg. It is understood that the imaginary part only is to be
taken at the end. It is convenient to introduce a new dimensionless
variable of integration as follows:
2x
(6.5.5) f = — or,
in terms of which (6.5.4) is readily found to take the form
(6.5.6) 0(x, yi t) --= - -' ( | <"** *'* <**+*> rff - f e™» e'*&~**> d£\
'-XX I Jo Jo )
with
op
(6.5.7) A:-- —
v 4cT
as a dimensionless parameter. The quantity m is of course also a
function of f, and exp {m(g)y} plays the role of the function ^(£) in
(6.5.1). When the parameter A: is large, we may approximate the
solution by using (6.5.2). For the phases </;(£) we have
(0.5.8) ?(£)-£2±2£
with stationary points given by
(0.5.0) ?/(£) = 2f ±2 = 0,
and we see that | ~ 1 is the only such point in the interval 0 < f < oo
over which the integrals are taken. Consequently only the second inte-
gral in (6.5.6) possesses a point of stationary phase, and at this point
we have
(6.5.10) ?/'(l) = 2, ?(!)= - 1-
We obtain therefore from (6.5.2) the approximate formula
(6.5.11 ) <t>(x, y\ t) c^ —
" nx
as one readily verifies, and this formula is a good approximation for
large values of k — gt*/&x. We can also calculate the free surface eleva-
166 WATER WAVES
tion 77 in the same way from rj = — (l/g)&t lv=o»* the result is easily
found to be
(6.5.12) ^.^
just as before (cf. (6.4.32)).
For the case of a concentrated impulse the method of stationary
phase as applied to (6.4.17) or (6.4.18) leads to the following approxi-
mation valid once again for large values of gt2/4>x:
(6.5.13) ^^^
and this coincides with the result given in (6.4.33).
In the case of an impulse distributed over a segment one obtains
from (6.4.16) the result
2P . gt*a . /{>t* n\
(6.5.14) „(,; 0 = - --^ «n ^ sm ^ - -J,
valid for large values of gt2/4tx.*
For the ring waves furnished by (6.4.23) the asymptotic formula is
gt* gt*
(6.5.15) ,(r; I) = - jff ™ —.
To obtain this formula it is necessary to replace the Bessel function
J0(sr) in (6.4.23) by its integral representation
2 ftt/2
cos (sr cos /?) dp
and then apply the method of stationary phase twice in succession.
Since such a procedure is discussed later on in dealing with the
simplified ship wave problem (cf. Chapter 8.1), we omit a discussion
of it here, except to remark that the approximate formula (6.5.15) is
valid for any r =£ 0 and g*2/4r sufficiently large.
* It may seem strange that this formula indicates that x = 0 is a singular
point for r), while x = 0 is not singular in the exact formula (6.4.16). This comes
about through the introduction of the new variable (6.5.5) and the parameter k
in (6.5.7) which were used to convert the original integral to the form (0.5.1).
However, the validity of the formula (6.5.2) is assured, as one can see from
section 6.8, only if x 96 0.
UNSTEADY MOTIONS 167
6.6. Discussion of the motion of the free surface due to disturbances
initiated when the water is at rest
We proceed to discuss the motions of the water surface in accord-
ance with the results given in the preceding section. The general
character of the motion is well given by the approximate formulas,
and we shall therefore confine our discussion to them. We observe
first that the oscillatory factors in the four approximate formulas
(6.5.12) — (6.5.15) do not differ essentially, but the slowly varying
nonoscillatory factors are different in the various cases: (a) at a fixed
point on the water surface the disturbance increases in amplitude
linearly in t in the case of an initial elevation concentrated at a point
(cf. (6.5.12)), while for a fixed time the amplitude becomes large for
small x like #-3/2; (b) in the case of an initial impulse concentrated
at a point the amplitude increases quadratically in t at a fixed point,
while for a fixed time the amplitude increases like ^~5/2 for small
x. (In these limit cases the approximate formulas are valid for x ^ 0,
since the only other requirement is that the quantity gfifax should be
large.) The behavior of these solutions near x — 0 is not very sur-
prising since there is a singularity there. The behavior at any fixed
point x as t -> oo is, however, somewhat startling: the amplitude is
seen to grow large without limit as the time increases in both of these
cases. This rather unrealistic result is a consequence of the fact that
the singularity at the origin is very strong. If the initial disturbance
were finite and spread over an area, the amplitude of the resulting
motion would always remain bounded with increasing time, as one
could show by an appeal to the general behavior of Fourier trans-
forms.* This fact is well shown in the special case of a distributed
impulse, as we see from (6.5.14), which is valid for all x ^= 0 and large
/: the amplitude remains bounded as t -> oo.
The general character of the waves generated by a point disturbance
is indicated schematically in the accompanying figures which show
the variation in surface elevation at a fixed point x when the time in-
creases, and at a fixed time for all x. These figures are based on the
formula (6.5.12) for the case of an initial elevation; the results for the
case of initial impulse would be of the same general nature.
* It is also a curious fact that the motion given by (6.5.14) for the case of
an impulse over a segment requires infinite energy input, since the amplitude
at any fixed point does not tend to y.ero. For the case of an initial elevation confined
to a segment, however, the wave amplitude would die out with increasing time.
168
WATER WAVES
It is worth while to discuss the character of the motion furnished
by (6.5.12) in still more detail. It has already been remarked that any
particular phase— such as a zero, or a maximum or minimum of r\— is
of necessity propagated with an acceleration since each such phase is
associated with a particular constant value of the quantity g*2/4#: if
the phase is fixed by setting g*2/4# = c, then this phase moves in
accordance with the relation x = g*2/4c. The formula (6.5.12) holds
only where the quantity gt2/4>x is large, and hence the individual pha-
ses are accelerated slowly in the region of validity of this formula; or,
Fig. 6.6.1a,b Propagation of waves due to an initial elevation
UNSTEADY MOTIONS 169
in other words, the phases move in such regions at nearly constant
velocity. Also, for not too great changes in x or t the waves behave
very nearly like simple harmonic waves of a certain fixed period and
wave length. This can be seen as follows. Suppose that we vary t
alone from / == tQ to t •= tQ + At. We may write for the phase <p:
as one readily verifies. Thus if At/tQ is small, i.e. if the change At in the
time is small compared with the total lapse of time since the motion
was initiated, we have for the change in phase:
Consequently the period T = At of the motion corresponding to the
change Aq> — 2n in the phase is given approximately by the formula
(6.6.1) T
0
The accuracy of this formula is good, as we know, if T/tQ c^L
is small, and this is the case since g/§M#o *s always assumed to be large.
Thus the period at any fixed point varies slowly in the time. In the
same way one finds for the local wave length A the approximate
formula
(6.6.2) A
go
by varying with respect to <r alone, and this is also easily seen to be
accurate if gt%/4xQ is large. Thus for a fixed position x the period and
wave length both vary slowly, and they decrease as the time increases,
while for a fixed time the same quantities increase with x, as is borne
out by the figures shown above.
It is of considerable interest next to compute the local phase velo-
city—the velocity of a zero of 77, for example— from gt2/4x = c
when x and t vary independently; the result is
dx 2x
(6-6-3> ¥ = T
for the velocity of any phase; thus for fixed x the phases move more
170 WATER WAVES
slowly as the time increases, but for fixed t more rapidly as x increases
—that is, the waves farther away from the source of the disturbance
move more rapidly, and they are also longer, as we know from (6.6.2).
The wave pattern is thus drawn out continually, and the waves as
they travel outward become longer and move faster. The last fact is
not too surprising since the waves in the vicinity of a particular point
have essentially the simple character of the sine or cosine waves of
fixed period that we have studied earlier, and such waves, as we have
seen in Chapter 3, propagate with speeds that increase with the wave
length. All of the above phenomena can be observed as the result of
throwing a stone into a pond; though the motion in this case is three-
dimensional it is qualitatively the same, as one can see by comparing
(6.5.15) with (6.5.12).
There is another way of looking at the whole matter which is
prompted by the last observations. Apparently, the disturbance at
the origin acts like a source which emits waves of all wave lengths and
frequencies. But since our medium is a dispersive medium in which
the propagation speed of a particular phase increases with its wave
length, it follows that the disturbance as a whole tends with increasing
time to break up into separate trains of waves each of which has ap-
proximately the same wave length, since waves whose lengths differ
move with different velocities. However, it would be a mistake to
think that such wave trains or groups of waves themselves move with
the phase speed corresponding to the wave length associated with the
group. If one fixes attention on the group as a whole rather than on
an individual wave of the group, the velocity of the group will be seen
to differ from that of its component waves. The phase velocity for the
present case can be obtained in terms of the local wave length readily
from the equation (6.6.3) by expressing its right hand side in terms
of the local wave length through use of (6.6.2); the result is
dx 2x i In ;
(6.6.4) — = — = l/^L.
V ' * * V 271
On the other hand, the position x of a group of waves of fixed wave
length A at time t is given closely by the formula
(6.6.5) « =
as we see directly from (6.6.2), so that the velocity of the group is
UNSTEADY MOTIONS 171
which is, evidently, just half the phase speed of its com-
ponent waves. In other words, the component waves in a particular
group move forward through the group with a speed twice that of the
group.
Finally, we observe that these results are in perfect accord with the
discussion in Chapter 3 concerning the notions of phase and group
velocity. The phase speed c for a simple harmonic wave of wave length
A in water of infinite depth is given (cf. (3.2.3)!), by c = VgA/2jr,
and this is also the phase speed of the waves whose wave length is
A — as we sec from (6.6.4). We have also defined in section 3.4 the
notion of group velocity for simple harmonic waves in water of infinite
depth, and found it to be just half the phase velocity. The kinematic
definition of the group velocity given in section 3.4 was obtained by
the superposition of trains of simple harmonic waves of slightly differ-
ent wave length and amplitude, while in the present case the waves
arc the result of a superposition of waves of all wave lengths and
periods. However, the principle of stationary phase, which furnishes
the approximate solution studied here, in effect says that the main
motion in certain regions is the result of the superposition of waves
whose wave lengths and amplitudes differ arbitrarily little from a
certain given value. The results of the analysis in the present case are
thus entirely consistent with the analysis of section 3.4.
At any time, therefore, the surface of the water is covered by groups
of waves arranged so that the groups having waves of greater length
are farther away from the source. These groups, therefore, tend to
separate, as one sees from (6.6.4). The waves in a given group do not
maintain their amplitude, however, as the group proceeds: one sees
readily from (6.5.12) in combination with (6.6.2) that their amplitude
is proportional to l/^/x for waves of fixed length L
The above interpretations of the results of the basic theory are all
borne out by experience. Figure 6.6.2 shows a time sequence of photo-
graphs of waves (given to the author by Prof. J. W. Johnson of the
University of California at Berkeley) created by a disturbance
concentrated in a small area: the decrease in wave length at a fixed
point with increasing time, the increase in the wave lengths near the
front of the outgoing disturbance as the time increases, the general
drawing out of the wave pattern with time, the occurrence of well-
defined groups, etc. are well depicted.
An interesting development in oceanography has been based on the
theory developed in the present section. Deacon [D. 6, 7] and his
172
WATER WAVES
Fig. 0.6.2. Waves due to a concentrated disturbance
UNSTEADY MOTIONS
173
Fig. 0.6/2. (Continued)
174 WATER WAVES
associates have carried out studies which correlate the occurrence of
storms in the Atlantic with the long waves which move out from the
storm areas and reach the coast of Cornwall in a relatively short time.
By analyzing the periods of the swell, as determined from actual
wave records, it has been possible to identify the swell as having been
caused by storms whose location is known from meteorological obser-
vations. Aside from the interest of researches of this kind from the
purely scientific point of view, it is clear that such hindcasts could, in
principle, be turned into methods of forecasting the course of storms
at sea in areas lacking meteorological observations.
6.7. Waves due to a periodic impulse applied to the water when
initially at rest. Derivation of the radiation condition for purely
periodic waves
In section 3 of Chapter 4 we have solved the problem of two-dimen-
sional waves in an infinite ocean when the motion was a simple
harmonic motion in the time that was maintained by an application
of a pressure at the surface which was also simple harmonic in the
time. In doing so, we were forced to prescribe radiation conditions
at oo— effectively, conditions requiring the waves to behave like out-
going progressing waves at oo— in order to have a complete formula-
tion of the problem with a uniquely determined solution. It was
remarked at the time that a different approach to the problem would
be discussed later on which would require the imposition of bounded-
ness conditions alone at oo, rather than the much more specific radia-
tion condition. In this section we shall obtain the solution worked out
in 4.3 without imposing a radiation condition by considering it as the
limit of an unsteady motion as the time tends to infinity. However, it
has a certain interest to make a few remarks about the question of
radiation conditions in unbounded domains from a more general point
of view (cf. [S. 21]).
In wave propagation problems for what will be called here, ex-
ceptionally, the steady state, i.e., a motion that is simple harmonic in
the time, it is in general not possible to characterize uniquely the
solutions having the desired physical characteristics by imposing only
boundedness conditions at infinity. It is, in fact, as we have seen in
special cases, necessary to impose sharper conditions. In the simplest
case in which the medium is such as to include a full neighborhood of
the point at infinity that is in addition made up of homogeneous matter,
UNSTEADY MOTIONS 175
the correct radiation condition is not difficult to guess. It is simply
that the wave at infinity behaves like an outgoing spherical wave
from an oscillatory point source, and such a condition is what is
commonly called the radiation, or Sommerfeld, condition. Among
other things this condition precludes the possibility that there might
be an incoming wave generated at infinity— which, if not ruled out,
would manifestly make a unique solution of the problem impossible.
If the refracting or reflecting obstacles to the propagation of waves
happen to extend to infinity— for example, if a rigid reflecting wall
should happen to go to infinity —it is by no means clear a priori what
conditions should be imposed at infinity in order to ensure the unique-
ness of a simple harmonic solution having appropriate properties
otherwise.* A point of view which seems to the author reasonable is
that the difficulty arises because the problem of determining simple
harmonic motions is an unnatural problem in mechanics. One should in
principle rather formulate and solve an initial value problem by
assuming the medium to be originally at rest everywhere outside a
sufficiently large sphere, say, and also assume that the periodic
disturbances are applied at the initial instant and then maintained
with a fixed frequency. As the time goes to infinity the solution of the
initial value problem will tend to the desired steady state solution
without the necessity to impose any but boundedness conditions at
infinity.**
The steady state problem is unnatural — in the author's view, at
least— -because a hypothesis is made about the motion that holds
for all time, while Newtonian mechanics is basically concerned with
the prediction — in a unique way, furthermore— of the motion of a
mechanical system from given initial conditions. Of course, in me-
chanics of continua that are unbounded it is necessary to impose con-
ditions at oo not derivable directly from Newton's laws, but for the
initial value problem it should suffice to impose only boundedness
conditions at infinity. In sec. 6.9. the relevant uniqueness theorem for
the special case to be considered later is proved.
* For a treatment of the radiation condition in such cases see Rellich [R.7],
John [J.5], and Chapter 5.5.
** The formulation of the usual radiation condition is doubtlessly motivated
by an instinctive consideration of the same sort of hypothesis combined with the
feeling that a homogeneous medium at infinity will have no power to reflect
anything back to the finite region. Evidently, we also have in mind here only
cases in which no free oscillations having finite energy occur — if such modes of
oscillation existed, clearly no uniqueness theorems of the type we have in mind
could be derived.
176 WATER WAVES
If one wished to be daring one might, on the basis of these remarks,
formulate the following general method of obtaining the appropriate
radiation condition: Consider any convenient problem in which the
part of the domain outside a large sphere is maintained intact and
initially at rest. (In other words, one might feel free to modify in any
convenient way any bounded part of the medium.) Next solve the
initial value problem for an oscillatory point source placed at any
convenient point. Afterwards a passage to the limit should be made in
allowing the time t to approach oo, and after that the space variables
should be allowed to approach infinity. The behavior at the far distant
portions of the domain should then furnish the appropriate radiation
conditions independent of the constitution of the finite part of the
domain. It might be worth pointing out specifically that this is a case
in which the order of the two limit processes cannot be interchanged:
obviously, if the time / is first held fixed while the space variables tend
to infinity the result would be that the motion would vanish at oo,
and no radiation conditions could be obtained.
The writer would not have set down these remarks — which are of a
character so obvious that they must also have occurred to many
others— if it were not for two considerations. Every reader will doubt-
lessly have said to himself: "That is all very well in principle, but will
it not be prohibitively difficult to carry out the solution of the initial
value problem and to make the subsequent passages to the limit?"
In general, such misgivings are probably all too well founded. How-
ever, the problem concerning water waves to be treated here happens
to be an interesting special case in which (1) the indicated program
can be carried out in all detail, and (2) it is slightly easier to solve the
initial value problem than it is to solve the steady state problem with
the Sommerfeld condition imposed.
We restrict ourselves to two-dimensional motion in an x, z/-plane,
with the y- axis taken vertically upward and the #-axis in the originally
undisturbed horizontal free surface. The velocity potential (p(x, y\ t)
is a harmonic function in the lower half-plane:
(6.7.1) (pxx + q>yv = 0, y < 0, t > 0.
The free surface boundary conditions are (cf. (6.1.1), (6.1.2)):
(6.7.2)
UNSTEADY MOTIONS 177
As usual, r\ = r\(x\ t) represents the vertical displacement of the free
surface measured from the #-axis, and p — p(x; t) represents the
pressure applied on the free surface. We suppose that 9? and its first
and second derivatives tend to zero at oo for any given time t— in fact
that they tend to zero in such a way that Fourier transforms exist—
but we do not, in accordance with our discussion above, make any
more specific assumptions about the behavior of our functions as
/ -> oo. At the time t — 0 we prescribe the following initial conditions
(6.7.4) <p(x, 0; 0) == <pt(x, 0; 0) = 0,
which state (cf. (6.1.6), (6.1.7)) that the free surface is initially at
rest in its horizontal equilibrium position.
In what follows we consider only the special case in which the sur-
face pressure p(x; t) is given by
(6.7.5) . p(x; t) = d(x)eimt, t > 0
in which d(x) is the Dirac d-function. We have not made explicit use
of the d-function until now, but we have used it implicitly in section
6.1 in dealing with concentrated impulses. It is to be interpreted in
the same way here, i.e. as a symbol for a limit process in which the
pressure is first distributed over a segment the length of which is
considered to grow small while the total pressure is maintained at the
constant value one. By inserting this expression for p in (6.7.3) and
eliminating the quantity rj by making use of (6.7.2) the free surface
condition is obtained in the form
i(*> . *
(6.7.6) gVy + Vii - - — 6(x)ewt9 t > 0.
Our problem now consists in finding a solution <fj(x, y\ t) of (6.7.1)
which behaves properly at oo, and which satisfies the free surface
condition (6.7.6) and the initial conditions (6.7.4).
We proceed to solve the initial value problem by making use of the
Fourier transform applied to the variable x. The result of transforming
(6.7.1) is
(6.7.7) -s*$ + ^,, = 0,
in which <p(s, y\ t) is the transform of y(x, y\ t) and use has been made
of the conditions at oo. The bounded solutions of (6.7.7) for y < 0,
* > 0 are all of the form
178 WATER WAVES
(6.7.8) y(s, y; t) = A(s; t)e'v.
The transform is now applied to the boundary condition (6.7.6),
with the result:
(6.7.9) g<pv + fc, = - -4= ^7 «"*, for j, = 0,
and on substitution of <p(s, 0; t) from (6.7.8) we find
1 ico
(6.7.10) Att+g,A = -^7<"".
The initial conditions (6.7.4) now furnish for A (s; t) the conditions
(6.7.11) ^(*;0) = ^(*;0) = 0.
The solution of (6.7.10) subject to the initial conditions (6.7.11) is
i fco r* £ta>(**T) ,—
(6.7.12) A(s; t)= =:— — -_^~ sin Vgs rdr.
VZn Q Jo Vgs
Finally, we insert the last expression for A(s; t) in (6.7.8) and apply
the inverse transform to obtain the following integral representation
for our solution <p(x9y9t):
ia) f00 /•^uoU-t)
(6.7.13) qp(x, y; t) = — — e*y cos sx ^_— sin Vgs rdrds.
0rcJo Jo Vgs
The fact that the solution is an even function of x has been used here.
Our object now is to study the behavior of this solution as / -> oo.
Since y is negative (we do not discuss here the limit as y -> 0,
i.e. the behavior on the free surface) the integral with respect to s
converges well and there is no singularity on the positive real axis of
the complex $-plane. However, the passage to the limit t -> oo is more
readily carried out by writing the solution in a different form in
which a singularity — a pole, in fact— then appears on the real axis
of the 0-plane. (It seerns, indeed, likely that such an occurrence would
be the rule in any considerations of the present kind since the limit
function as t -> oo would not usually be a function having a Fourier
transform, and one could expect that the limit function would some-
how appear as a contribution in the form of a residue at a pole. ) It is
UNSTEADY MOTIONS
179
convenient to deform the path of integration in the $-plane into the
path L indicated in the accompanying figure. The path L lies on the
s- plane
Fig. 0.7.1. Path of integration in s-plane
positive real axis except for a semicircle in the upper half-plane cen-
tered at the point s = o)2/g. By Cauchy's integral theorem this leaves
the function tp given in (6.7.13) unchanged.
We now replace sin Vgs r in (6.7.13) by exponentials and carry
out the integration on i to obtain
(6.7.14) p(a?,0;0 -
—~\e
*Q JL
cos sx
2 Vgs Vgs
2 Vgs Vgs
I " "«
cb.
We wish now to consider the three items in the bracket separately,
and, as we see, two of them do indeed have a singularity at s = o>2/g
which is by-passed through our choice of the path L. The first two
items arc rather obviously the result of the initial conditions and
hence could be expected to pro\ide transients which die out as
t -> oo. This is in fact the case, as can be seen easily in the following
way : That branch of \/s is taken which is positive on the positive real
axis, and we operate always in the right half-plane. If, in addition,
s is in the upper half-plane it follows that i(Vl&s db <*>) has its real part
negative (o> being real). Consider now the contribution furnished by
the uppermost item in the square brackets. Since the exponential has
a negative real part on the semi-circular portion of the path L it is
clear that as t -+ + oo this part of the path makes a contribution
that tends to zero. The remaining portions of L, which lie on the real
180 WATER WAVES
axis, are then readily seen to make contributions which die out
like 1/t: this can be seen easily by integration by parts, for example,
or by application of known results about Fourier transforms. The
middle item in the square brackets has no singularity on the real
axis, so that the path L can be taken entirely on the real axis; thus,
in accordance with the remarks just made concerning the similar
situation for the first item, it is clear that this contribution also dies
out like 1/t. Thus for large t we obtain the following asymptotic re-
presentation for 99:
(6.7.15)
L Ss ~
Actually, the right hand side is the solution of the steady state
problem— as obtained, for example, in the paper of Lamb [L.2] and
by a different method by us in section 4.3 (although in a different form)
—when the condition at oo is the radiation condition stating that cp
behaves like an out-going progressing wave. The steady state solution
as obtained in section 4.3 actually was a little more awkward to obtain
directly through use of the radiation condition than it was to obtain
the solution (6.7.13) of the initial value problem. In particular, the
asymptotic behavior of an integral representation had to be investi-
gated in the former case also before the radiation condition could be
used. Thus we have seen in this special case that the radiation condi-
tion can be replaced by boundedness conditions (in the space varia-
bles, that is) if one treats an appropriate initial value problem instead
of the steady state problem.
Even though not strictly necessary — since (6.7.15) is known to
furnish the desired steady state solution— it is perhaps of interest to
show directly that the right hand side of (6.7.15) has the behavior
one expects for an out-going progressing wave when x -> + °o. The
procedure is the same as that used in discussing (6.7.14): The factor
cos sx is replaced by exponentials to obtain
(6.7.16)
By the same argument as above one sees that the first integral makes
a contribution that tends to zero as x -> + oo. The second integral
is treated by deforming the path L over the pole s = o>2/g into a path
M which consists of the positive real axis except for a semi-circle
in the lower half-plane. The contribution of the second integral then
fn fir p*v eisx 1 C esv e~lKX "I
<p(x,y;t)~-e*«\-±-\ J_L_ cfc + _ I ? -— -<fr .
0 \*™JL&-<»* ^2mJLg*-a>* ]
UNSTEADY MOTIONS 181
consists of the residue at the pole plus the integral over the path M .
But the contribution of the latter integral is, once more, seen to tend
to zero as x -> + oo because of the factor e~i8X. Thus y(x, y; t) behaves
for large x as follows:
o>2 /o>2 \
(I) — V — t[ — X—(Ot ]
(6.7.17) <p(x, y)~ — — eg e \g ',
SQ
and this does in fact represent a progressing wave in the positive
^-direction which, in addition, has the wavelength 2jrg/o>2 appropriate
to a progressing sine wave with the frequency co in water of infinite
depth.
6.8. Justification of the Method of Stationary Phase
In section 6.5 the method of stationary phase was used (and it will
be used again later on) to obtain approximations of an asymptotic
character for the solutions of a variety of problems when these
solutions are given by means of integrals of the form
(6.8.1)
C
=
J a
and the object is to obtain an approximation valid when the real
constant k is large. Since we make use of such approximate formulas
in so many important cases, it seems worth while to give a mathema-
tical justification of the method of stationary phase, following a pro-
cedure due to Poincard. The presentation given here is based upon the
presentation given by Copson [C.5].
PoineareS's proof requires the assumption that <p(z) and y>(z) are
regular analytic functions of the complex variable z in a domain
containing the segment S: a <£ x ^ b of the real axis in its interior.
(In what follows, we assume a and b to be finite, but an extension to
the case of infinite limits would not be difficult.) In addition <p(z) is
assumed to be real when z is real. These conditions are more restric-
tive than is necessary for the validity of the final result. For example,
the function y might also depend on &, provided that y(x, k) is not
strongly oscillatory, or singular, for large values of k. The assumption
of analyticity is also not indispensable. However, these generaliza-
tions would complicate both the formulation and proof of our theorem
without changing their essentials; consequently we do not consider
them here.
182 WATER WAVES
It will be shown that the main contributions to I(k) arise from the
points of S near those values of x for which (p'(x) = 0— that is, near
the points of stationary phase. The term of lowest order in the asymp-
totic development of I(k) with respect to k will then be obtained on
the basis of this observation. Kelvin himself offered a heuristic argu-
ment (cf. sec. 5 above) indicating why such a procedure should yield
the desired result.
Since <p' (z) is regular in the domain containing S, it follows that its
zeros are isolated. Hence S can be divided into a finite number of
segments on which (p(z) has either one stationary point or no stationary
point. We shall show first that the contribution to I(k) from a segment
containing no stationary point is of order l/k. Next it will be shown
that a segment containing any given point of stationary phase can be
found such that the contribution to the integral furnished by the
segment is of lower order than 1 /k9 and a formula for this contribution
will be derived. It turns out that this contribution of lowest order
is independent of the length of the segment containing the point
of stationary phase, provided only that the segment has been chosen
short enough. Once these facts have been proved, it is clear that the
lowest order contributions to the integral arc to be found by adding
the contributions arising at each of the points of stationary phase.
Suppose, then, that (p(x) has no stationary point on a segment
c ^ x ^ d of S. We may write
- f
Jc
dx - - ^X]- —
since <p'(x) ^Qinc^x^dby hypothesis. Integration by parts
then leads to the result
ik<p'(d) ik(p'(c)
d
with tp^x) = — (y/y)- Since
_. i (
ikjc
C
<£
Jc
dx
because of the fact that kq>(x) is real, it follows that the integral in the
above expression is bounded. Thus /x is indeed of order I/A:, as stated
above. It might be noted that this argument really does not require
the analyticity of <p and \p9 but only that the integrands be integrable
and that integration by parts may be performed. Infinite limits for the
integrals could also be permitted if cp(x) and \p(x) behave appropriately
at oo.
UNSTEADY MOTIONS 183
Suppose now that <p(x) has one stationary value at x = a in the
segment a — el ^ x ^ a + £i> sl > 09 i.e., <p'(# ) vanishes only at
x = a in this interval. Suppose, in addition, that the second deriva-
tive y" (x) does not vanish at x — a, and indeed is positive there:
<p"(<x.) > 0. (The case in which 9/'(a) is negative and the more critical
case in which 9?"(a) = 0 will be discussed later.) We shall show that
a positive number e ^ sl exists such that
(6.8.2) /,(*) = (*)«'*« dx =
oe-,
In other words, we shall show that a fixed segment of length %e con-
taining a exists such that its contribution to / is independent of e and
is of order l/\/k, with an error of order I/A;.
To prove these statements we begin by introducing new variables
as follows:
(6.8.3) x = a + u, <p(x) = y(a) + w(u).
Consider first the integral /2(&, e^:
(6.8.4) /a(fc, cx) - ^^W ^fcM)(tt) ^(a + M) du = ^^(a) J.
J-«i
It is convenient to write the integral J as the sum of two terms:
f° rei
(6.8.5) J == f«^(«i) y(a + MI) duj + elkw(u*> y(a + ^2) ^2
J-ej Jo
- Ji + t/2.
Since 99(0?) has a minimum at x — a, it follows that w(ul) is a positive
monotonic function in the interval — e1 ^ % ^ 0, and likewise
w(t42) in the interval 0 ^ i/2 ^ £x. Hence we may introduce a new
integration variable t, which is furthermore real, in each of the in-
tegrals, defined as follows:
5 — w(ul) in — e1 ^ % 5^ 0, and
(6,8.6) .
\t2 =-- w(u2) in 0 ^ u2 <^
In each interval t is taken as the positive square root. The integrals
«/! and J2 become, as one readily sees:
(6.8.7)
184 WATER WAVES
with tfj == Vw(— fii), and t2 = Vwfa). The functions %(£)» w2(tf) are
solutions of w(ui) = t2. For w(u) we have the power series develop-
ment
(6.8.8) w(u) = <p(oc + u) — 9?(a) = a9u2 + a3w3 + . . .
since w?(0) = w'(Q) = 0 (cf. (6.8.3)); in addition 2a2 = g/'(«) > °>
by assumption. We suppose that this series converges in a circle which
contains the entire interval — £2 ^ u ^ £2 in its interior, with
e2 < fii- Since J2 = w>(w2) we may write
for 0 <S u2 ^ £ and
— £2 =
Since a2 7^ 0 we may express the square roots as power series in ui
and then invert the series to obtain u^ and u2 as power series in t, as
follows:
(6.8.10)
u, = c,« + c2<2 + . . ..
with Cj = + V/2/9?"(a). Hence we may write
in which P^(t) and P2(0 are convergent power series. It may be that
these series do not converge up to the values ^ and t2 of the upper li-
mits of the above integrals Jl and J2 in (6.8.7). In that case we simply
assume the length of the segment is taken to be still less than 2e2 so
that the inversion of the series (6.8.8) is permissible and the series
Pi(t) and P2(0 converge up to appropriate values ?x and £2. It is
clear that numbers ?x and 12 with these properties exist. The integrals
Jt and J2 may now be written in the form*
(6.8.11)
J2
= f h e™* {^(a) + tP^t)} dt, and
j°_
- clV(a) pV*'8<tt + (t*e™*tPt(t)dt = J3 + J4.
jo jo
* The requirement of analyticity for (p and y is used to permit this simple
introduction of t as variable of integration. However, the existence of a finite
number of derivatives would clearlv have sufficed.
UNSTEADY MOTIONS
185
We proceed to study the integrals J3 and «/4. Upon introducing 6 — kt2
as new variable in J3 we obtain
_ ci^(a) r »e__ dQ
2 \/k J o \/0
But we may write
piQ
(6.8.12) — d8
(*Jcta pi® /*oo piQ /*oo piQ
—. de = \ — dB - \ —
Jo V^ Jo \/^ J kt2 \/v
by a known formula. The last integral can now be shown to be of
order l/\/& by integration by parts, as follows:
The first contribution on the right hand side is obviously of order
since 12 is a fixed number; as for the second, we have
pv
L-dB
and hence the second contribution is also of order l/\/k. Thus for J3
we have the result
(6.8.13)
J =
The integral ./4 is first integrated by parts to obtain
t
Jo
- P,(0)
and hence
(6.8.14) | J4 | ^ 1
| P,(0) |
P't(t) | di
and the right hand side is thus of order 1/fc. The integral Jl can ob-
viously be treated in the same way as J2 and with an exactly analogous
186 WATER WAVES
result; consequently we have from (6.8.13), and (6.8.14) for the integral
given in (6.8.4) the result
(6.8.15) !,<*, e3) = /,(*) = y(a) «'**w+i + 0
once £3 has been chosen small enough. One observes how it comes
about that the lowest order term is independent of the values of ?x and
?2, and hence of the length of the segment: the entire argument re-
quires only that ?x and £2 be any fixed positive numbers since one needs
only the fact that the products kl\ and kl\ grow large with k.
If (p (x) had been assumed to have a maximum at x — a, with
9/'(<x) < 0, the only difference would be that — k<p"(u.) and — rc/4
would appear in the final formula instead of + k(p"(<x.) and + vt/4.
Consequently, in all cases in which <p" (a) ^ 0 we have
(6.8.16) I,(k) = y(«) (—^L
U \<p (a)
and the sign of the term w/4 should agree with the sign of <p"(oc).
Finally, in case <p"(oc) = 0, but 9?'"(oc) ^ 0 it is not difficult to
derive the appropriate asymptotic formula for /(ft). In fact, the steps
are nearly identical with those taken just now for the case <p"(oc) ^ 0.
One introduces x = a + u, <p(x) = y(a) + ^(w) as before and then
makes use of power series in the variable t defined by 23 = w(u ) in the
same way as above. The result is, for e sufficiently small:
(6.8.17) I2(k) = f a+V**<*) y(x) dx
= f a+
where F(\) refers to the gamma function. Hence the contribution
arising from the stationary point is now of a different order of magni-
tude, i.e., of order 1/&1/3 instead of I/ft1/2. This fact is of significance
in the case of the ship wave problem which will be treated later.
Naturally the lowest order terms in I(k) consist of a sum of terms
furnished by the contributions of all of the points of stationary phase
in the interval S. It is important enough to bear repetition that if no
such points exist, then I(k) is in general of order 1/fc.
In case a stationary point falls at an end point x = a or x = b of
the interval of integration, one sees readily that the contribution
furnished by such a point to I(k) is the same as that given above in
UNSTEADY MOTIONS 187
case <p" ^ 0 except that a factor 1/2 would appear in the final result.
On the other hand, if g/' = 0 but <p'" ^ 0 at an end point, then the
contribution differs in phase as well as in the numerical factor from
the contribution given above in (6.8.17).
6.9. A time-dependent Green's function. Uniqueness of unsteady
motions in unbounded domains when obstacles are present
In sec. 6.2 above the uniqueness of unsteady wave motions for
water confined to a vessel of finite dimensions was proved. More gener-
al results have been obtained by Kotik [K.17], Kamp£ de Feriet and
Kotik [K.l], and Finkelstein [F.3] with regard to such uniqueness
questions. In the present section a rather general uniqueness theorem
will be proved, following the methods of Finkelstein, who, unlike the
other authors mentioned, obtains uniqueness theorems when obstacles
arc present in the water. The essential tool for this purpose is a time-
dependent Green's function, which is in itself of interest and worth
while discussing for its own sake quite apart from its use in deriving
uniqueness theorems. With the aid of such a function, for example, all
of the problems solved in the preceding sections can be solved once
more in a different fashion, and still other and more complicated un-
solved problems can be reduced to solving an integral equation, as
we shall see.
We shall derive the time-dependent Green's function in question
for the case of three-dimensional motion in water of infinite depth,
although there would be no difficulty to obtain it in other cases as
well. The Green's function G in question is required to be a harmonic
function in the variables (x, y, z) with a singularity of appropriate
character at a certain point (£, 77, £) which is introduced at the time
t = r and maintained thereafter; thus G depends upon f , 77, f ; r and
x, y, z; t: G = G(£ , r], f ; r \ x, y, z; t). In fact, G is the velocity poten-
tial which yields the solution of the following water wave problem:
A certain disturbance is initiated at the point (£, 77, £) at the time
t — r. The pressure on the free surface of the water is assumed to be
zero always, and at the time t — r the water is assumed to have been
at rest in its equilibrium position. Since G is a harmonic function in
x, y, z it is reasonable to expect that the correct singularity to impose
at the point (£, rj, £) in order that it should have the properties one
likes a Green's function to have is that it behaves there like I/R, with
R ^ V^-r-(i?»)* -Hfa). Thus G should satisfy the
188 WATER WAVES
following conditions: It should be a solution of the Laplace equation
(6.9.1) Gxx + Gyy + GZZ = 0 for - oo < y < 0, t ^ T,
satisfying the free surface condition
(6.9.2) Gtt + gGy = 0, y = 0.
At oo we require G, Gt and their first derivatives to be uniformly
bounded at any given time t. (Actually, they will be seen to tend to
zero at oo.) At the point £, 77, f we require
(6.9.3) G — — to be bounded.
R
As initial conditions at the time t — r we have (cf. sec. 6.1)
(6.9.4) G = Gt = 0 for t = r, y = 0.
As we shall see later on, these conditions determine G uniquely.
We proceed to construct the function G explicitly. As a first step
we set
(6.9.5) G(f, 17, £; T | v, y, *; t) = A(£, 17, f | *, y, a) +
5(1, *?, C; T | a?, /y, z; «)
with ^f defined by
(6.9.6) A = 1 - -L with «' = V(f - V)»Tfo "+2/F+ G ^^-
/t /v
Thus ^4 contains the prescribed singularity, and we may require B
to be regular. Since A is a harmonic function, it follows that B is
harmonic; in addition, B satisfies the free surface condition
(6.9.7) Btt+gBy =
He — x)- -t- r)' -t- (t, — z)"]°"
d i
at « = 0'
— r
9*7 [(I - *)2 + *?2 + (C ~ *)2]1/2
as one can readily verify. To determine B from this and the other con-
ditions arising from those imposed on G it would be possible to employ
the Hankel transform in exactly the same way as the Fourier trans-
form was used in preceding sections. However, it seems better in the
present case to proceed directly by using the special, but well-known,
Hankel transform for the function e-b*/s (cf., for example, Sneddon
[S.ll], p. 528); this yields the formula
UNSTEADY MOTIONS 189
i r°°
(6.9.8) - = e~bs J0(as) ds,
Va2 + b* Jo
valid for b > 0. By means of this formula the right hand side of (6.9.7)
can be written in a different form to yield
(6.9.9) Btt + gBy = 2g JL f V< J0(sr) ds = 2g f *«>*• JQ(sr) ds
fyJo Jo
at y = 0
valid for r\ < 0 and with
(6.9.10) r = V(S - x)* + (C - *)2.
Since B is a harmonic function in x9 y, z, it would seem reasonable to
seek it among functions of the form
(6.9.11) B = I™ sT(t, s)e(*+ti' J0(sr) ds,
Jo
which are harmonic functions. The free surface condition (6.9.9) will
now be satisfied, as one can easily sec, if T(t) satisfies the differential
equation
(6.9.12) Ttt +g*T = 2g.
The function T is now uniquely determined from (6.9.12) and the
initial conditions T = Tt = 0 for t = T derived from (6.9.4); the
result is
(6.9.13) T(t, s) = 2 LT
Thus we have for G the function
(6.9.14)
+ 2 f "VtH-*) [i _ cos
Jo
and it clearly satisfies all of the conditions prescribed above, except
possibly the conditions at oo, which we shall presently investigate in
some detail because of later requirements. Before doing so, however,
we observe the important fact that G is symmetrical not only in the
space variables f , 77, £ and #, y, z9 but also in the time variables r and
/, i.e. that
190 WATER WAVES
(6.9.15) G(£, r), f ; r | a?, y, *; t) = G(x, y,z\t\ f, ??, f ; r) and
We turn next to the discussion of the behavior of G at oo. Consider
first the function A == l/R — 1/i?'. This function evidently will
behave at oo like a dipole; hence if a represents distance from the
origin it follows that A and its radial derivative Aa behave as follows
for large a:
A ~1(T2
On the free surface where y — 0 we have
A = 0 for y = 0,
l/a3 for y = 0 and large a.
To determine the behavior of B— i.e. of the integral in (6.9.14)— we
expand [1 — cos Vgs (r — t)] in a power series in r — t and write
(6.9.18) B = 2
It is clearly legitimate to integrate term-wise for y negative. The
formula (6.9.8) can be expressed in the form
(6.9.19)
4= (*
R Jo
and from it we obtain
= r $n
with // = cos 0, by a well-known formula for spherical harmonics.
It follows, since Pn(//) are bounded functions, that the leading term
in the asymptotic expansion of B arises from the first term in the
square bracket. Hence the behavior of B is seen from (6.9.20) for the
case- n = I to be given by
(6.9.21) l?~l/(r2,
for a large and any fixed values of r and t. The derivative By is seen,
also from (6.9.20), to behave like I/a3 and the derivative Br also can
be seen to behave like I/a3; thus the radial derivative Ba behaves in
the same way and we have
(6.9.22) Ba ~ I/a3, Bv ~ I/a3.
UNSTEADY MOTIONS 191
Summing up, we have for the Green's function G the following behavior
at oo:
G ~ I/a2
(6.9.23)
Ga ~ I/a3
Gy ~ I/a3.
All of these conditions hold uniformly for any fixed finite ranges in
the values of r and t.
We turn next to the consideration of a water wave problem of very
general character, as follows. The space y < 0 is filled with water and
in addition there are immersed surfaces St of finite dimensions having
a prescribed motion (which, of course, must of necessity be a motion of
small amplitude near to a rest position of equilibrium). The pressure
on the free surface Sf is prescribed for all time, and the initial position
and velocity of the particles on the free surface and the immersed
surfaces are given at the time t = 0. At infinity the displacement and
velocity of all particles are assumed to be bounded. The resulting
motion can be described for all times t > 0 in terms of a velocity
potential 0(x9 y, z; t) which satisfies conditions of the kind studied in
the first section of this chapter; these conditions are:
(6.9.24) V2,V(Z0 = 0
in the region R consisting of the half space y < 0 exterior to the im-
mersed surfaces S^ On the free surface the condition
(6.9.25) 0tt+g0v= ~-Pt=:P(x909z;t)9 t > 0, y = 0
Q
is prescribed, with p the given surface pressure (cf. (6.1.1 ) and (6.1.2)).
At the equilibrium position of the immersed surfaces the condition
(6.9.26) 0n = V on Si9 t ^ 0,
with V the normal velocity of Si9 is prescribed. The initial position of
St at t = 0 is, of course, assumed known, and for the initial conditions
otherwise we know (cf. 6.1) that it suffices to prescribe 0 and 0t on
the free surface at t — 0:
(6.9.27) j^O^O)^*,*)
I #«(*,0,*;0) = /,(*,*).
At oo we assume that 0, 0t and their first derivatives are uniformly
bounded.
192
WATER WAVES
We proceed now to set up a representation for the function 0 by
using the Green's function obtained above. In case there are no im-
mersed surfaces this representation furnishes an explicit solution of
the problem, and in the other cases it leads to an integral equation for
it. In all cases, however, a uniqueness theorem can be obtained.
To carry out this program we begin, in the usual fashion, by applying
Green's formula to the Green's function G and to 0t (rather than 0)
in a sphere centered at the origin of radius a large enough to include
the immersed surfaces and the singular point (|, 77, £) of the Green's
function minus a small sphere of radius e centered at the singular point.
Since G and 0t are both harmonic functions and G behaves like l/R
at the singular point, it follows by the usual arguments in potential
theory that 0t(x9 y> z; t) is obtained in the form of a surface integral,
as follows:
(6.9.28)
0t(x, y,
= -?- (T
4rc JJ
(G0tn ~ &tGn) dS.
The symmetry of G has been used at this point. The integration varia-
bles are f, 77, £. Even though 6? depends on the difference t — r the
integral in (6.9.28) depends only on t; that is, only the singular part of
the behavior of G matters in applying Green's formula, and the re-
sulting expression for 0t depends only on the time at which 0t and
0tn are measured. The surface integral is taken over the boundary
of the region just described (cf. Fig. 6.9.1), and n is the normal taken
Jk y
sfl
Fig.6.9.1. Domain for application of Green's formula
outward from the region. The boundary is composed of three different
parts: the portion of the sphere Sa of radius a lying below the plane
y = 0, the part Sf of the plane y = 0 cut out by the sphere Sa9 and
UNSTEADY MOTIONS
193
the immersed surfaces St (which might possibly cut out portions of the
plane y = 0).
It is important to show first of all that the contribution to the
surface integral provided by Sa tends to zero as a -> oo, and that the
integral over Sf exists as a -> oo. The second part is readily shown:
The integrand to be studied is G0ty — 0tGy. From the symmetry
of G and (6.9.23) we see that the above integrand behaves like I/a2
for large a since 0ty and 0t are assumed to be uniformly bounded at
oo ; hence the integral converges uniformly in t and r for any fixed
ranges of these variables. To show that the integral of G0to — 0tGa
over Sa tends to zero for a -> oo requires a lengthier argument. Con-
sider first the term 0tGa. Since Ga behaves like I/a3 for large a while
0t is bounded, it is clear that the integral of this term behaves like
I/a and hence tends to zero as a -> oo. The integral over the remaining
term is broken up into two parts, as follows:
(6.9.29) !(0iaGdS = f "f*
JJ Jo Jinl:
(a 12) +6
+
,Ga*sinOded(o
>iaGat sin 0 dd do).
f2n f(nl2)+6
I *
JO Jnj2
The integrations arc carried out in polar coordinates, and d is a small
angle (cf. Fig. 6.9.2); the second integral represents the contribution
from a thin strip of the sphere Sa adjacent to the free surface. Since
Fig. 6.9.2. The sphere Sa
0ta is bounded and G behaves like I/a2 for large a, it is clear that the
absolute value of the second contribution (i.e. that from the thin strip)
can be made less than e/2, say, if d is chosen small enough. Once d has
been fixed, it can be seen that the contribution of the remaining part
194 WATER WAVES
of Sa can also be made less than e/2 in absolute value if a is taken large
enough. If this is once shown it is then clear that the integral in ques-
tion vanishes in the limit as a -> oo. The proof of this fact is, however,
not difficult: we need only observe that 0t is by assumption bounded
at oo and it is a well-known fact* that 0ta then tends to zero uniformly
like 1 1 a along any ray from the origin which makes an angle ^ 6 with
the plane y = 0. Thus the integrand in the first term of (6.9.29)
behaves like I/a and it therefore can be made arbitrarily small by
taking a sufficiently large. Thus for 0t we now have the representation
(6.9.30) 0t(x, y, z; t) = i- [((G0tr) - 0,
Si
in which it is, of course, understood that any parts of the plane y — 0
cut out by St are omitted in the first integral. The next step is to
integrate both sides of (6.9.30) with respect to t from 0 to r. The result
is
(6.9.31) 0(x9 y, z; r) - 0(x9 y, z; 0)
• * If [J
»7=*0 S,
= 1 ff [ (CO*, + l 0tGt) T - f^G, + l0ttGt) dt\ d£ d'C + I
to JJ L g o Jo S J
»7=0
when Gtt + gGy = 0 for y = 0 is used (cf. (6.9.2)) and / is intended
as a symbol for the integral over S^ We have G = G t = 0 for t = r;
while for t = 0 we have 0t = /2, and 0y \ <=s0 uniquely determined by /j**
from the conditions (6.9.27). In addition, we know that 0y+(l/g)0tt
= ( l/g)P for * > 0 from (6.9.25). It follows that (6.9.31 ) can be written
in the form
* One way to prove it is to use the Poisson integral formula expressing <&t
at any point in terms of its values on the surface of a sphere centered at the point
in question. Differentiation of this formula yields for any first derivative of <Pt
an estimate of the form M/6 where M depends only on the bound for 3>t on the
sphere and b is the radius of the sphere. Finally, since our domain for 6 > (;r/2)-h<5
contains spheres of arbitrarily large radius at points arbitrarily far from the
origin, the result we need follows.
** Since <P(xt t/, z; 0) is harmonic, it is uniquely determined by its boundary
values on y'— 0 and the boundedness conditions at oo.
UNSTEADY MOTIONS 195
(6.9.32) 0(x,y,z;r) - 4>(x, y, z; 0)
1
»;=0
- /2G,) I + ("- GtPdt\ di- dl
S |«-o Jog J
We now sec that if there are no immersed surfaces *S\ an explicit
solution 0(x9 y, z\ r) is given at once by (6.9.32) in terms of the initial
conditions, which fix 0y |f=0 and /2, and the condition on the free
surface pressure fixing P — in fact, our general argument shows that
every solution having the required properties is representable in this
form. Consequently, the uniqueness theorem is proved for these cases.
In particular, the Green's function constructed above is therefore
uniquely determined since its regular part, B, satisfies the conditions
imposed above on 0.
In case there are immersed surfaces present the equation (6.9.32)
does not yield the solution 0, but it docs yield an integral equation
for it in the following way (which is the standard way of obtaining
an integral equation for a harmonic function satisfying various
boundary conditions): One goes back to the derivation of (6.9.30),
but considers that the singularity is at a point (x, y, z) of *S\. If St is
sufficiently smooth (and we assume that it is) the equation (6.9.30)
still holds, except that the factor l/4jt is replaced by I/2n, and 0 is
then of course given only on St. The integration on t from 0 to r is
once more performed, and an equation analogous to (6.9.32) is ob-
tained; it can be written in the form
(6.9.33) 0(x, i/, *; r) - F(x, y, z; r) - ~ JJ [" JT
dS
with F a known function obtained by adding together what corres-
ponds to the first two integrals on the right hand side of (6.9.32).
As we see, this is an integral equation for the determination of
0(x, y, z-9 r) on S{. If it were once solved, the value of 0 on St could
be used in (6.9.32) to furnish the values of 0 everywhere.
We may make use of (6.9.32) to obtain our uniqueness theorem in
the following fashion. Suppose there were two solutions 0l and $2.
Set 0 = 0l — 02. Then 0 satisfies all of the conditions imposed on
01 and 02 except that the nonhomogeneous boundary conditions and
196 WATER WAVES
initial conditions are now replaced by homogeneous conditions, i.e.
/2 = P = 0; fl = 0 and hence 0y \toQ = 0 since 0(x9 y, z; 0) is a
harmonic function which vanishes for y = 0, and 0tn = 0 since
0n = 0 on St. Thus for 0 we would have the integral representation:
(6.9.34) 0(x, y, z; r) = - i- ff f f**^
5,
Since Gn behaves at oo like I/or3 (cf. (6.9.23)) and values of0t on the
bounded surfaces Si are alone in question, it follows that 0 also be-
haves like I/a3 at oo for any fixed r since the surfaces Si are bounded.
The derivatives of 0 could also be shown to die out at oo at least as
rapidly as I/a3 since the derivatives of G could be shown to have this
property— for example, by proceeding in the fashion used to obtain
(6,9.23).
As a consequence the following function of t (essentially the energy
integral) exists:*
(6.9.35) E(t) = - [IT [01 +01+ 0\] dxdydz + -L jT#f dxdz.
s
,
Differentiation of both sides with respect to t yields
(6.9.36) E'(t) = [(*«).*. + (**),(*)» + (*«),(*).] dxdydz
R
- &t&tt
g
s
S,
by application of Green's first formula, with B — St + Sf the
boundary of R. But 0n = 0 on S^ and 0W = 0y = — ( 3 /g)<P« on S/.
It follows therefore that E'(t) = 0 and E = const. But 0 = 0 at
t = 0 and hence JB = 0 from (6.9.85). It follows that 0X9 0y, 0Z are
identically zero, and 0 thus also vanishes identically. Hence 0± = 02
and our uniqueness theorem is proved.
* It should perhaps be noted that the energy integral for the original motions
need not, and in general will not exist, since the velocity potential and its derivati-
ves are required only to be bounded at oo.
SUBDIVISION C
Waves on a Running Stream. Ship Waves
In this concluding section of Part II made up of Chapters 7, 8, and
9, we treat problems which involve small disturbances on a running
stream with a free surface; that is, motions which take place in the
neighborhood of a uniform flow, rather than in the neighborhood of
the state of rest, as has been the case in all of the preceding chapters
of Part II. In Chapter 7 the classical problems concerning steady
two-dimensional motions in water of uniform (finite or infinite) depth
are treated first. It is of considerable interest, however, to consider
also unsteady motions (which seem to have been neglected hitherto)
both because of their intrinsic interest and because such a study
throws some light on various aspects of the problems concerning
steady motions. In Chapter 8 the classical ship wave problem, in
which the ship is idealized as a disturbance concentrated at a point
on the surface of a running stream, is studied in considerable detail.
In particular, a method of justifying the asymptotic treatment of the
solution through the repeated use of the method of stationary phase is
given, and the description of the character of the waves for both
straight and curved courses is carried out at length. Finally, in Chapter
1) the problem of the motion of a ship of given hull shape is treated
under very general conditions: the ship is assumed to be a rigid body
having six degrees of freedom and to move in the water subject only
to the propeller thrust, gravity, and the pressure of the water, while
the motion of the water is not restricted in any way.
197
CHAPTER 7
Two-dimensional Waves on a Running Stream
in Water of Uniform Depth
As indicated in Fig. 7.0.1 we consider waves created in a channel
' y
y= -h
Fig. 7.0.1. Waves on a running stream
of constant depth h, when the stream has uniform velocity U in the
positive ^-direction in the undisturbed state. Such a uniform flow can
readily be seen to fulfill the conditions derived in Chapter 1 for a
potential flow with y = 0 as a free surface under constant pressure.
We assume that the motions arising from disturbances created in the
uniform stream have a velocity potential 0(x, y\ t)9 and we set
(7.0.1) 0(x9 y; t) = Ux + <p(x, y; t)9 - oo < x < oo, - h< y < r).
Since </>(#, y; t) is a harmonic function of x and y it follows that
<p(x9 y\ t) is also harmonic:
(7.0.2) W = <>•
The function <p(x, y;t) is assumed to yield a small disturbance on the
running stream, and we interpret this to mean that <p and its deriva-
tives are all small quantities and that quadratic and higher order
terms in them can be neglected in comparison with linear terips. We
assume also that the vertical displacement y = rj(x; t) of the free
198
TWO-DIMENSIONAL WAVES 199
surface, as measured from the undisturbed position y — 0, is also a
small quantity of the same order as <p(x, y; t). Under these circum-
stances the dynamic free surface condition as given by Bernoulli's
law (cf. (1.4.6)) and the kinematic free surface condition (cf.( 1.4.5))
take the forms
(7.0.3) L + gr, + <pt + Upx + - I/* = 0
at y = 0,
1 O I • I V • I M/ I f-
Q 2
(7.0.4) r,t
when quadratic terms in <p and 77 are neglected and an unessential
additive constant is ignored in (7.0.3).* At the same time, it is proper
and consistent in such an approximation to satisfy the free surface
conditions at y = 0 instead of at the displaced position y — r\. (The
reason for this is explained in Chapter 2 —actually only for the case
[7 = 0, but the discussion would be the same in the present case.)
At the bottom y = — h we have the condition
(7.0.5) (pv = 0 at y = — h.
In case the channel has infinite depth we replace (7.0.5) with
(7.0.5)' y and its derivatives up to second order are bounded at
y = — oo.
In addition to the conditions (7.0.2) to (7.0.5) it is necessary also to
postulate conditions at x = ± <x> and, unless the motion to be studied
is a steady** motion with <p independent of t, it is also necessary to
impose initial conditions at the time t = 0. The cases to be treated in
the remainder of this chapter differ with respect to these various
types of conditions, and we shall formulate them as they are needed.
7.1. Steady motions in water of infinite depth with p = 0 on the
free surface
If the disturbance potential <p is independent of t, and if p = 0 on
the free surface it follows that 99(0?, y) satisfies the conditions
* It is perhaps worth noting explicitly that it would be inappropriate to
assume that U, the velocity of the stream, is a small quantity of the same order
as r] and (p: to do so would lead to the elimination of the terms in U and the
resulting theory would not differ from that of the preceding chapters.
** In this chapter the term "steady motion" is used in the customary way to
describe a flow which is the same at each point in space for all times. In the
preceding chapters we have sometimes used this term (in conformity with esta-
blished custom in the literature dealing with wave propagation) in a different
sense.
200 WATER WAVES
(7.1.1) VV = 0, _ oo <y ^0,
J72
(7.1.2) <pv + — <pxx = o, y = °-
s
In addition, we require that
(7.1.3) 9? and its derivatives up to second order are bounded at oo,
though this condition is more restrictive than is necessary. The second
of these conditions was obtained from (7.0.3) and (7.0.4) by differen-
tiating (7.0.3) and eliminating 77.
It is interesting to find all functions q>(x, y) satisfying these condi-
tions, and it is easy to do so following the same arguments as were
used in Chapter 3.1. Using (7.1.1) we may re-write (7.1.2) in the form
U*
(7.1.4) v*-—v,i, = °> y = o-
o
(This of course makes use of the fact that <p is harmonic for y = 0,
which we assume to be true. One could easily show, in fact, that the
free surface condition (7.1,2) permits an analytic continuation of 9?
over y = 0, so that 9? is actually harmonic in a domain including
y = 0 in its interior.) We observe that (7.1.4) is the same condition
on <pv as was imposed on the function called <p in Chapter 3, and we
proceed as we did there by introducing a harmonic function y(x9 y)
through
U2
(7.1.5) V =<Pv - —<f>yy , 2/^0-
e
This function vanishes on y — 0, and can therefore be continued
analytically by reflection into the upper half plane. Since (p and its
derivatives were assumed to be bounded in the lower half plane, it
follows that \p is bounded in the entire plane and hence by Liouville's
theorem it is a constant; hence \p vanishes identically since \p = 0
for y = 0. Thus we have for y>y a differential equation given by (7.1.5)
with \f = 0, and it has as its only solutions the functions
(7.1.6) 9>v = c(
Since <py is also a harmonic function, it follows that c(x) is a solution
of the differential equation
/ a \ 2
c = 0.
TWO-DIMENSIONAL WAVES 201
Hence <p is given by
(7.1.8) <p(x, y) = Aeu*vcos /-L * + aj + c^x)
with A and a constants and c^x) an arbitrary function of x. By making
d2c
use of (7.1.2), however, one finds that — - = 0, and hence that
dx2
c± == const, since cp is bounded at oo. There is no loss of generality in
taking cl = 0. The only solutions of our problem are therefore given
by
(7.1.9) <p(x, y) = Aeu*vcos UL + A
Thus the only steady motions satisfying our conditions, aside from
a uniform flow, are periodic in x with the fixed wave length A given by
772
(7.1.10) A = 2n — .
g
The amplitude and phase of the motions are arbitrary. If we were to
observe these waves from a system of coordinates moving in the x-
direction with the constant velocity E7, we would see a train of pro-
gressing waves given by
<p = Aemv cos m (x -\~ Ut)
with
g 2n
m = — = — .
C/2 A
These waves are identical with those already studied in Chapter 3
(cf. sec. 3.2). The phase speed of these waves would of course be the
velocity U and the wave length A would, as it should, satisfy the rela-
tion (3.2.8) for waves having this propagation speed. In other words,
the only waves we find are identical (when observed from a coordinate
system moving with velocity U ) with the progressing waves that are
simple harmonic in the time and which have such a wave length that
they would travel at velocity U in still water.
7.2. Steady motions in water of infinite depth with a disturbing
pressure on the free surface
The same hypotheses are made as in the previous section, except
that we assume the pressure on the free surface to be a function
202
WATER WAVES
over the segment — a ^ x ^ a and zero otherwise, as indicat-
ed in Fig. 7.2.1. The free surface condition, as obtained from (7.0.8)
p«0
-o
-i-a
U
Fig. 7.2.1. Pressure disturbance on a running stream
and (7.0.4) by eliminating rj and assuming r\ and 9? to be independent
of t, is now given by
(7.2.1)
4. JL m = — J^l , on z/ = 0,
^ U^v UQ
as one readily verifies. We prescribe in addition that <p and its first
two derivatives are bounded at oo.
The solutions <p of our problems are conveniently derived by intro-
ducing the analytic function f(z) of the complex variable z = x + iy
whose real part is 9?:
(7.2.2) /(a) = <p(x, y) + iy(x9 y).
Since yy = — - yx, the condition (7.2.1) can be put in the form
(7.2.3)
0 p
— ~i V = — ~r- + const., on y = 0,
and the constant can be taken as zero without loss of generality, since
adding a constant to p can not affect the motion.
We consider now only the case in which the surface pressure p is a
constant p = p0 over the segment \x\ ^ a, and zero otherwise. Since
this surface pressure is discontinuous at x = ± a, it is necessary to
admit a singularity at these points; we shall see that a unique solution
of our problem is obtained if we require that q> is bounded at these
points while <px and <py behave like 1/r1"*, e > 0, with r the distance
TWO-DIMENSIONAL WAVES 203
from the points x = ± a on the free surface. (This singularity is
weaker than the logarithmic singularity of <p appropriate at a source
or sink.)
In terms of f(z), the free surface condition (7.2.3) clearly can be put
in the form
(7.2.4)
(ifz - A) / =
^ '
for Jm a = 0.
, \x\ > a
The device of applying the boundary condition in this form seems to
have been used first by Keldysh [K.21]. We now introduce the ana-
lytic function F(z) defined in the lower half plane by the equation
(7.2.5) F(*) = »y.- A/.
This function has the following properties: 1) Its imaginary part is
prescribed on the real axis. 2) The first derivatives of its imaginary
part are bounded at oo, since the first two derivatives of 99 are assumed
to have this property and hence fzz and fz are bounded in view of the
Cauchy-Riemann equations. 3) Near z = ±o its imaginary part be-
haves like l/\z ^r a\l~e, £ > 0, as one readily sees. It is now easy to
show that F(z) is uniquely determined,* within an additive real con-
stant, as follows: Let G — JPX — F2 be the difference of two functions
satisfying these three conditions. Jm G then vanishes on the entire
real axis, except possibly at the points (± #> 0), and G can therefore
be continued as a single-valued function into the whole plane except
at the points (± a, 0). However, the singularity prescribed at the
points (± a, 0) is weaker than that of a pole of first order, and hence
the singularities at these points are removable. Since the first deriva-
tives of Jm G are bounded at oo, it follows from the Cauchy-Riemann
equations that Gz is bounded at oo. Hence Gz is constant, by Liouville's
theorem, and G is the linear function: G = cz + d. Since Jm G = 0
on the real axis, it follows that c and d are real constants. However,
a term of the form cz + d on the left hand side of (7.2.5) leads to a
a
term of the form OLZ + 0, with — — a = c, in the solution of this
* In Chapter 4, the function F(z) given by (4.3.10) had a real part which
satisfied identical conditions except that the condition 2) is slightly more restric-
tive in the present case.
204 WATER WAVES
equation for /(*), and since f(z) is assumed to be bounded at oo. it
follows that c = 0.
We have here the identical situation that has been dealt with in
sec. 3 of Chapter 4, except that it was the real part of the function
F(z), rather than the imaginary part, that was prescribed on the real
axis, and we can take over for our present purposes a number of the
results obtained there. The function F(z), now known to be uniquely
determined within an additive real constant, is given by
(7.2.6) *(,
,
UQTt Z + a
which differs from F(z) as given by (4.3.12) essentially only in the
factor i— as it should. In any case, one can readily verify that F(z)
satisfies the conditions imposed above. We take that branch of the
logarithm that is real for z real and \z\ > a, and specify a branch cut
starting at z = — a and going to oo along the positive real axis. The
equation (7.2.5) is now an ordinary differential equation for the func-
tion f(z) which we are seeking.
The differential equation (7.2.5) has, of course, many solutions,
and this means that the free surface condition and the boundcdness
conditions at oo and at the points (± a, 0) are not sufficient to ensure
that a unique solution exists. In fact, it is clear that the non-vanishing
solution of the homogeneous problem found in the preceding section
could always be added to the solution of the problem formulated up
to now. A condition at oo is needed similar to the radiation condition
imposed in the analogous circumstances in Chapter 4. In the present
case, the solution can be made unique by requiring that the dis-
turbance created by the pressure over the segment | x \ fS a should
die out on the upstream side of the channel, i.e. at x — — oo. The only
justification for such an assumption— aside from the fact that it
makes the solution unique— is based on the observation that one never
sees anything else in nature.* In sec. 7.4 we shall give a more satis-
factory discussion of this point which is based on studying the un-
steady flow that arises when the motion is created by a disturbance
initiated at the time t = 0, and the steady state is obtained in the
limit as t -> oo. In this formulation, the condition that the motion
* Lamb [L.3], p. 399, makes use, once more, of the device of introducing
dissipative forces of a very artificial character which then lead to a steady state
problem with a unique solution when only boundedness conditions are prescribed
at oo.
TWO-DIMENSIONAL WAVES 205
should die out on the upstream side is not imposed; instead, it turns
out to be satisfied automatically.
A solution of the differential equation (7.2.5) (in dimensionless
form) has been obtained in Chapter 4 (cf. (4.3.13)) which has exactly
the properties desired in the present case; it is:
(7.2.7) /(*) = -2±-e-&[' & log t-^ dt,
t + a
The path of integration (cf. Fig. 4.3.1) comes from iao along the posi-
tive imaginary axis and encircles the origin in such a way as to leave
it and the point (— a, 0) to the left. That (7.2.7) yields a solution of
(7.2.5) is easily checked. One can also verify easily that q> = &ef(z)
satisfied all of the boundary and regularity conditions, except perhaps
the condition at oo on the upstream side. In Chapter 4, however, it
was found (cf. (4.3.15)) that f(z) behaves at oo as follows:
O /— \
for Ste z < 0,
,
Thus /(s) dies out as x -> — oo, but there are in general waves of
nonzero amplitude far downstream, i.e. at x = + oo. The uniquely
determined harmonic function 9? = 9te f(z) is now seen to satisfy all
conditions that were imposed.
The waves at x = + oo are identical (within a term of order I/a)
with the steady waves that we have found in the preceding section to
be possible when the stream is subject to no disturbance (cf. (7.1.9)),
and the wave far downstream has the wave length A = 2nU2/g.
However, we observe the curious and interesting fact (pointed out by
Lamb [L.3], p. 404) that this wave may also vanish: clearly if
ga/U2 = nn, n — 1, 2, . . ., 99 = Ste f(z) vanishes downstream as well
as upstream, and this occurs whenever 2a/A is an integer, i.e. whenever
the length of the segment over which the disturbing pressure is applied
is an integral multiple of the wave length of a steady wave in water of
velocity U (with no disturbance anywhere). This in turn gives rise to
the observation that there exist rigid bodies of such a shape that they
create only a local disturbance when immersed in a running stream:
one need only calculate the shape of the free surface— which is, of
course, a streamline— for ga/U2 = nn, take a rigid body having the
206 WATER WAVES
shape of a segment of this surface and put it into the water. (Involved
here is, as one sees, a uniqueness theorem for problems in which the
shape of the upper surface of the liquid, rather than the pressure, is
prescribed over a segment, but such a theorem could be proved along the
lines of the uniqueness proof of the analogous theorem for simple har-
monic waves given by F. John [J.5]. ) This fact has an interesting physi-
cal consequence, i.e., that such bodies are not subject to any wave re-
sistance (by which we mean that the resultant of the pressure forces
on the body has no horizontal component) while in general a resistance
would be felt. This can be seen as follows: Observe the motion from
a coordinate system moving with velocity U in the ^-direction. All
forces remain the same relative to this system, but the wave at + oo
would now be a progressing wave simple harmonic in the time and
having the propagation speed — U9 while at — oo the wave ampli-
tude is zero. Thus if we consider two vertical planes extending from
the free surface down into the water, one far upstream, the other far
downstream we know from the discussion in Chapter 3.3 that there is
a net flow of energy into the water through these planes since energy
streams in at the right, but no energy streams out at the left since
the wave amplitude at the left is zero. Consequently, work must be
done on the water by the disturbance pressure and this work is done
at the rate RU = F, where R represents the horizontal resistance
and F the net energy flux into the water through two planes contain-
ing the disturbing body between them. Thus if .F = 0— which is the
case if the wave amplitude dies out downstream as well as upstream—
then R = 0. This result might have practical applications. For exam-
ple, pontoon bridges lead to motions which are approximately two-
dimensional, and hence it might pay to shape the bottoms of the
pontoons in such a way as to decrease the wave resistance and hence
the required strength of the moorings. However, such a design would
yield an optimum result, as we have seen, only at a definite velocity of
the stream; in addition, the wave resistance is probably small com-
pared with the resistance due to friction, etc., except in a stream
flowing with high velocity.
We conclude this section by giving the solution of the problem of
determining the waves created in a stream when the disturbance is
concentrated at a point, i.e. in the case in which the length 2a of the
segment over which the pressure p0 is applied tends to zero but
lim 2p^a = P0. The desired solution is obtained at once from (7.2.7);
it is:
(7.2.9) /(«) = - e
TWO-DIMENSIONAL WAVES 207
ig.
f* I igt
\ L &* dt.
J ioo t
This solution behaves like I/* far upstream and like (— 2P0/C7g)
exp {— igz/U2} far downstream. Note that the amplitude downstream
does not vanish for any special values of U in this case. It is perhaps
also of interest to observe that f(z) behaves near the origin like i log z,
and hence the singularity at the point of disturbance has the character
of a vortex point; we recall that the singularity in the analogous case
of the waves created by an oscillatory point source that were studied in
Chapter 4 had the character of a source point, since f(z) behaved like
log z rather than like ilogz (cf. 4.8.28)), with a strength factor
oscillatory in the time. When one thinks of the physical circumstances
in these two different cases one sees that the present result fits the
physical intuition.
7.3. Steady waves in water of constant finite depth
In water of constant finite depth the circumstances are more com-
plicated, and in several respects more interesting, than in water of
infinite depth. This is already indicated in the simplest case, in which
the free surface pressure is assumed to be everywhere zero and the
motion is assumed to be steady. In this case we seek a function <p(x, y)
satisfying the conditions (7.0.2) to (7.0.5), with <pt and rjt both iden-
tically zero. The boundary conditions are thus
U2
(7.3.1) <py H -- (pxx = 0, y = 0,
g
and
(7.3.2) <p, = 0, y == - h.
A harmonic function which satisfies these conditions is given by:
(7.8.3) (p(x9 y) = A cosh m(y + h) cos (mx + a)
with A and a arbitrary constants, and m a root of the equation
(7.8.4) g^tanhmfe
gh mh
The condition (7.8.4) ensures that the boundary condition on the
free surface is satisfied, as one can easily verify. It is very important
for the discussion in this and the following section to study the roots
208
WATER WAVES
of the equation (7.8.4). The curves £ = tanh f and f = (U*/gh) f are
plotted in Fig. (7.3.1). The roots of (7.3.4) are of course furnished by
the intersections f = mh of these curves. One observes: 1) m = 0 is
always a root; 2) there are two real roots different from zero if U2/gh<l ;
Fig. 7.8.1. Roots of the transcendental equation (U2/gh < 1)
8) there are no real roots other than zero if U2/gh ^ 1; 4) if U2/gh = 1
the function U2m — g tanh mh vanishes at m = 0 like m3; 5) since
tan if = i tanh £, it follows that (7.8.4) has infinitely many pure
imaginary roots no matter what value is assigned to U2/gh.
On the basis of this discussion of the roots of (7.3.4) we therefore
expect that no motions other than the steady flow with no surface
disturbance (for which <p = const. ) will exist unless U2/gh < 1 . These
waves are then seen to have the wave length appropriate for simple
harmonic waves of propagation speed c = U in water of depth h, as
can be seen from (8.2.1), (3.2.2), and (3.2.8). It is possible to give a
rigorous proof of this uniqueness theorem— which holds when no con-
ditions at oo other than boundedness conditions are imposed— by
making use of an appropriate Green's function, or by making use of
the method devised by Weinstein [W.7] for simple harmonic waves in
water of finite* depth, but we will not do so here.
More interesting problems arise when we suppose that steady waves
$re created by disturbances on the free surface, or perhaps also on the
bottom. Mathematically this means that a nonhomogeneous boundary
condition would replace one, or perhaps both, of the homogeneous
TWO-DIMENSIONAL WAVES 209
boundary conditions (7.3.1) and (7.3.2). In addition, as we infer from
the discussion of the preceding section, it is also necessary in general
to prescribe a condition of "radiation" type at oo in addition to boun-
dedness conditions, and an appropriate such condition is that the
disturbance should die out upstream. In the present problem, how-
ever, the additional parameter furnished by the depth of the water
leads to some peculiarities that are conditioned in part by the differ-
ence in behavior of the solutions of the homogeneous problem in their
dependence on the parameter U*/gh: Since the only solution of the
homogeneous problem in the case U2/gh S> 1 is q> = 0, one expects
that the solution of the nonhomogeneous problem will be uniquely
determined in this case without the necessity of prescribing a radiation
condition at oo. However, if U*/gh < 1 it is clear that the nonhomo-
geneous problem can not have a unique solution unless a condition —
such as that requiring the disturbance to die out upstream— is
imposed that will rule out the otherwise possible addition of the non-
vanishing solution of the homogeneous problem. These cases have been
worked out (cf. Lamb [L.3], p. 407) with the expected results, as
outlined above, for U*/gh > 1 and U2/gh < 1, but the known re-
presentations of these solutions for the steady state make the wave
amplitudes large for U2jgh = 1 and \x\ large.
We shall not solve these steady state problems directly here be-
cause the peculiarities— not to say obscurities— indicated above can
all be clarified and understood by re-casting the formulation of the
problem in a way that has already been employed in the previous
chapter (cf. sec. 6.7)). The basic idea (cf. Stoker [S.22]) is to abandon
the formulation of the problem in terms of a steady motion in favor of
a formulation involving appropriate initial conditions at the time
t = 0, and afterwards to make a passage to the limit in the solutions
for the unsteady motion by allowing the time to tend to oo. As was
indicated in sec. 6.7, the advantage of such a procedure is that the
initial value problem, being the natural dynamical problem in New-
tonian mechanics (while the steady state is an artificial problem), has
a unique solution when no conditions other than boundedness con-
ditions are imposed at oo. If a steady state exists at all, it should then
result upon letting t -> oo, and the limit state would then automati-
cally have those properties at oo which satisfy what one calls radia-
tion conditions, and which one has to guess at if the steady state
problem is taken as the starting point of the investigation.
We shall proceed along these lines in the next section in attacking
210 WATER WAVES
the problem of the waves created in a stream of uniform depth when
a disturbance is created in the undisturbed uniform stream at the
time t = 0. The subsequent unsteady motion will be determined when
only boundedness conditions are imposed at oo. It will then be seen
that the behavior of the solutions as t -* oo is indeed as indicated
above, i.e. the waves die out at infinity both upstream and down-
stream when U2/gh > 1, that they die out upstream but not down-
stream when U2/gh < 1. One might be inclined to say: "Well, what of
it, since one guessed the correct condition on the upstream side any-
way?" However, we now get a further insight, which we did not
possess before, i.e. that for U2/gh = 1 there just simply is no steady
state when t -> oo although a uniquely determined unsteady motion
exists for every given value of the time t . In fact it will be shown that
the disturbance potential becomes infinite like J2/3 at all points of the
fluid when t -> oo and U2/gh = 1, and that the velocity also becomes
infinite everywhere when t -> oo.
7.4. Unsteady waves created by a disturbance on the surface of a
running stream
The boundary conditions on the disturbance potential <p(x, y\ t) at
the free surface (cf. Fig. 7.0.1 and equations (7.0.3) and (7.0.4)) are
v U2
(7.4.1) 11 + gr, + <pt + U<px + — - 0,
Q *
(7.4.2) ty + Ur,x -<py = 0,
to be satisfied at y = 0 for all times t > 0. The quantity p = p(x; t)
is the pressure prescribed on the free surface. At the bottom y = — h
we have, of course, the condition
(7.4.3) <py = 0, t ^ 0.
At the initial instant t = 0 we suppose the flow to be the undisturbed
uniform flow, and hence we prescribe the initial conditions:
(7.4.4) q>(x, y; 0) = ^(x; 0) = p(x; 0) = 0.
From (7.4.1), which we assume to hold at t = 0, we thus have the
condition
(7.4.5) (f>t(x, y; 0) = 0.
Finally, we prescribe the surface pressure p for t > 0:
(7.4.6) p = p(x)9 t > 0.
TWO-DIMENSIONAL WAVES 211
(The surface pressure is thus constant in time.) At oo we make no
assumptions other than boundedness assumptions. We shall not
formulate these boundedness conditions explicitly: instead, they are
used implicitly in what follows because of the fact that Fourier trans-
forms in x for — oo < x < oo are applied to q> and p and their
derivatives. Of course, this means that these quantities must not only
be bounded but also must tend to zero at oo, and this seems reasonable
since the initial conditions leave the water undisturbed at oo.
We have, therefore, the problem of finding the surface elevation
r)(x; t) and the velocity potential <p(x, y; t) in the strip — h ^ y 5^ 0,
— oo < x < oo, which satisfy the conditions (7.4.1) to (7.4.6). We
begin the solution of our problem by eliminating the surface elevation
77 from the first two boundary conditions to obtain:
(7.4.7) <ptt + t/Vxx + W<pxt +g<py=-- px, at y = 0.
Q
The Fourier transform with respect to x is now applied to (pxx +<pvv = 0
to yield (cf. sec. 6.3):
(7.4.8) ¥„ - s*y = 0,
where the bar over <tp refers to the transform ip = (p(s, y; t) of <p. From
(7.4.3) we have q>y = 0 for y = — h; hence ip, in view of (7.4.8) must
be of the form
(7.4.9) v(*> y> 0 = A(*> ') cosh s(y + A)>
with A(s; t) a function to be determined. The transform is next applied
to (7.4.7) with the result:
_ _ _ _ _
(7.4.10) Vii + 2isU<pt + g<pv - UWcp = -- p, at y = 0,
e
and this yields, from (7.4.9) for y == 0, the differential equation
isUp
(7.4.11) AU + 2isUAt + fe* tanh sh - s2U2]A = —
Q cosh sh
Here p(s) is of course the transform of p(x). As initial conditions at
t = 0 for A($; t) we have from (7.4.4) and (7.4.5) the conditions (again
in conjunction with (7.4.9)):
(7.4.12) A(a\ 0) = At(si 0) = 0.
The function A(s;t) is then easily found; it is
212 WATER WAVES
(7.4.13) A(s; t) = UVp
Q cosh sh
s2U2 — gs tanh sh
j e-it (sU + Vgs tanh sh)
2 Vgs tanh sh sU + Vgs tanh sh
1 e- it (sU - V
2 A/gs tanh 5/i st7 — Vgs tanh
The solution y(x> y; t) of our problem is of course now obtained by
inverting <p(s, y; t):
I f00
(7.4.14) w(x, y; t) == — — A(s; t) cosh s(y + h) elsx ds.
V%n J -oo
The path of integration is the real axis. One finds easily that the
integrand behaves for large s like e^v/s9 since the denominators of
the terms in the square brackets in (7.4.13) behave like s2, the ratio
cosh s(y + A)/cosh sh behaves like 0W v for large s, and p(s) tends to
zero at oo in general. Since y is negative (cf. Fig. 7.0.1) it is clear that
the integral converges uniformly. (We omit a discussion of the be-
havior on the free surface corresponding to y = 0, although such a
discussion would not present any real difficulties.) Upon examining
the function A(s; t) in (7.4.13) it might seem that it has singularities
at zeros of the denominators (and such zeros can occur, as we shall see)
but in reality one can easily verify that the function has no singulari-
ties when the three terms in the square brackets are taken together—
or, as one might also put it, any singularities in the individual terms
cancel each other. Thus the solution given by (7.4.14) is a regular
harmonic function in the strip — h ^ y < 0 for all time t, or, in
other words, a motion exists no matter what values are given to the
parameters. In addition, the fact that the integral exists ensures that
9? (and also its derivatives) tends to zero for any given time when
\x\ -> oo— this is the content of the so-called Riemann-Lebesgue
theorem. This means that the amplitude of the disturbance dies out
at infinity at any given time /—a not unexpected result since a certain
time must elapse before any appreciable effects of a disturbance are
felt at a distance from the seat of the disturbance.*
However, we know from our earlier discussion (and from everyday
* * It should be pointed out once more that disturbances propagate at infinite
speed since our medium is incompressible. Each Fourier component, however,
propagates with a finite speed.
TWO-DIMENSIONAL WAVES 213
observation of streams, for that matter) that as t -> oo it may happen
that a disturbance also propagates downstream as a wave with non-
vanishing amplitude. Our main interest here is to study such a passage
to the limit. It is clear that one cannot accomplish such a purpose
simply by letting t -> oo in (7.4.14), since, for one thing, the transform
y> of <p cannot exist if <p does not tend to zero at oo. What we wish to do
is to consider the contributions of the separate items in the brackets
in (7.4.13), and to avoid any singularities caused by zeros in their
denominators by regarding A (s; t) as an analytic function in the neigh-
borhood of the real axis of a complex $-plane and deforming the path
of integration in (7.4.14) by Cauchy's integral theorem in such a way
as to avoid such singularities. One can then study the limit situation
as t -> oo.
In carrying out this program it is essential to study the separate
terms defining the function A(s; t) given by (7.4.13). To begin with,
we observe that the function Vgs tanh sh can be defined as an analytic
and single- valued function in a neighborhood of the real axis since
the function s tanh sh has a power series development at s = 0 that
is valid for all s and begins with a term in s2, and, in addition, the
function has no real zero except s = 0. Once the function Vgs tanh sh
has been so defined, it follows that each of the terms in (7.4.13) is an
analytic function in a strip containing the real axis except at real zeros
of the denominators. It is important to take account of these zeros,
as we have already done in sec. 7.3. For our present purposes it is
useful to consider the function
sh
/U2 \
(7.4.15) W(s) = gs [ — . sh — tanh sh ) = s2U2 — gs tanh
\gh _ / _
= (sU + Vgs tanh sh)(sU — Vgs tanh sh)
= /+(')/-(*)•
With reference to Fig. 7.3.1 above and the accompanying discussion,
one sees that there are at most three real zeros of the function W(s):
s = 0 is in all cases a root, and there exist in addition two other real
roots if the dimensionless parameter gh/U2 is greater than unity. Also,
it is clear that if gh/U2 =£ 1 the origin is a double root of W(s), but is
a quadruple root if gh/U2 = 1. In case gh/U2 > I the real roots ± ft
of W(s) are simple roots. (It might be noted in passing that W(s) has
infinitely many pure imaginary zeros ±i/?n, n = 1, 2, . . ..)
It follows at once that if we deform the path of integration in
214
WATER WAVES
(7.4.14) from the real axis to the path P shown in Figure 7.4.1 we can
consider separately the contributions to the integral furnished by each
of the three items in the square brackets in (7.4.13), since the separate
+ 13
Fig. 7.4.1. The path P in the s-plane
integrals would then exist. This we proceed to do, except that we pre-
fer to consider the velocity components q>9 and <py of the disturbance
rather than 9? itself. For q>x we write*
(7.4.16) 9>« = ?if) + V®,
with <pW and <p® defined (in accordance with (7.4.13) and (7.4.14)) as
follows:
(7.4.17)
7.4.18) ?W -
p(s)s2 cosh s(y + h)
>t'«x ,
-u
W(s) cosh sh
p(s)s* cosh s(y + h)
[e ttf+S __ e ltf~(S]
cosh sh \/gs tanh sh
The functions W(s), f-(s), and f+(s) have been defined in (7.4.15).
Evidently the notation <p^\ (p® has been chosen to point to the fact
that <p^ should yield the steady part of the motion while q>® should
furnish "transients" which die out as t -> oo. This is indeed the case,
as we now show, at least when the parameter gh/U2 is not equal to
unity, its critical value.
Consider first the case gh/U* < 1. In this case there are no singu-
larities on the real axis, even at the origin (cf. (7.4.18)), since /+ and
/_ vanish to the first power and i/gs tanh sh vanishes to the first
power also at s = 0. Since p(s) is regular at s = 0 and $2 occurs in the
numerator of the integrand our statement follows. Consequently the
'path P can be deformed back again into the real axis. In this case the
* The discussion would differ in no essential way for <pv instead of <px.
TWO-DIMENSIONAL WAVES 215
behavior of <p(£ for large t can be obtained by the principle of station-
ary phase (cf. sec. 6.8). In the present case the functions f+(s) and
/_(s) have non-vanishing first derivatives for all s, and consequently
<p(£ -> 0 at least like l/t since there are no points where the phase is
stationary. (Here and in what follows no attempt is made to give the
asymptotic behavior with any more precision than is necessary for
the purposes in view. ) As t -> oo therefore we obtain the steady state
solution cpW. The behavior of q>W for \x\ large is also obtained at
once: one sees that the integrand in (7.4.17) has no singularities in
this case also, and it follows at once from the Riemann-Lebesgue
theorem that 9?^ -> 0 as \x\ -» oo. Thus a steady state exists, and
it has the property that the disturbances die out both upstream and
downstream.
We turn next to the more complicated case in which gh/U2 > 1.
The integrand for q>^ has no singularity at the origin, but it has
simple poles at s = db P (cf. Figure 7.4.1) furnished by simple zeros
of f_(s) at these points. Again we show that (pW -> 0 as t -> oo.
Consider first the contribution of the semicircles at s = ± /?. (Since
s = 0 is not a singularity, we deform the path back into the real axis
there.) In the lower half-plane near s = ± P one sees readily that
f-(s) has a negative imaginary part, and thus the exponent in
exp {— itf_(s)} has a negative real part, since /_($) is real on the real
axis and its first derivative f_(s) is positive there (so that /_($) be-
haves like c(s ^f /?) with c a positive constant). Thus for any closed
portion of the semicircles which excludes the end-points the contribu-
tion to the integral tends to zero as t -> oo, and hence also for the
whole of the semicircles. On the straight parts of the path the prin-
ciple of stationary phase can be used again to show that <p$ -> 0 as
t -> oo. In fact, this function behaves like l/^/t since one can easily
verify that /_(s) has exactly two points of stationary phase, i.e. two
points ± j80 where /-(± A>) = ° and /"(± /30) ^ 0. (The point s = 00
lies between the origin and the point s = 0 where /_($) vanishes.)
Thus the steady state is again given by <p^. However, unlike the pre-
ceding case, the steady state does not furnish a motion which dies
out both upstream and downstream. This can be seen as follows.
Consider first the behavior upstream, i.e. for x < 0. On the semicircu-
lar parts of the path P in the lower half-plane we see that the expo-
nent in ei8X in (7.4.17) has a negative real part, and therefore by the
same argument as above, these parts of P make contributions which
vanish as x -» — oo. The straight parts of P also make contributions
216 WATER WAVES
which vanish for large x (either positive or negative), by the Riemann-
Lebesgue theorem. Thus the disturbance vanishes upstream. On the
downstream side, i.e. for x > 0, we cannot conclude that the semi-
circular parts of P make vanishing contributions for large x since the
exponent in ei8X now has a positive real part. We therefore make use
of the standard procedure of deforming the path P through the poles
at s = ± /? and subtracting the residues at these poles. It is clear that
the semicircles in the upper half-plane yield vanishing contributions
to <pM when x -> + <x>: the argument is the same as was used above.
This leads to the following asymptotic representation (obtained from
the contributions at the poles), valid for x large and positive:
(7.4.19, ^.r. 00)=
..
Q cosh
Here W'(ft) ^ 0 is the value of the derivative of W (cf. (7.4.15)) at
s = /?, and the fact that W(/J) is an odd function has been used. In
particular, if the surface pressure p(x) were given by the delta func-
tion p(x) = d(x), i.e. if the disturbance were caused by a concentrated
pressure point at the origin, (7.4.19) would yield
,~, ^ / x
(7.4.19), ?.(*,„; «,).=
since the transform of d(x) is l/\/2n. Another interesting special
case is that in which p(x) is a constant p0 over the interval — a ^ x
^ a and zero over the rest of the free surface. In this case p =
(2p0/ \/2n) ( sin sa )/s and q>x behaves for large positive x and t as follows :
x 0 cosh fi(y + h) . D . 0
(7A19), „.(«, y; oo) - «n ^a sm /to.
This yields the curious result (mentioned above) that under the pro-
per circumstances the disturbance may die out downstream as well as
upstream; it will in fact do so if (ia = nn, i.e. if the length 2a of the
segment over which the disturbing pressure is applied is an integral
multiple of the wave length at oo— which is, in turn, fixed by the
velocity U and the depth h.
Finally we consider the critical case gh/U* = 1, and begin by dis-
cussing the behavior of the time dependent terms in <p as t -> oo. For
this purpose it is convenient to deal first with the time derivative of
this function:
TWO-DIMENSIONAL WAVES 217
(7.4.80) *,=
P cosh $A \/g$ tanh
The integrand has no singularities on the real axis and consequently
the path P can be deformed into the real axis. Thus the principle of
stationary phase can be employed once more. Since the derivative of
f+(s) = sU + \/gs tanh sh evidently does not vanish for any real s
while the derivative of /_(s) has one zero at s = 0, it follows that the
leading term in the asymptotic development of 9?^ for large t arises
from the term exp {— itf__(s)}. Since, in addition, £'(0) = 0 but /'"(O)
^ 0 we have (cf. sec. 6.8):
(7.4.21 ) 0f - Ap(0). - , A = const. ^ 0.
Since p(0) is in general different from zero, it follows that <pf^ behaves
like r1/3 and hence that q>(t) becomes infinite everywhere (for all x
and y, that is) like t*/3 as t -> oo.* Thus a steady state does not exist
if one considers it to be the limit as t -> oo. It might be thought that
the existence in practice of dissipative forces could lead to the vanish-
ing of the transients and thus still leave the steady state (p^ as given
by (7.4.17) as a representation of the final motion. That is, however,
also not satisfactory since <p^ becomes unbounded for x large when
gh/U2 = 1: at the origin there is a pole of order two since W($) be-
haves like s4 and consequently the term isx in the power series for eisx
leads to a contribution from this pole which is linear in x. In linear
theories based on assuming small disturbances one is reconciled to
singularities and infinities at isolated points, but hardly to arbitrarily
large disturbances in whole regions. All of this suggests that the
reasonable attitude to take in these circumstances is that the linear
theory, which assumes small disturbances, fails altogether for flows
at the critical speed U*/gh = I and that one should go over to a non-
* It might seem odd that we have chosen to discuss the function <p^ rather
than the function <p® (as we did in the other cases). The reason is that the asymp-
totic behavior of q>^ is not easily obtained directly by the method of stationary
phase in the present case since the coefficient of the leading term in this develop-
ment would be zero. However, one could show (by using Watson's lemma, for
example, which yields the complete asymptotic expansion of the integral) that
q>® behaves like f~^3, and hence that q>® behaves like *^3.
218 WATER WAVES
linear theory in order to obtain reasonable results from the physical
point of view. In Chapter 10.9, which deals with the solitary wave (an
essentially nonlinear phenomenon), we shall see that such a steady
wave exists for flows with velocities in the neighborhood of the
critical value.
CHAPTER 8
Waves Caused by a Moving Pressure Point. Kelvin's
Theory of the Wave Pattern Created by a Moving Ship
8.1. An idealized version of the ship wave problem. Treatment by the
method of stationary phase
The peculiar pattern of the waves created by objects moving over
the surface of the water on a straight course has been noticed by
everyone: the disturbance follows the moving object unchanged in
form and it is confined to a region behind the object that has the same
v-shape whether the moving object is a duck or a battleship. An ex-
planation and treatment of the phenomenon was first given by
Kelvin [K.ll], and this work deserves high rank among the many
imaginative things created by him. As was mentioned earlier, Kelvin
invented his method of stationary phase as a tool for approximating
the solution of this particular problem, and it is indeed a beautiful
and strikingly successful example of its usefulness.
It should be stated at once that there is no notion in this and the
next following section of solving the problem of the waves created by
an actual ship in the sense that the shape of the ship's hull is to be
taken into account; such problems will be considered in the next
chapter. For practical purposes an analysis of the waves in such cases
is very much desired, since a fraction— even a large fraction if the
speed of the ship is large— of the resistance to the forward motion of a
ship is due to the energy used up in maintaining the system of gravity
waves which accompanies it. The problem has of course been studied,
in particular, in a long series of notable papers by Havelock,* but the
difficulties in carrying out the discussion in terms of parameters which
fix the shape of the ship are very great. Indeed, a more or less com-
plete discussion of the solution to all orders of approximation even in
the very much idealized case to be studied in the present chapter, is
by no means an easy task— in fact, such a complete discussion, along
References to some of these papers will be given in the next chapter.
219
220 WATER WAVES
lines quite different from those of Kelvin, has been carried out only
rather recently by A. S. Peters [P. 4] (cf. also the earlier paper by
Hogner [H.16]). However, we shall follow Kelvin's procedure here in
a general way, but with many differences in detail.
The problem we have in mind to discuss as a primitive substitute
for the case of an actual ship is the problem of the surface waves
created by a point impulse which moves over the surface of the water
(assumed to be infinite in depth). We shall take the solution of section
6.5 for the wave motion due to a point impulse and integrate it along
the course of the "ship"— in effect, the surface waves caused by the
ship are considered to be the cumulative result of impulses delivered
at each point along its course. The result will be an integral represen-
tation for the solution, in the form of a triple integral, which can be
discussed by the method of stationary phase. However, it is necessary
to apply the method of stationary phase three times in succession, and
if this is not done with some care it is not clear that the approximation
is valid at all; or what is perhaps equally bad from the physical point
of view, it may not be clear where the approximation can be expected
to be accurate. Thus it seems worth while to consider the problem with
some attention to the mathematical details; this will be done in the
present section, and the interpretation of the results of the approxima-
tion will be carried out in the next section (which, it should be said,
can be read pretty much independently of the present section).
From section 6.4 the vertical displacement* r)(x, y, z; t) of the water
particles due to a point impulse applied on the surface at the origin
and at the time t = 0 can be put in the form
I /*oo /»w/2
(8.1.1) 77(0?, t/, z;t) = — - - I asmat'emvmdm cos(mrcos/J)d/J
in which a2 = gm and r2 = x2 + z2. We have replaced the Bessel
function J0(mr) by its integral representation
2 f*/2
J0(mr) = — cos (mr cos ft) df}
for reasons which will become clear in a moment. As we have
indicated, our intention is to sum up the effect of such impulses
as the "ship" moves along its course C. The notations to be used for
* Actually, we have considered only the displacement of the free surface in
that section, but it is readily seen that (8.1.1) furnishes the vertical displacement
of any points in the water.
WAVE PATTERN CREATED BY A MOVING SHIP
221
this purpose are indicated in Figure 8.1.1, which is to be considered as
a vertical projection of the free surface on any plane y = const. The
course of the ship is given in terms of a parameter t by the relations
(8.1.2)
0 ^ t ^ T,
and t is assumed to mean the time required for the ship to travel
from any point Q(xv z^ on its course to its present position at the
kP(x,z)
Fig. 8.1.1. Notation for the ship wave problem
origin. We seek the displacement of the water at (x, y, z) when the
ship is at the origin; it is therefore determined by the integral
(8.1.3) r](x,y,z)
I /»r /•» pjr/2
= k(t) dt a sin at emym dm cos (mr cos
%ngQ Jo Jo Jo
In this formula k(t) represents the strength of the impulse, which we
might reasonably assume to be constant if the speed of the ship is
constant; this constant is therefore the only parameter at our disposal
which might serve to represent the effect of the volume, shape, etc.
of a ship's hull. We write the last relation in the form
rooji/2
(8.1.4) ri(x,y>z)~K [[(amemv[ei(at~mrcos® + ei(at+mrcos®] d@ dm dt
ooo
with the understanding that the imaginary part of the integral is to be
taken. (K is a constant the value of which is not important for the
222 WATER WAVES
discussion to follow.) It should be noted that r2 = (x — x^2 +
(z — 3X)2. Since y < 0, the integral converges strongly because of the
exponential factor.
One of the puzzling features (to the author, at least) of existing
treatments of the problem by the method of stationary phase is that
it is not made clear what parameter is large in the exponentials as the
method is applied to each of the three integrals in turn, so that one is
not quite sure whether there might not be an inconsistency. The
ntatter is easily clarified by introduction of appropriate dimensionless
quantities, as follows (cf. Figure 8.1.1):
(8.1.5)
x = R cos a, xl = Rl cos ax , z = R sin a, z± — Rl sin al5
r = R V(A cos ax — cos a)2 + (Asinax — sin a)2 = R • /,
r = ct/R, R,/R = A, * = ^ , m = ^-l2.
4c2 4r2
Here the quantity c represents the speed of the ship in its course. It
should be noted that x, y, and z are held fixed— they represent the
point at which the displacement is to be observed—, but that xl9 zl
(and hence Rl and o^), and r all depend on t. We have also introduced
a new variable of integration f , replacing m, which depends on t. The
Jacobian 9(ra, t)/d(£, r) has the value gt2Rg/(2cr2) and hence vanishes
only for t = 0. In terms of the new quantities the integral (8.1.4) is
found to take the form:
(8.1.6) r)(x,y,z)
TO oojz/2
m-,3T5£4 xr^y f
w ^
000
where TO = cT//e.
Again we remark that the integral converges uniformly for y < 0.
However, the integrand has a singularity if the point (x, y, z) happens
to be vertically under a point on the course of the ship: in such a case
we have R = Rl (i.e. A == 1 ), and a = alf so that I = 0 for a certain
value r T£ 0 in the interval 0 fS r ^ TO, Because of the exponential
factor, the integral continues to exist, however. Indeed, one sees
from (8.1.4) that taking r = 0 does not make the integrand singular;
WAVE PATTERN CREATED BY A MOVING SHIP 228
the fact that a singularity crops up in (8.1.6) arises from our choice
of the variable | which replaces ra. This disadvantage caused by intro-
duction of the new variables is much more than outweighed by the
fact that we now can see that the approximation by the method of
stationary phase depends only on one parameter, i.e. the parameter
x = gjR/4r2 in the exponentials. We can expect the use of the method
of stationary phase to yield an accurate result if this parameter is
large, and that in turn is certainly the case if R is large, i.e. for points
not too near the vertical axis through the present location of the ship.
The application of the method of stationary phase to the integral
in (8.1.6) can now be justified by an appeal to the arguments used in
section 6.8. In doing so, the multiple integral is evaluated by inte-
grating with respect to each variable in turn; at the same time, the
integrands are replaced by their asymptotic representations as fur-
nished by the method of stationary phase. One need only observe, in
verifying the correctness of such a procedure, that the integrands
remain, after each integration, in a form such that the arguments of
that section apply— in particular that they remain analytic functions
of their arguments provided only that points (x, y, z) on or under the
ship's course are avoided*— and that an asymptotic series can be in-
tegrated termwise. It is not difficult to see that the contributions to
YI(X, y, z) of lowest order in \\x are made by arbitrarily small domains
containing in their interiors a point where the derivatives 9?^, <p^9 <pr of
the phase 9? = (2£ — £2 cos ^)r2/l(r) vanish simultaneously.
Even for points on the ship's course the argument of section 6.8
will still hold provided that no stationary point of the phase <p occurs
for a value of r such that l(r) = 0: the reason for this is that the
assumption of analyticity was used in section 6.8 only to treat a
neighborhood of a point of stationary phase, while for other segments
of the field of integration only the assumptions of integrability and
the possibility of integration by parts are needed. It happens that the
cases to be treated later on are such that l(r) does not vanish at any
points of stationary phase, and hence for them the asymptotic
approximation is valid also for points on the ship's course.
There is one further mathematical point to be mentioned. The
/• &
* In section 6.8 the integrals studied were of the form y(x) exp (ikq>(x)} dx,
/•b Jo
while here the integral is of the form y>(x, k) exp (ifop(x)} dx. However, one
Ja
can verify that the argument used in section 6.8 can easily be generalized to
include the present case.
224 WATER WAVES
above discussion requires that we take y < 0, and it is not entirely
clear that the passage to the limit y -> 0 is legitimate in the approxi-
mate formulas, so that the validity of the discussion might be thought
open to question for points on the free surface. Indeed, it would appear
to be difficult to justify such a limit procedure for the integral in
(8.1.1), for instance, since it certainly does not converge if we set
y = 0 since the integrand then does not even approach zero as m ->• oo.
However, this is a consequence of dealing with a point impulse. If we
had assumed as model for our ship a moving circular disk of radius a
over which a constant distribution of impulse is taken, the result for
the vertical displacement due to such a distributed impulse applied
at t = 0 could be shown to be given by
/*00
rj(x, y, z\ t) = Kl a sin at • ^mvJr0(mr)J1(ma) dm
j o
with Ji(ma) the Bessel function of order one and Kl a certain constant.
This integral converges uniformly for y ^ 0, as one can see from the
asymptotic behavior of JQ(mr) and J^(ma). Consequently rj(x, y, z; t)
is continuous for y ~ 0. On the other hand, if the radius a of the disk
is small the result cannot be much different from that for the point
impulse. Thus we might think of the results obtained in the next
section, which start with the formula (8.1.1 ) for a point impulse, as an
approximation on the free surface to the case of an impulse distributed
over a disk of small radius.
It has already been mentioned that the problem under discussion
here has been treated by A. S. Peters [P.4] by a different method.
Peters obtains a representation for the solution based on contour
integrals in the complex plane, which can then be treated by the
saddle point method to obtain the complete asymptotic development
of the solution with respect to the parameter x defined above, while
we obtain here only the term of lowest order in such a development.
However, the methods used by Peters lead to rather intricate deve-
lopments.
8.2. The classical ship wave problem. Details of the solution
In the preceding section we have justified the repeated application
of the method of stationary phase to obtain an approximate solution
for the problem of the waves created when a point impulse moves over
the surface of water of infinite depth. In particular, it was seen that
the approximation obtained in that way is valid at all points on the
WAVE PATTERN CREATED BY A MOVING SHIP
225
surface of the water not too near to the position of the "ship" at the
instant when the motion is to be determined (provided only that a
certain condition is satisfied at points on the ship's course). In this
section we carry out the calculations and discuss the results, returning
however to the original variables since no gain in simplicity would be
achieved from the use of the dimensionless variables of the preceding
section.
Kelvin carried out his solution of the ship wave problem for the
case of a straight line course traversed at constant speed. Up to a
certain point there is no difficulty in considering more general courses
\P(x,z)
Fig. 8.2.1. Notation for the ship wave problem
for the ship. In Figure 8.2.1 we indicate the course C as any curve
given in terms of a parameter t by the equations
(8.2.1)
x =
for 0 <, t < T.
The parameter t is taken to represent the time required for the ship
to pass from any point (#15 2X) to its present position at the origin O,
but it is convenient to take t = 0 to correspond to the origin so that
the point (xv yx) moves backward along the ship's course as t increases.
The shape of the waves on the free surface is to be determined at the
moment when the ship is at the origin. The #-axis is taken along the
tangent to the course C, but is taken positive in the direction opposite
to the direction of travel of the ship, Since we have taken t = 0 at the
origin the parameter t in (8.2.1) is really the negative of the time; as
a consequence the tangent vector t to C at a point Q(xl9 t/t) as given by
is in the direction opposite to that of the ship in traversing the course
226 WATER WAVES
C. The speed c(t) of the ship is the length of the vector t and is given by
•»
The point P(x, z) is the point at which the amplitude of the surface
waves is to be computed; it is located by means of the vector r:
(8.2.3) r = (x — xl9 z — %).
The angle 0 indicated on the figure is the angle (^ n) between the
vectors r and — t.
As we have stated earlier, the surface elevation r)(%9 z) at P(x, z)
is to be determined by integrating the elevations due to a point im-
pulse moving along C. The effect of an impulse at the point Q is ass-
sumed to be given by the approximate formula (6.5.15), in which,
however, we omit a constant multiplier which is unessential for the
discussion to follow:
— t3 tf?
(8.2.4) rj(r;t)~ --- sin 2_.
r4 4r
In other words, we assume that the formula (8.1.1) for the surface
elevation r) has been approximated by two successive applications
of the method of stationary phase. This formula yields the effect
at time t and at a point distant r from the point where the impulse
was applied at the time t = 0; it therefore applies in the present situa-
tion with
(8.2.5) r* = (x - ^)2 + (z - ^)2,
since t does indeed represent the length of time elapsed since the
"ship" passed the point Q on its way to its present position at O. The
integrated effect of all the point impulses is therefore given by
dt,
CTl3 et2
(8.2.6) n(x9 z) = A;0 - sin 51
Jo r4 4r
with fc0 a certain constant. For points on the ship's course, where
r = 0 for some value t = t0 in the interval 0 rg t ^ T, this integral
evidently does not exist. However, it has been shown in the preceding
section that neighborhoods of such points can be ignored in calculating
rj approximately provided that they are not points of stationary phase.
This condition will be met in general, and hence we may imagine that
a small interval about a point where r(t0) = 0 has been excluded from
WAVE PATTERN CREATED BY A MOVING SHIP 227
the range of integration in case we wish the wave amplitude at a point
on the ship's course. We write the integral in the form
fr
(8.2.7) ri(xt z) = V(*)*w) dt,
Jo
and take the imaginary part. The function \p(t) and the phase <p(t)
are given by
(8.2.8) y(t) = kQt*/r*
(8.2.9) <p(t) = gt*/4r.
We proceed to make the calculations called for in applying the
stationary phase method. In the integral given by (8.2.7) no large
parameter multiplying the phase is put explicitly in evidence; how-
ever, from the discussion of the preceding section we know that the
approximation will be good if the dimensionless quantity gJ?/4c2,
with R the distance from the ship, is large. It could also be verified
that (8.2.6) would result if the integrations in (8.1.6) on ft and £ were
first approximated by stationary phase followed by a re-introduction
of the original variables. We therefore begin by calculating dq>/dt:
(8.2.10) =
v ' dt 4\r
Hence the condition of stationary phase, dtpjdt = 0, leads to the im-
portant relation
(8.2.11) <^ = *.
dt t
The quantity dr/dt is next calculated for the ship's course using
(8.2.5); we find (cf. Figure 8.2.1):
= — r • t — cr cos 0,
in which c(t) is once more the speed of the ship. Thus
dr
(8.2.13) — = ccos0,
dt
which is a rather obvious result geometrically. Combining (8.2.11)
and (8.2.13) yields the stationary phase condition in the form
(8.2.14) r =
228
WATER WAVES
We recall once more the significance of this relation: for a fixed point
P(#, y) it yields those points Qf on C which are the sole points effective
(within the order of the approximation considered) in creating the
disturbance at P— the contributions from all other points being, in
effect, cancelled out through mutual interference. It is helpful to intro-
duce the term influence points for the points fy determined in this
way relative to a point P at which the surface elevation of the water
is to be calculated.
The last observation makes it possible to draw an interesting con-
clusion at once from (8.2.14), which can be interpreted in the following
way (cf. Figure 8.2.2): At point Q the speed c of the ship and / are
Fig. 8.2.2. Points influenced by a given point Q
known. The relation (8.2.14) then yields the polar coordinates (r, 0),
with respect to Q, of all points P for which Q is the influence point in
the sense of the stationary phase approximation. Such points P
evidently lie on a circle with a diameter tangent to the course C of
the ship at Q, and Q is at one end of the diameter. The center of the
circle is located on the tangent line from Q in the direction toward
which the ship moves (i.e. in the direction — t). We repeat that the
points P on the circle just described are the only points for which Q
is a point of stationary phase of the integral (8.2.7), and consequently
the contribution of the impulse applied at Q vanishes (within the
order considered by us) for all points except those on the circle. It
now becomes obvious that the disturbance created by the ship does
not affect the whole surface of the wpter, since only those points are
WAVE PATTERN CREATED BY A MOVING SHIP
229
affected which lie on one or more of the circles of influence of all points
Q on the ship's course. In other words, the surface waves created by
the moving ship will be confined to the region covered by all the in-
fluence circles, and thus to the region bounded by the envelope of this
one-parameter family of curves. This makes it possible to construct
graphically the outline of the disturbed region for any given course
traversed at any given speed: one need only draw the circles in the
manner indicated at a sufficient number of points Q and then sketch
the envelope. Two such cases, one of them a straight course traversed
at constant speed, the other a circular course, are shown in Figure
8.2.3. In the case of the straight course it is clear that the envelope
(a) (b)
Fig. 8.2.3. Region of disturbance (a) Circular course (b) Straight course
is a pair of straight lines; the disturbance is confined to a sector of
semi-angle r given by r = arc sin 1/3 = 19°28', as one readily sees
from Figure 8.2.3. This is already an interesting result: it says that
the waves following the ship not only are confined to such a sector
but that the angle of the sector is independent of the speed of the
ship as long as the speed is constant. If the speed were not constant
along a straight course, the region of disturbance would be bounded
by curved lines, and its shape would also change with the time. It is,
of course, not true that the disturbance is exactly zero outside the
region of disturbance as we have defined it here; but rather it is
small of a different order from the disturbance inside that region.
The observations of actual moving ships bear out this conclusion in
a quite startling way, as one sees from Figures 8.2.4 and 8.2.5.
The discussion of the region of disturbance has furnished us with a
certain amount of interesting information, but we wish to know a good
deal more. In particular, we wish to determine the character of the
230 WATER WAVES
wave pattern created by the ship and the amplitude of the waves.
For these purposes a more thoroughgoing analysis is necessary, and it
will be carried out later.
In the special case of a straight course traversed at constant speed
it is possible to draw quite a few additional conclusions through fur-
ther discussion of the condition (8.2.14) of stationary phase. In the
above discussion we asked for the points P influenced by a given
point Q on the ship's course. We now reverse the question and ask for
Fig. 8.2.4. Ships in a straight course
WAVE PATTERN CREATED BY A MOVING SHIP 231
the location of all influence points Q* that correspond to a given point
P. This question can be answered in our special case by making an-
other simple geometrical construction (cf. Lamb [L.3], p. 435), as
indicated in Figure 8.2.6. In this figure O represents the location of
Fig. 8.2.5a. A ship in a circular course
the ship, P the point for which the influence points are to be deter-
mined. The construction is made as follows: A circle through P with
center on OP and diameter half the length of OP is constructed; its
intersections with the ship's course are denoted by Sl and S2. From
the latter points lines are drawn to P and segments orthogonal to
them at P are drawn to their intersections Qt and Q2 on the ship's
course. The points Ql and Q2 are the desired influence points. To prove
that the construction yields the desired result requires only a verifica-
tion that P does indeed lie on the influence circles determined by the
points Q! and Q2 in the manner explained above. Consider the point
282
WATER WAVES
Q19 for example. Since the angle SlPQl = 90°, it follows that a
circle with S^ as diameter contains the point P. The segments RS1
and PQl are parallel since both are at right angles to S^; by con-
sidering the triangle OPQa one now sees that the segment OSl is just
Fig. 8.2.5b. Ships in curved courses
half the length of OQ1? and that is all that is necessary to show that the
circle having S^ as diameter is the influence circle for Qr Thus there
are in general two influence points or no influence points, the latter
case corresponding to points P outside the influence region; the tran-
sition occurs when P is on the boundary of the region of influence
(i.e. when the circle of Figure 8.2.6 having PR as diameter is tangent
to the course OQ2 of the ship), and one sees that in this limit case the
two influence points Qx and Q2 coalesce. Consequently one might well
expect that the amplitude of the waves at the boundary of the region
of disturbance will be higher than at other places, and this phenome-
non is indeed one of the prominent features always observed physical-
WAVE PATTERN CREATED BY A MOVING SHIP
233
ly. In addition, the direction of the curves of constant phase— a wave
crest, or trough, for example— can be determined graphically by the
above construction: one expects these curves to be orthogonal to the
Fig. 8.2.5c. Aircraft carriers maneuvering (from Life Magazine)
lines PQl and PQ2 drawn back from a point P to each of the points of
influence corresponding to P. That this is indeed the case will be seen
later, but it is evidently a consequence of the fact that the wave at P
is the sum of two circular waves, one generated at Qi and the other at
234
WATER WAVES
Q2- Thus we see that the wave pattern behind the ship is made up of
two different trains of waves— another fact that is a matter of com-
mon observation and which is well shown in Figures 8.2.4 and 8.2.5.
We have been able to draw a considerable number of interesting and
Fig. 8.2.6. Influence points corresponding to a given point
basic conclusions of a qualitative character through use of the condi-
tion of stationary phase (8.2.14). We proceed next to study analytic-
ally the shape of the disturbed water surface by determining the
curves of constant phase, and later on by determining the amplitude
of the waves. To calculate the curves of constant phase it is convenient
to express the basic condition (8.2.14) of stationary phase in other
forms through introduction of the following quantity a, which has the
dimension of length:
2c2 c2t2
(8.2.15) a = — w = — .
g 2r
From (8.2.14) one then finds
(8.2.16) ct = a cos 0, and
(8.2.17) r = £acos20,
as equivalent expressions of the stationary phase condition.
It would be possible to calculate the curves of constant phase for
any given course of the ship. We carry this out for the case of a cir-
cular course (this case has been treated by L. N. Sretenski [S.15] ) and
a straight course traversed at constant speed. The notation for the case
of the circular course is indicated in Figure 8.2.7, which should be com-
pared with Figure 8.2.1. For the past position (xv zl ) of the ship we have
WAVE PATTERN CREATED BY A MOVING SHIP
285
•P(x,z)
Fig. 8.2.7. Case of a circular course
{xl == R sin a
z1 = R(l — cos a)
(8.2.18)
with
(8.2.19) a = ct/R.
Here R is the turning radius of the ship, t the time required for it to
travel from Q to O, and c is the constant speed of the ship. The coor-
dinates of the point P, where the disturbance created by the ship is
to be found, are given by
{x = xl — r cos (a + 6 )
z^=zl-r^n (a + 0)
in which r and 0 are the distance and angle noted on the figure. In
these equations we replace xl and ^ from (8.2.18) and make use of
(8.2.17) to obtain
(8.2.20)
a
x — R sin a — - cos2 0 cos (a + 6)
2
(8.2.21)
We wish to find the locus of points (x* z) such that the phase <p remains
z = R(l — cos a) — - cos2 6 sin (a +0).
A
236
WATER WAVES
fixed, i.e. such that the quantity a in (8.2.15) is constant (cf. the
remarks following (8.2.9)). It is convenient to introduce the dimen-
sionless parameter K through
(8.2.22) K = a/R.
One then finds that the angle a (cf. (8.2.19)) is given by
(8.2.23) a = HCOS0,
through use of (8.2.16). In terms of these quantities the relations
(8.2.21) can be put in the following dimensionless form:
(8.2.24)
x/R = sin (x cos 6) — - cos2 6 cos (6 + K cos 0)
z/R = 1 — cos (x cos 0) — - cos2 0 sin (0 + « cos 0).
2
These equations furnish the curves of stationary phase in terms of 0
as parameter. Each fixed value of x furnishes one such curve, since
fixing K (for a fixed turning radius R) is equivalent to fixing the phase
(p. In Figure 8.2.8 a few curves of constant phase, as well as the
Fig. 8.2.8. Wave crests for a circular course
outline of the region of disturbance, as calculated from (8.2.24), arc
shown; the successive curves differ by 2n in phase. These curves should
be compared with the photographs of actual cases given in Figures
8.2.4 and 8.2.5. One sees that the wave pattern is given correctly by
the theory, at least qualitatively. The agreement between theory and
observation is particularly striking in view of the manner in which
the action of a ship has been idealized as a moving pressure point. In
particular there are two distinct sets of waves apparent, in conformity
with the fact that we expect each point in the disturbed region to
WAVE PATTERN CREATED BY A MOVING SHIP
237
correspond to two influence points: one set which seems to emanate
from the ship's bow, and another set which is arranged roughly at
right angles to the ship's course. These two systems of waves are called
the diverging and the transverse systems, respectively.
From (8.2.24) we can obtain the more important case of the ship
waves for a straight course by letting R -> oo while x -> 0 in such a
way that Rx -> a (cf. (8.2.22)). The result is
(8.2.25)
x = - (2 cos 0 — cos3 0)
z — — -- cos2 0 sin 0
2
for the curves of constant phase. In Figure 8.2.9 the results of cal-
culations from these equations are shown. These should once more be
compared with Figure 8.2.4, which shows an actual case. Again the
agreement is striking in a qualitative way. Actually, the agreement
Fig. 8.2.9. Wave crests for a straight course
would be still better if the two systems of waves —the diverging and
transverse systems— had been drawn in Figure 8.2.9 with a relative
phase difference: the photograph indicates that the crests of the two
systems do not join with a common tangent at the boundary of the
region of disturbance. We shall see shortly that a closer examination
of our approximate solution shows the two systems of waves to have a
phase difference there. It is worth while to verify in the present case a
general observation made earlier, i.e. that the curves of constant phase
238
WATER WAVES
are orthogonal to the lines drawn back to the corresponding influence
points. One finds from (8.2.25):
(8.2.26)
_ = _ -(3sin20 - I)sin0
dd 2
— = -(3sin20 - I)cos0.
dd 2
Hence dz/dx = — I/tan 0, which (cf. Figure 8.2.10) means that the
curves of constant phase are indeed orthogonal to the lines drawn to
z>
Fig. 8.2.10. Construction of curves of constant phase
the influence points. The values 0 = 0* at which 3 sin2 0 — 1 — 0
are singular points of the curves; they correspond to points P at the
boundary of the influence region where the influence points Ql and
Q2 coincide. Evidently there are cusps at these points. One sees also
that the diverging set of waves (for z > 0, say) is obtained when 0
varies in the range 0* ^ 0 fg jr/2, while the transverse waves corres-
pond to values of 0 in the range 0^0^ 0*. In addition, we observe
that to any point on the ship's course there corresponds (for 0 — 0°)
only one influence point (of type Q2) and it does not coincide with the
point P. (One sees, in fact, that the diverging wave does not occur on
the ship's course.) This is a fact that is needed to justify the applica-
tion of the method of stationary phase to points on the ship's course,
as we have remarked earlier in this section (cf. also the preceding
section).
In order to complete our discussion we must consider the amplitude
of the surface waves, as given by our approximation, as well as the
shape of the curves of constant phase. To this end we must calculate
y> and d2(p/dt2 (and even d3y>/dt*) for such values of t as satisfy the
WAVE PATTERN CREATED BY A MOVING SHIP
239
stationary phase condition dq>/dt — 0, as we know from the discussion
of section 6.5 and section 6.8. From (8.2.10) we find easily
(8.2.27) ^? = £ (l - — -
v dt2 2r \ 2r d
in view of (8.2.11). We shall also need the value of d3<p/dt3 at points
such that dqp/dt = d2(p/dt2 = 0; it is readily found to be given by
d3q> gt2 d3r
(8.2.28) — £ = — —2 — -
We wish to express our results in terms of the parameter 0 instead of
t. Since drjdt — c cos 6 from (8.2.13) we have
(8.2.29)
. n
— — — c smO —
d*2 dt
with c, the speed of the ship, now assumed to be constant. In order
to calculate dO/dt we introduce the angles /? and r indicated in Figure
8.2.11. We have 0 = n — (/? + T), and hence
2>
-t
P(x,z)
Fig. 8.2.11. The angles ft and T
in which s refers to the arc length of C. But dr/ds —- 1/18, with 12 the
radius of curvature of C; and since /? = arc tan (z — z^)l(x — x±) we
find
8.2.31
sin 9
d8 If/ v d%\ / v dB\\ gin
/ - - - « - ^ -i - (s - z,) -1 ^ = —
ds r2 L ^ dsjr
240 WATER WAVES
since the quantity in the square brackets is the vector product of r
and t/|t|. The expression for d2qp/dt2 given by (8.2.27) can now be
expressed in terms of 6 and r as follows:
(8.2.32) nZ = .L I 1 - 2 tan2 0 - - sin 01
V ' JJ° 2r [ R J
- 3 sin2 0 - — sin 0 (1 - sin2 0)
~ /t
2p .
COS2 0
as one can easily verify. The quantity a is defined by (8.2.15), and
the relation (8.2.17), in addition to those immediately above, has
been used. The points on the boundary of the region of disturbance
could be determined analytically, as follows: the set of all influence
points is the one-parameter family of circles given by dy/dt = %(x,z,t)
= 0, and the region of disturbance is bounded by the envelope of
these circles, i.e. by the points at which d2(p/dt2 = d%/dt = 0 in
addition to jf = 0. In the case of a straight course traversed at con-
stant speed, for example, we .see from (8.2.32) for R = oo that 0
then has the value 0* given by 1 — 3 sin2 0 = 0— a result found above,
where the value 0 = 0* also was seen to characterize cusps on the loci
of constant phase. From the form of the relation (8.2.32) one can con-
clude that the only courses for which the pattern of waves behind the
ship follows it without change (i.e. follows it like a rigid body) are
those for which R = const.; and thus only the straight and the cir-
cular courses have this property.
Finally, we have to consider the amplitude j\(x, z) of the waves given
by our approximate solution. The contribution of a point tQ of sta-
tionary phase to (8.2.7) is given by (cf. (6.5.2)):
(8.2.33) ,(«, ») = y(r, 0) I ?L_\* X*M)±i)
\ |?>"(r» 0)1 /
in which (r, 0) are polar coordinates which locate the point of sta-
tionary phase on the course C relative to the point (#, z) (cf. Figure
8.2.1). The sign of the term ± n/4t is to be taken the same as that of
q>" = d2q?/dt2. In principle, the surface elevation can be calculated for
any course, but the results are not very tractable except for the
simplest case; i.e. a straight course. We confine our discussion of
amplitudes, therefore, to this case in what follows. From (8.2.32) we
have
WAVE PATTERN CREATED BY A MOVING SHIP 241
(8.2.34)
dt* 2r \ cos
rin'OX
s2 0 /
We know that there are two values of 0— call them 8l and 02— at each
point in the disturbed region for which dcpjdt = 0: one belonging for
0 ^ Ol ^ 0* ~ arc sin l/\/8 to the transverse system, the other for
0* ^S 02 < nl% to the diverging system of waves. In the former case
d2<p/dt2 is positive; in the latter case negative. (At the boundary of
the region of disturbance, where 99" = 0, the formula (8.2.33) is not
valid, as we know. This case will be dealt with later.) For points in
the interior of the region of disturbance we have, therefore,
(8.2.35) ri(x, z)
V|?"(r2,02)|
Since ri = £ ctt cos 0t> rt- = ^ai cos2 0^, at- = 2c^p€/g = C2t2/2ri9 and
y> = fc^J/r} (cf. (8.2.15), (8.2.8)) at the points of stationary phase,
we may write (8.2.35) in the form
(8.2.36) \
I V 1 1 - 3 sin2 0X |
sec3 02
aj V|!~3 sin202|
The two systems of waves are thus seen, as was stated above, to have
a relative phase difference of n/2 at any point where ax = «2. Their
amplitudes die out like l/\/at on going away from the ship, and that
means that they die out like the inverse square root of the distance
from the ship. The wave amplitudes of both systems of waves become
infinite according to these formulas for 0 = 0*, i.e. for points at the
boundary of the disturbed region, but the asymptotic formula (8.2.33)
is not valid at such points since <p" = 0 there. We shall consider these
points in a moment. The diverging system also has infinite amplitude
for 02 = n/29 but this corresponds to the origin, and the infinite am-
plitude there results from our assumption of a moving point impulse
as a model for our ship.
To determine the amplitude of the waves along the boundary of the
disturbed region, we must calculate the value of ds(p/dt3 at such points
in order to evaluate the appropriate term in (6.5.2). (The problem of
242 WATER WAVES
the character of the waves in this region has been treated by Hogner
[H.13].) By differentiating (8.2.29) after replacing dO/dt by c sin 6/r
(cf. (8.2.30) and (8.2.31) for R = oo), one finds readily
(&MT) £ - - *5=»-*».
and from (8.2.28) in combination with r = \ct cos 0, r — £a cos2 0:
dzw 4>gc sin2 0
(8.2.38) — = — .
V ' dt* a2 cos*0
The amplitude of the waves along the boundary of the disturbed
region is given by (cf. (6.5.2)):
with all functions evaluated for 0 = 0* = arc sin l/\/f<*-
result is
(8.2.40) 77 ~ -i exp
The quantity A^ is a certain constant. We observe that the wave am-
plitudes now die out like I/a1/3 instead of like I/a1/2, as they do in the
interior of the disturbed region; i.e. the wave amplitudes are now of a
different, and higher, order of magnitude. As we have seen in all of
our illustrations of ship waves, the wave amplitudes are quite notice-
ably higher along the boundary of the disturbed region. The phase
also differs now by n/4> from the former values. On some of the photo-
graphs (cf. especially Fig. 8.2.4), there is some evidence of a rather
abrupt change of phase in the region of the boundary, though it may
be that one should interpret this effect as due rather to the finite
dimensions of the actual ship, which then acts as though several
moving point sources were acting simultaneously.
In the treatment of the present problem by A. S. Peters [P.4]
mentioned in the preceding section, the complete asymptotic develop-
ment of the solution was obtained.
The above developments hold only for the case of a point impulse
moving on the surface of water of infinite depth. It has some interest
to point out that there are considerable differences in the results if
the depth of the water is finite. Havelock [H.8] has carried out the
approximation to the solution by the method of stationary phase for
WAVE PATTERN CREATED BY A MOVING SHIP
243
the case of constant finite depth, with the following general results:
1 ) If the speed c of the ship and the depth h satisfy the inequality
c2/gh < 1, the general pattern of the waves is much the same as for
water of infinite depth except that the angle of the sector within
which the main part of the disturbance is found is now larger than
for water of infinite depth. 2) If c2/gk > I holds, the system of trans-
verse waves no longer occurs, but the diverging system is found.
Fig. 8.2.12. Speed boat in shallow water
Figure 8.2.12 is a photograph of a speed boat creating waves, presum-
ably in shallow water, in view of the difference in the wave pattern
when compared with Fig. 8.2.4. Finally, if c2/gh = 1 (i.e. for the case
of the critical speed), the method of stationary phase yields no rea-
sonable results; that this should be so is perhaps to be understood in
the light of the discussion of the corresponding two-dimensional
problem in Chapter 7.4.
CHAPTER 9
The Motion of a Ship, as a Floating Rigid Body, in a
Seaway
9.1. Introduction and summary
The purpose of this chapter is to develop a mathematical theory for
the motion of a ship, to be treated as a freely floating rigid body under
the action of given external forces (a propeller thrust, for example),
under the most general conditions compatible with a linear theory and
the assumption of an infinite ocean.* This of course requires the
amplitude of the surface waves to be small and, in general, that the
motion of the water should be a small oscillation near its rest position
of equilibrium; it also requires the ship to have the shape of a thin
disk so that it can have a translatory motion with finite velocity and
still create only small disturbances in the water. In addition, the mo-
tion of the ship itself must be assumed to consist of small oscillations
relative to a motion of translation with constant velocity. Within
these limitations, however, the theory presented is quite general in
the sense that no arbitrary assumptions about the interaction of the
ship with the water are made, nor about the character of the coupling
between the different degrees of freedom of the ship, nor about the
waves present on the surface of the sea: the combined system of ship
and sea is treated by using the basic mathematical theory of the
hydrodynamics of a non- turbulent perfect fluid. For example, the
theory presented here would make it possible in principle to deter-
mine the motion of a ship under given forces which is started with
arbitrary initial conditions on a sea subjected to given surface pres-
sures and initial conditions, or on a sea covered with waves of pre-
scribed character coming from infinity.
It is of course well known that such a linear theory for the non-
turbulent motion of a perfect fluid, complicated though it is, still does
not contain all of the important elements needed for a thoroughgoing
discussion of the practical problems involved. For example, it ignores
* The presentation of the theory given here is essentially the same as that
given in a report of Peters and Stoker [P.7].
245
246
WATER WAVES
the boundary-layer effects, turbulence effects, the existence in general
of a wake, and other important effects of a non-linear character. Good
discussions of these matters can be found in papers of Lunde and Wig-
ley [L.I 8], and Havelock [H.7], Nevertheless, it seems clear that an
approach to the problem of predicting mathematically the motion of
ships in a seaway under quite general conditions is a worthwhile enter-
prise, and that the problem should be attacked even though it is
recognized at the outset that all of the important physical factors can
not be taken into account. In fact, the theory presented here leads at
once to a number of important qualitative statements without the
necessity of producing actual solutions— for example, we shall see
that certain resonant frequencies appear quite naturally, and in
addition that they can be calculated solely with reference to the mass
distribution and the given shape of the hull of the ship. Interesting
observations about the character of the coupling between the various
degrees of freedom, and about the nature of the interaction between
the ship and the water, are also obtained simply by examining the
equations which the theory yields.
In order to describe the theory and results to be worked out in
later sections of this chapter, it is necessary to introduce our notation
and to go somewhat into details. In Fig. 9.1.1 the disposition of two of
the coordinate systems used is indicated. The system (X, Y, Z) is a
AY
-*. X
'2 *
Fig. 9.1.1. Fixed and moving coordinate systems
system fixed in space with the X, Z-plane in the undisturbed free
surface of the water and the F-axis vertically upward. A moving
system of coordinates (x, y, z) is introduced; in this system the #, z-
plane is assumed to coincide always with the X, Z-plane, and the
t/-axis is assumed to contain the center of gravity (abbreviated to e.g.
in the following) of the ship. The course of the ship is fixed by the
motion of the origin of the moving system, and the #-axis is taken along
THE MOTION OF A SHIP IN A SEAWAY
247
the tangent to the course. It is then convenient to introduce the
speed s(t) of the ship in its course: the speed s(t) is simply the magni-
tude of the vector representing the instantaneous velocity of this point.
At the same time we introduce the angular speed co(t) of the moving
system relative to the fixed system: one quantity fixes this rotation
because the vertical axes remain always parallel. The angle oc(£)
indicated in Fig. 9.1.1 is defined by
(9.1.1)
0)(t) (It,
implying that t — 0 corresponds to an instant when the #-axis and
JT-axis are parallel. In order to deal with the motion of the ship as a
rigid body it is convenient, as always, to introduce a system of coor-
dinates fixed in the body. Such a system (#', y\ z') is indicated in
Fig. D.I. 2. The #', t/'-plane is assumed to be in the fore-and-aft plane
(a)
(b)
Fig. 0.1. 2a, b. Another moving coordinate system
of symmetry of the ship's hull, and the ?/'-axis is assumed to contain
the e.g. of the ship. The moving system(tr', j/', z') is assumed to coin-
cide with the (iT, j/, z) system when the ship and the water are at rest
in their equilibrium positions. The e.g. of the ship will thus coincide
with the origin of the (#', j/', z') system only in case it is at the level
of the equilibrium water line on the ship; we therefore introduce the
constant y'c as the vertical coordinate of the e.g. in the primed coor-
dinate system.
The motion of the water is assumed to be given by a velocity poten-
tial 0(X, F, Z; t) which is therefore to be determined as a solution
of Laplace's equation satisfying appropriate boundary conditions at
the free surface of the water, on the hull of the ship, at infinity, and
also initial conditions at the time t = 0. The boundary conditions on
the hull of the ship clearly will depend on the motion of the ship,
248 WATER WAVES
which in its turn is fixed, through the differential equations for the
motion of a rigid body with six degrees of freedom, by the forces acting
on it —including the pressure of the water— and its position and veloc-
ity at the time t = 0. As was already stated, no further restrictive
assumptions except those needed to linearize the problem are made.
Before discussing methods of linearization we interpolate a brief
discussion of the relation of the theory presented here to that of other
writers who have discussed the problem of ship motions by means of
the linear theory of irrotational waves. The subject has a lengthy
history, beginning with Michell in 1898, and continuing over a long
period of years in a sequence of notable papers by Havelock, starting
m 1909. This work is, of course, included as a special case in what is
presented here. Extensive and up-to-date bibliographies can be found
in the papers by Weinblum [W.3] and Lunde [L.19]. Most of this work
considers the ship to be held fixed in space while the water streams
past; the question of interest is then the calculation of the wave
resistance in its dependence on the form of the ship. Of particular
interest to us here are papers of Krylov [K.20], St. Denis and Wein-
blum [S.I], Pierson and St. Denis [P.9] and Haskind [H.4], all of
whom deal with less restricted types of motion. Krylov seeks the
motion of the ship on the assumption that the pressure on its hull
is fixed by the prescribed motion of the water without reference to
the back effect on the motion of the water induced by the motion
of the ship. St. Denis and Weinblum, and Pierson and St. Denis,
employ a combined theoretical and empirical approach to the prob-
lem which involves writing down equations of motion of the ship
with coefficients which should be in part determined by model ex-
periments; it is assumed in addition that there is no coupling be-
tween the different degrees of freedom involved in the general mo-
tion of the ship. Haskind attacks the problem in the same degree
of generality, and under the same general assumptions, as are made
here; in the end, however, Haskind derives his theory completely only
in a certain special case. Haskind 's theory is also not the same as the
theory presented here, and this is caused by a fundamental difference
in the procedure used to derive the linear theory from the underlying,
basically nonlinear, theory. Haskind develops his theory by assuming
that he knows a priori the relative orders of magnitude of the various
quantities involved. The problem is attacked in this chapter by a
formal development with respect to a small parameter (essentially a
thickness-length ratio of the ship); in doing so every quantity is
THE MOTION OF A SHIP IN A SEAWAY 249
developed systematically in a formal series (for a similar type of
discussion see F. John [J.5]). In this way a correct theory should be
obtained, assuming the convergence of the series— and there would
seem to be no reason to doubt that the series would converge for
sufficiently small values of the parameter. Aside from the relative
safety of such a method— purchased, it is true, at the price of making
rather bulky calculations — it has an additional advantage, i.e., it
makes possible a consistent procedure for determining any desired
higher order corrections. It is not easy to compare Haskind's theory
in detail with the theory presented here. However, it can be stated
that certain terms, called damping terms by Haskind, are terms that
would be of higher order than any of those retained here. A more
precise statement on this point will be made later.
One of the possible procedures for linearizing the problem begins
by writing the equation of the hull of the ship relative to the coordinate
system fixed in the ship in the form
(9.1.2) z' - ± 0h(x'9 y')9 z'>0,
with ft a small dimensionless parameter.* This is the parameter with
respect to which all quantities will be developed. In particular, the
velocity potential 0(X, F, Z; /; ft) =. <p(x* y, z; t; ft) is assumed to
possess the development
(9.1.8) <p(x9 y, z; t; ft) = fafa y, z; t) + ft*<p2(x, &*;<)+••• •
The free surface elevation r](x9 z; t; ft) and the speed s(t; ft) and angu-
lar velocity a>(t; ft) (cf. (9.1.1 )) are assumed to have the developments
(9.1.4) ri(x, z\ t; ft) = /%(*, z; t) + ft*r]2(x. z;t) + ... 9
(9.1.5) s(t; ft) == So(t) + ftSl(t) + ... ,
(9.1.6) o>(f; ft) = o>0(0 + ftco^t) + . . . .
Finally, the vertical displacement yc(t) of the center of gravity and
the angular displacements** 01, 02, 03 of the ship with respect to the
#, t/, and z axes respectively are assumed given by
* It is important to consider other means of linearization, and we shall discuss
some of them later. However, it should be said here that the essential point is
that a linearization can be made for any body having the form of a thin disk:
it is not at all essential that the plane of the disk should be assumed to be vertical,
as we have done in writing equation (9.1.2).
** Since we consider only small displacements of the ship relative to a uniform
translation, it is convenient to assume at the outset that the angular displacement
can be given without ambiguity as a vector with the components 0lf 0f, 08 relative
to the #, t/, 2-coordinate system.
250 WATER WAVES
(9.1.7) OM ft) = ftO^t) + £»0<i(0 + • • - > i = 1,2,3,
(9.1.8) yc(t; ft) - y'e = ftVi(t)
These relations imply that the velocity of the water and the
elevation of its free surface are small of the same order as the "slender-
ness parameter" ft of the ship. On the other hand, the speed s(t) of the
ship is assumed to be of zero order. The other quantities fixing the
motion of the ship are assumed to be of first order, except for co(t),
but it turns out in the end that coQ(t) vanishes so that a) is also of first
order. The quantity y'c in (9.1.8) was defined in connection with the
description of Fig. 9.1.2; it is to be noted that we have chosen to
express all quantities with respect to the moving coordinate system
(x, y, z) indicated in that figure. The formulas for changes of coordi-
nates must be used, and they also are to be developed in powers of
ft; for example, the equation of the hull relative to the (x, y, z) co-
ordinate system is found to be
2l
x - ftOu(y - y'c) - fth(x, y) + . . . = o
after developing and rejecting second and higher order terms in ft.
In marine engineering there is an accepted terminology for describ-
ing the motion of a ship; we wish to put it into relation with the no-
tation just introduced. In doing so, the case of small deviations from
a straight course is the only one in question. The angular displace-
ments are named as follows: Ol is the rolling, 02 + a is the yawing, and
03 is the pitching oscillation. The quantity ft#i(t) in (9.1.5) is called
the surge (i.e., it is the small forc-arid-aft motion relative to the finite
speed s0(t) of the ship, which turns out to be necessarily a constant),
while yc — y'c fixes the heave. In addition there is the sidewisc dis-
placement dz referred to as the sway; this quantity, in lowest order,
can be calculated in terms of sQ(t) and the angle a defined by (9.1.1)
in terms of co(t) as follows:
(9.1.9) dz
f*
*,,« S0J^(
since a>Q(t) turns out to vanish.
In one of the problems of most practical interest, i.e. the problem
of a ship that has been moving for a long time (so that all transients
have disappeared) under a constant propeller thrust (considered to be
simply a force of constant magnitude parallel to the keel of the ship)
THE MOTION OF A SHIP IN A SEAWAY 251
into a seaway consisting of a given system of simple harmonic progres-
sing waves of given frequency, one expects that the displacement com-
ponents would in general be the sum of two terms, one independent of
the time and representing the displacements that would arise from
motion with uniform velocity through a calm sea, the other a term
simple harmonic in the time that has its origin in the forces arising
from the waves coming from infinity. On account of the symmetry of
the hull only two displacements of the first category would differ
from zero: one the vertical displacement, i.e. the heave, the other the
pitching angle, i.e. the angle 03. The latter two displacements apparent-
ly are referred to as the trim of the ship. In all, then, there would be
in this case nine quantities to be fixed as far as the motion of the ship
is concerned: the amplitudes of the oscillations in each of the six
degrees of freedom, the speed sQ9 and the two quantities determining
the trim. A procedure to determine all of them will next be outlined.
We proceed to give a summary of the theory obtained when the
scries (9.1.2) to (9.1.8) are inserted in all of the equations fixing the
motion of the system, which includes both the differential equations
and the boundary conditions, and any functions involving ft are in
turn developed in powers of ft. For example, one needs to evaluate (px
on the free surface y — rj in order to express the boundary conditions
there; one calculates it as follows (using (9.1.3) and (9.1.4)):
(9.1.10) Vm(x9 TI, z; t; ft) = ft[<pl9(x, 0, a; *) + rpp^(x, 0, z; t) + ...]
x, 0, z; t) + ^fotfW* °' z'> 0
We observe the important fact— to which reference will be made
later — that the coefficients of the powers of ft are evaluated at y = 0,
i.e. at the undisturbed equilibrium position of the free surface of the
water. In the same way, it turns out that the boundary conditions
for the hull of the ship arc automatically to be satisfied on the vertical
longitudinal mid-section of the hull. The end result of such calcula-
tions, carried out in such a way as to include all terms of first order in
ft is as follows: The differential equation for (p± is, of course, the La-
place equation:
(9.1.11) Vix*
in the domain y < 0, i.e. the lower half-space, excluding the plane
area A of the x, t/-plane which is the orthogonal projection of the
252 WATER WAVES
hull (cf. Fig. 9.1. 2b), in its equilibrium position, on the x9 r/-plane.
The boundary conditions on 9^ are
(9 1 12)
02i) - K + 02i)<* + Ou(y ~ 2/c)» on A-
in which A+ and .^__ refer to the two sides z = 0+ and 2 = 0_ of the
plane disk A. The boundary conditions on the free surface are
(9.1.13)
The first of these results from the condition that the pressure vanishes
on the free surface, the second arises from the kinematic free surface
condition. If sQ9 col9 021, and 0n were known functions of t, these boun-
dary conditions in conjunction with (9.1.11) and appropriate initial
conditions would serve to determine the functions cpl and rjl uniquely;
i.e. the velocity potential and the free surface elevation would be
known. Of course, the really interesting problems for us here are those
in which the quantities s0, col9 021, and Oll9 referring to the motion of
the ship, are not given in advance but are rather unknown functions
of the time to be determined as part of the solution of the boundary
problem. In principle, one method of approach would be to apply the
Laplace transform with respect to the time t to (9.1 .11), (9.1.12), and
(9.1.13)— of course taking account of initial conditions at the time
/ = 0— and then to solve the resulting boundary value problem
for the transform q>i(x9 y9 z; a) regarding s0 and the transforms oi^cr),
02i((T)> a]Qd fliifa) as parameters. However, for the purposes of this
introduction it is better to concentrate on the most important special
case (already mentioned above) in which the ship has a motion of
translation with uniform speed combined with small simple harmonic
oscillations of the ship and the sea having the same frequency.* In
this case we write the velocity potential ^(tf, j/, z; t)9 the surface
elevation rjl9 and the other dependent quantities in the form
(<?!(#, y9 z; t) = \pt(x9 y9 z) + ^(x9 y9 z)eiat
^(a?, z; t) = //„(*, z) + //!<#, z)eM
a*! - Q^\ 0n - eneM9 0ai - e^eM.
The functions y;0 and ^ are of course both harmonic functions. We
expect the functions g^ and rjl to have time-independent components
* It can be seen, however, that the discussion which follows would take much
the same course if more general motions were to be assumed.
THE MOTION OF A SHIP IN A SEAWAY 253
due to the forward motion of the ship; certainly they would appear
in the absence of any oscillatory components due, say, to a wave train
in the sea. Upon insertion of these expressions in equations (9.1.12),
and (9.1.13) we find for y>0 the conditions:
on A±9
at y = 0,
and for y>i the conditions
! — y)lz — — s0021 -\-(Ql-\-io02l)x — io&n(y—y') on A±
.
— g//! + *oVi.r — *aVi -'- Q I
We observe, in passing, that y>0 satisfies the same boundary conditions
as in the classical Michcll-Havelock theory. A little later we shall see,
in fact, that the wave resistance is indeed independent of all compo-
nents of the motion of the ship (to lowest order in /3, that is) except its
uniform forward motion with speed s0, and that the wave resistance
is determined in exactly the same way as in the Michell-Havelock
theory. We continue the description of the equations which determine
the motion of the ship, and which arise from developing the equations
of motion with respect to ft and retaining only the terms of order /?
and /ft2. (We observe that it is necessary to consider terms of both
orders.) In doing so the mass M of the ship is given by M = M^9
with Ml a constant, since we assume the average density of the ship
to be finite and its volume is of course of order 0. The moments of
inertia are then also of order (3. The propeller thrust is assumed to be
a force of magnitude T acting in the ^'-direction and in the x'9 y'-
plane at a point whose vertical distance from the e.g. is — /; the thrust
T is assumed to be of order /?2, since the mass is of order /? and accelera-
tions are also of order /?.* The propeller thrust could also, of course,
be called the wave resistance.
The terms of order /? yield the following conditions:
* We have in mind problems in which the motion of the ship is a small deviation
from a translatory motion with uniform finite speed. If it were desired to study
motions with finite accelerations — as would be necessary, for example, if the
ship were to be considered as starting from rest — it would clearly be necessary
to suppose the development of the propeller thrust T to begin with a term of first
order in /?, since the mass of the ship is of this order. In that case, the motion
of the ship at finite speed and acceleration would be determined independently
of the motion of the water: in other words, it would be conditioned solely by the
inert mass of the ship and the thrust of order 0.
254 WATER WAVES
(9.1.15) *0 = 0,
(9.1.16) 2eg f fihdA - MJg,
J A
(9.1.17) f xfihdA = 0,
J A
(9.1.18) f [(Vli - Wlc)]+ <L4 - 0,
J x
(9.1.19) f [a(Vlf - *tf>lx)]* cW - 0,
J A
(9.1.20) f [i/^, - 8<fflx)-]+ dA = 0.
J A
The symbol [ ]* occurring here means that the jump in the quan-
tity in brackets on going from the positive to the negative side of the
projected area A of the ship's hull is to be taken. The variables of in-
tegration are x and y. The equation (9.1.15) states that the term of
order zero in the speed is a constant, and hence the motion in the
^-direction is a small oscillation relative to a motion with uniform
velocity. (This really comes about because we assume the propeller
thrust T to be of order /?2.) Equation (9.1.16) is an expression of the
law of Archimedes: the rest position of equilibrium must be such that
the weight of the ship just equals the weight of the water it displaces.
Equation (9.1.17) expresses another law of equilibrium of a floating
body, i.e. that the center of buoyancy should be on the same vertical
line as the center of gravity of the ship. The remaining three equations
(9.1.18), (9.1.19), and (9.1.20) in the group serve to determine the dis-
placements 0U, 021, and cov which occur in the boundary condition
(9.1.12) for the velocity potential 9^. In the special case we consider
(cf. (9.1.14)) we observe that these three equations would determine
the values of the constants Q19 0n, and 0zl (the complex amplitudes
of certain displacements of the ship) which occur as parameters in the
boundary conditions for the harmonic function tpi(x, y, z) given in
(9.1.14)!.
We are now able to draw some interesting conclusions. Once the
speed s0 is fixed, it follows that the problem of determining the har-
monic function (jp1 is completely formulated through the equations
(9.1.14), (9.1.14)0, (9.1.14)!, and (9.1.18) to (9.1.20) inclusive (to-
gether with appropriate conditions at oo). In other words, the motion
of the water, which is fixed solely by <pv is entirely independent of the
THE MOTION OF A SHIP IN A SEAWAY 255
pitching displacement 031(0» the heave yi(t)9 and the surge s^t), i.e.
of all displacements in the vertical plane except the constant forward
speed SQ. A little reflection, however, makes this result quite plausible:
Our theory is based on the assumption that the ship is a thin disk
disposed vertically in the water, whose thickness is a quantity of
first order. Hence only finite displacements of the disk parallel to
this vertical plane could create oscillations in the water that are of
first order. On the other hand, displacements of first order of the disk
at right angles to itself will create motions in the water that are also
of first order. One might seek to describe the situation crudely in the
following fashion. Imagine a knife blade held vertically in the water.
Up-and-down motions of the knife evidently produce motions of the
water which arc of a quite different order of magnitude from motions
produced by displacements of the knife perpendicular to the plane of
its blade. Stress is laid on this phenomenon here because it helps to
promote understanding of other occurrences to be described later.
The terms of second order in /? yield, finally, the following conditions:
(9.1.21)
A/!*! - — p hx[(<Pit - s0<plxY + (<plt
J A
(9.1.22)
^i^~%f (2/i+^31)Mr-p hy[(q'it
JL JA
(9.1.23)
'si^ai^ -2eg°3i| (y-Vc)h*A-lQSyA xhdx
-2gg031f ,r2Mr+/T
J L
-Q\ \^hv-(y~yfc)hx][((plt~s^lx)+ + ((plt-
J A
We note that integrals over the projected water-line L of the ship on
the vertical plane when in its equilibrium position occur in addition
to integrals over the vertical projection A of the entire hull. The
quantity /31 arises from the relation / = /?/31 for the moment of
inertia / of the ship with respect to an axis through its e.g. parallel
to the s'-axis. The equation (9.1.21 ) determines the surge sv and also
the speed s0 (or, if one wishes, the thrust T is determined if SQ is
256 WATER WAVES
assumed to be given). Furthermore, the speed s0 is fixed solely by T
and the geometry of the ship's hull. This can be seen, with reference
to (9.1.14) and the discussion that accompanies it, in the following
way: The term y0(#, y, z) in (9.1.14) is the term in q>^ that is indepen-
dent of t. It therefore determines T upon insertion of ^ in (9.1.21).
This term, however, is obtained by finding the harmonic function
y0 as a solution of the boundary problem for \pQ formulated in (9.1.14)0.
In fact, the relation between s0 and T is now seen to be exactly the
same relation as was obtained by Michell. (It will be written down in
a later section. ) In other words, the wave resistance depends only on
the basic translatory motion with uniform speed of the ship, and not
at all on its small oscillations relative to that motion. If, then, effects
on the wave resistance due to the oscillation of the ship are to be
obtained from the theory, it will be necessary to take account of higher
order terms. Once the thrust T has been determined the equations
(9.1.22) and (9.1.23) form a coupled system for the determination of
yl and 031, since 9^ and 0n have presumably been determined previous-
ly. Thus our system is one in which there is a considerable amount of
cross-coupling. It might also be noted that the trim, i.e. the constant
values of yl and 031 about which the oscillations in these degrees of
freedom occur are determined from (9.1.22) and (9.1.23) by the time-
independent terms in these equations — including, for example, the
moment IT of the thrust about the e.g.
We proceed to the discussion of other conclusions arising from our
developments and concerning two questions which recur again and
again in the literature. These issues center around the question: what
is the correct manner of satisfying the boundary conditions on the
curved hull of the ship? Michell employed the condition (9.1.12),
naturally with 0n = 021 = coj = 0, on the basis of the physical argu-
ment that s0hx represents the component of the velocity of the water
normal to the hull, and since the hull is slender, a good approximation
would result by using as boundary condition the jump condition
furnished by (9.1.12). Havelock and others have usually followed the
same practice. However, one finds constant criticism of the resulting
theory in the literature (particularly in the engineering literature)
because of the fact that the boundary condition is not satisfied at the
actual position of the ship's hull, and various proposals have been
made to improve the approximation. This criticism would seem
to be beside the point, since the condition (9.1.12) is simply the con-
sequence of a reasonable linearization of the problem. To take account
THE MOTION OF A SHIP IN A SEAWAY
257
of the boundary condition at the actual position of the hull would, of
course, be more accurate —but then, it would be necessary to deal
with the full nonlinear problem and make sure that all of the essential
correction terms of a given order were obtained. In particular, it
would be necessary to examine the higher order terms in the condi-
tions at the free surface—after all, the conditions (9.1.13), which are
also used by Michell and Havelock (and everyone else, for that
matter), are satisfied at y = 0 and not on the actual displaced position
of the free surface. One way to obtain a more accurate theory would
be, of course, to carry out the perturbation scheme outlined here to
higher order terms.
Still another point has come up for frequent discussion (cf., for
example, Lunde and Wigley [L.18]) with reference to the boundary
condition on the hull. It is fairly common in the literature to refer to
ships of Miehell's type, by which is meant ships which are slender
not only in the fore-and-aft direction, but which are also slender in
the cross-sections at right angles to this direction (cf. Fig. 9.1.3) so
- y
(o) (b)
Fig. 9.1.3a, b. Ships with full and with narrow mid-sections
that hy, in our notation, is small. Thus ships with a rather broad
bottom (cf. Fig. 9.1.3a), or, as it is also put, with a full mid-section,
arc often considered as ships to which the present theory does not
apply. On the other hand, there are experimental results (cf. Havelock
[H.7]) which indicate that the theory is just as accurate for ships
with a full mid-section as it is for ships of Miehell's type. When the
problem is examined from the point of view taken here, i.e. as a
problem to be solved by a development with respect to a parameter
characterizing the slenderness of the ship, the difference in the two
cases would seem to be that ships with a full mid-section should be
regarded as slender in both draft and beam, (otherwise no lineariza-
tion based on assuming small disturbances in the water would be
258 WATER WAVES
reasonable), while a ship of Michell's type is one in which the draft is
finite and the beam is small. In the former case a development dif-
ferent from the one given above would result: the mass and moments
of inertia would be of second order, for instance, rather than first
order. Later on we shall have occasion to mention other possible ways
of introducing the development parameter.
We continue by pointing out a number of conclusions, in addition
to those already given, which can be inferred from our equations
without solving them. Consider, for example, the equations (9.1.22)
and (9.1.23) for the heave yl and the pitching oscillation 031, and make
the assumption that
(9.1.24) f xhdx = 0
(which means that the horizontal section of the ship at the water line
has the e.g. of its area on the same vertical as that of the whole ship).
If this condition is satisfied it is immediately seen that the oscillations
031 and yl are not coupled. Furthermore, these equations are seen to
have the form
(9.1.25) */i + AJft = p(0
(9.1.26) 031 + AJ081 - q(t)
with
(9.1.27) Af =
r~ r r ~\
%6£ (y ~ y' )hdA -\- x2hdx
(9.1.28) **= t-*-- L _Ji: -=L.
It follows that resonance* is possible ifp(t) has a harmonic component
of the form A cos (Aj + B) or q(t) a component of the form
A cos (A2< + 5): in other words, one could expect exceptionally
heavy oscillations if the speed of the ship and the seaway were to be
such as to lead to forced oscillations having frequencies close to these
values. One observes also that these resonant frequencies can be
computed without reference to the motion of the sea or the ship:
the quantities A15 A2 depend only on the shape of the hull.**
* The term resonance is used here in the strict sense, i.e. that an infinite
amplitude is theoretically possible at the resonant frequency.
** The equation (9.1.27) can be interpreted in the following way: it furnishes
the frequency of free vibration of a system with one degree of freedom in which
the restoring force is proportional to the weight of water displaced by a cylinder
of cross-section area 2§L hdx when it is immersed vertically in water to a depth yv.
THE MOTION OF A SHIP IN A SEAWAY 259
In spite of the fact that the linear theory presented here must be
used with caution in relation to the actual practical problems con-
cerning ships in motion, it still seems likely that such resonant fre-
quencies would be significant if they happened to occur as harmonic
components in the terms p(t) or q(t) with appreciable amplitudes.
Suppose, for instance, that the ship is moving in a sea-way that
consists of a single train of simple harmonic progressing plane waves
with circular frequency a which have their crests at right angles to
the course of the ship. If the speed of the ship is s0 one finds that the
circular excitation frequency of the disturbances caused by such
waves, as viewed from the moving coordinate system (#, t/, z) that is
used in the discussion here, is a + *0a2/g, since o2/g is 2n times the
reciprocal of the wave length of the wave train. Thus if Ax or A2 should
happen to lie near this value, a heavy oscillation might be expected.
One can also see that a change of course to one quartering the waves at
angle y would lead to a circular excitation frequency a+sQ cos y • a2/g
and naturally this would have an effect on the amplitude of the response.
It has already been stated that the theory presented here is closely
related to the theory published by Haskind [H.4] in 1946, and it was
indicated that the two theories differ in some respects. We have not
made a comparison of the two theories in the general case, which would
not be easy to do, but it is possible to make a comparison rather easily
in the special case treated by Haskind in detail. This is the special
case dealt with in the second of his two papers in which the ship is
assumed to oscillate only in the vertical plane— as would be possible
if the seaway consisted of trains of plane waves all having their crests
at right angles to the course of the ship. Thus only the quantities y^t)
and 031(/), which are denoted in Haskind's paper by £(0 and y>(t), are of
interest. Haskind finds differential equations of second order for these
quantities, but these equations are not the same as the corresponding
equations (9.1.22), (9.1.23) above. One observes that (9.1.22) con-
tains as its only derivative the second derivative/}^ and (9.1.23) con-
tains as its sole derivative a term with 031; in other words there are no
first derivative terms at all, and the coupling arises solely through
the undiffcrentiated terms. Haskind's equations are quite different
since first and second derivatives of both dependent functions occur
in both of the two equations; thus Haskind, on the basis of his theory,
can speak, for example, of damping terms, while the theory presented
here yields no such terms. On the basis of the theory presented so far
there should be no damping terms of this order for the following
260 WATER WAVES
reasons: In the absence of frictional resistances, the only way in
which energy can be dissipated is through the transport of energy to
infinity by means of out-going progressing waves. However, we have
already given valid reasons for the fact that those oscillations of the
ship which consist solely of displacements parallel to the vertical
plane produce waves in the water with amplitudes that are of higher
order than those considered in the first approximation. Thus no such
dissipation of energy should occur.* In any case, our theory has this
fact as one of its consequences. Haskind [H.4] also says, and we quote
from the translation of his paper (sec page 59): "Thus, for a ship
symmetric with respect to its midship section . . ., only in the absence
of translatory motion, i.e., for s0 = 0, are the heaving and pitching
oscillations independent." This statement does not hold in our version
of the theory. As one sees from (9.1.22) and (9.1.23) coupling occurs
if, and only if, ochdx ^ 0, whether SQ vanishes or not. In addition,
Haskind obtains no resonant frequencies in these displacements be-
cause of the presence of first-derivative terms in his equation; the
author feels that such resonant frequencies may well be an important
feature of the problem. Thus it seems likely that Haskind's theory
differs from that presented here because he includes a number of
terms which are of higher order than those retained here. Of course, it
does not matter too much if some terms of higher order are included
in a perturbation theory, at least if all the terms of lowest order are
really present: at worst, one might be deceived in giving too much
significance to such higher order terms.
The fact that the theory presented so far leads to the conclusion
that no damping of the pitching, surging, and heaving oscillations
occurs is naturally an important fact in relation to the practical pro-
blems. Unfortunately, actual hulls of ships seem in many cases to be
designed in such a way that damping terms in the heaving and pitch-
ing oscillations are numerically of the same order as other terms in
the equations of motion of a ship. (At least, there seems to be experi-
mental evidence from model studies— see the paper by Korvin-
Krukovsky and Lewis [K.I 6]— which bears out this statement.)
Consequently, one must conclude that either actual ships are not
* It is, however, important to state explicitly that there would be damping
of the rolling, yawing, and swaying oscillations, since these motions create waves
having amplitudes of the order retained in the first approximation, and thus
energy would be carried off to infinity as a consequence of such motions.
THE MOTION OF A SHIP IN A SKA WAY 261
sufficiently slender for the lowest order theory developed here to apply
with accuracy, or that important physical factors such as turbulence,
viscosity, etc., have effects so large that they cannot be safely neg-
lected. If it is the second factor that is decisive, rather than the loss
of energy due to the creation of waves through pitching and heaving,
it is clear that only a basic theory different from the one proposed
here would s6rve to include such effects. If, however, the damping
has its origin in the creation of gravity waves we need not be entirely
helpless in dealing with it in terms of the sort of theory contemplated
here. It would not be helpful, though, to try to overcome the difficulty
by carrying the development to terms of higher order, for example,
even though there would certainly then be damping effects in pitching
and heaving: such damping effects of higher order could evidently not
introduce damping into the motions of lower order. This is fortunately
not the only way in which the difficulty can be attacked. One rather
obvious procedure would be to retain the present theory, and simply
add damping terms with coefficients to be fixed empirically, in some-
what the same fashion as has been proposed by St. Denis and Wein-
blum [S.I], for example.
There are still other possibilities for the derivation of theories which
would include damping effects without requiring a semi-empirical
treatment, but rather a different development with respect to a slen-
derness parameter. One such possibility has already been hinted at
above in the course of the discussion of ships of broad mid-section
compared with ships of MichelPs type. If the ship is considered to be
slender in both draft and beam the waves due to oscillations of the
ship would be of the same order with respect to all of the degrees of
freedom; a theory utilizing this observation is being investigated.
Another possibility would be to regard the draft as small while the
beam is finite (thus the ship is thought of as a flat body with a
planing motion over the water), i.e. to base the perturbation scheme
on the following equation for the hull (instead of (9.1.2)):
and to carry out the development with respect to /?, This theory has
been worked out in all detail, though it has not yet been published.
With respect to damping effects the situation is now in some respects
just the reverse of that described above: now it is the oscillations in
the vertical plane, together with the rolling oscillation, that are
damped to lowest order, while the yawing and swaying oscillations
262
WATER WAVES
are undamped. It would seem reasonable therefore to investigate the
results of such a theory for conventional hulls and make comparisons
with model experiments. This still does not exhaust all of the possibili-
ties with respect to various types of perturbation schemes, particu-
larly if hulls of special shape are introduced. Consider, for example,
a hull of the kind used for some types of sailing yachts, and shown
schematically in Fig. 9.1.4. Such a hull has the property that its beam
Fig. 9.1.4. Cross section of hull of a yacht
and draft are both finite, but the hull cross section consists of two
thin disks joined at right angles like a T. In this case an appropriate
development with respect to a slenderness parameter can also be
made in regarding both disks as being slender of the same order. The
result is a theory in which all oscillations, except the surge, would be
damped; this theory has been worked out too but not yet published.
It would take up an inordinate amount of space in this book to deal
in detail with all of the various types of possible perturbation schemes
mentioned above. In addition, only one of them seems so far to permit
explicit solutions even in special cases, and that is the generalization
of the Michell theory which was explained at some length above. Con-
sequently, only this theory (in fact, only a special case of it) will be
developed in detail in the remainder of the chapter. In all other
theories, it seems necessary to solve certain integral equations before
the motion of the ship can be determined even under the most restric-
tive hypotheses— such as a motion of pure translation with no oscilla-
tions whatever, for example. Even in the case of the generalized
Michell theory (i.e. the case of a ship regarded as a thin disk disposed
vertically) an explicit solution of the problem for the lowest order
approximation ^ to the velocity potential— in terms of an integral
THE MOTION OF A SHIP IN A SEAWAY 268
representation, say— seems out of the question. In fact, as soon as
rolling or yawing motions occur, explicit solutions are unlikely to be
found. The best that has been done so far in such cases has been to
formulate an integral equation for the values of 9^ over the vertical
projection A of the ship's hull; this method of attack, which looks
possible and somewhat hopeful for numerical purposes since the
motion of the ship requires the knowledge of (pl only over the area A,
is under investigation. However, if the motion of the ship is confined
to a vertical plane, so that co1 — 6n = 02i — 0> ft is possible to solve
the problems explicitly. This can be seen with reference to the bound-
ary conditions (9.1.12) and (9.1.13) which in this case are identical
with those of the classical theory of Michell and Havelock, and hence
permit an explicit solution for q>± which is given later on in section
9.4. After 9?! is determined, it can be inserted in (9.1.21), (9.1.22), and
(9.1.23) to find the forward speed $Q corresponding to the thrust T,
the two quantities fixing the trim, and the surge, pitching, and heav-
ing oscillations.* In all, six quantities fixing the motion of the ship
can be determined explicitly. Only this version of the theory will be
presented in detail in the remainder of the chapter.
The theory discussed here is very general, and it therefore could be
applied to the study of a wide variety of different problems. For exam-
ple, the stability of the oscillations of a ship could be in principle
investigated on a rational dynamical basis, rather than as at present
by assuming the water to remain at rest when the ship oscillates. It
would be possible to investigate theoretically how a ship would move
with a given rudder setting, and find the turning radius, angle of heel,
etc. The problem of stabilization of a ship by gyroscopes or other de-
vices could be attacked in a very general way: the dynamical equa-
tions for the stabilizers would simply be included in the formulation
of the problem together with the forces arising from the interactions
of the water with the hull of the ship.
In sec. 9.2 the general formulation of the problem is given; in
sec. 9.3 the details of the linearization process are carried out for the
case of a ship which is slender in beam (i.e. under the condition
implied in the classical Michell-Havelock theory); and in sec. 9.4 a
solution of the problem is given for the case of motion confined to the
vertical plane, including a verification of the fact that the wave
resistance is given by the same formula as was found by Michell.
* These free undamped vibrations are uniquely determined only when initial
conditions are given.
264
WATER WAVES
9.2. General formulation of the problem
We derive here a theory for the most general motion of a rigid body
through water of infinite depth which is in its turn also in motion in
any manner. As always we assume that a velocity potential exists.
Since we deal with a moving rigid body it is convenient to refer the
motion to various types of moving coordinate systems as well as to a
fixed coordinate system. The fixed coordinate system is denoted by
O — X, F, Z and has the disposition used throughout this book: The
X, Z-plane is in the equilibrium position of the free surface of the
water, and the Y-axis is positive upwards. The first of the two moving
coordinate systems we use (the second will be introduced later) is
denoted by o — x9 y, z and is specified as follows (cf. Fig. 9.2.1):
Fig. 9.2.1. Fixed and moving coordinate system
The x, 2-plane coincides with the X, Z-plane (i.e. it lies in the undis-
turbed free surface), the y-axis is vertically upward and contains the
center of gravity of the ship. The #-axis has always the direction of the
horizontal component of the velocity of the center of gravity of the
ship. (If we define the course of the ship as the vertical projection of
the path of its center of gravity on the X, Z-plane, then our conven-
tion about the a?-axis means that this axis is taken tangent to the
ship's course.) Thus if Rc = (Xe, YC9 Zc) is the position vector of the
center of gravity of the ship relative to the fixed coordinate system
and hence Rc = (XC9 yc, Zc) is the velocity of the e.g., it follows that
the #-axis has the direction of the vector u given by
(9-2.1) u = X€I + ZCK
with I and K unit vectors along the X-axis and the Z-axis. If i is a
unit vector along the <r-axis we may write
(9.2.2) s(t)i = u,
THE MOTION OF A SHIP IN A SEAWAY 265
thus introducing the speed s(t) of the ship relative to a horizontal
plane. For later purposes we also introduce the angular velocity
vector <o of the moving coordinate system:
(9.2.3) <o = a)(t)J,
and the angle a (cf. Fig. 9.2.1) by
(9.2.4) a(0 = I aj(r)dr.
J o
The equations of transformation from one coordinate system to the
other are
!X=x cos a +z sin a+Jtc ;x=(X— Xc) cos a— (Z— Zc) sin a
Y=y ;y=Y
Z— —x sin a +z cos a -\-Zc; z=(X—Xc) sin a -\-(Z— Zc) cos a.
By 0(X, Y, Z; t) we denote the velocity potential and write
(9.2.6) 0(X,Y,Z;t)
~ 0(x cos a + 2 sin a + Xc, y, — x sin a + z cos a + Zc; t)
= <p(x, y, z; t).
In addition to the transformation formulas for the coordinates, we
also need the formulas for the transformation of various derivatives.
One finds without difficulty the following formulas:
&x — <Px cos <* + y>z sin a
(9.2.7) 0Y = <py
0Z = — (px sin a -f- q>z cos a.
It is clear that grad2 0(X, F, Z; t) = grad2 <p(x, y, z; t) and that y is
a harmonic function in a?, y, z since 0 is harmonic in X, F, Z. To cal-
culate 0t is a little more difficult; the result is
(9.2.8) 0t = - (s + coz)(px + cox<pz + <pt.
(To verify this formula, one uses 0t — (fyxi + <pvyt + <pzzt + <Pt and
the relations (9.2.5) together with s cos a = Xc> « sin a = — ZC0 The
last two sets of equations make it possible to express Bernoulli's law
in terms of (p(x, y, z; t); one has:
v I
(9.2.9) -f- + gy + — (grad <p)2 — (s + <oz)<px + cox<pz + <pt = 0.
Q 2
Suppose now that F(X, F, Z; J) — 0 is a boundary surface (fixed
or moving) and set
(9.2.10) F(x cos a + . . ., j/, — x sin a + . . .; t) = f(x, y, z; t),
266
WATER WAVES
so that /(a?, t/, z; t) = 0 is the equation of the boundary surface rela-
tive to the moving coordinate system. The kinematic condition to be
satisfied on such a boundary surface is that the particle derivative
dF/dt vanishes, and this leads to the boundary condition
(9.2.11 ) <pJ9 + <pyfv + <pj, - (* + a*)/. + <*>*/* + ft = 0
relative to the moving coordinate system upon using the appropriate
transformation formulas. In particular, if y — rj(x, z\ t) = 0 is the
equation of the free surface of the water, the appropriate kinematic
condition is
(9.2.12) - yjj9 +<PV- <Ptfz + (9 + coztyx - 0*09, - fy = 0
to be satisfied for y = 77. (The dynamic free surface condition is of
course obtained for y = ?? from (9.2.9) by setting p — 0.)
We turn next to the derivation of the appropriate conditions, both
kinematic and dynamic, on the ship's hull. To this end it is convenient
to introduce another moving coordinate system 0' — #', y'9 z' which
is rigidly attached to the ship. It is assumed that the hull of the ship
has a vertical plane of symmetry (which also contains the center of
gravity of the ship); we locate the x'9 y' -plane in it (cf. Fig. 9.2.2) and
suppose that the t/'-axis contains the center of gravity. The o' — #',
t/', z' system, like the other moving system, is supposed to coincide
(a) (b)
Fig. 9.2.2a, b. Another moving coordinate system
with the fixed system in the rest position of equilibrium. The center
of gravity of the ship will thus be located at a definite point on the
t/'-axis, say at distance y'0 from the origin o': in other words, the system
of coordinates attached rigidly to the ship is such that the center of
gravity has the coordinate (0, yc, 0).
In the present section we do not wish in general to carry out lineari-
zations. However, since we shall in the end deal only with motions
THE MOTION OF A SHIP IN A SEAWAY 267
which involve small oscillations of the ship relative to the first moving
coordinate system o — x, y, z9 it is convenient and saves time and
space to suppose even at this point that the angular displacement
of the ship relative to the o — #, y, z system is so small that it can
be treated as a vector 8:
(9.2.13) 8-0^ +0J + 03k.
The transformation formulas, correct up to first order terms in the
components 0i of 8, are then given by:
' x9 = x + 03(t/ — yc) - 02*
y y \y c y § / i i 3
z' = z + 62? — O^y - yc)
with yc of course representing the ^-coordinate of the center of
gravity in the unprimed system. It is assumed that yc — y'e is a small
quantity of the same order as Qt and only linear terms in this quantity
have been retained. (The verification of (9.2.14) is easily carried out
by making use of the vector-product formula 8 = 8 X r, for the
small displacement 8 of a rigid body under a small rotation 8.)
The equation of the hull of the ship (assumed to be symmetrical
with respect to the #', j/'-plane) is now supposed given relative to the
primed system of coordinates in the form:
/a o i K\ yf — r /Yr' ii'\ y' > O
\*Jȣt,\.*Jj A ^T S V4^ 5 a /> <
The equation of the hull relative to the o — x, y, s-system can now
be written in the form
(9.2.16) z + Bv-OAy - y'e) - f (a, y) + [0# - 03(j/ - ^)]C. (*,y)
+ [(yc - y'e) - 0i* + OrtM*, y) = o, *' > o,
when higher order terms in (yc — y'c) and 0t are neglected. The left
hand side of this equation could now be inserted for / in (9.2.11) to
yield the kinematic boundary condition on the hull of the ship, but
we postpone this step until the next section.
The dynamical conditions on the ship's hull are obtained from the
assumption that the ship is a rigid body in motion under the action
of the propeller thrust T, its weight Mg, and the pressure p of the
water on its hull. The principle of the motion of the center of gravity
yields the condition
(9.2.17) M~ (*i + yj) - J pn dS + T - Mgj.
268 WATER WAVES
By n we mean the inward unit normal on the hull. Our moving
coordinate system o — #, y, z is such that di/dt = — cok and dj/dt =
0, so that (9.2.17) can be written in the form
(9.2.18) Msi - Mscok + Mycj = ( pn dS + T - Mgj,
with p defined by (9.2.9). The law of conservation of angular momen-
tum is taken in the form:
(9.2.19) 1 ( (R - Rc) X (R - Rc)dm
dt J M
= ( p(R - Rc) n dS + (RT - Rc) x T.
J s
The crosses all indicate vector products. By R is meant the position
vector of the element of mass dm relative to the fixed coordinate
system. Rc (cf. Fig. 9.2.1) fixes the position of the e.g. and RT
locates the point of application of the propeller thrust T. We introduce
r = (#, y, z) as the position vector of any point in the ship in the
moving coordinate system and set
(9.2.20) q = r - ycj,
so that q is a vector from the e.g. to any point in the ship. The relation
(9.2.21) R = Rc + (co + 6) X q
holds, since co + 8 is the angular velocity of the ship; thus (9.2.21)
is simply the statement of a basic kinematic property of rigid bodies.
By using the last two relations the dynamical condition (9.2.19) can
be expressed in terms of quantities measured with respect to the
moving coordinate system o — x9 y, z, as follows:
(9.2.22) ~ J (r - ycj) X [(co + 9) X (r - ycj)]dn
r
0(r — J/cJ) X n dS + (RT — Rc) x T.
We have now derived the basic equations for the motion of the
ship. What would be wanted in general would be a velocity potential
<p(x, y, z; t) satisfying (9.2.11) on the hull of the ship, conditions
(9.2.9) (with p = 0) and (9.2.12) on the free surface of the water;
and conditions (9.2.17) and (9.2.22), which involve 9? under integral
signs through the pressure p as given by (9.2.9). Of course, the quan-
THE MOTION OF A SHIP IN A SEAWAY 269
tities fixing the motion of the ship must also be determined in such a
way that all of the conditions are satisfied. In addition, there would
be initial conditions and conditions at oo to be satisfied. Detailed
consideration of these conditions, and the complete formulation of the
problem of determining y(x, y, z; t) under various conditions will be
postponed until later on since we wish to carry out a linearization
of all of the conditions formulated here.
9.3. Linearization by a formal perturbation procedure
Because of the complicated nature of our conditions, it seems wise
(as was indicated in sec. 1 of this chapter) to carry out the lineari-
zation by a formal development in order to make sure that all terms of
a given order are retained; this is all the more necessary since terms
of different orders must be considered. The linearization carried out
here is based on the assumption that the motion of the water relative
to the fixed coordinate system is a small oscillation about the rest
position of equilibrium. It follows, in particular, that the elevation of
the free surface of the water should be assumed to be small and, of
course, that the motion of the ship relative to the first moving coor-
dinate system o — x, y> z should be treated as a small oscillation.We
do not, however, wish to consider the speed of the ship with respect
to the fixed coordinate system to be a small quantity: it should rather
be considered a finite quantity. This brings with it the necessity to
restrict the form of the ship so that its motion through the water does
not cause disturbances so large as to violate our basic assumption;
in other words, we must assume the ship to have the form of a thin
disk. In addition, it is clear that the velocity of such a disk-like ship
must of necessity maintain a direction that does not depart too much
from the plane of the thin disk if small disturbances only are to be
created. Thus we assume that the equation of the ship's hull is given by
(9.3.1) z' =ph(x'9y'), z' > 0,
with ft a small dimensionless parameter, so that the ship is a thin
disk symmetrical with respect to the xl \ j/'-plane, and @h takes the
place of f in (9.2.15). (It has already been noted in the introduction
to this chapter that this is not the most general way to describe the
shape of a disk that would be suitable for a linearization of the type
carried out here. ) We have already assumed that the motion of the
ship is a small oscillation relative to the moving coordinate system
270 WATER WAVES
o — tc,y, z. It seems reasonable, therefore, to develop all our basic
quantities (taken as functions of x, y,z;t) in powers of /?, as follows:
(9.8.2) y(x, y, 2; t; /?) = ^
(9.8.8) 17(0, 2; <; 0) = fa
(9.8.4) <(<; 0) = *0 + fo + /»•«, + . . .,
(9.8.5) «(<; j8) = o>0 + /toj + /Wo, + . . .,
(9.8.6) 0,(*; /?) = /30« + /W« + . . ., 1 = 1, 2, 8
The first and second conditions state that the velocity potential and
the surface wave amplitudes, as seen from the moving system, are
small of order /3. The speed of the ship, on the other hand, and the
angular velocity of the moving coordinate system about the vertical
axis of the fixed coordinate system, are assumed to be of order zero.
(It will turn out, however, that cu0 must vanish— a not unexpected
result.) The relations (9.3.6) and (9.3.7) serve to make precise our
assumption that the motion of the ship is a small oscillation relative
to the system o — x, t/, z.
We must now insert these developments in the conditions derived
in the previous section. The free surface conditions are treated first.
As a preliminary step we observe that
(9.3.8) <px(x, rj, z; t; ft) - P[<pl9(x, 0, z; t) + rff^(x9 0, z; t) + . . .]
*, 0, z; t) + P*[fli<pixv(x> °» *; ') + ?i«(*» °» *5 0]
with similar formulas for other quantities when they are evaluated
on the free surface y = rj. Here we have used the fact that r\ is small
of order ft and have developed in Taylor series. Consequently, the
dynamic free surface condition fory==ri arising from (9.2.9) with
p = 0 can be expressed in the form
(9.3.9) gfot + (3*r)2 + ...] +tf«[(grad ^)2 + . . .]
- Oo + fo + • • • + *K +#»! + •• OHM* +
+ <P*x) + . . •]
and this condition is to be satisfied for y = 0. In fact, as always in
THE MOTION OF A SHIP IN A SEAWAY 271
problems of small oscillations of continuous media, the boundary
conditions are satisfied in general at the equilibrium position of the
boundaries. Upon equating the coefficient of the lowest order term
to zero we obtain the dynamical free surface condition
(9.3.10) — g^ + (s0 + co<p)(plx — a)<p(plz — (plt = 0 for y = 0,
and it is clear that conditions on the higher order terms could also be
obtained if desired. In a similar fashion the kinematic free surface
condition can be derived from (9.2.12); the lowest order term in ft
yields this condition in the form:
(9.3.11) q>ly + (SQ + co^)rjlx - a)^rjlz - i?lt = 0 for y = 0.
We turn next to the derivation of the linearized boundary condi-
tions on the ship's hull. In view of (9.3.6) and (9.3.7), the transforma-
tion formulas (9.2.14) can be put in the form
(x'=x +06^-9'.) -fin*
(9.3.12) ly' = y - fa + ftQ^z - ftB^x
(z'=z+p02lx-(30n(y-y'c)
when terms involving second and higher powers of ft are rejected.
Consequently, the equation (9.2.16) of the ship's hull, up to terms in
ft2, can be written as follows:
z + fi62lx - peu(y - y'c) - fth[x + ftQ^y - y'c) - ftQ^z,
y - Pvi + flu* - &**] = o,
and, upon expanding the function fe, the equation becomes
(9.3.13) z + ftQ2lx - 00n(y - y'c) - ph(x, y) + . . . = 0,
the dots representing higher order terms in ft. We can now obtain the
kinematic boundary condition for the hull by inserting the left hand
side of (9.3.13) for the function / in (9.2.11); the result is
(9314) l*00^0
l^i* = *<>(02i - *«)
when the terms of zero and first order only arc taken into account.
It is clear that these conditions are to be satisfied over the domain
A of the x, t/-plane that is covered by the projection of the hull on the
plane when the ship is in the rest position of equilibrium. As was
mentioned earlier, it turns out that o>0 = 0, i.e., that the angular
velocity about the t/-axis must be small of first order, or, as it could
also be put, the curvature of the ship's course must be small since the
272 WATER WAVES
speed in the course is finite. The quantity s^t) in (9.3.4) evidently
yields the oscillation of the ship in the direction of the #-axis (the so-
called "surge").
It should also be noted that if we use z' = — f}h(x', y') we find,
corresponding to (9.3.14), that
9>i* = *o(02i + hx) - K + 02i)* + bu(y - y'c).
This means that A must be regarded as two sided, and that the last
equation is to be satisfied on the side of A which faces the negative
2-axis. The last equation and (9.3.14) imply that <p may have discon-
tinuities at the disk A.
The next step in the procedure is to substitute the developments
with respect to /?, (9.3.2) — (9.3.7), in the conditions for the ship's
hull given by (9.2.18) and (9.2.22). Let us begin with the integral
1
pn dS which appears in (9.2.18). In this integral S is the immersed
s
surface of the hull, n is the inward unit normal to this surface and p
is the pressure on it which is to be calculated from (9.2.9). With
respect to the o — x , y, z coordinate system the last equations of the
symmetrical halves of the hull are
02" Z ==
where
<Q q im
(9.3.16)
We can now write
pn dS = pnt dS1 + pn2 dS2
Js Js1 Js2
in which nx and n2 are given by
H1 + H ~ k
We can also write
pn dS = - eg \ yn dS + \ Pln dS
Js Js Js
= —68] Vn ds + I Pinidsi +
Js Jsi Jsa
where pl9 from (9.2.9), is given by
(9.3.17) pl = - 0[£(grad <p)2 - (s + (oz)(px + xcoy, + yt].
THE MOTION OF A SHIP IN A SEAWAY 273
If S0 is the hull surface below the #, 2-plane, the surface area SQ — S
is of order ft and in this area each of the quantities t/, Hv H2 is of
order ft. Hence one finds the following to hold:
- f yn dS = - f yndS + (l+ j)O(^) + kO(^).
Js Js0
From the divergence theorem we have
- f yn dS = VI
JS9
where V is the volume bounded by SQ and the x9 s-plane. With an
accuracy of order /?3, V is given by
V=20 f hdA- f ft(yl+0Blx)dB=2ft f hdA-2p* ( (
JA JB JA JL
Here A is the projection of the hull on the vertical plane when the
hull is in the equilibrium position, B is the equilibrium water line
area, and L is the projection of the equilibrium water line on the
If Wv W2 are the respective projections of the immersed surfaces
Sl9 S2 on the x, j/-plane we have
f Pln dS = 1 ( f Pl(jr, y, HI; t)HlrdW,- ! Pl(xt y, H2;
Js \Jwl Jwz
+J ( f Pi(*> *J< » i5 0# ndH'i - f Pi(*> y> H*
U \YI J \\-2
-k ( f Pi(*> y> »i; t)d\\\- f Pl(x, y, H2; t)d
I J ^ J ira
Neither Wl nor W2 is identical with A. Each of the differences
ll\ — A, W2 — A is, however, an area of order ft. From this and the
fact that each of the quantities p, Hlx, Hly, H2X9 H2y is of order ft, it
follows that
(9.3.18)
J Pln dS=l j f [Pl(^, y, II,; t)IIlx-Pl(x, y, II2;t)H2x]dA+O(p*)\
+j( f [Pi(^ y,Hi;t)Hly -Pl (cr,i/,//2; t)H2y]
\JA
y, HI; 0-Pite 2/» W
274
WATER WAVES
(9.3.19)
It was pointed out above that <p may be discontinuous on A. Hence
from (9.8.17), (9.8.2), (9.8.4) we write
.(*,!,, ffi;<) =
(x,y,H»t) = t
Here the + and — superscripts denote values at the positive and
negative sides of the disk A whose positive side is regarded as the side
which faces the positive s-axis. If we substitute the developments of
HI*, Hly, ff2a!, ff2y, and (9.3.19) in (9.3.18), then collect the previous
results, we find
f pn dS=i I 491 f [(A.-fln)(t^.-^)
Js \ JA
(9.3.20)
! [(hy+Ou)(sQ<plx-<plt)++(hy^^^^^
JA
f
JA
f
The integral p(r — yc\) X ndS which appears in (9.2.22) can
Js
be written
dS
f p(r-t/cj)xndS=-eg| y(r-yc])xn
Js Js
If we use the same procedure as was used above for the expansion of
pn dS we find
-^^
f W«cft.-^)+-
JA
THE MOTION OF A SHIP IN A SEAWAY 275
(9.3.21)
8gp f xhdA-*QgpOu f (y-y'cWA-2Qgpyi ! xhfa-2QgpoJ x*hdx
JA JA JL JL
+e/?2 f [^(^+eii)(Wix~9ie)++^(Av-0n)(Wix-^ie)1^
JA
-Qn[(y-y*)(h*-^i)(*^^
JA
We now assume that the propeller thrust T is of order fP and is
directed parallel to the #'-axis: that is
T = ]WY
where i' is the unit vector along the #'-axis. We also assume that T is
applied at a point in the longitudinal plane of symmetry of the ship /
units below the center of mass. Thus we have the relations
(9.3.22) T = P*Ti + 0(/J3),
and
(9.3.23) (Rr - Rc) X T = - JJ X T
The mass of the ship is of order /?. If we write M — M^ and ex-
pand the left hand side of (9.2.18) in powers of ft it becomes
P/^o + MJtit + 003*)] + Jflf^ +0(jJ>)] -
(9.3.24) = f pn dS + T - Af^gJ.
The expansion of the left hand side of (9.2.22) gives
(9.3.25)
where /J/31 is the moment of inertia of the ship about the axis which is
perpendicular to the longitudinal plane of symmetry of the ship and
which passes through the center of mass.
If we replace the pressure integrals and thrust terms in the last two
equations by (9.3.20), (9.3.21), (9.3.22), (9.3.23), and then equate the
coefficients of like powers of ft in (9.3.24) and (9.3.25) we obtain the
following linearized equations of motion of the ship. From the first
order terms we find
276 WATER WAVES
(9.8.26) *„ = 0
(9.8.27) 2gg f phdA^MJg
JA
(9.8.28) I xphdA=0
JA
(9.3.29) I [(s<fl>lx-<plt)+-(s(fplx-<plt)-]dA=0
JA
(9.8.80) f [x(s^lx-(plt)+-x(s0<plx-(plt)-]dA^O
JA
(9.3.31) f [(y-y'e)(s<fplx-<plt)+-(y-y'e)(s0<plx-<plt)-]dA=o
JA
or by (9.8.29)
(9.3.32) | [y(s</plx-(f>lt)+-y(s0<jplx-<plt)-]dA=0.
JA
From the second order terms we find
(9.3.33)
Mi*i=e f [(hx-0Zi
JA
=6 [
JA
(9.3.34)
Miy^-lqn
JL
+Q [(hv+Ou)(stf>lx-(plt)++(hy-On)(s0(plx-(plt)-]dA
JA
= -20g (Vi+*9n)Ma:+Q \ [hv
JL JA
iJn =-2pgeSl f (y-y'e)hdA-2egyi ( xhdx-'2eg031 (
JA JL J
+6 [y(hv+^i)(^i
JA
-e l(y-yc)(kx-0n
JA
THE MOTION OF A SHIP IN A SEAWAY 277
or by (9.8.30), (9.3.31)
. . r , /• r
^ai^ai — "-2?^! I (y—ye^hdA—ZQgyt I xhdx—2QgQ3l I x2hdx+lT
JA JL JL
(9.8.35)
f
JA °
Equation (9.3.26) states that the motion in the ^-direction is a
small oscillation relative to a motion with uniform speed SQ = const.
Equation (9.3.27) is an expression of Archimedes' law: the rest position
of equilibrium must be such that the weight of the water displaced
by the ship just equals the weight of the ship. The center of buoyancy
of the ship is in the plane of symmetry, and equation (9.8.28) is an
expression of the second law of equilibrium of a floating body; namely
that the center of buoyancy for the equilibrium position is on the
same vertical line, the t/'-axis, as the center of gravity of the ship.
The function <pl must satisfy
<Plxx + <Plw + <Pizz = °
in the domain D — A where D is the half space y< 0, and A is the
plane disk defined by the projection of the submerged hull on the
x, y-plane when the ship is in the equilibrium position. We assume that
A intersects the x, 2-plane. The boundary conditions at eacli side of A
are
(9.3.36)
= + sofa + 02i) - K + #21)* +
The boundary condition at y = 0 is found by eliminating rjl from
(9.3.10) and (9.3.11). Since co0 = 0 these equations are
— <Pit =
— <Piv ~ Mix + ^i< = °
and they yield
(9.3.37) sfolxx - 2^9?!^ + g<fiy + <Pw = 0
for y = 0. The boundary conditions (9.8.36) and (9.8.87) show that
Pi depends on co^t), On(t) and 021(0- The problem in potential theory
for (pl can in principle be solved in the form
without using (9.3.29), (9.3.30), (9.3.32). The significance of this has
278 WATER WAVES
already been discussed in sec. 9.1 in relation to equations (9.1.14).
The general procedure to be followed in solving all problems was also
discussed there.
The remainder of this chapter is concerned with the special case of
a ship which moves along a straight course into waves whose crests are
at right angles to the course. In this case there are surging, heaving
and pitching motions, but we have dl = 0, 02 = 0, co = 0; in addition
we note that the potential function <p can be assumed to be an even
function of z. Under these conditions the equations of motion are
much simpler. They are
(9.8.38) Mj^gl hx(s<fpix~<Pit)dA+T
JA
(9.3.39) M^yl=-2Qgy1 \hdx-2eg6^ \xhdx+2e hy(sQ<plx-<plt)dA
JL JL JA
(9.3.40) /3i03i = -2eg03if (y-y'e}hdA-2QgyA xhdx
JA JL
i f
JL
x*hdx+lT
+2Q [xhy-(y-y'c)hx}(s<fplx-(plt)dA.
JA
It will be shown in the next section that an explicit integral represen-
tation can be found for the corresponding potential function and that
this leads to integral representations for the surge sv the heave yl
and the pitching oscillation 031.
9.4. Method of solution of the problem of pitching and heaving of
a ship in a seaway having normal incidence
In this section we derive a method of solution of the problem of
calculating the pitching, surging, and heaving motions in a seaway
consisting of a train of waves with crests at right angles to the course
of the ship, which is assumed to be a straight line (i.e., co = 0). The
propeller thrust is assumed to be a constant vector.
The harmonic function 9^ and the surface elevation iyt therefore
satisfy the following free surface conditions (cf. (9.8.10) and (9.8.11),
with co0 = 0):
THE MOTION OF A SHIP IN A SEAWAY 279
(9.4.1)
= 0
The kinematic condition arising from the hull of the ship is (cf.
(9.3.14) with 021 = 0n = o>! = 0):
(9.4.2) <plz = - V**-
Before writing down other conditions, including conditions at oo,
we express <pl as a sum of two harmonic functions, as follows
(9.4.3) 9^ (x, y, z; t) = ^0(a?, y, 2) + Xl(x, y, z; t).
Here Xo is a harmonic function independent of t which is also an even
function of z. We now suppose that the motion of the ship is a steady
simple harmonic motion in the time when observed from the moving
coordinate system o — x, y, z. (Presumably such a state would result
after a long time upon starting from rest under a constant propeller
thrust.) Consequently we interpret XQ(X, J/» z) as the disturbance
caused by the ship, which therefore dies out at oo; while %i(x9 t/, z; t)
represents a train of simple harmonic plane waves covering the whole
surface of the water. Thus fa is given, with respect to the fixed coor-
dinate system O— X, Y, Z by the well-known formula (cf. Chapter 8):
~Y /a2 \
Xl = Ce" sin lat + -X +yl,
with a the frequency of the waves. In the o — x, y, z system we have,
therefore:
o r" 2 / 2\ *n
(9.4.4) Xl(x, y, z; t) = Ce^ sin — x + la +^J < + Y •
We observe that the frequency, relative to the ship, is increased above
the value a if $0 is positive — i.e. if the ship is heading into the waves
—and this is, of course, to be expected. With this choice of £lf it is
easy to verify that %Q satisfies the following conditions:
(9.4.5) S$XQXX + gXov = 0 at y = 0,
obtained after eliminating r/l from (9.4.1), and
(9.4.6) fa, = — s^hx on A,
with A, as above, the projection of the ship's hull (for z > 0) on its
vertical mid-section. In addition, we require that XQ -> 0 at oo.
280 WATER WAVES
It should be remarked at this point that the classical problem con-
cerning the waves created by the hull of a ship, first treated by Michell
[M.9], Havelock [H.7], and many others, is exactly the problem of
determining #0 from the conditions (9.4.5) and (9.4.6). Afterwards,
the insertion of ^ = %Q in (9.3.38), with ix = 0, <plt = 0, leads to the
formula for the wave resistance of the ship— i.e. the propeller thrust T
is determined. Since yl and 03 are independent of the time in this case,
one sees that the other dynamical equations, (9.3.39) and (9.3.40),
yield the displacement of the e.g. relative to the rest position of
equilibrium (the heave), and the longitudinal tilt angle (the pitching
angle). However, in the literature cited, the latter two quantities are
taken to be zero, which implies that appropriate constraints would
be needed to hold the ship in such a position relative to the water.
The main quantity of interest, though, is the wave resistance, and it
is not affected (in the first order theory, at least) by the heave and pitch.
We proceed to the determination of #0, using a method different
from the classical method and following, rather, a course which it is
hoped can be generalized in such a way as to yield solutions in more
difficult cases.
Suppose that we know the Green's function G*(f , 77, £; x,y,z) such that
G* is a harmonic function for rj < 0, f > 0 except at (#, y, z) where
it has the singularity 1/r; and G* satisfies the boundary conditions
(9.4.7) G£ + kG* = 0 on 77 = 0
G* = 0 on C = 0
where k = g/s§« We shall obtain this function explicitly in a moment,
and will proceed here to indicate how it is used. Let 27 denote the half
plane r\ = 0, £ > 0; and let Q denote the half plane f — 0, r] < 0.
From Green's formula and the classical argument involving the
singularity 1/r we have
Then, since
=0,
THE MOTION OF A SHIP IN A SEAWAY 281
we have an explicit representation of the solution in the form
or
(*> 2/> «)= - — J j Xo*G*d£dr),
(9.4.8) £,(*, y, z)= A«(£, ^ )<?*(£, q, 0; *, y, z
^
upon using (9.4.6).
In order to determine G* consider the Green's function G(£, r], C;
x,y,z) for the half space 77 < 0 which satisfies
on 77 = 0. This function can be written as
where
and g is a potential function in 77 < 0 which satisfies
., a i
on r) = 0. The formula
_. d 1
-^^ -2A;
(obtained from the well-known analogous representation for l/r) in
which the Besscl function J0 can be expressed as
. ___ 2 f n/2
-)2]= - oos [p(f-#) cos 0] cos [p(C-s) sin 0] d0,
^Jo
allows us to write
4& r°° r/2
cos ~~a? cos cos ~~
0
4A; f00 f
— —
^ J 0 J
for r\ = 0 and j/ < 0. It is now easy to see that
282 WATER WAVES
4Jfe f °° f*/2
,= —
n Jo Jo
gtf +*&,= — pc'Wrt cos [p(f-a?) cos 6] cos [p(C-a) sin 9] dB dp
n Jo Jo
is a potential function in 77 < 0 which satisfies the boundary condition.
An interchange of the order of integration gives
dp
4Jfc M* f *
= — dQ 9te p cos [p(f-*) sin
ft Jo Jo
where &e denotes the real part. If we think of p as a complex variable,
the path from 0 to oo in the last result can be replaced by any equi-
valent path L, to be chosen later:
4fk f ^2 f
,= —\ dOMei
n Jo JL
p cos [p(t-*) sin 0]e»[(m)+«£-) cos o] dp
Since the right hand side of this differential equation for g is expressed
as a superposition of exponentials in | and rj it is to be expected that
a solution of it can be found in the form
L &p-p2cos20
provided the path L can be properly chosen. The path L, which will
be fixed by a condition given below, must, of course, avoid the pole
p = A/cos2 6.
It can now be seen that the function G*(f, 17, £; #, y, z) —
G(f, rj, £; x, y, z) + G(|, ?j, £; a?, y, — 2) satisfies all the conditions
imposed on the Green's function employed in (9.4.8): the sum on the
right has the proper singularity in 77 < 0, £ > 0, it satisfies the
boundary condition (9.4.7) and
Gc(f, ??, C; x, y, z) + Gc(£, ??, £; x, y, - z)
is zero at £ = 0. Thus we have for G* the representation:
-r '
LA/-a;i-
f=o
8ft f*/2 - f ens (<nz sin 0^ ^(v+i?)+<(e-*) cos a] ,
+ ^"Jo JL
ft— p cos2
The substitution of this in (9.4.8) gives finally
)=^ ff A««,
2*JJ
THE MOTION OF A SHIP IN A SEAWAY
283
A condition imposed on %Q(x, y, z) is that #0(#, y, z) -* 0 as x -> + oo.
This condition is satisfied if we take L to be the path shown in Fig. 9.4.1 .
(P)
c/cos20
> i — • — i *
Fig. 9.4.1. The path^L in the p-plane, with c = k
The function g^ is given by
and therefore the important quantity s<fplx — <plt is given by
— ra2jc I s a2\ 1
(9.4.9) Wu - Vlt - - Cce 9 cos — + I a + -±-\t + y + StfQx.
If this is substituted in the equation (9.3.38) for the surge we have
T— + |(T+ —
The last equation shows that in order to keep s1 bounded for all t we
must take for T the value
(9.4.10)
where
T = -
igp cos 0
284 WATER WAVES
In effect, T is determined by the other time-independent term in the
equation of motion. Equation (9.4.10) gives the thrust necessary to
maintain the speed s0, or inversely it gives the speed s0 which corres-
ponds to a given thrust. The integral in (9.4.10) is called the wave
resistance integral. As one sees, it does not depend on the seaway. The
integral can be expressed in a simpler form as follows.
The function fax(x, t/, 0) is a sum of integrals of the type
?;*,
If an integral of this type is substituted in the wave resistance integral
we have
4 A
say. This is the same as
(I IJ
A A
and we see that / = 0 if
/(£,??;#, t/) = —f(x,y;£,
Therefore
, r,)/,
A A
where
/ = f "I2d0 &e ( igP C°S ° ^^ °OS [p(*-~x"> cos 0] dp
1 Jo JL g-s§pcos20
Since 3te I is zero except for the residue from the integration along
JL
the semi-circular path centered at the point
g _ &
*J cos2 6 cos2 6 '
we find from the evaluation of this residue that
/1== f*! r sec8
*Q J 0
6 e^+rt "^ e cos [k(£-x) cos 6] d6.
THE MOTION OF A SHIP IN A SEAWAY 285
We introduce MichelPs notation:
= ((hx(x, y)ekv***« cos (kx sec 6) dxdy
Q(0) = hx(x, y)ekv ^ e sin (kx sec 0) dxdy
A
and can then write
This is the familiar formula of Michcll for the wave resistance.
The surge is given by
-*°-\ (P2+Q2)sec30d0.
g
A
Hereafter we will suppose for simplicity that there is no coupling
between (9.3.39) and (9.3.40), so that xhdx = 0. The substitution
JL
of (9.4.9) in (9.3.39) therefore gives the following equation for the
heave:
2e« J
\\
A
The time independent part of yr the heave component of the trim,
we denote by r/f; it is given by
(9.4.11 ) g h
A
Here y* is the vertical displacement of the center of gravity of the
ship from its rest position when moving in calm water. The integral
on the right hand side of (9.4.11) is even more difficult to evaluate
than the wave resistance integral.
The response to the seaway in the heave component is given by
286 WATER WAVES
/y — [~a2T I o2\ 1
— 2gCa \\hye° cos — - -f|cr+s0 — h+y
,**
~~
For the case under consideration, the theory predicts that resonance
in the heave occurs when
g
The equation for the pitching angle is
I" f (y-y'e)hdA+ \ a*hfa\
LJA JL J
cos — + L + *-t+y dxdy
g \ e
The time independent part of 081, which we denote by 0*j_ is given by
2fig[ f (y-y'c)hdA+ [x^hdx\Q^
LJA JL J
f [xh.-
JA
The angle 0*! is called the angle of trim; it is the angular displacement
of a ship which moves with the speed SQ in calm water.
The oscillatory part of the heave 031 to the sea is
dxdy
ff [xhy-(y-y'e)hx-] cos ( — + lo+ 9^]t+y \
JJ I 8 \ g / )
268
and we see that the theory predicts resonance when
THE MOTION OF A SHIP IN A SEAWAY 287
Of course, the differential equations for yl and 031 permit also
solutions of the type of free undamped oscillations of a definite fre-
quency (in fact, having the resonant frequencies just discussed) but
with arbitrary amplitudes which could be fixed by appropriate initial
conditions. This point has been discussed at length in the introduction
to this chapter.
PART III
CHAPTER 10
Long Waves in Shallow Water
10.1. Introductory Remarks and Recapitulation of the Basic Equations
The basic theory for waves in shallow water has already been de-
rived at length in Chapter 2 in two different ways: one derivation,
along conventional lines, proceeded on the basis of assuming the
pressure to be determined by the hydrostatic pressure law p =
&Q(n — y) (see Fig. 10.1.1), the other by making a formal develop-
ment in powers of a parameter a; the two theories are the same in
y -
.Free Surface
h(x) > 0
Bottom
Fig. 10.1.1. Long waves in shallow water
lowest order. With one exception, the present chapter will make use
only of the theory to lowest order and consequently the derivation of
it given in sections 2 and 3 of Chapter 2 suffices for all sections oi this
chapter except section 9.
We recapitulate the basic equations. In terms of the horizontal
velocity component u = u(x, t), and the free surface elevation
rj =rj(x9t) the differential equations (cf. (2.2.11), (2.2.12)) are
(10.1.1) ut +uux = -gr)x,
(10.1.2) [wfo + h)]x = - fit.
291
292 WATER WAVES
It is sometimes useful and interesting to make reference to the gas
dynamics analogy, by introducing the "density" Q through
(10.1.3) Q = Q(r) + A),
and the "pressure" p by p = \ pdy, which in view of the hydrostatic
pressure law yields the relation
(10.1.4) Pr==/-£8'
This is an "adiabatic law" with "adiabatic exponent" 2 connecting
pressure and density. As one sees, it is the depth of the water, essen-
tially, which plays the role of the density in a gas. In terms of these
quantities, the equations (10.1.1) and (10.1.2) take the form
(10.1.5) g(ut + uux) = - px + gQhX9
(10.1.6) (QU)X = - Qt.
These equations, together with (10.1.4), correspond exactly to the
equations of compressible gas dynamics for a one-dimensional flow if
hx = 0, i.e. if the depth of the undisturbed stream is constant. It
follows that a "sound speed" or propagation speed c for the pheno-
mena governed by these equations is defined by c = V dp/dp, as in
acoustics, and this quantity in our case has the value
(10.1.7) c = — = Vg(7? +h)9
as we see from (10.1.4) and (10.1.3). Later on, we shall see that it is
indeed justified to call the quantity c the propagation speed since it
represents the local speed of propagation of "small disturbances"
relative to the moving stream. We observe the important fact that c
(which obviously is a function ofx and t) is proportional to the square
root of the depth of the water.
The propagation speed c(x, t) is a quantity of such importance that
it is worthwhile to reformulate the basic equations (10.1.1 ) and (10.1.2)
with c in place of 77. Since cx = (grjx + ghx)/2c and ct = gr)tj2c one
finds readily
(10.1.8) ut + uux + 2ccx -Hx = 0,
(10.1.9) 2ct + 2ucx + cux = 0,
with
(10.1.10) H = gh.
LONG WAVES IN SHALLOW WATER . 293
The verification in the general case that the quantity c represents
a wave propagation speed requires a rather thorough study of certain
basic properties of the differential equations. However, if we restrict
ourselves to motions which depart only slightly from the rest position
of equilibrium (i.e. the state with rj = 0, u = 0) it is easy to verify
that the quantity c then is indeed the propagation speed. From (10.1.7)
we would have in this case c = c0 + e(x, t), with c0 = Vgh and s a
small quantity of first order. We assume u and its derivatives also to
be small of first order and, in addition, take the case in which the
depth h is constant. Under these circumstances the equations (10.1.8)
and (10.1.9) yield
(10.1.11) ut + 2^ = 0,
(10.1.12) 2et+cQux = 0
if first order terms only are retained. By eliminating s we obtain for
u the differential equation
(10.1.13) utt - c*uxx - 0.
This is the classical linear wave equation all solutions of which are
functions of the form u — u(x i c0t) and this means that the motions
arc superpositions of waves with constant propagation speed c0= Vgfe.
The role of the quantity c as a propagation speed (together with
many other pertinent facts) can be understood most readily by dis-
cussing the underlying integration theory of equations (10.1.8) and
(10.1.9) by using what is called the method of characteristics; we
turn therefore to a discussion of this method in the next section.
10.2. Integration of the Differential Equations by the Method of
Characteristics
The theory of our basic differential equations (10.1.8) and (10.1.9),
which are of the same form as those in compressible gas dynamics,
has been very extensively developed because of the practical necessity
for dealing with the flow of compressible gases. The purpose of the
present section is to summarize those features of this theory which
can be made useful for discussing the propagation of surface waves
in shallow water. In doing so, extensive use has been made of the
presentation given in the book by Courant and Friedrichs [C.9]; in
fact, a good deal of the material in sections 10.2 to 10.7, inclusive,
follows the presentation given there.
294 WATER WAVES
The essential point is that the partial differential equations (10.1.8)
and (10.1.9) are of such a form that the initial value problems asso-
ciated with them admit of a rather simple discussion in terms of a
pair of ordinary differential equations called the characteristic differ-
ential equations. We proceed to derive the characteristic equations
for the special case in which [cf. (10.1.10)]
(10.2.1) Hx = m = const.
i.e. the case in which the bottom slope is constant. In fact, this is the
only case we consider in this chapter. If we add equations (10.1.8)
and (10.1.9) it is readily seen that the result can be written in the
form:
9 . . a
(10.2.2) ^- + (u+c) — ^.(u + 2c- mt) = 0.
The expression in brackets is, of course, to be understood as a dif-
ferential operator. Similarly, a subtraction of (10.1.9) from (10.1.8)
yields
id d \
(10.2.3) 1 _ + (u _ c) — 1 . (tt _ 2c - mt) = 0.
[ ot ox]
But the interpretation of the operations defined in (10.2.2) and
(10.2.3) is well known (cf. (1.1.3)): the relation (10.2.2), for example,
states that the function (u + 2c — mt) is constant for a point moving
through the fluid with the velocity (u + c), or, as we may also put it,
for a point whose motion is characterized by the ordinary differential
equation dx/dt = u + c. Equation (10.2.3) can be similarly interpre-
ted. That is, we have the following situation in the x, /-plane: There are
two sets of curves, Cl and C2, called characteristics, which are the
solution curves of the ordinary differential equations
dx
Cx:
(10.2.4)
Cx : — = u + c, and
dx
£ . = U C
and we have the relations
u + 2c — mt = Aj = const, along a curve Cl and
1 u — 2c — mt = &2 = const, along a curve C2.
Of course the constants k^ and k2 will be different on different curves
in general. It should also be observed that the two families of charac-
LONG WAVES IN SHALLOW WATER 295
teristics determined by (10.2.4) are really distinct because of the fact
that c = Vg(r) + h) ^ 0 since we suppose that rj > — h, i.e. that
the water surface never touches the bottom.
By reversing the above procedure it can be seen rather easily that
the system of relations (10.2.4) and (10.2.5) is completely equivalent
to the system of equations (10.1.8) and (10.1.9) for the case of con-
stant bottom slope, so that a solution of either system yields a solu-
tion of the other. In fact, if we set f(x, t) = u + 2c — - mt and ob-
serve that f(x, t) = &! = const, along any curve x = x(t) for which
dxfdt = u + c it follows that along such curves
dx
(10.2.6) /, + /. — - h + (u + c)f, = 0.
In the same way the function g(x, t) = u — 2c — mt satisfies the
relation
(10.2.7) ft + (u - c)gx = 0
along the curves for which dx/dt — u — c. Thus wherever the curve
families C1 and C2 cover the r, J-plane in such a way as to form a non-
singular curvilinear coordinate system the relations (10.2.6) and
(10.2.7) hold. If now equations (10.2.6) and (10.2.7) are added and
the definitions of /(#, t ) and g(x, t) are recalled it is readily seen that
equation (10.1.8) results. By subtracting (10.2.7) from (10.2.6)
equation (10.1.9) is obtained. In other words, any functions u and c
which satisfy the relations (10.2.4) and (10.2.5) will also satisfy
(10.1.8) and (10.1.9) and the two systems of equations are therefore
now seen to be completely equivalent.
As we would expect on physical grounds, a solution of the original
dynamical equations (10.1.8) and (10.1.9) could be shown to be
uniquely determined when appropriate initial conditions (for t = 0,
say) are prescribed; it follows that a solution of (10.2.4) and (10.2.
5) is also uniquely determined when initial conditions are prescribed
since we know that the two systems of equations are equivalent.
At first sight one might be inclined to regard the relations (10.2.4)
and (10.2.5) as more complicated to deal with than the original dif-
ferential equations, particularly since the right hand sides of (10.2.4)
are not known in advance and hence the characteristic curves are also
not known: they must, in fact, be determined in the course of deter-
mining the unknown functions u and c which constitute the desired
solution. Nevertheless, the formulation of our problems in terms of
296
WATER WAVES
the characteristic form is quite useful in studying properties of the
solutions and also in studying questions referring to the appropriate-
ness of various boundary and initial conditions. It is useful to begin
by describing briefly a method of determining the characteristics and
thus the solution of a given problem by a method of successive approx-
imation which at the same time makes possible a number of useful
observations and interpretations regarding the role played by the
characteristics in general. Let us for this purpose consider a problem
in which the values of the velocity u and the surface elevation rj
(or, what amounts to the same thing, the propagation or wave speed
c = Vg(?7 + h)) are prescribed for all values of a? at the initial instant
t = 0. We wish to calculate the solution for t > 0 by determining u
and c through use of (10.2.4) and (10.2.5) and the given initial condi-
tions. At t = 0 we assume that
(10.2.8)
u(x, 0) = u(x)
(x, 0) = c(x)
in which u(x) and ~c(x) are given functions. We can approximate the
values 01 u and c for small values of t as follows: consider a scries of
points on the #-axis (cf. Fig. 10.2.1) a small distance 6x apart. At all
of these points the values of u and c are known from (10.2.8). Conse-
quently the slopes of the characteristics Cl and C2 at these points are
t
Fig. 10.2.1. Integration by finite differences
known from (10.2.4). From the points 1, 2, 3, 4 straight line segments
with these slopes are drawn until they intersect at points 5, 6, and 7,
and if dx is chosen sufficiently small it is reasonable to expect that
LONG WAVES IN SHALLOW WATER 297
the positions of these points will be good approximations to the inter-
sections of the characteristics issuing from the points 1, 2, 3, 4 since
we are simply replacing short segments of these curves by their
tangents. The values of both x and t at points 5, 6, and 7 are now
known— they can be determined graphically for example— and
through the use of (10.2.5) and the initial conditions we can also
determine the approximate values of u and c at these points. For this
purpose we observe that along any particular segment issuing from
the points 1, 2, 3 or 4 the values of u + 2c •— mt and u — 2c — rnt
are known constants since the values of u and c are fixed by (10.2.8)
for t = 0; hence we have
I along Cx: u + 2c — mt = u • -f- 2c, and
\ along C2: u — 2c — mt = u — 2c.
At the points 5, 6, and 7 we know the values of t and hence (10.2.9)
furnishes two independent linear equations for the determination of
the values of u and c at each of these points. Once u and c are known
at points 5, 6, and 7 the slopes of the characteristics issuing from these
points can be determined once more from (10.2.4) and the entire pro-
cess can be carried out again to yield the additional points 8 and 9
and the approximate values of u and c at these points. In this way
we can approximate the values of u and c at the points of a net over
a certain region of the x, <-plane, and can then obtain approximate
values for u and c at any points in the same region either by inter-
polation or by refining the net inside the region. It is quite plausible
and could be proved mathematically that the above process would
converge as dx ->Q to the unique solution of (10.2.4) and (10.2.5)
corresponding to the given initial conditions for sufficiently small
values of t (i.e. for a region of the #, /-plane not too far from the #-axis)
provided that the prescribed initial values of u and c are sufficiently
regular functions of x— for example, if they have piecewise con-
tinuous first derivatives.
It should be clear that once the characteristics are known the values
of u and c for all points of the x9 t -plane covered by them are also
known, since the constants A:x and k2 in (10.2.5) are known on each
characteristic through the initial data and hence the values of u and c
for any point (x, t) can be calculated by solving the linear equation
(10.2.5) for the characteristics through that point. This statement of
course implies that each one of the two families of characteristics
covers a region of the x, /-plane simply and that no two members of
298
WATER WAVES
different families are tangent to each other— in other words it is
implied that the two families of characteristics form a regular curvi-
linear coordinate system over the region of the x, J-plane in question.
One of the points of major interest in the later discussion centers
around the question of determining where and when the character-
istics cease to have this property, and of interpreting the physical
meaning of such occurrences.
The method of finite differences used above to determine the cha-
racteristics can be interpreted in such a way as to throw a strong light
on the physical properties of the solution. Consider the point 10 of
Fig. 10.2.1 for example. We recall that the approximate values ulo
and c10 of u and c at point 10 were obtained through making use of the
initial values of u and c at points 1, 2, 3, 4 on the #-axis only, and
furthermore that the values ulo and c10 required the use of points con-
fined solely to the region within the approximate characteristics join-
ing point 10 with points 1 and 4. Since the finite difference scheme
outlined above converges as dx -> 0 to yield the exact characteristics
we are led to make the following important statement: the values of u
and c at any point P(x, t) within the region of existence of the solution
are determined solely by the initial values prescribed on the segment of
the x-axis which is subtended by the two characteristics issuing from P.
Range of influence of Q
C, EL
Domain of
\ determinacy
J
Domain of dependence of P
Fig. 10.2.2. Domain of dependence and range of influence
In addition, the two characteristics issuing from P are also determined
solely by the initial values on the segment subtended by them. Such
a segment of the #-axis is often called the domain of dependence of the
LONG WAVES IN SHALLOW WATER 299
point P. Correspondingly we may define the range of influence of a
point Q on the #-axis as the region of the x, £-plane in which the values
of u and c are influenced by the initial values assigned to point Q. In
Fig. 10.2.2 we indicate these two regions. It is also useful on occasion
to introduce the notion of domain of determinacy relative to a given
domain of dependence. It is the region in which the motion is deter-
mined solely by the data over a certain segment of the #-axis. These
regions arc outlined by characteristic curves, as indicated in Fig.
10.2.2, in an easily understandable fashion in view of the discussion
above.
We are now in a position to understand why it is appropriate to
call the quantity c the propagation or wave speed. To this end we
suppose that a certain motion of water exists at a definite time, which
we take to be t — 0. This means, of course, that u and c are known at
that time, and, as we have just seen, the motion would be uniquely
determined for t > 0. However, we raise the question: what difference
would there be in the subsequent motion if we created a disturbance
in some part of the fluid, say over a segment QxQ2 °f *he #-axis (cf.
Fig. 10.2.3)? This amounts to asking for a comparison of two solutions
of our equations which differ only because of a difference in the initial
0, Q2
Fig. 10.2.3. Propagation of disturbances
conditions over the segment Q1Q2. Our whole discussion shows, that the
two solutions in question would differ only in the shaded region of
Fig. 10.2.3, which comprises all points of the x, J-plane influenced by
the data on the segment QxQ^ and which is bounded by characteristics
C2 and Cx issuing from the endpoints Qt and Q2 of the segment. These
curves, however, satisfy the differential equations dx/dt = u — c,
dx/dt = u + c. Since u represents the velocity of the water, it is then
300 WATER WAVES
clear that c represents the speed relative to the flowing stream at
which the disturbance on the segment QXQ2 spreads over the water.
This implies that the data in our two problems really differ at points
Qx and Q2 and that these differences persist along the characteristics
issuing from these points. Actually, only discontinuities in derivatives
at Ql and Q2 (and not of the functions themselves) are permitted in the
above theory, and it could be shown that such discontinuities would
never smooth out entirely along the characteristics Cl and C2. We are
therefore justified in referring to the quantity c = Vg(rj + h) as the
(local) propagation speed of small disturbances— that is, small in the
sense that only discontinuities in derivatives occur at the front of a
disturbance.
10.3. The Notion of a Simple Wave
There is an important class of problems in which the theory of
characteristics as presented in the preceding section becomes parti-
cularly simple. These are the problems in which (1) the initial un-
disturbed depth h of the water is constant so that the quantity m in
(10.2.1) (cf. also (10.1.10)) is zero, (2) the water extends from the
origin to infinity at least in one direction, say in the positive direction
of the a?-axis, and (3) the water is either at rest or moves with constant
velocity and the elevation of its free surface is zero at the time t = 0.
In other words, the water is in a uniform state at time t = 0 such that
u = u0 = const, and c = CQ = Vgh = const, at that instant. Our
discussion from here on is modeled closely on the discussion given by
Courant and Friedrichs [C.9], Chapter III.
We now suppose that a disturbance is initiated at the origin x = 0
so that either the particle velocity u, or the surface elevation r/ (or the
wave velocity c = Vg(7/ + h)) changes with the time in a prescribed
manner.* That is, a disturbance at one point in the water propagates
into water of constant depth and uniform velocity. Under these
circumstances we show that one of the two families of characteristics
furnished by (10.2.4) consists entirely of straight lines along each of which
u and c are constant. The corresponding motion we call a simple wave.
* One might accomplish this experimentally in a tank as follows: To obtain
a prescribed velocity u at one point it would only be necesary to place a vertical
plate in the water extending from the surface of the water to the bottom of the
tank and to move it with the prescribed velocity. To change r] at one point water
might be either poured into the tank or pumped out of it at that point at an
appropriate rate.
LONG WAVES IN SHALLOW WATER
301
Our statement is an immediate consequence of the following funda-
mental fact: if the values of u and c on any characteristic curve, C® say
(i.e. a solution curve of the first of the two ordinary differential equations
(10.2.4)), are constant, then CJ is a straight line and furthermore it is
embedded in a family of straight line characteristics along each of which
u and c are constant, at least in a region of the x, f -plane where u(x, t)
and c(x, t) are without singularities and which is covered by the
two distinct families of characteristics. The proof is easily given. To
begin with, the curve CJ is a straight line if u and c are constant along
it, since the slope of the curve is constant in that case from (10.2.4).
Next, let Cl be another characteristic near to CJ. We consider any two
points A$ and BQ on CJ together with the characteristics of the family
C2 through AQ and BQ and suppose that the latter characteristics
intersect Cl at points A and B (cf. Fig. 10.3.1 ): To prove our statement
Fig. 10.3.1. Region containing a straight characteristic
we need only show that u(A )~u(B) and c(A ) = c(B) since then u and c
would be constant on Cl (because of the fact that A and B are any
arbitrary points on CJ and hence the slope of the curve Cr would be
constant, just as was argued for Cj. We have u(A0) = u(BQ) and
c(AQ) = c(BQ) and consequently we may write
(10 81)
u A — 2cA — UA — 2cA ,
UB - 2cB = UBQ - 2<?Bo =
by making use of the second relation of (10.2.5) (which holds along
the characteristics C2) and observing that m = 0 since the original
802
WATER WAVES
depth of the water is assumed to be constant. Next we make use of
the first relation of (10.2.5) for Cl to obtain
(10.3.2) UA + 2cA = UB + 2cB.
But from (10.3.1) we have
(10.3.3) UA - 2cA = UB - 2cB,
and (10.3.2) and (10.3.3) are obviously satisfied only if UA — UB
and CA = CB . Our statement is therefore proved.
The problems formulated in the first paragraph of this section are
at once seen to have solutions (at least in certain regions of the
x9 /-plane) of the type we have just defined as simple waves since
there is a region near the #-axis in the x, /-plane throughout which the
particle velocity u and wave speed c arc constant, and in which there-
fore the characteristics are two sets of parallel straight lines. The cir-
cumstances are illustrated in Fig. 10.3.2 below: There is a zone / along
Fig. 10.3.2. A simple wave
the 07-axis which might be called the zone of quiet* inside which the
characteristics are obviously straight lines x ± c0t = const. (These
lines are not drawn in the figure). This region is terminated on the
upper side by an "initial characteristic" x — erf which divides the
* In a "zone of quiet" we permit the particle velocity u to be a non zero
constant, but the free surface elevation r\ is taken to be zero in such a region.
In case u — UQ = const. 7^ 0 initially, the motion can be thought of as observed
from a coordinate system moving with that velocity; thus there is no real loss
of generality in assuming UQ = 0, as we frequently do in the following.
LONG WAVES IN SHALLOW WATER 303
region of quiet from the disturbed region above it. The physical inter-
pretation of this is of course that the disturbance initiated at the
time / = 0 propagates into the region of quiet, and the water at any
point remains unaffected until sufficient time has elapsed to allow
the disturbance to reach that point. The exact nature of the motion
in the disturbed region is determined, of course, by the character of
the disturbance created at the point x = 0, i.e., by appropriate data
prescribed along the /-axis.* One set of characteristics, i.e., the set
containing the initial characteristic C?, therefore consists of straight
lines. (That the characteristics C2 in the zone// are necessarily curved
lines and not straight lines can be seen from the fact that they would
otherwise be the continuations of the straight characteristics from the
zone / of quiet and hence the zone // would also be a zone of quiet, as
one sees immediately). Furthermore, the set of straight characteristics
C\ in zone // is completely determined by appropriate conditions pre-
scribed at x — 0 for all /, i.e., along the /-axis. What these conditions
should be can be inferred from the following discussion. Consider any
straight characteristic issuing from a point / = T on the /-axis. We
know that the slope dx/dt of this straight line is given in view of
(10.2.4), by
dx
(10.3.4) — = u(r) + c(r).
Suppose now that there is a curved characteristic C2 going back
from / = T on the /-axis to the initial characteristic C° (see the dotted
curve in Fig. 10.3.2). We have the following relation from (10.2.5):
(10.3.5) u(r) - 2c(r) - u0 - 2c0,
in which u0 and CQ are the known values of u and c in the zone of quiet.
Hence the slope of any of the straight characteristics issuing from the
/-axis can be given in either of the two forms:
dx 1 r
-=;-[3u(T)-uQ] + cQ, or
(10.3.6)
(UV
- . 8c(r)
as one sees from (10.3.4) and (10.3.5). Thus if either u(r) or c(r) is
* Our discussion in the preceding section centered about the initial value
problem for the case in which the initial data are prescribed on the #-axis, but one
sees readily that the same discussion would apply with only slight modifications
to the present case, in which what is commonly called a boundary condition (i.e.
at the boundary point x — 0), rather than an initial condition, is prescribed.
304
WATER WAVES
given, i.e. if either u or c is prescribed along the J-axis, then the slopes
of the straight characteristics Cl and with them the characteristics Cl
themselves are determined. Since we know, from (10.3.5), the values
of both u and c along the /-axis if either one is given, and since u and c
are clearly constant along the straight characteristics, it follows that we
know the values of u and c throughout the entire disturbed region— in
other words, the motion is completely determined.
So far, we have considered only the case in which the curved cha-
racteristics (i.e., those of the type C2) which issue from the boundary
x = c0J of the disturbed region actually reach the /-axis. This, however,
need not be the case. Suppose, for example, that UQ is positive and
u0 > c0 = Vg/T. In this case the slope dxjdt of the curves C2 is positive,
and we cannot expect that they will turn to the left, as in Fig. 10.3.2.
Indeed, in such a case one does not expect that a disturbance will pro-
pagate upstream (that is, to the left in our case) since the stream velo-
city is greater than the propagation speed. In gas dynamics one would
say that the flow is supersonic, while in hydraulics the flow is said
to be supercritical. One could also look at the matter in another way:
For not too large values of t the velocity u can be expected to remain
supersonic and hence for such values of t both sets of characteristics
issuing from the f-axis would go into the right half plane (u being
again taken positive). Thus we would have the situation indicated
in Fig. 10.3.3, in which a segment of the /-axis is subtended by two
t
*»<V».c0)t
Fig. 10.8.3. The supercritical case
LONG WAVES IN SHALLOW WATEE 805
characteristics drawn backward from P. In this case, as in the case of
the initial value problem treated in the preceding section, we must
prescribe the values of both u and c along the /-axis. If we do so, then
the solution is once more determined through (10.8.4) and the fact
that u + 2c is constant along one set of characteristics and u — 2c is
constant along the other.
In either of our two cases, i.e. of subcritical or supercritical flow,
we see therefore that the simple wave can be determined. One sees
also how useful the formulation in terms of the characteristics can be
in determining appropriate subsidiary conditions such as boundary
conditions.
If we wish to know the values of u and c for any particular time
t = tQ9 once the simple wave configuration is determined, we need
only draw the line t = J0 and observe its intersections with the
straight characteristics since the values of u and c are presumably
known on each one of the latter. Thus u and c would be known as
functions of x for that particular time. Of course, the surface elevation
j\ would also be known from
c = Vg(h+fi).
10.4. Propagation of disturbances into still water of constant depth
In the preceding section we have seen how the method of charac-
teristics leads to the notion of a simple wave in terms of which we can
describe with surprising ease the propagation of a disturbance initiat-
ed at a point into water of constant depth moving with uniform speed.
In the present section we consider in more detail the character of the
simple waves which occur in two important special cases. We assume
always that the pulse is initiated at x = 0 and that it then propagates
in the positive ^-direction into still water. Thus we are considering
cases in which the flow is subcritical at the outset.
One of the most striking and important features of our whole dis-
cussion is that there is an essential difference between the propagation
of a pulse which is created by steadily decreasing the surface elevation
rf at x = 0 and of a pulse which results by steadily increasing the ele-
vation at x = 0. If the pulse is created by initiating a change in the
particle velocity u at x = 0 (which might be achieved simply by
moving a vertical barrier at x = 0 with the prescribed particle velo-
city) instead of by changing the surface elevation rj the same typical
differences will result if u is in the first case decreased from zero
through negative values, and in the other case is gradually increased
306 WATER WAVES
so that it becomes positive (i.e. if the particles at x = 0 are given in
the first case a negative acceleration and in the second case a positive
acceleration.) The qualitative difference between the two cases from
the physical point of view is of course that in the first case it is a
depression in the water surface and in the second case an elevation
above the undisturbed surface— sometimes referred to later on as a
hump— which propagates into still water.
If we were to consider waves of very small amplitude so that we might
linearize our equations (as was done in deriving equation (10.1.13))
there would be no essential qualitative distinction between the motions
in the two cases; that there is actually a distinction between the two
is a consequence of the nonlinearity of the differential equations.
In the preceding section we have seen that the motions in either of
our two cases can be described in the #, /-plane by means of a family of
straight characteristics which issue from the /-axis. In Figure 10.4.1
we show these characteristics together with a curve indicating a set
of prescribed values for c — Vg(A+^) = <*(0 at oc = 0, which in
turn result from prescribed values of rj at that point. We assume that
u — UQ = 0 in the zone of quiet /. Hence the slope docjdt of any straight
characteristic issuing from a point t ~ r on the /-axis is given, in ac-
cordance with (10.3.6) by
dx
(10.4.1) _=8c(T)-2cc.
When r is varied (10.4.1 ) yields the complete set of straight characteris-
tics in the zone //. The values oft/ and c along the same characteristic
are constant (as we have seen in the preceding section), so that the
value of u along a characteristic is determined, from (10.3.5) by
(10.4.2) u(r) - 2[c(r) - c0],
since UQ is assumed to be zero and c(r) is given.
We are now in a position to note a crucial difference between the
two cases described above. In the first of the two cases — i.e. that of a
depression moving into still water— the elevation rj(t) at x — 0 is
assumed to be a decreasing function of / so that c(t) also decreases
with increase of /. It follows that the slopes dxjdt of the straight line
characteristics as given by (10.4.1) decrease as / increases* so that the
family of straight characteristics diverge on moving out from the
* One should observe that decreasing values of dx/dt mean that the charac-
teristics make increasing angles with the #-axis, i.e. that they become steeper with
respect to the horizontal.
LONG WAVES IN SHALLOW WATER
307
tf-axis. (This is the case indicated in Fig. 10.4.1). In the second case,
however, the value of 77 and thus of c is assumed to be an increasing
function of t at x = 0 so that the straight characteristics must cven-
c(0,t)
Fig. 10.4.1. Propagation of pulses into still water
tually intersect — in fact, they will have an envelope in general— and
this in turn means that our problem can riot be expected to have a
continuous solution for values of ^r and t beyond those for which
such intersections exist. In the first case the motion is continuous
throughout. What happens in the second case beyond the point
where the solution is continuous can not be discussed mathematically
until we have widened our basic theory, but in terms of the physical
behavior of the water we might expect the wave to break, or to devel-
op what is called a bore,* some time after the solution ceases to be
continuous. In later sections we propose to discuss the question of the
development of breakers and bores in some detail.
The two cases discussed above are the exact analogues of two cases
well known in gas dynamics: Consider a long tube filled with gas at
rest and closed by a piston at one section. If the piston is moved away
from the gas with increasing speed in such a way as to cause a
rarefaction wave to move into the quiet gas, then a continuous motion
results. However, if the piston is pushed with increasing speed into
the gas so as to create a compression wave, then such a wave always
* In certain estuaries in various parts of the world the incoming tides from the
ocean are sometimes observed to result in the formation of a nearly vertical wall
of water, called a bore, which advances more or less unaltered in form over
quite large distances. What is called a hydraulic jump is another phenomenon
of the same sort. Such phenomena will be discussed in detail later on.
308
WATER WAVES
develops eventually into a shock wave. That is, the development of
a shock in gas dynamics is analogous to the development of a bore
(and also of a hydraulic jump) in water.
10.5. Propagation of depression waves into still water of constant depth
In this section we give a detailed treatment of the first type of
motion in which a depression of the water surface propagates into
still water. However, it is interesting and instructive to prescribe the
disturbance in terms of the velocity of the water rather than in terms
of the surface elevation. We assume, in addition, that the velocity is
prescribed by giving the displacement x = x(t) of the water particles
originally in the vertical plane at x = 0,* and this, as we have remark-
ed before, could be achieved experimentally simply by moving a ver-
tical plate at the end of a tank in such a way that its displacement
is x(t).** Figure 10.5.1 indicates the straight characteristics which
t= const.
Fig. 10.5.1. A depression wave
initiate on the "piston curve" x = x(t). The piston is assumed to
start from rest and move in the negative direction with increasing
speed until it reaches a certain speed w < 0, after which the speed
remains constant. That is, xt decreases monotonically from zero at
t = 0 until it attains the value w, after which it stays constant at that
value. In Fig. 10.5.1 this point is marked B; clearly the piston curve is
* In our theory, it should be recalled, the particles originally in a vertical
plane remain always in a vertical plane.
** Moving such a plate at the end of a tank of course corresponds in gas dyna-
mics to moving a piston in a gas-filled tube.
LONG WAVES IN SHALLOW WATER 309
a straight line from there one. At any point A on the piston curve we
have UA = xt(t), corresponding to the physical assumption that the
water particles in contact with the piston remain in contact with it
and thus have the same velocity. If we consider the curved character-
istic drawn from A back to the initial characteristic Cj which termina-
tes the zone / of rest we obtain from (10.3.5) the relation
(10.5.1) CA = Ki + *o.
since in our case u0 — 0. The slope of the straight characteristic at A
is thus given by (cf. (10.4.1)):
dx 3
—
(10.5.2) — = -UA
Since we have assumed that UA — xt(t) always decreases as t increases
until xt ~ w it follows from (10.5.2) that dx/dt also decreases as t in-
creases in this range of values of t so that the characteristics diverge
as they go outward from the piston curve. Beyond the point B the
straight characteristics are parallel straight lines, since UA = w —
const, on that part of the piston curve, and the state of the water is
therefore constant in the zone marked /// in Fig. 10.5.1. The zone //
is thus a region of non-constant state connecting two regions of differ-
ent constant states. Since CA = ^/g(h-\-r]A), where r\A refers to the
elevation of the water surface at the piston, it follows from (10.5.1)
that t]A decreases in the zone // as t increases, i.e. the water surface at
the "piston" moves downward as the piston moves to the left, since we
assume* that UA decreases as A moves out along the piston curve. Since
u and c are constant along any straight characteristic it is not difficult
to describe the character of the motion corresponding to the disturbed
zone // at any time t: Consider any straight line t = const. Its inter-
section with a characteristic yields the values of u and c at that point
—they are the values of u and c which are attached to that character-
istic. Since the characteristics diverge from the piston curve one sees
that the elevation rj steadily increases upon moving from the piston to
the right and the particle velocity decreases in magnitude, until the
initial characteristic Cj is reached after which the water is undisturbed.
On the other hand, if attention is fixed on a definite point x > 0 in the
water and the motion is observed as the time increases it is clear— once
more because the characteristics diverge— that the water remains un-
disturbed until the time reaches the value determined by x = c0£, after
which the water surface falls steadily while the water particles passing
310
WATER WAVES
that point move more and more rapidly in the negative ^-direction.
In the foregoing discussion of a depression we have made an assump-
tion without saying so explicitly, i.e. that the speed UA of the piston is
such that CA = \UA + CQ (cf. (10.5.1)) is not negative, and this in
turn requires that
(10.5.3) - UA ^ 2c0.
Since — UA increases monotonically to the terminal value — w it
follows that — w must be assumed in the above discussion to have at
Fig. 10.5.2. A limit case
most the value 2cQ. The limit case in which — w just equals 2c0 is
interesting. Since the straight characteristics have the slope dxfdt =
u + c and since CA = 0 from (10.5.1) when UA = — 2c0, it follows in
this case that dx/dt = UA on the straight part of the piston curve.
But this means that the straight characteristics have all coalesced
into the piston curve itself in this region, or in other words that the
zone /// has disappeared in this limit case. The circumstances are
indicated in Fig. 10.5.2. At the front of the wave for values of x to the
left of B the elevation r\A of the water is equal to — h from CA =
+ V!A ) == 0> which means that the water surface just touches the
bottom at the advancing front of the wave.
It is now clear what would happen if the terminal speed — w of the
piston were greater than 2c0: The zone // would terminate on the
tangent to the piston curve drawn from the point where the piston
speed — xt just equals 2c0. The region between this terminal charac-
teristic and the remainder of the piston curve beyond it might be
called the zone of cavitation, since no water would exist for (x, t)
LONG WAVES IN SHALLOW WATER
311
values in such a region. In other words, the piston eventually pulls
itself completely free from the water in this case. Quite generally we
see that the piston will lose contact with the water (under the cir-
cumstances postulated in this section, of course) if, and only if, it
finally exceeds the speed 2c0. Once this happens it is clear that the
piston has no further effect on the motion of the water. These circum-
stances are indicated in Fig. 10.5.3.
Covitotion
Zone
Fig. 10.5.3. Case of cavitation
If the acceleration of the piston is assumed to be infinite so that
its speed changes instantly from zero to the constant terminal value
— w, the motion which results can be described very simply by ex-
plicit formulas. The general situation in the x, 2-plane is indicated in
Fig. 10.5.4. This case might be considered a limit case of the one
indicated in Fig. 10.5.1 which results when the portion of the piston
curve extending from the origin to point B shrinks to a point. The
u=w
Fig. 10.5.4. Centered simple wave
812 WATER WAVES
consequence is that the straight characteristics in zone // all pass
through the origin. The zone /// is again one of constant state. In
the zone // we have obviously for the slopes of the characteristics
dx x
(10.5.4) Tt=7.
At the same time we have from (10.5.2) dxjdt — f u + c0 so that
(10.5.5) ?- = ^u+c0.
It follows that the zone // is terminated on the upper side by the line
(10.5.6) *
From (10.5.5) and (10.5.1) we can obtain the values of u and c within
zone Hi
2 (x \
(10.5.7) u = — ( — - cA and
1 1 Ix
(10.5.8) C = _tt+Co==_^_
Since c ^ 0 we must have — x/t ^ 2c0 so that — w must be ^ 2c0
from (10.5.6) in conformity with a similar result above. If w — — 2c0,
the terminal characteristic of zone // is given, from (10.5.6), by
x = — 2cQt = wt and this line falls on the piston curve since the slope
of the piston curve is w. In this limit case, therefore, the zone ///
collapses into the piston curve. If the piston is moved at still higher
speed, then cavitation occurs as in the cases discussed above since
c = 0 at the front of the wave, or in other words, the water surface
touches the bottom.
From (10.5.8) we can calculate the elevation rj of the water surface
since c = Vg(h + 77) ;
(10.5.9) >? + h = + 2c
In the case of incipient cavitation, i.e. — w = 2c0, we have r\ = — h
at the front of the wave. The curve of the water surface at any time t
is a parabola from the front of the wave to the point x = cQt (cor-
responding to the characteristic which delimits the zone of quiet),
after which it is horizontal. In Fig. 10.5.5 the total depth rj + h of
LONG WAVES IN SHALLOW WATER
313
the water is plotted against x for a fixed time t. The surface of the
water is tangent to the bottom at the front x — — 2cQt of the moving
water. The region in which the water is in motion extends from this
point back to the point x = cQt. From (10.5.7) we can draw the follow-
ing somewhat unexpected conclusion in this case: Since t may be
given arbitrarily large values it follows that the velocity u of the water
at any fixed point x tends to the values — §c0 as t grows large.
The case of cavitation may have a certain interest in practice: the
motion of the water might be considered as an approximation to the
flow which would result from the sudden destruction of a dam built
in a valley with very steep sides and not too great bottom slope (cf.
Water
x=-2c0t
xscol
Fig. 10.5.5. Breaking of a dam
the paper of Re [R-5])- If the water behind the dam were 200 feet
high, for example, our results indicate that the front of the wave
would move down the valley at a speed of about 110 miles per hour.
By setting x = 0 in (10.5.9) we observe that the depth of the water
at the site of the dam is always constant and has the value \h,
i.e. four-ninths of the original depth of the water behind the dam.
The velocity of the water at this point is also constant and has the
value u = — f CQ = — f Vgh, as we see from (10.5.7). The volume
rate of discharge of water at the original location of the dam is thus
constant.
So far we have not considered the motion of the individual water
particles. However, that is readily done in all cases once the velocity
u(x, t) is known: We have only to integrate the ordinary differential
equation
dx
(10.5.10)
In zone // in our present case we have
314 WATER WAVES
dx 2 Ix
By setting £ — x + 2c0t one finds readily that £ satisfies the differen-
tial equation d$/dt = 2£/3t, from which £ == At2t3 with A an arbitrary
constant. Hence we have for the position x(t) of any particle in zone //
(10.5.12) x = t{At~11* — 2c0}.
In the case of cavitation this formula holds for arbitrarily large t so
that we have for large t the asymptotic expression for x:
(10.5.13) x~ - 2cQt.
(This is not in contradiction with our above result that u ~ — f c0
for large t and fixed x since in that case different particles pass the
point in question at different times, while (10.5.13) refers always to
the same particle).
In the first section of Chapter 12 this same problem of the breaking
of a dam will be treated by using the exact nonlinear theory in such
a manner as to determine the motion during its early stages after the
dam has been broken— in other words, at the times when the shallow
water theory is most likely to be inaccurate.
10.6. Discontinuity, or shock, conditions
The difference in behavior of a depression which propagates into
still water as compared with the behavior of a hump has already been
pointed out: in the first case the motion is continuous throughout, but
in the second case the motion can not be continuous after a certain
time. The general situation is indicated in Fig. 10.6.1, which shows
the characteristics in the x, £-plane for the motion which results when
a "piston" at the end of a tank is pushed into the water with steadily
increased speed. As before, the slope dxjdt of a straight characteristic
issuing from the "piston curve" x = x(t) is given (cf. (10.5.2)) by
dx/dt = ^UA + CD» in which UA = xt(t) is the velocity of the piston.
Since UA is assumed to increase with t it is clear that the characteris-
tics will cut each other. In general, they have an envelope as indi-
cated by the heavy line in the figure. The continuous solutions
furnished by our theory, which have been the only ones under con-
sideration so far, are thus valid in the region of the x, 2-plane between
-the initial characteristic and the piston curve up to the curved
characteristic (indicated by the curve segment ED) through the
LONG WAVES IN SHALLOW WATER
315
"first" point E on the envelope of the straight characteristics, but
not beyond ED.
What happens "beyond the envelope" can in principle therefore
Fig. 10.6.1. Initial point of breaking
not be studied by the theory presented up to now. However, it seems
very likely that discontinuous solutions may develop as the time in-
creases beyond the value corresponding to the point E9 which are
then to be interpreted physically as motions involving the gradual
development of bores and breakers in the water.
x*<Ut) £(t) x»a,(t)
O I
Fig. 10.6.2. Discontinuity conditions
There is a particularly simple limit case of the situation indicated
in Fig. 10.6.2 for which a discontinuous solution can be found once
we have obtained the discontinuity conditions that result from the
816 WATER WAVES
fundamental laws of mechanics. That is the case in which the "piston"
is accelerated instantaneously from rest to a constant forward velocity
so that the piston curve is a straight line issuing from the origin in
the x, J-plane. It is the exact counterpart of the case discussed at the
end of the preceding section in which the piston was withdrawn from
the water at a uniform speed.
To obtain the conditions at a discontinuity we consider a region
made up of the water lying between two vertical planes x — aQ(t)
and x = a>i(t) with a^ > a0 and such that these planes contain always
the same particles. Such an assumption can be made, we recall from
Chapter 2, since in our theory the particles which are in a vertical
plane at any instant always remain in a vertical plane. Hence the
horizontal particle velocity component u is the same throughout any
vertical plane. We now suppose that there is a finite discontinuity in
the surface elevation rj at a point x = £(t) within the column of water
between x = aQ(t) and x = a1(^), as indicated in Fig. 10.6.2.
The laws of conservation of mass and of momentum as applied to
our column of water yield the relations
d
(10.6.1)
and
d f°iW po pi
nnftftl A S(r,+h)udx = \ p,dy-\ Pldy
(10.6.2) ^JaQ(t) J-n J-h
when the formula p = gQ(r) — y) for the pressure in the water is
used. The second relation states that the change in momentum of the
water column is equal to the difference of the resultant forces over the
end sections of the column.
The integrals in these relations have the form
/•«h(«
=
Ja0(t)
(x, t) dx
in which \p(x, t) has a discontinuity at x = g(t). Differentiation of this
integral yields the relation
dl d f*« d
yidx
(10.6.3)
- dx
LONG WAVES IN SHALLOW WATER 317
The quantities UQ and u± are the velocities a0(t) and a^t) at the ends
of the column, f is the velocity of the discontinuity, and y;(£_, t) and
y(£+, t) mean that the limit values of y to the left and to the right of
x = f respectively are to be taken. We wish to consider the limit case
in which the length of the column tends to zero in such a way that
the discontinuity remains inside the column. When we do so the
integral on the right-hand side of (10.6.3) tends to zero and we obtain
dl
(10.6.4) lim — = y^ - Wo
in which vl and VQ are the relative velocities given by
(10.6.5)
and if?! and y0 refer to the limit values of \p to the right and to the left
of the discontinuity, respectively. The important quantities VQ and vl
are obviously the flow velocities relative to the moving discontinuity.
Upon making use of (10.6.4) and (10.6.5) for the limit cases which
arise from (10.6.1) and (10.6.2) we obtain the following conditions
(10.6.6) gfo + hfa - Q(rh + h)v0 - 0
and
(io.6.7) eOh+AK^-efoo+A^
If we introduce, as in section 10.1, the quantities £ and p (which are
the analogues of the density and pressure in gas dynamics) by the
relations (of. (10.1.3) and (10.1.4))
(10.6.8) Q = e(i, + h)
and
(10.6.9) p = ^(r/+A)i= Igi,
we obtain in place of (10.6.6) and (10.6.7) the discontinuity conditions
(10.6.10) QM = QOVO,
and
(10.6.11) e&iVi ~ (?(Wo = Po — Pi-
The last two relations are identical in form with the mechanical con-
ditions for a shock wave in gas dynamics when the latter are expressed
in terms of velocity, density and pressure changes.
318 WATER WAVES
Henceforth we shall often refer to a discontinuity satisfying (10.6.
10) and (10.6.11) as a shock wave or simply as a shock even though
such an occurrence is better known in fluid mechanics as a bore, or
if it is stationary as a hydraulic jump.
Since u± — UQ — vl — VQ from (10.6.5) it is easily seen that the
shock conditions (10.6.10) and (10.6.11) can be written in the form
(10.6.12) _
I m(^ - VQ) = po - plf
in which m represents the mass flux across the shock front.
To fix the motion on both sides of the shock five quantities are
needed; i.e. the particle velocities UQ, uv the elevations 77 0 and rjl (or,
what is the same, the "pressures" p or the "densities" g as given by
(10.6.8) and (10.6.9) on both sides of the shock), and the velocity |
of the shock. Evidently the relative velocities VQ and vl would then
be determined. Since the five quantities satisfy the two relations
(10.6.12) we see that in general only three of the five quantities could
be prescribed arbitrarily. Since the equations to be satisfied are not
linear it is not a priori clear whether solutions can be found for two
of the quantities when any other three are arbitrarily prescribed or
whether such solutions would be unique. We want to investigate this
question in a number of important special cases.
Before doing so, however, it is important to consider the energy
balance across a shock. The fact is, as we shall see shortly, that the
law of conservation of energy does not hold across a shock, but rather
the particles crossing* the shock must either lose or gain in energy.
Since we do not wish to postulate the existence of energy sources at
the shock front capable of increasing the energy of the water particles
as they pass through it, we assume from now on that the water
particles do not gain energy upon crossing a shock front. This will in
effect furnish us with an inequality which in conjunction with the
two shock relations (10.6.12) leads in all of our cases to unique solu-
tions of the physical problems. We turn, then, to a consideration of
the energy balance across a shock, which we can easily do by following
* It is important to observe that the water particles always do cross a shock
front: the quantity m in (10.6.12), the mass flux through the shock front, is
different from zero if there is an actual discontinuity since otherwise vl = VQ = 0,
tij = UQ = £, and p0 = pl and hence @0 = ^ — in other words the motion is
continuous. There is thus no analogue in our theory of what is called a contact
discontinuity in gas dynamics in which velocity and pressure are continuous,
but the density and temperature may be discontinuous.
LONG WAVES IN SHALLOW WATER 319
the same procedure that was used to derive the shock relations (10.6.
10) and (10.6.11 ). For the rate of change dE/dt of the energy E in the
water column of Fig. 10.6.2 we have, as one can readily verify:
dE d
(10.6.1,3) <*o«)
Pi
J _h
l-h*
and this in turn yields in the limit when a0 -+ av through use of
(10.6.5), (10.6.8), (10.6.9), and the hydrostatic pressure law, the
relation
dE
(10.6.14) ~ ~
for the rate at which energy is created or destroyed at the shock front.
If we multiply (10.6.11) by | on both sides and then subtract from
(10.6.14) the result is an equation which can be written after some
manipulation and use of (10.6.5) in the form
dE
(10.6.15) — -= m {%(v* — vl) f 2(~pi/ei — p0/Q0)}
dt
in which m is the mass flux through the shock front defined in
(10.6.12). In this way we express dE/dt entirely in terms of the
relative velocities VQ and vl and the change in depth. By eliminating
vl and v0 through use of v1 = m/gx and v0 = M/{JO and replacing pl and
p0 in terms of QI and QO we can express dE/dt in terms of QO and g^;
the result is readily found to be expressible in the simple form
dE mg (PA — pi)3
(10.6.16) -7- — — , _ — .
dt Q *Qi6Q
We sec therefore that energy is not conserved unless g0 = Q19 i.e. unless
the motion is continuous. Since QQ — Qi = p(^o ~~ ^i) ^ follows from
(10.6.16) that the rate of change of the energy of the particles crossing
the shock is proportional to the cube of the difference in the depth of
the water on the two sides of the shock, or as we could also put it in
case rjQ — r/l is considered to be a small quantity: the rate of change
of energy is of third order in the "jump" of elevation of the water
surface.
The statement that the law of conservation of energy does not hold
in the case of a bore in water must be taken cum grano salis. What we
mean is of course that the energy balance can not be maintained
320
WATER WAVES
through the sole action of the mechanical forces postulated in the
above theory. The results of our theory of the bore and the hydraulic
jump are therefore to be interpreted as an idealization of the actual
occurrences in which the losses in mechanical energy are accounted
for through the production of heat due to turbulence at the front of
the shock (cf. the photograph of the bore in the Tsien-Tang river
shown in Fig. 10.6). 8). In compressible gas dynamics the theory used
Fig. 10.6.3. Bore in the Tsien-Tang River
allows for the conversion of mechanical energy into heat so that the
law of conservation of energy holds across a shock in that theory.
The analogue of the loss in mechanical energy across a shock in water
is the increase in entropy across a shock in gas dynamics; furthermore,
both of these discontinuous changes are of third order in the differen-
ces of "density" on the two sides of the shock.
We have tacitly chosen as the positive direction of the #-axis, and
hence of all velocities, the direction from the side 0 toward the side 1
(cf. Fig. 10.6.2). Suppose now that the mass flux m is assumed to be
positive; it follows from (10.6.12) and the fact that j50 and QI are posi-
tive that VQ and vl are also positive and hence that the water particles
cross the shock front in the direction from the side 0 toward the side 1.
Our condition that the water particles can not gain in energy on cross-
ing the shock then requires, as we see at once from (10.6.16) since w,
LONG WAVES IN SHALLOW WATER 321
g> £>» {>o» anc^ ^i are a^ positive, that g0 < gla In other words, our energy
condition requires that the particles always move across the shock from
a region of lower total depth to one of higher total depth.* Since the mass
flux m is not zero unless the flow is continuous, and hence there is no
shock, it is possible to define uniquely the two sides of the shock by the
following useful convention: the front and back sides of the shock are
distinguished by the fact that the mass flux passes through the shock
from front to back, or, as one could also put it, the water crosses the
shock from the front side toward the back side. Our conclusion based
on the assumed loss of energy across the shock can be interpreted in
terms of this convention as follows: the water level is always lower on
the front side of the shock than on the back side.
For the further discussion of the shock relations it is important to
observe that all of them, including the relation (10.6.16) for the
energy loss, can be written in such a way as to involve only the velocities
VQ and vl of the water particles relative to the shock front and not the abso-
lute velocities UQ and ur It follows that we may always assume one of
the three velocities u0, ul9 £ to be zero if we wish, with no essential loss
of generality, because the laws of mechanics are in any case invariant
with respect to axes moving with constant velocity, and adding the
same constant to MO, % and £ does not affect the values of v0 and i^.
Let us assume then that uQ — 0, i.e. that the water is at rest on one
side of the shock. Also, we write the second of the shock conditions
(10.6.12) in the form
(10.6.17) ^PQ =?- "-?1,
<?o — 61
which follows from mvl = QOVOVI and mv0 = pi^t'o and (10.6.12).
From UQ = 0 we have v^ = — f and v± = ul — £ (cf. (10.6.5)) so that
(10.6.17) takes the form
(10.6.18) - fK - f ) = :~ (go + ei)
AQ
upon making use of p = gg2/2? (cf- (10.6.9)). The first shock condition
now takes the form
(10.6.19) giK _ {) = -
so that (10.6.18) can be written
(10.6.20, f'=
This conclusion was first stated by Rayleigh [R.3J.
322 WATER WAVES
if u± is eliminated, or it may be written in the form
if g0 is eliminated. Thus (10.6.19) together with either (10.6.20) or
(10.6.21) are ways of expressing the shock conditions when u0 = 0.
We are now in a position to discuss some important special cases.
Having fixed the value of w0, i.e. UQ = 0, at most two of the remaining
quantities |, g0, QV and wx can be prescribed arbitrarily. For our later
discussion it is useful to single out the following two cases: Case 1. JDJ.
and g0 are given, i.e. the depth of the water on both sides of the shock
and the velocity on one side are given. Case 2. ^ and u± are given, i.e.
the velocity of the water on both sides of the shock and the depth of
the water on one side are given. We proceed to discuss these cases in
detail.
Case 1. From (10.6.20) we see that |2 is determined for any arbitrary
values (positive, of course) of £0 and gl9 i.e. of the water depths. Hence
£ is determined by (10.6.20) only within sign. Suppose now that
61 -^ (?<)• ^n ^is case ^e side 0 is, as we have seen above, the front
side of the shock, and since u0 = 0 the shock front must move in the
direction from the side 1 toward the side 0 in order that the mass flux
should pass through the shock from front to back.
Hence if it is once decided whether the side 0 is to the left or to the
right of the side 1 the sign of £ is uniquely fixed. If, as in Fig. 10.6.4,
€ *-
0
1
-u. <0
X
Fig. 10.0.4. Bore advancing into still water
the side 0 is chosen to the left of the side 1, and the ^-direction is posi-
tive to the right, it follows that f is negative, as indicated. It is useful
to introduce the depths A0 and Ax of the water on the two sides of the
shock:
(10.6.22)
LONG WAVES IN SHALLOW WATER 328
and to express (10.6.20) in terms of these quantities. The result for f
in our case is
(10.6.28)
I/" hi
« - - K « *
as one readily sees from gt = ght. From (10.6.23) we draw the im-
portant conclusion: Since h^ > A0, the shock speed | f | is greater than
Vgh0 since h0 < (Ax + A0)/2 < /&x. Also, in the case w0 = 0 we have
from (10.6.19)
(10.6.24)
u, = | (l - JsJ,
so that the velocity of the water behind the shock has the same sign as
£ (since hQ /hl< 1 ) but is less than f numerically.
Finally, it is very important to consider the speed vl of the shock
front relative to the water particles behind it: from (10.6.24) we have
(10.6.25) vl=ul^S=-'T^
HI
and this in turn can be expressed through use of (10.6.23) in the
form
^
so that vl < Vghv In other words, the speed of the shock relative to
the water particles behind the shock is less than the ivave propagation
speed Vg/?! in the water behind the shock. Hence a small disturbance
created behind a shock will eventually catch up with it. Although the
conclusion was drawn for the special case u0 = 0 it holds quite
generally for the shock velocities relative to the motion of the water
on both sides of a shock, in view of earlier remarks on the dependence
of the shock relations on these relative velocities.
The case illustrated by Fig. 10.6.4 is that of a shock advancing into
still water. The fact that f is in this case of necessity negative is a
consequence of the assumption of an energy loss across the shock. It
is worth while to restate this conclusion in the negative sense, as
follows: a depression shock can not exist, i.e. a shock wave which
leaves still water at reduced depth behind it should not be observed
in nature.* The observations bear out this conclusion. Bores advancing
* In gas dynamics the analogous situation occurs: only compression shocks
and not rarefaction shocks can exist.
324
WATER WAVES
Rigid ^
Woll *^
- *•
u0'0
h
0
h « — ui
Fig. 1Q.6.5. Reflection from a rigid wall
! .1
u,—
go »
1
Fig. 10.6.6. Hydraulic jump
into still water are well known, but depression waves are always
smooth.
Instead of assuming that gx > g0 (or that At > h0) as in the case of
Fig. 10.6.4 we may assume ^ < £0 (or /&x < hQ), so that the side 1 is
the front side. In other words the water is at rest on the baek side of
the shock in this case. If the front side is taken on the right, the situa-
tion is as indicated in Fig. 10.6.5. In this case | must be positive
and % negative in order that the mass flux should take place from the
side 1 to the side 0. The value of t^ is given by (10.6.24) in this case
also. The case of Fig. 10.6.5 might be realized in practice as the result
of reflection of a stream of water from a rigid wall so that the water in
contact with the wall is brought to rest. We shall return to this case
later.
In the above two cases we considered u0 to be zero. However, we
know that we may add any constant velocity to the whole system
without invalidating the shock conditions. It is of interest to consider
the motion which arises when the velocity — £ is added to uQ9 ur and
| in the case shown in Fig. 10.6.4. The result is the motion indicated
by Fig. 10.6.6 in which the shock front is stationary. This case— one
of frequent occurrence in nature-— is commonly referred to as the
hydraulic jump. From our preceding discussion we see that the water
always moves from the side of lower elevation to the side of higher
elevation. The velocities UQ and u± are both positive, and UQ > uv
LONG WAVES IN SHALLOW WATER
325
Also the velocity u0 on the incoming side is greater than the wave
propagation speed Vgh0 on that side while the velocity u± is less than
Vghlt This follows at once from the known facts concerning the re-
lative shock velocities and the fact that u0 and % are the velocities
relative to the shock front in this case. The hydraulic engineers refer
to this as a transition from supercritical to subcritical speed.
Case 2. We recall that in this case u0 = 0, u^ and pL (or h^) are
assumed given and f and h0 are to be determined. The value of f is
to be determined from (10.6.21). To study this relation it is conve-
nient to set x = — f and y = u^ — £ so that (10.6.21) can be replaced
by
(10.6.27)
y = k2x/(x2
y = HI + x.
In Fig. 10.6.7 we have indicated these two curves, whose intersections
yield the solutions f = — x of (10.6.21). The first equation is re-
presented by a curve with three branches having two asymptotes
% — ± k. As one sees readily, there are always three different real
roots for — £ no matter what values arc chosen for the positive quan-
tity k2 = g/4/2 and for the velocity uv Furthermore, one root £+ =
x_ is always positive, another £_ = — x+ is negative, while the
third | ~ — x lies between the other two. However, the third root
| = — x must be rejected because it is not compatible with (10.6.19):
Since QI and QO are both positive it follows that x = — - f and y =
Fig. 10.6.7. Graphical solution of shock conditions
326 WATER WAVES
M! — f must have the same sign. But the sign ofy — y corresponding
to x = x is always the negative of x as one sees from Fig. 10.6.7.
(If U-L = 0 ,then x = y = 0, but there is no shock discontinuity in
this case.) The other two roots, however, are such that the signs of
— £ and ux — - f are the same. In the case 2, therefore, equation
(10.6.21) furnishes two different values of f which have opposite
signs and these values when inserted in (10.6.19) furnish two values
of the depth /20. The two cases are again those illustrated in Figs.
10.6.4 and 10.6.5. An appropriate choice of one of the two roots must
be made in accordance with the given physical situation, as will be
illustrated in one of the problems to be discussed in the next section.
Before proceeding to the detailed discussion of special problems
involving shocks it is perhaps worth while to sum up briefly the main
facts derived in this section concerning them: the five essential quan-
tities defining a shock wave— f, MO, u^\ g0, gx (or, what is the same,
h0 and h^)— must satisfy the shock conditions (10.6.12). If it is as-
sumed in addition that the water particles may lose energy on crossing
the shock but not gain it, then it is found that the shock wave travels
always in such a direction that the water particles crossing it pass from
the side of lower depth to the side of higher depth. If hQ < hl9 so that the
side 0 is the front side of the shock, the speeds \ v0 \ and \ v± \ of the water
relative to the shock front satisfy the inequalities
(10.6.28)
In hydraulics it is customary to say that the velocity relative to the
shock is supercritical on the front side (i.e. greater than the wave
propagation speed corresponding to the water depth on that side)
and subcritical on the back side of the shock*
10.7. Constant shocks: bore, hydraulic jump, reflection from a
rigid wall
In the preceding section shock discontinuities were studied for the
purpose of obtaining the relations which must hold on the two sides
of the shock, and nothing was specified about the motion otherwise
except that the shock under discussion should be the only disconti-
* In gas dynamics the analogous inequalities lead to the statement that the
flow velocity relative to the shock front is supersonic with respect to the gas
on the side of lower density and subsonic with respect to the gas on the other side.
LONG WAVES IN SHALLOW WATER 327
nuity in a small portion of the fluid on both sides of it. In the present
and following sections we wish to consider motions which are conti-
nuous except for the occurrence of a single shock. Furthermore we
shall limit our investigations in this section to cases in which the mo-
tion on each of the two sides of the shock has constant velocity and
depth. These motions, or flows, are evidently of a very special cha-
racter, but they are easy to describe and also of frequent occurrence
in nature.
It is perhaps not without interest in this connection to observe that
the only steady and continuous wave motions (i.e., motions in which
the velocity u and wave propagation speed c = Vg(& +77) are in-
dependent of the time ) furnished by our theory for the case of constant
depth h are the constant states u = const., c = const. This follows
from the original dynamical equations (10.1.8) and (10.1.9). In fact,
when u and c are assumed to be functions of x alone these equa-
tions reduce to
du dc
u — + 2c — = 0, and
ax ax
dc du
2u~ + c— = 0
ax ax
for the case in which h = const, (and so Hx — 0). These equations are
immediately integrable to yield u2 + 2c2 = const, and uc2 = const,
and these two relations are simultaneously satisfied only for constant
values of u and c. On the other hand, any constant values whatever
could be taken for u and c. The cases we discuss in this section are
motions which result by piecing together two such steady motions
(each with a different constant value for the depth and velocity)
through a shock which moves with constant velocity. In this case the
motion as a whole would be steady if observed from a coordinate
system attached to the moving shock front. In view of our above dis-
cussion it is clear that such a motion with a single shock discontinuity
is the most general progressing wave which propagates unchanged in
form that could be obtained from our theory.*
Let us consider now the problem referred to at the beginning of the
preceding section: a vertical plate— or piston, as we have called it— at
* This result should not be taken to mean that the so-called "solitary wave'*
does not exist. (By a solitary wave is meant a continuous wave in the form of a
single elevation which propagates unaltered in form.) It means only that our
approximate theory is not accurate enough to furnish such a solitary wave. This
is a point which will be discussed more fully in section 10.9.
328 WATER WAVES
the left end of a tank full of water at rest is suddenly pushed into the
water at constant velocity w. As we could infer from the discussion
at the beginning of the preceding section the motion must be dis-
continuous from the very beginning— or, as we could also put it, the
"first" point on the envelope of the characteristics would occur at
t = 0. Since the piston moves with constant velocity we might expect
the resulting motion to be a shock wave advancing into the still water
and leaving a constant state behind such that the water particles move
with the piston velocity w. The circumstances for such an assumed
motion are indicated in Fig. 10.7.1, which shows the x, J-plane together
with the water surface at a certain time tQ. We know that the constant
states on each side of the shock satisfy our differential equations. In
addition, we show that they can always be "connected" through a
shock discontinuity which satisfies the shock relations derived in the
preceding section. In fact, the relations (10.6.18) and (10.6.19) yield
through elimination of ^ == p/^ the relation
(10.7.1) _flw_f)
for | in terms of w and the depth hQ in the still water, when we set
go = Qh0. Equation (10.7.1) is the same as (10.6.21) except that g0
replaces glf and the discussion of its roots | follows exactly the same
lines as for (10.6.21): for each A0 > 0 and any w ^ 0 the cubic equa-
tion (10.7.1) has three roots for f : one negative, another positive, and
a third which has a value between these two. In the present case the
positive root for | must be taken in order to satisfy our energy con-
dition (cf. the discussion based on (10.6.27) of the preceding section)
since the side 0 is the front side of the shock. Once |+ has been calcu-
lated from (10.7.1) we can determine the depth of the water h^ behind
the shock from the first shock condition
(10.7.2) h,(w -£+)=- AO|+.
It is therefore clear that a motion of the sort indicated in Fig. 10.7.1
can be determined in a way which is compatible with all of our con-
ditions.*
A few further remarks about the above motion are of interest. In
* It should be pointed out that our discussion yields a discontinuous solution
of the differential equations, but does not prove that it is the only one which
might exist. However, it has been shown by Goldner [G.6] that our solution would
be unique under rather general assumptions regarding the type of functions
admitted as possible solutions.
LONG WAVES IN SHALLOW WATER
829
wt
Fig. 10.7.1. A bore with constant speed and height
sxs ft
Reflected.
Shock
(2)
i
^_
CD
Inclden
Shock
t=t,
"2=0
I
t=t
* 0
tlto
Fig. 10.7.2. Reflection of a bore from a rigid wall
330 WATER WAVES
Fig. 10.7.1 we have indicated the line x = cQt, c0 = VghQ> which
would be the initial characteristic terminating the state of rest if the
motion were continuous, i.e. if the disturbance proceeded into still
water with the wave speed c0 for water of the depth A0. We know,
however, from our discussion of the preceding section that the shock
speed | is greater than c0, which accounts for the position of the shock
line x = |J below the line x = cQt in Fig. 10.7.1. On the other hand
we know that the velocity w — £ of the water particles behind the
shock relative to the shock is less than the wave speed c± = VgAx in
the water on that side. It follows, therefore, that a new disturbance
created in the water behind the shock should catch up with it since
the front of such a disturbance would always move relative to the
water particles with a velocity at least equal to cr For example, if the
piston were to be decelerated at a certain moment a continuous de-
pression wave would be created at the piston which would finally
catch up with the shock front, and a complicated interaction process
would then occur.
The case we have treated above corresponds to the propagation of
a bore into still water. If we were to superimpose a constant velocity
— | on the water in the motion illustrated by Fig. 10.7.1 the result
would be the motion called a hydraulic jump in which the shock front
is stationary. We need not consider this case further.
. We treat next the problem of the reflection of a shock wave from
a rigid vertical wall by following essentially the same procedure as
above. The circumstances are shown in Fig. 10.7.2. We have an
incoming shock moving toward the rigid wall from the left into still
water of depth hQ. The shock is reflected from the wall leaving still
water of depth h2 behind it. Since the water in contact with the wall
should be at rest, such an assumed motion is at least a plausible one.
We proceed to show that the motion is compatible with our shock
conditions and we calculate the height h2 of the reflected wave.
We assume that h^ and uv = 10, the depth and the velocity of the
water behind the shock, are known. The shock speed f + is then de-
termined by taking the largest of the three roots of the cubic equation
(10.6.21), which we write down again in the form
Once |+ has been determined, the depth h0 in front of the shock is
fixed from the first shock condition, which is in the present case
LONG WAVES IN SHALLOW WATER
381
(10.7.4) (W - S+fa = - |A-
To determine the reflected shock we may once more evidently make
I I
r • 1.4
Ol L
I I I I I I I I
0 2 4 6 8 10 12 14 16 18
(a)
ho
100
80
60
40
20
1 I I I I I I
8 • 10 12 14 16
(b)
Fig. l().7.3a, b. Reflection of a bore from a rigid wall
use of (10.7.4), since At and u^ = w remain the same on one side of the
shock, but we must now choose the smallest of the three roots of
382
WATER WAVES
(10.7.3) as the shock speed f_ since the side (1) is now obviously the
front side of the shock. The depth h2 of the water behind the shock
after the reflection— that is, of the water in contact with the wall
after reflection— is then obtained in the same way as h0 by using
(10.7.4) with |_ in place of £+ and h2 in place of hQ:
(10.7.4)! (w — £_)&! = — £_/*2.
By taking a series of values for w we have determined the ratios
/&2/&J and A2/A0 as functions of /?1/A0. That is, the height h2 of the
reflected wave has been determined as a function of the ratio of the
depth AJ of the incoming wave to the initial depth h0 at the wall. The
results of such a calculation are shown in Figs. 10.7.3a and 10.7. 3b:
In Fig. 10.7.4 we give a curve showing (h2 — A0)/^o as a function of
(Aj — AO)/AO> that is, we give a curve showing the increase in depth
after reflection as a function of the relative height (Ax — /*0)/^o °* t'ie
incoming wave.
100
80
60
40
20
*\>
1
I
1
•i o
2 4 6 8 10 12 14 16
Fig. 10.7.4. Height of the reflected bore
hrho
For A!/AO near to unity, i.e. for (/^ — hQ)/h0 small, it is not difficult
to show that
fin fin fl-t ' lle\
(1(\ 7 K\ _* - r*u 9 .
^AvF.i.tjy - £t .
f&Q f?/Q
From this relation we may write h2 — h0 ~ 2(7^ — h0) if (Aj — A0) is
small, i.e. the increase in the depth of the water after reflection is
LONG WAVES IN SHALLOW WATER
333
twice the height of the incoming wave when the latter is small. This
is what one might expect in analogy with the reflection of acoustic
waves of small amplitude. However, if h^h^ is not small, the water
increases in depth after reflection by a factor larger than 2. For
instance, if AX/A0 is 2, then h2 — h0 ~ 3(/&x — A0); while if h^h^ is 10,
then A2 — A0^ 35(At — A0), as one sees from the graph of Fig.
10.7.4. In other words, the reflection of a shock or bore from a rigid
wall results in a considerable increase in height and hence also in pres-
sure against the wall if the incoming wave is high. In fact, for very
high waves the total pressure p per unit width of the wall could be
shown to vary as the cube of the depth ratio hjh^.
In the upper curve of Fig. 10.7.3a we have drawn a curve for the
analogous problem in gas dynamics, i.e. the reflection of a shock from
the stopped end of a tube. In the case of air with an adiabatic expo-
nent y — 1.4 the density ratio Q2/6i as a function of Q^QQ (in an
obvious notation) is plotted as a dotted curve in the figure. As we
see, the density in air on reflection is higher than the corresponding
quantity, the depth, in the analogous case in water. However, the
curve for air ends at Ql/QQ = 6, since it is not possible to have a shock
wave in a gas with y = 1.4 which has a higher density ratio. In water
there is no such restriction. The explanation for this difference lies
in the fact that the energy law is assumed to hold across a shock in gas
dynamics, but not in our theory for water waves.
10.8. The breaking of a dam
At the end of section 10.5 we gave the solution to an idealized ver-
sion of the problem of determining the flow which results from the sud-
den destruction of a dam if it is assumed that the downstream side
Dam
Fig. 10.8.1. Breaking of a dam
384
WATER WAVES
of the dam is initially free of water. In the present section we consider
the more general problem which arises when it is assumed that there
is water of constant depth on the downstream as well as the upstream
side of the dam. Or, as the situation could also be described: a hori-
zontal tank of constant cross section extending to infinity in both
directions has a thin partition at the section x = 0. For x > 0 the
water has the depth h0 and for x < 0 the depth hv with A0 < hl9 as
indicated in Fig. 10.8.1. The water is assumed to be at rest on both
sides of the dam initially. At the time t = 0 the dam is suddenly des-
troyed, and our problem is to determine the subsequent motion of
the water for all x and t.
The special case h0 = 0— the cavitation case— was treated, as we
have already mentioned, at the end of section 10.5. We found there
that the discontinuity for x = 0 and t — 0 was instantly wiped out
and that the surface of that portion of the water in motion took the
form of a parabola tangent to the #-axis (i.e. to the bottom) at the
point x = — 2\/ghlt = — 2o1<, in which t is the time and cx the wave
x=-c,t
(I)
Free surface at t * t
(3)
(2)
Fig. 10.8.2. Breaking of a dam
LONG WAVES IN SHALLOW WATER 335
speed in water of depth hv If A0 is different from zero we might there-
fore reasonably expect (on the basis of the discussion at the beginning
of section 10.6) that a shock wave would develop sooner or later on
the downstream side, since the water pushing down from above acts
somewhat like a piston being pushed downstream with an accelera-
tion. In fact, since the water at x = 0 seems likely to acquire instan-
taneously a velocity different from zero it is plausible that a shock
would be created instantly on the downstream side. The simplest
assumption to make would be that the shock then moves downstream
with constant velocity £ (cf. Fig. 10.8.2). If this were so, the state of
the water immediately behind the shock (i.e. on the upstream side
of it) would be constant for all time, since the velocity u2 and depth
A2 on the upstream side of the shock would have the constant values
determined from the shock relations for the fixed values UQ = 0 and
h — h0 for the velocity and depth on the downstream side and the
assumed constant value | for the shock velocity. However, it is clear
that the constant state behind the shock could not extend indefinitely
upstream since u2 ^ 0 while the velocity of the water far upstream is
zero. Since we undoubtedly are dealing with a depression wave be-
hind the shock it seems plausible to expect that the constant state
behind the shock changes eventually at some point upstream into a
centered simple wave of the type discussed in section 10.5. In Fig.
10.8.2 we indicate in an #, 2-plane a motion which seems plausible as a
solution of our problem. In the following we shall show that such a
motion can be determined in a manner compatible with our theory
for every value of the ratio A0//ir
As indicated in Fig. 10.8.2, we consider four different regions in the
fluid at any time t = /0: the zone (0) is the zone of quiet downstream
which is terminated on the upstream side by the shock wave, or bore;
the zone (2) is a zone of constant state in which the water, however, is
not at rest; the zone (3) is a centered simple wave which connects the
constant state (2) with the constant state (1 ) of the undisturbed water
upstream. We proceed to show that such a motion exists and to deter-
mine it explicitly for all values of the ratio h^h^ between zero and one.
For this purpose it is convenient to write the shock conditions for
the passage from the state (0) to the state (2) in the form
(10.8.1) -!(«*•-{)=«<$+*;);
(10.8.2) cl(u2 - f) = - cfe
which are the same conditions as (10.6.18) and (10.6.19; with
386 WATER WAVES
replaced by c\ = ghi9 i.e. by the square of the wave propagation speed
in water of depth ht. By eliminating c\ from (10.8.1 ) by use of (10.8.2)
and then solving the resulting quadratic for u2 one readily obtains
(10.8.3) u2/c0 = #cc - - 1 + (Vl
(The plus sign before the radical was taken in order that u2 — f and
— - f should have the same sign. We observe also that only positive
values of | and u2 are in question throughout our entire discussion
since the side of (0) is the front side of the shock and the positive
^-direction is taken to the right.) It is also useful to eliminate u2
from (10.8.3) by using (10.8.2); the result is easily put into the form
(10.8.4) = (i (VT+8tfM* - 1)}*.
co
The relations (10.8.3) and (10.8.4) yield the velocity u2 and the wave
speed c2 behind the shock as functions of f and the wave speed c0 in
the undisturbed water on the downstream side of the dam. We pro-
ceed to connect the state (2) by a centered simple wave (cf. the dis-
cussion in section 10.5) with the state (1). In the present case the
straight characteristics in the zone (3) (cf. Fig. 10.8.2) are those with
the slope u — c (rather than the slope u + c as in section 10.5);
hence the straight characteristics which delimit the zone (3) are the
lines x = — cj on the left and x — (u2 — c2)t on the right. Along
each of the curved characteristics in zone (3)— one of these is indicated
schematically by a dotted curve in Fig. 10. 8. 2 —the quantity u + 2c
is a constant; it follows therefore that on the one hand
(10.8.5) u + 2c = 2cx
since u± = 0, while on the other hand
(10.8.6) u + 2c = u2 + 2c2
throughout the zone (3). The relation
(10.8.7) u2/c0 + 2c2/c0 - 2^/Co
must therefore hold. Our statement that a motion of the type shown
in Fig. 10.8.2 exists for every value of the depth ratio h^h^— or, what
amounts to the same thing, the ratio cj/cjj — is equivalent to the state-
ment that the relation (10.8.7) furnishes through (10.8.3) and (10.8.4)
an equation for f/c0 which has a real positive root for every value of
C^CQ larger than one. This is actually the case. In Fig. 10.8.3 we have
plotted curves for u2/cQ9 2c2/cQ, and u2/cQ + 2c2/cQ as functions of f /c0.
Once the curves of Fig. 10.8.3 have been obtained, our problem can
LONG WAVES IN SHALLOW WATER 337
be considered solved in principle: From the given value of
cilco we can determine £/c0 from the graph (or, by solving (10.8.7)).
The values of u2/cQ and c2/c0 are then also determined, either from the
graph or by use of (10.8.3) and (10.8.4). The constant state in the
zone (2) would therefore be known. In zone (3) the motion can now
be determined exactly as in section 10.5; we would have along the
straight characteristics in this zone the relations
dx x ,
- = - = U-c = 2c1-3c = ^-c1
from which
i
(10.8.8) C2 = j2cl -l , and
(10.8.9) "
Thus the water surface in the zone (3) is curved in the form of a
parabola in all cases.* At the junctions with both zones (1) and (2)
the parabola does not have a horizontal tangent, so that the slope of
the water surface is discontinuous at these points.
Some interesting conclusions can be drawn from (10.8.8) and
(10.8.9). By comparison with Fig. 10.8.2 we observe that the J-axis,
i.e. the line x = 0, is a characteristic belonging to the zone (3) pro-
vided that u2 ^ c2 since the terminal characteristic of the zone (3)
on the right lies on the J-axis or to the right of it in this case. If this
condition is satisfied we observe from (10.8.8) and (10.8.9)— which
are then valid on the /-axis— that c and u are both independent of t
at x = 0, which means that the depth of the water and its velocity u
arc both independent of t at this point, i.e. at the original location of
the dam, and hence that the volume of water crossing the original
dam site per unit of time (and unit of width) dQ/dt = uh is independ-
ent of time although the motion as a whole is not a steady motion.
In fact, h — \hl and u = f cx for all time t at this point. In
addition, u and c, and thus also dQ/dt, are not only independent of t
as long as u2 ^ c2, but also independent of the undisturbed depth A0
on the lower side of the dam if Ax is held fixed. Of course, it is clear that
AO/A! must be kept under a certain value (which from section 10.5
evidently must be less than 4/9) or the condition u2 I> c2 could not be
* Relations (10.8.8) and (10.8.9) are exactly the same as (10.5.8) and (10.5.7)
except for a change of sign which arises from a different choice of the positive
aj-direction.
888
WATER WAVES
7,
I
I
V'o
I
I
I
I
234567
Fig. 10.8.3. Graphical solution for £
fulfilled. In fact, the critical value of the ratio /^//^ at which u2 •= c2
can be determined easily by equating the right hand sides of (10.8.3)
and (10.8.4) and determining the value of |/c0 for this case, after
which c2/c0 = Vh2/h0 is known from (10.8.4). Since c2 ~ f q in
the critical case— either from the known fact that we still have
h2 = |Aj_ in this case, or from (10.8.8) with x — 0— we thus are
able to compute the critical value of cj/cj = A1/A0. A numerical cal-
culation yields for the critical value of the ratio h^h^ the value 7.225,
or for AQ/A! the value .1384. Thus if the water depth on the lower side
of the dam is less than 13.8 percent of the depth above the dam the
discharge rate on breaking the dam will be independent of the original
depth on the lower side as well as independent of the time. However,
if AQ/AJ exceeds the critical value .1384, the depth, velocity, and dis-
charge rate will depend on h0; but they continue to be independent of
LONG WAVES IN SHALLOW WATER
330
the time since the line x = 0 in the x> J-plane is under the latter cir-
cumstances contained in the zone (2), which is one of constant state.
The above results, which at first perhaps seem strange, can be
made understandable rather easily from the physical point of view,
as follows. If the zone (3) includes the /-axis (i.e. if A0/At is below the
critical value) we may apply (10.8.8) and (10.8.9) for x = 0 to obtain
at this point c = u — fcj. In other words, the flow velocity at the
dam site is in this case just equal to the wave propagation speed
there. For x > 0, i.e. downstream from the dam, we observe from
(10.8.8) and (10.8.9) that u is greater than c. Since c is the speed at
which the front of a disturbance propagates relative to the moving
water we see that changes in conditions below the dam can have no
effect on the flow above the dam since the flow velocity at all points
below the dam is greater than the wave propagation speed at these
points and hence disturbances can not travel upstream. However,
once AQ//^ is taken higher than the critical value, the flow velocity
at the dam will be less than the wave propagation speed at this point,
as one can readily prove, and we could no longer expect the flow at
that point to be independent of the initial depth assumed on the
downstream side.
The discharge rate dQ/dt — hu per unit width at the dam, i.e. at
x = 0, is plotted in Fig. 10.8.4 as a function of the depth A0. In accord-
ance with our discussion above we observe that dQ/dt remains con-
0.3
0.2
0.1
0.2 0.4 0.6 0.8
1.0
Fig. 10.8.4. Discharge rate at the dam
stant at the value dQ/dt = .296hlcl until h^h^ reaches the critical
value .138, after which it decreases steadily to the value zero when
h0 = /ir i.e. when the initial depth of the water below the dam is the
same as that above the dam.
340
WATER WAVES
Another feature of interest in the present problem is the height of
the bore, i.e. the quantity h2 — A0, as a function of the original depth
ratio h^jhv When A0 = 0 we know that there is no bore and the water
surface (as we found in section 10.5) appears as in Fig. 10.8.5. The
water surface at the front of the wave on the downstream side is
tangent to the bottom and moves with the speed 2clB On the other
hand, when h^h^ approaches the other extreme value, i.e. unity, it is
clear that the height h2 — h0 of the bore must again approach zero.
Hence the height of the bore must attain a maximum for a certain
-Parabola
x = -c,t x = 2c,t x
Fig. 10.8.5. Motion down the dry bed of a stream
value of hQ/hv In Fig. 10.8.6 we give the result of our calculations for
A2 — - A0 as a function of h^h^. The curve rises very steeply to its
maximum A2 — A0 = .32^ for h^h^ = .176 and then falls to zero
again when hQ = hv It is rather remarkable that the bore can attain
a height which is nearly 1/3 as great as the original depth of the water
behind the dam.
0.1
1
1
0 0.2 04 0.6 0.8 IX)
Fig. 10.8.6. Maximum height of the bore
LONG WAVES IN SHALLOW WATER
341
It is instructive to describe the motion by means of the #, £-plane
when AO/AJ is near its two limit values unity and zero. These two cases
are schematically shown in Fig. 10.8.7. When h0 ~ hv we note that
the zone (3) is very narrow and that the shock speed f approaches cl9
i.e. the propagation speed of small disturbances in water of depth hl9
corresponding to the fact that the height of the shock wave tends to
zero as A0-> h^ (cf. Fig. 10.8.6). The other limit situation, i.e. h0 c± 0,
is more interesting. Since we tacitly consider h± to remain fixed in our
present discussion, and hence that h2 is also fixed since we are in the
supercritical case, it follows (for example from (10.6.23) with A2 in
place of hi) that f ->oo as A0-> 0. On the other hand as we see from
Free
(I)
surface for t«tQ
(3)
(2)
(0)
Fig. 10.8.7. Limit cases
Fig. 10.8.6, the height h2 — A0 of the shock wave tends to zero rather
slowly as A0 -> 0. In the limit, point P becomes the front of the wave
in accordance with the motion indicated by Fig. 10.8.5. Thus as
A0 -> 0 the shock wave becomes very small in height but moves down-
stream with great speed; or, as we could also say, in the limit the water
in front of the point P is pinched out and P is the front of the wave.
342 WATER WAVES
10.9. The solitary wave
It has long been a matter of observation that wave forms of a per-
manent type other than the uniform flows with an undeformed free
surface occur in nature; for example, Scott Russell [R.14] reported
in 1844 his observations on what has since been called the solitary
wave, which is a wave having a symmetrical form with a single hump
and which propagates at uniform velocity without change of form.
Later on, Boussinesq [B.16] and Rayleigh [R.3] studied this problem
mathematically and found approximations for the form and speed of
such a solitary wave. Korteweg and de Vries [K.15] modified the
method of Rayleigh in such a way as to obtain waves that arc periodic
in form— called cnoidal waves by them— and which tend to the soli-
tary wave found by Rayleigh in the limiting case of long wave lengths.
A systematic procedure for determining the velocity of the solitary
wave has been developed by Weinstein [W.6],
At the beginning of section 10.7 we have shown that the only con-
tinuous waves furnished by the theory used so far in this chapter
which progress unchanged in form are of a very special and rather
uninteresting character, i.e., they are the motions with uniform velo-
city and horizontal free surface.* This would seem to be in crass con-
tradiction with our intention to discuss the solitary wave in terms of
the shallow water theory, and it has been regarded by some writers as a
paradox. **The author's view is that this paradox— like most others —
becomes not at all paradoxical when properly examined. What is
involved is a matter of the range of accuracy of a given approximate
theory, and also the fact that a perturbation or iteration scheme of
universal applicability does not exist: one must always modify such
schemes in accordance with the character of the problem. In the pre-
sent case, the salient fact is that the theory used so far in the present
chapter represents the result of taking only the lowest order terms in
the shallow water theory as developed in section 4 of Chapter 2, and
it is necessary to carry out the theory to include terms of higher order
* If motions with a discontinuity are included in the discussion, then the motion
of a bore is the only other possibility up to now in this chapter with regard to
waves propagating unchanged in form.
** Birkhoff [B.ll, p. 23], is concerned more about the fact that the shallow
water theory predicts that all disturbances eventually lead to a wave which
breaks when on the other hand Struik [S.29] has proved that periodic progressing
waves of finite amplitude exist in shallow water. In the next section the problem
of the breaking of waves is discussed. Ursell [U.ll] casts doubt on the validity
of the shallow water theory in general because it supposedly does not give rise
to the solitary wave.
LONG WAVES IN SHALLOW WATER 348
if one wishes to obtain an approximation to the solution of the problem
of the solitary wave. This has been done by Keller [K.6], who finds
that the theory of Friedrichs [F.ll] presented in Chapter 2, when car-
ried out to second order,* yields both the solitary wave and cnoidal
waves of the type found by Korteweg and de Vries [K.15] (thus the
shallow water theory is capable of yielding periodic progressing waves
of finite amplitude). As lowest order approximation to the solution
of the problem, Keller finds (as he must in view of the remarks above),
that the only possibility is the uniform flow with undeformed free
surface, but if the speed U of the flow is taken at the critical value
U = \/gk with h the undisturbed depth, then a bifurcation phenome-
non occurs (that is, among the set of uniform flows of all depths and
velocities, the solitary wave occurs as a bifurcation from the special
flow with the critical velocity) and the second order terms in the de-
velopment of Friedrichs lead to solitary and cnoidal waves with
speeds in the neighborhood of this value. To clinch the matter, it has
been found by Friedrichs and Hycrs [F.13] that the existence of the
solitary wave can be proved rigorously by a scheme which starts with
the solution of Keller as the term of lowest order and proceeds by
iterations with respect to a parameter in essentially the same manner
as in the general shallow water theory.** In the following, we shall
derive the approximation to the solution of the solitary Wave problem
following the method of Friedrichs and Hyers rather than the general
expansion scheme which was used by Keller, and we can then state
the connection between the two in more detail.
The author thus regards the nonlinear shallow water theory to be
well founded and not at all paradoxical. Indeed, the linear theory of
waves of small amplitude treated at such length in Part II of this
book is in essentially the same position as regards rigorous justifica-
tion as is the shallow water theory: we have only one or two cases so
far in which the linear theory of waves of small amplitude is shown to
be the lowest order term in a convergent development with respect to
amplitude. We refer, in particular, to the theory of Levi-Civita [L.7]
and Struik [S.29] in which the former shows the existence of periodic
progressing waves in water of infinite depth and the latter the same
thing (and by the same method) for waves in water of finite constant
* In order to fix all terms of second order, Keller found it necessary to make
use of certain relations which result from carrying the development of some of
the equations up to terms of third order.
** W. Littman, in a thesis to appear in Communs. Pure and AppJ. Math.,
has proved rigorously in the same way the existence also of cnoidal waves.
844 WATER WAVES
depth.* This theory will be developed in detail in Chapter 12. It might
be added that those who find the nonlinear shallow water theory
paradoxical in relation to the solitary wave phenomenon should by
the same type of reasoning also find the linear theory paradoxical,
since it too fails to yield any approximation to the solitary wave, even
when carried out to terms of arbitrarily high order in the amplitude,
except the uniform flow with undisturbed free surface. In fact, if one
were to assume that a development exists for the solitary wave which
proceeds in powers of the amplitude as in the theory discussed in the
first part of Chapter 2, it is easily proved that the terms of all orders
in the amplitude are identically zero. There is no paradox here, how-
ever; rather, the problem of the solitary wave is one in which the
solution is not analytic in the amplitude in the neighborhood of its
zero value, but rather has a singularity —possibly of the type of a
branch point— there. Thus a different kind of development is needed,
and, as we have seen, one such possibility is a development of the type
of the shallow water theory starting with a nonlinear approximation.
Another possibility has been exploited by Lavrentieff [L.4] in a
difficult paper; Lavrentieff proves the existence of the solitary wave
by starting from the solutions of the type found by Struik for periodic
waves of finite amplitude and then making a passage to the limit by
allowing the wave length to become large and, presumably, in such a
way that the parameter gh/U2 tends to unity. This procedure of
Lavrentieff thus also starts with a nonlinear first approximation.
The problem thus furnishes another good example of the well-known
fact that it is not always easy to guess how to set up an approximation
scheme for solving nonlinear boundary value problems, since the
solution may behave in quite unexpected ways for particular values of
the parameters. Hindsight, however, can help to make the necessity
for procedures like those of Friedrichs and Hycrs and of Lavrentieff
in the present case more apparent: we have seen in Chapter 7.4 that
a steady flow with the critical speed U = VgA is in a certain sense
highly unstable since the slightest disturbance would lead, in terms
of the linear theory for waves of small amplitude, to a motion in which
infinite elevations of the free surface would occur everywhere; thus
the linear theory of waves of small amplitude seems quite inappro-
priate as the starting point for a development which begins with a
' * L. Nirenberg [N.2] has recently proved the existence of steady waves of
finite amplitude caused by flows over obstacles in the bed of a stream.
LONG WAVES IN SHALLOW WATER
345
uniform flow at the critical speed, and one should consequently use
a basically nonlinear treatment from the outset.
We turn now to the discussion of the solution of the solitary wave
problem. The theory of Friedrichs and Hyers begins with a formula-
tion of the general problem that is the same as that devised by Levi-
Civita for treating the problem of existence of periodic waves of finite
amplitude, and which was motivated by the desire to reformulate the
problem in terms of the velocity potential <p and stream function \p as
independent variables in order to work in the fixed domain between
the two stream lines \p = const, corresponding to the bottom and the
free surface instead of in the partially unknown domain in the physi-
cal plane. We therefore begin with the general theory of irrotational
waves in water when a free surface exists. The wave is assumed to
be observed from a coordinate system which moves with the same
velocity as the wave, and hence the flow can be regarded as a steady
flow in this coordinate system.
Fig. 10.9.1. The solitary wave
A complex velocity potential %'(x', y') =%'(z'):
(10.9.1 ) x' = ?' + *V> *' = x' + {y'
is sought in an #', i/'-plane (cf. Fig. 10.9.1) such that at infinity the
velocity is U and the depth of the water is h. %' is of course an analytic
function of z'. The real harmonic functions y' and \p' represent the
velocity potential and the stream function. The complex velocity
wf = d%'/dz' is given by
(10.9.2) w' = u' — iv'9
in which u' and v' are the velocity components. This follows by virtue
of the Cauchy-Riemann equations:
(10.9.3) <p'x< = v>V» VV = ~ V>'*''
846 WATER WAVES
since w' ••= q>'x> + i\p'x>. It is convenient to introduce new dimensionless
variables:
(10.9.4) z = z'/h, w = w'/U, x = V + *V = X'/(hU)>
and a parameter y:
(10.9.5) y = gW2-
In terms of these quantities the free surface corresponds to \p — 1
if the bottom is assumed given by y> = 0, since the total flow over a
curve extending from the bottom to the free surface is Uh. The
boundary conditions are now formulated as follows:
(10.9.6) v = — Jmw = 0 at y> = 0,
(10.9.7) \ \w\2 + yy = const. at \p = 1.
The second condition results from Bernoulli's law on taking the
pressure to be constant at the free surface and the density to be unity,
as one sees from equation (1.3.4) of Ch. 1. At oo we have the condition
(10.9.8) w -> 1 as | x | -* oo.
We assume now that the physical plane (i.e. the #, i/-plane) is mapped
by means of %(z) into the 99, y-plane in such a way that the entire flow
is mapped in a one-to-one way on the strip bounded by \p — 0 and
\p = 1.* In this case the inverse mapping function z(%) exists, and we
could regard the complex velocity w as a function of # defined in the
strip bounded by \p = 0, \p — 1 in the ^-plane. We then determine the
analytic function w(%) in that strip from the boundary conditions
(10.9.6), (10.9.7), (10.9.8), after which %(z) can be found by an inte-
gration and the free surface results as the curve given by \p = Jm # =1.
It is convenient, however, again following Levi-Civita to replace
the dependent variable w by another (essentially its logarithm)
through the equation
(10.9.9) w = <rl(*+tA).
It follows that
(10.9.10) \w\ = ex, 0 = argiei,
and thus A = log |io|, with \w\ the magnitude of the velocity vector,
* Our assumption that the mapping of the flow on the /-plane is one-to-one
can be shown rather easily to follow from the other assumptions and Levi-Civita
carries it out. The equivalence of the various formulations of the problem is then
readily seen. In Chapter 12.2 these facts are proved.
LONG WAVES IN SHALLOW WATER 847
and 0 is the inclination relative to the #-axis of the velocity vector.
We proceed to formulate the conditions for the determination of 6
and A in the <p, y plane. The condition (10.9.6) becomes, of course,
6 = 0 at \p = 0. To transform the condition (10.9.7) we first differen-
tiate with respect to q> along the line ip = 1 to obtain
d \w\ dy
(10.9.11) | w | — ^ — - + y-^ = 0 on w = 1.
dcp f dq> r
Since x and y are conjugate harmonic functions of <p and \p we may
write
doc dy (px u
dv ~ dw ~~ <pl + cpl ~~ j w |2
(10.9.12) { 7 r Vx-rVv
dy ox v
d<p d\p \w |2
in accordance with well-known rules for calculating the derivatives
of functions determined implicitly, or from
dz 1 1 u + iv
d>X d% u — iv u2 + v2 '
dz
As a consequence we have from (10.9.11):
or, since | w \ ~ ex and v = — J>w e~i(P+iK} — e* sinO:
6^A ^— = -- w~2A sin 0,
t/9?
and since ^A/S^ = —dO/dy because A and 6 are harmonic conjugates
it follows finally that
dO
(10.9.13) — = w-3Asin0 atw = 1.
oy
The boundary conditions 0 = 0 for \f = 0 and (10.9.13) at \p = 1 are
Lcvi-Civita's conditions, but the condition at oo imposed here is re-
placed in Levi-Civita's and Struik's work by a periodicity condition in
x, —and this makes a great difference. Levi-Civita and Struik proceed
on the assumption that a disturbance of small amplitude is created
relative to the uniform flow in which w = const.; this is interpreted to
848 WATER WAVES
mean that 6 + iX is a quantity which can be developed in powers of
a small parameter e, and the convergence of the series for sufficiently
small values of e is then proved. In Chapter 12.2 we shall give a proof
of the convergence of this expansion. (In lowest order, we note that
the condition (10.9.13) leads for small A and 6 to the condition dO/dy —
yd = 0 at \f — 1 — in agreement with what we have seen in Part II.)
In the case of the solitary wave such a procedure will not succeed, as
was explained above, or rather it would not yield anything but a
uniform flow. The procedure to be adopted here consists in developing,
roughly speaking, with respect to the parameter y near y — 1; but,
as in the shallow water theory in general in the version presented in
section 4 of Chapter 2, we introduce a stretching of the horizontal
coordinate <p which depends on y while leaving the vertical coordinate
unaltered (see equation (10.9.19)). This stretching of only one of the
coordinates is the characteristic feature of the shallow water theory.
(The approximating functions are then no longer harmonic in the new
independent variables.) Specifically, we introduce the real parameter
x by means of the equation
(10.9.14) e~*»* = y = gh/U*.
This implies that gh/U2 < 1, but that seems reasonable since all of
the approximate theories for the solitary wave lead to such an
inequality. We also introduce a new function r, replacing A, by the
relation
(10.9.15) r = A + x2.
For 6((f>9 \p) and r(y, y>) we then have the boundary conditions
(10.9.16) 6 = 0, \p = 0
90
(10.9.17) g- = e~3x sin 0, y> = 1.
For <p -> ± oo we have the conditions imposed by the physical pro-
blem:
(10.9.18) 0->0, r->*2,
the latter resulting since A -> 0 at oo from | w \ = eK and | w \ -> 1
at oo. As we have already indicated, the development we use requires
stretching the variable <p so that it grows large relative to \p when x
is small; this is done in the present case by introducing the new in-
dependent variables
LONG WAVES IN SHALLOW WATER 349
(10.9.19) y = xy, ijp = y.
The dependent variables 6 and r are now regarded as functions of y
and ip and they are then expanded in powers of x:
(10 9 20}
{ '' '
(We have omitted writing down a number of terms which in the
course of the calculation would turn out to have zero coefficients.)
Friedrichs and Hyers have proved that the lowest order terms in
these series, as obtained formally through the use of the boundary
conditions, are the lowest order terms in a convergent iteration scheme
using x as small parameter. Their convergence proof also involves the
explicit use of the stretching process. However, the proof of this
theorem is quite complicated, and consequently we content ourselves
here with the determination of the lowest order terms: we remark,
however, that higher order terms could also be obtained explicitly
from the formal expansion.
The series in (10.9.20) are now inserted in all of the equations which
serve to determine 6 and r9 and relations for the coefficient functions
ri(*P> yO and Oiiy* V) are obtained. The Cauchy-Riemann equations
for 0 and r lead to the equations
(10.9.21) 0^ = -XT-, Ty = xOj
in terms of the variables <p and \p, and the series (10.9.20) then yield
the equations
(10.9.22) Tl5 - 0, 0<- = - T^ , T2^ = 0* .
Thus TJ = TI(IJ>) is independent of y, and integration of the remaining
equations gives the following results:
(10.9.23) T, = - 4y«Tj' +
The primes refer to differentiation with respect to <p. An additive
arbitrary function of (p in the first of these equations was taken to be
zero because of the boundary condition 6l = 0 for \p = 0.
Upon substitution of (10.9.20) into the boundary condition (10.9.
17) we find
350 WATER WAVES
and consequently we have the equations
(10.9.24)
The first equation is automatically satisfied because of the first equa-
tion of (10.9.23). The second equation leads through (10.9.23) to the
condition
(10.9.25) r[" = 9^,
for TV as one readily verifies. Once rl has been determined, one sees
that 0X is also immediately fixed by the first equation in (10.9.23).
Boundary conditions are needed for the third order nonlinear differ-
ential equation given by (10.9.25); we assume these conditions to be
(o) -o,
(10.9.26) TJ, (oo) = 1,
These conditions result from our assumed physical situation: the first
is taken since a symmetrical form of the wave about its crest is ex-
pected and hence 0X(0) = 0, the second arises from (10.9.18), while
the third is a reasonable condition that is taken in place of what looks
like the more natural condition ri(oo) = 0 since the latter condition
is automatically satisfied, in view of the first equation of (10.9.23)
and 0i(oo) = 0, and thus docs not help in fixing r± uniquely.
An integral of (10.9.25) is readily found; it is:
Ti =lri + const.,
and the boundary conditions yield
From this one obtains, finally, the solution:
(10.9.27) T!(<P) = 1-3 sech2 (3<p/2),
and 6l is then fixed by (10.9.23). From these one finds for the shape of
the wave— that is, the value of y corresponding to y = 1 — , and for
the horizontal component u of the velocity the equations
(10.9.28) y = 1 + 3*2 sech2 — ,
2
(10.9.29) u = 1 — 3*2sech2-— .
2
LONG WAVES IN SHALLOW WATER
351
Fig. 10.U.2. A solitary wave
In calculating these quantities,
higher order terms in x have been
neglected. The expression for the
wave profile is identical with those
found by Boussinesq, Rayleigh,
and Keller. For the velocity u,
the two former authors give u = I
while Keller gives the same expres-
sion as above except that the factor
3#2 is replaced by another which
differs from it by terms of order
tt4 or higher.
Thus a solitary wave of sym-
metrical form has been found with
an amplitude which increases with
its speed 17. Careful experiments
to determine the wave profile and
speed of the solitary wave have
been carried out by Daily and
Stephan [D.I], who find the wave
profile and velocity to be closely
approximated by the above formu-
las with a maximum error in the
latter of 2.5 % at the highest ampli-
tude-depth ratio tested. Fig. 10.9.2
is a picture of a solitary wave taken
by Daily and Stephan; three frames
from a motion picture film are
shown.
10.10. The breaking of waves in shallow water. Development of bores
In sections 10.4 and 10.6 above it has already been seen that the
shallow water theory, which is mathematically analogous to the
theory of compressible flows in a gas, leads to a highly interesting
and significant result in cases involving the propagation of disturb-
ances into still water that are the exact counterparts of the corre-
sponding cases in gas dynamics involving the motions due to the action
of a piston in a tube filled with gas. These cases, which are very easily
352 WATER WAVES
described in terms of the concept of a simple wave (cf. sec. 10.3),
lead, in fact to the following qualitative results (cf. sec. 10.4): there is
a great difference in the mode of propagation of a depression wave
and of a hump with an elevation above the undisturbed water line;
in the first case the depression wave gradually smooths out, but in
the second case the front of the wave becomes progressively steeper
until finally its slope becomes infinite. In the latter case, the mathe-
matical theory ceases to be valid for times larger than those at which
the discontinuity first appears, but one expects in such a case that the
wave will continue to steepen in front and will eventually break. This
is the correct qualitative explanation, from the point of view of
hydrodynamical theory, for the breaking of waves on shallow beaches.
It was advanced by Jeffreys in an appendix to a book by Cornish
[C.7] published in 1934. Jeffreys based his discussion on the fact that
the propagation speed of a wave increases with increase in the height
of a wave above the undisturbed level. Consequently, if a wave is
created in such a way as to cause a rise in the water surface it follows
that the higher points on the wave surface will propagate at higher
speed than the lower points in front of them— in other words there is
a tendency for the higher portions of the wave to overtake and to
crowd the lower portions in front so that the front of the wave be-
comes steep and eventually curls over and breaks; the same argument
indicates that a depression wave tends to flatten out and become
smoother as it advances.
It is of interest to recall how waves break on a shallow beach.
Figures 10.10.1, 10.10.2, and 10.10.3 are photographs* of waves on
the California coast. Figure 10.10.1 is a photograph from the air,
taken by the Bureau of Aeronautics of the U.S. Navy, which shows
how the waves coming from deep water are modified as they move
toward shore. The waves are so smooth some distance off shore that
they can be seen only vaguely in the photograph, but as they move
inshore the front of the waves steepens noticeably until, finally,
breaking occurs. Figures 10.10.2 and 10.10.3 are pictures of the same
wave, with the picture of Figure 10.10.3 taken at a slightly later time
than the previous picture. The steepening and curling over of the
wave are very strikingly shown.
At this point it is useful to refer back to the beginning of section
10.6 and especially to Fig. 10.6.1. This figure, which is repeated here
* These photographs were very kindly given to the author by Dr. Walter Munk
of the Sctipps Institution of Oceanography.
LONG WAVES IN SHALLOW WATER
358
Fig. 10.10.1. Waves on a beach
for the sake of convenience, indicates in terms of the theory of
characteristics what happens when a wave of elevation is created by
Fig. 10.10.2. Wave beginning to break
354
WATER WAVES
Fig. 10.10.3. Wave breaking
pushing the moveable end of a tank of water into it so that a disturb-
ance propagates into still water of constant depth: the straight
characteristics issuing from the "piston curve" AD, along each of
Fig. 10.10.4. Initial point of breaking
LONG WAVES IN SHALLOW WATER 355
which the velocity u and the quantity c = Vg(h + rj) are constant,
eventually intersect at the point E. The point E is a cusp on the enve-
lope of the characteristics, and represents also the point at which the
slope of the wave surface first becomes infinite. The point E might
thus — somewhat arbitrarily, it is true— -be taken as defining the break-
ing point (xb9 tb) of the wave, since one expects the wave to start
curling over after this point is reached. It is possible to fix the values
of xb and tb without difficulty once the surface elevation r\ = 77(0, t) is
prescribed at x = 0; we carry out the calculation for the interesting
case of a pulse in the form of a sine wave:
(10.10.1) 17(0, t) = A sin cot.
For t = 0, x > 0 we assume the elevation r\ of the water to be zero
and its velocity UQ to be constant (though not necessarily zero, since
it is of interest to consider the effect of a current on the time and place
of breaking).
As we know, the resulting motion is easily described in terms of the
characteristics in the x, f-plane, which arc straight lines emanating
from the J-axis, as indicated in Figure 10.10.6. The values of u and c
are constant along each such straight line. The slope dx/dt of any
straight characteristic through the point (0, r) is given by
dx
(10.10.2) — = 9c - 2r0 + MO>
at
which is the same as (10.3.6). The quantity CQ has the value c0 = Vgh,
while c = Vg(A +17), as always. On the other hand, the slope of this
characteristic is clearly also given in terms of a point (x, t) on it by
xj(t — r) so that (10.10.2) can be written in the form
(10.10.3) x = (t — r)[8c(r) - 2<?0 + MO]
in which we have indicated explicitly that c depends only on r since
it (as well as all other quantities) is constant along any straight
characteristic. Thus (10.10.3) furnishes the solution of our problem,
once c(r ) is given, throughout a region of the x, £-plane which is cover-
ed by the straight lines (10.10.3) without overlapping. However, the
interesting cases for us are just those in which overlapping occurs,
i.e. those for which the characteristics converge and eventually cut
each other, and this always happens if an elevation is created at
x — 0. In fact, if c is an increasing function of r, then dx/dt as given
by (10.10.2) increases with r and hence the characteristics for x > 0
856 WATER WAVES
must intersect. In this case, furthermore, the family of straight cha-
racteristics has an envelope beginning at a point (xb9 tb), which we
have defined to be the point of breaking.
We proceed to determine the envelope of the straight lines (10.10.3).
As is well known, the envelope can be obtained as the locus resulting
from (10.10.3) and the relation
(10.10.4) 0 = - [8c(r) - 2c0 + u0] + 3(t - T)C'(T)
obtained from it by differentiation with respect to T. For the points
(xC9 tc) on the envelope we then obtain the parametric equations
MAinKt [3c(r) - 2C0 + utf
(10.10.5) *c = - — - ,
and
(10.10.6) te = r
We are interested mainly in the "first" point on the envelope, that
is, the point (xb9 tb) for which te has its smallest value since we iden-
tify this point as the point of breaking. To do so really requires a
proof that the water surface has infinite slope at this point. Such a
proof could be easily given, but we omit it here with the observation
that an infinite slope is to be expected since the characteristics which
intersect in the neighborhood of the first point on the envelope all
carry different values for c.
We have assumed that 77(0, t) is given by (10.10.1 ) and consequently
the quantity c(r) in (10.10.5) and (10.10.6) is given by
(10.10.7) c(r) = Vg(h + A sin COT).
If we assume A > 0 we see that C'(T) is a positive decreasing function
of T for small positive values of T. Since c(r) increases for small posi-
tive values of r it follows that both xc and tc in (10.10.5) and (10.10.6)
are increasing functions of T near T = 0. A minimum value of xc and
te must therefore occur for T = 0, so that the breaking point is given
by
noin«i * 2<?o(c° + "o)2
(10.10.8) x, =
and
••-
as one can readily verify. We note that the point (xb, tb) lies on the
LONG WAVES IN SHALLOW WATER 357
initial characteristic x = (CQ + w0)f, as it should since r = 0 for this
characteristic. From the formulas we can draw a number of interesting
conclusions. Since c0 = Vgh we see that breaking occurs earlier in
shallower water for a pulse of given amplitude A and frequency a).
Breaking also occurs earlier when the amplitude and frequency are
larger. It follows that short waves will break sooner than long waves,
since longer waves are correlated with lower frequencies. Finally we
notice that early breaking of a wave is favored by small values for
uQ, the initial uniform velocity of the quiet water. In fact, if u$ is
negative, i.e. if the water is flowing initially toward the point where
the pulse originates, the breaking can be made to occur more quickly.
Everyone has observed this phenomenon at the beach, where the break-
ing of an incoming wave is often observed to be hastened by water
rushing down the beach from the breaking of a preceding wave.
It is of some importance to draw another conclusion from our theory
for waves moving into water of constant depth: an inescapable con-
sequence of our theory is that the maxima and minima of the surface
elevation propagate into quiet water unchanged in magnitude with re-
spect to both distance and time. This follows immediately from the fact
that the values of the surface elevation are constant along the straight
characteristics so that if 7? has a relative maximum for x = 0, t = r,
say, then this value of rj will be a relative maximum all along the
characteristic which issues from x — 0, t = r. The waves change
their form and break, but they do so without changes in amplitude.
In a report of the Hydrographic Office by Sverdrup and Munk
[S.36] some results of observations of breakers on sloping beaches are
given in the form of graphs showing the ratio of breaker height to
deep water amplitude and the ratio of undisturbed depth at the break-
ing point to the deep water amplitude as functions of the "initial
steepness" in deep water, the latter being defined as the ratio of
amplitude to wave length in deep water. The "initial steepness" is thus
essentially the quantity Ao> in our above discussion, and our results
indicate that it is a reasonable parameter to choose for discussion of
breaking phenomena. The graphs given in the report — reproduced here
in Figures 10.10.5a and 10.10.5b— show very considerable scattering
of the observational data, and this is attributed in the report to errors
in the observations, which are apparently difficult to make with
accuracy. On the basis of our above conclusion— that the breaking of
a wave in water of uniform depth occurs no matter what the amplitude
of the wave may be in relation to the undisturbed depth— we could
358
WATER WAVES
offer another explanation for the scatter of the points in Figures
10.10.5a and 10.10.5b, i.e. that the amplitude ratios are relatively
independent of the initial steepness. Of course, the curves of Figures
• W.H.0.1.
A B.E.B ~ 1:6.3
A B.E.B — 1:20.4
o B.E.B — r.33.3
© THEORY
0.003 0.005
0.01 0.02 0.03
INITIAL STEEPNESS H'
0.05
0.15
Fig. 10.10.5a. Ratio of breaker height to wave height in deep water, //»/
assuming no refraction
•f.V
d /H1 « 6 66
' SLOPE '
1
•*"*" b
0
0 S.I.O
3.5
•
• W.H.0.1.
•
A B.E.B — 1-63
3-0
A B.E B
- 1:20.4
1
c
o
i B E B
- 1:33.3
l!
x° 2.5
A
•o
2.0
•
V
!V
•o
^
09
n 0
II
o° !
f
o 6&v
A t6° o
1.5
1.0
-
0
I
0
1
1 I
:*:«*°
^
a
^
i i
\
__ XHEORE^TICA
LIMIT
» c
*>
0.003 0.005 0.01 0.02 0.03 0.05 O.I 0.15
INITIAL STEEPNESS HQ /LQ
Fig. 10.10.5b. Ratio of depth of water at point of breaking to wave height in
deep water, d6/H0', as function of steepness in deep water, H07L0, assuming
no refraction
LONG WAVES IN SHALLOW WATER
859
10.10.5a and 10.10.5b refer to sloping beaches and hence to cases in
which the wave amplitudes increase as the wave moves toward shore;
but still it would seem rather likely that the amplitude ratios would be
relatively independent of the initial steepness in these cases also since
the beach slopes are small. The detailed investigation of breaking of
waves by Hamada [H.2], which is both theoretical and experimental
in character, should be consulted for still further analysis of this and
other related questions. The papers by Iversen [1.6] and Suquet
[S.31 ] also give experimental results concerning the breaking of waves.
We continue by giving the results of numerical computations for
three cases of propagation of sine pulses into still water of constant
depth. The cases calculated are indicated in the following table:
Case
Type of pulse
Case 1 is a half-sine pulse in the form of a positive elevation, case 2
is a full sine wave which starts with a depression phase, and case 3
consists of several full sine waves.
Figure 10.10.6 shows the straight characteristics in the x, f-plane
for case 1. (In all of these cases, the quantities x and y are now certain
dimensionless quantities, the definitions of which are given in [S.19].)
We observe that the envelope begins on the initial characteristic in
this case, in accord with earlier developments in this section. The
envelope has two distinct branches which meet in a cusp at the
breaking point (xb, tb). Figure 10.10.7 gives the shape of the wave for
two different times. As we see, the front of the wave steepens until it
finally becomes vertical for x = tcb, t =- tb, while the back of the wave
flattens out. The solution given by the characteristics in Figure
360
WATER WAVES
Region of
Constant State
Region of Constant State
Fig. 10.10.6. Characteristic diagram in the x, /-plane
Fig. 10.10.7. Wave height versus distance for a half-sine wave of amplitude
hQ/5 in water of constant depth at two instants, where hn is the height of the
still water level
LONG WAVES IN SHALLOW WATER
361
Fig. 10.1O.8. Wave profile after breaking
Fig. 1O.1O.9. Characteristic diagram in the a?, £-plane
362
WATER WAVES
10.10.6 is not valid for x > x& t > t^ and we expect breaking to
ensue. However, we observe that the region between the two branches
of the envelope is quite narrow, so that the influence of the developing
Fig. 10.10.10. Wave height versus distance for a full negative sine wave with
amplitude /*0/5 in water of constant depth at t = 3.0, t = 5.0, and t = 6.28
breaker may not seriously affect the motion of the water behind it.
Thus we might feel justified in considering the solution by characteris-
tics given by Figure 10.10.6 as being approximately valid for values of
LONG WAVES IN SHALLOW WATDR
863
t slightly greater than tb. (This also seems to the writer to be intuitive-
ly rather plausible from the mechanical point of view.) Figure 10.10.8
was drawn on this basis for a time considerably greater than tb. The
full portion of the curve was obtained from the characteristics outside
the region between the branches of the envelope, while the dotted
portion— which is of doubtful validity— was obtained by using the
characteristics between the branches of the envelope in an obvious
manner. In this way one is able to approximate the early stages of the
curling over of a wave.
Figures 10.10.9, 10.10.10, and 10.10.11 refer to case 2, in which a
depression phase precedes a positive elevation. In this case the enve-
lope of the characteristics does not begin, of course, on the initial
characteristic but rather in the interior of the simple wave region, as
indicated in Figure 10.10.9. Figure 10.10.10 shows three stages in the
progress of the pulse into still water. The steepening of the wave front
is very marked by the time the breaking point is reached —much more
marked than in the preceding case for which no depression phase oc-
curs in front. Figure 10.10.11 shows the shape of the wave a short time
after passing the braking point. This curve was obtained, as in the
preceding case, by using the characteristics between the branches of
the envelope. Although this can yield only a rough approximation, still
Fig. 10.10.11. r\ versus x at t = 7 for non-sloping bottom where the pulse is an
entire negative sine-wave. The dotted part of the curve represents r\ in the region
between the branches of the envelope
one is rather convinced that the wave really would break very soon
after the point we have somewhat arbitrarily defined as the breaking
point.
364 WATER WAVES
Figure 10.10.12 shows the water surface in case 3 for a time well
beyond the breaking point.
Fig. 10.10.12. Water profile after breaking
In gas dynamics where u and c represent the velocity and sound
speed throughout an entire cross section of a tube containing the gas,
it clearly is not possible to give a physical interpretation to the region
between the two branches of the envelope in the cases analogous to
that shown in Figure 10.10.6, since the velocity and propagation
speed must of necessity be single-valued functions of x. However, in our
case of water waves u and c refer to values on the water surface so that
there is no reason a priori to reject solutions for u and c which are not
single- valued in x. Thus we might be tempted to think that the dotted
part of the curve in Figure 10.10.8 is valid within the general frame-
work of our theory, but this is, unfortunately, not the case: our fun-
damental differential equations are not valid in the "overhanging"
part of the wave, simply because that part is not resting on a rigid
bottom. It may be that one could pursue the solutions beyond the
point where the breaking begins by using the appropriate differential
equations in the overhanging part of the wave and then piecing to-
gether solutions of the two sets of differential equations so that con-
tinuity is preserved, but this would be a problem of considerable
difficulty. In this connection, however, it is of interest to report the
results of a calculation by Biesel [B.10] for the change of form of
progressing waves over a beach of small slope. Not the least interesting
aspect of Biesel's results is the fact that they are based essentially on
the theory of waves of small amplitude, i.e. on the type of theory
which forms the basis for the discussions in Part II of this book.
However, in Part II only the so-called Eulerian representation was
used, in which the dependent quantities such as velocity, pressure,
LONG WAVES IN SHALLOW WATER 365
etc., are all obtained at fixed points in space. As a result, when lineari-
zations are introduced the free surface elevation 77, for example, is
a function of x and t and must of necessity be single-valued. Biesel,
however, observes that one can also use the Lagrangian representa-
tion* just about as conveniently as the Eulerian when a development
with respect to amplitude is contemplated. In this approach, all quan-
tities are fixed in terms of the initial positions of the water particles
(and the time, of course). In particular, the displacements (£, 77) of
the water particles on the free surface would be given as functions of
a parameter, i.e. £ = f (a, t), r\ = r\(a, t), and there would be no necessity
a priori to require that rj should be a single-valued function of x.
Biesel has carried out this program with the results shown in Figs.
10.10.13 to 10.10.16 inclusive. A sinusoidal progressing wave in
deep water is assumed. The first two figures refer to the theory when
carried out only to first order terms in the displacements relative to the
rest position of equilibrium. The second figure is a detail of the motion
in a neighborhood of the location shown by the dotted circle in the
first figure. Fig. 10.10.15 and Fig. 10.10.16 treat the same problem,
but the solution is carried to second order terms. In both cases the
development of a breaker is strikingly shown. A comparison of the
results of the first order and second order theories is of interest; the
main conclusions are: if second order corrections are made the break-
ing is seen to occur earlier (i.e. in deeper water), the height of the wave
at breaking is much greater, and the tendency of the wave to plunge
downward after curling over at the top is considerably lessened.
Actually, our shallow water theory cannot be expected to yield a
good approximation near the breaking point since the curvature of
the water surface is likely to be large there. However, since the motion
should be given with good accuracy at points outside the immediate
vicinity of the breaking point it might be possible to refine the treat-
ment of the breaking problem along the following lines: consider the
motion of a fixed portion of the water between a pair of planes located
some distance in front and in back of the breaking point. The motion
of the water particles outside the bounding planes can be considered as
given by our shallow water theory. We might then seek to determine
the motion of the water between these two planes by making use of a
refinement of the shallow water theory or by reverting to the original
exact formulation of the problem in terms of a potential function with
* In Chapter 12.1 this representation is explained and used to solve other
problems involving unsteady motions.
366
WATER WAVES
Fig. 10.10.13. Progression and breaking of a wave on a beach of 1 in 10 slope.
First-order theory
Fig. 10.10.14. Details of breaking of wave shown, in Fig. 10.10.18. First-order
theory
LONG WAVES IN SHALLOW WATER
367
Fig. 10.10.15. Progression and breaking of a wave on a beach of 1 in 10 slope.
Second-order theory
Fig. 10.10.16. Details of breaking of wave shown in Fig. 10.10.15. Second-order
theory
368 WATER WAVES
the nonlinear free surface condition and determine it by using finite
difference methods in a bounded region.
It is of interest now to return to the problem with which we opened
the discussion of the present section, i.e. to the problem of a tank with
a moveable end which is pushed into the water. As we have seen, the
wave which arises will eventually break. Suppose now we assume
that the end of the tank continues to move into the water with a uni-
form velocity. The end result after the initial curling over and break-
ing will be the creation of a steady progressing wave front which is
steep and turbulent, behind which the water level is constant and the
Fig. 10.10.17. The bore in the Tsicn Tang River
water has everywhere the constant velocity imparted to it by the end
of the tank. Such a steady progressing wave with a steep front is
called a bore. It is the exact analogue of a steady progressing shock
wave in a gas. In Figure 10.10.17 we show a photograph, taken from
the book by Thorade [T.4], of the bore which occurs in the Tsien-
Tang River as a result of the rising tide, which pushes the water into
a narrowing estuary at the mouth of the river. The height of this bore
apparently is as much as 20 to 30 feet. According to the theory pre-
sented above, this bore should have been preceded by an unsteady
phase during which the smooth tidal wave entering the estuary first
curled over and broke. Methods for the treatment of problems in-
volving the gradual development of a bore in an otherwise smooth
flow have been worked out by A. Lax [L.5] (see also Whitham [W.I 2] ).
We have, so far, used our basic theory— the nonlinear shallow
LONG WAVES IN SHALLOW WATER 369
water theory —to interpret the solutions of only one type of problem,
i.e. the problem of the change of form of a pulse moving into still
water of constant depth. The theory, however, can be used to study
the propagation of a wave over a beach with decreasing depth just
as well (cf. the author's paper [S.19]), but the calculations are made
much more difficult because of the fact that no family of straight
characteristics exists unless the depth is constant. This problem, in
fact, brings to the fore the difficulties of a computational nature
which occur in important problems involving the propagation of flood
waves and other surges in rivers and open channels in general. Such
problems will be discussed in the next chapter.
On an actual beach on which waves are breaking, the motion of
the water, of course, does not consist in the propagation of a single
pulse into still water, but rather in the occurrence of an approximately
periodic train of waves. However, experiments indicate that only a
slight reflection of the wave motion from the shore occurs. The in-
coming wave energy seems to be destroyed in turbulence due to break-
ing or to be converted into the energy of flow of the undertow. In
other words, each wave propagates, to a considerable degree, in a
manner unaffected by the waves which preceded it. Consequently the
above treatment of breaking, in which propagation of a wave into still
water was assumed, should be at least qualitatively reasonable. An-
other objection to our theory has already been mentioned, i.e. that
large curvatures of the water surface near the breaking point seem sure
to make the results inaccurate. Nevertheless, the theory should be
valid, except near this point, in many cases of waves on sloping bea-
ches, since the wave lengths arc usually at least 10 to 20 or more times
the depth of the water in the breaker zone, hence the theory presented
above should certainly yield correct qualitative results and perhaps
also reasonably accurate quantitative results.
Waves do not by any means always break in the manner described
up to this point. In Fig. 10.10.1 8a, b we show photographs (given to
the author by Dr. Walter Munk ) of waves breaking in a fashion con-
siderably at variance with the results of the theory presented here. We
observe that the waves break, in this instance, by curling over slightly at
the crest, but that the wave remains, as a whole, symmetrical in shape,
while the theory presented here yields a marked steepening of the wave
front and a very unsymmetrical shape for the wave at breaking.
Observation of cases like that shown in Figure 10.10.18 doubtlessly
led to the formulation of the theory of breaking due to Sverdrup and
370
WATER WAVES
(a)
(b)
Fig. l().l().18a, b. Waves breaking at crests
Munk [S.33]; their theory is based on results taken from the study of
the solitary wave, which has been discussed in the preceding section.*
The solitary wave is, by definition, a wave of finite amplitude con-
* An interesting mathematical treatment of breaking phenomena from this
point of view was given some time ago by Kculegan and Patterson K.lllj.
LONG WAVES IN SHALLOW WATER 871
sisting of a single elevation of such a shape that it can propagate un-
changed in form. At first sight, this would seem to be a rather curious
wave form to take as a basis for a discussion of the phenomena of
breaking, since it is precisely the change in form resulting in breaking
that is in question. On the other hand, the waves often look as in
Figure 10.10.18 and do retain, on the whole, a symmetrical shape,*
with some breaking at the crest. Actually, the situation regarding the
two different theories of breaking from the mathematical point of
view is the following, as we can infer from the discussion of the pre-
ceding section: Both theories are shallow water theories. In fact, as
Keller [K.6], and Friedrichs and Hyers [F.13], have shown, the theory
of the solitary wave can be obtained from the approximation of next higher
order above that used in the present section, if the assumption is made that
the motion is a steady motion. In other words, the theory used by
Sverdrup and Munk is a shallow water theory of higher order than
the theory used in this section, which furnishes in principle the con-
stant state as the only continuous wave which can propagate un-
altered in form. On the other hand, the theory presented here makes
Fig. 10.10.10. Symmetrical waves breaking at crests
it possible to deal directly with the unsteady motions, while Sverdrup
and Munk are forced to approximate these motions by a series of
different steady motions. One could perhaps sum up the whole matter
by saying that waves break in different ways depending upon the
individual circumstances (in particular, the depth of the water com-
pared with the wave length is very important), and the theory which
should be used to describe the phenomena should be chosen accord-
ingly. In fact, Figures 10.10.17 and 10.10.18 depicting a bore and
* Sverdrup and Munk, like the author, assume that, when considering breaking
phenomena, each wave in a train can be treated with reasonable accuracy as
though it were uninfluenced by the presence of the others.
872
WATER WAVES
waves breaking only at the crests of otherwise symmetrical waves
perhaps represent extremes in a whole series of cases which include
the breaker shown in Figures 10.10.2 and 10.10.3 as an intermediate
case. Some pertinent observations on this point have been made by
Mason [M.4]. A theory has been developed by Ursell [U.ll] which
differs from the theories discussed here and which may well be appro-
priate in cases not amenable to treatment by the shallow water theory.
The paper by Hamada [H.2] referred to above should also be men-
tioned again in this connection. In particular, Fig. 10.10.19, taken
from that paper, shows waves created in a tank which break by curling
at the crest but still preserving a symmetrical form. It is interesting
to observe that the wave length in this case is almost the same as the
depth of the water. It is also interesting to add that in this case a
current of air was blown over the water in the direction of travel of
the waves. Fig. 10.10.20 shows a similar case, but with somewhat
greater wave length. The two waves were both generated by a wave
Fig. 10.10.20. Breaking induced by wind action
making apparatus at the right; the only difference is that a current
of air was blown from right to left in the case shown by the lower
photograph. The breaking thus seems due entirely to wind action in
LONG WAVES IN SHALLOW WATER
878
this ease. Finally, Fig. 10.10.21 shows two stages in the breaking of a
wave in shallow water, when marked dissymmetry and the formation
of what looks like a jet at the summit of the wave are seen to occur.
It is of interest, historically and otherwise, to refer once more to the
case of symmetrical waves breaking at their crests. The wave crests
in such cases are quite sharp, as can be seen in the photograph shown
in Fig. 10.10.18. Stokes [S.28] long ago gave an argument, based on
quite reasonable assumptions, that steady progressing waves with an
angular crest of angle 120° could occur; in fact, this follows almost at
once from the Bernoulli law at the free surface when the free surface is
assumed to be a stream line with an angular point. There is another
fact pertinent to the present discussion, i.e. that the exact theory for
steady periodic progressing waves of finite amplitude, as developed
in Chapter 12.2, shows that with increasing amplitude the waves
flatten more and more in the troughs, but sharpen at the crests.
Fig. 10.10.21. Breaking of a long wave in shallow water
In fact, the terms of lowest order in the development of the free surface
amplitude 77 as given by that theory can easily be found; the result is
rf(x) = — a cos x + a2 cos 2x
for a wave of wave length 2n. Fig. 10.10.22 shows the result of super-
874
WATER WAVES
imposing the second-order term a2 cos 2x on the wave — a cos x
which would be given by the linear theory; as one sees, the effect is as
indicated. It would be a most interesting achievement to show rigor-
ously that the wave form with a sharp crest of angle 120° is attained
with increase in amplitude. An interesting approximate treatment of
the problem has been given by Davies [D.5], However, the problem
thus posed is not likely to be easy to solve; certainly the method of
Levi-Civita as developed in Ch. 12.2 does not yield such a wave form
since it is shown there that the free surface is analytic. Presumably,
Fig. 10.10.22. Sharpening of waves at the crest
any further increase in amplitude would lead to breaking at the crests
—hence no solutions of the exact problem would exist for amplitudes
greater than a certain value.
10.11. Gravity waves in the atmosphere. Simplified version of the
problem of the motion of cold and warm fronts
In practically all of this book we assume that the medium in which
waves propagate is water. It is, however, a notable fact that some
motions of the atmosphere, such as tidal oscillations due to the same
cause as the tides in the oceans, i.e. gravitational effects of the sun
and moon, as well as certain large scale disturbances in the atmosphere
such as wave disturbances in the prevailing westerlies of the middle
latitudes, and motions associated with disturbances on certain dis-
continuity surfaces called fronts, are all phenomena in which the air
can be treated as a gravitating incompressible fluid. In addition, one
of the best-founded laws in dynamic meteorology is the hydrostatic
pressure law, which states that the pressure at any point in the at-
mosphere is very accurately given by the static weight of the column
pf air above it. When we add that the types of motions enumerated
above are all such that a typical wave length is large compared with
LONG WAVES IN SHALLOW WATER 375
an average thickness (on the basis of an average density, that is) of
the layer of air over the earth, it becomes clear that these problems fall
into the general class of problems treated in the present chapter. Of
course, this means that thermodynamic effects are ignored, and
with them the ingredients which go to make up the local weather,
but it seems that these effects can be regarded with a fair approxima-
tion as small perturbations on the large scale motions in question.
There are many interesting problems, including very interesting un-
solved problems, in the theory of tidal oscillations in the atmosphere.
These problems have been treated at length in the book by Wilkes [W. 2] ;
we shall not attempt to discuss them here. The problems involved in
studying wave propagation in the prevailing westerlies will also not
be discussed here, though this interesting theory, for which papers by
Charney [C.15] and Thompson [T.10] should be consulted, is being
used as a basis for forecasting the pressure in the atmosphere by nu-
merical means. In other words, the dynamical theory is being used for
the first time in meteorology, in conjunction with modern high speed
digital computing equipment, to predict at least one of the elements
which enter into the making of weather forecasts.
In this section we discuss only one class of meteorological problems,
i.e. motions associated with frontal discontinuities, or, rather, it
would be better to say that we discuss certain problems in fluid
dynamics which are in some sense at least rough approximations to the
actual situations and from which one might hope to learn something
about the dynamics of frontal motions. The problems to be treated
here —unlike the problems of the type treated by Charney and
Thompson referred to above — are such as to fit well with the preceding
material in this chapter; it was therefore thought worthwhile to in-
clude them in this book in spite of their somewhat speculative charac-
ter from the point of view of meteorology. Actually, the idea of using
methods of the kind described in this chapter for treating certain special
types of motions in the atmosphere has been explored by a number of
meteorologists (cf. Abdullah [A.7], Freeman [F.10], Tepper [T.ll]).
One of the most characteristic features of the motion of the atmos-
phere in middle latitudes and also one which is of basic importance
in determining the weather there is the motion of wave-like disturb-
ances which propagate on a discontinuity surface between a thin
wedge-shaped layer of cold air on the ground and an overlying layer
of warmer air. In addition to a temperature discontinuity there is also
in general a discontinuity in the tangential component of the wind
376 WATER WAVES
velocity in the two layers. The study of such phenomena was initiated
long ago by Bjerknes and Solberg [B.20] and has been continued
since by many others. In considering wave motions on discontinuity
surfaces it was natural to begin by considering motions which depart
so little from some constant steady motion (in which the discontinuity
surface remains fixed in space) that linearizations can be performed,
thus bringing the problems into the realm of the classical linear
mathematical physics. Such studies have led to valuable insights,
particularly with respect to the question of stability of wave motions
in relation to the wave length of the perturbations. (The problems
being linear, the motions in general can be built up as a combination,
roughly speaking, of simple sine and cosine waves and it is the
wave length of such components that is meant here, cf. Haurwitz
[H.5, p. 234].) One conjecture is that the cyclones of the middle lati-
tudes are initiated because of the occurrence of such unstable waves
on a discontinuity surface.
A glance at a weather map, or, still better, an examination of weath-
er maps over a period of a few days, shows clearly that the wave
motions on the discontinuity surfaces (which manifest themselves as
the so-called fronts on the ground ) develop amplitudes so rapidly and
of such a magnitude that a description of the wave motions over a
period of, say, a day or two, by a linearization seems not feasible with
any accuracy. The object of the present discussion is to make a first
step in the direction of a nonlinear theory, based on the exact hydro-
dynamical equations, for the description of these motions, that can be
attacked by numerical or other methods. No claim is made that the
problem is solved here in any general sense. What is done is to start
with the general hydrodynamical equations and make a series of
assumptions regarding the flow; in this way a sequence of three non-
linear problems (we call them Problems I, II, III), each one furnishing
a consistent and complete mathematical problem, is formulated.
One can see then the effect of each additional assumption in
simplifying the mathematical problem. The first two problems result
from a series of assumptions which would probably be generally
accepted by meteorologists as reasonable, but unfortunately even
Problem II is still pretty much unmanageable from the point of view
of numerical analysis. Further, and more drastic, assumptions lead
to a still simpler Problem III which is formulated in terms of three
first order partial differential equations in three dependent and three
independent variables (as contrasted with eight differential equations
LONG WAVES IN SHALLOW WATER 377
in four independent variables in Problem I). The three differential
equations of Problem III are probably capable of yielding reasonably
accurate approximations to the frontal motions under consideration,
but they are still rather difficult to deal with, even numerically,
principally because they involve three independent variables*: such
equations are well known to be beyond the scope of even the most
modern digital computing machines as a rule. Consequently, still
further simplifying assumptions are made in order to obtain a theory
capable of yielding some concrete results through calculation.
At this point, two different approaches to the problem are proposed.
One of them, by Whitham [W.12], deals rather directly with the
three differential equations of Problem III. Two of these equations
are essentially the same as those of the nonlinear shallow water theory
treated in the preceding sections of this chapter. These two equations
—which refer to motions in vertical planes— -can therefore be inte-
grated. Afterwards the transverse component of the velocity is found
by integrating a linear first order partial differential equation. In this
way a quite reasonable qualitative description of the dynamics of
frontal motions can be achieved, at least in special cases, which is
in good agreement with many of the observed phenomena. However,
this theory has a disadvantage in that it docs not permit a complete
numerieal integration because of a peculiar difficulty at cold fronts.
(The difficulty stems from the fact that a cold front corresponds in
this theory to what amounts to the propagation of a bore down the
dry bed of a stream— a mathematical impossibility. If one had a
means of taking care of turbulence and friction at the ground, it would
perhaps be possible to overcome this difficulty.) Nevertheless, the
qualitative agreement with the observed phenomena is an indication
that the three differential equations furnishing the basic approximate
theory from which we start— i.e. those of our Problem III— have in
them the possibility of furnishing reasonable results.
The author's method (ef. [S.24]) of treating the three basic differ-
ential equations is quite different from that of Whitham, but it un-
fortunately involves a further assumption which has the effect of
limiting the applicability of the theory. The guiding principle was that
* The work of Freeman [F.9, 10] is based on a theory which could be con-
sidered as a special case of Problem III in which the Coriolis terms due to the
rotation of the earth are neglected and the motion is assumed at the outset to
depend on only one space variable and the time. The idea of deriving the theory
resulting in Problem III occurred to the author while reading Freeman's paper
and, indeed, Freeman indicates the desirability of generalizing his theory.
378
WATER WAVES
differential equations in only two independent variables should be
found, but that the number of dependent variables need not be so
ruthlessly limited. Finally, it is highly desirable to obtain differential
equations of hyperbolic type in order that the theory embodied in the
method of characteristics becomes available in formulating and solv-
ing concrete problems. These objectives can be attained by making
quite a few further simplifying assumptions with respect to the me-
chanics of the situation. The result is what might be called Problem
IV. The theory formulated in Problem IV is embodied in a system of
four nonlinear first order partial differential equations of hyperbolic
type in four dependent and two independent variables. A numerical
integration of these equations is possible, but the labor of integrating
the equations is so great that only meagre results are so far available.
Once Whitham's theory and Problem IV have been formulated,
one is led once more to consider dealing with Problem III numerically
in spite of the fact that there are three independent variables in this
case; in Problem IV, and also in the theory by Whitham, for that
matter, the basic idea is that variations in the ^/-direction are less
rapid than those in the ^-direction, and thus a finite difference scheme
in two space variables and the time might be possible under such
special circumstances.
Worm
Worm
Ground
Fig. 10.11.1. A stationary front
Cold
TT7?
We proceed to the derivation of the basic approximate theory. To
begin with, a certain steady motion (called a stationary front) is taken
as an initial state, and this consists of a uniform flow of two super-
imposed layers of cold and warm air, as indicated in Figure 10.11.1.
The s-axis is taken positive upward* and the x, t/-plane is a tangent
* Here we deviate from our standard practice of taking the t/-axis as the
vertical axis, in order to conform to the usual practice in dynamic meteorology.
This should cause no confusion, since this section can be read to a large extent
independently of the rest of the book.
LONG WAVES IN SHALLOW WATER 379
plane to the earth. The rotation of the earth is to be taken into
account but, for the sake of simplicity, not its sphericity —a common
practice in dynamic meteorology. The coordinate system is assumed
to be rotating about the js-axis with a constant angular velocity
Q = co sin (p, with co the angular velocity of the earth and <p the lati-
tude of the origin of our coordinate system. (The motivation for this is
that the main effects one cares about are found if the Coriolis terms
are included, and that neglect of the curvature of the earth has no
serious qualitative effect.) As indicated in Figure 10.11.1, the cold air
lies in a wedge under the warm air and the discontinuity surface
between the two layers is inclined at angle a to the horizontal. The
term "front" is always applied to the intersection of the discontinuity
surface with the ground, and in the present case we have therefore as
initial state a stationary front running along the iT-axis. The wind
velocity in the two layers is parallel to the #-axis (otherwise the dis-
continuity surface could not be stationary), but it will in general be
different in magnitude and perhaps even opposite in direction in the
two layers. The situation shown in Figure 10.11.1 is not uncommon.
For instance, the j?-axis might be in the eastward direction, the t/-axis
in the northward direction and the warm air would be moving in the
direction of the prevailing westerlies. The origin of the cold air at the
ground is, of course, the eold polar regions. We shall see later that
sueh configurations are dynamically eorreet and that the angle a
is uniquely determined (and quite small, of the order of £°) once
the state of the warm air and eold air is given. (The discontinuity
surface is not horizontal because of the Coriolis force arising from
the rotation of the earth.)
We proceed next to describe what is observed to happen in many
eases once sueh a stationary front starts moving. In Figure 10.11.2 a
sequence of diagrammatic sketches is given which indicate in a general
way what can happen. All of the sketches show the intersection of the
moving discontinuity surface (cf. Figure 10.11.1) with the ground (the
x, jy-plane with the j/-axis taken northward, theo?-axis taken eastward).
The shaded area indicates the region on the ground covered by cold
air, while the unshaded region is covered at the ground by warm air.
Of course, the cold air always lies in a thin wedge under a thick layer
of warm air. In Figure 10.11. 2a the development of a bulge in the
stationary front toward the north is indicated.* Such a bulge then
* What agency serves to initiate and to maintain such motions appears to
be a mystery. Naturally such an important matter has been the subject of a great
(footnote continued)
880
WATER WAVES
frequently deepens and at the same time propagates eastward with a
velocity of the order of 500 miles per day. It now becomes possible to
define the terms cold front and warm front. As indicated in Figure 10.
11.26, the cold front is that part of the whole front at which cold air is
taking the place of warm air at the ground, and the warm front is the
Cold /
Front
Warm
Front
(o)
(b)
Occluded
Front
(c) (d)
Fig. 10.11.2. Stages in the motion of a frontal disturbance
portion of the whole front where cold air is retreating with warm air
taking its place at the ground. Since such cold and warm fronts are
accompanied by winds, and by precipitation in various forms — in
fact, by all of the ingredients that go to make up what one calls
deal of discussion and speculation, but there seems to be no consistent view about
it among meteorologists. In applying the theory derived here no attempt is made
to settle this question a priori: we would simply take our dynamical model,
assume an initial condition which in effect states that a bulge of the kind just
described is initiated, and then study the subsequent motion by integrating the
differential equations subject to appropriate initial and boundary conditions.
However, if the approximate theory is really valid, such studies might perhaps
be used, or could be modified, in such a way as to throw some light on this im-
portant and vexing question.
LONG WAVES IN SHALLOW WATER 381
weather— it follows that the weather at a given locality in the middle
latitudes is largely conditioned by the passage of such frontal dis-
turbances. Cold fronts and warm fronts behave differently in many
ways. For example, the cold front in general moves faster than the
warm front and steepens relative to it, so that an originally symme-
trical disturbance or wave gradually becomes distorted in the manner
indicated in Figure 10.11.2c. This process sometimes— though by no
means always— continues until the greater portion of the cold front
has overrun the warm front; an occluded front, as indicated in Figure
10.11.2d, is then said to occur. The prime object of what follows is to
derive a theory— or perhaps better, to invent a simplified dynamical
model— capable of dealing with fluid motions of this type that is not
on the one hand so crude as to fail to yield at least roughly the observed
motions, and on the other hand is not impossibly difficult to use
for the purpose of mathematical discussion and numerical calculation.
Since it is desired that this section should be as much as possible
self contained, we do not lean on the basic theory developed earlier
in this book. Thus we begin with the classical hydrodynamical equa-
tions. The equations of motion in the Eulerian form arc taken:
du dv
dv
(10.11.1)
dw dp
} "77 == ~ a~ + 6FM ~ OS
at OZ
with d/dt (the1 particle derivative) defineel by the operator d/dt +
u dfdx + v d/dy + w d/dz. In these equations u9 v9 re are the velocity
components relative to our rotating coordinate system, p is the pres-
sure, Q the density, £>/<%) etc. the components of the Coriolis force
due to the rotation of the coordinate system, anei pg is the force of
gravity (assumed to be constant). These equations hold in both the
warm air and the cold air, but it is preferable to distinguish the eie-
pendent quantities in the two different layers; this is done here
throughout by writing u'9 v', w' for the velocity components in the
warm air and similarly for the other elependent quantities.
We n6w introduce an assumption which is commonly made in
dynamic meteorology in discussing large-scale motions of the atmos-
phere, i.e. that the air is incompressible. In spite of the fact that such
382 WATEE WAVES
an assumption rules out thermodynamic processes, it does seem rather
reasonable since the pressure gradients which operate to create the
flows of interest to us are quite small and, what is perhaps the decisive
point, the propagation speed of the disturbances to be studied is very
small compared with the speed of sound in air (i.e. with disturbances
governed by compressibility effects). It would be possible to consider
the atmosphere, though incompressible, to be of variable density.
However, for the purpose of obtaining as simple a dynamical model as
possible it seems reasonable to begin with an atmosphere having a
constant density in each of the two layers. As a consequence of these
assumptions we have the following equation of continuity:
(10.11.2) u*+vy+w, = 0.
The equations (10.11.1) and (10.11.2) together with the conditions
of continuity of the pressure and of the normal velocity components
on the discontinuity surface, the condition w — 0 at the ground,
appropriate initial conditions, etc. doubtlessly yield a mathematical
problem— call it Problem I— the solution of which would furnish a
reasonably good approximation to the observed phenomena. Unfor-
tunately, such a problem is still so difficult as to be far beyond the
scope of known methods of analysis— including analysis by numerical
computation. Thus still further simplifications arc in order.
One of the best-founded empirical laws in dynamic meteorology is
the hydrostatic pressure law, which states that the pressure at any
point in the atmosphere is very closely equal to the static weight of
the column of air vertically above it. This is equivalent to saying that
the vertical acceleration terms and the Coriolis force in the third
equation of (10.11.1) can be ignored with the result
(10.11.3) £ = - &.
This is also the basis of the long-wave or shallow water theory of
surface gravity waves, as was already mentioned above. Since the
vertical component of the acceleration of the particles is thus ignored,
it follows on purely kinematical grounds that the horizontal compo-
nents of the velocity will remain independent of the vertical coordinate
z for all time if that was the case at the initial instant t = 0. Since we
do in fact assume an initial motion with such a property, it follows
that we have
LONG WAVES IN SHALLOW WATER 883
(10.11.4) u = u(x, y, t), v = v(x, y, t), w = 0.*
The first two of the equations of motion (10.11.1) and the equation of
continuity (10.11.2) therefore reduce to
(10.11.5)
vuy = — — px + F(x)
vvy = py + F(y)
ux+vy = 0,
where we use subscripts to denote partial derivatives and subscripts
enclosed in parentheses to indicate components of a vector. The
Coriolis acceleration terms are now given by
f F(x) = 2co sin q> • v = Av
(10.11.6) '*' \
[ F(y} = — 2w sin (p • u = — Aw
when use is again made of the fact that w = 0. (The latitude angle 9?,
as was indicated earlier, is assumed to be constant.) We observe once
more that all of these relations hold in both the warm and cold layers,
and we distinguish between the two when necessary by a prime on the
symbols for quantities in the warm air. It is perhaps also worth men-
tioning that the equations (10.11.5) with F(x} and F(y) defined by
(10.11.6) arc valid for all orientations of the x, j/-axes; thus it is not
necessary to assume (as we did earlier, for example) that the original
stationary front runs in the east-west direction.
We have not so far made full use of the hydrostatic pressure law
(10.11.3). To this end it is useful to introduce the vertical height
h — /i(o% y, t) of the discontinuity surface between the warm and cold
layers and the height h' = h'(x, y,t) of the warm layer itself (see
Figure 10.11.3). Assuming that the pressure p' is zero at the top of
the warm layer we find by integrating (10.11.3):
(10.11.7) p'Or, y, z, t) = e'g(h' - z)
for the pressure at any point in the warm air. In the cold air we have,
in similar fashion:
(10.11.8) p(xt y, z, t) == q'g(h' - h) + Qg(h - z)
* It would be wrong, however, to infer that we assume the vertical displacements
to be zero. This is a peculiarity of the shallow water theory in general which
results, when a formal perturbation series is used, because of the manner in
which the independent variables are made to depend on the depth (cf. Ch. 2 and
early parts of the present chapter).
384
WATER WAVES
when the condition of continuity of pressure, p' = p for z = h, is
used. (The formula (10.11.8) is the starting point of the paper by
Freeman [F.10] which was mentioned earlier.) Insertion of (10.11.8)
h'(xty,t)
Worm
Ps P>
h(x,y,t) Cold
''•^ X
Fig. 10.11.3. Vertical height of the two layers
in (10.11.5) and of (10.11.7) in (10.11.5)' yields the following six
equations for the six quantities u9 v9 h, ur, v\ h':
(10.11.9)
(cold air)
(10.11.10)
(warm air)
u
vt
u
uu
4
u'u
uv
wv = —
0
vuy =
vv =
= 0.
gh'x
gh'y
These equations together with the kinematic conditions appropriate
at the surfaces z = h and z — h', and initial conditions at t =- 0,
would again constitute a reasonable mathematical problem— call it
Problem II— which could be used to study the dynamics of frontal
motions. The Problem II is much simpler than the Problem I formu-
lated above in that the number of dependent quantities is reduced
from eight to six and, probably still more important, the number of
independent variables is reduced from four to three. These simplifica-
tions, it should be noted, come about solely as a consequence of assum-
ing the hydrostatic pressure law, and since meteorologists have much
LONG WAVES IN SHALLOW WATER 385
evidence supporting the validity of such an assumption, the Problem
II should then furnish a reasonable basis for discussing the problem
of frontal motions. Unfortunately, Problem II is just about as in-
accessible as Problem I from the point of view of mathematical and
numerical analysis. Consequently, we make still further hypotheses
leading to a simpler theory.
As a preliminary to the formulation of Problem III we write down
the kinematic free surface conditions at z = h and z = h' (the dyna-
mical free surface conditions, p = 0 at z = h' and p = p' at z = h,
have already been used.) These conditions state simply that the
particle derivatives of the functions z — h(x, y, t) and z — h'(z, y, t)
vanish, since any particle on the surface z — h = 0 or the surface
z — h' = 0 remains on it. We have therefore the conditions
uhx + vkv + ht = 0
(10.11.11) u'hr + v'hy + ht = 0
> *x + »'*; + tit - o,
in view of the fact that w vanishes everywhere. It is convenient to
replace the third equations (the continuity equations) in the sets
(10.11.9) and (10.11.10) by
(10.11.12) (uh)x + (vh)v + ht = 0, and
(10.11.13) [u'(hf - h)]x + [v'(h' - h)]y + (h' - h)t = 0,
which are readily seen to hold because of (10.11.11). In fact, the last
two equations simply state the continuity conditions for a vertical
column of air extending (in the cold air) from the ground up to z = h,
and (in the warm air) from z = h to z = h'.
We now make a really trenchant assumption, i.e. that the motion
of the warm air layer is not affected by the motion of the cold air layer.
This assumption has a rather reasonable physical basis, as might be
argued in the following way: Imagine the stationary front to have
developed a bulge in the ^-direction, say, as in Figure 10.1 1.4a. The
warm air can adjust itself to the new condition simply through a
slight change in its vertical component, without any need for a change
in u' and v'9 the horizontal components. This is indicated in Figure
10, 11. 46, which is a vertical section of the air taken along the line
AB in Figure 10. 11.40; in this figure the cold layer is shown with a
quite small height— which is what one always assumes. Since we
ignore changes in the vertical velocity components in any case, it thus
seems reasonable to make our assumption of unaltered flow conditions
386
WATER WAVES
in the warm air. However, in the cold air one sees readily —as indicat-
ed in Figure 10.11.4c— that quite large changes in the components
u, v of the velocity in the cold air may be needed when a frontal dis-
turbance is created. Thus we assume from now on that u', v', h' have
for all time the known values they had in the initial steady state in
//// B
7/m/
(o)
Fig. 10.11.4. Flows in warm and cold air layers
which v' — 0, u' — const. The differential equations for our Problem
III can now be written as follows:
(10.11.14)
ut + uux
= - g f— h'x + il - ^-\ hA +
= - g P-A; + (i - ^~\ AJ -
(uk)x + (vh)v = 0,
LONG WAVES IN SHALLOW WATER
387
in which h'x and h'y are known functions given in terms of the initial
state in the warm air. The initial state, in which v' = v = 0, u =
const., u1 = const., must satisfy the equations (10. 11. 9) and (10.11.10);
this leads at once to the conditions
*;-
A
g1
^
g'
*(e' , \
-I — u — U I
g\0 /
(10.11.15)
for the slopes of the free surfaces initially. The slope hy of the dis-
continuity surface between the two layers is nearly proportional to
the velocity difference u' — u since g'/p differs only slightly from
unity, and it is made quite small under the conditions normally en-
countered because of the factor A, which is a fraction of the angular
velocity of the earth. The relation for the slope hy of the stationary
discontinuity surface is an expression of the law of Margulcs in meteor-
ology. The differential equations for Problem III can, finally, be ex-
pressed in the form:
(10.11.16)
Problem III
wt + t<ux + ruv + g J 1 - — |
vt + uvf + vvv + (> ( 1 l/t,
/?, -f (U/t), + (!'/*)„ = 0.
by using the formulas for h'x and h'v given in (10.11.15). We note that
the influence of the warm air expresses itself through its density g'
and its velocity u'. The three equations (10.11.16) undoubtedly have
uniquely determined solutions once the values of u, v, and h are given
at the initial instant t =- 0, together with appropriate boundary con-
ditions if the domain in #, y is not the whole space, and such solutions
might reasonably be expected to furnish an approximate descrip-
tion of the dynamics of frontal motions.* Unfortunately, these equa-
tions are still quite complicated. They could be integrated numerically
* These equations are in fact quite similar to the equations for two-dimen-
sional unsteady motion of a compressible fluid with h playing the role of the
density of the fluid.
388 WATER WAVES
only with great difficulty even with the aid of the most modern high-
speed digital computers— -mostly because there are three independent
variables.
Consequently, one casts about for still other possibilities, either of
specialization or simplification, which might yield a manageable
theory. One possibility of specialization has already been mentioned:
if one assumes no Coriolis force and also assumes that the motion is
independent of the ^-coordinate, one obtains the pair of equations
ut + uux + g l - — \hx =
(10.11.17)
(Hfc). = 0
which are identical with the equations of the one-dimensional shallow
water gravity wave theory. These equations contain in them the
possibility of the development of discontinuous motions— called
bores in sec. 10.7— and this fact lies at the basis of the discussions by
Freeman [F.10] and Abdullah [A.7]. In such one-dimensional treat-
ments, it is clear that it is in principle not possible to deal with the
bulges on fronts and their deformation in time and space, since such
problems depend essentially on both space variables x and y. Another
possibility would be a linearization of the differential equations
(10.11.16) based on assuming small perturbations of the frontal sur-
face and of the velocities from the initial uniform state. This procedure
might be of some interest, since such a formulation would take care of
the boundary condition at the ground, while the existing linear treat-
ments of this problem do not. However, our interest here is in a non-
linear treatment which permits of large displacements of the fronts.
One such possibility, devised by Whitham [W.12], involves essentially
the integration of the first and third equations for u and h as functions
of x and t, regarding y as a parameter, and derivatives with respect to
y as negligible compared with derivatives with respect to x, and assum-
ing initial values for v; this is feasible by the method of characteristics.
Afterwards, v would be determined by integrating the second equa-
tion considering u and h as known, and this can in principle be done
because the equation is a linear first order equation under these con-
ditions. As stated earlier, this procedure furnishes qualitative results
which agree with observations. In addition, the discussion can be
carried through explicitly in certain cases, by making use of solutions
of the type called simple waves, along exactly the same lines as in
sec. 10.8 above. We turn, therefore, to this first of two proposed
LONG WAVES IN SHALLOW WATER 389
approximate treatments of Problem III, as embodied in equations
(10.11.16).
The basic fact from which Whitham starts is that the slope a = hy
of the discontinuity surface is small initially, as we have already seen
in connection with the second equation of (10.11.15), and the fact that
A is a fraction of the earth's angular velocity, and is expected to remain
in general small throughout the motions considered. Since the Coriolis
forces are of order a also (since they are proportional to A) it seems
clear that derivatives of all quantities with respect to y will be small of
a different order from those with respect to x\ it is assumed therefore
that uy9 hy and vy are all small of order a, but that ux and hx are finite.
Furthermore we can expect that the main motion will continue to be
a motion in the ^-direction, so that the i/-component v of the velocity
will be small of order a while the ^-component u remains of course
finite. Under these circumstances, the equations (10.11.16) can be
replaced by simpler equations through neglect of all but the lowest
order terms in a in each equation; the result is the set of equations
ht + uhx + hux = 0
« + uux + khx = 0
IQ' \
vt + uvx = — khy + A ( — u' — u\
with the constant k defined by
(10.11.19) * =
A considerable simplification has been achieved by this process, since
the variable y enters into the first two equations of (10.11.18) only as
a parameter and these two equations are identical with the equations
of the shallow water theory developed in the preceding sections of this
chapter if k is identified with g and h with 77. This means that the
theory developed for these equations now becomes available to dis-
cuss our meteorological problems. Of course, the solutions for h and
u will depend on the variable y through the agency of initial and
boundary conditions. Once u(x9 y, t) and h(x, y> t) have been obtained,
they can be inserted in the third equation of (10.11.18), which then
is a first order linear partial differential equation which, in principle
at least, can be integrated to obtain v when arbitrary initial conditions
v = v(x9 y, 0) are prescribed. The procedure contemplated can thus be
summed up as follows: the motion is to be studied first in each vertical
390 WATER WAVES
plane y = constant by the same methods as in the shallow water
theory for two-dimensional motions (which means gas dynamics
methods for one-dimensional unsteady motions), to be followed by
a determination of the "cross-component" v of the velocity through
integration of a first order linear equation which also contains the
variable y, but only as a parameter.
This is in principle a feasible program, but it presents problems too
complicated to be solved in general without using numerical com-
putations. On the other hand we know from the earlier parts of this
chapter that interesting special solutions of the first two equations of
(10.11.18) exist in the form of what were called simple waves, and
these solutions lend themselves to an easy discussion of a variety of
motions in an explicit way through the use of the characteristic form
of the equations. In order to preserve the continuity of the discussion
it is necessary to repeat here some of the facts about the characteristic
theory and the theory of simple waves; for details, sees. 10.2 and 10.3
should be consulted.
By introducing the new function c2 — kh, replacing h, we obtain
instead of the first two equations in (10.11.18) the following equations:
( 2ct + 2ucx + cux == 0
(10.11.20)
I ut + uux + 2ccx == 0.
Thus the quantity c = Vkh, which has the dimensions of a velocity,
is the propagation speed of small disturbances, or wavelets — in ana-
logy with the facts derived in sec. 10.2. These equations can in turn
be written in the form
which can be interpreted to mean that the quantities u± 2c are con-
stant along curves C± in the #, 2-plane such that dx/dt = u ± c:
„ dx
u -\- 2c = const, along C+: — = u + c
(10.11.21) <
ax
u — 2c = const, along C_: — = u — c.
These relations hold in general for any solutions of (10.11.20). Under
special circumstances it may happen that u — • 2c9 for example, has the
same constant value on all C_ characteristics in a certain region; in
LONG WAVES IN SHALLOW WATER 391
that case since u + 2c is constant along each C+ characteristic it
follows that u and c would separately by constant along each of the
C+ characteristics, which means that these curves would all be straight
lines. Such a region of the flow (the term region here being applied
with respect to some portion of an «r, £-plane) is called a simple wave.
It is then a very important general fact that any flow region adjacent
to a region in which the flow is uniform, i.e. in which both c and u are
everywhere constant (in both space and time, that is), is a simple
wave, provided that u and c are continuous in the region in question.
It is reasonable to suppose that simple waves would occur in cases
of interest to us in our study of the dynamics of frontal motions,
simply because we do actually begin with a flow in which u and h
(hence also c) are constant in space and time, and it seems reasonable
to suppose that disturbances are initiated, not everywhere in the flow
region, but only in certain areas. In other words, flows adjacent to
uniform flows would occur in the nature of things. Just how in detail
initial or boundary conditions, or both, should be prescribed in order
to conform with what actually occurs in nature is, as has already been
pointed out, something of a mystery; in fact one of the principal
objects of the ideas presented here could be to make a comparison of
calculated motions under prescribed initial and boundary conditions
with observed motions in the hope of learning something by inference
concerning the causes for the initiation and development of frontal
disturbances as seen in nature.
One fairly obvious and rather reasonable assumption to begin with
might be that u> v, and h are prescribed at the time t = 0 to have
values over a certain bounded region of the upper half (y > 0) of
the x, y-plane (cf. Fig. 10.11.1 ) in such a fashion that they differ from
the constant values in the original uniform flow with a stationary
front. According to the approximate theory based on equations (10.11.
18), this means, in particular, that in each vertical plane y = t/0— an
x, J-plane — initial conditions for u(x,y09t) and h(x,y0,t) would be
prescribed over the entire #-axis, but in such a way that u and h are
constant with values u = UQ > 0, h = A0 ^ 0 (hence c = c0 =
Vkh0)* everywhere except over a certain segment xl ^ x ^ #2, as
indicated in Fig. 10.11.5. The positive characteristics C+ are drawn in
full lines, the characteristics C_ with dashed lines in this diagram,
* It should, however, always be kept in mind in the discussion to follow
that c0, particularly, will usually have different values in different vertical planes
y = const.
892
WATER WAVES
which is to be interpreted as follows. Simple waves exist everywhere
in the x, J-plane except in the triangular region bounded by the C+
characteristic through A and the C_ characteristic through B and
terminating at point C; in this region the flow could be determined
A B
Fig. 10.11.5. Simple waves arising from initial conditions
numerically, for example by the method indicated in sec. 10.2 above
(in connection with Fig. 10.2.1). The disturbance created over the
segment AB propagates both "upstream" and "downstream" after a
certain time in the form of two simple waves, which cover the regions
bounded by the straight (and parallel) characteristics issuing from
A, B9 and C. In other words the disturbance eventually results in two
distinct simple waves, one propagating upstream, the other down-
stream, and separated by a uniform flow identical with the initial
state. In our diagram it is tacitly assumed that c > \ u |, i.e. that the
flow is subcritical in the terminology of water waves (subsonic in
gas dynamics)— otherwise no propagation upstream could occur. We
have supposed u to be positive, i.e. that the ^-component of the flow
velocity in the cold air layer has the same direction as the velocity in
the warm air, which in general flows from the west, but it can be (and
not infrequently is) in the westward rather than the eastward direc-
tion. Since the observed fronts seem to move almost invariably to the
eastward, it follows, for example, that it would be the wave moving
upstream which would be important in the case of a wind to the west-
ward in the cold layer, and a model of the type considered here — in
which the disturbance is prescribed by means of an initial condition
LONG WAVES IN SHALLOW WATER 393
and the flow is subcritical— implies that the initial disturbances are
always of such a special character that the downstream wave has a
negligible amplitude. For a wind to the eastward, the reverse would
be the case. All of this is, naturally, of an extremely hypothetical
character, but nevertheless one sees that certain important elements
pertinent to a discussion of possible motions are put in evidence.
The last remarks indicate that a model based on such an initial
disturbance may not be the most appropriate in the majority of cases.
In fact, such a formulation of the problem is open to an objection
which is probably rather serious. The objection is that such a motion
has its origin in an initial impulse, and this provides no mechanism by
which energy could be constantly fed into the system to "drive" the
wave. Of course, it would be possible to introduce external body forces
in various ways to achieve such a purpose, but it is not easy to see how
to do that in a rational way from the point of view of mechanics.
Another way to introduce energy into the system would be to feed it
in through a boundary —in other words formulate appropriate bound-
ary conditions as well as initial conditions. For the case of fronts
moving eastward across the United States, a boundary condition
might be reasonably applied at some point to the east of the high
mountain system bordering the west coast of the continent, since
these mountain ranges form a rather effective north-south barrier
between the motions at the ground on its two sides. In fact, a cold
front is not infrequently seen running nearly parallel to the moun-
tains and to the cast of them— as though cold air had been deflected
southward at this barrier. Hence a boundary condition applied at
some point on the west seems not entirely without reason. In any case,
we seek models from which knowledge about the dynamics of fronts
might be obtained, and a model making use of boundary conditions
should be studied. We suppose, therefore, that a boundary condition
is applied at x = 0, and that the initial condition for t — 0, x > 0 is
that the flow is undisturbed, i.e. u = UQ = const., c = c0 = const..
(Again we remark that we are considering the motion in a definite
vertical plane y = yQ. ) In this case we would have only a wave propa-
gating eastward— in effect, we replace the influence of the air to the
westward by an assumed boundary condition. The general situation
is indicated in Fig. 10.11.6. There is again a simple wave in the region
of the #, $-plane above the straight line x — (UQ + cQ)t which marks
the boundary between the undisturbed flow and the wave arising
from disturbances created at x = 0. This is exactly the situation which
394
WATER WAVES
is treated at length in sec. 10.3; in particular, an explicit solution of
the problem is easily obtained (cf. the discussion in sec. 10.4) for
arbitrarily prescribed disturbances in the values of either of the two
quantities u or c. Through various choices of boundary conditions it
is possible to supply energy to the system in a variety of ways.
Fig. 10.11.6. Wave arising from conditions applied at a boundary
We proceed next to discuss qualitatively a few consequences which
result if it is assumed that frontal disturbances can be described in
terms of simple waves in all vertical planes y = yQ = const, at least
over some ranges in the values of the ^-coordinate. (We shall see later
that simple waves are not possible for all values of y. ) In this discussion
we do not specify how the simple wave was originated — we simply
assume it to exist. Since we consider only waves moving eastward
(i.e. in the positive ^-direction) it follows that the straight character-
istics are C+ characteristics, and hence that u — 2c is constant (in
each plane y =• const.) throughout the wave; we have therefore
(10.11.22) u - 2c = A(y),
with A(y) fixed by the values UQ and c0(y) in the undisturbed flow:
(10.11.22^ A(y) = UQ - 2c0(y).
In addition, as explained before, we know that u + 2c is a function of
y alone on each positive characteristic dx/dt = u + c; hence u and c
are individually functions of y on each of these characteristics. There-
fore, the characteristic equation may be integrated to yield
(10.11.23) x == £ + (u + c)t,
where | is the value of x at t = 0. (The time t = 0 should be thought
of as corresponding to an arbitrary instant at which simple waves
exist in certain planes y = const. ) Now, the values of u and c on the
LONG WAVES IN SHALLOW WATER 395
characteristic given by (10.11.23) are exactly the same as the values
(for the same value of y) at the point t = 0, x = f ; therefore, if we
suppose, for example, that c is a given function C(#, y) at t = 0, the
value of c in (10.11.23) is C(f, y) and the value of u is, from (10.11.22),
A(y) + 2C(f, t/). Thus the simple wave solution can be described by
the equations
( c = C(£, y),
(10.11.24) • u = A(y) +2C(f,y),
, a? = £ + {^(y) + 8C(f , y)}t.
(Although the arbitrary function occurring in a simple wave could be
specified in other ways, it is convenient for our purposes to give the
value of h, and hence c, at t = 0.)
We could write down the solution for the "cross component", or
north-south component, v of the velocity in this case; by standard
methods (cf. the report of Whitham [W.12]) it can be obtained by
integrating the linear first order partial differential equation which
occurs third in the basic equations (10.11.18). To specify the solution
of this equation uniquely an initial condition is needed; this might
reasonably be furnished by the values v = v(x, y) at the time t = 0.
The result is a rather complicated expression from which not much
can be said in a general way. One of the weaknesses of the present
attack on our problem through the use of simple waves now becomes
apparent: it is necessary to know values of v some time subsequent to
the initiation of a disturbance in order to predict them for the future.
It is possible, however, to draw some interesting conclusions from
the simple wave motions without considering the north-south com-
ponent of the velocity. For example, suppose we consider a motion
after a bulge to the northward in an initially stationary front had
developed as indicated schematically in Fig. 10.11.2. In a plane y =
const, somewhat to the north of the bulge we could expect the top of
the cold air layer (the discontinuity surface, that is) as given by h(x, t)
to appear, for t = 0 say, as indicated in Fig. 10.11.7. The main fea-
tures of the graph arc that there is a depression in the discontinuity
surface, but that h > 0 so that this surface does not touch the ground.
(The latter possibility will be discussed later.) Assuming that the
motion is described as a simple wave, we see from (10.11.24) that the
value of c = Vkh at the point x = xl is equal to the value of c which
was at the point x = ^ at t = 0, where f 1 — xl — (A(y) + 3C(f1,t/)}/1.
That is, the value c = C(flf y) has been displaced to the right by an
896
WATER WAVES
amount (A(y) + BC(^V y)}tr Since this quantity is greater for greater
values of C, the graph of h becomes distorted in the manner shown in
Fig. 10.11.7: the "negative region" (where hx < 0) steepens whilst the
"positive region" (where hx > 0) flattens out. The positive region
continues to smooth out, but, if the steepening of the negative region
Fig. 10.11.7. Deformation of the discontinuity surface
were to continue indefinitely, there would ultimately be more than
one value of C at the same point, and the wave, as in our discussion
of water waves (cf. sec. 10.6 and 10.7), starts to break. Clearly the
latter event occurs when the tangent at a point of the curve in
Fig. 10.11.7 first becomes vertical. At this time, the continuous solu-
tion breaks down (since c and u would cease to be single-valued
functions) and a discontinuous jump in height and velocity must be
permitted. In terms of the description of the wave by means of the
characteristics, what happens is that the straight line characteristics
converge and eventually form a region with a fold. Such a disconti-
nuous "bore" propagates faster than the wavelets ahead of it (the
paths of the wavelets in the x, /-plane are the characteristics) in a
manner analogous to the propagation of shock waves in gas dynamics
and bores in water.
In the above paragraph we supposed that h and c were different
from zero, and hence the discussion does not apply to the fronts, which
LONG WAVES IN SHALLOW WATER 397
are by definition the intersection of the discontinuity surface with
the ground. When c = 0 there are difficulties, especially at cold
fronts, but nevertheless a few pertinent observations can be made,
assuming the motion to be a simple compression wave with u — 2c
constant. When c = 0, it follows that u = UQ — 2c0, and since c0 =
VkhQ and hQ = at/0 with a the initial inclination of the top of the cold
air layer, it follows that u = UQ — 2\/<x.ky in this case. But u then
measures the speed of the front itself in the ^-direction, since a particle
once on the front stays there; consequently for the speed uf of the
front we have
(10.11.25) uf = UQ — 2\/o%.
Thus the speed of the front decreases with y, and on this basis it
follows that a northward bulge would become distorted in the fashion
indicated by Fig. 10.11.8, and this coincides qualitatively with obser-
vations of actual fronts.
Actually, things are not quite as simple as this. If c = 0, it follows
from the first equation of (10.11.20) that ct + ucx = 0 on such a
locus, and this in turn means that c = 0 on the particle path defined
by dx/dt ~- u. At the same time the C+ and C_ characteristics have
the same direction, since they are given by dx/dt — u ± c. On the
other hand, we have, again from (10.11.20), ut + uux — — khx and
we see that the relation u = const, along a characteristic for which
c — 0 cannot be satisfied unless hx = 0. In connection with Fig.
10.11.7 we have seen that the rising portion toward the east of a de-
pression in the discontinuity surface tends to flatten out, while the
falling part from the west tends to steepen and break because the
higher portions tend to move faster and crowd the lower portions.
Thus when /?, and hence r, tends to zero the tendency will be for
breaking to occur at the cold front, but not at the warm front. The
slope of the discontinuity surface at the cold front will then be infinite.
However, a bore in the sense described above cannot occur since there
must always be a mass flux through a bore: the motion of the cold
front is analogous to what would happen if a dam were broken and
water rushed down the dry bed of a stream. Without considering in
some special way what happens in the turbulent motion caused by such
continuous breaking at the ground, it is not possible to continue our
discussion of the motion of a cold front along the present lines, al-
though such a problem is susceptible to an approximate treatment.
Nevertheless, this discussion has led in a rational way to a qualitative
398
WATER WAVES
explanation for the well-known fact that a warm front does indeed
progress in a relatively smooth fashion as compared with the turbu-
lence which is commonly observed at cold fronts. Thus near a cold
front the height of the cold air layer may be considerably greater than
in the vicinity of the warm front, where h ~ 0; consequently the speed
of propagation of the cold front could be expected to be greater than
Cold
Warm
t = 0
Warm
t= t, > 0
Cold
Warm
tst2>t,
Fig. 10.11.8. Deformation of a moving front
near the warm front (as indicated by the dotted modifications of the
shape of the cold front in Fig. 10.11.8), with the consequence that the
gap between the two tends to close, and this hints at a possible ex-
planation for the occlusion process. One might also look at the matter
in this way: Suppose c ^ 0, but is small in the trough of the wave
shown in Fig. 10.11.7. If breaking once begins, it is well known that
LONG WAVES IN SHALLOW WATER 399
the resulting bore moves with a speed that is greater than the propa-
gation speed of wavelets in the medium in front (to the right) of it.
although slower than the propagation speed in the medium behind it.
Again one sees that the tendency for the wave on the cold front side
to catch up with the wave on the warm front side is to be expected on
the basis of the theory presented here.
Finally we observe that the velocity of the wave near the undisturbed
stationary front is UQ, but well to the north it is given roughly by
uf — UQ ~~ % vaky, which is less than UQ. There is thus a tendency to
produce what is called in meteorology a cyclonic rotation around the
center of the wave disturbance.
To sum up, it seems fair to say that the approximate theory embo-
died in equations (10.11.18), even when applied to a very special type
of motions (i.e. simple waves in each plane y =. const.), yields a
variety of results which are at least qualitatively in accord with
observations of actual fronts in the atmosphere. Among the pheno-
mena given correctly in a qualitative way are: the change in shape of
a wave as it progresses eastward, the occurrence of a smooth wave at
a warm front but a turbulent wave at a cold front, and a tendency to
produce the type of motion called a cyclone.
It therefore seems reasonable to suppose that the differential
equations of our Problem III, which were the starting point of the
discussion just concluded, contain in them the possibility of dealing
with motions which have the general characteristics of frontal motions
in the atmosphere, and that numerical solutions of the equations of
Problem III might well furnish valuable insights. This is a difficult
task, as has already been mentioned. However, an approximate theory
different from that of Whitham is possible, which has the advantage
that no especial difficulty arises at cold fronts, and which would per-
mit a numerical treatment. This approximate theory might be con-
sidered as a new Problem IV.
The formulation of Problem IV was motivated by the following
considerations. If one looks at a sequence of weather maps and
thinks of the wave motion in our thin wedge of cold air, the re-
semblance to the motion of waves in water which deform into brea-
kers (especially in the case of frontal disturbances which develop into
occluded fronts) is very strong. The great difference is that the wave
motion in water takes place in the vertical plane while the wave mo-
tion in our thin layer of cold air takes place essentially in the horizon-
tal plane. When the hydrostatic pressure assumption is made in the
400 WATER WAVES
case of water waves the result is a theory in exact analogy to gas
dynamics, and thus wave motions with an appropriate "sound speed"
become possible even though the fluid is incompressible— the free
surface permits the introduction of the depth of the water as a de-
pendent quantity, this quantity plays the role of the density in gas
dynamics, and thus a dynamical model in the form of a compressible
fluid is obtained. It would seem therefore reasonable to try to invent
a similar theory for frontal motions in the form of a long- wave theory
suitable for waves which move essentially in the horizontal, rather
than the vertical, plane, and in which the waves propagate essentially
parallel to the edge of the original stationary front, i.e. the #-axis. In
this way one might hope to be rid of the dependence on the variable y
at right angles to the stationary front, thus reducing the independent
variables to two, x and t; and if one still could obtain a hyperbolic
system of differential equations then numerical treatments by finite
differences would be feasible. This program can, in fact, be carried out
in such a way as to yield a system of four first order nonlinear differ-
ential equations in two independent and four dependent variables
which are of the hyperbolic type.
Once having decided to obtain a long-wave theory for the horizontal
plane, the procedure to be followed can be inferred to a large extent
from what one does in developing the same type of theory for gravity
waves in water, as we have seen in Chapter 2 and at the beginning of
the present chapter. To begin with it seems clear that the displace-
ment r)(x,t) of the front itself in the ^-direction should be introduced
as one of the dependent quantities— all the more since this quantity
is anyway the most obvious one on the weather maps. To have such a
"shallow water" theory in the horizontal plane requires— unfortuna-
tely—a rigid "bottom" somewhere (which is, of course, vertical in
this case), and this we simply postulate, i.e. we assume that the
t/-component v of the velocity vanishes for all time on a vertical plane
y = d = const, parallel to the stationary front along the #-axis (see
Figure 10.11.9). The velocity v(x, j/, t) is then assumed to vary
linearly* in y, and its value at the front, y = r\(x, t), is called v (x, t).
The intersection of the discontinuity surface z = h(x9 1/, t) with the
plane y = d is a curve given by z = Ti(x, t), and we assume that the
discontinuity surface is a ruled surface having straight line generators
running from the front, y = rj(x9 1), to the curve z — Ti(x, t), and
* The analogous statement holds also in the long-wave theory in water (to
lowest order in the development parameter, that is).
LONG WAVES IN SHALLOW WATER
401
parallel to the t/, 2-plane. Finally, we assume (as in the shallow water
theory) that u9 the ^-component of the velocity, depends on x and /
a z
Fig. 10.11.9. Notations for Problem IV
only: u =• u(x, t). The effect of these assumptions is to yield the re-
lations
^ y -*?(*»*)
(10.11.26)
(10.11.27)
V(JT, y, t) =
d — YI(X, t)
d-y
h(x, t),
d-f,(x,t) ~v~'""
as one readily sees. In addition, we assume that a particle that is once
on the front y - r)(x, t) = 0 always remains on it, so that the relation:
(10.11.28) v(x, t) = r\t + ur\x
must hold. The four quantities u(x9 /), rj(x, t), h(x, t), and v(x, t) are
our new dependent variables. Differential equations for them will be
obtained by integrating the basic equations (10.11.16) of Problem III
with respect to y from y = rj to y = d —which can be done since the
dependence of w, u, and h on y is now explicitly given— and these three
equations together with (10.11.28) will yield the four equations we
want.
Before writing these equations down it should be said that the
most trenchant assumption made here is the assumption concerning
the existence of the rigid boundary y = d. One might think that as
402 WATER WAVES
long as the velocity component v dies out with sufficient rapidity in
the ^-direction such an assumption would yield a good approximation,
but the facts in the case of water waves indicate this to be not suffi-
cient for the accuracy of the approximation: with water waves in very
deep water the vertical component of the velocity (corresponding to
our v here) dies out very rapidly in the depth, but it is nevertheless
essential for a good approximation to assume that the ratio of the
depth down to a rigid bottom to the wave length is small. However,
such a rigid vertical barrier to the winds does exist in some cases of
interest to us in the form of mountain ranges, which are often much
higher than the top of the cold surface layer (i.e. higher than the
curve z = h(x, t) in Figure 10.11.9). In any case, severe though this
restriction is, it still seems to the author to be worth while to study
the motions which are compatible with it since something about the
dynamics of frontal motions with large deformations may be learned
in the process. In particular, one might learn something about the
kind of perturbations that are necessary to initiate motions of the
type observed, and under what circumstances the motions can be
maintained.
In carrying out the derivation of the differential equations of our
theory according to the plan outlined above, we calculate first a
number of integrals. The first of these arise from (10.11.26) and
(10.11.27):
r6 h r* i _
hdy = - - (y ~
J n o —rj J ^
From these we derive by differentiations with respect to x and t an-
other set of relations:
J
C8 1
J vxdy = -vx(d
= oM<5 -n) -*%*>
n i *
6 i i_
2
1
05??«'
r* i i_
J r\ " "
LONG WAVES IN SHALLOW WATER
403
(In deriving these relations, it is necessary to observe that the lower
limit T] is a function of x and t.) One additional relation is needed, as
follows:
r*
I
(hu)xdy ^ —
1 \ I - 1 _
h dy | = - (hxu + hux)(6 -»?)-- hur}x.
T) vyu \ %} ri 1 & ~
We now integrate both sides of the equations (10.11.16) with re-
spect to y from 77 to d, make use of the above integrals, note that
u = u(x, t) is independent of j/, and divide by d — TJ. The result is the
equations
1 _ 1 kh 1
*t + MM, + - khx - - i -nx - -Av,
(10.11.29)
.-
<5— 77
jo _ v2 2kh ^ I Q' \
-
uhx
u
h, - -
= 0.
with k a constant replacing the quantity g(l — Q'/'Q). These equations,
together with (10.11.28), form a system of four partial differential
equations for the four functions u, 77. £, and 7L By analogy with gas
dynamics and the nonlinear shallow water theory, it is convenient to
introduce a new dependent quantity c (which will turn out to be the
propagation speed of wavelets) through the relation
(10.11.30)
--\h.
The quantity c is real if Q' is less than Q, and this holds since the warm
air is lighter than the cold air. In terms of this new quantity the
equations (10.11.28) and (10.11.29) take the form
ut + uux + 2rr, — TJX = •• - Ai",
(10.11.31)
vt + uvx = —
6 —
/ Q' \
2A 1 u — u \ ,
cv
cux + 2ucx =
d — r)
rjt + ur/x = i".
It is now easy to write the equations (10.11.31) in the characteristic
404 WATER WAVES
form simply by replacing the first and third equations by their sum
and by their difference. The result is:
(10 11 32)
4c2 / o' \
x = -- -- 2A [u - — u'\,
o — ri \ Q J
t + urjx = v.
As one sees, the equations are in characteristic form: the characteristic
curves satisfy the differential equations
dx dx dx
(10.11.33) — = u +c, — = u — c, — = u,
dt dt dt
and each of the equations (10.11.32) contains only derivatives in the
direction of one of these curves. The characteristic curves defined by
dxjdt — u are taken twice. Thus one sees that the quantity c is indeed
entitled to be called a propagation speed, and small disturbances can
be expected to propagate with this speed in both directions relative to
the stream of velocity u. (In the theory by Whitham, in which the
motion in each vertical plane y = const, is treated separately, the
propagation or sound speed of small disturbances is given by Vkh.
The sound speed in the theory given here thus represents a kind of
average with respect to y of the sound speeds of Whitham's theory. )
Since the propagation speed depends on the height of the disconti-
nuity surface, it is clear that the possibility of motions leading to
breaking is inherent in this theory.
Once the dynamical equations have been formulated in character-
istic form it becomes possible to see rather easily what sort of subsi-
diary initial and boundary conditions are reasonable. In fact, there
are many possibilities in this respect. One such possibility is the
following. At time t — 0 it is assumed that u = const., r) = 0, h =
const, (as in a stationary front), but that rjt = f(x) over a segment of
the «r-axis. In other words, it is assumed that a transverse impulse is
given to the stationary front over a portion of its length. The sub-
'sequent motion is uniquely determined and can be calculated nume-
rically. Another possibility is to prescribe a stationary front at t = 0
LONG WAVES IN SHALLOW WATER 405
for x > 0, say, and then to give the values of all dependent quantities*
at x = 0 as arbitrary functions of the time; i.e. to prescribe a boundary
condition which allows energy to be introduced gradually into the
system. One might visualize this case as one in which, for example,
cold air is being added or withdrawn at a particular point (x = 0 in
the present case). This again yields a problem with a uniquely deter-
mined solution, and various possibilities are being explored numeri-
cally.
It was stated above that the most objectionable feature of the
present theory is the assumption of a fixed vertical barrier back of the
front. There is, however, a different way of looking at the problem as
a whole which may mitigate this restriction somewhat. One might try
to consider the motion of the entire cap of cold air that lies over the
polar region, using polar coordinates (0, y) (with 0 the latitude angle,
say). One might then consider motions once more that depend
essentially only on q> and t by getting rid of the dependence on 6
through use of the same type of assumptions (linear behavior in 0,
say) as above. Here the North Pole itself would take the place of the
vertical barrier (v — 0!). The result is again a system of nonlinear
equations— this time with variable coefficients. Of course, it would
be necessary to begin with a stationary flow in which the motion takes
place along the parallels of latitude.
All in all, the ideas presented here would seem to yield theories
flexible enough to permit a good deal of freedom with regard to initial
and other conditions, so that one might hope to gain some insight into
the complicated dynamics of frontal motions by carrying out numeri-
cal solutions in well-chosen special cases.
10.12. Supercritical steady flows in two dimensions. Flow around bends.
Aerodynamic applications
The title of this section is a slight misnomer, since the flows in
question are really three-dimensional in nature; however, since we
consider them here only in terms of the shallow water theory, the
depth dimension is left out. Thus the velocity is characterized by the
two components (u, w) in the horizontal plane (the x, 2-plane); and
these quantities, together with the depth h of the water at any point
constitute the quantities to be determined in any given problem. By
* In the numerical cases so far considered we have had | c \ < | u \ so that
even on the /-axis all four dependent quantities can be prescribed.
406 WATER WAVES
specializing the general equations (2.4.18), (2.4.19), (2.4.20), of the
shallow water theory as derived in Chapter 2 for the case of a steady
flow, the differential equations relevant for this section result. They
can also be derived readily from first principles, as follows: Assuming
that the hydrostatic pressure law holds and that the fluid starts from
rest (or any other motion in which the vertical component of the
velocity of the water is zero) it follows that the vertical component of
the velocity remains zero and that u and w are independent of the ver-
tical coordinate. The law of continuity can thus be readily derived for
a vertical column; for a steady flow it is
(10.12.1) (hu)x + (hw)z = 0.
We assume that the flows we study are irrotational, and hence that
(10.12.2) uz — wx = 0.
The Bernoulli law then holds and can be written in the form
(10.12.3) (u2 + w2) + 2gh = const.
In these equations h is the depth of the water at any point. By using
(10.12.3) to express h in terms of u and w, and introducing the quan-
tity c by the relation
(10.12.4) c2 = gh
we obtain the equation
(10.12.5) (c2 - u2)ux — uw(wx + uz) + (c2 — w2)wz =- 0,
and this equation together with (10.12.2), with c defined in terms of
u and w through (10.12.4) and (10.12.3), constitute a pair of first
order partial differential equations for the determination of u(x, z)
and w(x, z).
The theory of these latter equations can be developed, as in the
cases treated previously in this chapter, by using the method of
characteristics, provided that the quantity c remains always less than
the flow speed everywhere, i.e. provided that
(10.12.6) c2 < u2 + w2.
The flow is then said to be supercritical. (In hydraulics the contrast
subcritical— supercritical is commonly expressed as tranquil-shoot-
ing.) Only then do real characteristics exist. We shall not develop
this theory here, but rather indicate some of the problems which have
been treated by using the theory. Complete expositions of the char-
LONG WAVES IN SHALLOW WATER
407
acteristic theory can be found in the paper by Preiswerk [P.16], and
in Chapter IV of the book by Courant and Friedrichs [C.9]. The theory
is, of course, again perfectly analogous to the theory of steady two-
dimensional supersonic flows in gas dynamics.
Fig. 10.12.1. Hydraulic jump
We have already encountered an interesting example of a flow
which is in part supercritical, in part subcritical, i.e. the case of a
hydraulic jump in which the character of the flow changes on passage
through the discontinuity. Figure 10.12.1 is a photograph, taken
from the paper of Preiswerk, of such a hydraulic jump. Figure 10.12.2,
also taken from the paper of Preiswerk, shows a more complicated
case in which hydraulic jumps occur at oblique angles to the direction
of the flow. The picture shows a flow through a sluice in a dam, with
conditions (i.e. depth differences above and below the dam) such that
supercritical flow develops in the sluice, and changes in level take place
so abruptly that they might well be treated as discontinuities (as was
done in earlier sections in the treatment of bores). The two disconti-
nuities at the sides of the sluice (marked 1 and 2 in the figure) are
turned toward each other and eventually intersect to form a still
408
WATER WAVES
higher one (marked 1+2). Such oblique discontinuities can be
treated mathematically; the details can be found in the works cited
above.
Another interesting problem of the category considered here is the
1+2
Fig. 10.12.2. Hydraulic jumps at oblique angles to the direction of the flow
problem of supercritical flow around a bend in a stream. This type of
problem is relevant not only for flows in water, but also for certain
flows in the atmosphere (for which see Freeman [F.9]). It is possible
in these cases to have flows of the type which are mathematically of
the kind called simple waves in earlier sections. This means that one
of the families of characteristics is a set of straight lines along each of
which wand w (hence also h) are constant. Even the notion of a cen-
tered simple wave can be realized in these cases. Suppose that the
flow comes with constant supercritical velocity along a straight wall
LONG WAVES IN SHALLOW WATER 409
(cf. Fig. 10.12.3) until a smooth bend begins at point A. The straight
characteristics are denoted by C+ in the figure; they form a set of
Fig. 10.12.3. Supercritical flow around a smooth bend
parallel lines in the region of constant flow, which then terminates
along the C+ characteristic through the point A, where the bend be-
gins; beyond that characteristic a variable regime begins. The straight
characteristics themselves are called Mach lines; they have physical
significance and would be visible to the eye: the Mach lines are lines
along which infinitesimal disturbances of a supercritical flow are
propagated; in an actual flow they would be made visible because of
the existence of small irregularities on the surface of the wall of
the bend. If the bend contracts into a sharp corner, the straight
characteristics, or Mach lines, which lie in the region in which the
flow is variable, all emanate from the corner, as indicated in Fig.
10.12.4; the flow as a whole consists of two different uniform flows
Fig. 10.12.4. Supercritical flow around a sharp corner
connected through a centered simple wave. If the bend in the stream
is concave toward the flow, rather than convex as in the preceding
two cases, the circumstances are quite different, since the Mach lines
would now converge, rather than diverge, in certain portions of the
flow, as indicated in Fig. 10.12.5. Overlapping of the characteristics
would mean mathematically that the depth and velocity would be
multi- valued at some points in the flow; this being physically impos-
410 WATER WAVES
sible it is to be expected that something new happens and, in fact,
the development of a hydraulic jump is to be expected. If the bend
Fig. 10.12.5. Mach lines for a supercritical flow around a concave bend
is a sharp angle, as in Fig. 10.12.6, the configuration consisting of two
uniform flows parallel to the walls of the bend and connected by an
oblique hydraulic jump is mathematically possible, and it occurs in
practice.
Having considered flows delimited on one side only by a wall, it is
natural to consider next flows between two walls as in a sluice or
channel of variable breadth. (Such flows are analogous to two-dimen-
Fig. 10.12.6. Oblique hydraulic jump
sional steady flows through nozzles in gas dynamics.) The possibilities
here are very numerous, and most of them lead to cases not describ-
able solely in terms of simple waves. They are of considerable import-
ance in practice. For example, v. Karman [K.2] was led to the study
of particular flows of this type because of their occurrence in bends in
the concrete spillways designed to carry the flows of the Los Angeles
LONG WAVES IN SHALLOW WATER
411
river basin through the city of Los Angeles; the seasonal rainfall is so
heavy and the terrain so steep that supercritical flows are the rule
rather than the exception during the rainy season. Experiments for
sluices of special form were carried out by Preiswerk; Fig. 10.12.7,
b)
gemessen bei ha=31,1mm
Fig. 10.12.7. Laval nozzle a) Mach lines b) contour lines of the water surface
for example, shows the result of an experiment in a particular case.
The upper figure shows the Mach lines, the lower figure shows the
contour lines of the water surface as given by the theory as well as by
experiment; as one sees, the agreement is quite good.
Finally, we discuss briefly some applications of interest because of
their connection with aerodynamics. Because of the analogy of the
shallow water theory with compressible gas dynamics, it is of course
possible to interpret experiments with flows in shallow water in terms
of the analogous flows in gases. Since it is much cheaper and simpler
412 WATER WAVES
to obtain supercritical flows experimentally in water than it is to
obtain supersonic flows in gases, it follows that "water table" experi-
ments (as they are sometimes called) may have considerable import-
ance for those whose principle interest is in aerodynamics. There is a
considerable literature devoted to this subject; we mention, for exam-
ple, papers by Crossley [C.12], Einstein and Baird [E.5], Harleman
Fig. 10.12.8. Photogram of hydraulic- jump intersection
[H.8], Laitone [L.I], Bruman [B.19]. Figure 10.12.8 is a photograph,
taken from the paper by Crossley, showing the interaction of two
hydraulic jumps; this is a case essentially the same as that shown by
Fig. 10.12.2. The ripples with short wave lengths constitute an effect
due to surface tension, and the discontinuities are smoothed out so that
a hydraulic jump does not really occur; the changes in depth are quite
abrupt, however. Another important case that has been studied by
means of the hydraulic analogy is, as a matter of course, the flow
LONG WAVES IN SHALLOW WATER 413
pattern which results when a rigid body (simulating a projectile or an
airfoil) is immersed in a stream. Figure 10.12.9 shows a photograph of
Fig. 10.12.9. Shock wave in front of a projectile
Fig. 10.12.10. Flow pattern of a projectile
such a flow (taken from the paper by Laitone). The shock wave in
front of the projectile is well shown. Figure 10.12.10 is another photo-
graph made by Preiswerk; here, Mach lines are clearly visible.
414
WATER WAVES
10.13. Linear shallow water theory. Tides. Seiches. Oscillations in
harbors. Floating breakwaters
Up to now in this chapter we have considered problems of wave
motion in water sufficiently shallow to permit of an approximation in
terms of what we call the shallow water theory. This theory is non-
linear in character, and consequently presents difficulties which are
often quite formidable. By making the assumption that the wave
amplitudes in the motions under study are small in addition to the
assumption that the water is shallow, it is possible to obtain a theory
which is linear— and thus attackable by many known methods —
and which is also applicable with good approximation in a variety
of interesting physical situations. We begin by deriving the linear
shallow water theory under conditions sufficiently general to permit
us to discuss the cases indicated in the heading of this section. (A
brief mention of the linear shallow water theory was made in Chapter
2 and in Chapter 10.1.)
The linear shallow water theory could of course be derived by
appropriate linearizations of the nonlinear shallow water theory. It is,
however, more convenient— and perhaps also interesting from the
standpoint of method— to proceed by linearizing first the basic general
theory as developed in Chapter 1, and afterwards making the ap-
proximations arising from the assumption that the water is shallow.
In other words, we shall begin with the exact linear theory of Chapter
2.1, and proceed to derive the linear shallow water theory from it. One
of the advantages of this procedure is that the error terms involved
in the shallow water approximation can be exhibited explicitly.
We suppose the water to fill a region lying above a fixed surface
Fig. 10.13.1. Linear shallow water theory
(the bottom) y = — h(z, z)9 and beneath a surface y — Y(x9 z\ t),
the motion of which is for the time being supposed known (cf. Fig.
LONG WAVES IN SHALLOW WATER 415
10.13.1). The t/-axis is taken vertically upward, and the x, s-plane is
horizontal. The upper surface of the water given by y = Y(x, z; t)
will consist partly of the free surface (to be determined, for example,
by the condition that the pressure vanishes there) and partly of the
surfaces of immersed bodies; it is, however, not necessary to specify
more about this surface for the present than that it should represent
always a small displacement fro?n a rest position of equilibrium of the
combined system consisting of water and immersed bodies.
We recapitulate the equations of the exact linear theory as derived
in Chapter 2.1. The velocity components are determined as the deri-
vatives of the velocity potential 0(x, y. z; t) which satisfies the Laplace
equation
(10.13.1) 0xx+0yv+0zz==Q
in the space filled by the water. It is legitimate to assume that all
boundary conditions at the upper surface of the water are to be satis-
fied at the equilibrium position; this position is supposed given by
(10.13.2) y^fj(x,z).
(The bar over the quantity rj points to the fact that fj could also be
interpreted as the average position of the water in the important
special case in which the motion is a simple harmonic motion in the
time.) The x, z-planc is taken in the undisturbed position of the free
surface, and this in turn means that fj in (10.13.2) has the value zero
there. Under any immersed bodies the value of fj will be fixed by the
static equilibrium position of the given bodies. Thus fj is in all cases
a given function of x and z; for a floating rigid body, for example, it
would be determined by hydrostatics.
The condition to be satisfied at the upper surface is the kinematic
condition :
(10.13.3) 0XYX + 0ZYZ - 0y + Yt = 0,
which states that a particle once on the surface remains on it. At the
bottom surface, the condition to be satisfied is
(10.13.3)! 0xhx + 0zhz + 0V = 0 at y = - h(x, z).
Bernoulli's law for determining the pressure at any point in the water is
(10.13.4) - + 0t + gy - W = 0.
Q
Here we have assumed that there may be other external forces beside
416 WATER WAVES
gravity, and these forces are assumed to be determined by a work
function W(x9 j/, z; t) whose space derivatives furnish the force com-
ponents; in this case it is known that the motion can be irrotational
and that Bernoulli's law holds in the above form (cf. the derivations
in Chapter 1 ). We now write the equation of the moving upper surface
in the form
(10.13.5) y = Y(x, z; t) = ij(x9 z) + *?(#, z; t)
and assume in accordance with our statement above that r)(x, z; t) re-
presents a small vertical displacement from the equilibrium position
given by y = q. Upon insertion in (10.13.3) and (10.13.4) we find
after ignoring quadratic terms in r/ and 0 and their derivatives:
(10.13.6) 0xfjx + 0efjz - 0y + r,t = 0
at y = fj(x, z)
(10.13.7)
as boundary conditions to be satisfied at the equilibrium position of
the upper surface of the water. At points corresponding to a free sur-
face where p = 0 we would have, for example, fj = 0 and hence
(10.13.8) _0y+^==o
(10.13.9) '
A special case might be that in which the motion of a portion of the
upper surface is prescribed, i.e. rj(x, z; t) as well as fj would be pre-
sumed known; in such a case the condition (10.13.6) alone would
suffice as a boundary condition for the harmonic function 0. In some
of the problems to be treated here, however, we do not wish to assume
that the motion of some immersed body, for example, is known in
advance; rather, it is to be determined by the interaction with the
water which exerts a pressure p(x, z; t) on it in accordance with (10.
13.7). Thus the exact formulation of our problems would require
the determination of a harmonic function 0(x, t/, z; t) in the space
between y = — h(x, z) and y = fj which satisfies the conditions (10.
13.6) and (10.13.7) at the upper surface (in particular the conditions
(10.13.8) and (10.13.9) on the free surface) and (10.13.3)! at the
bottom. Additional conditions where immersed bodies occur (to be
obtained from the appropriate dynamical conditions for such bodies)
would be necessary to determine the pressure p, which provides the
"coupling" between the water on the one hand and the immersed
bodies on the other. Finally, appropriate initial conditions for the
LONG WAVES IN SHALLOW WATER 417
water and the immersed bodies at the initial instant would be needed
if one were to study non-steady motions, or— as will be the case here
—conditions at oo of the radiation type would be needed if simple
harmonic motions (that is, steady vibrations instead of transients)
are studied. It need hardly be said that the difficulties of carrying
out the solutions of such problems are very great indeed (cf. Chapter
9, for example)— so much so that we turn to an approximate theory
which is based on the assumption that the depth of the water is
sufficiently small and that the immersed bodies are rather flat.*
In the derivation of the shallow water theory we start from the
Laplace equation (10.13.1) for 0 and integrate it with respect to y
from the bottom to the equilibrium position** of the top surface
y = fj(x, z) to obtain, after integration by parts:
(10.13.10) "_&yvdy = 0y - 0y = - (0XX + 0zz)dy
Here, and in what follows, a bar over the quantity 0 means that it is
to be evaluated at the equilibrium position of the upper surface of the
water, i.e. for y = ?/(#, 2), and a bar under the quantity means that it
is to be evaluated at the bottom y = — h(x, z). From the kinematic
surface condition (10.13.6) and the condition (10.13.3 )t at the bottom,
we have therefore (due regard being paid to the fact that a bar should
now be put over 0 in (10.13.6) and under 0 in (10.13.3)!):
(10.13.11) nt=-
This condition— really a continuity condition— expresses the fact
that the water is incompressible. Consider next the result of integrat-
ing by parts the right hand side of (10.13.11); in particular:
(10.13.12) 0X dy = fj0x + h0x - y0xy dy.
Since we have
* In the course of the derivation the terms neglected are given explicitly so
that a precise statement about them can be made.
** One sees readily that carrying out the integration to y ~ 7] rather than
to y = rj + // yields the same results within terms of second order in small
quantities.
418 WATER WAVES
(10.13.18) h$xv dy = h$x - h&x
J —h
we may eliminate 0X from (10.13.12) to obtain:
(10.13.14) 0X dy=(rj + h)0x - (h + y)0xy dy.
Indeed, we have quite generally for any function F(x, y,z;t) the
formula:
(10.13.14)! (* Fdy=(fj + h)F - P (h + y)Fy dy.
J —/i J —n
Making use of the analogous expression for the integral of 0Z we
obtain from (10.13.11) the relation
(10.13.15) r,t = - [(fj + h)0x]x - [(fj + h)0z]z + IX + Jz
in which
(10.13.16) / = * (h + y)0xy dy, J = " (h + y)0zy dy.
In addition, we have from (10.13.10) in combination with (10.13.14)
the condition:
(10.13.17) 0y = - (fj + h)[0xx + 0ZZ] - hx0x - hz0z +IX+ J29
as one can readily verify.
Up to this point we have made no approximations other than those
arising from linearizing. The essential step in obtaining our approxi-
mate theory is now taken in neglecting the terms Ix and Jz. This in
turn is justified if it is assumed that certain second and third deriva-
tives of 0 are bounded when h is small and that 77 and its first deriva-
tives are small* of the same order as h: one sees that the terms Ix
and Jz in the right hand sides of (10.13.15) and (10.13.17) are then of
order h2 while the remaining terms are of order h. Under the free sur-
face in the case of a simple harmonic oscillation one can show that this
approximation requires the depth to be small in comparison with the
wave length.
Upon differentiating the relation (10.13.7) for the pressure at the
upper surface of the water with respect to t (again noting that a bar
should be placed over the term 0t in (10.13.7)) and using (10.13.15)
we find the equation
* This means that the theory developed here applies to immersed bodies
which are flat.
LONG WAVES IN SHALLOW WATER 419
(10.13.18) &tt+L-wt=[(jj+ h)0x]x +
after dropping the terms Ix and Jz. This is the basic differential equa-
tion for the function 0(x, z; t) which holds everywhere on the upper
surface of the water. In particular, we have at the free surface where
p — 0 and fj ~ 0 the equation
(10.13.19) (h$x)x + (A0,), - -$tt = - Wt.
o o
We recall that W(x, y, z; t) represents the work function for any
external forces in addition to gravity (tide generating forces, for
example), so that W9 its value on the free surface, would be given by
W(x, 0, 2; t). If, in addition, it is assumed that h is a constant, i.e. that
the depth of the water is uniform, and that gravity is the only external
force, we would have the equation
(10.13.19)! &xx + $„- — $it = 0,
that is, the linear wave equation in the two space variables #, z and
the time /. As a consequence, all disturbances propagate in such a
case with the constant speed c = Vgh, as is well known for this
equation.
If there is an immersed object in the water, the equation (10.13.19)
holds everywhere in the x, z-plane exterior to the curve C which
defines the water line on the immersed body in its equilibrium posi-
tion. The curve C is supposed given by the equations
(10.13.20) x = x(s\ z = z(s)
in terms of a parameter s. We must have boundary, or perhaps it is
better to say, transition conditions at the curve C which connect the
solutions of (10.13.19) in the exterior of C in an appropriate manner
with the motion of the water under the immersed body. Reasonable
conditions for this purpose can be obtained from the laws of conser-
vation of mass and energy. In deriving these conditions we assume
W = 0, since we wish to deal only with gravity as the external force
when considering problems involving immersed bodies. Consider an
element of length ds of the curve C representing the water line of the
immersed body (cf. Fig. 10.13.2). The expression
420
WATER WAVES
represents the mass flux through a vertical strip having the normal n
C(x(s),y(s))
Fig. 10.18.2. Boundary at water line of an immersed body
and extending from the bottom to the top of the water. From (10.13.
x applied for F = 0n we have
(10.13.21 ) e 0n dy = e(ij + h)0n - Q (h + y)0nv dy
^ Q(fj + h)0n
where the second term is ignored because it is of order h2. Thus it
would be reasonable to require that (77 + h)0n should be continuous
on C since this is the same as requiring that the mass of the water is con-
served within terms of the order retained otherwise in our theory. For
the flux of energy across a vertical strip with the normal n we have
(10.13.22) (J^ p0n dy ) ds = (- Q J^0«#n dy - gQ J^ y0n dy) ds
upon making use of the Bernoulli law (10.13.4) for the pressure p
(when W = 0). Once more we may ignore the second term in the
brackets since it is of order h2. Upon applying (10.13.14)! with
F = 0t0n and again ignoring a term of order h2 we find
(10.13.23) j F p0n dy)ds = {-
\ J —n i
0t} ds.
Since we have already required that (fj + h)0n should be continuous,
we see that the additional requirement, 0t continuous, ensures the
continuity of the energy flux.
As reasonable transition conditions on the curve C delimiting the
immersed body at its water line we have therefore
(10.13.24) (fj + h)0n, 0t continuous on C.
LONG WAVES IN SHALLOW WATER 421
Of course, if fj is continuous (e.g. if the sides of the body in contact
with the water do not extend vertically below the undisturbed free
surface) it follows that <pn would then be continuous.
In order to make further progress it would be necessary to specify
the properties of the immersed body. However, we have succeeded in
obtaining the equation (10.13.18) which is generally valid and of basic
importance for our theory together with the transition conditions
(10.13.24) valid at the edge of immersed bodies; in particular, we have
the definitive equation for the free surface itself in the form of the
linear wave equation (10.13.19). The idea behind this method of ap-
proximation is to get rid of the depth variable by an integration over
the depth so that the problems then arc considered only in the x, z-
plane. As a result, the problems are no longer problems in potential
theory in three space variables, but rather problems involving the wave
cq nation with only two space variables, and hence they are more open
to attack by known methods. How this comes about will be seen in
special cases in the following.
As a first example of the application of the above theory we con-
sider briefly the problem of the tides in the oceans, with a view to
indicating where this theory fits into the theory of gravity waves in
general, but not with the purpose of giving a detailed exposition. (For
details, the long Chapter VIII in Lamb [L.3] should be consulted.)
To begin with, it might seem incredible at first sight that the shallow
water theory could possibly be accurate for the oceans, since depths of
five miles or more occur. However, it is the depth in relation to the
wave length of the motions under consideration which is relevant.
The tides are forced oscillations caused by the tide-generating at-
tractions of the moon and the sun, and hence have the same periods
as the motions of the sun and moon relative to the earth. These periods
are measured in hours, and consequently the tidal motions in the
water result in waves having wave lengths of hundreds of miles;*
the depth-wave length ratio is thus quite small and the shallow
water theory should be amply accurate to describe the tides. This
means, in effect, that the differential equation (10.13.19), or rather, its
analogue for the case of water lying on a rotating spheroid (with Cori-
olis terms put in if a coordinate system rotating with the earth is
* For example, in water of depth 10,000 feet (perhaps a fairly reasonable
average value for the depth of the oceans) a steady progressing wave having
a length of 10,000 feet has a period of only 44.2 sec. (cf. Lamb [L.3], p. 369).
Since the wave length varies as the square of the period, the correctness of our
statement is obvious.
422 WATER WAVES
used), would serve as a basis for calculating tidal motions. Of course,
the function Wt would be defined in terms of the gravitational forces
due to the attraction of the sun and moon. The variable depth of the
water in the oceans would come into play, as well as boundary condi-
tions at the shore lines of the continents. Presumably, Wt would be
analyzed into its harmonic components (which could be obtained
from astronomical data), the response to each such harmonic would
be calculated, and the results superimposed. Such a problem consti-
tutes a linear vibration problem of classical type— it is essentially the
same as the problem of transverse forced oscillations of a tightly
stretched non-uniform membrane with an irregular boundary. If it
were not for one essential difficulty, to be mentioned in a moment,
such a problem would in all likelihood be solvable numerically by
using modern high speed computational equipment. The difficulty
mentioned was pointed out to the author in a conversation with
H. Jeffreys, and it is that there are difficulties in prescribing an ap-
propriate boundary condition in coastal regions where there is dissi-
pation of energy in the tidal motions (in the bay of Fundy, for exam-
ple, to take what is probably an extreme case). At other eoastal re-
gions the correct boundary condition would of course often be simply
that the component of the velocity normal to the coast line vanishes.
Of course, there would also be a difficulty in using a differential
equation like (10.13.19) near any shores where h = 0, since the dif-
ferential equation becomes singular at such points. Nevertheless, a
computation of the tides on a dynamical basis would seem to be a
worthwhile problem— perhaps it could be managed in such a way as
to help, in conjunction with observations of the actual tides, in
providing information concerning the dissipation of energy in such
motions.
These remarks might be taken to imply that the dynamical theory
is not at present used to compute the tides. This is not entirely correet,
since the tide tables for predicting the tides in various parts of the
world are based on fundamental consequences of the assumption that
the tides are indeed governed by a differential equation of the same
general type as (10.13.19). The point is that the oceans are regarded as
a linear vibrating system under forced oscillations due to excitation
from the periodic forces of attraction of the sun and moon. It is
assumed that all free vibrations of the oceans were long ago damped
out, and hence, as remarked above, that the tidal motions now exist-
ing in the oceans are a superposition of simple harmonic oscillations
LONG WAVES IN SHALLOW WATER 423
having periods which are very accurately known from astronomical
observations. To obtain tide tables for any given point a superposition
of oscillations of these frequencies is taken with undetermined ampli-
tudes and phases which are then fixed by comparing them with a
harmonic analysis of actual tidal observations made at the point in
question. The tide predictions are then made by using the result of
such a calculation to prepare tables for future times. The dynamical
theory is thus used only in a qualitative way. An interesting addition-
al point might be mentioned, i.e. that tides of observable amplitudes
are sometimes measured which have as frequency the sums or differ-
ences (or also other linear combinations with integers) of certain of
the astronomical frequencies, which means from the point of view of
vibration theory that observable nonlinear effects must be present.
Another type of phenomenon in nature which can be treated by
the theory derived here concerns periodic motions of rather long
period, called seiches, which occur in lakes in various parts of the
world. The first observations of this kind seem to have been made by
Forel [F.7] in the lake at Geneva in Switzerland, in which oscillations
having a period of the order of an hour and amplitudes of up to six
feet have been observed. In larger lakes still larger periods of oscilla-
tion arc observed— about fifteen hours in Lake Erie, for example.
A rather destructive oscillation, generally supposed to be of the type
of a seiche, occurred in Lake Michigan in June 1054; a wave with an
amplitude of the order of ten feet occurred and swept away a number
of people who were fishing from piers and breakwaters. What the
mechanism is that gives rise to seiches in lakes has been the object of
considerable discussion, but it seems rather clear that the motions
represent free vibrations of the water in a lake which are excited by
external forces of an impulsive character, the most likely type arising
from sudden differences in atmospheric pressure over various portions
of the water surface. Bouasse [B.I 5, p. 158] reports, however, that
the Lisbon earthquake of 1755 caused oscillations in Loch Lomond
with a period of about 5 minutes and amplitudes of several feet.
In any case, the periods observed seem to correspond to those calcu-
lated on the basis of the linear shallow water theory, which should be
quite accurate for the study of seiches because of their long periods and
small amplitudes. It follows, therefore, that the differential equation
(10.13.18) is applicable; we suppose that Wt = 0 (since tidal forces
play no role in this case), and also set fj = 0 since there are no immersed
bodies to be considered. The differential equation for 0(x, z\ t ) is thus
424 WATER WAVES
(10.18.25)
& e«r
The free natural vibrations of the lake are investigated by setting
pt = 0 and 0(x, z; t) = 99(01, *)£*" in (10.13.25) with the result
a2
(10.13.26) (hpx)x + (%)2 + -<p = 0.
As boundary condition along the shore of the lake we would have
(10.13.27) (pn = 0.
The problem thus posed is one of the classical eigenvalue problems of
mathematical physics. Solutions <p other than the trivial solution
q> = 0 of (10.13.26) under the homogeneous boundary condition
(10.13.27) are wanted; such solutions exist only for special values of
the circular frequency a, and these values yield the natural frequencies
corresponding to the natural modes y(x, z) which are correlated with
them. In general, an infinite set of such natural frequencies occurs.
For particular shapes and depths h— rectangular or circular lakes of
constant depth, for example —it is possible to solve such problems
more or less explicitly. In practice however, lakes have such irregular
outlines and depths that the determination of the natural frequencies
and modes requires numerical computation. A reasonable and gener-
ally applicable method of carrying out such computations is furnished
here, as in other instances in this and the subsequent chapter, by
the method of finite differences.* In this method, the derivatives in
the differential equation and boundary conditions are replaced by
difference quotients defined by means of the values of the function
at the discrete points of a net in the domain of the independent varia-
bles. The resulting finite equations are then solved to yield approxi-
mate values for the unknown function at the net points. The difference
approximation will be more accurate for a closer spacing of the net
points. We proceed to illustrate the method for the case of a lake of
constant depth in the form of a square of length / on each side, with
a view to comparing the result with the exact solution which is easy
to write down in this case. The differential equation (10.13.26) can
be written in the form
* A different method was used by Chrystal [C.2] to calculate the periods of
the free oscillations of Loch Earn; he found good agreement with the obser-
vations for the first six modes of oscillation.
LONG WAVES IN SHALLOW WATER
(10.13.26)! (pxx + <pgz + m2<p = 0, m2 = a2/gh
425
in this case. A division of the square in a mesh with mesh width
8 = J/7 is taken, as indicated in Fig. 10.13.3. In numbering the net
points it has been assumed that only modes of oscillation that are
symmetrical with respect to the center lines parallel to the sides and to
to the diagonals are sought, which, however, is not the case for the
y
4
8 S
7 9
>
0 9
. . .
3
2
§ 8
5 6
9 8
7 6
i
i
2 3
4
i
i
? .
X
°t
Fig. 10.13.3. Finite differences for a seiche
mode having the lowest frequency. The boundary condition q>n = 0 is
satisfied approximately by supposing that the solution is reflected
over the boundaries in such a way as to yield values which arc equal at
the mirror images in the boundaries, as is also indicated in Fig. 10.13.3.
The formulas used for approximating the derivatives are defined as
follows (cf. Fig. 10.13.3):
dz
ffm. n+l -" ^n?, w-1
%9m,n + 9V n+l + Vm.n-l
Consequently, the differential equation (10.13.26)1 is replaced at each
net point (m , n) by the difference equation
(10.13.28) -4<pmfn+<pm,n+1+<pm^
Such an equation is written for each of the net points in Fig. 10.13.3.
The results for points 1,6,9, for example, are:
426 WATER WAVES
(10.13.29)
(<5w)Vi - 0
6: — 4<pe + g?3 + <p7 + 9?8 + <p5 + (dw)2^ = 0
9: — 4<p9 + 2y8 + <p7 + <p10 + (5m)29?9 = °-
These homogeneous linear equations of course have always the solu-
tion <fi : = 0, i = 1, 2, . . ., 10 unless their determinant vanishes, and
this condition is a tenth degree equation in the quantity (dm)2, the
smallest root of which furnishes an approximation to the lowest
frequency. The exact solution of the differential equation (10.13.26)!
which satisfies the boundary condition is, in the present case, <p =
A cos (knx/l) cos (jnz/l), with k and / any integers, provided that
m2 = n2(k2 + j2)/l2. A numerical comparison of the lowest value of
in for the mode having double symmetry— i.e. the value for k = 1,
j = 1 — with the value computed from the determinantal equation
shows the approximate value of m to be too low by 6.5 %.
However, this mode corresponds to one of the higher eigenvalues,
so that the accuracy of the finite difference method is rather good.
The error for the lowest mode is very much smaller, but because of the
lack of symmetries the amount of calculation needed to determine
the corresponding frequency would be much greater for the present
case. If one were to treat a long narrow lake, the calculation would
be simpler. It could also be advantageous to employ the Rayleigh-Ritz
method. In principle, similar calculations could be made in more
complicated cases (for many examples of problems solved along these
lines see the book by Southwell [S.14]).
Wave motions in harbors are often of a type suitable for discussion
in terms of the linear shallow water theory: they are indeed often of
the type called seiches above. In these cases oscillations of the water
in the harbor are also commonly excited by the motion at the harbor
mouth, which in its turn is due, of course, to wave motions generated
in the open sea. An experimental and theoretical investigation of such
waves in a model has been carried out by McNown [M.7]. The model
was in the form of a circle 3.2 meters in diameter with vertical walls.
The depth of the water in this idealized harbor was 16 cm. An opening
of angle n/8 radians in the harbor wall permitted a connection with a
large tank in which waves (simulating the open sea) were produced.
Figures 10.13.4 and 10.13.5 are photographs of the model (taken from
the paper by McNown), which also show two specific cases of symme-
trical oscillations. The free vibrations again are governed by equation
(10.13.26). (It might be noted that McNown makes use of the exact
LONG WAVES IN SHALLOW WATER
427
linear theory rather than the shallow water theory. The only difference
is that the relation between cr2 and m is a2 = gm tanh mh, instead of
(T2 = ghm2, as given above: the differential equation for the velocity
potential 0(x, y, z; t) in the exact linear theory treated in Part I is
Fig. 10.13.4. and 10.13.5. Waves in a harbor model
written in the form 0 = A cosh m(y + h)eiat(p(x, z), and (p(x, z) then
satisfies V2<p + m2<p = 0.) Solutions of the differential equation are
sought in the form
(10.13.30) <p(r, 0) = Jn(mr) cos nO
in polar coordinates (r, 0), under the assumption that the port is
closed, i.e. that its boundary is the whole circle r = R. As is well
428
WATER WAVES
known, <p(r, 6) is a solution of (10.13.26 )x only if Jn(mr) is a Bessel
function of order w, and since it is reasonable to look only for solutions
that are bounded we choose the Bessel functions of the first kind
which are regular at the origin. The boundary condition requires that
(10.13.31) -^- = 0 at r = R
and this in turn leads to the condition dJJdr = 0 for r = R. For
each n this transcendental equation has infinitely many roots m^ ,
each corresponding to a mode of oscillation with various nodal dia-
meters and circles, and with a definite frequency which is fixed by
a2 = gm^ (or, more accurately, by a2 — gm^ tanh hm(^ ). Figure
10,13.6, obtained by McNown, shows a comparison of observed and
calculated amplitudes for two modes of oscillation; the upper curve
is drawn for a motion having no diametral nodes and two nodal circles,
while the lower is for a motion having two nodal diameters and one
-2
-3
-4
Y_
(kr)
entrance
o Observed
— Theoretical
center
Fig. 10.13.6. Comparison of results of experiment and theory for resonant move-
ments in a circular port
LONG WAVES IN SHALLOW WATER
429
nodal circle. The motions were excited by making waves in the tank,
and providing an opening for communication with the harbor, as
noted above. The figures were drawn assuming that the amplitudes
would agree at the entrance to the harbor— the experimental check
10.18.7. of a
Fig. 10.13.8. Model of a harbor with breakwater
thus applies only to the shapes of the curves. As one sees, the ex-
perimental and theoretical values are remarkably close. The ampli-
tudes used were large enough so that nonlinear effects were observed:
the troughs are flatter than the crests by measurable amounts. Of
course, having an opening in the harbor wall violates the boundary
condition assumed, but this effect apparently is slight: changing the
angle of the opening at the harbor mouth had practically no effect on
430
WATER WAVES
the waves produced, and, in addition, it was found that very little
wave energy radiates outward through the harbor entrance.
Problems of harbor design, involving construction of breakwaters,
location of docks, etc. are commonly studied by constructing models.
Figs. 10.13.7 and 10.13.8 show two photographs of a model of a har-
bor,* the first before a breakwater was constructed, the second
afterward. As one sees, the breakwater has a quite noticeable effect.
Fig. 10.13.9. shows the same model, with the waves approaching the
harbor mouth at a different angle, however; as one sees the break-
Fig. 10.13.9. Model of a harbor with breakwater
water seems to be on the whole less effective when the wave fronts
are less oblique to the breakwater. The diamond-shaped pattern, due
to reflection, of the waves on the sea side of the breakwater is inter-
Y////////7*
/////////A
Fig. 10.13.10. Floating plane slab
* These photographs were given to the author by the Hydrodynamics Labora-
tory at California Institute of Technology.
LONG WAVES IN SHALLOW WATER 481
esting. Model studies are rather expensive, and consequently it might
well be reasonable to explore the possibilities of numerical solution
of the problems, perhaps by using appropriate modifications of the
method of finite differences outlined above for a simple case.
We turn next to a discussion of the effect of floating bodies on
waves in shallow water, on the basis of the theory presented in this
section. Only two-dimensional motions will be considered (so that all
quantities are independent of the variable z). The first case to be
studied is that of the motion of a floating rigid body in the form of a
thin plane slab (cf. Fig. 10.13.10) in water of uniform depth. Such
problems have been treated by F. John [J.5]. The ends of the slab
arc at jc — i a. In accordance with the theory presented above we
must determine the surface value 3>(tT, t) and the displacement r](x9 t)
of the board from the differential equations (cf. (10.13.19), (10.13.15)
with rj = 0, and dropping /^ and «/2):
(10.13.32) #,,-^0,,, |*| >a
(10.13.33) rjt = - h&xx, \ * \ < a.
We have dropped the bar over the quantity 0. We have also assumed
that fj(x) for \ x \ < a, the rest position of equilibrium of the board, is
zero; this is an approximation that is justified because we assume
that the board is so light that it does not sink appreciably below the
water surface when in equilibrium. (This assumption is by no means
necessary — it would not be difficult to deal with the problem if this
simplifying assumption were not made.)
Since fj is zero, it follows (cf. (10.13.24)) that the transition con-
ditions at the ends of the board are
(10.13.34) 0X, 0t continuous at x — ± a.
We are interested in the problem of the effectiveness of the floating
board as a barrier to a train of waves coming from the right (x =
+ 00). The equation (10.13.32) has as its general solution
0(x, 1) = F(x - ct) + G(x + ct), c = VgA
in terms of two arbitrary functions F and G (as one can readily verify )
which clearly represent a superposition of two progressing waves
moving to the right and to the left, respectively, with the speed Vgh.
It is natural, in our present problem, to expect that for x > a there
would exist in general both an incoming and an outgoing wave be-
cause of reflection from the barrier, while for x < — a we would pre-
432 WATER WAVES
scribe only a wave going outward (i.e. to the left). We shall see that
these qualitative requirements lead to a unique solution of our problem.
We consider only simple harmonic waves; it is thus natural to write
(10.13.35) &(x, t) = (p(x)eiat, \ x \ > a,
(10.13.36) ri(x, t) = v(x)eia\ - a < x < a,
with the stipulation that the real part is to be taken at the end. (It is
necessary also to permit <p(x) and v(x) to be complex-valued func-
tions of the real variable x.) The conditions (10.13.32) and (10.13.33)
now become
d2cp a2
(10.13.37) I + = 0. 1*1 >a
a
(10.13.38) Tl + T^0' 1*1 <«•
*
The equation (10.13.37) has as general solution
(10.13.39) y(x) = Ae~ikx + Beik*
with k given by
(10.13.40) k = a/Vgh.
For 0(x, t) we have therefore
(10.13.41) &(x, t) = Ae-Wx-a» + Be'l***"*,
the first term representing a progressing wave moving to the right,
the second a wave moving to the left. In our problems we prescribe
the incoming wave from the right, and hence for cp(x) we write
(10.13.42) <p(x) = Beikx + Re~tkx, x > a,
in which B is prescribed, while .K— the amplitude of the reflected
wave (more precisely, | R \ is its amplitude) — is to be determined.
At the left we write
(10.13.43) <p(x) = Teikx,
with T— the amplitude of the transmitted wave —to be determined.
To complete the formulation of the problem it is necessary to con-
sider the dynamics of our floating rigid body. We shall treat two cases:
a) the board is held rigidly fixed in a horizontal position, b) the board
floats freely in the water.
a) Rigidly Fixed Board.
If the board is rigidly fixed we have 77(0?, t) = 0, and hence (cf.
(10.13.36)) v(x) =0. It follows from (10.13.88) that <pxx vanishes
LONG WAVES IN SHALLOW WATER
433
identically under the board and hence that <p(x) is a linear function:
(10.13.44) <p(x) = yx + d, — a < x < + a.
Since 0x(z, t) furnishes the horizontal velocity component of the
water, it follows from (10.13.44) and (10.13.35) that the velocity under
the board is given by yeiat, i.e. it is constant everywhere under the
board at each instant— a not unexpected result.
We now write down the transition conditions at x = ± « from
(10.13.34), making use of (10.13.35) and of (10.13.42) at x = + a
and (10.13.43) at x = - a; the result is:
Betka + Re~lka = ya + 6
Bezka - Re~ika = y/ffc
Te-ika = _ ya _|
Te-ika =
(10.13.45)
Once the real number B— which fixes the amplitude of the incoming
wave — has been prescribed, these four equations serve to fix the con-
stants R, T, y, and 6 and hence the functions 0(x, t) and rj(x, t). The
pressure under the board can then be determined (cf. the expression
(10.13.4) for Bernoulli's law) from
(10.13.46) p(x, /.) = - Q&t - — Qia(p(x)eiat.
(Observe that the quantity y in (10.13.4) is zero in the present case.)
In terms of the dimensionless parameter
(10.13.47) 0 = 2a/A,
the ratio of the length of the board to the wave length A on the free
surface, given by (cf. (10.13.41))
(10.13.48) A = 2^/fc,
the solution of (10.13.45) is
R =
(10.13.49)
Oni + 1
_
7
dni
a 07i-l
d = Beeni.
The reflection and transmission coefficients are obtained at once:
434
WATER WAVES
(10.13.50)
Cr =
Qn
Vi +
c* c*
0.54
0.85
0.30
0.95
0.157
0.986
They depend only upon the ratio 0 =• 2a/A, as one would expect. They
also satisfy the relation C* + Cf — 1, as they should: this is an ex-
pression of the fact that the incoming and outgoing energies are the
same. The following table gives a few specific values for these co-
efficients:
0
0.5
1.0
2.0
Thus a fixed board whose length is half the incoming wave length has
the effect of reducing the amplitude behind it by about 50 percent and
of reflecting about 72 percent of the incoming energy. One should,
however, remember that the theory is only for long waves in shallow
water, and, in addition, it seems likely that the length of the board
will also play a role in determining the accuracy of the approximation.
This question has been investigated by Wells [W.10] by deriving the
shallow water theory in such a way as to include all terms of third order
in the depth h and studying the magnitude of the neglected terms in
special cases; in particular, the present case of a floating rigid body
is investigated. Wells finds that if A/A is small and if a/A (the ratio of
the half-length of the board to the wave length) is not smaller than 1,
the neglected higher order terms are indeed negligible, but if a/A is less
than 1, the higher order terms need not be small. In other words,
floating obstacles ought to have lengths of the order of the wave
length of the incoming waves if the shallow water theory to lowest
order in h is expected to furnish a good approximation.
It is of interest to study the pressure variation under the board.
This is given in the present case (cf. (10.13.46)) by
(10.13.51)
(x, t) = —
d)e
dot
the real part only to be taken. Thus the pressure varies linearly in x>
but it is a different linear function at different times since y and d are
LONG WAVES IN SHALLOW WATER
435
complex constants. The result of taking the real part of the right hand
side of (10.13.51) can be readily put in the form
(10.13.52) p(x, t) = Pi(x) cos at — p2(x) sin at
with
Pi(x) = agBfiifa) sin r + b2(x) cos r)
p2(x) = agB(b2(x) sin r — b^x) cos r)
and
(10.13.53)
(10.13.54)
-i -5 0 5 */0 i
Fig. 10.13.11. Pressure variations for a stationary board. 0 = 1, p in poundsl(ft)2
436 WATER WAVES
We have assumed in making these calculations, as stated above, that
B. which represents the amplitude of the incoming wave, is a real
number.
In Fig. 10.13.11 the results of computations for the pressure distri-
bution for time intervals of 1/4 cycle over the full period are given for
a special numerical case in which the parameter 6 has the value 0 = 1,
i.e. the length of the board is the same as the wave length. One ob-
serves that the pressure variation is greater at the right end than at
the left, which is not surprising since the board has a damping effect on
the waves. One observes also that the pressure is sometimes less than
atmospheric (i.e. it is negative at times, while p — 0 is the assumed
pressure at the free surface).
b) Freely Floating Board.
In Fig. 10.13.12 the notation for the present case is indicated:
u(t), v(t) represent the coordinates of the center of gravity of the
board in the displaced position, and co(t) the angular displacement.
As before, we consider only simple harmonic oscillations and thus
take u9 v, and CD in the form
(10.13.55) u = xeiQ\ v = yeiat, CD = we™*,
Fig. 10.13.12. A freely floating board
in which x, y, and w arc constants representing the complex ampli-
tudes of these components of the oscillation. For rj(x, t) we have,
therefore
(10.13.56) ri(x9 1) = [y + (x - x)w]eiat
= (y + xw)eiat
when terms of first order in x, y, and w only are considered. (The hori-
zontal component of the oscillation is thus seen to yield only a second
order effect.) The relation (10.13.56) now yields (cf. (10.13.38)):
(10.13.57) <pxx=-^(y +
LONG WAVES IN SHALLOW WATEE
487
in which (p is the complex amplitude of the velocity potential 0(x, t) =
(p(x)eiat. Hence <p is the following cubic polynomial:
(10.13.58) <p(x) = - !
Since the pressure is given by p = — g0f — gg^ we have in the
present case
(10.13.59) p(x) = [- icre0>(0) - Qg(y + xw)]eiat.
The transition conditions (10.13.34) at x = ± fl now lead, in the
same way as above from (10.13.42) and (10.13.43), to the equations
Beika + Re~n
(10.13.60)
Te
tffei
~tka
i — o \ 4*
= ( — ya\ V.
*A\ 2 9 J kY
These four equations are not sufficient to determine the six constants
ff, T, w, y< y, and 6. We must make use of the dynamical equations of
motion of the floating rigid body for this purpose. We have the
equations of motion:
(10.13.61) F - Alv, and L = /a
at our disposal. In the first equation F and M are the total vertical
force on the board and its mass, per unit width, and v is the vertical
acceleration of its center of gravity, / the moment of inertia, L the
torque, and a the angular acceleration. These dynamical conditions
then yield the following relations:
(10.13.62) p dx = Mv, px dx = lib,
J-o J-a
and these in turn lead to the equations
0.13.63)
1"
J_a
--!««.
488 WATER WAVES
In the first equation we have ignored the weight of the board, since
it is balanced by the hydrostatic pressure. The equations (10.13.60)
and (10.13.63) now determine all of the unknown complex amplitu-
des.
We omit the details of the calculations, which can be found in the
paper by Fleishman [F.5], In Fig. 10.13.13 the results of calculations
for the pressure distribution in a numerical case are given. The para-
meters were chosen as follows:
0 = 1, h = 1 ft, B = 1 ft*/sec, a = 4 ft,
M = 18.72 pounds/ft, a = 4.46 rad/sec, A = 8 ft.
It might be added that the value chosen for M is such that the struc-
ture sinks down 0.0375 feet when in equilibrium.
A few observations should be made. First of all, we note that in both
cases the pressure variation at the right end (x = + a), where the
incoming wave is incident, is greater than at the left end. This is to
be expected, since the barrier exercises a damping effect on the wave
going under it. The pressure distribution in the case of the floating
board is quadratic in #, in contrast with the case of the fixed board in
which the distribution of pressure was linear in x. Next, we note that
the pressure variation near the right end of the stationary board is
greater than at the same end of the floating one; this too might be
expected since the fixed board receives the full impact of the incident
wave, while the floating one yields somewhat. Finally, we see that at
the left end the opposite effect occurs: there the pressure variation
under the stationary barrier is less than that under the floating barrier.
This is not surprising either, since the fixed board should damp the
wave more successfully than the movable board.
Finally, we take up the case of a floating elastic beam (cf. [F.5] ).
The beam is assumed to extend from x ~ — ltox = Q and, as in the
above cases, to be in simple harmonic motion due to an incoming
wave from x = + oo. The basic relations for <£(#, t) on the free sur-
face, and for rf(x, t) under the beam are the same as before:
(10.13.64) 0XX = -1 0tt9 x > 0, x < - I,
gh
(10.13.65) ijt = - h0xx, - / < x < 0.
We assume once more that the beam sinks very little below the water
surface when in equilibrium (i.e. very little in relation to the depth of
LONG WAVES IN SHALLOW WATER
439
p-
200
0
-200
•500
<T 1 = IT/4
— i —
-5
x/o
-500
P
400
200
•200
•5
.5
x/a
P
400
200
0
•200
-1-5 0 5 *'° J
Fig. 10.13.13. Pressure variations for floating board. 6 = 1, p in pounds I (ft)*
440 WATER WAVES
the water), so that the coefficient of <&xx in (10.13.65) can be taken as
h rather than (h +r)) (cf. (10.13.15)), and also the transition condi-
tions at the ends of the beam are
(10.13.66) 0X and <Pt continuous at x = 0, x = — /.
After writing
(10.13.67) <P(x, t) = (p(x)eiat9 17(0, /) = V(X)CM
we find, as before:
a2
(10.13.68) <pxx H -- <p = 0, x > 0, x < ~ I
gh
(10.13.69) <pxx + ^ v = 0, - I < x < 0.
h
The conditions at oo have the effect that (cf. (10.13.41) et seq.):
(10.13.70) <p(x) = Beik* + Ife-"* x > 0,
(10.13.71) <p(x) - 2V** x < - Z,
with A: = cr/Vgfe- All of this is the same as for the previous cases. We
turn now to the conditions which result from the assumption that the
floating body is a beam.
The differential equation governing small transverse oscillations
of a beam is
(10.13.72) EIr,xxxx + mr,tt = p,
in which E is the modulus of elasticity, / the moment of inertia of a
cross section of unit breadth (or, perhaps better, El is the bending
stiffness factor), m the mass per unit area, and p is the pressure. We
ignore the weight of the beam and at the same time disregard the
contribution of the hydrostatic pressure term in p corresponding to
the equilibrium position of the beam— i.e. the pressure here is that due
entirely to the dynamics of the situation. Thus
(10.13.73) p = ~ Q&t
Insertion of this relation in (10.13.72) and use of (10.13.69) leads at
once to the differential equation for <p(cc):
(10.18.T4)
El dx* Elh
LONG WAVES INT SHALLOW WATER 441
that is valid under the beam. The case of greatest importance for us—
that of a floating beam used as a breakwater— leads obviously to the
boundary conditions for the beam which correspond to free ends, i.e.
to the conditions that the shear and bending moments should vanish
at the ends of the beam. These conditions in turn mean that rjxx and
r]xxx should vanish at the ends of the beam, and from (10.13.67) and
(10.13.69) we thus have for q> the boundary conditions
(10.13.75) -J? = - = 0 at x = 0, x = - I.
ax* ax5
The transition conditions (10.13.66) require that <p and q>x be conti-
nuous at x = 0, x = — I, and this, in view of (10.18.70) and (10.13.
71), requires that
(10 13 76) f
* ' ' ' \<p(- 1) = Ter*», <pm(-l) = ikTe~ikl.
We remark once more that the constant B is assumed to be real, but
that R and T will in general be complex constants, and that the real
parts of 0 and rj as given by (10.13.67) are to be taken at the end.
In order to solve our problem we must solve the differential equa-
tion (10.13.74) subject to the conditions (10.13.75) and (10.13.76).
A count of the relations available to determine the solution should be
made: The general solution of (10.13.74) contains six arbitrary con-
stants, and we wish to determine the constants R and T (the am-
plitudes of the reflected and transmitted waves) occurring in (10.13.
76) once the constant B (the amplitude of the incoming wave) has been
fixed. In all there are thus eight constants to be found, and we have in
(10.18.75) and (10.13.76) eight relations to determine them. Once
these constants have been determined, the reflection and transmission
coefficients are known, and the deflection of the beam can be found
from (10.13.69). The maximum bending stresses in the beam can then
be calculated from the usual formula: s = Me//, with M = Elrfxx
and c the distance from the neutral axis to the extreme outer fibres of
the beam.
In principle, therefore, the solution of the problem is straightfor-
ward. However, the carrying out of the details in the case of the beam
of finite length is very tedious, involving as it does a system of eight
linear equations for eight unknowns with complex coefficients. In
addition, one must determine the roots of a sixth degree algebraic
equation in order to find the general solution of (10.13.74). These
442 WATER WAVES
roots are in general complex numbers and they involve the essential
parameters of the mechanical system. Thus it is clear that a dis-
cussion of the behavior of the system in general terms with respect to
arbitrary values of the parameters of the system is not feasible, and
one must turn rather to concrete cases in which most of the parameters
have been given specific numerical values. The results of some calcula-
tions of this kind, for a case proposed as a practical possibility, will be
given a little later.
The case of a semi-infinite beam— i.e. a beam extending from x = 0
to x = — oo — is simpler to deal with in that the conditions in the
second line of (10.13.76) fall away, and the conditions (10.13.75) at
x — — oo can be replaced by the requirement that q> be bounded at
x — — oo. The number of constants to be fixed then reduces to four
instead of eight, but the determination of the deflection of the beam
still remains a formidable problem; we shall consider this case as well
as the case of a beam of finite length.
We begin the program indicated with a discussion of the general
solution of the differential equation (10.13.74). Since it is a linear
differential equation with constant coefficients we proceed in the
standard fashion by setting q> = £*, inserting in (10.13.74), to find
for K the equation
(10'.13.77) x« + ax2 + b --= 0
with
This is a cubic equation in x2 = /?, which happens to be in the standard
form to which the Cardan formula for the roots of a cubic applies
directly. For the roots /^ of this equation one has therefore
(10.13.79)
= u + v
s2v
H CT
with u and v defined by
/ b /b2 as\i\* / b /b2 a3\i\i
(10.18.80) u=l- — + (— + —I | , 0 = 1 _+_ ))
\ 2T\4 27//' \ 2 \4 27/ /
and e the following cube root of unity:
LONG WAVES IN SHALLOW WATER 443
The constant a is positive, since or, the frequency of the incoming
wave, is so small in the cases of interest in practice that qg is much
larger than ma2. The constant b is obviously positive. Consequently
the root ^ is real and negative since \u\ <\v\ and v is negative.
Thus the roots »x = + /?J'2, x2 = — £J/2 are pure imaginary. The
quantities f}2 and /?3 are complex conjugates, and their square roots
yield two pairs of complex conjugates
For /92 and /?3 we have
(10.13.82) & - - - (it + v) + i - (u - v),
2 2
(10.13.83) /?, = - 1 (u + ») - i ~ (u - v).
& £
Thus f}2 and ^3 both have positive real parts. We suppose the roots
x& #4, H5, x6 to be numbered to that x3 and x5 are taken to have posi-
tive real parts, while x4 and x6 have negative real parts. The general
solution of (10.13.74) thus is
(10.13.84) <p(x) - a^i* + a<£**x + atf**x + a4^4* + a^x + a^x.
In the case of a beam covering the whole surface of the water, i.e.
extending from — oo to + oo, the condition that <p be bounded at
cr = : ^ oo would require that «3 = «4 = «5 = «6 = 0 since the ex-
ponentials in the corresponding terms have non-vanishing real parts.
The remaining terms yield progressing waves traveling in opposite
directions; their wave lengths are given by A = 2n/\ K± \ = 2n/\ x2 I
and thus by
(10.13.85) A = 2n/V\ u + v |,
with u + v defined by (10.13.80). The wave length and frequency are
thus connected by a rather complicated relation, and, unlike the case
of waves in shallow water with no immersed bodies or constraints on
the free surface, the wave length is not independent of the frequency
and the wave phenomena are subject to dispersion.
In the case of a beam extending from the origin to — oo while the
water surface is free for x > 0, the boundedness conditions for <p at
— oo requires that we take a4 = ae = 0 since x4 and x6 have negative
real parts and consequently «*«* and e*«* would yield exponentially
unbounded contributions to <p at x = — oo. We know that «j and x2
444
WATER WAVES
are pure imaginary with opposite signs, with x2, say, negative imagin-
ary. Since no progressing wave is assumed to come from the left, we
must then take a2 = 0. Thus the term a^x yields the transmitted
wave and the terms involving a3 and a5 yield disturbances which die
out exponentially at oo. The conditions (10.13.70) and (10.13.71 ) at
x = 0 now yield the following four linear equations:
(10.13.86)
= 0
a + "
- B + R
x6at~ik(B-R)
for the constants av a3, #5, R. For the amplitude R of the reflected
wave one finds
0
(10.13.87)
- 1 - 1 - 1 -f 1 ,
K! *3 *5 /*
1 3 5
Even in this relatively simple case of the semi-infinite beam the re-
flection coefficient is so complicated a function of the parameters
(even though it is algebraic in them) that it seems not worthwhile to
write it down explicitly. The results of numerical calculations based
on (10.13.87) will be given shortly.
In the case of the beam of finite length extending from <r — /
to x = 0 the eight conditions given by (10.13.75) and (10.13.76) must
be satisfied by the solution (10.13.84) of the differential equation
(10.13.74), and these conditions serve to determine the six constants
of integration and the amplitudes R and T of the reflected and trans-
mitted waves. The problem thus posed is quite straightforward but
extremely tedious as it involves solving eight linear eq nations for
eight complex constants. For details reference is again made to the
work of Wells [F.5].
This case of a floating beam was suggested to the author by J. H.
LONG WAVES IN SHALLOW WATER 445
Carr of the Hydraulics Structures Laboratory at the California Insti-
tute of Technology as one having practical possibilities; at his sug-
gestion calculations in specific numerical cases were carried out
in order to determine the effectiveness of such a breakwater. The
reason for considering such a structure for a breakwater as a
means of creating relatively calm water between it and the shore
is the following: a structure which floats on the surface without sink-
ing far into the water need not be subjected to large horizontal forces
and hence would not necessarily require a massive anchorage. How-
ever, in order to be effective as a reflector of waves such a floating
structure would probably have to be built with a fairly large dimen-
sion in the direction of travel of the incoming waves. As a consequence
of the length of the structure, it would be bent like a beam under the
action of the waves and hence could not in general be treated with
accuracy as a rigid body in determining its effectiveness as a barrier.
This brings with it the possibility that the structure might be bent
so much that the stresses set up would be a limiting feature in the
design. The specifications (as suggested by Carr) for a beam having
a width of one foot (parallel to the wave crest, that is) were:
Weight: 85 pounds •/ 'ft2
Moment of inertia (of area of cross-section): 0.2 /24
Modulus of elasticity: 437 X 107 pounds I ft2.
The depth of the water is taken as 40 feet. Simple harmonic progres-
sing waves having periods of 8 and of 15 sees, were to be considered,
and these correspond to wave lengths of 287 and 539 feet, and to cir-
cular frequencies a of 785 x 10~3 and 418 x 10~3 cycles per second,
respectively. The problem is to determine the reflecting power of the
beam under these circumstances when the length of the beam is varied.
In other words, we assume a wave train to come from the right hand
side of the beam and that it is partly transmitted under the beam to
the left hand side and partly reflected back to the right hand side.
The ratio R/B of the amplitude R of the reflected wave and the am-
plitude B of the incoming wave is a measure of the effectiveness of
the beam as a breakwater.
Before discussing the case of beams of finite length it is interesting
and worthwhile to consider semi-infinite beams first. Since the calcu-
lations are easier than for beams of finite length it was found possible
to consider a larger range of values of the parameters than was given
446 WATER WAVES
above. The results are summarized in the following tables (taken
from [F.5]):
TABLE A
X (ft) a l—\ W (pounds) I (ft) E^^-} h (ft) R/B
\secj \ ft2 J
539 0.418 85 0.20 437 X 107 40 0.14
287 0.785 85 0.20 437 X 107 40 0.19
225 1.0 85 0.20 437 X 107 40 0.23
150 1.5 85 0.20 437 X 107 40 0.32
113 2.0 85 0.20 437 X 107 40 0.43
In Table A the beam design data are as given above. At the two speci-
fied circular frequencies of 0.418 and 0.785 one sees that the floating
beam is quite ineffective as a breakwater since the reflected wave has
an amplitude of less than 1/5 of the amplitude of the incoming wave,
even for the higher frequency (and hence shorter wave length), which
means that less than 4 % of the incoming energy is reflected back. At
higher frequencies, and hence smaller wave lengths, the breakwater
is more effective, as one would expect. However the approximate
theory used to calculate the reflection coefficient R/B can be expected
to be accurate only if the ratio Xjh of wave length to depth is suffi-
ciently large, and even for the case A — 287 ft. (a =- .785) the re-
flection coefficient of value 0.19 may be in error by perhaps 10 % or
more since A/A is only about 7, and the errors for the shorter wave
lengths would be greater. Calculations for still other values of the
parameters are shown in Table B. The only change as compared with
TABLE B
W I E h RIB
539 0.418 384 0.20 437 X 107 40 0.51
287 0.785 384 0.20 437 X 107 40 0.75
the first two rows of Table A is that the weight per foot of the beam
has been increased by a factor of more than 4 from a value of 85
pounds] ft* to a value of 384 pounds] 'ft2. The result is a decided increase
in the effectiveness of the breakwater, especially at the shorter wave
length, since more than half (i.e. (.75)2) of the incoming energy would
be reflected back. However, this beneficial effect is coupled with a
decided disadvantage, since quadrupling the weight of the beam
LONG WAVES IN SHALLOW WATER 447
would cause it to sink deeper in the water in like proportion and hence
might make heavy anchorages necessary. Table C is the same as the
TABLE C
A
(7
W
/
E
h
R/B
539
.418
85
2.0
437 X 107
40
.26
287
.785
85
2.0
437 X 107
40
.32
00
1
first two rows of Table A except that the bending stiffness has been
increased by a factor of 10 by increasing the moment of inertia of the
beam cross-section from 0.2 //4 to 2.0 //4. Such an increase in stiffness
results in a noticeable increase in the effectiveness of the breakwater,
but by far not as great an increase as is achieved by multiplying the
weight by a factor of four. If the stiffness were to be made infinite
(i.e. if the beam were made rigid) the reflection coefficient could be
made unity, and no wave motion would be transmitted. This is
evidently true for a semi-infinite beam, but would not be true for a
rigid body of finite length.
TABLE D
A
a
W
7
E
h
RIB
539
.418
85 ~
(T
~Q
~ 40
. ooT
287
.785
85
0
0
40
.007
In Table D the difference as compared with Table A is that the
beam stiffness is taken to be zero. This means that the surface of the
water is assumed to be covered by a distribution of inert particles
weighing 85 pounds per foot. (Such cases have been studied by Gold-
stein and Keller [G.I].) As we observe, there is practically no reflec-
tion and this is perhaps not surprising since the mass distribution per
unit length has such a value that the beam sinks down into the water
only slightly.
One might perhaps summarize the above results as follows: A very
long beam can be effective as a floating breakwater if it is stiff enough.
However, a reasonable value for the stiffness (the value 0.2 given
above) leads to an ineffective breakwater unless the weight of the
beam per square foot is a fairly large multiple (say 8 or 10) of the
weight of water.
In practice it seems unlikely that beams long enough to be considered
448 WATER WAVES
semi-infinite would be practicable as breakwaters. (The term "long
enough" might be interpreted to mean a sufficiently large multiple of
the wave length, but since the wave lengths are of the order of 200
feet or more the correctness of this statement seems obvious. ) It there-
fore is necessary to investigate the effectiveness of beams of finite
length. Such an investigation requires extremely tedious calculations
—so much so that only a certain number of numerical cases have been
treated. These are summarized in the following tables.
a = .785, A = 287 a = .418, A = 539
I (ft) R/B I (ft) R/B
17.5 0 145.9 .17
49.2 .93 196.9 .53
72.9 0 291.8 .13
98.5 .75 443.0 .90
145.9 .10 583.6 .74
196.9 0 656.2 .62
291.8 .33 874.9 .07
450.4 .32 875.4 .08
583.6 .12 948.3 .54
656.3 .13 oo .14
875.4 .32
oo .19
In these tables the parameters have values the same as in the first
two rows of Table A, except that now lengths other than infinite
length are considered. The most noticeable feature of the results given
in the tables is their irregularity and the fact that at certain lengths
—even certain rather short lengths— the beam proposed by Carr
seems to be quite effective. For example, when the wave length is
287 ft. a beam less than 50 ft. long reflects more than 80 % of the
incoming energy. A beam of length 443 ft. is also equally effective at
the longer wave length of 539 ft.*
* It might not be amiss to consider the physical reason why it is possible that
a beam of finite length could be more effective as a breakwater than a beam of
infinite length. Such a phenomenon comes about, of course, through multiple
reflections that take place at the ends of the beam. Apparently the phases some-
times arrange themselves in the course of these complicated interactions in such
a way as to yield a small amplitude for the transmitted wave. That such a process
might well be sensitive to small changes in the parameters, as is noted in the
discussion, cannot be wondered at.
LONG WAVES IN SHALLOW WATER 449
However, the maximum effectiveness of any such breakwater
occurs for a specific wave length within a certain range of wave
lengths; thus the reflection of a given percentage of the incoming wave
energy would involve changing the length (or some other parameter) of
the structure in accordance with changes in the wave length of the
incoming waves. Also, the reflection coefficient seems to be rather
sensitive to changes in the parameters, particularly for the shorter
structures (a relatively slight change in length from an optimum value,
or a slight change in frequency, leads to a sharp decrease in the re-
flection coefficient). It is also probable —as was indicated earlier on the
basis of calculations by Wells [W.10] — that the shallow water approx-
imation used here as a basis for the theory is not sufficiently accurate
for a floating beam whose length is too much less than the wave length.
Nevertheless, it does seem possible to design floating breakwaters of
reasonable length which would be effective at a given wave length.
Perhaps it is not too far-fetched to imagine that sections could be
added to or taken away from the breakwater in accordance with
changing conditions.
Another consequence of the theory— which is also obvious on
general grounds— is that there is always the chance of creating a
large standing wave between the shore and the breakwater because of
reflection from the shore, unless the waves break at the shore; this
effect is perhaps not important if the main interest is in breakwaters
off beaches of not too large slope, since breaking at the shore line then
always occurs. (The theory developed here could be extended to cases
in which the shore reflects all of the incoming energy, it might be
noted.) In principle, the calculation of the deflection curve of the
structure, and hence also of the bending stresses in it, as given by the
theory is straightforward, but it is very tedious; consequently only
the reflection coefficients have been calculated.
CHAPTER 11
Mathematical Hydraulics
In this chapter the problems to be treated are, from the mathema-
tical point of view, much like the problems of the preceding chapter,
but the emphasis is on problems of rather concrete practical signifi-
cance. Aside from this, the essential difference is that external forces
other than gravity, such as friction, for example, play a major role in
the phenomena. Problems of various types concerning flows and
wave motions in open channels form the contents of the chapter. The
basic differential equations suitable for dealing with such flows under
rather general circumstances are first derived. This is followed by a
study of steady motions in uniform channels, and of progressing waves
of uniform shape, including roll waves in inclined channels. Flood
waves in rivers are next taken up, including a discussion of numerical
methods appropriate in such cases; the results of such calculations
toy a flood wave in a simplified model of the Ohio River and for a
model of its junction with the Mississippi are given. This discussion
follows rather closely the two reports made to the Corps of Engineers
of the U.S. Army by Stoker [S.23] and by Isaacson, Stoker, and
Trocsch [1.4], These methods of dealing with flood waves have been
applied, with good results, to a 400-mile stretch of the Ohio as it
actually is for the case of the big flood of 1945, and also to a flood
through the junction of the Ohio and the Mississippi; these results
will be discussed toward the end of this chapter.
There is an extensive literature devoted to the subject of flow in
open channels. We mention here only a few items more or less directly
connected with the material of this chapter: the famous Essai of
Boussinesq [B.17], the books of Bakhmeteff [B.3] and Rouse [R.10,
11] (in particular, the article by Gilcrest in [R.ll]), the Enzyklopadie
article of Forchheimer [F.6] and the booklet by Thomas [T.2].
451
452
WATER WAVES
11.1. Differential equations of flow in open channels
It has already been stated that the basic mathematical theory to
be used in this chapter does not differ essentially from the theory
derived in the preceding chapter. However, there are additional
complications due to the existence of significant forces beside gravity,
and we wish to permit the occurrence of variable cross-sections in the
channels. Consequently the theory is derived here again, and a some-
what different notation from that used in previous chapters is em-
ployed both for the sake of convenience and also to conform somewhat
with notations used in the engineering literature.
The theory is one-dimensional, i.e. the actual flow in the channel is
assumed to be well approximated by a flow with uniform velocity over
each cross-section, and the free surface is taken to be a level line in
each cross-section. The channel is assumed also to be straight enough
so that its course can be thought of as developed into a straight line
without causing serious errors in the flow. The flow velocity is denoted
by v, the depth of the stream (commonly called the stage in the
engineering literature) by y, and these quantities are functions of the
Fig. 11.1.1. River cross-section and profile
distance x down the stream and of the time t (cf. Fig. 11.1.1). The
vertical coordinates of the bottom and of the free surface of the stream,
as measured from the horizontal axis x9 are denoted by z(x) and
MATHEMATICAL HYDRAULICS 453
h(x, t), with z positive downward, h positive upward; thus y =• h + z.
The slope of the bed is therefore counted positive in the positive
^-direction, i.e. downward. The breadth of the free surface at any
section of the stream is denoted by B.
The differential equations governing the flow are expressions of the
laws of conservation of mass and momentum. In deriving them the
following assumptions, in addition to those mentioned above, are
made *: 1) the pressure in the water obeys the hydrostatic pressure
law, 2 ) the slope of the bed of the river is small, 3) the effects of friction
and turbulence can be accounted for through the introduction of a
resistance force depending on the square of the velocity v and also, in
a certain way to be specified, on the depth y.
We first derive the equation of continuity from the fact that the
mass gAAx included in a layer of water of density p, thickness Ax,
and cross-section area A, changes in its flow along the stream only
through a possible inflow along the banks of the stream, say at the
rate qq per unit length along the river. The total flow out of the ele-
ment of volume A Ax is given by the net contributions Q(Av)xAx from
the flow through the vertical faces plus the contribution @BhtAx due
to the rise of the free surface, with B the width of the channel; since
Bht represents the area change At it follows that the sum [(Av)x +
At}Ax equals the volume influx qAx over the sides of the channel,
with q the influx per unit length of channel. The subscripts x and t
refer, of course, to partial derivatives with respect to these variables.
Tfye equation of continuity therefore has the form
(11.1.1) (Av)x+At = q.
It should be observed that A — A(y(x, t)9 x) is in the nature of
things a given function of y and x, although y(x, t) is an unknown
function to be determined; in addition, q = q(x, t) depends in general
on both x and / in a way that is supposed given. In the important
special case of a rectangular channel of constant breadth B, so that
A — By, the equation of continuity takes the form
(11.1.2) vxy + vyx + yt = q/B.
The equation of motion is next derived for the same slice of mass
m = qAAx by equating the rate of change of momentum d(mv)jdt
* These assumptions are not the minimum number necessary: for example,
assumption 1 ) has as a consequence the independence of the velocity on the vertical
coordinate if that were true at any one instant (cf. the remarks on this point
in Ch. 2 and Ch. 10).
454 WATER WAVES
to the net force on the element. We write the equation of motion for
the horizontal direction:
(11.1.8) Q — (AvAx) = HAx—FfAx cos 9? +QgA Ax sin q>.
dt
In this equation H represents the unbalanced horizontal pressure
force at the surface of the element. The angle 9? is the slope angle of the
bed of the channel, reckoned positive downward. The quantity Ff re-
presents the friction force along the sides and bottom of the channel,
and the term QgAAx sin <p represents the effect of gravity in accelerat-
ing the slice down-hill as manifested through the normal reaction of
the stream bed. Since 9? was assumed small we may replace sin (p by
the slope S — dz/dx and cos 99 by 1. In the frictional resistance term
we set
This is an empirical formula called Manning's formula. The resistance
is thus proportional to the square of the velocity and is opposite to its
direction; in addition, the friction is inversely proportional to the
4/3-power of the hydraulic radius R, defined as the ratio of the cross-
section area A to the wetted perimeter (thus R = Byj(B -\- 2y) for a
rectangular channel and R = y for a very wide rectangular channel),
and inversely proportional to y, a roughness coefficient.
We calculate next the momentum change Qd(AvAx)/dt. In doing so,
we observe that the symbol d/dt must be interpreted as the particle
derivative (cf. Chapter 1.1 and equation (1.1.8)) d/dt + vd/dx since
Newton's law must be applied in following a given mass particle along
its path x = x(t). However, the law of continuity (11.1.1 ) derived above
is clearly equivalent to writing d(AAx)ldt — qAx, with d/dt again in-
terpreted as the particle derivative. Since
— (AvAx)=v — (AAx)+AAx —
dt dt dt
it follows that
— (AvAx)—AAx(vvx+vt)-}-qvAx.
dt
Finally, the net contribution HAx of the pressure forces over the
surface of the slice is calculated as follows: The total pressure over a
vertical face of the slab is given by f v Qg[y(x, t) — £]b(x, f ) d£ from
the hydrostatic pressure law (cf. Fig. 11.1.1); while the component
MATHEMATICAL HYDRAULICS 455
in the ^-direction of the total pressure over the part of the slice in
contact with the banks of the river is given by
try \
j I 6S[y "~ S]bx(x9 f ) d£ Ax, we have for HAx the following equation:
(11.1.5)
/o
my
-t-f f
WO
[1
= -- I @gyxb(%, %)d!;~ ~ogAyx.
Jo
In this calculation the integrals involving bx cancel out, and we have
used the fact that yx is independent of f .
Adding all of the various contributions we have
(11.1.6) vt+wx+ - v=Sg-S,g-gyx
scL
upon defining what is called the friction slope Sf by the formula
(11-1.7) S,
with Ff defined by (11.1.4). It should perhaps be mentioned that the
term qv/A on the left hand side of (11.1.6) arises because of the
tacit assumption that flows enter the main stream from tributaries
or by flow over the banks at zero velocity in the direction of the main
stream; if such flows were assumed to enter with the velocity of the
main stream, the term would not be present— it is, in any case, a
term which is quite small. If we introduce A = A(y(x, t), x) in
(11.1.1) the result is
(11.1.8) Ayyxv + Axv + Avx + Ayyt — q.
The two differential equations (11.1.6) and (11.1.8), which serve
to determine the two unknown functions, the depth y(x, t) and the
velocity v(x, t), are the basic equations for the study of flood waves in
rivers and flows in open channels generally. For any given river or
channel it is thus necessary to have data available for determining
the cross-section area A and the quantities y and R in the resistance
term Ff as functions of x and j/, and of the slope S of its bed as a
function of x in order to have the coefficients in the differential equa-
tions (11.1.6) and (11.1.8) defined. Three of these quantities are
purely geometrical in character and could in principle be determined
456 WATER WAVES
from an accurate contour map of the river valley, but the determina-
tion of the roughness coefficient y of course requires measurements of
actual flows for its determination.
11.2. Steady flows. A junction problem
We define a steady flow in the usual fashion to be one for which the
velocity v and depth y are independent of the time, that is, vt— yt — 0.
In this section channels of constant rectangular cross-section and
constant slope will be considered for the most part. It follows from the
equation of continuity (cf. (11.1.2)):
Vt + vyx + yvx = 0,
that for steady flow
(11.2.1) (vy)x = 0 whence vy — D (Da constant),
when no flow into the channel from its sides occurs (i.e. q = 0 in
(11.1.2)). Similarly, the equation of motion (cf. (11.1.6))
^ + vvx + gyx + g(S, -S) = 0
yields
(11.2.2) vvx + gy, + g(St -S) = Q.
It follows from equation (11.2.1) that
D A D
v = _ and vx = -- - yx,
y y2
so that equation (11.2.2) becomes
Here the hydraulic radius is given by R = y/(l + 2y/B) because the
channel is assumed to be rectangular in cross-section.
For a channel with given physical parameters such as cross-section,
resistance coefficient, etc. the steady flows would provide what are
called backwater curves. In general, one could in principle always
find steady solutions y = y(x) and v = v(x) for a non-uniform chan-
nel. The explicit determination of the stage y and discharge rate BD
as functions of x would be possible by numerical integration of the
pair of first order ordinary differential equations arising from (11.1.6)
and (11.1.8) when time derivatives are assumed to vanish.
MATHEMATICAL HYDRAULICS 457
We note that equation (11.2.3) has the simple solution y = constant
for y satisfying
This means that we can find a flow of uniform depth and velocity
having a constant discharge rate BD (B is, as in the preceding section,
the width of the channel). Conversely, by fixing the depth y we can
find the discharge from (11.2.4) appropriate to the corresponding
uniform flow. Physically this means that the flow velocity is chosen
so that the resistance due to turbulence and friction and the effect of
gravity down the slope of the stream just balance each other. We re-
mark that if (11.2.4) is satisfied at any point where the coefficient
g — D2lyz of yx in (11.2.3) does not vanish, then y = constant is the
only solution of (11.2.3) because of the fact that the solution is then
uniquely determined by giving the value of y at any point x. We note
that g — D2/y3 = 0 corresponds to v = VliJ/> *-e- to a fl°w at critical
speed (a term to be discussed in the next section), since D = vy.
Furthermore, the differential equation (11.2.3) can be integrated to
yield x as a function of y:
(11.2.5) a? =
when x — 0 for y — yQ.
We proceed to make use of (11.2.5) in order to study a problem
involving a steady flow at the junction of two rivers each having a
rectangular channel. Later on, the same problem will be treated but
for an unsteady motion resulting from a flood wave traveling down
one of the branches, and such that the steady flow to be treated here
is expected to result as a limit state after a long time. The numerical
data arc chosen here for the problem in such a way as to correspond
roughly with the actual data for the junction of the Ohio River with
the Mississippi River. Thus the Ohio is assumed to have a rectangular
channel 1000 feet in width and a constant slope of .5 feet/mile. In
Manning's formula for the resistance the constant y is assumed given
by y =. (1.49/n)2 in terms of Manning's roughness coefficient n, and
n is given the value 0.03. The upstream branch of the Mississippi was
taken the same in all respects as the Ohio, but the downstream branch
is assumed to have twice the breadth, i.e. 2000 feet, and its slope to
458 WATER WAVES
have a slightly smaller value, i.e. 0.49 feet/mile instead of 0.5 feet/mile.
With these values of the parameters, a flow having the same uniform
depth of 20 feet in all three branches is possible— the choice of the
Lower
Mississippi
Fig. 11.2.1. Junction of Ohio and Mississippi Rivers
value 0.49 feet/mile for the slope of the downstream branch of the
Mississippi River was in fact made in order to ensure this. Later on
we intend to calculate the progress of a flood which originates at a
moment when the flow is such a uniform flow of depth 20 feet. The
flood wave will be supposed to initiate at a point 50 miles up the Ohio
from the junction and to be such that the Ohio rises rapidly at that
point from the initial depth of 20 feet to a depth of 40 feet in 4 hours.
A wave then moves down the Ohio to the junction and creates waves
which travel both upstream and downstream in the Mississippi as well
as a reflected wave which travels back up the Ohio. After a long time
we would expect a steady state to develop in which the depth at the
point 50 miles up the Ohio is 40 feet, while the depth far upstream in
the Mississippi would be the original value, i.e. 20 feet (since we would
not expect a retardation of the flow far upstream because of an inflow at
the junction). Downstream in the Mississippi we expect a change in the
flow extending to infinity. It is the steady flow with these latter charac-
teristics that we wish to calculate in the present section. ( See Fig. 11.2.1)
We remark first of all that the stream velocities in all of the three
branches will always be subcritical— in fact, they are of the order of
a few miles per hour while the critical velocities \/gt/ are of the order
of 15 to 25 miles per hour. It follows that the quantity g — D2/t/3 in the
integrand of the basic formula (11.2.5) for the river profiles (i.e. the
curve of the free surface) is always positive. The integrand I(y) in
MATHEMATICAL HYDRAULICS
459
that formula has the general form indicated by Fig. 11.2.2 in the case
of flows at subcritical velocities. The vertical asymptote corresponds
to the value of y for which a steady flow of constant depth exists
Fig. 11.2.2. The integrand iu the wave profile formula
(cf. (11.2.4)), since the square bracket (the denominator in the inte-
grand ) vanishes for this value. It follows that x can become positive
infinite for finite values of y only if y takes on somewhere this value;
but in that case we have seen that the whole flow is then one with
constant depth everywhere. Consequently the downstream side of the
Mississippi carries a flow of constant speed and depth, though the
values of these quantities are not known in advance. However, in the
upstream branch of the Mississippi the flow need not be constant, and
of course we do not expect it to be constant in the Ohio: in these
branches x must be taken to be decreasing on going upstream and
consequently the negative portion of I(y) indicated in Fig. 11.2.2
comes into use since we may, and do, set x = 0 at the junction.
For the sake of convenience we use subscripts 1, 2, and 3 to refer to
all quantities in the Ohio, the upstream branch of the Mississippi, and
the downstream branch of the Mississippi respectively. The conditions
to be satisfied at the junction are chosen to be
(11.2.6) * = y, = y, = y,
(11.2.7) D1+D2
460 WATER WAVES
The first condition simply requires the water level to have the same
value yj (which is, however, not known in advance) in all three bran-
ches, while the second states, upon taking account of the first condition,
that the combined discharge of the two tributaries makes up the total
discharge in the main stream. The quantity Z)2, the discharge in the
upper Mississippi, is known since the flow far upstream in this branch
is supposed known— i.e. it is a uniform flow of depth 20 feet.
By using (11.2.7) in (11.2.4) as applied to the lower branch of the
Mississippi (in which the flow is known to be constant) we have
Next, we write equation (11.2.5) for the 50-mile stretch of the Ohio
which ends at the point where the depth in that branch was prescribed
to be 40 feet (and which was the point of initiation of a flood wave);
the result is
(11.2.9) 50- \ I(y,Dl,Bl)dy
in which it is indicated that D and B (as well as all other parameters)
are to be evaluated for the Ohio; the quantity y has the value 40/5280
in miles. Equations (11.2.8) and (11.2.9) are two equations containing
j/;- and D! as unknowns, since Z)2 is known. They were solved by an
iterative process, i.e. by taking for Dl an estimated value, determining
a value for yj from (11.2.9), reinserting this value in (11.2.8) to deter-
mine a new value for Z)1, etc. The results obtained by such a calcula-
tion are as follows:
y\ = 2/2 = 2/3 = Vt = 81-2 feet
vl = 4.83 miles/hour, v2 = 1.53 miles/hour, v3 ~ 3.18 miles/hour.
The profiles of the river surface can now be computed from (11.2.5);
the results are given in Fig. 11.2.3.
The solution of the mathematical problem has the features we
would expect in the physical problem. The flow velocity and stage are
increased at the junction, even quite noticeably, by the influx from
the Ohio. Upstream in the Mississippi the stage decreases rather
rapidly on going away from the junction, and very little backwater
effect is noticeable at distances greater than 50 miles from the junc-
tion. This illustrates a fact of general importance, i.e. that backwater
MATHEMATICAL HYDRAULICS
461
effects in long rivers arising from even fairly large discharges of tri-
butaries into the main stream do not persist very far upstream,
but such an influx has an influence on the flow far downstream.
For unsteady motions this general observation also holds, and is in
fact one of the basic assumptions used by hydraulics engineers in
Ohio
Upstream
Mississippi
y feet
40
Downstream
Mississippi
4-
, -50 Junction 50 100 miles
Fig. 11.2.3. Steady flow profile in a model of the Ohio and Mississippi Rivers
computing the passage of flood waves down rivers (a process called
flood routing by them). Later on, in sec. 6 of this chapter, we shall
deal with the unsteady motion described above in our model of the
Ohio-Mississippi system, and we will see that the unsteady motion
approaches the steady motion found here as the time increases.
11.3. Progressing waves of fixed shape. Roll waves
In addition to the uniform steady flows treated above there also
exist a variety of possible flows in uniform channels in the form of
progressing waves moving downstream at constant speed without
change in shape. Such waves arc expressed mathematically by depths
y(x, t) and velocities v(x9t) in the form
(11.8.1) y(x, t) = y(x — Ut)< v(x, t) = v(x — Ut), U = const.
462 WATEE WAVES
The constant U is of course the propagation speed of the wave as
viewed from a fixed coordinate system; if viewed from a coordinate
system moving downstream with constant velocity U the wave profile
would appear fixed, and the flow would appear to be a steady flow
relative to the moving system. It is convenient to introduce the va-
riable £ by setting
(11.3.2) C = x - Ut
so that y and v are functions of £ only. In this case the equations of
continuity and motion given by (11.1.6) and (11.1.8) become, for a
rectangular channel of fixed breadth and slope:
m oo^ f (*> - U)y: + yvc = 0,
(11.3.3) | ^ ^ ^
with «S the slope of the channel and Sf defined, as before, by
(11.3.4) Sf=
The first equation of (11.3.3) can be integrated to yield
(11.3.5) (v — U)y = D = const.
as one readily verifies, and the second equation then takes the form
The first order differential equation (11.3.6) has a great variety of
solutions, which have been studied extensively, for example by Tho-
mas [T.I], but most of them are not very interesting from the physical
point of view. However, one type of solution of (11.3.6) is particularly
interesting from the point of view of the applications, and we there-
fore proceed to discuss it briefly. The solution in question furnishes
the so-called monoclinal rising flood wave in a uniform channel (see
the article by Gilcrest in the book of Rouse [R.ll, p. 644]). This, as
its name suggests, is a progressing wave the profile of which tends to
different constant values (and the flow velocity v also to different con-
stant values) downstream and upstream, with the lower depth down-
stream, connected by a steadily falling portion, as indicated schema-
tically in Fig. 11.3.1. In this wave the propagation speed U is larger
MATHEMATICAL HYDRAULICS
463
than the flow velocity v. It is always a possible type of solution of
(11 .3.6) if the speed of propagation of the wave relative to the flow is
Fig. 11.3.1. Monoclinal rising flood wave
subcritical, i.e. if (£7— v)2 is less than gy, in which case the coefficient
of the first derivative term in (11.3.6) is seen to be positive. This can
be verified along the following lines. The differential equation can be
solved explicitly for f as a function of y\
(11.3.7)
-f
Jv*
with the integrand I(y) defined in the obvious manner; here y* is the
value of y corresponding to £ — 0. The function I(y) has the general
Fig. 11.8.2. The integrand I(y) for a monoclinal wave
464 WATER WAVES
form shown in Fig. 11.3.2 if the propagation speed U and the constant
D in (11.8.5) are chosen properly. The main point is that the curve has
two vertical asymptotes at y — yQ and y = yl between which I(y)
is negative. By choosing y* between y0 and yl we can hope that
f -* + oo as y -* t/0, while £ -> — oo as y -> yx: all that is necessary
is that I(y) becomes infinite at j/0 and yl of sufficiently high order.
This is, in fact, the case; we can select values of D and U in such a
way that I(y) becomes infinite at t/0 and yl through setting the quan-
tity Sj — S in (11.3.6) equal to zero, i.e. by choosing D and U such that
(11.3.8)
For given positive values of z/0 and yl these are a pair of linear equa-
tions (after taking a square root) which determine U and D uniquely.
An elementary discussion of the possible solutions of these equations
shows that U must be positive and D negative, and this means, as wo
see from (11.3.5), that U is larger than v9 i.e. the propagation speed of
the wave is greater than the flow speed.
By taking the numerical data for the model of the Ohio given in the
preceding section and assuming the depth far upstream to be 40 feet,
far downstream 20 feet, it was found that the corresponding mono-
clinal flood wave in the Ohio would propagate with a speed of 5
miles/hour. The shape of the wave will be given later in sec. 6 of this
chapter, where it will be compared with an unsteady wave obtained by
gradually raising the level in the Ohio at one point from 20 feet to
40 feet, then holding the level fixed there at the latter value, and cal-
culating the downstream motion which results. We shall see that the
motion tends to the monoclinal flood wave obtained in the manner
just now described. Thus the unsteady wave tends to move eventually
at a speed of about 5 miles/hour, while on the other hand, as we know
from Chapter 10 (and will discuss again later on in this chapter), the
propagation speed of small disturbances relative to the stream is <\/gy
and hence is considerably larger in the present case, i.e. of the order
of 15 to 25 miles/hour. This important and interesting point will be
discussed in sec. 6 below.
MATHEMATICAL HYDRAULICS
465
Fig. 11.8.8. Roll waves, looking down stream (The Grunnbach, Switzerland)
466 WATER WAVES
We turn next to another type of progressing waves in a uniform
channel which can be described with the aid of the differential equa-
tion (11.3.6), i.e. the type of wave called a roll wave. A famous exam-
ple of such waves is shown in Fig. 11.3.3, which is a photograph taken
from a book of Cornish [C.7], and printed here by the courtesy of the
Cambridge University Press. As one sees, these waves consist of a
series of bores (cf. Chapter 10.7) separated by stretches of smooth
flow. The sketch of Fig. 11.3.4 indicates this more specifically. Such
Fig. 11.3.4. Roll waves
waves frequently occur in sufficiently steep channels as, for example,
spill-ways in dams or in open channels such as that of Fig. 11.3.3.
Roll-waves sometimes manifest themselves in quite unwanted places,
as for example in the Los Angeles River in California. The run-off from
the steep drainage area of this river is carried through the city of Los
Angeles by a concrete spill- way; in the brief rainy season a large
amount of water is carried off at high velocity. It sometimes happens
that roll waves occur with amplitudes high enough to cause spilling
over the banks, though a uniform flow carrying the same total amount
of water would be confined to the banks. The phenomenon of roll
waves thus has some interest from a practical as well as from a theore-
tical point of view; we proceed to give a brief treatment of it in the
remainder of this section following the paper of Dressier [D.12]. In
doing so, we follow Dressier in taking what is called the Ch^zy for-
mula for the resistance rather than Manning's formula, as has been
done up to now. The Ch£zy formula gives the quantity Sf the following
definition:
in which r2 is a "roughness coefficient" and R is, as before, the hy-
draulic radius. For a very broad rectangular channel, the only case
we consider, R = y. Under these circumstances the differential
equation (11.3.6) takes the form
MATHEMATICAL HYDRAULICS 467
sS_^(Uy+D)\Uy+D\
0 o "
dy
Z)2
- _
q
as can be readily seen.
It is natural to inquire first of all whether (11.3.10) admits of solu-
tions which are continuous periodic functions of £ since this is the
general type of motion we seek. There are, however, no such periodic
and continuous solutions (cf. the previously cited paper of Thomas
[T.I]) of the equations; in fact, since the right hand side of (11.3.10)
can be expressed as the quotient of cubic polynomials in y the
types of functions which arise on integrating it are linear combina-
tions of the powers, the logarithm, and the arc tangent function and
one hardly expects to find periodic functions on inverting solutions
C(j/) of this type. This fact, together with observations of roll waves of
the kind shown in Fig. 11.3.3, leads one to wonder whether there might
not be discontinuous periodic solutions of (11.3.10) with discontinui-
ties in the form of bores, which should be fitted in so that the discon-
tinuity or shock conditions described in sec. 6 of the preceding chap-
ter * are satisfied. This Dressier shows to be the case; he also gives a
complete quantitative analysis of the various possibilities.
The starting point of the investigation is the observation, due to
Thomas [T.I], that only quite special types of solutions of (11.3.10)
come in question once the roll wave problem has been formulated in
terms of a periodic distribution of bores. In fact, we know from Chap-
ter 10 that the flow relative to a bore must be subcritical behind a
bore but supercritical in front of it; consequently there must be an in-
termediate point of depth j/0, say, (cf. Fig. 11.3.4) where the smooth
flow has the critical speed, i.e. where
(11.3.11) K- tf)2 = g2/0>
since C7, the speed of the bore, coincides with the propagation speed
of the wave. At such a point the denominator on the right hand side
of (11.3.10) vanishes, since D = (v — U)y9 and hence dy/d£ would be
infinite there— contrary to the observations— unless the numerator
of the right hand side also vanishes at that point. The right hand side
can now be written as a quotient of cubic polynomials, and we know
* The shock conditions were derived in Chapter 10 under the assumption
that no resistances were present. As one would expect, the resistance terms play
no role in shock conditions, as Dressier [D.12] verifies in his paper.
468 WATER WAVES
that numerator and denominator have yQ as a common root; it follows
that a factor y — yQ can be cancelled and the differential equation
then can be put in the form
after a little algebraic manipulation. Since the denominator on the
right hand side is positive and since we seek solutions for which
dy/d£ is everywhere (cf. Fig. 11.3.4) positive, it follows in particular
that the quadratic in the numerator must be positive for y = yQ. This
leads to the following necessary condition for the formation of roll
waves
(11.3.13) 4r2 < 5,
obtained by using (11.3.11 ) and other relations. A practically identical
inequality was derived by Thomas on the basis of the same type of
reasoning. The inequality states that the channel roughness, which is
larger or smaller with r2, must not be too great in relation to the steep-
ness of the channel, and this corroborates observations by Rouse
[R.10] that roll waves can be prevented by making a channel suffi-
ciently rough. Dressier also shows in his paper that it is important for
the formation of roll waves that the friction force for the same rough-
ness coefficient and velocity should increase when the depth decreases;
he finds, in fact, that roll waves would not occur if the hydraulic ra-
dius R in the Ch£zy formula (11.3.9) were to be assumed independent
of the depth y.
Dressier goes on in his paper to show that smooth solutions of
(11.3.12) can be pieced together through bores in such a way that the
conditions referring to continuity of mass and momentum across the
discontinuity are satisfied as well as the inequality requiring a loss
rather than a gain in energy. For the details of the calculations and a
quantitative analysis in terms of the parameters, the paper of Dressier
should be consulted, but a few of the results might be mentioned here.
Once the values of the slope S and the roughness coefficient r2 are
prescribed by the physical situation, and the wave propagation speed
U is arbitrarily given, there exists a one-parameter family of possible
roll-waves. As parameter the wave length A, i.e. the distance between
two successive bores, can be chosen; if this parameter is also fixed,
the roll wave solution is uniquely determined. A specific solution
MATHEMATICAL HYDRAULICS 469
would also be fixed if the time period of the oscillation were to be
fixed together with one other parameter— the average discharge rate,
say. Perhaps it is in this fashion that the roll waves are definitely
fixed in some cases — for example, the roll waves down the spill- way
of a dam are perhaps fixed by the period of surface waves in the dam
at the crest of the spill-way. Schonfeld [S.4a] discusses the problem
from the point of view of stability and arrives at the conclusion that
only one of the solutions obtained by Dressier would be stable, and
hence it would be the one likely to be observed.
11.4. Unsteady flows in open channels. The method of characteristics
In treating unsteady flows it becomes necessary to integrate the
nonlinear partial differential equations (11.1.1) and (11.1.6) for pre-
scribed initial and boundary conditions. It has already been mentioned
that such problems fall into the same category as the problems treated
in the preceding chapter, since they are of hyperbolic type in two
independent variables and thus amenable to solution by the method
of characteristics. It is true that the equations (11.1.1) and (11.1.6)
are more complicated than those of Chapter 10 because of the occur-
rence of the variable coefficient A and of the resistance term, so that
solutions of the type called simple waves (cf. Ch. 10.3) do not exist for
these equations. Nevertheless the theory of characteristics is still
available and leads to a variety of valuable and pertinent observa-
tions regarding the integration theory of equations (11.1.1) and (11.1.
6) which are very important. The essential facts have already been
stated in Chapter 10.2, but we repeat them briefly here for the sake
of preserving the continuity of the discussion. Our emphasis in this
chapter is on numerical solutions, which can be obtained by operating
with the characteristic form of the differential equations, but since we
shall not actually use the characteristic form for such purposes we
shall base the discussion immediately following on a special case, al-
though the results and observations are applicable in the most general
case. The special case in question is that of a river of constant rectan-
gular section and uniform slope, with no flow into the river from the
banks (i.e. q = 0 in (11.1.2) and (11.1.6)). In this case the differential
equations can be written as follows:
(11.4.1) vxy +vyx + yt = 0,
(11.4.2) vt + vvx + gyx + E = 0.
470 WATER WAVES
We have introduced the symbol E for the external forces per unit
mass:
(11.4.3) E = - gS + gSf9 S = const.
The term E differs from the others in that it contains no derivatives
of y or v.
The theory of characteristics for these equations can be approached
very directly * in the present special case by introducing a new
quantity c to replace y, as follows:
(11.4.4) c - ^gy.
This quantity has great physical significance, since it represents— as
we have seen in Chapter 10— the propagation speed of small disturb-
ances in the river. From (11.4.4) we obtain at once the relations
(11-4.5) 2ccx = gyX9 2cct = gyt,
and the differential equations (11.4.1) and (11.4.2) take the form
2cc x + vt + vvx + E = 0,
(11.4.6)
' 2w?a + 2ct = 0.
These equations are next added, then subtracted, to obtain the follow-
ing equivalent pair of equations:
a
— — I — — — >
dx dt j
(11.4.7)
We observe that the derivatives in these equations now have the form
of directional derivatives— indeed, to achieve that was the purpose of
the transformation— so that c and v in the first equation, for example,
are both subject to the operator (c + v)d/dx + d/dt, which means
that these functions are differentiated along curves in the a% £-plane
which satisfy the differential equation dx/dt = c + v. In similar
fashion, the functions c and v in the second equation are both subject
to differentiation along curves satisfying the differential equation
dx/dt = — c + fl-
it is entirely feasible to develop the integration theory of equations
(11.4.7) quite generally on the basis of these observations (as is done,
for example, in Courant-Friedrichs [C.9, Ch. 2]), but it is simpler, and
leads to the same general results, to describe it for the special case in
* For a treatment which shows quite generally how to arrive at the for-
mulation of the characteristic equations, see Courant-Friedrichs [C.9, Ch. 2].
MATHEMATICAL HYDRAULICS 471
which the resistance force Ff is neglected so that the quantity E in
(11.4.7) is a constant (see (11.4.3)). In this case the equations (11.4.7)
can be written in the form
(11.4.7 )t
as one can readily verify. But the interpretation of the operations de-
fined in (11.4.7)! has just been mentioned: the relations state that the
functions (v ± 2c + Et) are constant for a pbint moving through the
fluid with the velocity (v ± c), or, as we may also put it, for a point
whose motion in the x9 /-plane is characterized by the ordinary dif-
ferential equations dxjdt = v ± c. That is, we have the following
situation in the tr, /-plane: There are two sets of curves, Cl and C2,
called characteristics, which are the solution curves of the ordinary
differential equations
dx _
— =t;-4-c, and
dt
(11.4.8)
"c- dX -v t
L/2- — " — '
and we have the relations
'•v-\-2c-}-Et=kl= const, along a curve Cx and
(11.4.9)
1 v — 2c+Et =k2=^ const, along a curve C2.
Of course the constants k^ and k2 will be different on different curves
in general. It should also be observed that the two families of charac-
teristics determined by (11.4.8) arc really distinct because of the fact
that c — \/gy ^ 0 since we suppose that y > 0, i.e. that the water
surface never touches the bottom.
By reversing the above procedure it can be seen rather easily that the
system of relations (11.4.8) and (11.4.9) is completely equivalent to
the system of equations (11.4.6) for the case of constant bottom slope
and zero resistance, so that a solution of either system yields a solution
of the other. In fact, if we set /(#, /) = v + 2c + Et and observe that
f(x, t) = k± = const, along any curve x — x(t) for which dxjdt =
v + c it follows that along such curves
(11.4.10)
dt
472 WATER WAVES
In the same way the function g(x, t) = v — 2c + Et satisfies relation
(11.4.11) ft + (v - c)gx = 0
along the curves for which dx/dt = v — c. Thus wherever the curve
families Cl and C2 cover the as, J-plane in such a way as to furnish a
curvilinear coordinate system the relations (11.4.10) and (11.4.11)
hold. If now equations (11.4.10) and (11.4.11) are added and the
definitions of f(x9 1) and g(x, t) are recalled it is readily seen that the
first of equations (11.4.6) results. By subtracting (11.4.11) from
(11.4.10) the second of equations (11.4.6) is obtained. In other words,
any functions v and c which satisfy the relations (11.4.8) and (11.4.9)
will also satisfy (11.4.6) and the two systems of equations are there-
fore now seen to be completely equivalent.
As we would expect on physical grounds, a solution of the original
dynamical equations (11.4.6) could be shown to be uniquely deter-
mined when appropriate initial conditions (for t = 0, say) and boun-
dary conditions are prescribed; it follows that any solutions of (11.4.
8) and (11.4.9) are also uniquely determined when such conditions
are prescribed since we know that the two systems of equations are
equivalent.
At first sight one might be inclined to regard the relations (11.4.8)
and (11.4.9) as more complicated than the original pair of partial
differential equations, particularly since the right hand sides of (11.4.8)
are not known and hence the characteristic curves are also not known.
Nevertheless, the formulation in terms of the characteristics is quite
useful in studying properties of the solutions and also in studying
questions referring to the appropriateness of various boundary and
initial conditions. In Chapter 10.2 a detailed discussion along these
lines is given; we shall not repeat it here, but will summarize the con-
clusions. The description of the properties of the solution is given in
the x9 J-plane, as indicated in Fig. 11.4.1. In the first place, the values
of v and c at any point P(x9 t) within the region of existence of the solu-
tion are determined solely by the initial values prescribed on the segment
of the x-axis which is subtended by the two characteristics issuing from P.
In addition, the two characteristics issuing from P are themselves also
determined solely by the initial values on the segment subtended by
them. Such a segment of the ff-axis is often called the domain of de-
pendence of the point P. Correspondingly we may define the range of
influence of a point Q on the #-axis as the region of the x, £-plane in
which the values of v and c are influenced by the initial values assigned
MATHEMATICAL HYDRAULICS
473
to point Q, i.e., it is the region between the two characteristics issuing
from Q. In Fig. 11.4.1 we indicate these two regions.
t
Range of influence of Q
Domain of
determmacy
J
Domain of dependence of P
Fig. 11.4.1. The role of the characteristics
We are now in a position to understand the role of the charac-
teristics as curves along which discontinuities in the first and higher
derivatives of the initial data are propagated, since it is reasonable to
expect (and could be proved) that those points P whose domains of
dependence do not contain such discontinuities are points at which
the solutions v and c also have continuous derivatives. On the other
hand, it could be shown that a discontinuity in the initial data at a
certain point docs not in general die out along the characteristic
issuing from that point. Such a discontinuity (or disturbance in the
water) therefore spreads in both directions over the surface of the
water with the speed v + c in one direction and v — c in the other in
view of the interpretation given to the characteristics through the
relations (11 .4.7 )r Since v is the velocity of the water particles we see
that c represents quite generally the speed at which a discontinuity
in a derivative of the initial data propagates relative to the moving
water. We are therefore justified in referring to the quantity c = ^/gy
as the wave speed or propagation speed.
We considered above a problem in which only initial conditions,
and no boundary conditions, were prescribed. In the problems we
consider later, however, such boundary conditions will occur in the
form of conditions prescribed at a certain fixed point of the river in
terms of the time: for example, the height, or stage, of the river might
be given at a certain station as a function of the time. In other words,
474 WATER WAVES
conditions would be prescribed not only along the #-axis of our x, t-
plane, but also along the £-axis (in general only for t > 0) for a certain
fixed value of x. The method of finite differences used in Chapter 10.2
to discuss the initial value problem, with the general result given above,
can be modified in an obvious way to deal with cases in which bound-
ary conditions are also imposed. In doing so, it would also become
clear just what kind of boundary conditions could and should be im-
posed. For example, in the great majority of rivers— in fact, for all
in which the flow is subcritical, i.e. such that v is everywhere less than
the wave speed \/^jy—it is possible to prescribe only one condition
along the J-axis, which might be either the velocity v or the depth y,
in contrast with the necessity to impose two conditions along the ay-
axis. This fact would become obvious on setting up the finite differ-
ence scheme, and examples of it will be seen later on.
Finally, it should be stated that the role of the characteristics, and
also the method of finite differences applied to them could be used
with reference to the general case of the characteristic equations as
embodied in the equations (11.4.7) and (11.4.8) in essentially the
same way as was sketched out above for the system comprised of
(11.4.8) and (11.4.9) which referred to a special case. In particular,
the role of the characteristics as curves along which small disturbances
propagate, and their role in determining the domain of dependence,
range of influence, etc. remain the same.
11.5. Numerical methods for calculating solutions of the differential
equations for flow in open channels
It has already been stated that while the formulation of our pro-
blems by the method of characteristics is most valuable for studying
many questions concerned with general properties of the solutions
of the differential equations, it is in most cases not the best formula-
tion to use for the purpose of calculating the solutions numerically.
That is not to say that the device of replacing derivatives by differ-
ence quotients should be given up, but rather that this device should
be used in a different manner. The basic idea is to operate with finite
differences by using a fixed rectangular net in the x9 /-plane, in con-
trast with the method outlined in Chapter 10.2, in which the net of
points in the x9 J-plane at which the solution is to be approximated is
determined only gradually in the course of the computation. In the
latter procedure it is thus necessary to calculate not only the values
MATHEMATICAL HYDRAULICS
475
of the unknown functions v and c, but also the values of the coordinates
x, t of the net points themselves, whereas a procedure making use of a
fixed net would require the calculation of v and c only, and it would
also have the advantage of furnishing these values at a convenient set
of points.
However, the question of the convergence of the approximate
solution to the exact solution when the mesh width of a rectangular
net is made to approach zero is more delicate than it is when the meth-
od of characteristics is used. For example, it is not correct, in general,
to choose a net in which the ratio of the mesh width At along the /-axis
and the mesh width Ax along the «r-axis is kept constant independent
of the solution: such a procedure would not in general yield approxi-
mations converging to the solution of the differential equation pro-
blem. The reason for this can be understood with reference to one of
the basic facts about the solution of the differential equations which
was brought out in the discussion of the preceding section. The basic
fact in question is the existence of what was called there the domain
of dependence of the solution. For example, suppose the solution were
to be approximated at the points of the net of Fig. 11. 5. la by advanc-
ing from one row parallel to the tT-axis to the next row a distance At
from it. In addition, suppose this were to be done by determining the
approximate values of v and c at any point such as P (cf. Fig. 11.5.1b)
P\C2
I
2 x
0 b
11.5.1. Approximation by using a rectangular net
by using the values of these quantities at the nearest three points
0, 1, 2 in the next line below, replacing derivatives in the two different-
ial equations by difference quotients, and then solving the resulting
algebraic equations for the two unknowns vp and cp. It seems reason-
able to suppose that such a scheme would be appropriate only if P
were in the triangular region bounded by the characteristics drawn
from points 0 and 2 to form the region within which the solution is de-
476 WATER WAVES
termined solely by the data given on the segment 0—2: otherwise it
seems clear that the initial values at additional points on the #-axis
ought to be utilized since our basic theory tells us that the initial data
at some of them would indeed influence the solution at point P. On the
other hand, the characteristic curves themselves depend upon the
values of the unknown functions v and c— their slopes, in fact, are
given (cf. (11.4.8)) by dxjdt — v ± c and thus the interval At must be
chosen small enough in relation to a fixed choice of the interval Ax
so that the points such as P will fall within the appropriate domains of
determinacy relative to the points used in calculating the solution at
P. In other words, the theory of characteristics, even if it is not used
directly, comes into play in deciding the relative values of At and Ax
which will insure convergence (for rigorous treatments of these
questions see the papers by Courant, Isaacson, and Rees [C.ll], and
by Keller and Lax [K.4]).
We shall introduce two different schemes employing the method of
finite differences in a fixed rectangular net of the x, J-plane. The first
of these makes use of the differential equations in the form given by
(11.4.7), and we no longer suppose that the function E is restricted in
any way. (It might be noted that the slopes of the characteristics as
given by (11.4.8) are determined by the quantities v dr c, no matter
how the function E is defined, and in fact also for the most general
case of a river having a variable cross section A9 etc., and hence we are
in a position to determine appropriate lengths for the ^-intervals, in
accord with the above discussion, in the most general case. This is
also a good reason for working with the quantity c in place of y.)
At the same time, the calculation is based on assuming that the ap-
Fig. 11.5.2. A rectangular net
proximate values of c and v have been calculated at the net points
L, M, R (cf. Fig. 11.5.2) and that the differential equations are to be
MATHEMATICAL HYDRAULICS 477
used to advance the approximate solution to the point P. The differ-
ential equations to be solved are thus
(11.5.1) 2{(c + v)cx + ct} + {(c + v)vx +vt}+E = 0,
(11.5.2) - 2{(- c + v)cx + ct} + {(- c + v)vx + vt} + E = 0,
and the characteristic directions are determined by dx/dt = v ± c.
The characteristic with slope v + c we call the forward characteristic,
and that with slope v — c the backward characteristic. We shall re-
place the derivatives in the equations by difference quotients which
approximate the values of the derivatives at the point M. In order to
advance the values of v and c from the points L, M9 R to the point P
by using (11.5.1) and (11.5.2) it is natural to replace the time deriva-
tives vt and ct by the following difference quotients
niK«x y, Vp — vM cp — CM
(11.5.3) „, = -__, ci = —zr
in both equations. However, in order to insure the convergence of
the approximations to the exact solution when Ax ->• 0 and At -> 0
(see Courant, Isaacson, and Rees [C.ll] for a proof of this fact) it is
necessary to replace the derivatives vx and cx by difference quotients
which are defined differently for (11.5.1) than for (11.5.2), as follows:
(11.5.4) vx = V^L^9 Cx = ^_I_^ in (11.5.1),
Ax Ax
(11.5.5) „. = *-*=!* , c. = C-*^-** in (11.5.2).
Ax Ax
The reason for this procedure is, at bottom, that (11.5.1 ) is an equation
associated with the forward characteristic, while (11.5.2) is associated
with the backward characteristic. The coefficients of the derivatives
and the function E arc, of course, to be evaluated at the point M. The
difference equations replacing (11.5.1) and (11.5.2) are thus given by
«n.,e, ,,.„
+ <„„ +„„) !*-p + "J^!L )+£(„„, cu) = 0,
Ax At }
(-
, = o.
478 WATEE WAVES
We observe that the two unknowns, vp and cp, occur linearly in these
equations; hence they are easily found by solving the equations. The
result is
(11.5.8) VP^VM+ —
(11.5.9) Cp-cM+|—
In accordance with the remarks made above, we must also require that
the ratio of At to Ax be taken small enough so that P lies within the
triangle formed by drawing lines from L and R in the directions of the
forward and backward characteristics respectively, i.e. lines with the
slopes VL + CL at L and VR — CR at jK: a condition that is well-de-
termined since the values of v and c are presumably known at L and R.
One can now see in general terms how the initial value problem
starting at t = 0 can be solved approximately: One starts with a net
along the #-axis with spacing Ax. Since c and v arc known at all of
these points, the values of c and v can be advanced through use of
(11.5.8) and (11.5.9) to a parallel row of points on a line distant At
along the 2-axis from the a?-axis. However, the mesh width At must
be chosen small enough so that the convergence condition is satisfied
at all net points where new values of v and c are computed.
We can now see also how to take care of boundary conditions, i.e.
of conditions imposed at a fixed point (say at the origin, x = 0) as
given functions of the time. For example, the depth y (corresponding
to the stage of the river) or the velocity v (which together with the
cross-section area A fixes the rate of discharge) might be given in
terms of the time. Initial conditions downstream from this point (i.e.
for x > 0) might also be prescribed. Suppose, for example, that the
stage of the river is prescribed at x -= 0, i.e. that j/(0, t) is known, and
that the calculation had already progressed so far as to yield values of
v and c at net points along a certain line parallel to the #-axis and
containing the points L, M, JB, as indicated in Fig. 11.5.3. It is clear
that the determination of the values of v and c at point P can be ob-
tained from their values at L, M, R by using (11.5.8) and (11.5.9),
as in the above discussion of the initial value problem, and similarly
MATHEMATICAL HYDRAULICS 479
for points Pv P2, etc. However, the value of v at Q must be deter-
mined in a different manner; for this purpose we use the equation
(11.5.7) with the subscript Q replacing P, L replacing Af, and M
replacing R. Since VM, CM, VR, CR are supposed known, and CQ is also
t '
-P
M
Fig. 11.5.3. Satisfying boundary conditions
known since the values of y arc proscribed on the /-axis, it follows that
equation (11.5.7) contains VQ as the only unknown; in fact it is given
by the equation
(11.5.10) vQ^vL+At\ — (cL
The reason for using (11.5.7) instead of (11.5.0) is, of course, that the
points M and Q are associated with the backward characteristic, and
hence (11.5.2) should be used to approximate the ^-derivatives at
.. point L (where the differential equations are replaced by difference
equations). It is quite clear that the same general procedure could be
used to calculate CQ if the values of v had been assumed given along
the /-axis. If, on the other hand, we had a boundary condition on the
right of our domain instead of on the left, as above, we could make use
of (11.5.6) for the forward characteristic as a basis for obtaining the
formula for advancing the solution along the /-axis.
The above discussion would seem to imply that under all circum-
stances only one boundary condition could be imposed— that is, that
either v or c could be prescribed at a fixed point on the river, but not
both— since prescribing one of these quantities leads to a unique de-
termination of the other. This is, indeed, true in any ordinary river,
but not necessarily in all cases. In fact, we made a tacit assumption
in the above discussion, i.e. that of the two characteristics issuing
from any point of the /-axis only the forward characteristic goes into
480
WATER WAVES
the region x > 0 to the right of the J-axis, and this in turn implies that
v + c and v — - c, which fix the slopes of the characteristics, are op-
posite in sign. The physical interpretation of this is that the value of
v (which is positive here) must be less than c = ^gy, i.e. that the
flow must be what is called tranquil, or subcritical.* Otherwise, as
we see from Fig. 11.5.4, we should expect to determine the values of
v and c at points close to and to the right of the J-axis, say at K9 by
•M
Fig. 11.5.4. A case of super-critical flow
utilizing values for both v and c along the segment LQ, its domain of
dependence. The scheme outlined above would therefore have to be
modified in a proper way under such circumstances. One sees, how-
ever, how useful the theory based on the characteristics can be even
though no direct use of it is made in the numerical calculations (aside
from decisions regarding the maximum permissible size of the ^-inter-
val).
The procedure sketched out above, while it is recommended for use
.P
M
Fig. 11.5.5. A staggered net
In gas dynamics the flow in an analogous case would be called subsonic.
MATHEMATICAL HYDRAULICS
481
when a boundary condition is to be satisfied, is not always the best
one to use for advancing the solution to such points as P, Pl9 P2, . . .
in Fig. 11.5.3. For such "interior points" a staggered rectangular net,
as indicated in Fig. 11.5.5, and a difference equation scheme based on
the original differential equations (11.4.6) may be preferable (cf.
Keller and Lax [K.4] for a discussion of this scheme). The equations
(11.4.6) were
(11.5.11)
cv
2vc
2ct = 0.
The values VM and CM at the mid-point M (which is, however, not a
net point) of the segment LR are defined by the averages:
(11.5.12)
after which the derivatives at M are approximated in a quite natural
way by the difference quotients
(11.5.13)
Ax '
Ax '
u M
LM
At
Upon substitution of these quantities into (11.5.11), evaluation of the
coefficients c, u, and E at point M9 and subsequent solution of the
two equations for vp and cp, the result is
(11.5.14)
7-
Ax
—
Ax
As we see on comparison with (11.5.8) and (11.5. 9\ these equations
are simpler than the earlier ones. The criterion for convergence re-
mains the same as before, i.e. that P should lie within a triangle formed
by the segment LR and the two characteristics issuing from its
ends.
482 WATER WAVES
11.6. Flood prediction in rivers. Floods in models of the Ohio River
and its junction with the Mississippi River
The theory developed in the preceding sections can be used to make
predictions of floods in rivers on the basis of the observed, or estimat-
ed, flow into the river from its tributaries and from the local run-off,
together with the state of the river at some initial instant. Hydraulics
engineers have developed a procedure, called flood-routing, to accom-
plish the same purpose. The flood-routing procedure can be deduced
as an approximation in some sense to the solution of the basic differ-
ential equations for flow in open channels (cf. the article by B. R.
Gilcrest in the book by Rouse [R.ll] ), but it makes no direct use of the
differential equations. However, the flood-routing procedure in ques-
tion seems not to give entirely satisfactory results in cases other than
that of determining the progress of a flood down a long river — for
example, the problem of what happens at a junction, such as that of
the Ohio and Mississippi Rivers, or the problem of calculating the
transient effects resulting from regulation at a dam, such as the
Kentucky dam at the mouth of the Tennessee River, seem to be diffi-
cult to treat by methods that are modifications of the more or less
standard flood-routing procedures. Even for a long river like the
Ohio, the usual procedure fails occasionally to yield the observed river
stages at some places. On the other hand, the basic differential equa-
tions for flow in open channels are in principle applicable in all cases
and can be used to solve the problems once the appropriate data de-
scribing the physical characteristics of the river and the appropriate
initial and boundary conditions are known.
The idea of using the differential equations directly as a means of
treating problems of flow in open channels is not at all new. In fact,
it goes back to Massau [M.5] as long ago as 1889. Since then the idea
has been taken up by many others (mostly in ignorance of the work
of Massau)— for example, by Preiswerk [P.16], von Karman [K.2],
Thomas [T.2], and Stoker [S.19]. Thomas, in particular, attacked the
flood-routing problem in his noteworthy and pioneering paper and
outlined a numerical procedure for its solution based on the idea of us-
ing the method of finite differences. However, his method is very la-
borious to apply and would also not necessarily furnish a good
approximation to the desired solution even if a large number of
divisions of the river into sections were to be taken. In general, the
amount of numerical work to be done in a direct integration of the
MATHEMATICAL HYDRAULICS 483
differential equations looked too formidable for practical purposes
until rather recently.
During and since the late war new developments have taken place
which make the idea of tackling flood prediction and other similar
problems by numerical solution of the relevant differential equations
quite tempting. There have been, in fact, developments in two differ-
ent directions, both motivated by the desire to solve difficult problems
in compressible gas dynamics: 1) development of appropriate nu-
merical procedures— for the most part methods using finite differences
— for solving the differential equations, and 2) development of com-
puting machines of widely varying characteristics suitable for carry-
ing out the numerical calculations. As we have seen, the differential
equations for flood control problems are of the same type as those for
compressible gas dynamics, and consequently the experience and cal-
culating equipment developed for solving problems in gas dynamics
can be used, or suitably modified, for solving flood control problems.
In carrying out such a study of an actual river it is necessary to
make use of a considerable bulk of observational data— cross-sections
and slopes of the channels, measurements of river depths and dis-
charges as functions of time and distance down the river, drainage
areas, observed flows from tributaries, etc.— in order to obtain the
information necessary to fix the coefficients of the differential equa-
tions and to fix the initial and boundary conditions. This is a task
with many complexities. For the purposes of this book it is more
reasonable to carry out numerical solutions for problems which are
simplified versions of actual problems. The present section has as its
purpose the presentation of the solutions in a few such special cases,
together with an analysis of their bearing on the concrete problems
for actual rivers. In any case, the general methods for an actual river
would be the same— there would simply be greater numerical compli-
cations.
The simplified models chosen correspond in a rough general way
(a) to two types of flow for the Ohio River and (b) to the Ohio and
Mississippi Rivers at their junction. Rivers of constant slope, with
rectangular cross-sections having a uniform breadth, and with con-
stant roughness coefficients are assumed. In this way differential
equations with constant coefficients result. The values of these quan-
tities are, however, taken to correspond in order of magnitude with
those for the actual rivers. In the model of the Ohio, for example, the
slope of the channel was assumed to be 0.5 ft/mile, the quantity n
484 WATER WAVES
(the roughness coefficient in Manning's formula) was given the value
0.03, and the breadth of the river was taken as 1000 feet. It is assumed
that a steady uniform flow with a depth of 20 ft existed at the initial
instant t = 0, and that for t > 0 the depth of the water was increased
at a uniform rate at the point x = 0 from 20 ft to 40 ft within 4
hours and was then held fixed at the latter value. (These depths are
the same as for the problem of a steady progressing wave treated in
sec. 11.2 above.) The problem is to determine the flow downstream,
i.e. the depth y and the flow velocity v as functions of x (for x > 0)
and t.
The methods used to obtain the solution of this problem of a flood in
a model of the Ohio River, together with a discussion of the results,
will be given in detail later on in this section. Before doing so, a few
general remarks and observations about them should be made at this
point. In the first place, it was found possible to carry out the solution
numerically by hand computation over a considerable range of dis-
tances and times (values at 900 net points in the #, 2-plane were de-
termined by finite differences), and this in itself shows that the
problems are well within the capacity of modern calculating equip-
ment. It might be added that the special case chosen for a flood in the
Ohio was one in which the rate of rise at the starting point upstream
was extremely high (5 feet per hour, in comparison with the rate of
rise during the flood of 1945— one of the biggest ever recorded in the
Ohio— which was never larger than 0.7 feet per hour at Wheeling,
West Virginia), so that a rather severe test of the finite difference
method was made in view of the rapid changes of the basic quantities
in space and time. The decisive point in estimating the magnitude of
the computational work in using finite differences is the number of
net points needed; for a river such as the Ohio it is indicated that an
interval Ax of the order of 10 miles along the river and an interval At
of the order of 0.3 hours in time in a rectangular net in the x, J-plane
will yield results that are sufficiently accurate. (Of course, a problem
for the Ohio in its actual state involves empirical coefficients in the
differential equations and other empirical data, which must be coded
for calculating machines, but this would have no great effect on these
estimates for Ax and might under extreme flood conditions reduce At
by a factor of 1/2.)
As we know from sec. 11.3 above, there is a case in which an exact
solution of the differential equations is known, i.e. the case of a
steady progressing wave with two different depths at great distances
MATHEMATICAL HYDRAULICS 485
upstream and downstream. The exact solution obtained in sec. 11.3
for the case of a wave of depth 20 ft far downstream and 40 ft far
upstream was taken as furnishing the initial conditions at t — 0 for
a wave motion in the river. With the initial conditions prescribed in
this way the finite difference method was used to determine the mo-
tion at later times; of course the calculation,if accurate, should fur-
nish a wave profile and velocity distribution which is the same at
time t as at the initial instant t = 0 except that all quantities are dis-
placed downstream a distance Ut, with U the speed of the steady
progressing wave. In this way an opportunity arises to compare the
approximate solution with an exact solution. In the present case the
phase velocity U is approximately 5 mph. Interval sizes of Ax = 5
miles in a "staggered" finite difference scheme (cf. equations (11.5.14))
with At = .08 hr were taken and a numerical solution was worked
out. We report the results here. After 12 hours, the calculated values
for the stage y agreed to within .5 per cent with the exact values.
The discharge and the velocity deviated by less than .8 per cent
from the exact values.
One of the valuable insights gained from working out the solution
of the flood problem in a model of the Ohio was an insight into the
relation between the methods used by engineers— for example, by
the engineers of the Ohio River Division of the Corps of Engineers in
Cincinnati — for predicting flood stages, and the methods explained
here, which make use of the basic differential equations. At first sight
the two methods seem to have very little in common, though both, in
' the last analysis, must be based on the laws of conservation of mass
and momentum; indeed, in one important respect they even seem to
be somewhat contradictory. The methods used in engineering prac-
tice (which make no direct use of our differential equations) tacitly
assume that a flood wave in a long river such as the Ohio propagates
only in the downstream direction, while the basic theory of the dif-
ferential equations we use tells us that a disturbance at any point in
a river flowing at subcritical speed (the normal case in general and
always the case for such a river as the Ohio) will propagate as a wave
traveling upstream as well as downstream. Not only that, the speed
of propagation of small disturbances relative to the flowing stream, as
defined by the differential equations, is \/gy for small disturbances
and this is a good deal larger (by a factor of about 4 in our model of
the Ohio) than the propagation speed used by the engineers for their
flood wave traveling downstream. There is, however, no real dis-
486 WATER WAVES
crepancy. The method used by the engineers can be interpreted as a
method which yields solutions of the differential equations, with cer-
tain terms neglected, that are good approximations (though not under
all circumstances, it seems) to the actual solutions in some cases,
among them that of flood waves in a river such as the Ohio. The
neglect of terms in the differential equations in this approximate
theory is so drastic as to make the theory of characteristics, from
which the properties of the solutions of the differential equations were
derived here, no longer available. The numerical solution presented
here of the differential equations for a flood wave in a model of the
Ohio yields, as we have said, a wave the front of which travels down-
stream at the speed \/^y; but the amplitude of this forerunner is
quite small,* while the portion of the wave with an amplitude in the
range of practical interest is found by this method to travel with
essentially the same speed as would be determined by the engineers'
approximate method. What seems to happen is the following: small
forerunners of a disturbance travel with the speed \/gy relative to the
flowing stream, but the resistance forces act in such a way as to de-
crease the speed of the main portion of the disturbance far below the
values given by i/gy, i.e. to a value corresponding closely to the speed
of a steady progressing wave that travels unchanged in form. (One
could also interpret the engineering method as one based on the as-
sumption that the waves encountered in practice differ but little from
steady progressing waves). As we shall see a little later, our unsteady
flow tends to the configuration of a steady progressing wave of depth
40 ft upstream and 20 ft downstream.
This analysis of the relation between the methods discussed here
and those commonly used in engineering practice indicated why it
may be that the latter methods, while they furnish good results in
many important cases, fail to mirror the observed occurrences in other
cases. For example, the problem of what happens at a junction of two
major streams, and various problems arising in connection with the
operation of such a dam as the Kentucky Dam in the Tennessee River
seem to be cases in which the engineering methods do not furnish
accurate results. These would seem to be eases in which the motions
of interest depart too much from those of steady progressing waves,
and cases in which the propagation of waves upstream is as vital as the
propagation downstream. Thus at a major junction it is clear that
In an appendix to this chapter an exact statement on this point is made.
MATHEMATICAL HYDRAULICS 487
considerable effects on the upstream side of a main stream are to be
expected when a large flow from a tributary occurs. In the same way,
a dam in a stream (or any obstruction, or change in cross-section, etc.)
causes reflection of waves upstream, and neglect of such reflections
might well cause serious errors on some occasions.
The above general description of what happens when a flood wave
starts down a long stream— in particular, that it has a lengthy front
portion which travels fast, but has a small amplitude, while the main
part of the disturbance moves much more slowly— has an important
bearing on the question of the proper approach to the numerical solu-
tion by the method of finite differences. It is, as we shall see shortly,
necessary to calculate— or else estimate in some way— the motion up
to the front of the disturbance in order to be in a position to calculate
it at the places and times where the disturbances are large enough to
be of practical interest. This means that a large number of net points
in the finite difference mesh in the #, J-plane lie in regions where the
solution is not of much practical interest. Since the fixing of the solu-
tion in these regions costs as much effort as for the regions of greater
interest, the differential equation method is at a certain disadvantage
by comparison with the conventional method in such a case. However,
it is possible in simple cases to determine analytically the character
of the front of the wave and thus estimate accurately the places and
times at which the wave amplitude is so small as to be negligible;
these regions can then be regarded as belonging to the regions of the
x, f-planc where the flow is undisturbed, with a corresponding re-
duction in the number of net points at which the solutions must be
calculated. A method which can be used for this purpose has been
derived by G. Whitham and A. Troesch, and a description of it is
given in an appendix to this chapter. If a modern high speed digital
computer were to be used to carry out the numerical work, however,
it would not matter very much whether the extra net points in the
front portion of the wave were to be included or not: many such
machines have ample capacity to carry out the necessary calculations.
We proceed to give a description of the calculations made for our
model of the Ohio, including a discussion of various difficulties which
occurred for the flood wave problem near the front of the disturbance,
and particularly at the beginning of the wave motion (i.e. near x = 0,
/ = 0), and an enumeration of the features of the calculation which
must play a similar role in the more complicated cases presented by
rivers in their actual state. This will be followed by a description of
488 WATER WAVES
the method used and the calculations made for a problem simulating
a flood coming down the Ohio and its effect on passing into the
Mississippi. This problem and its solution give rise to further general
observations which will be made later on.
The differential equations to be solved are
(11.6.1)
2ct + 2vcx + cvx = 0,
with v(x9t) the velocity, and c = Vgy the propagation speed of
small disturbances. The assumption of a uniform cross-section and
the assumption that no flow over the banks occurs (i.e. q = 0 in the
basic differential equations (11.1.1) and (11.1.6)) have already been
used. The quantity E is given by
E = - gS + gSf,
with S the slope of the river bed and Sf, the friction slope, given by
Manning's formula
Here we assume the channel to be rectangular with breadth B.
The numerical data for the problem of a flood in a model of the
Ohio River are as follows. For the slope S a value of 0.5 ft/mi was
chosen, and B is given the value 1000 ft. For y a value of 2500 was
taken (in foot-sec units), corresponding to a value of Manning's
constant n (in the formula y — (1.49/n)2) of 0.03. The special pro-
blem considered was then the following: At time t = 0, a steady flow
of depth 20 ft is assumed. At the "headwaters" of the river, corres-
ponding to x = 0, we impose a linear increase of depth with time which
brings the level to 40 ft in 4 hours. For subsequent times the level of
40 ft at x = 0 is maintained. The initial velocity of the water cor-
responding to a uniform flow of depth yQ = 20 ft is calculated from
Sf = S to be
i>0 = 2.38 mph;
the^propagation speed of small disturbances corresponding to the
depth of 20 ft is
MATHEMATICAL HYDRAULICS
489
C =
= 17.3 mph.
The problem then is to determine the solution of (11.6.1) for v(x, t),
c(x, t) for all later times t ^ 0 along the river x ^ 0. Figures 11.6.1
and 11.6.2 present the result of the computation in the form of stage
and discharge curves plotted as functions of distance along the river
at various times.
In order to indicate how the solution was calculated it is conven-
ient to refer to diagrams in the (x, t) plane given by Figs. 11.6.3
and 11.6.4. According to the basic theory, we know that for x ^
fao + co)t = 19-7*, called region O in Fig. 11.6.3, the solution is given
10-
Legend
t- time m hours after start of flood
y - stage in feet
x - distance along Ohio m miles
^sTJG
-•<£;| ;:% ^-A«%3;e$ A^^ ^^ x
20 4O 60 8O 100 I2O
Fig. 11.6.1. Stage profiles for a flood in the Ohio River
by the unchanged initial data, v(x9 t) = 00, c(x, t) = C0 (since the
forerunner of the disturbance travels at the speed w0 + CQ = 19.7 mph).
Experiments were made with various interval sizes and finite
difference schemes in order to try to determine the most efficient way
to calculate the progress of the flood. We proceed to describe the
various schemes tried and the regions in which they were used on the
basis of Figs. 11.6.3 and 11.6.4.
Region I, 0 ^ x ^ 19.7J, 0 ^ t ^ .4. Quite small intervals of
490
WATER WAVES
300.
260.
220.
Legend
t * time in hours after start of flood
Q» discharge in 1000 c.fs.
x 3 distance along Ohio in miles
t«a>
0 20 40 60 80 100 120 140 160
Fig. 11.6.2. Discharge records for a flood in the Ohio River
1.25
Fig. 11.6.3. Regions in which various computational methods were tried
Ax = l mile and At = .048 hours were required owing to the sudden
increase of depth at x = 0, t = 0. The finite difference formulas given
above in equations (11.5.8), (11.5.9) were used.
In Region II, 0 ^ x ^ 19.7*, .4 ^ t <^ .7, with Ax = 1 mile,
MATHEMATICAL HYDRAULICS
491
t
2.0-
1.5 J
1.0 J
Legend
1 s time in hours
x a distance in miles
° 10 20 30
Fig. 11.6.4. Net points used in the finite difference schemes
At — .024 hr, the "staggered" scheme was used. The formulas for this
scheme have been given above in equations (11.5.14). In order to
calculate 0(0, 2), the velocity at the upstream boundary of the river,
the formula associated with the backward characteristic, namely
equation (11.5.10), has to be used twice in succession: for the triangles
FBM and MRP (cf. Figs. 11.5.3 and 11.6.5). The values CB and VB
arc simply determined by linear interpolation from the values at the
points F and G.
492 WATER WAVES
Region III, 0 ^ x ^ 5, .7 ^ t^ 1.25, with Ax = 1 mile, z^ = .024 hr.
The same procedure was used as in Region II.
Region IV, 5 ^ x ^ 19.7J, .7 ^ * ^ 1.25, with Ax = 2 miles,
At = .048 hr. The values at the boundary between Regions III and
.G .H
x
Fig. 11.6.5. Net point arrangement used at boundary in "staggered" scheme
IV were obtained by linear interpolation from the neighboring values.
Other quantities were computed by the " staggered" scheme as in
Regions II and III.
Region V, 0 ^ x ^ Ut, 1.25 ^ t <, 10, Ax = 5 miles, J* = .17 hr.
U represents a variable speed which marks the downstream end of
what might be called the observable disturbance (U & 10 mph).
That is, by using an expansion scheme (see the appendix to this
chapter) we obtain the solution in
Region VI, defined by Ut ^ as ^ 19.72, back of the forerunner
of the disturbance, in which the flow is essentially undisturbed
for all practical purposes. The expansion valid near the front of the
wave and referred to above was used to calculate the various quantities
in Region VI, and a staggered scheme was used to compute the values
in Region V.
A number of conclusions reached on the basis of the experience
gained from these calculations of a flood in a model of the Ohio River
can be summarized as follows:
(a) The rate of rise of the flood —5 feet per hour— is extreme, and
such a case exaggerates the way in which errors in the finite
difference methods are propagated. For example, slight inaccu-
racies at the head, x ~ 0, were found to develop upon increasing
the size of the Ax interval. In spite of the exceptionally high rate
of rise of the flood, the fluctuations created by using finite dif-
ference methods were damped out rather strongly (in about
8 — 10 time steps). It is possible to control these inaccuracies
MATHEMATICAL HYDRAULICS 493
simply by using small interval sizes. The process by which the
small errors of the finite difference scheme are caused to die out
may be described as follows: A value of v which is too large
produces a correspondingly larger friction force which slows down
the motion and produces at a later time a smaller velocity. The
lower velocity in a similar way then operates through the resistance
to create a larger velocity and the process repeats in an oscillatory
fashion with a steady decrease in the amplitude of variation.
(b) The accuracy of our computation (as a function of the interval
size) was checked by repeating the calculation for two different
interval sizes over the same region in space and time.
(c) A linearized theory of wave propagation, obtained by assuming
a small perturbation about the uniform flow with 20 ft depth, is
easily obtained, and the problem was solved using such a theory.
However, it does not give an accurate description of the solution
of our problem. It was found that the stage was predicted too low
by the linear theory by as much as 2 feet after only 2 hours— a
very large error.
(d) It would be convenient to be in possession of a safe estimate for
the maximum value of the particle velocity, in order to select an
appropriate safe value for the time interval At, since we must
have At 5g Ax/(v -)- c) in order to make sure that the finite dif-
ference scheme converges. The calculations in our special case
indicate that this may not be easy to obtain in a theoretical way,
since the maximum velocity at x — 0, for example, greatly ex-
reeds its asymptotic value, as indicated in Fig. 11.6.6. In a
5.4-
24
v(0,t)
mph
velocity for
40ft steady flow
t
4 hours
Fig. 11.6.6. Water velocity obtained at "head" of river
computation for an actual river, however, no real difficulty is
likely to result, since c is in general much larger than v and is
determined by the depth alone.
494
WATER WAVES
(e)
As was already indicated above, the curves of constant stage
turn out to have slopes which are closer to 5 mph (the speed with
which a steady progressing flow, 40 ft upstream and 20 ft down-
stream, moves) than they are to the 19.7 mph speed of pro-
pagation of small disturbances. This is shown by Fig. 11.6.7.
10.
5.
t hours
region of
practically
undisturbed
flow
Smph-slope I97mph-slope
50
100
x miles
Fig. 11.6.7. Curves of constant stage — comparison with first characteristic and
steady progressing flow velocity
The region of practically undisturbed flow (determined by an
expansion about the "first" characteristic x = 19.7J, for which
see the appendix to this chapter) is shown above. In an actual
river, we would of course expect the local runoff discharges and the
non-uniform flow conditions to eliminate largely the region of
practically undisturbed flow. For this reason it is not feasible
to use analytic expansion schemes as a means of avoiding
computational labor.
We turn next to our model of the junction of the Ohio and Missis-
sippi Rivers and the problem of what happens when a flood wave
comes down the Ohio and passes through the junction.* The physical
data chosen are the same as were used above in sec. 11.2 in discussing
the problem of a steady flow at a junction.
We suppose the upstream side of the Mississippi to be identical
with the Ohio River— i.e. that it has a rectangular cross-section
1000 ft wide, a slope of .5 ft/mile, and that Manning's constant n has
the value .03. The downstream Mississippi is also taken to be rectan-
gular, but twice as wide, i.e. 2000 ft in width, Manning's constant is
again assumed to have the value .03, but the slope of this branch is
given the value .49 ft/mile. This modification of the slope was made
* The analogous problem in gas dynamics would be concerned with the pro-
pagation of a wave at the junction of two pipes containing a compressible gas.
MATHEMATICAL HYDRAULICS
495
in order to make possible an initial solution corresponding to a uni-
form flow of 20 ft depth in all three branches. (Such a change is
necessary in order to overcome the decrease in wetted perimeter
which occurs on going downstream through the junction.) Figure
11.6.8 shows a schematic plan of the junction. The concrete problem
to be solved is formulated as follows. A flood is initiated in the Ohio
L3J
Downstream
Mississippi
Fig. 11.6.8. Schematic plan of junction
at a point 50 miles above the junction by prescribing a rise in depth of
the stream at that point from 20 ft to 40 ft in 4 hours — in other
words, the same initial and boundary conditions were assumed as for
the case of the flood in the Ohio treated in detail above. After about
2.5 hours the forerunner, or front, of the wave in the Ohio caused by
the disturbance 50 miles upstream reaches the junction; up to this
instant nothing will have happened to disturb the Mississippi, and
the numerical calculations made above for the Ohio remain valid
during the first 2.5 hours. Once the disturbance created in the Ohio
reaches the junction, it will cause disturbances which travel both
upstream and downstream in the Mississippi, and of course also a
reflected wave will start backward up the Ohio. The finite difference
calculations therefore were begun in all three branches from the
moment that the junction was reached by the forerunner of the Ohio
flood, and the solution was calculated for a period of 10 hours.
We proceed to describe the method of determining the numerical
solution. Let r(1), c(1), i>(2), c(2), 0(8), c(3) represent the velocity v and the
propagation speed c for the Ohio, upstream Mississippi, and down-
496
WATER WAVES
stream Mississippi, respectively. A "staggered" scheme was used with
intervals Ax = 5 miles and At — .17 hr as indicated in Fig. 11.6.9.
The junction point is denoted by x = 0, the region of the Ohio and
x -P x
• L *M .R
.K XA .F XB .G
Ohio [II
Upstream Mississippi C2J Junction
Downstream Mississippi [31
Fig. 11.6.9. Junction net point scheme
the upstream Mississippi are represented by x ^ 0, while the down-
stream Mississippi is described for x S> 0. The time t = 2.5 hrs, as
explained above, corresponds to the instant that the forerunner of
the flood reaches the junction.
The values of the quantities v and c at the junction were determined
as follows: Assume that the values of v and c have been obtained at all
net points for times preceding that of the boundary net point P, which
represents a point at the junction. We use at this point the relations
since c = \/gy and the water level is the same in the three branches at
the junction. In addition, we have
since what flows into the junction from the upstream side of the Mis-
sissippi and from the Ohio must flow out of the junction into the down-
stream branch of the Mississippi. If the values of v and c were known
at the point M in Fig. 11.6.9 in the respective branches of the rivers,
we could find the values at P from equation (11.5.6) for the Ohio and
the upstream side of the Mississippi, and equation (11.5.7) for the
MATHEMATICAL HYDRAULICS 497
downstream side of the Mississippi. We rewrite the equations for
convenience, as follows:
CP(l) •= CP(2) =" <V(3)> (with C =
+ IJP(2) = 2^P(3)» (since y(l) = y(2) = j/(3)),
nifi9\
(11.6.2),
and
, •>./,. ... \ VL(,)\\
+ { ----- -- + (Of(J)+PMU))l — - --- I j
A
( At Ax
l)
I """27 ---- M(3)"A/(3) — - ^ - j
The above system of six linear equations determines uniquely the
values u(1), c(1), u(2), c(2), z;{3), c(3) at P in terms of their values at the
preceding points L, M and R. The equations can be solved explicitly.
The values of the relevant quantities at M are determined in the same
way from the preceding values at A, F and B. The values at A and B
arc determined by interpolation between the neighboring points
(K, F) and (F, G) respectively (sec Fig. 11.6.9). Of course, it is ne-
cessary to treat the motions in each of the branches away from the
junction by the same methods as were described for the problem of
the Ohio treated above, and this is feasible once the values of v and c
have been obtained at the junction.
The results of the calculations are shown in Fig. 11.6.10, which
furnishes the river profiles, i.e. the depths as functions of the location
in each of the three branches, for times t = (K 2.5, 4, and 10 hours
after the beginning of the flood 50 miles up the Ohio. The curves for
t = oo are those for the steady flow which was calculated above in
sec*. 11.2 (cf. Fig. 11.2.3). The calculations indicate that the unsteady
flow does tend to the steady flow as the time increases. Another no-
ticeable effect is the backwater effect in the upper branch of the
Mississippi. For example, the stage is increased by about 2 feet at a
point in the Mississippi 20 miles above the junction and 7.5 hours after
the flood wave from the Ohio first reaches the junction.
498
WATER WAVES
It might be mentioned that the forerunners of the flood in all three
branches were computed by using the expansion scheme which is
explained in the appendix to this chapter.
x = distonce in miles
measured from junction
y = stage measured in feet
t= time in hours after start
of flood
40'
//// i i i i i i i 1 1 i i i i i i i i 1 1 i i
10'
Fig. 11.6.10 River profiles for the junction
11.7. Numerical prediction of an actual flood in the Ohio, and at its
junction with the Mississippi. Comparison of the predicted with
the observed floods
The methods for numerical analysis of flood wave problems in
rivers developed above and applied to simplified models of the Ohio
and its junction with the Mississippi have been used to predict the
progress of a flood in the Ohio as it actually is, and likewise to predict
the progress of a flood coming from the Ohio and passing through the
junction with the Mississippi. The data for the flood in the Ohio were
taken for the case of the big flood of 1945, and predictions were made
numerically for periods up to sixteen days for the 400-mile long
stretch of the Ohio extending from Wheeling, West Virginia, to
Cincinnati, Ohio. For the flood through the junction, the data for
the 1947 flood were used, and predictions were made in all three
branches for distances of roughly 40 miles from the junction along
MATHEMATICAL HYDRAULICS 499
each branch. In each case the state of the river, or river system, was
taken from the observed flood at a certain time t = 0; for subsequent
times the inflows from tributaries and the local run-off in the main
river valley were taken from the actual records, and then the differ-
ential equations were integrated numerically with the use of the
UNIVAC digital computer in order to obtain the river stages and dis-
charges at future times. The flood predictions made in this way were
then compared with the actual records of the flood.
A comparison of observed with calculated flood stages will be given
later on; however, it can be said in general that there is no doubt that
this method of dealing with flood waves in rivers is entirely feasible
since it gives accurate results without the necessity for unduly large
amounts of expensive computing time on a machine such as the
UNIVAC. For example, a prediction for six days in the 400-mile
stretch of the Ohio requires less than three hours of machine time.
This amount of calculating time — which is anyway not unreasonably
large — could almost certainly be materially reduced by modifying
appropriately the basic methods; so far. no attention has been given
to this aspect of the problem, since it was thought most important
first of all to find out whether the basic idea of predicting floods by
integrating the complete differential equations is sound. The fact that
such problems can be solved successfully in this way is, of course, a
matter of considerable practical importance from various points of
view. For example, this method of dealing with flood problems in
rivers is far less expensive than it is to build models of a long river or
a river system, and it appears to be accurate. Actually, the two
methods— empirically by a model, or by calculation from the theory
- -are in the present case basically similar, since the models are really
huge and expensive calculating machines of the type called analogue
computers, and the processes used in both methods are at bottom the
same, even in details. An amplification of these remarks will be made
later on.
It would require an inordinate amount of space in this book to deal
in detail with the methods used to convert the empirical data for a
river into a form suitable for computations of the type under discussion
here, and with the details of coding for the calculating machine; for
this, reference is made to a report [1.4]. Instead, only a brief outline
of the procedures used will be given here.
In the first place, it is necessary to have records of past floods with
stages up to the maximum of any to be predicted. It would be ideal
500 WATER WAVES
to have records of flood stages and discharges (or, what comes to the
same thing, of average velocities over a cross-section) at points closely
spaced along the river— at ten mile intervals, say. Unfortunately,
measurements of this kind are available only at much wider inter-
vals * — of the order of 50 to 80 miles or more— even in the Ohio
River, for which the data are more extensive than for most rivers in
the United States. From such records, it is possible to obtain the co-
efficient of the all-important resistance term in the differential equa-
tion expressing the law of conservation of momentum. This coefficient
depends on both the location of the point along the river and the
stage. The other essential quantity, the cross-section area, also as a
function of location along the river and of stage, could in principle
be determined from contour maps of the river valley; this is, in fact,
the method used in building models, and it could have been used in
setting up the problem for numerical calculation in the manner under
discussion here. If that had been done, the results obtained would
probably have been more accurate; however, such a procedure is
extremely laborious and time consuming, and since the other equally
important empirical element, i.e. the resistance coefficient, is known
only as an average over each of the reaches (this applies equally to
the models of a river), it seems reasonable to make use of an average
cross-section area over each reach also. Such an average cross-section
area was obtained by analyzing data from past floods in such a way
as to determine the water storage volumes in each reach, and from them
an average cross-section area as a function of the river stages was
calculated. In this way the coefficients of the differential equations are
obtained as numerically tabulated functions of x and y. (It might
perhaps be reasonable to remark at this point that the carrying out
of this program is a fairly heavy task, which requires close cooperation
with the engineers who are familiar with the data and who understand
also what is needed in order to operate with the differential equations).
In Fig. 11.7.1 a diagrammatic sketch of the Ohio River between
Wheeling and Cincinnati is shown, together with the reaches and
observation stations at their ends. What we now have are resistance
coefficients and cross-section areas that represent averages over any
given reach. However, the reaches are too long to serve as intervals
for the method of finite differences— which is basic for the numerical
integration of the differential equations. Rather, an interval between
* Each such interval is called a reach by those who work practically with river
regulation problems.
MATHEMATICAL HYDRAULICS 501
net points (in the staggered scheme described in the preceding section)
of 10 miles was taken in order to obtain a sufficiently accurate approx-
imation to the exact solution of the problem. A time interval of
Cincinnati
Huntmgton
Fig. 11.7.1. Reaches in the Ohio
9 minutes was used. Actually, calculations were first made using a
5-mile interval along the river, but it was found on doubling the inter-
val to 10 miles that no appreciable loss in accuracy resulted.
To begin with, flood predictions for the 1945 flood were made, start-
ing at a time when the river was low and the flow was practically a
steady flow. Calculations were first made for a 36 hour period during
which the flood was rising; as stated earlier, these were made using
the measured inflows from tributaries, and the estimated run-off
in the main valley. Upon comparison with the actual records, it was
found that the predicted flood stages were systematically higher than
the observed flood stages, and that the discrepancy increased steadily
with increase in the time. It seemed reasonable to suppose that the
error was probably due to an error in the resistance coefficient. Con-
sequently a series of calculations was made on the UNIVAC in which
this coefficient was varied in different ways; from these results, cor-
rected coefficients were estimated for each one of the reaches. Actually
502 WATER WAVES
this was done rather roughly, with no attempt to make corrections
that would require a modification in the shape of these curves in their
dependence on the stage. The new coefficients, thus corrected on the
basis of 36-hour predictions (and thus for flood stages far under the
maximum), were then used to make predictions for various 6-day
periods, as well as some 16-day periods, with quite good results, on
the whole.
It might be said at this point that making such a correction of the
resistance oceff icient on the basis of a comparison with an actual flood
corresponds exactly to what is done in making model studies. There,
it is always necessary to make a number of verification runs after the
model is built in order to compare the observed floods in the model
with actual floods. In doing so, the first run is normally made without
making any effort to have the resistance correct — in fact, the rough-
ness of the concrete of the model furnishes the only resistance at the
start. Of course it is then observed that the flood stages arc too low
because the water runs off too fast. Brass knobs are then screwed
into the bed of the model, and wire screen is placed at some parts of
the model, until it is found that the flood stages given by the model
agree with the observations. This is, in effect, what was done in
making numerical calculations. In other words, the resistance cannot
be scaled properly in a model, but must be taken care of in an empi-
rical way. The model is thus not a true model, but, as was stated earlier,
it is rather a calculating machine of the class called analogue com-
puters. It is, however, a very expensive calculating machine which can,
in addition, solve only one very restricted problem. A model of two
fair sized rivers, for example, consisting of two branches perhaps 200
miles in length upwards from their junction, together with a short
portion below the junctions, could cost more than a UNIVAC.
It has already been stated that average cross-section areas for the
individual reaches were used in making the numerical computations,
while in the model the cross-sections arc obtained from the contour
maps. In operating numerically it is possible to change the local cross-
section areas without any difficulty, and this might be necessary at
certain places along the river.
Some idea of the results of the calculations for the 1945 flood in the
Ohio is given by Fig. 11.7.2. The graph shows the river stage at Po-
meroy as a function of the time. At the other stations the results were
on the whole more accurate. The graph marked "computation with
original data", and which covers a 36 hour period, was computed on
MATHEMATICAL HYDRAULICS
503
the basis of the resistance coefficients as estimated from the basic
flow data for the river. As one sees, these coefficients resulted in much
too high stages, and corrections to them were made along the river
Computed hydrogroph
(resistance adjusted)
554
12 0 12
Feb 27 Feb 28
Fig. 1 1 .7.2. Comparison of calculated with observed stages at Pomeroy for the
1945 flood in the Ohio River
on the basis of the results of this computation. Afterwards, flood
predictions wore made for periods up to 16 days without further
Thebes
Metropolis
MISSISSIPPI
Hickmon
32m.
Fig. 11.7.3. The junction of the Ohio and the Mississippi
504
WATER WAVES
correction of these coefficients. The graph indicates results for a 6 day
period during which the flood was rising. Evidently, the calculated
and observed stages agree very well.
300
296
292-
288
Stage at Hickman
Jan 15 18 21 24 27 30
304-
300
Observed stages
Computed hydrograph
Stage at Cairo
Jan 15 18 21 24 27 30
Fig. 11.7.4. Calculated and observed stages at Cairo and Hickman
In Fig. 11.7.3 a diagrammatic sketch of the junction of the Ohio
and the Mississippi is shown indicating the portions of these rivers
which entered into the calculation of a flood coming down the Ohio
and passing through the junction. The flood in question was that of
MATHEMATICAL HYDRAULICS 505
1947. It was assumed that the stages at Metropolis in the Ohio (about
40 miles above Cairo) and at Thebes in the upper Mississippi (also
about 40 miles above Cairo) were given as a function of the time. At
Hickman in the lower Mississippi (about 40 miles below Cairo) the
stage-discharge relation at this point, as known from observations,
was used as a boundary condition. The results of a calculation for a
16 day period are shown in Fig. 11.7.4, which gives the stages at
Cairo, and at the terminating point in the lower Mississippi, i.e. at
Hickman. As one sees, the accuracy of the prediction is very high,
the error never exceeding 0.6 foot. It might be mentioned that a
prediction for 6 days requires about one hour of calculating time
on the UNIVAC, so that the calculating time for the 16 day period
was under 8 hours, which seems reasonable. This problem of rout-
ing a flood through a junction is, as has been mentioned before,
one which has not been dealt with successfully by the engineering
methods used for flood routing in long rivers.*
Appendix to Chapter 11
Expansion in the neighborhood of the first characteristic
It has been mentioned already that whereas the forerunner of a
disturbance initiated at a certain point in a river at a moment when
the flow is uniform travels downstream with the speed v + \/gy, the
main part of the flood wave travels more slowly (cf. Deymte [D.9]),
depending strongly on the resistance of the river bed. An investigation
of the motion near the head of the wave, i.e. near the first characteris-
tic (cf. the first part of sec. 11.6) with the equation x = (VQ + <?0)<,
shows immediately why the main part of the disturbance will in
general fall behind the forerunners of the wave.
The motion is investigated in this Appendix by means of an ex-
pansion in terms of a parameter that has been devised by G. Whitham
and A. Troesch and carried out to terms of the two first orders for the
model of the Ohio River, and to the lowest order in the much more
* Added in proof: In the meantime, calculations have been completed ( see [I.4a] )
for the case of floods through the Kentucky Reservoir at the mouth of the
Tennessee River. The calculated and observed stages differed only by inches for
a flood period of three weeks over the 186 miles of the resevoir.
506
WATER WAVES
complicated case of the junction problem. The results obtained make
it possible to improve the accuracy of the solution near the first
characteristic which separates the region of undisturbed flow from
that of the flood wave. It turns out that the finite difference scheme
yields river depths which are too large, as indicated by Fig. 11. A.I.
vprofile computed by
finite differences
wove
front
' Region of
I undisturbed
' uniform flow
Fig. 11. A.I. Error introduced by finite difference scheme in neighborhood of
first characteristic of a rapidly rising flood wave
In order to expand the solution in the neighborhood of the wave
front, we introduce new coordinates f and r as follows:
£ = x and r = (v0 + c0)t — x
such that the £-axis (i.e. r = 0) coincides with the first characteristic.
Near the front of the wave T will be small, and the expansion will be
carried out by developing v and c in powers of r. The basic system of
equations is restated for convenience:
%ccx +vt+ vvx - gS + gSf = 0,
cvx + 2vcx + 2ct = 0.
Upon substitution of the new variables | and r we find
cT) + v(vt-vr) + (v0 + c0)vr- gS + gS, - 0,
0 + c(0£-0T) + 2(»0 + cQ)cr = 0.
(11.A.1)
where the friction slope Sf for a rectangular channel of width B is given
by
4/3
MATHEMATICAL HYDRAULICS
507
We expand v and c as power series in r with coefficients that are func-
tions of £ as follows:
v =
This expansion is to be used for r > 0 only, since for r < 0 we are in
the undisturbed region and all the functions ^(1), u2(£), . . ., cx(f ),
c2(£), • • • vanish identically. If we insert the series for v and c into
equations (ll.A.l) and collect terms of the same order in T, we get
ordinary differential equations for ^(f ), cx(f ), . . .. The equations
resulting from the terms of zero order in r yield vl = 2Cj . The
first order terms become, after thus eliminating vl .
(11.A.2)
dc±
1
*>o
By adding these two equations and removing the common factor 4,
we find the differential equation for ^(1) is:
1 2 1
- 0.
.wove front
= 0
Although the solution of this differential equation for ^(f ) is easily
obtained, the result expressed in general terms is complicated, and it
is preferable to give it only for the case of the model of the Ohio River
Fig. 11.A.2. Behavior near the front of a wave
508
WATER WAVES
using the parameter values introduced above. In this case we find:
Cl = (1.05 + 8.06 e0'146*)-1, with ^ and f in miles and hours. This re-
sult has the following physical meaning: The angle a of the profile
measured between the wave front and the undisturbed water surface
dies out exponentially: oc~ 1/(1 + aebx), with a and b constants de-
pending on the river and the boundary condition at x = 0. Theore-
tically, a could also increase exponentially downstream so that a bore
would eventually develop, but only if the increase in level at x = 0 is
extremely fast; in our example no bore will develop unless the water
rises at the extremely rapid rate of at least 1 ft per minute.
Unfortunately, the evaluation of c2(£), which yields the curvature of
t '
hours
10
= !9.7t
50
100 150 x,
miles
Fig. 11.A.3. Region of practically undisturbed flow
the profile at the wave front, is already very cumbersome. The curva-
ture is found to decrease for large x like xc~bx, b being a positive con-
stant. With the two highest order terms in the expansion known, it is
possible to estimate the region adjacent to the first characteristic
where the flow is practically undisturbed. It is remarkable how far
behind the forerunner the first measurable disturbance travels (see
Fig. 11.A.3).
In a similar way, an expansion as a power series in T has been carried
out for the problem of the junction of the Ohio and Mississippi, as
described in earlier sections. Here even the lowest order term was
obtained only after a complicated computation, since it was necessary
to work simultaneously in three different x, J-planes, with boundary
conditions at the junction. The differential equations for cl are, in all
three branches, of the same type as for the Ohio, and their solution
for the junction problem with the parameters of section 11.6
MATHEMATICAL HYDRAULICS 509
are cx = .00084 c-145* for the upstream branch of the Mississippi,
and c1 = .00084 <r~-229* for the downstream branch of the Mississippi,
cl and f both being given in miles and hours. This means that the
angle a also dies out exponentially in the Mississippi, a little faster
downstream than upstream, as might have been expected, since the
oncoming water in the upstream branch has the affect of making the
wave front steeper.
In the problem of the idealized Ohio River and of the idealized
problem of its junction with the Mississippi River the expansions
were carried out numerically in full detail and were used to avoid
computation by finite differences in a region of practically undisturb-
ed flow.*
This would become more and more important if the flow were to be com-
puted beyond 10 hours.
PART IV
CHAPTER 12
Problems in which Free Surface Conditions are Satisfied
Exactly. The Breaking of a Dam. Levi-Civita's Theory
This concluding chapter constitutes Part IV of the book. In Part I
the basic general theory and the two principal approximate theories
were derived. Part II deals with problems treated by means of the
linearized theory arising from the assumption that the motion is a
small deviation from a state of rest or from a uniform flow. Part III
is concerned with the approximate nonlinear theory which arises
when the depth of the water is small, but the amplitude of the waves
need not be small. Finally, in this chapter we deal with a few problems
in which no assumptions other than those involved in the basic general
theory are made. In particular, the nonlinear free surface conditions
arc satisfied exactly.
The first type of problem considered in this chapter belongs in the
category of problems concerned with motions in their early stages
after initial impulses have been applied. A typical example is the
motion of the water in a dam when the dam is suddenly broken. This
problem will be treated along lines worked out by Pohle [P.ll],
[P.12], Similar problems involving the collapse of a column of liquid
in the form of a circular half-cylinder or of a hemisphere resting on a
rigid bottom have been treated by Penney and Thornhill [P.2] by
a method different from that used by Pohle.
The second section of the chapter deals with the theory of steady
progressing waves of finite amplitude. The existence of exact solutions
of this type is proved, following in the main the theory worked out
by Levi-Civita [L.7].
12.1. Motion of water due to breaking of a dam, and related problems
With the exception of the present section we employ throughout
this book the so-called Euler representation in which the velocity and
pressure fields are determined as functions of the space variables and
513
514 WATER WAVES
the time. In this section it is convenient to make use of what is com-
monly called the Lagrange representation, in which the displacements
of the individual fluid particles are determined with respect to the
time and to parameters which serve to identify the particles. Usually
the parameters used to specify individual particles are the initial
positions of the particles, and we shall conform here to that practice.
Only a two-dimensional problem will be treated in detail here; con-
sequently we choose the quantities a, 6, and t as independent variables,
with a and b representing Cartesian coordinates of the initial positions
of the particles at the time t — 0. The displacements of the particles
are denoted by X(a, b; t) and Y(a, b; t), and the pressure by p(a, b; t).
The equations of motion are
Xtt = --- Px
e>
Ytt = - - PY - S
Q
in accord with Newton's second law. We assume gravity to be the
only external force. These equations are somewhat peculiar because
of the fact that derivatives of the pressure p with respect to the de-
pendent variables X and Y occur. To eliminate them we multiply by
Xa and Fa, respectively, and add, then also by Xb, Yb< and add;
the result is
XttXa + (Yti
XttXb + (Ytt + g)Yb
(12.1.1) e
Q
and these are the equations of motion in the Lagrangian form. These
equations are not often used because the nonlinearities occur in an
awkward way; however, they have the great advantage that a solu-
tion is to be found in a fixed domain of the a, 6-plane even though
a free surface exists. For an incompressible fluid— the only case
considered here— the continuity condition is expressed by requiring
that the Jacobian of X and Y with respect to a and b should remain
unchanged during the flow (since an area element composed always
of the same particles has this property); but since X — a and Y =- b
initially, it follows that
(12.1.2) XaYb - XbYa = I
THE BREAKING OF A DAM 515
is the condition of continuity. If the pressure p is eliminated from
(12.1.1) by differentiation the result is
(12.1.3) (XaXbt + YaY,t)t =. (XbXat + YbYat)t.
Integration with respect to / leads to
(12.1.4) (XaXtt + YaYbl) - (XbXaf + Y6Yat) - f(a, b)
with / an arbitrary function. It can easily be shown by a calculation
using the Eulcrian representation that the left hand side of this equa-
tion represents the vorticity; consequently the equation is a verifi-
cation of the law of conservation of vorticity. If the fluid starts from
rest, or from any other state with vanishing vorticity, the function
j(a, b) would be zero.
The method used by Pohlc [P.ll], [P.12] to solve the equations
(12.1.1) and (12.1.2)— which furnish the necessary three equations for
the three functions X, F, and p consists in assuming that solutions
exist in the form of power series developments in the time, with co-
efficients which depend on a and b:
(12.1.5)
In these expansions we observe that the terms of order zero in X and Y
are a and b — in accordance with the basic assumption that these
quantities fix the initial positions of the particles. It should also be
noted that X(l) and F(1) are the components of the initial velocity,
and X(2) and F(2) similarly for the acceleration; in general, we would
therefore expect that X(l) and F(1) would be prescribed in advance as
part of the initial conditions. Of course, boundary conditions imposed
on .Y, F, and p would lead to boundary conditions for the coefficient
functions in the series developments. The convergence of the series
for the cases discussed below has not been studied, but it seems likely
that the scries would converge at least for sufficiently small values of
the time. The convergence of developments of this kind in some simp-
ler problems in hydrodynamics has been proved by Lichtenstein
[L.12].
The series (12.1.5) are inserted first in equation (12.1.2) and the
coefficient of each power of / is equated to zero with the following
result for the first two terms:
X(a9 b; t) =-- a + X™(a. b) • 1 + X^(a, b) • 1* + . . .,
Y(a, ft; 0 - b + Y™(a. b)-t + F<2>(0, b) • *2 + . . .,
p(a. ft; 0 - p(0)(fl, b) + p(l)(a. b) • t + p<2)(0, b) • t2 +
516 WATER WAVES
F<2> = -
We observe that X(l} and F(1) are subject to the above relation and
hence cannot both be prescribed arbitrarily; however, if the fluid
starts from rest so that X(l} = F(1) — 0, the condition is automatic-
ally satisfied. The equation for X(2) and F(2) is linear in these quanti-
ties, but nonlinear in X (1) and F(1). This would be the situation in
general: X(n) and F(n) would satisfy an equation of the form
jqw + y<«) = F(X<»9 y(1), *<2), F<2>, . . ., JC<»-i>, F<"-»),
with F a nonlinear function in X(i\ Y(l\ i = 1, 2, . . ., n — 1. In
the following we shall consider only motions starting from rest. Con-
sequently, we have X(l) = F(1) = 0, and equation (12.1.4) holds
with / = 0; a substitution of the series in powers of t in equation
(12.1.4) yields (for the lowest order term):
(12.1.7) X?> - Ff - 0.
The higher order coefficients satisfy an equation of the form
X(n) _ y(n) = G(JC<«, F<2>, . . ., Xi*-u9 F<w~1>), with G a nonlinear
function of X(i\ F(t), i = 2, 3, . . ., n - 1. Thus we observe that X™
and F(2) satisfy the Cauchy-Riemann equations and arc therefore
conjugate harmonic functions of a and b. The higher order coefficients
would satisfy Poisson's equation with a right hand side a known
function fixed by the coefficient functions of lower order. Thus the
coefficients in the series for X and F can be determined step-wise by
solving a sequence of Poisson equations. Once the functions X(t) and
Y(i) have been determined, the coefficients in the series for the
pressure p can also be determined successively by solving a sequence
of Poisson equations. To this end we of course make use of equations
(12.1.1); the result for p(o)(a,b) is
(12.1.8) p(* + pfl = - 2Q(XP + Ff ) = 0,
from (12.1.6) and JC(1) = F^> = 0. Thus p(0}(a, b) is a harmonic
function. For p(n)(a, b) one would find a Poisson equation with a right
hand side determined by X(i) and F(t) for i = 2, 3, . . ., n + 2.
It would be possible to consider boundary conditions in a general
way, but such a procedure would not be very useful because of its
complexity. Instead, we proceed to formulate boundary conditions
for the special problem of breaking of a dam, which is in any case
typical for the type of problems for which the present procedure is
THE BREAKING OF A DAM
517
recommended. We assume therefore that the region occupied initially
by the water (or rather, a vertical plane section of that region) is the
half-strip 0 ^ a < oo, 0 ^ b ^ h, as indicated in Fig. 12.1.1. The
b= h
b=0 o
Fig. 12.1.1. The breaking of a dam
damris of course located at a = 0. Since we assume that the water
is initially at rest when filling the half-strip we have the conditions
(12.1.9) X(a.b;0) = a, Y(a,b;0) = b,
and
(12.1.10) Xt(a, b; 0) = 0. Yt(a, b; 0) = 0.
When the dam is broken, the pressure along it will be changed
suddenly from hydrostatic pressure to zero; it will of course be pre-
scribed to be zero on the free surface. This leads to the following
boundary conditions for the pressure:
(12.1.11)
( p(a, h; t) = 0,
\
= 0,
0
0
a < GO,
b < h,
t > 0,
Finally the boundary condition at the bottom b = 0 results from the
assumption that the water particles originally at the bottom remain
in contact with it; as a result we have the boundary condition
0
a < oo, t > 0.
(12.1.12) Y(a, 0;*)=0.
The conditions (12.1.9) are automatically satisfied because of the
form (cf. (12.1.5)) chosen for the scries expansion. The conditions
(12.1.10) are satisfied by taking X(l)(a, b) = F(1)(a, b) = 0.
In order to determine the functions X(2)(a, b) and F(2)(a, 6), it is
necessary to obtain boundary conditions in addition to the differential
equations given for them by (12.1.6) and (12.1.7). Such boundary
conditions can be obtained by using (12.1.11) and (12.1.12) in con-
junction with (12.1.1) and the power series developments. Thus from
518 WATER WAVES
(12.1.12) we find F<2)(0, 0) ±= 0 for 0 ^ a < oo (indeed, F<">(a, 0)
would be zero for all n). Insertion of the series (12.1.5) and use of the
boundary conditions for 6 = h yields
(12.1.13) XW(a, h) = 0,
upon using the first of the equations in (12.1.1). The second equation
of (12.1.1) leads to the condition
(12.1.14) F<2>(0,&)=: -1
2
We know that Z(z) == F<2) + iX™ is an analytic function of the
complex variable z = a + ib in the half-strip, and we now have
prescribed values for either its real or its imaginary part on each of the
three sides of the strip; it follows that the function Z can be deter-
mined by standard methods— for example by mapping conformally
on a halfplane. In fact, the solution can be given in closed form, as
follows: Since X™(a, h) = 0, we see that X^(a9 h) = 0, and hence
that F£2)(a, h) = Q since X(2) and F(2) arc harmonic conjugates.
Therefore the harmonic function F(2)(a, b) can be continued over the
line b = h by reflection into a strip of width 2/i, as indicated in Fig.
12.1.2; the boundary values for F(2) are also shown. Thus a complete-
Y(2)-b b:2h
b=0
Fig. 12.1.2. Boundary value problem for F(2)(a, b)
ly formulated boundary value problem for F(2)(0, b) in a half-strip
has been derived. To solve this problem we map the half-strip on the
upper half of a w-plane by means of the function w = cosh (nz/2h)
—either by inspection or by using the Schwarz-Christoffel mapping
formula— and observe that the vertices z = 0 and z = 2ih of the half-
strip map into the points w = ± 1 of the wj-plane, as indicated in
Fig. 12.1.3. The appropriate boundary values for Y(2}(w) on the real
axis of the w-plane are indicated. The solution for Y(2)(w) under
THE BREAKING OF A DAM
519
w- plane
-H
Fig. 12.1.3. Mapping on the w-plane
these conditions is well known; it is the function Y(2)(P) ~
— (g/2n)(62 — 0J, with 0X and 02 the angles marked in Fig. 12.1.3.
The analytic function of which this is the real part is well known; it is
y<2>
+ 1
as can in any case be easily verified. Transferring back to the z-plane
we have
u
cosh — — 1
2h
Z(z) = Y(2) + iX™ = - log
,
cosh
and upon separation into real and imaginary parts we have finally:
cos2 — + sinh2 —
(12.1.15)
X<*>(a9b) = - -*-
sn
sinh2 —
F<2>(a, 6) = -
n
. nb
sin —
2h
. , na
smh —
One checks easily that the boundary conditions X(2>(a, h) — 0,
F<2>(a, 0) = 0 are satisfied, and that F(2)(0, 6) = — g/2. The initial
pressure distribution p(0)(a, 6) can be calculated, now that X(2)(a. b)
is known, by using the first equation of (12.1.1), which yields
520 WATER WAVES
(12.1 16) pj» -
In the present case there are advantages in working first with the
pressure p(a, b; t) and determining the coefficient of the series for it
directly by solving appropriate boundary value problems; afterwards
the coefficients of the series for X and Y are easily found. The main
reason for basing the calculation on the pressure in the first instance
is that the boundary conditions at b = h and a = 0 are very simple,
i.e. p = 0 and hence p<*> = 0 for all indices i. The boundary conditions
at the bottom 6 = 0 involve the displacements Y. For instance, one
finds readily in the same general way as above that p^ = — gg,
pW =. 0, and pj,2) _ __ QgY^ as boundary conditions at 6 = 0.
Since p<0) is harmonic, it is found at once without reference to dis-
placements—an interesting fact in itself. Once p(o) is found, X(2) and
F(2) can be calculated without integrations (cf. (12.1.16), for example).
Since p^ = 0 for b = 0, and p(1) is also harmonic, it follows that
p(1)(a, b) = 0. Since F(2) is now known,it follows that a complete set
of boundary conditions for p(2)(a, b) is known, and p(2)(a, 6) is then
determined by solving the differential equation
(12.1.17)
rd(x(z) F(2M
= e ^ST-'-M— - V2 {(Jf(2))2 + (F(2))2
L d(a, b)
L9 / I nCL n^\
h* I cosh — — cos — I
\ h hj
whose right hand side is obtained after a certain amount of mani-
pulation. This process can be continued. One would find next that
XW = F<3) = o, and that X(^ and F(4) can be found once p(2) is
known. However, the boundary condition at the bottom, and the right
hand sides in the Poisson equations for the functions p(i)(a,b) become
more and more complicated.
The initial pressure p(o)(a, b) can be discussed more easily on the
basis of a Fourier series representation than from the solution in
closed form obtainable from (12.1.16); this representation is
(12.1.18) p<°>(a, 6)
.,
= oglh — b) -- —- V - e --- zh — cos
^ }
THE BREAKING OF A DAM
521
We note that the first term represents the hydrostatic pressure, and
that the deviation from hydrostatic pressure dies out exponentially
as a -> oo and also as h -> 0, i.e. on going far away from the dam and
also on considering the water behind the dam to be shallow (or, better,
considering a/h to be large). This is at least some slight evidence of
the validity of the shallow water theory used in Chapter 10 to discuss
this same problem of the breaking of a dam— at least at points not
too close to the site of the dam.
The shape of the free surface of the water can be obtained for small
times from the equations
(12.1.19)
X = a
= b
evaluated for a — 0 (for the particles at the face of the dam) and for
b — h on the upper free surface. The results of such a calculation for
the specific case of a dam 200 feet high are shown in Fig. 12.1.4.
x
miles
50 40 30 20 10 100
Fig. 12.1.4. Free water surface after the breaking of a dam
One of the peculiarities of the solution is a singularity at the origin
a = 0, 6 — 0 which is brought about by the discontinuity in the
pressure there. In fact, X(2} has a logarithmic singularity for a = 0,
b = 0, as one sees from (12.1.15) and X is negative infinite for all
t ^ 0. This, of course, indicates that the approximation is not good
at this point; in fact, there would be turbulence and continuous
breaking at the front of the wave anyway so that any solution
ignoring these factors would be unrealistic for that part of the flow.
In the thesis by Pohle [P.ll], the solution of the problem of the
collapse of a liquid half-cylinder and of a hemisphere on a rigid plane
are treated by essentially the same method as has been explained for
522 WATER WAVES
the problem of the breaking of a dam. These problems have also been
treated by Penney and Thornhill [P.2], who also use power series in
the time but work with the Eulerian rather than the Lagrangian re-
presentation, which leads to what seem to the author to be more com-
plicated calculations than are needed when the Lagrangian represen-
tation is used.
12.2. The existence of periodic waves of finite amplitude
In this section a proof, in detail, of the existence of two-dimensional
periodic progressing waves of finite amplitude in water of infinite
depth will be given. This problem was first solved by Nekrassov
[N.I, la] and later independently by Levi-Civita [L.7]; Struik
[S.29] extended the proof of Levi-Civita to the same problem for
water of finite constant depth. A generalization of the same theory
to liquids of variable density has been given by Dubreuil-Jacotin
[D.I 5, 15a], Lichtenstein [L.ll] has given a different method of
solution based on E. Schmidt's theory of nonlinear integral equations.
Davies [D.5] has considered the problem from still a different point of
view. Gerber [G.5] has recently derived theorems on steady flows in
water of variable depth by making use of the Schauder-Leray theory.
We shall start from the formulation of the problem given by Levi-
Civita (and already derived in 10.9 above), but, instead of proving
directly, as he does, the convergence of a power series in the ampli-
tude to the solution of the problem, an iteration procedure devised
by W. Littman and L. Nirenberg will be used to establish the existence
of the solution. The two procedures are not, however, essentially
different.
It is convenient to break up this rather long section into sub-sec-
tions as a means of focusing attention on separate phases of the
existence proof.
12.2a. Formulation of the problem
As in sec. 10.9, the problem of treating a progressing wave which
moves unchanged in form and with constant velocity is reduced to a
problem of steady flow by observing the motion from a coordinate
system which moves with the wave. A complex velocity potential (see
sec. 10.9 for details) %(z) is therefore to be found in the #, t/-plane
(cf. Fig. 12.2.1):
LEVI-C1 VITA'S THEORY
523
Fig. 12.2.1. Periodic waves of finite amplitude
(12.2.1) x = <p + i\p = jf(z), z = x + iy.
The velocity at y = — oo should be [7. The real harmonic functions
<p(x,y) and y(x,y) represent the velocity potential and the stream
function. The complex velocity w is given by
(12.2/2)
w = — — u — iv
dz
with u, v the velocity components. This follows at once from the
Cauchy-Riemann equations:
(12.2.3) (px = yy = w, <pv = - yx = ^
since w — 993. -f iy^.
We proceed to formulate the boundary conditions at the free sur-
face. The kinematic free surface condition can be expressed easily be-
cause the free surface is a stream line, and we may choose y(x> y) = 0
along it. The dynamic condition expressed in Bernoulli's law is given
by
(12.2.4) | | w |2 + gy = const. at y = 0,
as one can readily verify. The problem of satisfying this nonlinear
condition is of course the source of the difficulties in deriving an
existence proof. At oo the boundary condition is
(12.2.5) w -> U uniformly as y -> — oo,
and w is in addition supposed to be nowhere zero and to be uniformly
bounded. We seek waves which are periodic in the ^-coordinate and
thus we require ^ to satisfy the condition
(12.2.6) X(z + h) - x(z) = 0,
with h a real constant.
524 WATER WAVES
Following Levi-Civita, we assume that the region of flow in the
s-plane is mapped into the 99, ^-plane by means of %(z). The free sur-
face in the physical plane corresponds to the real axis \p — 0 of the
^-plane, and we assume that the entire region of the flow in the 2-plane
is mapped in a one-to-one way on the lower half of the ^-plane. (We
shall prove shortly that a function %(z) satisfying the conditions given
above would have this property.) In this case the inverse mapping
z(%) exists, and we may regard the complex velocity w(z) as an ana-
lytic function of # defined in the lower half of the ^-plane. In this
way we are enabled to work with a domain in the <p, y-plane that is
fixed in advance instead of with an unknown domain of the x, j/-planc.
Levi-Civita goes a step further by introducing a new dependent varia-
ble co, replacing w,\ by the relation
(12.2.7) w = Ue~l°>< co = 6 + ir;
so that (jo is an analytic function of cp -(- iy. Consequently we have
(cf. (12.2.2))
(12.2.8) I w | = Uer, 0 = argw.
Thus r — log (\w\/U), while 0 is the inclination of the velocity vector.
In the same way as in sec. 10.9 (cf. the equations following (10.9.11))
the boundary condition (12.2.4) can be put in the form
(12.2.9) 0V = AV3T sin 0, for y> = 0,
with A' defined by
(12.2.10) A' = g/f/3.
Our problem now is to determine an analytic function a>(#) —
0(<p9 ip) + if(<p, ty) in the lower halfplane y < 0 and a constant A' in
(12.2.9) such that a) o> is analytic for \p < 0, continuous for \p ^ 0,
b) Ov is continuous for y ^ 0 and the nonlinear boundary condition
(12.2.9) is satisfied, c) co has the period Uh in 99, d) co(%) -> 0 as
yj -> — oo, c) | a)(%) | fg £. The last two conditions are motivated by
the conditions imposed on w at oo and the condition w 3= 0: the con-
dition d) from (12.2.5) and (12.2.7), while the condition e) is imposed
in order to ensure that w is uniformly bounded away from both zero
and infinity. As we shall see, the condition c) leads to the periodicity
condition (12.2.6) on #.
We proceed to show briefly (again following Levi-Civita) that a
solution of the problem we have formulated for co would lead through
(12.2.7) to a function w(%) and then to a function %(z) through the
LEVI-CI VITA'S THEORY 525
differential equation d%(z)/dz = w(%) which satisfies all of the condi-
tions formulated above. The essential items requiring verification
are the periodicity condition and the one-to-one character of the
mapping z(%) defined by
over the half plane ^ < 0.
We proceed to investigate the second property. From (12.2.7) we
have
l/w(%) — — e~r (cos 0 + i sin 0).
and hence that
fM = i«-*
W t/
Since \ co \ ^ ^, it follows that w is bounded away from zero, so that
rxci«,
the integral — converges. Since both | r | ^ J and | 0 | ^ J, it
Jo »
follows that &e(I/w) is positive (we assume (7 to be positive) and
bounded away from 0 and oo. We have ^ + iyv = i/w9 oc^ + iy^ = l/w9
so that yv = ^(l/w)and x9 — &te(\\w}\ since <#e(I/w) > 0 it follows
therefore that y is a strictly monotonic increasing function of y, and
x similarly in (p. Consequently the mapping z(%) is one-to-one, and
in addition y -> — oo when y -> — oo, while # -> i oo when
9^ -> i °° since &e(I/w) is positive and bounded away from zero; thus
the flow is mapped onto the entire halfplane \p < 0.
&
dz
We consider the periodicity condition next. We have —
since a) has the period Uh by assumption. This implies that z(% + Uh)
— %(%) = const. This constant is easily seen to have the value h by
letting ip -» oo in the formula
dx
I
J
= z(x + Uh) -
since w -> U uniformly when y> -»• — oo. Consequently we have
*(* + UA) - x(X) = h,
y(X + Uh) - y(x) = 0.
526 WATER WAVES
We know from (12.2.8) and | 0 | ^ J that the stream lines \p = const,
have no vertical tangents, hence they can be represented in the form
y :=- y(v), and the last two equations show that they are periodic in
x of period h. The problem of determining co(%) subject to the condi-
tions a)— e) is therefore equivalent to the problem formulated for %(z).
1 2.2b. Outline of the procedure to be followed in proving the existence
of the function
The proof of the existence of the analytic function co(%) which
solves our problem will be carried out as follows. First of all, we ob-
serve that the problem has always the solution co(%) =0, correspond-
ing to the uniform flow w = U with undisturbed free surface. We
shall begin by assuming that a solution a>(%) ^ 0 exists, and will then
proceed, through the use of the properties assumed for co, to derive a
functional equation for the values £0(99, 0) of co on the boundary ip — 0,
— oo < <p < oo. It will then be shown that the functional equation
has a solution co(<p9 0) ^= 0 in the form of a complex- valued continuous
function &>(<p), and this function will be used to determine an analytic
function co(<p, y}) in — oo < y> < 0, — oo < (p < oo, with co(<p) as
boundary values, which is then shown to satisfy all of the conditions
a)-e).
It will occasion no surprise to remark at this point that the solution
we obtain will give a motion in a neighborhood of the uniform flow
with horizontal free surface, i.e. with an amplitude in a neighborhood
of the zero amplitude. Also, it should be remarked that the problem
in perturbation theory which thus arises involves a bifurcation pheno-
menon, since the desired solution of the nonlinear problem, once the
wave length is fixed, requires that the perturbations take place in the
neighborhood of a definite value of the velocity U. In other words, the
desired solution bifurcates from a definite one of the infinitely many
possible flows with uniform velocity which are exact solutions of the
nonlinear problem.
The decisive relation in the process just outlined is the nonlinear
boundary condition (12.2.9). It is convenient to introduce at this point
some notations which refer to it, to recast it in a different and more
convenient form, and also to derive a number of consequences which
flow out of it. At the same time, some factors which motivate all that
follows will be put in evidence.
LEVI-CIVITA'S THEORY 527
Since we wish to concentrate attention on the boundary values
co (<p, 0) of a>, it is useful to introduce the notation
(12.2.11) eo(p, 0) = £>(<p) = 6(<p) + ii(<p),
and then to introduce the operator f[co] defined by
(12.2.12) /[£] - )i(e~*~ sin 0 - 0) + ee~^ sin 0 = F(<p)
with e defined by
(12.2.13) e - V - L
The constant A will be given an arbitrary but fixed value; the quantity
27T/A will then be the period of the function co(%). The constant e, and
with it A' through (12.2.13), will be fixed by the solution a)(%) in a
manner to be indicated below, and the propagation speed U is then
determined by the formula (12.2.10). As can be seen at once, the
boundary condition (12.2.9) now takes the form
(12.2.14) 0V - A0 - F(<p), y = 0.
The reasons for writing the free surface condition in the form
(12.2.14) are as follows: As remarked above, we seek a motion in the
neighborhood of a uniform flow, so that co as defined by (12.2.7) should
be small in some sense. It would seem reasonable to set up an iteration
procedure which starts with that solution (o^y, y) of the problem
which results when F(<p), which contains the nonlinear terms in the
free surface condition, vanishes identically. Afterwards the successive
approximations will be inserted in F(cp) to obtain a sequence of linear
problems whose solutions cok converge to the desired solution of our
problem.
The problem of determining co^y, ^>), when F = 0, is exactly the
problem posed by the linear theory which was discussed at length in
Chapter 3; in fact, if F(q>) vanishes, we know from the discussion in
Chapter 3 that the only bounded conjugate harmonic functions
0i (^» V>)» ri(V> V0> °ther than 01 — TI = 0, in the lower half plane
ip < 0 which satisfy the homogeneous free surface condition
QIV — A0! = 0 are the functions
01(^, y>) = a^ sin A<p,
ri(<P) V) — aie^ cos ^P»
once (p is taken to be zero at a crest or trough of the wave. Thus the
boundedness condition at oo and the homogeneous free surface con-
dition lead automatically to waves which are sines or cosines of <p.
528 WATER WAVES
The ' 'amplitude" at is, of course, arbitrary on account of the homo-
geneity of the problem. The corresponding function co1(^) is then
We suppose, naturally, that the "amplitude" | ax | is small and hence
that | ct^ | is also small of the same order. The basic parameter in the
iteration procedure will be the quantity av and the procedure will
be so arranged that the quantity s in (12.2.13) as well as the iterates
a)k will be of order | ox |. It is then easily seen that F((p) will always
be of order | ax |2, which indicates that such a scheme of iteration is
reasonable. We shall show that it does indeed lead to a sequence COA
which converges to the desired solution co for all sufficiently small
values of | % |, and that the solution co fixes a value of e, and hence of
A' since A is once for all fixed, in a manner to be explained in a moment.
It might be mentioned that it is not difficult to verify that the
corresponding motion furnished in the physical plane by Xi(z) would,
up to terms of first order in | ax |, be given by
and this coincides with what was found in Chapter 3 by a more direct
procedure.
Iterations, as we have indicated, are to be performed, starting with
the solution a)^ = iale*(v~i(p] of the linearized problem, with ax regarded
as a small parameter. This is then inserted in the right hand side of
(12.2.14); a bounded harmonic function 02(g9, y) in the half plane
ip < 0 is then determined through this nonhomogeneous boundary
condition and the corresponding analytic function co2(%) — ^2 + 7'T2
with it. In order to solve the boundary problem for 02 (and through it
o>2), however, it is necessary to dispose of the parameter e in (12.2.13)
appropriately. This comes about because, as we have just seen, the
homogeneous linear boundary value problem for 01 has a non-trivial
solution, 6l — a^ sin h<p, and hence an orthogonality condition on
F(<p) is needed which will ensure the existence of the solution of the
nonhomogeneous problem for 02. This condition is well known to be
that the integral F(<p) sin hydy should vanish. It turns out that
the value of e so determined really is of the same order as av Con-
tinuing the iterations in this fashion, the result is a sequence of
functions eon(#), and a sequence of corresponding values en of s sueh
LE VI -CI VITA'S THEORY 529
that | a)n | and | en \ are all of order | ax |. It is to be shown that both
sequences converge to yield a function CD (/) and a number e which solve
the problem, the quantity A' in (12.2.10) being fixed by e = lim en and
the arbitrarily chosen value of A through (12.2.13).
We observe that this whole procedure is in marked contrast with
the method of solution of the problem of the solitary wave given
by Friedrichs and Hyers [F.13] and explained in sec. 10.9; in the latter
case the iteration procedure is quite different and it is carried out
with respect to a parameter which has an entirely different signific-
ance from the parameter ax which is used here.
The procedure outlined here also differs from the procedure followed
by Liechtenstein [L.I 1 J in solving the same problem. Lichtenstcin applies
E. Schmidt's bifurcation theory to an appropriate nonlinear integral
equation (essentially the counterpart of the functional equation to be
used here). In this procedure, the basic idea is to modify what cor-
responds to the function F(q>) in (12.2.14) in such a way that the
modified problem (which is arranged to contain one or more parame-
ters) can always be solved. Afterwards, conditions are written down
to ensure that the modified problem is identical with the original
problem; these conditions arc called the bifurcation conditions. Such
a process could have been used here in conjunction with an iteration
scheme, as a substitute for the process of fixing the parameters en at
t-cich stage of the iteration procedure in the manner indicated above.
Basically the method of solution of our nonlinear problem just out-
lined requires the solution of a sequence of linear problems. We turn
next, therefore, in sec. 12.2c to the derivation of the solution of these
linear problems, and afterwards, in sec. 12.2d we shall prove that an
appropriate sequence of solutions of the linear problems converges to
the desired solution of the nonlinear problem.
12.2c. The solution of a class of linear problems
The linear problems we have in mind to solve, in accordance with
the above discussion, are problems for co(<p, \p] = 0(<p, y) -f- ir((p, \p)
when F(q>) in (12.2.14) is regarded as a given function. That is, u>
should satisfy all of the conditions formulated above, except for the
free surface condition. Later on we shall begin our iteration process
with a function o^ which has the period 2n/A in <p; F(q>), all sub-
sequent iterates, and the solution itself, will have the same period.
Since we expect the waves to be symmetrical about a crest or trough
530 WATER WAVES
(indeed, our existence proof will yield only waves with this property),
we suppose that the origin is taken at such a point, and hence that at
any stage of the iteration process 6(<p, 0) = 6(<p) would be an odd func-
tion of 9?, while r(<p, 0) — r((p) would be an even function of 9?, and
both would have the period 2n/L Thus F(q>) in (12.2.14) as defined
by (12.2.12) should be taken for our purposes as an odd function of y
with the real period 2n/L
If we were to work at the outset with Fourier series, it follows that
6 would be represented as a sine series, and F(<p) also. However, we
wish later on to carry out an iteration process in which only contin-
uous, and not necessarily differentiable, functions of q> are employed,
and in which the existence of a certain continuous periodic function
ft) (9?) of period 2;rc/A is first proved; this function will furnish the
boundary values of the solution a)(q>9 ip). (Afterwards, the question of
the existence of the normal derivative 6^(99, 0) in (12.2.14), and of
other derivatives, will be dealt with separately. ) In doing so, we shall
have occasion to approximate such continuous periodic functions by
finite Fourier series, or Fourier polynomials, a process justified by the
Weierstrass approximation theorem which states that such a poly-
nomial can always be constructed to yield a uniform approximation
for all values of <p and any arbitrary degree of approximation.
Suppose, therefore, that F(<p) in (12.2.14) had been approximated
at some stage in the iteration procedure by a function g((p] in the
form of the following finite sine series:
(12.2.15) g(<p) = J bv sin
v=l
and we seek the bounded harmonic function 6(99, y) which satisfies
the boundary condition (12.2.14) with F = g. For this purpose we
write the solution 6(<p, y>) = Ste 00(99, \p) also as a finite Fourier sum:
(12.2.16) 0(q>9 \p) = J avev** sin vhp.
v=l
Insertion of this sum in (12.2.14), with F = g, leads to the following
equations for the determination of the coefficients av:
v v, .....
7>-l)a, = 6,
and thus to the conditions
LEVI-CIVITA'S THEORY 531
[ at arbitrary, fcj — 0
(12.2.17) a = fc
It is very important for the following to observe that the Fourier sine
series for F(<p) must lack the first order term: otherwise, as we have
remarked above, our problem would have no solution: a term of the
form &! sin T^p in F(q>) is a "resonance" term, the presence of which
would preclude the existence of the solution of the nonhomogeneous
problem. The unique solution for 0(<p, y) is
n ft
(12.2.18) 0(<p. \p) = ^ -— e^ sin vty + «i^sin A<p,
v=2^(v ~ 1)
once ax is prescribed (cf. Chapter 3) and 0 is assumed to be an odd
function of (p. The harmonic conjugate r(cp9 y>) of 0(<p, \p) is obtained by
integrating the Cauchy-Riemann equation 6V — — r^, with the result
n b
(12.2.19) T(Q?, w) = V - - ~ ev^ cos vhy + a^ cos Ay.
vf2A(v — 1)
(A possible additive integration constant is set equal to /ero since
r -> 0 when ip -> — oo.) Thus we would have for co(y, \p) under the
assumed circumstances the expression
(12.2.20) co(o>, V) = * 2 ~~V e^-^
£2X(v - 1)
In other words, if F(<p) is given as in (12.2.15) by a finite Fourier
series of sines which lacks its lowest order term, i.e. is such that
(12.2.21 ) I* 2n/*F((p ) sin Xydy = 0,
j o
then, as we see from (12.2.20) and the discussion preceding it, the
function a)(<p) — co((p, 0) — 6((p) + ii((p) given by
(12.2.22) a>(<p) - i 2 -A -- e-<"** + ia^""
yields the boundary values of an analytic function co(<p, ip) which
would satisfy the boundary condition (12.2.14). Evidently, o) would
also satisfy all of the conditions a) to e) formulated above, if the
amplitude ax of the first order term of the Fourier series is chosen
582 WATER WAVES
small enough, except that the boundary condition b) is replaced by
a linear condition.
It is clear that the insertion of a function a)(<p) as given by (12.2.22)
in (12.2.12) to determine a new function F(cp) in order to continue the
iteration process would not yield in general a function representable
as a finite Fourier sum, but rather to one representable only by a
Fourier series. However, we have already stated that we wish to
carry out our iteration scheme in the nonlinear problem within the
class of continuous functions, which need not possess convergent
Fourier series. Nevertheless, the general scheme outlined above for
determining the successive iterations can still be used once it has been
extended in an appropriate way to the wider class of functions. For
this purpose, and later purposes as well, it is convenient to introduce
the terminology of functional analysis. Thus we speak of the linear
vector space of elements which are complex-valued functions
g(q>) — <x.((p) + ifi(<p)9 continuous for all <p and of period 2rc/A, such that
a is an odd function and ft an even function of <p. The scalars are the
real numbers. This space is made into a normed linear space by intro-
ducing as the distance from the origin to the "point" g the following
norm, written || g ||:
|| g || = || a + ift || - max A/a2 + ]82 - max | g |,
9
and as the distance between two elements or points gl5 g2, the norm of
their difference, i.e. || gl — g2 ||. This space, which we shall call the
space JB, is complete, i.e. it has the property that every Cauchy se-
quence in the space converges to an element in the space.* By a Cau-
chy sequence gn we mean a sequence such that || gm — gn II -* 0
when m, n -> oo. Since the norm is the maximum of the absolute
value of g, it follows that a sequence gn which is a Cauchy sequence is
uniformly convergent and hence has a continuous function as a limit.
We remark also that the notion of distance thus introduced in our
space has the usual properties required for the distance function in a
metric space, i.e., the distance is positive definite:
|| g || ^ 0, and || g || = 0 implies g = 0,
and the triangle inequality
ll*i+ foil ^11 gill +11 foil
holds.
* We remark that a complete linear normed space is called a Banach space ;
however, such properties of these spaces as are needed will be developed here.
LEVI-CIVITA'S THEORY 533
We introduce next the subspaee Bl of our Banach space B which
consists of all real functions g(<p) given by finite Fourier sums of sines
lacking the term of first order:
(12.2.23) g(<p) = J 6vsinvA<p, bv real.
v=2
With respect to this set Bl of functions we define a transformation
S as follows:
(12.2.24) Sg(<p) = i J - - - #** e-i9**, - oo < \p ^ 0.
Since ip ^ 0 we have for the norm* of Sg the bound
From Cauchy's inequality the following inequality for || Sg \\ is then
obtained :
the last step resulting from (12.2.23) because of the orthogonality
of the functions sin vA,q>. Since
it follows that a constant K exists which is independent of g and n
(though not of A), such that
(12.2.25) \\Sg\\ <K \\&\\, torgCB,.
Thus S is what is called a bounded transformation in Bl since it
transforms each element of Bl into an element of B with a norm bound-
ed by a constant times the norm of the original element. Clearly, S
transforms a certain class of boundary data given in terms of the real
function g(<p) into an analytic function defined in the lower half plane.
We proceed next to extend the domain of definition of the trans-
formation S in such a way that it applies to a certain set of real
* By the norm of a function of two variables we mean the least upper bound of
its absolute value.
584 WATER WAVES
functions in B which contains the set B19 i.e., to the set JB2 of continu-
ous real functions g in B with vanishing first Fourier coefficients, that
is, to functions such that j * g((p) sin h<pdy = 0; this subspace B2 is
j o
also complete, with the same norm. The extension of the definition of
S is made in the following rather natural way: Take any function g
in B2 and let gn be a sequence of functions in B^ (i.e., a set of real tri-
gonometric polynomials lacking first order terms) which approximate
g uniformly. That such a sequence exists is known from the Weier-
strass approximation theorem. We then form the sequence Sgn— which
is possible since S is applicable to these functions— and observe that
because S is obviously a linear transformation, and hence
\\Sgm-Sgn\\ ^K\\gm~ gj|
from (12.2.25) since gm — gn is an clement of J5X. Thus the sequence
Sgn is a Cauchy sequence, for 1 1 gm — gn \ \ -> 0 because the functions
gn are assumed to furnish uniform approximations to g; hence the se-
quence Sgn has a unique continuous limit function which we define
as Sg. The transformation S thus extended will be referred to by the
symbol S. S is easily seen to be a linear transformation and the
inequality (12.2.25) holds for it with the same value of K since it
holds for all the functions gn, independent of n.
Once the definition of the transformation S is extended so that it
applies to functions in B2, it becomes possible to widen the class of
functions within which a (generalized ) solution of our linear problem
can be sought, and at the same time to reformulate the boundary
problem in terms of the the following functional equation:
(12.2.26) o>(0>) = Sg(<p) + ia^r***,
in which Sg(<p) refers to the above extension of Sg((p), with g(q>) C 1?2,
evaluated on the boundary \p — 0. By virtue of (12.2.20) and the
definition of Sg(y>) for g(q>) in J52, one might expect that <*>(<p) would
yield correct boundary values for the solution a>(q>9 y). We proceed
to show that this is indeed the case, i.e. that any continuous function
<*>($) which is given by (12.2.26) for gCB2 furnishes the boundary
values of a function c*)((p,y)) defined and continuous for ip ^ 0 which is
analytic in the lower half plane, has a real part Q(<p,y) with a continuous
normal derivative Qv in the closed half plane, and such that the boundary
condition (12.2.14) with F((p) == g(<p) is satisfied.
LEVI-CIVITA'S THEORY 535
The regularity properties of the function co((p9 y) on the boundary
y — 0 come about because of certain smoothing properties of the
transformation S, which we proceed to discuss. Consider first the
special case in which g(<p) is given, as in (12.2.23), by a finite Fourier
sum. The function a(<p) + $(<?) = Sg:
Z _L. .-* ?„ ^ b* sin yfo . v v 6* cos
a + ?p = o£ = 7 -- ~r ^ 7
^ p 8
has in this case the following property:
- & - g + A5.
The validity of this formula for *S follows from the fact that Sg(<p) —
a(9?> V) + ^(9?, y;) as given by (12.2.24) is an analytic function in the
closed half plane y ^ 0, and that Sg satisfies the nonhomogeneous
boundary condition (12.2.14) with F = g, when 6(<p, \p) is identified
with a(<p, \p). This implies, because of (12.2.26) and the triangle in-
equality, the inequality
|| £,11 ^AMIgH, #! = constant,
as one easily sees. If g is any function in J32, it now can be proved that
Sg— defined for functions g in B2 in the manner described above— is
such that its imaginary part /9 has a continuous derivative with respect
to (p. This is done by approximating g uniformly by a sequence gn of
finite Fourier sums in B^ The corresponding derivatives f$ntp form a
Cauchy sequence because of the above inequality and hence would
converge to a continuous function. The relation — ^ — g+Aoc also
would hold in the limit for the derivative /^; thus ^ is again seen to be
continuous. It follows, therefore, that a continuous function co((p) =
0(<p)+&(v) giyen by (12.2.26) has the property that r((p) has a conti-
nuous derivative, and in addition —^(9?) = g(q>)+AO(q>)- We observe
next that 6+ir furnishes the boundary values of an analytic function
00(9?, y>) defined for — oo < \p < 0 and continuous in the closed half
plane: this follows again by approximating g(<p) by functions gn(<p) in
J?!, as in (12.2.23), defining the corresponding o>n(<p, y) by(12.2.20),
and making the passage to the limit to obtain a>(<p,\p )=0(<p,y) ) +ir((p,ip );
that the functions con((p, ip) converge to a continuous function for
y> ^ 0 we know, and that the limit is analytic at interior points follows
since it is the uniform limit of analytic functions. Since — r^((p, ip) =
Ov(<p, y))fory) < 0, and since r^ is itself a harmonic function with con-
586 WATER WAVES
tinuous boundary values r^ it follows that rv(<p9 y>) -> r^ as y -> 0,
and hence that 0V is also continuous for \p = 0, i.e. 6v((jp9 0) = 0V;
hence the condition — rv = g(<p) + A0, which we have proved above
to hold becomes 0V — A0 = g(g?), and this is our boundary condition.
We have therefore shown that a continuous function a)(<f>) which is
given by (12.2.26), with g(^) any function in B2, furnishes the boun-
dary values of an analytic function co(^, y) = 6 + it in \p < 0 which
is continuous for y ^ 0, whose real part 0(^, ip) has a continuous deri-
vative 0V in the closed lower half plane with 0V(^, 0) — A0(<£, 0) = g(^)
or, as we also write it: 0V — A0 = g. In the subsection immediately
following we shall establish the existence of a continuous solution
o}(<f>) of (12.2.26) when g(^) is not given a priori, but depends in a non-
linear way on c5(<£) = 0(g) + ir(<f>), i.e. when g(<£) = F(<f>) =
A(tf-3*sin0 - 0) + «?-3*sin 0 (cf. (12.2.12)). Assuming this to have
been proved, we proceed to draw at once further conclusions regarding
the properties of oj(<f>) and its continuation a)(<f>9 y>) as an analytic
function in the lower half plane ip ^ 0. We show, in fact, that the
solution a>(<f>) in B of our nonlinear functional equation will not only
furnish the boundary values of an analytic function a)(<f>9 ip ) in \p < 0,
with co as boundary values, but that a) has continuous first derivatives
in the closed half plane ip ^ 0, and is as a consequence then seen
actually to be analytic for \p — 0. Thus, in particular, cfl(^) would
possess a convergent Fourier series. Consider the analytic function
F(x) defined in the lower half plane y < 0 with boundary values
<%e fi(%) = g(y) for \p = 0 and with &(%) bounded at oo. The boun-
dary condition 0V — W — — r^ — A0 = g(^) satisfied by our solu-
tion a)((p<y>) is also extended analytically into the lower half plane
if < 0 by means of the relation 3le(a)^ — Aco) = 3te &(%) in the
manner used frequently in Chapters 3 and 4; hence we have
ia> — Aco =
since the imaginary parts of cox and co both vanish for y = — oo. We
have just seen above that co has an imaginary part r with a continuous
derivative TV on the boundary y = 0. The fact that r^ is continuous
then makes it possible to show that co((p, y) is Holder continuous for
y> = 0. This follows, in fact, from a classical theorem of Privaloff (Bull.
Soc. Math. France, Vol. 44, 1916) which states that a function which is
defined and continuous in the unit circle, analytic in the interior of
the circle, and has an imaginary part which is Holder continuous on
LEVI-CIVITA'S THEORY 537
the boundary of the circle, is itself Holder continuous in the closed
unit circle: in other words, Holder continuity of the imaginary part
brings with it the Holder continuity of the real part of the function.
This theorem is made applicable in the present case by mapping one
of the period strips of the solution a)(%) in the #-plane conformally on
the unit circle of a £-plane, say: we know, in fact, that CD has the real
period 2n/L The part of the boundary of the strip given by y) = 0
(i.e. a full period interval on the boundary) is mapped on the boundary
of the unit circle. (Since | co | -> 0 as \p -+ — oo, the infinity of the
strip is mapped on the center of the circle.) Thus co — 0 + iv has an
imaginary part with a continuous derivative rv on y> = 0, and it
follows that r is certainly Holder continuous for \p = 0. Consequently
the real part of co(£), hence co(£) itself, is Holder continuous in the
closed unit circle, since this property is not destroyed by the conformal
mapping. The real part of F(%) on the boundary \p = 0, which is given
by g(<p), is now seen to be Holder continuous, simply because of the
way g(<f>) is given in terms of 0 and r, and the Holder continuity of
the latter functions. A second application of Privaloff 's theorem, this
time to F(%), then leads to the Holder continuity of P(%) for \p ^ 0.
The relation icox— Aco = %*(%) thus holds for y> = 0 and it shows that
cox is continuous for \p = 0, since both co and F have this property. In
other words co^ and cov are both continuous for \p = 0. Finally, once
co(%)is shown to have a continuous derivative with respect to % on the
boundary, we could make use of a theorem of H. Lewy [L.9] to show
that co(%) is actually analytic on the boundary.
12.2d. The solution of the nonlinear boundary value problem
The nonlinear problem to be solved here differs from the linear
problems discussed above because of the fact that the function g(<p)9
the nonhomogeneous term in the free surface condition, is not given
a priori, but rather becomes known only when the solution co(q>, \p)
itself is determined. On the other hand, we have seen that the equa-
tion (12.2.26) furnishes the boundary values <o(<p) for co(<p, y), in case
g((jp) is a known function in the space B2. To solve the nonlinear pro-
blem we now reverse this process: we regard the equation (12.2.26) as
a functional equation for the determination of the function 6>(<p) when
g(y>) is identified with the function F(<p) in equation (12.2.12), i.e.
when g(<p) itself depends in an explicitly given way on co(<p). The dis-
cussion of the preceding subsection shows that we have to prove
538 WATER WAVES
only that the functional equation has a solution o> (y) in the Banach
space B.
The existence of the solution 6>((p) of the functional equation will
be carried out, as we have stated earlier, by an iteration process
applied to the functional equation. To this end it is convenient to
introduce a nonlinear transformation R defined for any function
g = a + ifi in the whole space B by means of the relation
(12.2.27) Rg(<p) = X(e-M sin a - a) + ee~^ sin a.
In order that the transformation S defined in the preceding subsection
should be applicable to Rg(<p) we require, as part of the definition of
R, that e should be so determined that Rg(<p) lies in B2 and thus lacks
its first Fourier coefficient, i.e. such that J " Rg((p) sin h<pd(p = 0;
this leads to the following condition on e:
I " A(£~3^ sin a — a) sin hpdcp
(12.2.28) * = -
e-w sin a sin /.<pd<p
This implies that R is defined only if the denominator of (12.2.28)
does not vanish. Clearly, this nonlinear transformation yields always
real odd functions.
Consider now the functional equation
(12.2.29) <b(<p) = SRco((p) + iaf**>
in which Sg((p) refers to the extension of Sg(y>)9 as defined in the pre-
ceding subsection for functions g(<p) in the space I?2, on the boundary
y = 0 and ax is a given real constant. Because of (12.2.12), (12.2.14),
and the discussion of the preceding paragraph, it follows that a solu-
tion co(%) of the nonlinear boundary value problem will be established
once a function c5(<p) in B is found which satisfies the functional equa-
tion (12.2.29).
In carrying out the existence proof it is convenient to introduce a
few new notations. The function r(<p) is introduced:
(12.2.30) r(<p) = °^ - &-<*,
and a new transformation T on r is defined by
(12.2.31) Tr = 1 SRfafr + t>
LEVI-CIVITA'S THEORY 539
In other words, T is applied only to those functions r such that
ai(r + ie****) is in the domain of definition of R. The functional
equation (12.2.29) is now seen to be equivalent to the equation
(12.2.32) r = Tr,
and we seek a solution of it in the space B.
We shall solve (12.2.32) by an iteration process which starts with
a function i\ in B such that the corresponding function /?ft>1 is in Bz
(i.e. such that R is applicable to o^), inserts it in r2 = Tr^ etc., thus
obtaining a sequence rk with rk = Trk_v In order to make sure that
(12.2.28) holds for the solution we stipulate that the parameter e in
(12.2.12) be fixed at each stage of the iteration process so that (12.2.28)
is satisfied; this is done by setting
I n X(
(12.2.33) sk =
I e~~k sn a — a sn
f2*M Q* . - ,
e~*p* sin <x.k sin Acpckp
At the same time, this ensures that the transformation T is really
applicable to the members of the sequence rk. Of course, it will be
necessary to show that the denominators in the equations (12.2.33)
are bounded away from zero and that the sequence ek converges. The
existence of a sequence rk converging to a solution r of (12.2.32) will
be shown by disposing properly of the arbitrary constant % (the
amplitude of the wave in the linearized solution of the problem),
i.e. by. showing that «t 7^ 0 can be chosen small enough so that the
sequences rk and ek, each of which is a function of ar converges. We
note in passing that if d)k = <xfc -f- ijjk is small of order %, then ek as
given by (12.2.33) is also of order al— later on, we give an explicit
estimate for it— so that the quantities ek should not turn out to be of
the wrong order.
The convergence of the sequence of iterates rk to a solution r of
r = Tr will be shown by proving that all of the functions rk, for
values of ax less than a certain fixed constant, satisfy the following
conditions: for some real positive constant r] and real positive « < 1
I) \\r\\^r, implies \\Tr\\ ^r],
II) llrJMIr.H^ implies || JVj -2V.H £x || 1^-r, ||,
for any pair of functions rv r2. Condition I) says that the transforma-
tion T carries any function in the closed "sphere" of radius 77 into
540 WATER WAVES
another function in the same "sphere", and Condition II) is a Lip-
schitz condition.
The iteration scheme for solving the functional equation Tr = r
proceeds in the following standard fashion. Take any function rQ(<p)
in B with || r0 || < r\ to which T is applicable and consider the iterates
rn defined by rw = Trn_±. From I) we see that all such functions r
have a bounded norm. We have, evidently:
rn+i -rn = Trn - Trn_v
Since II) holds we may write
II fVu - rw || = || Trn - Trn^ || ^ * \\ rn - rn_, \\.
and hence
We consider next the norm of rm — rn, m ^ n:
II rm - rn || = || (rm - r^) + (rm_^ - rm_2) + . . . + (rn^ - rn) \\
- r
1 -
(The triangle inequality is of course used here.) Since x < 1 it is clear
that the sequence rn is a Cauchy sequence and hence it converges to
a unique limit function r in B with norm less than r). (The uniqueness
statement holds of course only for functions r with norm less than rj. )
That the limit function r satisfies Tr = r is clear, since the sequence
rn is identical with the sequence Trn_± and hence both converge to
the same limit r.
In order to establish conditions I) and II) for the functions in the
sequence rk9 and hence to complete our existence proof, it is conve-
nient to introduce certain continuous functions Fl(N)9 F2(N), . . .,
which are defined for real N 2> 0, bounded near N = 0, and increasing
with N.
Suppose that rl C B is such that || rl \\ ^rj. We set OJ1 — Ot + irl
and (cf. (12.2.30)) recall that rl = l + lTl - ie-*<p so that || o>, || ^
<*i
I ai I (1 + 1?) = N. In what follows, however, we omit the circumflex
over 0 and r, and we shall also omit the subscript on a.
The following inequalities hold when | a \ is sufficiently small:
LEVI-CIVITA'S THEORY 541
1. | |<r**i sin ,| |
(12284) 2' ll*-3T'sin0i-
8. \\e-^smei-e-^sinem\\^\a\\\
4. 1 1 e~3T« sin 0Z - e-3r»» sin dm - (Bl - 6m)\ \
^\a\\\ri-rm\\NFt(N).
00
These inequalities are all based on the fact that if A(f ) = J &nfn is
o
an absolutely convergent power series for all real f , then 1 f | ^ N
00
implies | A(|) \ ^ ^\ hn\ Nn. We derive the second and third of the
o
above inequalities as typical cases— the others are derived in a similar
way. Consider the second inequality; we write
|| e-* sin 0 - 0 || = || 0(*-3T - 1) + e-3T (sin 0 - 0) ||
..
with F2(Ar) = 3^3N + Ate4Ar. Consider next the third inequality. From
the mean value theorem we have
--3e-T snT^rm+^r cos ,-m.
in which 0*, T* are some values on the segment joining (0j, rt) and
(0m, rm). From this we have
|| *-*. sin 0, -e-**m sin 0m || ^ || 0t-0m
with F3(A^) = 4e3JV, in view of the definition of rl9 rm given in (12.2.
30).
It is also essential to give an estimate for ek in (12.2.33). First we
obtain a lower bound for the denominator. We have, from the defini-
tion of r and the second inequality above:
r
sin 0 sin
/•2^1/A /»2
— (e~^r sin 0 — 0) sin Ag? dgp + 0 sin
Jo Jo
-f | a | (sin Arc + ^?* ^) sin
Jo
~ [| a | (1 - 27?) -
542 WATER WAVES
Subscripts have been dropped in the above. Since N = \ a \(l + q)
it is clear that for r/ ^ J and | a \ sufficiently small, say | a \ ^ a(1),
the resulting expression is greater than k \ a |, with k a positive con-
stant depending on a(1). Use of this fact together with the second in-
equality above in the definition (12.2.33) of e leads at once to the
inequality:
5. |.|
a
Thus e is of order | a \ if TJ ^ J, since N is of order | a \. Thus the
quantities ek, as defined by (12.2.33), are in fact of the correct order.
In the same fashion, by using all four of the above inequalities, one
obtains
(12.2.34) 6. \sl-£m\^\\rl- rm
a
We are now in a position to show that the conditions I) and II)
hold once proper choices of 17 and a have been made. We suppose that
|| r || ^ r\ ^ J and choose a such that 0 < | a \ ^ a(1); any value x
in the range 0 < K < 1 is taken. As before, the norm of the function
a) defined by r satisfies || o> || ^ | a |(1 + 77) = N ^ || a \. Our next
objective is to give an estimate for Tr as defined by (12.2.31). We
have, in view of (12.2.31), and (12.2.27) and (12.2.25):
| a I I a I | a
and this in turn yields:
| Tr < A
\a\ \a\
with K a fixed constant, upon using the first, second, and fifth of our
inequalities, together with the fact that N is of order a. Thus if
a(2) ^ a(1) is a positive constant such that Ka(2)F7(^a(2}) ^ r\ it
follows that
|| Tr || ^ r\ if || r \\ <^ r\ ^ J and | a \ ^ a(2).
This establishes the condition I). The proof that II) holds is carried
out in much the same way. Suppose that rl9 r2 are such that || rl ||,
II ^2 II ^il- We have, upon using the inequalities 1. to 6.:
LEVI-CIVITA'S THEORY 543
—
a
sin ei-f9r, sin 02-0! +02)
\a\
+e1(tf~3Ti sin dl— <r3T2 sin 02)-f e~*T2 sin 6^—e^)
K JV3
I al X 2 4 3 5 Tal l 8
with ^ a fixed positive constant. If «(3) ^ <z(2) is a fixed positive con-
stant such that 3?a(a>Fe(£a<8)) ^ *, then
and the condition II) is verified.
It follows that an iteration process starting with an arbitrary
function r0 in B, such that Ra)0 lies in B2, with \\rQ\\ ^rj ^ ± will
converge to a solution r of Tr = r if 0 < | a \ ^ a(3). The function
a> — a(r + te""*^) is then a solution of the functional equation
(12.2.29) which lies in /?, and which is furthermore not the "trivial"
solution a) = 0 (which always exists), since || cb \\ ^ a(l — || r \\) ^
fa since \\ r \\ fg J. This concludes the proof for the existence of a
continuous solution co(^) of the nonlinear functional equation. Once
this has been done we have seen at the end of the preceding sub-
section that d>(<f>) is actually analytic in <£.
It is also clear that the quantities ek assigned to each cbk and rk
exist, and that they converge since the ek form a Cauchy sequence in
view of the sixth inequality above and the fact that 1 1 rm — rn \ \ -> 0.
If we set e — lim ek, it is clear that the resulting value of A' obtained
from (12.2.13), in conjunction with the arbitrarily prescribed value of
A, yields the propagation speed U through (12.2.10) as a function of
the amplitude parameter a. Since co(#) has the period 2yr/A, it follows
from the discussion at the beginning of this section that the motion
in the physical plane has the period, or wave length, 2jr/AC7.
Bibliography
Arthur, R. S.
[A.I] Revised wave forecasting graphs and procedure. Scripps Institution of
Oceanography of the University of California. Wave Report 78, 1947.
i.A.2] Variability in direction of wave travel. Annals of the New York Academy
of Sciences, Vol. 51, Art. 3, 1949, pp. 511-522.
[A. 3] Refraction of shallow water .... Transactions of the American Geophysi-
cal Union, Vol. 31, 1950, pp. 549-552.
A.4] See [M.14].
A.5] See [P.18].
Atkinson, F. V.
A. 6] On SommerfeWs "radiation condition." London, Dublin and Edinburgh
Philosophical Magazine, Ser. 7, Vol. 40, 1949, pp. 641-651.
Abdullah, A. J.
rA.7] Cyclogenesis by a purely mechanical process. Journal of Meteorology,
Vol. 6, 1949, pp. 86-97.
Baird, E. G.
[B.I] See [E.5].
Baker, B. B., and E. T.Copson
[B.2] The Mathematical Theory of Huygens' Principle. The Clarendon Press,
Oxford, 1939.
Bakhmeteff, B. A.
[B.3] Hydraulics of Open Channels. McGraw-Hill, New York, 1932.
Barber, N. F., and F. Ursell
[B.4] The generation and propagation of ocean waves and swell. 1. Wave
periods and velocity. Philosophical Transactions of the Royal Society
. of London, 240, 1948, pp. 527-560.
Batcman, H.
[B.5] Partial Differential Equations of Mathematical Physics. The University
Press, Cambridge (Eng.), 1932. Chapter 11, Diffraction problems.
Bates, C. C.
; B.6] Utilization of wave forecasting in the invasions of Normandy, Burma, and
Japan. Annals of the New York Academy of Sciences, Vol. 51, Art. 3,
1949.
Beach Erosion Board
[B.7] Technical Report No. 2. A summary of the theory of oscillatory waves.
U. S. Government Printing Office, 1942.
Bernard, P.
[B.8] Sur certaines propriet^s de la houle 6tudi6es a Vaide des enregistrements
stismographes. Bulletin Institut Oc£anographique, Monaco, Vol. 38,
1941.
Biesel, F.
[B.9] Le filtre a houle. La Houille Blanche, Vol. 4, 1949, pp. 373-375.
[B.10] Study of wave propagation in water of gradually varying depth. U.S.
National Bureau of Standards. Gravity Waves. NBS Circular 521, 1952.
Birkhoff, G.
[B.I I] Hydrodynamics. Princeton University Press, 1950.
545
546 WATER WAVES
Birkhoff, Garrett, and Jack Kotik
[B.12] Fourier analysis of wave trains. Proc. of the NBS Semicentennial
Symposium on Gravity Waves, 1951.
Blue, F. L.f and J. W. Johnson
[B.I 3] Diffraction of waves through a breakwater gap. Transactions of the
American Geophysical Union, Vol. 30, 1949, pp. 705-718.
Bondi, H.
[B.14] On the problem of breakers. Great Britain, Admiralty Computing Service,
WA-2304-13, 1943.
Bouasse, H.
[B.I 5] Houle, rides, seiches et mare'es. Librairie Delagrave, Paris, 1924, pp.
92-145.
Boussinesq, J.
[B.I 6] Thtorie de V intumescence liquide appeUe onde solitaire ou de translation
se propageant dans un canal rectangulaire. Institut de France, Academic
des Sciences, Comptes Rendus, .June 19, 1871, p. 755.
[B.17] Essai sur la the*orie des eaux courantes. Institut de France, Academic
des Sciences, Memoires present6s par divers savants, Vol. 23, 1877.
Broer, L. J. F.
[B.I 8] On the propagation of energy in linear conservative waves. Applied
Scientific Research, Vol. A 2, 1951, pp. 447-468.
Bruman, J. R.
[B.19] Application of the water channel — compressible gas analogy. North Ameri-
can Aviation, Inc., Report NA-47-87, 1947.
Bjerknes, J., and H. Solberg
[B.20] Life cycle of cyclones and the polar front theory of atmospheric circulation.
Geofysiske Publikationer (Kristiania), Vol. 3, 1922, pp. 3-18.
Carr, J. H., and M. E. Stelzriede
[C.I] Diffraction of water waves by breakwaters. U.S. National Bureau of
Standards, Gravity Waves, NBS Circular 521, 1952.
Chrystal, G.
[C.2] An investigation of the seiches of the Loch Earn by the Scottish lake
survey. Transactions of the Royal Society of Edinburgh, Vol. 45, 1906,
pp. 361-396.
Cooper, R. I. B., and M. S. Longuet-Higgins
[C.3] An experimental study of the pressure variations in standing water waves.
Proceedings of the Royal Society of London, Ser. A, Vol. 206, 1951,
pp. 424-435.
Copson, E. T.
[C.4] On an integral equation arising in the theory of diffraction. Quarterly
Journal of Mathematics, Oxford Scries, Vol. 17, 1946, pp. 19-34.
[C.5] The asymptotic expansion of a function defined by a definite integral or
contour integral. British Admiralty Computing Service, Report No.
S.R.E./ACS 106, 1946.
Also see [B.2]
Cornish, Vaughan
[C.6] Waves of the Sea, and Other Water Waves. T. Fisher Unwin, London, 1910.
[C.7] Ocean Waves and Kindred Geophysical Phenomena. Cambridge University
Press, London, 1934.
Coulson, C. A.
[C.8] Waves. Fourth Edition, Oliver and Boyd, Edinburgh, 1947.
Courant, R., and K. O. Friedrichs
[C.9] Supersonic Flow and Shock Waves. Interscience Publishers, Inc., New
York, 1948.
BIBLIOGRAPHY 547
Courant, R., and D. Hilbert
[C.10] Methoden der Mathematischen Physik. 2 Volumes, Interscience Publish-
ers, Inc., New York, 1944.
Courant, R., E. Isaacson, and M. Rees
[C.ll] On the solution of nonlinear hyperbolic differential equations by finite
differences. Communications on Pure and Applied Mathematics, Vol. 5,
1952, pp. 243-255.
Crossley, H. E., Jr.
[C.I 2] The analogy between surface shock waves in a liquid and shocks in com-
pressible gases. Experimental study of hydraulic jump interactions.
California Institute of Technology Hydrodynamics Laboratory Report
No. N-54, 1, 1949.
Cray a, A.
C.13] Calcul graphiques des regimes variables dans les canaux. La Houille
Blanche, N.S. Annee 1, 1946.
C.14] The criterion for the possibility of roll wave formation. U.S. National
Bureau of Standards, Gravity Waves, NBS Circular 521, 1952, pp.
141-151.
Charney, J. G.
I C.I 5] Dynamic forecasting by numerical process. Compendium of Meteorolo-
gy, American Meteorological Society Boston, 1951, pp. 470-482.
Carson, Rachel L.
[C.16] The Sea Around Us. Oxford University Press, New York, 1951.
Daily, J. W., and S. C. Stephan Jr.
[D.I] The solitary wave. Massachusetts Institute of Technology, Hydro-
dynamics Laboratory, Technical Report No. 8, 1952.
Dane!, Pierre
'D.2J On the limiting clapotis. U.S. National Bureau of Standards, Gravity
Waves, NBS Circular 521, 1952, pp. 35-38.
[D.3] Hydraulic Research Work. Edited by La Houille Blanche.
Darbyshire, J.
[D.4] Identification of microseismic activity with sea waves. Proceedings of the
Royal Society of London, Vol. 202, Ser. A., 1950, pp. 439-448.
Davies, T. V.
[D.5] Symmetrical, finite amplitude gravity waves. U.S. National Bureau of
Standards, Gravity \Vaves, NBS Circular 521, 1952, pp. 55-60.
Deacon, G. E. R.
[D.6] Recent studies of waves and swell. Annals of the New York Academy of
Sciences, Vol. 51, Art. 3, 1949, pp. 475-482.
[D.7] Analysis of sea waves. U.S. National Bureau of Standards, Gravity
Waves, NBS Circular 521, 1952, pp. 209-214.
Dean, W. R.
[D.8] Note on waves on the surface of running water. Proceedings of the Cam-
bridge Philosophical Society, Vol. 43, 1947, pp. 96-99.
Deymie, P.
[D.9] Propagation d'une intumesence allonge'e (probleme aval). International
Congress for Applied Mechanics, 5th Proceedings, September 1938,
John Wiley and Sons, New York, 1939.
Dirichlet, P. L.
[D.10] Untersuchungen uber ein Problem der Hydrodynamik. Journal fur die
reine und angewandte Mathematik, Vol. 58, 1861, pp. 181-228.
Dixon, G. F.
[D.ll] Theses prteenttes a la Faculty des Sciences de VUniversitt de Grenoble,
1949.
548 WATER WAVES
Dressier, R. F.
[D.I 2] Mathematical solution of the problem of roll-waves in inclined open
channels. Communications on Pure and Applied Mathematics, Vol. 2,
1949, pp. 149-194.
[D.I 3] Stability of uniform flow and roll-wave formation. U.S. National Bureau
of Standards, Gravity Waves, NBS Circular 521, 1952, pp. 237-241.
Dressier, R. F., and F. V. Pohle
[D.14] Resistance effects on hydraulic instability. Communications on Pure and
Applied Mathematics, Vol. 6, 1953, pp. 93-96.
Dubreuil-Jacotin, M. L.
[D.15] Sur les ondes de type permanent dans les liquides he'te'rogenes. Rendiconti
della Accadcmia Nazionale dei Lincei, Ser. 6, Vol. 15, 1932, pp. 814-819.
|p.l5aj£wr la determination rigoureuse des ondes permanentes pdriodiques
d'ampleur finie. Journal de Math£matiques Pures et Appliqu^es, Ser. 9,
Vol. 13, 1934, pp. 217-291.
Dean, \V. R.
[D.I 6] On some cases of the reflection of surface ivaves by an infinite plane
barrier. Proceedings of the Cambridge Philosophical Society, Vol. 42,
1945, pp. 24-28.
[D.17] On the reflection of surface waves by a submerged plane barrier. Proceed-
ings of the Cambridge Philosophical Society, Vol. 41, 1945, pp. 231-238.
Eckart, C.
[E.I] The ray-particle analogy. Journal of Marine Research, Vol. 9, 1950,
pp. 139-144.
[E.2] Surface waves on water of variable depth. Scripps Institution of Oceano-
graphy, Lecture Notes, Fall Semester 1950-51. Ref. 51-12. Wave
Report No. 100, 1951.
[E.3] The propagation of gravity waves from deep to shallow water. U.S.
National Bureau of Standards, Gravity Waves, NBS Circular 521, 1952,
pp. 165-173.
Egiazarov, I.
[E.4] On the Daily Regulation of Hdryoelectric Stations (Experimental In-
vestigations of the Negative Wave). Leningrad, 1931. (In Russian, with
an English summary. Cited in Favre (F.2J.)
Einstein, H. A., and E. G. Baird
[E.5] Progress report of the analogy between surface shock waves on liquids
and shocks in compressible gases. California Institute of Technology,
Hydrodynamics Laboratory, Report No. N-54, 1946.
Emde, F.
[E.6J See [J.I].
Favre, H.
[F.I] Contribution a Vttude des courants liquides. Publication du Laboratoire
de Recherches hydrauliques annex£ a 1'Ecole Polytechnique F6d6rale
de Zurich, 1933.
[F.2] Etude thtorique et exptrimentale des ondes de translation dans les canaux
dtcouverts. Dunod, Paris, 1935.
Finkelstein, A.
[F.3] The initial value problem for transient water waves. Dissertation, New
York University, 1953.
Fjeldstad, J. E.
[F.4] Observations of the internal tidal waves. U.S. National Bureau of Stand-
ards, Gravity Waves, NBS Circular 521, 1952.
BIBLIOGRAPHY 549
Fleishman, B., J. J. Stoker, and L. Weliczker,
[F.5] Floating breakwaters in shallow water. New York University, Institute
of Mathematical Sciences, Report No. IMM-192, 1953.
Forchheimer, P.
[F.6] Hydraulik. Encyklopadie der mathematischen Wissenschaften, Vol. IV,
Teil 8, pp. 324-472, B. G. Teubner, Leipzig, 1901-1908.
Forel, F. A.
[F.7] Le Leman, monographic limnologique. Bibliotheque Nationale, Vol. 2,
F. Rouge, Lausanne, 1895.
Frank, P., and R. v. Mises,
[F.8] Die Differential- und Integralgleichungen der Mechanik und Physik.
F. Vieweg und Sohn, Braunschweig, 1925-27.
Freeman, J. C., Jr.
[F.9] An analogy between the equatorial easterlies and supersonic gas flows.
Journal of Meteorology, Vol. 5, 1948, pp. 138 — 146.
[F.10] The solution of nonlinear meteorological problems by the method of charac-
teristics. Compendium of Meteorology, American Meteorological Society,
1951, pp. 421-4.33.
Friedrichs, K. O.
[F.ll] On the derivation of the shallow water theory. Appendix to The formation
of breakers and bores by J. J. Stoker, Communications on Pure and
Applied Mathematics, Vol. 1, 1948, pp. 81-85.
Friedrichs, K. O., and H. Lewy
[F.I 2] The dock problem. Communications on Pure and Applied Mathematics,
Vol. 1, 1948, pp. 135-148.
Friedrichs, K. O., and 1). II. Ilycrs
[F.I 3] The existence of solitary waves. Communications on Pure and Applied
Mathematics, Vol. 7, 1954, pp. 517-550.
Friedrichs, K. O.
[F.14] Water waves on a shallow sloping beach. Communications on Pure and
Applied Mathematics, Vol. 1, 1948, pp. 109-134.
[F.15] See [C.9].
Fleming, R. H.
[F.16] Sec [S.32a].
Goldstein, E., and J. B. Keller
[G.I] Reflection of waves in shall wv water. (Unpublished.) Sec E. Goldstein,
Master's Thesis, New York University, 1950.
Guthrie, F.
[G.2] On stationary liquid waves. London, Edinburgh and Dublin, Philosophi-
cal Magazine, Ser. 4, Vol. 50, 1875.
Gwyther, R. F.
[G.8J The classes of progressive long waves. London, Edinburgh and Dublin,
Philosophical Magazine, Ser." 5, Vol. 1, 1900. pp. 213, 308, 349.
Goldstein, E.
[G.4] See [K.8].
Gerber, R.
[G.5] Sur les solutions exactes des Equations du mouvement avec surface libre
d'un liquide pesant. (Thesis, Univ. of Grenoble, 1955.) To appear in
Journal de Mathematique.
Goldner, S. R.
[G.6] Existence and uniqueness theorems for two systems of nonlinear hyperbolic
differential equations for functions of two independent variables. Doctor's
thesis, New York University, 1949.
550 WATER WAVES
Hadamard, J.
[H.I] Sur les ondefi liquides. Institut de France. Academic des Sciences,
Comptes Rendus, Vol. 150, 1910, pp. 609-611, 772-774.
Hamada, T.
[H.2] Breakers and beach erosion. Report of Transportation Technical
Research Institute (Report No. 1) Tokyo, 1951.
Harleman, D. R. F.
[H.8] Studies on the validity of the hydraulic analogy to supersonic flow. Parts I
and II. Massachusetts Institute of Technology, 1950. (Technical report
to U.S. Air Force).
Haskind, M. D.
[H.4] Waves arising from oscillation of bodies in shallow water. Prikladnaya
Matematika i Mechanika, Vol. 10, 1940, pp. 475-480.
[H.5] The hydrodynamic theory of the oscillation of a ship in waves (in Russian).
Prikladnaya Matematika i Mechanika, Vol. 10, 1946, pp. 39-66.
Oscillation of a ship on a calm sea (in Russian), Izv. Ak. Nauk, Ot. Tex.
Nauk, No. 1, 1946, pp. 23-34. Translated by V. Wehausen as "Two
papers on the hydrodynamic theory of heaving and pitching of u ship",
Technical Research Bulletin 1-12, Society of Naval Architects and
Marine Engineers, New York, 1953.
Haurwitz, B.
[H.6] Dynamic meteorology. McGraw-Hill, New York, 1941.
Havelock, T. H.
[H.7] The wave-making resistance of ships. Proceedings of the Royal Society
of London, Ser. A, Vol. 82, 1909, pp. 276-300.
[H.8] Wave resistance theory and its application to ship problems. Society of
Naval Architects and Marine Engineers, New York, 1950.
[H.9] The propagation of groups of waves in dispersive media. The University
Press, Cambridge (Eng.), (Cambridge Tracts, No. 17), 1914.
Hayashi, T.
[H.10] Mathematical theory of flood waves. Japan National Congress for Applied
Mechanics, First. Proceedings. 1951.
Heins, A. E.
[H.ll] Water waves over a channel of finite depth with a submerged plane barrier.
Canadian Journal of Mathematics, Vol. 2, 1950, pp. 210-222.
[H.12] Water waves over a channel of finite depth with a dock. American Journal
of Mathematics, Vol. 70, 1948, pp. 730-748.
Henry, Marc
[H.13] Note sur la propagation des intumescences dans un canal rectangulaire.
Communique a la seance du 5 nov. 1937 du comit£ technique de la
soci^te" hydrotechnique de France.
Hinze, J. O.
[H.14] Die Erzeugung von Ringwellen auf einer FlUssigkeitsoberfldche durch
periodisch wirkende Druckkrdfte. Zeitschrift fur angewandte Mathematik
und Mechanik, Bd. 16, 1986.
Hogner, E.
[H.15] A contribution to the theory of ship waves. Arkiv for Matematik, Astro-
nomi, och Fysik, Vol. 17, 1922.
Hopf, E.
[H.16] On S. Bernstein's theorem on surfaces z(x, y) of nonpositive curvature.
Proceedings of the American Mathematical Society, Vol. 1, 1950,
pp. 80-85.
Hyers, D. H.
See [F.18]
BIBLIOGRAPHY
551
Hanson, E. T.
[H.17] The theory of ship waves. Proceedings of the Royal Society of London,
Ser. A, Vol. Ill, 1926, pp. 491-529.
Isaacson, E.
[I.I ] Water waves over a sloping bottom. Communications on Pure and Applied
Mathematics, Vol. 3, 1950, pp. 1-32.
[1.2] Waves against an overhanging cliff. Communications on Pure and
Applied Mathematics, Vol. 1, 1948, pp. 201-209.
[I.3J See [C.ll].
Isaacson, E., J. J. Stoker, and B. A. Troesch
[1.4] Numerical solution of flood prediction and river regulation problems.
Report 2. Numerical solution of flood problems in simplified models of the
Ohio River and the junction of the Ohio and Mississippi Rivers. New
York University, Institute of Mathematical Sciences, Report No.
IMM-205, 1954.
[I.4al Numerical solution of flood prediction and river regulation problems.
Report 3. Results of the numerical prediction of the 1945 and 1948 floods
in the Ohio River, of the 1947 flood through the junction of the Ohio and
Mississippi Rivers, and of the Jtoods of 1950 and 1948 through Kentucky
Reservoir. New York University, Institute of Mathematical Sciences,
Report No. IMM-NYU-235, 1956.
Isaacs, J. D.
Ll.31 See M.7J.
Iverson, H. W.
ij.6] Laboratory study of breakers. U.S. National Bureau of Standards,
Gravity Waves', NBS Circular 521, 1952.
Jahnkc, K., and F. Kmde
[J.1J Tables of Functions. Dover, New York, 1945, p. 138.
.Jeffreys, Harold
[J.2] The flow of water in an inclined channel of rectangular section. London,
Edinburgh and Dublin, Philosophical Magazine, Ser. 6, Vol. 49, 1925,
pp. 793-807.
Jeffreys, H. and Bertha S. Jeffreys
!J.3j Methods of Mathematical Physics. Cambridge (Eng.), The University
Press, 1948.
John, F.
["J.4J \Vavett in the presence of an inclined barrier. Communications on Pure
and Applied Mathematics, Vol. 1, 1948, pp. 149-200.
[J.5] On the motion of floating bodies I, II. Communications on Pure and
Applied Mathematics, Vol. 2, 1949, pp. 13-57; Vol. 3, 1950, pp. 45-100.
Johnson, J. VV. and M. P. O'Brien
[J.6] The engineering aspects of waves in harbor design. International Asso-
ciation of Hyd. Str. Research, Grenoble, 1949.
Johnson, J. VV., M. P. O'Brien, and J. D. Isaacs
[J.7] Graphical construction of wave refraction diagrams. Hydrographic Office,
Navy Department, Publication no. 605, 1948.
Johnson, J. W.
[J.8] See (B.I 3].
Johnson. M. YV.
[J.9] See [S.32a].
552 WATER WAVES
Kampe de Feriet, J., and J. Kotik
[K.I] Surface waves of finite energy. Journal of Rational Mechanics and
Analysis, Vol. 2, 1953, pp. 577-685.
von Karman, T.
[K.2] Eine praktische Anwendung der Analogic zwischen Ubcrschallstromung
in Gasen und uberkritischer Stromung in offenen Gerinnen. Zentralblatt
fiir angewandte Mathematik und Mechanik, Vol. 18, 1939.
Karp, S. N.
[K.3] Wiener-Hopf techniques and mixed boundary value problems. Com-
munications on Pure and Applied Mathematics, Vol. 3, 1950, pp.
411-426.
Keller, J. B., and P. Lax
[K.4] Finite difference schemes for hyperbolic systems. LAMS 1201, 1950.
Keller, J. B.
[K.5] The solitary wave and periodic waves in shallow water. Annals of the New
York Academy of Sciences, Vol. 51, Art. 3. 1949, pp. 345-350.
[K.6] The solitary wave and periodic waves in shallow water. Communications
on Pure and Applied Mathematics, Vol. 1, 1948, pp. 323-339.
[K.7] Scattering of water waves treated by the variational vncthod. U.S. National
Bureau of Standards, Gravity Waves, NBS Circular 521, 1952.
Also see [W.8]
Keller, J. B., and E. Goldstein
[K.8] Water wave reflection due to surface tension and floating ice. Transact ioiks
of the American Geophysical Union, Vol. 34, 1953, pp. 43-48.
Keller, J. B. and M. Weitz
[K.9] Reflection and transmission coefficients for waves entering or leaving an
icefield. Communications on Pure and Applied Mathematics, Vol. 6,
1953, pp. 415-417.
Kellog, O. D.
[K.10] Foundations of potential theory. J. Springer, Berlin, 1929.
Kelvin, Lord (Sir W. Thomson)
[K.ll] On the waves produced by a single impulse in water of any depth, or in a
dispersive medium. Proceedings of the Royal Society of London, Ser. A.
Vol. 42, 1887, pp. 80-85.
Keulegan, G. H.
[K.12] The characteristics of internal solitary ivaves. (Abstract only.) U.S.
National Bureau of Standards, Gravity Waves, NBS Circular 521 , 1952.
Keulegan, G. H., and G. W. Patterson
[K.13] Mathematical theory of irrotational translation waves. U.S. National
Bureau of Standards Journal of Research, Vol. 24, 1940.
Kotschin, N. J., I. A. Kibel, and N. W. Rose
[K.14] Theoretische Hydrodynamik. Akad. Verlag, Berlin, 1954.
Korteweg, D. J., and G. de Vries
[K.I 5] On the change of form of long waves advancing in a rectangular canal and
on a new type of long stationary waves. London, Dublin and Edinburgh,
Philosophical Magazine, Ser. 5, Vol. 39, 1895, p. 422.
Korvin-Krukovsky, B. V., and E. V. Lewis
[K.I 6] Ship motions in regular and irregular seas. Stevens Institute of Technol-
ogy, Experimental Towing Tank, Technical Memorandum no. 106, 1954.
Kotik, J.
[K.I 7] Existence and uniqueness theorems for linear water waves. Massachusetts
Institute of Technology, 1952.
[K.18] See [K.I], [B.12].
Kreisel, G.
[K.19] Surface waves. Quarterly of Applied Mathematics, Vol. 7, 1 949, pp. 21-44.
BIBLIOGRAPHY 553
Krylov, A. N.
[K.20] A general theory of the oscillations of a ship on waves. Transactions of the
Institute of Naval Architects and Engineers, Vol. 37, 1896.
Keldysh, M. V.
[K.21] Remarks on certain motions of a heavy fluid. (Russian) Tckhnich.
Zametki Tsagi, no. 52, 1935.
Laitone, E. V.
[L.I] A study of transonic gas dynamics by the hydraulic analogy. University of
California, Berkeley, California, Institute of Engineering Research, 1951.
Lamb, H.
[L.2] On deep water waves. Proceedings of the London Mathematical Society,
Ser. 2, Vol. 2, 1904, p. 388.
[L.3] Hydrodynamics. Dover Publications, New York, 1945; Cambridge
University Press, 1932.
Lavrcnticff, M.
[L.4] On the theory of long waves. Akad. Nauk Ukrain. R.S.R., Zb. Proc. Just.
Math., Vol. 1, 1948. In Ukrainian.
Lax, Anneli
[L.5] Decaying shocks. Communications on Pure and Applied Mathematics,
Vol. 1, 1948, pp. 247-257.
Lax, Peter
[L.6] See [K.4J.
Levi-Civita, T.
[L.7j Determination rigoureuse des ondes permanentes d'ampleur finie. Mathe-
matische Annalen, Vol. 93, 1925, pp. 264-314.
Lcwy, H.
^L.8] Water waves on sloping beaches. Bulletin of the American Mathematical
Society, Vol. 52, 1946.
[L.9J A note on harmonic functions and a hydrodynamical application. Proceed-
ings of the American Mathematical Society, Vol. 3, No. 1, 1952.
[L.9al On steady free surface flwv in a gravity field. Communications on Pure
and Applied Mathematics, Vol. 5, 1952.
[L.10] See [F.12],
Lichteustein, L.
[L.ll] Vorlesungen uber einige Klassen nichtlinearer lutegralgleichungen und
Integro-Differentialgleichungen nebst Anwendungen. J. Springer, Berlin,
1931.
[L.12] Grundlagen der Hydromechanik. J. Springer, Berlin, 1929.
Longuet-Higgins, M. S.
[L.I 8] A theory of the generation of microseisms. Philosophical Transactions of
the Royal Society of London, Ser. A, Vol. 243, 1950, pp. 1-35.
Longuet-Higgins, M. S., and F. Ursell
[L.14J Seawaves and microseisms. Nature, Vol. 162, 1948.
Longuet-Higgins, M. S.
[L.15] See [C.3].
Lowell, S. C.
[L.16] The propagation of waves in shallow water. Communications on Pure and
Applied Mathematics, Vol. 2, 1949, pp. 275-291.
[L.17J The Propagation of Waves in Shallow Water. Ph. D. Thesis, New York
University, June 1949.
Lunde, J. K., and W. C. S. Wigley
[L.18] Calculated and observed wave resistances for a series of forms of fuller
midsection. Quarterly Transactions of the Institute of Naval Architects,
London, Vol. 90, 1948.
554 WATER WAVES
Lunde, J. K.
[L.19] On the linearized: theory of wave resistance for displacement ships in
steady and accelerated motion. Society of Naval Architects and Marine
Engineers, 1951.
Lewis, E. V.
[L.20] See [K.16].
MacDonald, H. M.
[M.I] A class of diffraction problems. Proceeding of the London Mathematical
Society, Ser. 2, Vol. 14, 1915, pp. 410-427.
MacRobert, T. M.
[M.2] Functions of a Complex Variable. MacMillan, London, 1947.
Martin, J. C., W. J. Moyce, W. G. Penney, A. T. Price, and C. K. Thornhill
[M.3] Some gravity wave problems in the motion of perfect liquids. Philosophical
Transactions of the Royal Society of London, Ser. A, Vol. 244, 1952.
Mason, Martin A.
[M.4] Some observations of breaking waves. U.S. National Bureau of Standards,
Gravity Waves, N. B. S. Circular 521, 1952.
Massau, J.
[M.5] Me'moire sur V integration graphique des equations aux dcrive'es partielles.
Annales des ing6nieurs sortis de Gand, Tome XII, 1889.
McGowan, J.
[M.6] On the solitary wave. London, Dublin and Edinburgh, Philosophical
Magazine, Ser. 5, Vol. 32, p. 45, 1891. On the highest wave of permanent
type. Ser. 5, Vol. 38, p. 351, 1894.
McNown, J. S.
[M.7] Waves and seiches in idealized ports. U.S. National Bureau of Standards,
Gravity Waves, N. B. S. Circular 521, 1952.
Miche, A.
[M.8] Mouvements ondulatoires de la mer en profondeur constante ou dfaroissante.
Annales des ponts et chaussees, 1944, pp. 25-78, 131-164, 270-292,
369-406.
Michell, J. H.
[M.9] The wave resistance of a ship. London, Dublin and Edinburgh, Philoso-
phical Magazine, Ser. 5, Vol. 45, 1898.
Milne-Thompson, L. M.
[M.10] Theoretical Hydrodynamics. MacMillan, London, 1938.
Moyce, W. J.
[M.ll] See [M.3J.
Munk, W. H.
[M.12] The solitary wave theory and its application to surf problems. Annals of
the New York Academy of Sciences, Vol. 51, Art. 3, 1949, pp. 376-424.
[M.I 3] Supplement to breakers and surf, principles in forecasting. Scripps
Institution Oceanography. Wave Report No. 49.
Munk, W. H., and R. S. Arthur
[M.14] Wave intensity along a refracted ray. U. S. National Bureau of Standards,
Gravity Waves, N. B. S. Circular 521, 1952.
Munk, W. H., and M. A. Traylor
[M.15] Height of breakers and depth of breaking. Scripps Institution of Oceano-
graphy, Wave Report No. 47, 1945.
[M.I 6] Refraction of ocean waves: a process linking underwater topography to
beach erosion. Journal of Geology, Vol. 15, 1947.
Munk, W. H.
[M.17] See [S.33], [S.84], [S.35], [S.36], [S.37].
BIBLIOGRAPHY 555
Nekrassov, A. I.
[N.I ] On steady waves. Izv. Ivanovo- Voznesensk. Politekhn. In-ta, No. 3, 1 921 .
[N.la] The exact theory of steady waves on the surface of a heavy fluid. Izdat.
Akad. Nauk, SSSR, Moscow, 1951.
Nirenberg, L.
[N.2] Waves of finite amplitude created by an obstacle in a stream. Unpublished
manuscript (New York University).
Nakano, N., S. Unoki, and Y. Kuga
[N.3] On the result of wave observations at Jogashima Island and its application
to wave forecasting. Meteorological Research Institute, Tokyo, 1953.
O'Brien, M. P.
[O.I] See [J.Q].
[O.2] See [J.7].
Patterson, G. W.
[P.I] See [K.131.
Penney, W. G.
[P.2] See (M.3]
Penney, W. G. and A. T. Price
[P.3] Diffraction of sea waves by breakwaters. Directorate of Miscellaneous
Weapons Development History No. 26-Artif icial Harbors, Sec. 3D, 1 944.
Peters, A. S.
fP.4] A new treatment of the ship wave problem. Communications on Pure and
Applied Mathematics, Vol. 2, 1949, pp. 123-148.
[P. 5] The effect of a floating mat on water waves. Communications on Pure
and Applied Mathematics, Vol. 3, 1950, pp. 319-354.
[P.6] Water waves over sloping beaches and the solution of a mixed boundary
value problem for A(p — k*<p — 0 in a sector. Communications on Pure
and Applied Mathematics, Vol. 5, 1952, pp. 87-108.
Peters, A. S., and J. J. Stoker
[P.7] The motion of a ship, as a floating rigid body, in a seaway. Institute of
Mathematical Sciences, Report No. IMM-203, New York University,
1954.
Pierson* W. J., Jr.
[P.8] The interpretation of crossed orthogonals in wave refraction phenomena.
Technical Memorandum No. 21, Beach Erosion Board, Corps of
Engineers.
Picrson, W. J., Jr., and M. St. Denis
[P.9] On the motion of ships in confused seas. Society of Naval Architects and
Marine Engineers, 1953.
Pierson, W. J., Jr.
[P. 1 0 ] A unified mathematical theory for the analysis, propagation , and refraction
of storm generated ocean surface waves. Part I, Part II, Reports of
College of Engineering, New York University, 1952.
Pohle, F. V.
[P.ll] The Lagrangian equations of hydrodynamics: solutions which are analytic
functions of the time. Thesis, New York University, 1950.
[P.12J Motions of water due to breaking of a dam, and related problems. U. S.
National Bureau of Standards, Gravity Waves, N. B. S. Circular 521,
1952.
[P.13] See [D.14J.
Price, A. T.
[P.14] See [P.8].
[P.15] See [M.3].
556 WATER WAVES
Preiswerk, E.
[P.16] Anwendung gasdynamischer Methoden auf Wasserstromungen mil freier
Oberflache. Mitteilungen aus dem Institut fur Aerodynamik, Eidg. Tech-
nische Hochschule, Ztirich, 1038.
Putman, H. J.
[P.17J Unsteady flow in open channels. Transactions of the American Geophy-
sical Union, Vol. 29, 1948.
Putnam, J. A., and R. S. Arthur
[P. 18] Diffraction of water waves by breakwaters. Transactions of the American
Geophysical Union, 29 (4), 1948.
Peters, A. S., and J. J. Stoker
[P. 19] A uniqueness theorem and a new solution for Sommerf eld's and other
diffraction problems. Communications on Pure and Applied Mathema-
tics. Vol. 7, 1954, pp. 565-585.
Rayleigh, Lord
[R.I] On waves. London, Dublin and Edinburgh, Philosophical Magazine,
Ser. 5, Vol. 1, 1876.
[R.2 j The form of standing waves on the surface of running water. Proceedings
of the London Mathematical Society, Vol. 15, 1883-4, pp. 69-78.
[R.3] On the theory of long waves and bores. Proceedings of the Royal Society
of London, Ser. A, Vol. 90, 1914.
IR.4] The Theory Of Sound. Dover Publications, New York, 1945.
Re R.
[R.5] titude du lacher instantane d'une retenue d'eau dans un canal par la
m&hode graphique. La Houille Blanche N. S. Vol. 1, 1946.
Rees, M.
[R.6] See [C.ll].
Rellich F.
[R.7] Vber das asymptotische Verhalten der Losungen von Au -f- u = 0 in
unendlichen Gebieten. Jahresbericht der deutschen Mathematiker Ver-
einigung, Vol. 53, 1943, pp. 57-65.
Riabouchinsky, D.
[R.8] Sur V analogic hydraulique des mouvemente d'un fluide compressible.
Institut de France, Academic des Sciences, Comptes Rendus, Vol. 195,
1932, p. 998.
Rose, N. W.
See [K.14]
Roseau, M.
[R.9] Contribution a la the'orie des ondes liquides de gramle en profondeur
variable. Publications Scicntifiques et Techniques du Ministere de
1'Air, No. 275, Paris, 1952.
Rouse, H.
[R.10] Fluid mechanics for hydraulic engineers. McGraw-Hill, New York, 1938.
Rouse, H., editor
[R. 1 1 ] Engineering Hydraulics. John Wiley and Sons, New York, and Chapman
and Hall, Ltd., London, 1950. (Especially Gilcrest, B. R., Flood rout-
ing, pp. 635-676.)
Ruellan, F., and A. Wallet
[R.12J Trajectoires internes dans un clapotis partiel. La Houille Blanche, Vol. 5,
1950.
Rubin, H.
[R.13] The dock of finite extent. Thesis, New York University, 1953.
Russell, S.
[R.14] Report on waves. British Association Reports, 1844.
BIBLIOGRAPHY 557
St. Denis, M., and G. Weinblum
[S.I] On the motions of ships at sea. Society of Naval Architects and Marine
Engineers, 1950.
[S.2] See fP-91.
Saint Venant, B. de
[S.3] Thtorie du mouvement non permanent des eaux. Institut de France,
Academic des Sciences, Comptes Rendus, Paris, July 1871, Vol. 73,
pp. 147, 237.
Schonfeld, .J. C.
fS.4] Propagation of Tides and Similar Waves. Doctor's Thesis, Delft,
Holland, 1951.
FS.4a] Distortion of long waves; equilibrium and stability. Union Geod£sique et
Geophysique Internationale, Vol. 4, 1951.
Sch winger, J. S.
[S.5] Fourier transform solution of integral equations. M.I.T. Radiation
Laboratory Report.
Scripps Institution of Oceanography
[S.6 j Waves in shallow water (I). Change in wave height along Scripps Institu-
tion pier with special emphasis on breaker cliaracteristics. Wave Report
No. 13.
[S.7' Effect of bottom slope on breaker characteristics as observed along the
Scripps Institution pier. Wave Report No. 24, 1944.
Seiwell, H. R.
[S.8J Military oceanography in World War II. Mil. Eng. 39 (259), 1947.
I^S.9] The principles of time series analyses applied to ocean wave data. Proceed-
ings of the National Academy of Sciences, Vol. 35, 1949, pp. 518-528.
S.10] Sea surface roughness measurements in theory and practise. Annals of the
New York Academy of Sciences, Vol. 51, Art. 3, 1949, pp. 483-501.
Sneddon, I. N.
[S.11J Fourier Transforms. McGraw-Hill, New York, 1951.
Solberg, II.
See [B.20]
Sommerfeld, A.
[S.I 2] Mathematische Theorie der Diffraktion. Mathematische Annalen, Ud.
47, p. 317, 1896.
[S.13] Vorlcsungen iiber theoretische Physik, II, Mechanik der deformierbaren
Medien. Akademie Verlag, Leipzig, 1949.
Southwell, R.
[S.14] Relaxation Methods. Oxford Press, 1940.
Sretenski, L. N.
[S.15J Ship waves for circular courses (in Russian). Bulletin dc TAcaddmie des
Sciences de TURSS, 1946.
Stelzriede, M. E.
[S.16] See [C.I].
Stephan, S. C., Jr.
[S.17] See [D.I].
Stoker, J. J.
[S.I 8] Surface waves in water of variable depth. Quarterly of Applied Mathe-
matics, Vol. 5, 1947, pp. 1-54.
,S.19] The formation of breakers and bores. Communications on Pure and
Applied Mathematics, Vol. 1, 1948, pp. 1-87.
[S.20] The breaking of waves in shallow water. Annals of the New York Academy
of Sciences, Vol. 51, Art. 3, 1949, pp. 360-375.
S.21] On radiation conditions. Sixth Symposium on Applied Mathematics.
American Mathematical Society, 1953.
558 WATER WAVES
[S.22] Unsteady waves on a running stream. Communications on Pure and
Applied Mathematics, Vol. 6, 1953, pp. 471-481.
[S.23] Numerical solution of flood prediction and river regulation problems I :
Derivation of basic theory and formulation of numerical methods of attack.
New York University, Institute of Mathematical Sciences, Report
No. IMM-200, 1953.
[S.24] Dynamical theory for treating the motion of cold and warm fronts in the
atmosphere. Institute of Mathematical Sciences, Report No. IMM-195,
1953, New York University.
[S.25] See [F.5].
[S.26] See [1.4].
[S.27] See [P.7].
[S.27a] See [I.4a].
Stokes, G. G.
[S.28] On the theory of oscillatory waves. Transactions of the Cambridge Philo-
sophical Society, Vol. 8, 1847, and Supplement. Scientific Papers,
Vol. 1.
Struik, D. J.
[S.29] Determination rigoureuse des ondes irrotationnelles periodiques dam un
canal a profondeur finie. Mathematische Annalcn, Vol. 95, 1926, pp.
595-634.
Suquet, F.
[S.30] Remarks on graphical computation of wave refraction. International
Association for Hydraulic Research, Grenoble, 1949.
[S.31] Etude experimental du deferlement dc la houle. La Houille Blanche,
Vol. 5, 1950.
Sverdrup, H. U.
[S.32] Oceanography for Meteorologists. Prentice- Hall, Inc., X. Y., 1948.
Sverdrup, H. U., M. W. Johnson, and R. H. Fleming
|S.82n] The Oceans. Prentice-Hall, New York, 1946.
Sverdrup, H. U., and W. H. Munk
[S.33] Theoretical and empirical relations in forecasting breakers and surf.
Transactions of the American Geophysical Union, Vol. 27, 1946.
[S.34] A study of progressive oscillatory waves in water; a summary of the theory
of water waves. Technical reports No. 1, 2, Beach Erosion Board,
Office of Chief of Engineering, U.S. War Department.
[S.35] Breakers and surf. U.S. Navy Department, Hydrographic Office,
Publication No. 234, 1944.
[S.36] Wind, sea9 and swell: Theory of relations for forecasting. U.S. Xavy
Department, Hydrographic Office, 1947.
[S.37] Empirical and theoretical relations between zvind, sea and swell. Trans-
actions of the American Geophysical Union, Vol. 27, 1946.
Sretenski, L. N.
[S.38] Waves. In: Mekhanika v SSSR za tridtsat'let 1917-1947. Moscow, 1950,
pp. 279-299.
Thomas, H. A.
[T.I] The propagation of stable wave configurations in steep channels. Carnegie
Institute of Technology, Pittsburgh, Pa.
[T.2] Hydraulics of flood movements in rivers. Carnegie Institute of Tech-
nology, Pittsburgh, Pa., 1937.
[T.8] Propagation of waves in steep prismatic conduits. Proceedings of Hydrau-
lics Conference, University of Iowa, Studies in engineering, Bulletin 20,
1940.
BIBLIOGRAPHY 559
Thorade, H. F.
[T.4] Probleme der Wasserwellen. 1931, from Probleme der kosmischen Physik,
Bd. 13-14.
Thornhill, C. K.
[T.5] See [M.3].
Titchmarsh, E. C.
[T.6] Introduction to the Theory of Fourier Integrals. Second Edition, The
Clarendon Press, Oxford, 1948.
Tray lor, M. A.
[T.7] See [M.15].
[T.8] See [M.16].
Troesch, B. A.
[T.9] See [1.4].
[T.9a] See [I.4a].
Thompson, P. D.
[T.10] Notes on the theory of large-scale disturbances in atmospheric flow with
applications to numerical weather prediction. Airforce Cambridge Re-
search Center, Geophysical Research Papers, No. 16, 1952.
Tepper, M.
[T.ll] A proposed mechanism of squall lines: the pressure jump line. Journal of
Meteorology, Vol. 7, 1950, pp. 21-29.
U.S. Navy Hydrographic Office
[U.1J Wind, ivaves and swell: Principles in forecasting. H. O. Misc. 11, 1944.
[U.2] Breakers and surf, principles in forecasting. H. O. 234 1944.
Ursell, F.
[U.3] The effect of a vertical barrier on surface waves in deep water. Proceedings
of the Cambridge Philosophical Society, Vol. 43, 1947. pp. 374-382.
[U.4] On the waves due to the rolling of a ship. Quarterly Journal of Mechanics
and Applied Mathematics, Vol. 1, 1948, pp. 246-252.
[U.5J Surface waves on deep water in the presence of a submerged circular
cylinder, I, II. Cambridge Philosophical Society, Vol. 46, 1950, pp.
141-152, 153-163.
[1T.6] On the application of harmonic analysis to ocean wave research. Science,
Vol. Ill, 1950.
[LI. 7] Discrete and continuous spectra in the theory of gravity waves. U.S.
National Bureau of Standards, Gravity waves, N.B.S. Circular 521,
1952, pp. 1-5.
[U.8J Trapping modes in the theory of surf ace leaves. Proceedings of the Cam-
bridge Philosophical Society, Vol. 47, 1951.
[U.9] On the rolling motion of cylinders on the surface of a fluid. Quarterly
Journal of Mechanics and Applied Mathematics, Vol. 2, 1949.
[U.10J On the heaving motion of a circular cylinder on the surface of a fluid.
Quarterly Journal of Mechanics and Applied Mathematics, Vol. 2, 1949.
[U.11J The long-wave paradox in the theory of gravity leaves. Proceedings of the
Cambridge Philosophical Society, Vol. 49, 1953.
[U.12J See [B.4J.
[U.13J See [L.14].
Unaki, G., and M. Nakano
[U.14] On the Cauchy-Poisson waves caused by the eruption of a submarine
volcano, (I) and (II). Oceanograph. Mag. 4,5; 1953.
Vedernikov, V. V.
[V. 1 ] Conditions at the front of a translation wave disturbing a steady motion of a
real fluid. Doklady (Comptes rendus) Akademiya Nauk SSSR, Vol. 48,
1945.
560 WATER WAVES
[V.2] Characteristic features of a liquid flow in an open channel. Doklady
(Comptes rendus), Vol. 52, 1946, pp. 207-210.
Vergne, Henri
[V.3] Ondes liquides de gravite*. Actualites Scientifiques et Industrielles,
Hermann et Cie., Paris, 1928.
de Vries, G.
[V.4] See [K.15].
Wallet, A.
See [R.12]
Watson, G. N.
[W.I] A Treatise on the Theory of Bessel Functions. The University Press, Cam-
bridge, 1944.
Wilkes, M. V.
[W.2] Oscillations of the Earth's Atmosphere. The University Press, Cambridge,
1949.
Weinblum, G. P.
[W.3] Analysis of wave resistance. David W. Taylor Model Basin (Washington,
D. C.), Report 710, 1950.
[W.4] See [S.I].
Weinstein, A.
[W.5] On surface waves. Canadian Journal of Mathematics, Vol. 1, 1949,
pp. 271-278.
[W.6] Sur la vitesse de propagation de Vonde solitaire. Reale Acadcmia dci
Lincei, Rendiconti, Classe di scienze fisiche, matematiche, naturali,
Ser. 6, Vol. 3, 1926, pp. 463.
[W.7J Sur un probleme aux limites dans une bande inde'finie. Institut de France.
Comptes Rendus de 1' Academic des Sciences. Vol. 184, 1927, p. 497.
Weitz, M., and J. B. Keller
[W.8] Reflection of water waves from floating ice in water of finite depth. Com-
munications on Pure and Applied Mathematics. Vol. 3, 1950, pp.
305-318.
Also see [K.9]
Weliczker, L.
[W.9] See [F.5].
Wells, Leon W. (formerly L. Weliczker)
[W.10] Some remarks on shallow water theory. New York University, Institute
of Mathematical Sciences, Report No. IMM-198, 1953.
Westergaard, H. M.
fW.ll] Water pressure on dams during earthquakes. Transactions of American
Society of Civil Engineers, Vol. 93, 1933.
Whitham, G. B.
[W.12] Dynamics of meteorological fronts. New York University, Institute of
Mathematical Sciences, Report No. IMM-195, 1953.
Wigley, W. C. S.
[W.13] See [L.18].
Author Index
Abdullah, A. J., 373, 388 Freeman, J. C., Jr. 375, 377, 384, 388,
Arthur, R. S., 132, 133 408
Friedriclis, K. O., 27, 32, 73, 74, 75, 76,
Baird, E. G., 412 108, 293, 300, 343, 344, 345, 849, 371,
Bakhmeteff, B. A., 451 407, 470, 529
Bateman, II., 117
Bates, C. C., 138 Gerber, R., 522
Biesel, F., 364, 305 Gilcrest, B. R., 451, 462, 482
Birkhoff, G., 342 Goldstein, E., 447
Bjerknes, J., 376
Boridi, H., 72 Hamada, T., 359, 372
Bouasse, H., 423 Hanson, E. T., 72, 89
Boussinesq, J., 342, 351, 451 Harleman, D. R. F., 412
Broer, L. J. F., 53 Haskind, M. D., 248, 259
Bruman, J. R., 412 Haurwitz, B., 376
Haveloek, T. II., 219, 242, 246, 248,
Carr, J. H., 132, 445, 448 253, 256
Carson, R. L., XIV Heins, A. E., 74, 108, 141
Cauchy, A. L., 35, 154 Hinze, J. O., 160
Charney, J. G., 375 Hogner, E., 220, 242
Chrystul, G., 424 Hopf, E., 44
Cooper, R. I. B., 138 Hyers, D. H., 32, 343, 345, 349, 371, 529
Copson, E. T., 117, 181
Cornish, V., 466 Isaacs, J. D., 133
Courant, R., 293, 300, 407, 470, 476, 477 Isaaeson, E., 73, 74, 451, 476, 477
Crossley, H. E., 412 Iversen, H. W., 359
Daily, J. \V., 351 Jeffreys, II., 352, 422
Danel, P., 138 John, F., 76, 113, 146, 175, 206, 249,
Darby shire, J., 137 431
Davies, T. V., 374, 522 Johnson, J. W., 133
Deacon, G. E. R., 137, 171 Johnson, M. \V., 133
Deymie, P., 505
Dressier, R. F., 466, 467, 468 Kampd de Feriet, J., 187
Dubreuil-Jaeotin, M. I,., 522 von Kdrman, T., 410, 482
Karp, S. N., 117, 141
Eekart, C., 73, 136 Keldysh, M. V., 203
Einstein, II. A., 412 Keller, J. B., 32, 141, 146, 343, 351, 371,
447, 476, 481
Finkelstein, A., 153, 187 Kelvin, W. T., 163, 219
Fjeldstad, J. E., 147 Keulegan, G. H., 370
Fleishman, B., 438 Korteweg, D. J., 342, 343
Fleming, R. H., 133 Korvin-Krukovsky, B. V., 260
Forchheimer, P., 451 Kotik, J., 153, 187
Forel, F. A., 423 Kotsehin, N. J., 61
561
562 WATEE WAVES
Kreisel, G., 146 Re, R., 818
Krylov, A. N., 248 Rees, M., 476, 477
Rellich, F., 113, 175
Laitone, E. V., 412, 418 Riabouchinsky, D., 25
Lamb, H., 28, 27, 57, 58, 180, 421 Roseau, M., 74, 75, 76, 78, 95, 146
Lavrentieff, M., 344 Rouse, H., 451, 462, 468, 482
Lax, A., 368 Rubin, H., 146
Lax, P., 476, 481 Ruellan, F.. 41
Leray, J., 522 Russell, S., 342
Levi-Civita, T., 17, 21, 343, 345, 346, 347,
374, 513, 522, 524 St. Denis, M., 248, 261
Lewis, E. V., 260 Schwinger, J. S., 117, 146
Lewy, H., 61, 72, 73, 74, 76, 79, 108, 537 Seiwell, H. R., 138
Lichtenstein, L., 515, 522, 529 Solberg, H., 376
Littman, W., 522 Sommerfeld, A., 52, 77, 116, 117
Longuet-Higgins, M. S., 137, 138 Southwell, R., 426
Lowell, S. C., 136 Sretenski, L. N., 234
Lunde, J. K., 246, 248, 257 Stelzriede, M. E., 132
Stoker, J. J., Ill, 451, 482
Macdonald, H. M., 117 Stokes, G. G., 96, 373
McNown, J. S., 426, 428 Struik, D. J., 17, 21, 342, 343, 344, 347,
Massau, J., 482 522
Mason, M. A., 372 Suquet, F., 133
Miche, A., 72, 137 Sverdrup, II. U., 133, 357, 369, 371
Michell, J. H., 248, 253, 256, 263, 285
Munk, W. H., 69, 133, 352, 357, 369, Tepper, M., 375
870, 371 Thomas, II. A., 451, 462, 467, 468, 482
Thompson, P. D., 375
Nekrassov, A. I., 522 Thorade, H. F., 368
Nirenberg, L., 344, 522 Thornhill, C. K., 513, 522
Traylor, M. A., 133
O'Brien, M. P., 133 Troesch, B. A., 451, 487, 505
Patterson, G. W., 370 Ursell, F., 96, 137, 146, 342, 372
Penney, W. G., 513, 522
Peters, A. S., 74, 75, 76, 78, 95, 96, 98, de Vries, G., 342, 343
102, 103, 111, 124, 224, 242, 245
Pierson, W. J., Jr., 133, 138, 248 Wallet, A., 41
Pohle, F. V., 513, 515 Weinblum, G. P., 248, 261
Poincare, H., 181 Weinstein, A., 40, 87, 208, ,342
Poisson, S., 35, 154 Weitz, M., 141
Preiswerk, E., 407, 411, 413, 482 Wells, L. W., 434, 444, 449
Putnam, J. A., 132 Whitham, G. B., 368, 377, 378, 388,
389, 395, 399, 404, 487, 505
Rankine, W. J. M., 76 Wigley, W. C. S., 246, 257
Rayleigh, J. W., 49, 321, 342, 351 Wilkes, M. V., 147, 375
Subject Index
Aerodynamics, 411
Angle of trim, 286
Archimedes' law, 254, 277
Atmosphere, gravity waves, 374
tidal oscillations, 374
waves on discontinuity surfaces, 375
Backwater curves, 456
Backwater effects in long rivers, 461
Beaches. (See also Sloping beaches.)
waves breaking on shallow, 352
Bernoulli's law, 9
Bernstein, S., theorem, 43
Bifurcation conditions, 529
Bifurcation phenomenon, 343
board, as fixed breakwater, 432
as floating breakwater, 436
Bore 307, 315, 326, 368
development 351
Tsien-Tang River, 320, 368
Boundary conditions, 10, 19
dynamical, 55
fixed boundary surface, 11
free surface, 11
kinematic, 16, 56
small amplitude theory, 19
tidal theory, 422
Breaking of a dam, 313, 333, 513
discharge rate, 338
resulting bore, 334
Breaking of waves, 69, 307, 315
at crests, 369
in shallow water, 351
induced by wind action, 372
on shallow beaches, 352
Breakwaters, 429
dispersion induced, 443
fixed board, 432
floating board, 436
floating elastic beam, 438
reflection of energy, 446
Cauchy-Riemann equations, 345
Cavitation, 310
Characteristic^ ), curves, 294
differential equations, 294
envelope, 307, 314, 355
intersection, 355
method, 293, 469
propagation of discontinuities along,
473
Chezy formula, 466
Circulation, 7
Cliff, waves against a vertical, 84
Cnoidal waves, 342
Cold front, 380
Comparison of predicted and observed
floods, 498
Compressibility, 3
Contact discontinuity, 318
Continuity equation, 453
Convolution theorem, 143
Coriolis acceleration, 383
force, 381
Crests, breaking of waves at, 369
Critical speed, inappropriateness of
linear theory at, 217, 344
instability of steady flow with, 344
Cyclone, 376, 399
Dam, breaking, 313, 333, 513
discharge rate on breaking, 338
shock resulting from breaking, 334
Diffraction around a vertical wedge, 109
problem of Sommerfeld, 109
theory, physical verification of, 132
Dipoles, 13
Discontinuities, propagated along char-
acteristics, 473
Discontinuity conditions, 314
surfaces in the atmosphere, 375
Dispersion, 51
Divergence theorem, 6
Diverging system of waves, 237
Dock problem, 74
two-dimensional, 108
Domain of dependence, 298
of determinancy, 299
Dynamic boundary condition, 16, 55
563
564
WATER WAVES
Eigenvalue problems, 424
Elastic beam, used as floating break-
water, 438
Energy, 13
average, 50
balance across a shock, 318
flux, 13
rate of change, 13
reflected by a breakwater, 446
transmission by progressing waves, 15
transmission by simple harmonic
waves, 47
velocity of the flow of, 49
Envelope of characteristics, 307, 314,
355
Engineering methods in flood wave
problems, 485
Equation of continuity, 7, 453
Equations of flow in open channels, 452
Equations of motion, 4
Eulerian form, 6
Lagrangian form, 4
Equations of shallow water theory,
nonlinear, 24
validity beyond the breaking point,
362
Euler variables, 5
Exact free surface condition, 513
Experimental, wave tanks, 71
solitary wave, 351
waves on sloping beaches, 71
Finite difference methods, 296, 424, 474
convergence of, 477, 481
Floating, bodies in shallow water, 431
breakwaters, 414
elastic beam, 438
rigid body, 245
Flood prediction, 482
Flood routing, 461
Flood waves, in the Mississippi and
Ohio Rivers, 458, 483, 494
monoclinal, 462
Flow, around bends, 405
between two walls, 410
in open channels, 451
of energy, 13
over obstacles, 344
through a sluice, 407
Forced oscillations, 55
Fourier integral theorem, 153
Fourier transform, 35, 155
Free natural vibrations of a lake, 424
Free surface, 11. (See also Surface.)
Free surface condition, 20
exact, 11, 513
linearized, 11, 12, 35
Free surface elevation, 16
Friction, 451
Friction slope, 455
Front, 378
cold, 374, 380
occluded, 381
stationary, 378
warm, 374, 380
Front of shock, 320
Gas dynamics analogy, 25
Geometrical optics, 133
Gravity waves in the atmosphere, 374
Green's function, 280
time-dependent, 187
Group of waves, 51
Group velocity, 51, 170
Harbors, design of, 420
model studies, 429
oscillations, 414
Heave, 250, 255
Heaving, 278
HelmhoUVs theorem, 7
Higher-order approximations in shallow
water theory, 28, 32
Hump, 352
Hydraulic analogy, 412
Hydraulic jumps, 307, 324, 407. (See
also Bore.)
interaction of, 412
Hydraulic radius, 454
Hydraulics, mathematical, 451
Hydrostatic pressure law, 24, 31
in meteorology, 374, 382
Influence point, 228
Initial characteristic, 302
Initial steepness of a wave, 357
Instability of steady flow with critical
speed, 344
Interaction of two hydraulic jumps, 412
Internal waves, 147
Intersection of characteristics, 855
Irrotational flow, 9
Iteration process, 539
Jump, hydraulic. See Hydraulic jumps.
Junction of the Ohio and Mississippi
Rivers, 457, 509
flood wave through, 494
Kelvin's theory of ship waves, 219
Kinematic boundary condition, 16, 56
SUBJECT INDEX
565
Lagrangian form of the equations of
motion, 4
Lake, free natural vibrations of, 424
Levi-Civita's theory, 513, 522
Linear theory, 35
derivation of, 19
free surface condition, 21
Local speed of small disturbance. See
Wave speed.
Long waves, theory of, 23, 291
Much lines, 40U
Manning's formula, 454
Manning's roughness coefficient, 457
Margules' law, 387
Mass flux, 0
across a shock front, 318
Mathematical hydraulics, 451
Meteorology, 374
hydrostatic pressure law in, 374, 382
Michell's type ship, 257
Microscisms, origin of, 137
Mississippi Hivcr, 509
flood waves, 458, 484
model, 482, 509
junction with the Ohio River, 457, 509
Models of the Ohio and Mississippi
Rivers, 482, 508
Model studies of harbors, 429
Momentum, conservation of, 3
Monoclinal flood wave, 4(52
Motions. (See also Flow; Wares.)
steady, 199, 201
uniqueness of unsteady, 187
unsteady, 149
Moving pressure point, 217
Non-existence of depression shock, 323
Nonlinear free surface condition, 11, 513
Non linearity of breaking phenomena, 71
Nonlinear shallow water theory equa-
tions, 24
Numerical solutions for sloping beaches,
73, 75
Obstacles, flows over, 344
waves due to, 35
Occluded front, 381
Oceanography, 133
Ocean tides, 421
Ohio River, 505
flood waves, 458, 484
junction with the Mississippi River,
457
model of, 482
Open channel flows, 451
unsteady, 409
Optics, geometrical, 133
Oscillations, forced, 55
free, 55
in harbors, 414
of a lake, 423
of the atmosphere, 374
pitching, 250
rolling, 250
simple harmonic, 37
small, 35
yawing, 250
Overhanging cliff, 73
Particle derivative, 5
Periodic impulse, waves due to, 17 1
Periodic surface pressure, 57
Periodic waves. (See also Waves.)
existence of, 522
Perturbation procedure, 19, 269
Phase speed, 170
Pitching oscillation, 250, 278
Point source (or sink), 12
Potential, singularities of, 12
Potential flow, 9
Pressure, 3. (Sec also Surface pressure.)
periodic, 57
waves caused by moving, 219
Privaloff's theorem, 536
Profile, of a river, 458
Progressing waves, 57. (See also Wares.)
of fixed shape, 461
simple harmonic, 45
Propagation of discontinuities along
characteristics, 474
Propagation speed. See Wave speed.
Radiation condition, 174, 209. (See also
Sommerfeld entries. )
Range of influence, 299
Rnyleigh-Uitz method, 426
Reflection of energy, 446
of shock, 330
of waves, 71, 95
Refraction along a coast, 133
Resistance force, 453
Resonance, 58
Rigid body, floating, 245
River profile, 458
Rivers, backwater effects in, 461
Rolling oscillations, 250
Roll waves, 466
Roughness coefficient, 454, 466
Manning, 457
566
WATER WAVES
Running stream, waves on a, 198
Schauder-Leray theory, 522
Schmidt, E., bifurcation theory of, 529
Seiche, 423
Seismology, 137
Shallow water, floating bodies, 431
long waves, 291
Shallow water theory, 22, 291
accuracy, 27
equations of, 24
for sloping beaches, 75
higher-order approximation in, 28, 32
linear, 25, 75, 414
linear, compared with numerical
solution, 75
linear sound speed, 419
mathematical justification, 31
reformulation of equations, 292
systematic derivation of, 27
validity beyond the breaking point,
362
Ship, as a floating rigid body, 245
Ship wave problem, 219, 224
Ship waves, diverging system, 237
in water of finite depth, 243
method of stationary phase, 219
transverse system, 237
Shock, 317
advancing into still water, 323
back of, 321
conditions, 314
constant, 326
energy balance across, 318
front of, 321
mass flux across, 318
non-existence of depression, 323
reflected from a rigid wall, 330
resulting from the breaking of a dam,
333
turbulence at front of, 320
Simple harmonic waves, 37
energy transmission, 47
progressing, 45
Simple wave, 300, 469
applications to problems of meteoro-
logy, 391
centered, 311
Singularities of the velocity potential,
12, 13
Sink, 12
Slenderness parameter, 250
Sloping beaches, 69, 369. (See also
Beaches.)
experiments on, 71, 73, 75, 373
numerical solutions, 73, 75
Sluice in a dam, flow through, 407
Small amplitude theory, 19. (See also
Linear theory.)
Small oscillations. See Oscillations.
Solitary wave, 327, 342, 370
approximation, 343
experimental work, 351
Sommerfeld's diffraction problem, 109
Sommerfeld's radiation condition, 59,
65, 111, 113, 175
Sound speed, 26. (See also Wave speed.)
Source, 12
Standing waves, cylindrically sym-
metric, 41
simple harmonic, 37
three-dimensional, 41
two-dimensional, 38
Stationary front, 378
Stationary phase, 163, 219
justification, 181
Steady flow, supercritical, 405
with critical speed, 344
Steady motions, 199
Steady state problems, unnaturalness,
175
Stoke's phenomenon, 117
Stoke's theorem, 8
Stream, waves on a running, 198
Surge, 250
Sway, 250
Subcritical flow, 305, 406
Supercritical flow, 304, 406
supersonic flow, 304
Surface, 11. (See also Free surface.)
condition, exact, 11, 513
disturbance, motions due to, 156
pressure, confined to a segment, 58
periodic, 57
simple harmonic, 55
Surfaces of discontinuity in the at-
mosphere, 375
Surface waves, 18
typical problem, 15
Tidal oscillations of the atmosphere,
375
Tidal theory, 22
boundary conditions, 422
Tides in the oceans, 421
Transverse system of ship waves, 237
Trim of a ship, 251
Turbulence, 453
at a shock, 320
Undertow, 71
Uniqueness, 150, 187
SUBJECT INDEX
567
Velocity, group, 170
of flow of energy, 49
Velocity potential, 9
singularities of, 12
Vertical cliff, three-dimensional waves
against a, 84
Vertical wedge. See Diffraction around
a vertical wedge.
Viscosity, 3
Vibrations, of a lake, 424
Warm front, 380
Water table experiments, 412
Wave motions. (See also Flow; Motions;
and Waves. )
in open channels, 451
on discontinuity surfaces in the
atmosphere, 375
Wave refraction along a coast, 133
Wave resistance integral, 284
Waves. (See also Flew; Motions.)
against a vertical cliff, 67, 84
breaking of, 69, 307
breaking of at crests, 369
breaking of on shallow beaches, 352
breaking point, 354
centered simple, 311
cnoidal, 342
depression, 306, 352
diverging system, 237
due to a moving pressure point, 219
due to disturbances from rest, 35
due to harmonic surface pressure, 49
due to obstacles in a running stream,
35
due to periodic impulse, 174
energy transmission, 47
existence of periodic, 522
experimental work on solitary, 351
experiments on sloping beaches, 71,
73, 75
group, 51
initial, steepness, 357
in open channels, 451
internal, 147
in the atmosphere, 374
of small amplitude, 19
on sloping beaches, 69, 369
past obstacles, 69
progressing, 57, 67
progressing, of fixed shape, 461
progressing, simple harmonic, 45
reflection from shore, 71, 95
roll, 466
ship. See Ship waves.
simple, 300, 469
simple harmonic standing, 37
solitary, 327, 342, 870
standing. See Standing waves.
steady, 207
transverse system, 237
unsteady, 210
Wave speed, 26, 293, 299, 473
in linear shallow water theory, 419
in meteorology, 404
Wave tanks, experiments in, 71, 73
Wetted perimeter, 454
Wiener-Hopf technique, 108, 141
Wine glass effect, 74
Yawing oscillation, 250
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11