J.J. Stoker - Water waves. The mathematical theory with applications

TEXT FLY WITHINTHE BOOK ONLYDOu< OU 160917>m^GOSTOKERWATER WAVESOSMANIA UNIVERSITY LIBRARYCall No.AuthorTitle^VL-'SjS-Jfc fcfcT,S7 lOTAccession No.-CO<X^L/LThis book should be returned on or before the date last marked below.WATER WAVESThe Mathematical Theory with ApplicationsPURE AND APPLIED MATHEMATICSASeries of Textsand MonographsEdited byR.COURANTL.BERS.J.

J.VOLUME IVSTOKERWaves about a harborWATER WAVESThe Mathematical Theorywith ApplicationsJ.STOKERJ.INSTITUTE OF MATHEMATICAL SCIENCESNEW YORK UNIVERSITY, NEW YORK19ffill57INTERSCIENCE PUBLISHERS,INTERSCIENCE PUBLISHERSINC.,LTD.,NEW YORKLONDONAll Rights ReservedLIBRARY OF CONGRESS CATALOG CARD NUMBER56-8228INTERSCIENCE PUBLISHERS INC.250 Fifth Avenue, New York 1, N.Y.For Great Britain and Northern Ireland:INTERSCIENCE PUBLISHERS LTD.88/90 Chancery Lane, LondonW.C. 2PRINTED IN THE NETHERLANDSBY LATE HOITSEMA BROTHERS, GRONINGENToNANCYIntroductionIntroduction1.The purposeof thisbookisto present a connected account of themotion in liquids with a free surfacemathematical theory of waveand 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 ofmathematicians beginning apparently with Lagrange, and continuing with Cauchy and Poisson in France.* Later the British schoolof 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.

Inthe latter part of the nineteenth century the French once more tookup the subject vigorously, and the work done by St. Venant andBoussinesq in this field has had a lasting effect: to this day theFrench have remained active and successful in the field, and particularly in that part of it which might be called mathematicalhydraulics.

Later, Poincar^ 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 earlierby Liapounoff. One of the most outstanding accomplishmentsin thethe proof of thefrom the purely mathematical point of viewoffinitewas made byexistence of progressing wavesamplitudeNckrassov [N.I], [N.lajf in 1921 and independently by a differentfieldmeans 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], almosta third of which is concerned with gravity wave problems.

Thereare books by Bouasse [B.15], Thorade [T.4], and Sverdrup [S.39]* Thistwould be considerably extended(to include Euler, the Bernoullis,hydrostatics were to be regarded as an essential part of our subject.Numbers in square brackets refer to the bibliography at the end of the book.listand others)ifXINTRODUCTIONdevoted exclusively to the subject. The book by Thorade consistsalmost entirely of relatively brief reviews of the literature up to1931an indication of the extent and volume of the literatureon the subject.

The book by Sverdrup was written with the specialneeds of oceanographers in mind. One of the main purposes of thepresent book is to treat some of the more recent additions to ourknowledge in the field of surface wave problems. In fact, a large partof the book deals with problems the solutions of which have beenfound during and since World War II; this material is not availablein the books just now mentioned.The subject of surface gravity waves has great variety whetherregarded from the point of view of the types of physical problemswhich occur, or from the point of view of the mathematical ideasand methods needed to attack them. The physical problems rangefrom discussion of wave motion over sloping beaches to flood wavesin rivers, the motion of ships in a sea-way, free oscillations of enclosedbodies of water such as lakes and harbors, and the propagation offrontal discontinuities in the atmosphere, to mention just a few.The mathematical tools employed comprise just about the whole ofthe tools developed in the classical linear mathematical physicsconcerned with partial differential equations, as well as a good partof what has been learned about the nonlinear problems of mathematical physics.

Thus potential theory and the theory of the linearwave equation, together with such tools as conformal mapping andcomplex variable methods in general, the Laplace and Fouriertransform techniques, methods employing a Green's function, integralequations, etc. are used. The nonlinear problems arc of both ellipticand hyperbolic type.In spite of the diversity of the material, the book is not a collectionof disconnected topics, written for specialists, and lacking unity andcoherence.

Instead, considerable pains have been taken to supplythe fundamental background in hydrodynamicsand also in someof the mathematics neededand to plan the book in order that itshould be asmuchas possible a self-containedand readable whole.Though the contents of the book are outlined in detail below, it hassome point to indicate briefly here its general plan. There arc fourmain parts of the book:andpresents the derivation ofthe basic hydrodynamic theory for non-viscous incompressible fluids,and also describes the two principal approximate theories which formPartI,comprising Chapters12,INTRODUCTIONXIthe 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 wavemotions considered is small. The result is a linear theory which fromthe mathematical point of view is a highly interesting chapter inpotential theory.

On the physical side the problems treated includethe propagation of waves from storms at sea, waves on slopingbeaches, diffraction of waves around a breakwater, waves on arunning stream, the motion of ships as floating rigid bodies in a sea-way. Although this theory was known to Lagrange, it is often referredto 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 problemsinvolving waves in shallow water. The approximate theory whichresults from assuming the water to be shallow is not.

a linear theory,and wave motions with amplitudes which are not necessarily smallcan be studied by its aid. The theory is often attributed to Stokesand Airy, but was really known to Lagrange. If linearized by makingthe additional assumption that the wave amplitudes are small, thetheory becomes the same as that employed as the mathematicalbasis for the theory of the tides in the oceans. In the lowest orderof approximation the nonlinear shallow water theory results in asystem of hyperbolic partial differential equations, which in important special cases can be treated in a most illuminating way withthe aid of themethodof characteristics.The mathematical methodsare treated in detail in Chapter 10.

The physical problems treated inChapter 10 are quite varied; they include the propagation of unsteadywaves due towater, the breaking ofwaves, the solitary wave, floating breakwaters in shallow water. Alengthy section on the motions of frontal discontinuities in thelocal disturbances intostillatmosphere is included also in Chapter 10. In Chapter 11, entitledMathematical Hydraulics, the shallow water theory is employed tostudy wave motions in rivers and other open channels which, unlikethe problems of the preceding chapter, are largely conditioned bythe necessity to consider resistances to the flow due to the roughand bottom of the channel.

Steady flows, and steady progressingwaves, including the problem of roll waves in steep channels, arefirst studied. This is followed by a treatment of numerical methodsof solving problems concerning flood-waves in rivers, with the objectof making flood predictions through the use of modern high speedsidesINTRODUCTIONXIIThat such methods can be used to furnish accuratebeenverified for a flood in a 400-mile stretch of thehaspredictionsOhio River, and for a flood coming down the Ohio River and passingdigital computers.through its junction with the Mississippi River.Part IV, consisting of Chapter 12, is concerned with problemssolved in terms of the exact theory, in particular, with the use of theAproof of the existence ofperiodic waves of finite amplitude, following Levi-Civita in a generalexact nonlinear free surface conditions.way,isincluded.The amount of mathematical knowledge needed to read the bookvaries in different parts.

For considerable portions of Part II theelements of the theory of functions of a complex variable are assumedknown, together with some of the standard facts in potential theory.On the other hand Part III requires much less in the way of specificknowledge, and, as was mentioned above, the basic theory of thehyperbolic differential equations used there is developed in all detailhope that this part would thus be made accessible to engineers,in thewho have anmathematical treatment ofproblems concerning flows and wave motions in open channels.In general, the author has made considerable efforts to try toachieve a reasonable balance between the mathematics and themechanics of the problems treated.