Water Waves: The Mathematical Theory with Applications

by James Johnston Stoker

Overview

First published in 1957, this is a classic monograph in the area of applied mathematics. It offers a connected account of the mathematical theory of wave motion in a liquid with a free surface and subjected to gravitational and other forces, together with applications to a wide variety of concrete physical problems. A never-surpassed text, it remains of permanent value to a wide range of scientists and engineers concerned with problems in fluid mechanics.
The four-part treatment begins with a presentation of the derivation of the basic hydrodynamic theory for non-viscous incompressible fluids and a description of the two principal approximate theories that form the basis for the rest of the book. The second section centers on the approximate theory that results from small-amplitude wave motions. A consideration of problems involving waves in shallow water follows, and the text concludes with a selection of problems solved in terms of the exact theory. Despite the diversity of its topics, this text offers a unified, readable, and largely self-contained treatment. 

Product Details

ISBN-13:	9780486839929
Publisher:	Dover Publications
Publication date:	04/17/2019
Series:	Dover Books on Physics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	592
File size:	36 MB
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About the Author

James Johnston Stoker (1905–92) was born in Pittsburgh, Pennsylvania, and received his Ph.D. from the Federal Institute of Technology in Zurich. The following year he became a member of the faculty of the Department of Mathematics at New York University. He later succeeded Richard Courant as Director of the Courant Institute of Mathematical Sciences at NYU. In 1970 Stoker received the Timoshenko Medal in recognition of his contributions to the field of applied mechanics.

Read an Excerpt

INTRODUCTION

I. 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 applications to a wide variety of concrete physical problems.

Surface wave problems have interested a considerable number of mathematicians beginning apparently with Lagrange, and continuing with Cauchy and Poisson in France. Later the British school of mathematical physicists gave the problems a good deal of attention, 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 particularly in that part of it which might be called mathematical hydraulics. Later, Poincare made outstanding contributions particularly with regard to figures of equilibrium of rotating and gravitating 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 Nekrassov [N.1], [N.1a] 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] 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 mathematical 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 are 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 arc outlined in detail below, it has some point to indicate briefly here its general plan. There are 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 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 approximate 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 mathematicians 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 important 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 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 deductions — so often shadowy and vague in the literature concerned with the mechanics of continuous media - has always been maintained. 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 introduction to the subject, but in the course of writing it many gaps and inadequacies in the literature were noticed and some of them have 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 such 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 Tyndall'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 foreknowledge 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 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 hydrodynamics. 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 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 dimensionless 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 constitute 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.

(Continues…)

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Table of Contents

Part 1
Introduction ix
Acknowledgments 
1. Basic Hydrodynamics
1.1 The laws of conservation of momentum and mass 
1.2 Helmholtz's theorem 
1.3 Potential flow and Bernoulli's law
1.4 Boundary conditions 
1.5 Singularities of the velocity potential 
1.6 Notions concerning energy and energy flux 
1.7 Formulation of a surface wave problem 
2. The Two Basic Approximate Theories 
2.1 Theory of waves of small amplitude 
2.2 Shallow water theory to lowest order. Tidal theory .
2.3 Gas dynamics analogy 
2.4 Systematic derivation of the shallow water theory 
PART II
Subdivision 
Waves Simple Harmonic in the Time
3. Simple Harmonic Oscillations in Water of Constant Depth 
3.1 Standing waves
3.2 Simple harmonic progressing waves 
3.3 Energy transmission for simple harmonic waves of small amplitude
3.4 Group velocity. Dispersion
4. Waves Maintained by Simple Harmonic Surface Pressure in
Water of Uniform Depth. Forced Oscillations 
4.1 Introduction 
4.2 The surface pressure is periodic for all values of x 
4.3 The variable surface pressure is confined to a segment of the surface 
4.4 Periodic progressing waves against a vertical cliff 
5. Waves on Sloping Beaches and Past Obstacles
5.1 Introduction and summary 
5.2 Two-dimensional waves over beaches sloping at angles u> — nßn
5.3 Three-dimensional waves against a vertical cliff
5.4 Waves on sloping beaches. General case 
5.5 Diffraction of waves around a vertical wedge. Sommerfeld's diffraction problem 
5.6 Brief discussions of additional applications and of other methods of solution 
Subdivision B
Motions Starting from Rest. Transients
6. Unsteady Motions 
6.1 General formulation of the problem of unsteady motions 
6.2 Uniqueness of the unsteady motions in bounded domains
6.3 Outline of the Fourier transform technique 
6.4 Motions due to disturbances originating at the surface
6.5 Application of Kelvin's method of stationary phase
6.6 Discussion of the motion of the free surface due to disturbances
initiated when the water is at rest 
6.7 Waves due to a periodic impulse applied to the water wheninitially at rest. Derivation of the radiation condition for purely periodic waves 
6.8 Justification of the method of stationary phase 
6.9 A time-dependent Green's function. Uniqueness of unsteady
motions in unbounded domains when obstacles are present . 
Subdivision C
Waves on a Running Stream. Ship Waves
7. Two-dimensional Waves on a Running Stream in Water of Uniform Depth 
7.1 Steady motions in water of infinite depth with p = 0 on the free surface 
CHAPTER PAGE
7.2 Steady motions in water of infinite depth with a disturbing pressure on the free surface 
7.3 Steady waves in water of constant finite depth 
7.4 Unsteady waves created by a disturbance on the surface of a
running stream 
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
8.2 The classical ship wave problem. Details of the solution
9. The Motion of a Ship, as a Floating Rigid Body, in a Seaway
9.1 Introduction and summary 
9.2 General formulation of the problem 
9.3 Linearization by a formal perturbation procedure 
9.4 Method of solution of the problem of pitching and heaving of a
ship in a seaway having normal incidence
PART III
10. Long Waves in Shallow Water 
10.1 Introductory remarks and recapitulation of the basic equations 
10.2 Integration of the differential equations by the method of characteristics
10.3 The notion of a simple wave 
10.4 Propagation of disturbances into still water of constant depth 
10.5 Propagation of depression waves into still water of constant depth
10.6 Discontinuity, or shock, conditions 
10.7 Constant shocks: bore, hydraulic jump, reflection from a rigid wall
10.8 The breaking of a dam
10.9 The solitary wave 
10.10 The breaking of waves in shallow water. Development of bores
10.11 Gravity waves in the atmosphere. Simplified version of the problem of the motion of cold and warm fronts
10.12 Supercritical steady flows in two dimensions. Flow around bends. Aerodynamic applications 
10.13 Linear shallow water theory. Tides. Seiches. Oscillations in harbors. Floating breakwaters
11. Mathematical Hydraulics 
11.1 Differential equations of flow in open channels
11.2 Steady flows. A junction problem 
11.3 Progressing waves of fixed shape. Roll waves
11.4 Unsteady flows in open channels. The method of characteristics
11.5 Numerical methods for calculating solutions of the differential equations for flow in open channels 
11.6 Flood prediction in rivers. Floods in models of the Ohio River
and its junction with the Mississippi River 
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 
Appendix to Chapter 11. Expansion in the neighborhood of the first characteristic 
PART IV
12. Problems in which Free Surface Conditions are Satisfied Exactly.
The Breaking of a Dam. Levi-Civita's Theory 
12.1 Motion of water due to breaking of a dam, and related problems
12.2 The existence of periodic waves of finite amplitude 
12.2a Formulation of the problem 
12.2b Outline of the procedure to be followed in proving the existence
of the function <o(x) 
12.2c The solution of a class of linear problems 
12.2(1 The solution of the nonlinear boundary value problem 
Bibliography 
Author Index
Subject Index