J.J. Stoker - Water waves. The mathematical theory with applications, страница 2

Usually a discussion of the physicalfactors and of the reasons for making simplified assumptions in eachfor example,interest in thenew typeof concrete problem precedes the precise formulation of themathematical problems. On the other hand, it is hoped that a cleardistinction between physical assumptions and mathematical deducso often shadowy and vague in the literature concernedhas always been mainwith the mechanics of continuous mediatained. Efforts also have been made to present important portionsof the book in such a way that they can be read to a large extenttionsindependently of the rest of the book; this was done in some casesat the expense of a certain amount of repetition, but it seemed tothe author more reasonable to save the time and efforts of the readerthan to save paper. Thus the portion of Chapter 10 concerned withthe dynamics of the motion of fronts in meteorology is largelyself-contained.

The same is true of Chapter 11 on mathematicaland 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 andinadequacies in the literature were noticed and some of them havehydraulics,INTRODUCTIONXIIIbeenfilled in; thus a fair share of the material presented representsthe result of researches carried out quite recently. A few topics whichare even rather speculative have been dealt with at some length(the theory of the motion of fronts in dynamic meteorology, givenin Chapter 10.12, for example); others (like the theory of waves onsloping beaches) have been treated at some length as much becausethe author had a special fondness for the material as for their intrinsicmathematical interest.

Thus the author has written a book which israther personal in character, and which contains a selection ofmaterial chosen, very often, simply because it interested him, andhe has allowed his predilections and tastes free rein. In addition,the book has a personal flavor from still another point of view sincea quite large proportion of the material presented is based on the workof individualmembersof the Institute of Mathematical Sciences ofNew YorkUniversity, and on theses and reports written by studentstheInstitute. No attempt at completeness in citing theattendingeventhe more recent literature, was made by the author;literature,on the other hand, a glance at the Bibliography (which includesonly works actually cited in the book) will indicate that the recentliterature has not by any means been neglected.In early youth by good luck the author came upon the writingsof scientists of the British school of the latter half of the nineteenthcentury.

The works of Tyndall, Huxley, and Darwin, in particular,made a lasting impression on him. This could happen, of course, onlybecause the books were written in an understandablein sucli awayas to create interestand enthusiasm:way andbutthisalsowasone of the principal objects of this school of British scientists.Naturally it is easier to write books on biological subjects for nonspecialists than it is to write them on subjects concerned with themathematical sciencesjust because the time and effort needed toofmathematical tools is very great.modernaknowledgeacquireThat the task is not entirely hopeless, however, is indicated by JohnTyndaU's book on sound, which should be regarded as a great classicof scientific exposition.

On the whole, the British school of popularizersof science wrote for people presumed to have little or no foreknowledge of the subjects treated. Now-a-days there exists a quite largepotential audience for books on subjects requiring some knowledgeof mathematics and physics, since a large number of specialists ofall kinds must have a basic training in these disciplines. The authorhopes that this book, which deals with so many phenomena of everyINTRODUCTIONXIVday occurrence in nature, might perhaps be found interesting, andunderstandable in some parts at least, by readers who have somemathematical training but lack specific knowledge of hydrodynamics.* For example, the introductory discussion of waves onsloping beaches in Chapter 5, the purely geometrical discussion ofthe wave patterns created by moving ships in Chapter 8, great partsof Chapters 10 and 11 on waves in shallow water and flood waves inrivers, as well as the general discussion in Chapter 10 concerningthe motion of fronts in the atmosphere, are in this category.2.Outline of contentsIt has already been stated that this book is planned as a coherentand unified whole in spite of the variety and diversity of its contentson both the mathematical and the physical sides.

The possibility ofachieving such a purpose lies in the fortunate fact that the materialcan be classified rather readily in terms of the types of mathematicalproblems which occur, and this classification also leads to a reasonablyconsistent ordering of the material with respect to the various typesof physical problems. The book is divided into four main parts.Part I begins with a brief, but it is hoped adequate, developmentof the hydrodynamics of perfect incompressible fluids in irrotationalflow without viscosity, with emphasis on those aspects of the subjectrelevant to flows with a free surface. Unfortunately, the basic generaltheory is unmanageable for the most part as a basis for the solutionof concrete problems because the nonlinear free surface conditionsmake for insurmountable difficulties from the mathematical pointof view.

Itistherefore necessary to make restrictive assumptionseffect of yielding more tractable mathematicalwhich have theformulations. Fortunately there are at least two possibilities in thisrespect which are not so restrictive as to limit too drastically thesame time they are such as to leadmathematical problems about which a great deal of knowledgephysical interest, while at thetoisavailable.Onetwo approximate theories results from the assumptionthat the wave amplitudes are small, the other from the assumption*of theshould be referred to here. This book ishighly recommended for supplementaryreading. Parts of it are particularly relevant to some of the material inChapter 6 of the present book.The book by Rachel Carsonentirelynonmathematical, but[C.I 6]itisXVINTRODUCTIONit is the depth of the liquid which is smallin both cases, ofcourse, the relevant quantities are supposed small in relation to someother significant length, such as a wave length, for example.

Both ofthatthese approximate theories are derived as the lowest order termsof formal developments with respect to an appropriate small dimen-by proceeding in this way, however, it can bethe approximations could be carried out to include higherorder terms. The remainder of the book is largely devoted to thesionless parameter;seenhowworking out of consequences of these two theories, based on concretephysical 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 upPart IV, in which a few problems are solved in terms of the basicgeneral theory and the nonlinear boundary conditions are satisfiedexactly; this includes a proof along lines due to Levi-Civita, of theexistence, from the rigorous mathematical point of view, of progressingwaves 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 obtainedfrom the basic exact theory simply by linearizing the free surfaceconditions on the assumption that the wave motions studied constitute a small deviation from a constant flow with a horizontal freesurface.

Since we deal only with irrotational flows, the result is atheory based on the determination of a velocity potential in the spacevariables (containing the time as a parameter, however) as a solutionof the Laplace equation satisfying certain linear boundary and initialconditions. This linear theory thus belongs, generally speaking, topotential theory.such a variety of material to be treated in Part II, whichcomprises Chapters 3 to 9, that a further division of it into subdivisions is useful, as follows: 1) subdivision A, dealing with wavemotions that arc simple harmonic oscillations in the time; 2) subdivision B, dealing with unsteady, or transient, motions that ariseTherefromisinitialdisturbances 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 areassumed to be small oscillations near the rest position of equilibriumof the fluid.SubdivisionAismade upof Chapters3, 4,and5.In Chapter 3the basically important standing and progressing waves in liquidsINTRODUCTIONXVIof uniform depth and infinite lateral extent are treated; the importantfact that these waves are subject to dispersion comes to light, andthe notion of group velocity thus arises.

The problem of the uniquenessof the solutionsconsideredin fact, uniqueness questions areintentionally stressed throughout Part II because they are interestingmathematically and because they have been neglected for the mostispart until rather recently. It might seem strange that there could beany interesting unresolved uniqueness questions left in potentialtheory at this late date; the reason for it is that the boundary condition at a free surface is of the mixed type, i.e.