J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 28
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5.6.6.?,WAVES ON SLOPING BEACHES AND PAST OBSTACLES(5.6.6)=y)<p(x,frifo>]Jowhen the radiusdG+5d?/oof the circle+ y sinexp {ik(x cosC 2 is allowed to tend tooo,143)}and the boun-dary condition (5.6.2), the regularity conditions, and conditions at ooof the screenare used. (A mild singularity at the edge x0, y==mustalso be permitted.) The symbol [9] under the integral sign rejump in <p across the screen, which is of course not knownpresents theadvance. 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 differenandtiate both sides of (5.6.6) with respect to y, then set y --=inconfine attention to positive values of x\ in view of the boundarycondition (5.6.2) we obtain in this way the integral equation/:- ik sin(5.6.7)QeikxrQSo+\(p(.r Qxx^dx^)]K(x>0.JoThekernel#K(x)of the integral equation^-*o)-(5.6.8)d2isgiven byGv,-oyoy,<)a typical example of an integral equation solvableEquation (5.6.7)theWiener-Hopf technique; its earmarks are that the kernel is abyisxfunction of (xaxis.)and the range of integrationisthe positive real,Thestarting point of themethodisthe observation that the integralreminiscent of the convolution type of integralin the theory of the Fourier transform.
In fact, if the limits of integraoo and the equation were validoo totion in (5.6.7) were fromin (5.6.7) is strongly+could be solved at once by making use of theconvolution theorem. This theorem states that iffor all values of x,/(<*)=fJi.e. if00/fM ex PandAT arethen /(a)^(a)=it{iouc Q }dcc QK(cn)=fJK(xx^dx^)exp{00the Fourier transforms of / and kf* i(x Q )K(xJand(cf.Chapter 6)in other words, thetransform-OCof the integral on the right is the product of the Fourier transformsof the function f(x Q ) and K(x ) (cf.
Sneddon [S.ll], p. 24). Conse-quently if (5.6.7) held in the wider domain indicated,used to yielditcould beWATER WAVES144with*(a)+[<p(a)]tf (a),the transform of the nonhomogeneous term in the integral/t(a)equation. This relation in turn defines the transform [<p(a)j ofsince /&(a) and K(<x.) are the transforms of known functions,[(p(x )]and hence [<P(MQ)] itself.are, of course, not in a position to proceedWeat once in this fashion; but the idea of the Wiener- Hopf method is toextend the definitions of the functions involved in such a way thatone can do so. To this end the following definitions are madeg(x)-(5.6.9)h(x)h(x)Equation(5.6.10)===(5.6.7)=g(x)0,x0,x> 0;< 0;ik sin 6 Qe ikx/(a?f(xcos=))\<p],XQ0,XQ\x><> 0.can now be replaced by the equivalent equation*fx )dKQ,oo < j < oo.h(x)f(x )K(x+J00the function we seek- is,unknown for x < and f(x Qunknown for x > 0; thus we have only one equation fortwo unknown functions.
Nevertheless, both functions can bedetermined by making use of complex variable methods applied tothe Fourier transform of (5.6.10); we proceed to outline the method.Here g(x)is)of course,Wehave, to begin with, from (5.6.10):g(a)-Ma)(5.6.11)+ /(a)jT(a),with A(a) and /?(a) known functions given by(5.6.12)fe(oc)(5.6.13)#(a)k sin='a=k cos^-(k*-a2)*.next shown to be valid in a strip of the comreal axis in its interior. We omit thedetails of the discussion required to establish this fact; it follows in anelementary way from the assumption that the constant k has a positive imaginary part, and from the conditions of regularity and boundedness imposed on the solution 99 of the basic problem. K(<x.) isfactored* in the form (i/2)JSL(oO JP+(a) with J?_(a) = (k - a) 1/21/2 withK__ anda)(A;+ (QL)+ regular in lower and upper half-The equation(5.6.11)isplex a-plane which contains the,K=+K* Such amanipulation occurs in general in using this technique; usually acontinued product expansion of the transform of the kernel is required.WAVES ON SLOPING BEACHES AND PAST OBSTACLES145planes, respectively.
The equation (5.6.11) can then be expressed,after some manipulation, in the form(5.6.14)_(k+ a)1/ 2a-k cos[(*~~(A:+ k cos1/2)(a-+ a)1/2+ fccos(ki~~k cos'a0^)where the symbols g + and /_ refer to the fact that g(oc) and /(a) can beshown to be regular in upper and lower half-planes of the complexoc-plane, respectively, each of which overlaps the real axis. In fact, theentire left side of (5.6.14) is regular in such an upper half-plane, andsimilarly for the right hand side in a lower half-plane. Thus the twosides of the equation define a function which is regular in the entireplane, or, in other words, each side of the equation furnishes theanalytic continuation of the function defined by the other side.Finally,it israther easy to show, by studying the behavior ofg(oc)and/(a) at oo, that the entire function thus defined tends uniformlyto zero at oo; it is therefore identically zero.
Thus (5.6.14) defines bothand/(a) since they can be obtained by equating both sides separately to zero. Thus g(x) and /(#) arc determined, and the problemis, in principle, solved.g(a)The Wiener-Hopf methodevidently, a most amusing and inalso has somewhat the air of a tour deis,genious procedure.
However, itforce which uses a good many tools from function theory (whilethe problem itself can be solved very nicely without going into thecomplex domain atit also employs thewe haveseen in the preceding section) andartificial device of assuming a positive imaginaryall,aspart for the wave number k. (This brings with it, we observe fromoo, it be(5.6.4), that while the primary wave dies out as x->Intheasinfinitecomes exponentiallyxaddition,oo.)problem+by a wedge, rather than by a plane barrier, can not besolved by the Wiener-Hopf method, but yields readily to solution bythe simple method presented in the preceding section.
The authorof diffractionhazards the opinion that problems solvable by the Wiener-Hopftechnique will in general prove to be solvable more easily by othermethodsfor example,by moredirect applications ofcomplexin-tegral representations, perhaps along the lines used to solve thedifficult mixed boundary problem treated in section 5.4 above.We mentionnext two other papers in which integral equations areWATER WAVES146employed to solve interesting water wave problems. The first of theseis the paper by Kreisel [K.19] in which two-dimensional simpleharmonic progressing waves in a channel of finite depth containingrigid reflecting obstacles are treated.
Integral equations areobtainedby using an appropriate Green's function; Kreisel then shows thatthey can be solved by an iteration method provided that the domainoccupied by the water does not differ too much from an infinite stripwith parallelsides. (Roseau [R.9] has solved similar problems fordomains which are not restricted in this way. ) It is remarkablethat Kreisel is able to obtain in some important cases good and useablcupper and lower bounds for the reflection and transmission coefficients.
References have already been made to the papers by John[J.5] on the motion of floating bodies. In the second of these papersthe problem of the creation of waves by a prescribed simple harmonicmotion of a floating body is formulated as an integral equation. Thiscertainintegral equation does not fall immediately into the category of thosewhich can be treated by the Fredholm theory; in fact, its theory hasanumberof interestingand unusual features sinceitturns out thatthe homogeneous integral equation has non-trivial solutions which,however, are of such a nature that the nonhomogeneous problemnevertheless always possesses solutions.Various problems concerning the effect of obstacles on waves, andwave motions created by immersed oscillating bodies, havebeen treated in a series of notable papers by Ursell [U.3, 4, 5 andU.8, 9, 10]. Ursell usually employs the method of expansions in termsof theof orthogonal functions, or representations by integrals of the Fouriertype, as tools for the solution of the problems.Finally, it should be mentioned that the approximate variationalmethods devised by Schwinger [S.5] to treat difficult problems in thetheory of electromagnetic waves can also be used to treat problems inwater waves (cf.
Keller [K.7]). A notable feature of Schwinger'smethod is that it is a technique which concentrates attention on thequantities 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 formulated the problem of the finite dock which has so far defied allefforts to obtain an explicit integral representation for its solutionas a variational problem of a somewhat unconventional type, andproved the existence, on the basis of this formulation, of solutions be-having at oo like progressing waves.WAVES ON SLOPING BEACHES AND PAST OBSTACLES147Aninteresting type of problem which might well have been discussed at length in this book is the problem of internal waves. Thisrefers to the occurrence of gravity waves at an interface between twoliquids of different density.
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