J.J. Stoker - Water waves. The mathematical theory with applications, страница 3

it involves a linearcombination of the potential function and its normal derivative, andthis combination is such as to lead to the occurrence of non-trivialsolutions of the homogeneous problems in cases which would in themore conventional problems ofpotential theory possess only identically constant solutions. In fact, it is this mixed boundary condition at a free surface which makes Part II a highly interestingchapter in potential theoryquite apart from the interest of theproblems on the physical side. Chapter 4 goes on to treat certainsimple harmonic forced oscillations, in contrast with the free oscillations treated in Chapter 3. Chapter 5 is a long chapter which dealswith simple harmonic waves in cases in which the depth of the wateris not constant.

A large part of the chapter concerns the propagationof progressing waves over a uniformly sloping beach; various methodsof treating the problem are explainedin part with the object ofillustrating recentlyproblems (both fordeveloped techniques useful for solving boundaryharmonic functions and functions satisfying thereduced wave equation) in which mixed boundary conditions occur.Another problem treated (in Chapter 5.5) is the diffraction of wavesaround a vertical wedge. This leads to a problem identical with theclassical diffraction problem first solved by Sommerfeld [S.I 2] forthe special case of a rigid half-plane barrier. Here again the uniquenessquestion comes to the fore, and, as in many of the problems of Part II,involves consideration of so-called radiation conditions at infinity.

Auniqueness theorem is derived and also a new, and quite simple anditelementary, solution for Sommerfeld's diffraction problem is given.It is a curious fact that these gravity wave problems, the solutionsof which are given in terms of functions satisfying the Laplaceequation, nevertheless require for the uniqueness of the solutionsthat conditions at infinity of the radiation type, just as in the morefamiliar problems basedon thelinearwave equation, be imposed;IINTRODUCTIONXVIIordinarily in potential theory it is sufficient to require only boundednessconditions at infinity to ensure uniqueness.In subdivisionBof Part II, comprised of Chapter 6, a variety ofproblems involving transient motions is treated.

Here initial conditions at the time tare imposed. The technique of the Fouriertransformisexplained and used to obtain solutions in the form ofintegral representations.The importantclassical cases (treated firstby Cauchy and Poisson) of thecircular waves due to disturbances ata point of the free surface in an infinite ocean are studied in detail.For this purpose it is very useful to discuss the integral representationsby using an asymptotic approximation due to Kelvin (and, indeed,developed by him for the purpose of discussing the solutions of justsuch surface wave problems) and called the principle, or method, ofstationary phase. These results then can be interpreted in a strikingin terms of the notion of group velocity. Recently there havebeen important applications of these results in oceanography: oneof them concerns the type of waves called tsunamis, which aredestructive waves in the ocean caused by earthquakes, anotherconcerns the location of storms at sea by analyzing wave recordson shore in the light of the theory at present under discussion.

Theagain a problemquestion of uniqueness of the transient solutionswaysolved only recentlyAnopportunityisistreated in the final section of Chapter 6.also afforded for a discussion of radiation con-ditions (for simple harmonic waves) as limits asproblemsconcerning transients, int-> oo in appropriatewhich boundedness conditions atinfinity suffice to ensure uniqueness.The final subdivision of Part II, subdivision C, deals with smalldisturbances created in a stream flowing initially with uniformvelocity and with a horizontal free surface.

Chapter 7 treats waves instreams 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 circumstances with respect to radiation conditions depend radically on the2and h the velocity and depth at infinity, resparameter U /g/*, withUpectively.ThusifU2/gh>1,no radiation conditions need be im-=21 somethingthey are needed, while if U jghtheirareandThesemattersoccurs.studied,physicalquite exceptionalinandIn7.37.4.discussedareChapter 8ChapterinterpretationsafortheidealizedcaseofwavesofKelvin's theoryship regardedshipas a point disturbance moving over the surface of the water is treatedposed,ifU2/gh<1INTRODUCTIONXVIIIprinciple of stationary phase leads to abeautiful and elegant treatment of the nature of ship waves that ispurely geometrical in character.

The cases of curved as well asin considerable detail.Thestraight courses are considered, and photographs of ship waves takenfrom airplanes are reproduced to indicate the good accord withobservations. Finally, in Chapter 9 a general theory (once more theresult of quite recent investigations) for the motion of ships, regardedas floating rigid bodies, is presented. In this theory no restrictiveregarding, for example, the coupling (or lack ofasinanold theory due to Krylov [K.20] between thecoupling,motion of the sea and the motion of the ship, or between the variousassumptionsare made other than those needed todegrees of freedom of the shiplinearize the problem. This means essentially that the ship must beregarded 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 stillwaves which do not have large amplitudes; in addition, itnecessary to suppose that the motion of the ship is a small oscillation relative to a motion of translation with uniform velocity.

Thecreateistheory is obtained by making a formal development of all conditionsof the complete nonlinear boundary problem with respect to a parameter which is a thickness-length ratio of the ship. The resultingtheory contains the classical Michell-IIavelock theory for the waveresistance of a ship in terms of the shape ofspecial case.itshull as the simplestWeturn next to Part III, which deals with applications of theapproximate theory which results from the assumption that it is thedepth of the liquid which is small, rather than the amplitude of thesurface waves as in Part II.

The theory, called here the shallowwater theory, leads to a system of nonlinear partial differentialequations which are analogous to the differential equations for themotion of compressible gases in certain cases. We proceed to outlinethe contents of Part III, which is composed of two long chapters.In Chapter 10 the mathematical methods based on the theory ofcharacteristics are developed in detail since they furnish the basisfor the discussion of practically all problems in Part III; it is hopedthat this preparatory discussion of the mathematical tools will makePart III of the book accessible to engineers and others who have nothad advanced training in mathematical analysis and in the methodsof mathematical physics. In preparing this part of the book theauthor's task was made relatively easy because of the existence of theINTRODUCTIONXIXbook by Courant and Friedrichs[C.9], which deals with gas dynamics;the presentation of the basic theory given here is largely modeledon the presentation given in that book.

The concrete problems dealtwith in Chapter 10 are quite varied in character, including thepropagation of disturbances into still water, conditions for theoccurrence of a bore and a hydraulic jump (phenomena analogous tothe occurrence of shock waves in gas dynamics), the motion resultingfrom the breaking of a dam, steady two dimensional motions atsupercritical velocity, and the breaking of waves in shallow water.The famous problem of the solitary wave is discussed along the linesused recently by Friedrichs and Hyers [F.13] to prove rigorouslythe existence of the solitary wave from the mathematical point ofview; this problem requires carrying the perturbation series whichformulate the shallow water theory to terms of higher order. Theproblem of the motion of frontal discontinuities in the atmosphere,which lead to the development of cyclonic disturbances in middleon the basis of hypotheses whichlatitudes, is given a formulationwhich brings it within the scopesimplify the physical situationof a more general "shallow water theory".

Admittedly (as has alreadybeen noted earlier) this theory is somewhat speculative, but it isnevertheless believed to have potentialities for clarifying some ofthe mysteries concerning the dynamical causes for the developmentand deepening of frontal disturbances in the atmosphere, especiallyif modern high speed digital computing machines are used as an aidin solving concreteproblems numerically.concludeswith the discussion of a few applications of10Chapterthe linearized version of the shallow water theory.