J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 23
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Once anyo 2 /g,bottomt/=Asatisfiedisreal value for the frequency a is prescribed,whichto determine the real constantmequation (5.5.3) is usedooand weatwill turn out to be the wave number of the wavesnote that (5.5.3) has exactly one real solution of in except for sign;2if the water is infinitely deep we have in<7 /#, and the functionmvcosh m(yh) in (5.5.1) is replaced by eThus the function /(r, 0) is to be determined as a solution of the,+.reduced wave equation(5.5.4)V^+w/=20) /0,<r <oo,< <v,subject to the boundary conditions (5.5.2). Actually, we shall in theend carry out the solution in detail only for the case of a reflectingfor the special case v2jr), but it will be seenmethod would furnish the result for any wedge.
It isrigid plane stripthat the same(i.e.convenient to introduce aby the equation r=new independentvariablep,replacingr,Q/m; in this variable equation (5.5.4) has theform(5.5.5)V^ /+/ =0)0,0< e<oo,0<0<*,and we assume this equation as the basis for the discussion to follow.So far we have not formulated conditions at oo, except for thevague statement that we want to consider the effect of our wedgeShaped barrier on an incoming plane wave from infinity. Of course,we then expect a reflected wave from the barrier and also diffractioneffects from the sharp corner at the origin.
In conformity with ourWAVES ON SLOPING BEACHES AND PAST OBSTACLES111general practice we wish to formulate these conditions at oo in sucha way that the solution of the problem will be uniquely determined.Ithas some point to consider the question of reasonable conditionsat oo which determine unique solutions of the reducedwave equationunder more general circumstances than those considered in thephysical problem formulated above. For general domains it is notknown how to formulate these conditions at oo, and, in fact, it wouldseem to be a very difficult task to do so since such a formulationwould almost certainly require consideration of many special cases.In one special case, however, the appropriate condition to be imposed at infinity has been known for a long time. This is the casein which any reflecting or refracting obstacles lie in a bounded domainof the plane, or, stated otherwise, it is the case in which a full neighborhood of the point at infinity is made up entirely of the homogeneousmedium in which the waves propagate.
In this case, the conditionat oo which determines the "secondary" waves uniquely is Sommerfeld's radiation condition, which states, roughly speaking, that thesewaves behave like a cylindrical outgoing progressing wave at oo.However, if the reflecting or refracting obstacles extend to infinity,the Soinmorfeld conditionfor example, the case in(i.e.willnot be appropriate at all.
Consider,tT-axis is a reflecting barrierand the primary wave is an incoming plane waveon physical grounds that the secondary wavebe the reflected plane wave, which certainly does not behavethe casefrommaywhich the entirevJT),infinity. It is clearat oo like a cylindrical wave since, for example, its amplitude doesnot even tend to zero at oo. Another case is that of Sommerfeld'sproblem in which an incoming plane wave isfrom a barrier consisting of the positive half of the ^-axis.In this case, the secondary wave has both a reflected componentwhich has a non-zero amplitude at oo, and a diffracted part whichdies out at oo.
A uniqueness theorem has been derived by Peters andStoker [P.10] which includes these special cases; we proceed to givethis proof both for its own sake and also because it points the wayto a straightforward and elementary solution of the special problemformulated above. In Chapters 6 and 7 a different way of lookingclassical di (Tractionreflectedat the problem of determining appropriate radiation conditions isproposed; it involves considering simple harmonic waves (Chapter 6),or steady waves (Chapter 7) as limits when t -> oo in appropriatelyformulated initial value problems which correspond to unsteadymotions.WATER WAVES112The uniqueness theorem, whichis general enough to include theintheformulatedfollowing way: We assume thatproblem above,/(#, y) is a complex- valued solution* of the equationis+/=V 2/(5.5.6)Dwith boundary jT, part of which may extend to infinity.
It is supposed that any circle C in the x, j/-plane cuts out ofa domain in which the application of Green's formula is legitimate,and, in addition, that the boundary curve F outside a sufficientlyina domainDThe domainFig. 5.5.2.DR going to oothe conditionlarge circle consists of a single half-ray5.5.2).**Onthe boundaryFfn(5.5.7)(cf.Figure=imposed, i.e. the normal derivative of / vanishes, correspondingto a reflecting barrier. (We could also replace this condition on part,0.
)now write the solutions ofor all, ofby the condition /is=Ff inDwhichsatisfy (5.5.6) in the(5.5.8)fWeform= g+h,in order to formulate the conditions atWhat we havemindoo ina convenient way.to separate the solution into a part hwhich satisfies a radiation condition and a part g which contains,inis* It is natural to consider suchcomplex solutions, since, for example, a plane'wave is obtained by taking f(q, 6) = exp { IQ cos (0 -f- a)}.** Our theorem also holds ifis the more general domain in which the ray Kis replaced at oo by a sector, and the uniqueness proof given below holds withDinsignificant modifications for this case also.WAVES ON SLOPING BEACHES AND PAST OBSTACLES118roughly speaking, the prescribed incoming wave together with anysecondary reflected or refracted waves which also do not satisfy aradiation condition.
More precisely, we require h to satisfy thefollowing radiation condition:dh(5.5.9)lim=+^h,,.0.Here C is taken to be a circle, with its center O (cf. Figure 5.5.2)on the ray R going to infinity, and with radius Q so large that allRobstacle curves except a part oflie in its interior. This conditionthefollowsfromwell-knownSommerfeld radiation condition,clearlywhich requires that(5.5.9),lim p*^oo(V +ih\\OQ->Juniformly in 0, and, incidentally, this is a condition independent ofthe particular point from which Q is measured; we observe that if hbehaves at oo like e~~ iQ /\/Q, i.e.
like an outgoing cylindrical wave,then condition (5.5.9)! is satisfied. We shall make use of the radiationcondition in the form (5.5.9) in much the same way as F. John [J.5]who used it to obtain uniqueness theorems for (5.5.6) in cases otherthan those treated here; his methods were in turn modeled on thoseof Rcllich [R.7].The behavior of the function g at infinity(5.5.10)g~g! +g aisprescribed as follows:at oo,with g l a function that is once for all prescribed,* while g 2 is a functionsatisfying the same radiation condition as /i, i.e.
the condition(5.5.9). (That the behavior of g at oo is fixed only within an additivefunction satisfying the radiation condition is natural and inevitable. )Finally, we prescribe regularity conditions at re-entrant points(such as A, B, C in Figure 5.5.2) of the boundary of D; these conditions are that(5.5.11)f(Q,0)~c vfQ(Q*0)~^>kk<l>Qwith(g,6) polar coordinates centered at the particular singular point,arid c l and c 2 constants. (These conditions on / mean physically thatthe radial velocity component may be infinite at a corner, but notHowthe function g l should be chosenisa matter for later discussion.WATER WAVES114as strongly asitwould befor a source or sink.)At other boundarywerequire continuity of / and its normal derivative.pointscan now state our theorem as follows:WeUniqueness theorem:A solution / of (5.5.6) in D is uniquely determinedboundary condition (5.5.9); 2) admits of a decomformofthe(5.5.8) with h a function satisfying (5.5.9), g apositionfunction behaving as prescribed by (5.5.10) at oo; and 3) satisfies theregularity conditions at the boundary of D.if it1) satisfies thewill be given shortly, but we proceedThe theorem is at first sight somewhathere.implicationsunsatisfactory since it involves the assumption that every solutionThe proof ofto discussthistheoremitsconsidered can be decomposed according to (5.5.8), with g(p, 6) acertain function the behavior of which at oo, in so far as the leadingterm g l(cf.not given a(5.5.10)) in its asymptoticconcerned, isnot difficult in some instances atdevelopmentisHowever, it ison the basis of physical arguments, how the functiong 1 (p, 6) should be defined.
For example, suppose the domain Dconsisted of the exterior of bounded obstacles only. In such a case itseems clear that gi(g, 6) should be defined as the function describingthe incoming wave either as a plane wave from infinity, say, or awave originating from an oscillatory source -since bounded obstaclesgive rise only to reflected and diffracted components which die outat oo and which could be expected to satisfy the radiation condition.Even if there is a ray in the boundary that goes to oo (as was postulatedabove), it still would seem appropriate to take g t (p, 0) as the functiondescribing the incoming wave, provided that it arises from an oscillatory point source,* since such a source would hardly lead to reflctedor refracted secondary waves that would violate the radiation condition. However, if the incoming wave is a plane wave and anobstacle extends to oo, one expects an outgoing reflected wave tooccur which would in general not satisfy the radiation condition;in this case the function g 1 (g, 0) should be taken as the sum of theincoming plane wave and an outgoing reflected wave.
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