J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 24
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For example,one might consider the case in which the entire #-axis is a reflectingbarrier, as in Figure 5.5.3. In this case one would in an altogethernatural way define gi((), 6) as the sum of the incoming and of thepriori.least to guess,reflectedwaveas follows:*The same statement would doubtlessly hold if the disturbance originateda bounded region, since this case could be treated by making use of a distribu-intion of oscillatory point sources.WAVES ON SLOPING BEACHES AND PAST OBSTACLESg^g, 0) ==(5.5.12)e iQcos(*- a)+eiQ115"with a the angle of incidence of the incoming plane wave. If we werethen to set /g l + h (i.e. we set gg l everywhere) and prescribethat h should satisfy the radiation condition, it is clear that we would==Fig. 5.5.3. Infinite straight line barrier=have a unique solution by taking h0.
Our uniqueness theoremdoes not apply directly here since there are two infinite reflectingrays going to oo, but it could be easily modified so that it wouldapply to this case. Thus we have for the first time, it seems a///Fig. 5.5.4.Sommerfeld'sdiffractionproblemuniqueness theorem for this particularly simple problem of thereflection of a plane wave by a rigid plane. A less trivial example isthe classical Sommerfeld diffraction problem in effect, a specialWATER WAVES116inease of the problem with which our present discussion beganwhich a plane wave coming from infinity at angle a to the #-axis isand diffracted by arigid half-plane barrier along theindicatedinasFigure 5.5.4.
In this case it seemspositivetodefinethefunctiong 1 (g, 6) as follows:plausiblereflectedir-axis,e ie**!0,cos (0-oe)+ecos(e - a),ig cos(0+<^< <rc-a<0<7r+an + a < < 2n.JTOCThis functionof course, discontinuous, corresponding to theis,division of the plane into the regions in which a) the incoming waveand its reflection from the barrier coexist, b) the region in whichonly the wave transmitted past the edge of the barrier exists, andc) the region in the shadow created by the barrier.
Again we would==be inclined to takegg l (cf. (5.5.8) and (5.5.10)) and set/g l + h,with h satisfying the radiation condition. Of course, the functionh(Q 6) in (5.5.8) representing the diffracted wave would then alsobe discontinuous in that case since the sum g lh is everywherecontinuous. It will be seen that the well-known solution given bySommerfeld can be decomposed in this way and that h then satisfiesthe radiation condition.
Our uniqueness theorem will thus be shownto be applicable in at least the important special case of particular9+interest in this section.One might hazard a guess regarding the right way to determinethe function g in all cases involving unbounded domains: it seemshighly plausible that it would always be correctly given by themethods of geometrical optics. By this we mean, from the mathematical point of view, that g would be the lowest order term in an asymptotic expansion of the solution / with respect to the frequency of themotion that is valid for large frequencies; the methods of geometricalwould thus be available for determining g. However, to provea theorem of such generality would seem to be a very difficult tasksince it would probably require some sort of representation for thesolution of wave propagation problems when more or less arbitrarydomains and boundary data are prescribed.Once having proved that the solution of Sommerfeld's diffractionproblem could be decomposed in the way indicated above into the.sum of two discontinuous functions, one of which satisfies theradiation condition, it was observed that the latter fact opens theway to a new solution of the diffraction problem which is entirelyopticsWAVES ON SLOPING BEACHES AND PAST OBSTACLES117elementary, straightforward, and which can be written down in a fewlines.
In other words, once the reluctance to work with discontinuousovercome, the solution of the problem is reduced tosomething quite elementary by comparison with other methods ofsolution. The problem was solved long ago by Sommerfeld [S.12],and afterwards by many others, including Macdonald [M.I], Batemanfunctions[B.5],isCopson[C.4],Schwingerand Karp[S.5],[K.3].We shall first prove the uniqueness theorem. Afterwards, the simplesolution ofSommerfeld's problemjust referred to will be derived;form of a Fourier series. The Fourier seriesnext transformed to furnish a variety of solutions giventhis solutionsolutionbyisisin theincluding the familiar representationnew representations are particularlyintegral representations,given by Sommerfeld.
Theconvenient for the purpose of discussing a number of properties ofthe solution. In particular, two such representations can be used toshow that the function h in the decomposition /h (cf.gl(5.5.13)) satisfies the radiation condition, and that our solution /satisfies the regularity conditions at the origin; thus the solution isshown, by virtue of our uniqueness theorem, to be the only one whichbehaves at oo like g l plus a function satisfying the radiation condition.=TheStokes'phenomenon encounteredtinuity of the functions g l and his+in crossing the lines of discon-also discussed.The uniqueness theorem formulated above is proved in the followingway. Suppose there were two solutions/* given byHe,(5.5.U)We=)*(e 0)+/and/* (cf.
(5.5.8)) with**(e *)introduce the difference ^(^, 0) of these solutions:(5.5.15)*(e,0)=/(M)-/*(M)=8(Q, 0)-g*(Q, 0)+ h( e,0)-A*( e 0),and observe that %(Q, 0) satisfies the radiation condition (5.5.9), byvirtue of the Schwarz inequality, since h, A*, and the difference__gg* a il satisfy it by hypothesis; thus we have2(5.5.16)limim->ooThe complex-valued,Jcfunction #imaginary parts:(5.5.17)and Green's formula+*xis=0.decomposed into= Xi + *XitsrealandWATER WAVES118(5.5.18)applied to # x and # 2 in the domain D* indicated in Figure 5.5.5.The domain JD* is bounded by a circle C so large as to include all ofisthe obstacles initsinterior except R,Fig.
5.5.5.by curves which exclude theThe domain D*prolongation of R into the interior of C, and by curves excludingthe other bounded obstacles. By (5.5.15), # is a solution of (5.5.6)which also clearly satisfies the boundary condition (5.5.7). Since==2Xi an d V # 2# 2 it follows that the integrand of theleft-hand side of (5.5.18) vanishes. Because of the regularity conditionsV 2Xi>boundary points we are permitted to deform the boundary curveand it then follows from the boundaryJat1* into the obstaclecurves,condition (5.5.7) that(5.5.19)since the contributions at the obstacles all vanish.use of the easily verified identityWe now makeWAVES ON SLOPING BEACHES AND PAST OBSTACLES119(5.5.20)to deduce from (5.5.19) the condition2(5.5.21)+J c (l*nlfrom which we obtain,limf->oo * c(5.5.22)2\X\ )d*~jc\Xn+ ix\*ds =view of (5.5.16) and % nin\%\*ds=0,=dx/dg on C:limf->oo ^Fromthe boundary condition (5.5.7), as applied on R, we see that%(Q, 0) can be continued as a periodic function of period 4jr in 6on C; hence # can be represented for all sufficiently large values ofQ by the Fourier series00*(5.5.23)= I^n/nO2(0)cos-,*with A n i 2 (()) the Fourier coefficient, a certain linear combinationof the Bessel functions J w / 2 (g) and F n/2 (g), since % is a solution of(5.5.6).
The Fourier coefficients are given by9(5.5.24)'</,(<?)=1-f 2*n JonOX (e,0) C os-?.dO,2and consequently we havele*^(5.5.25)dOIt follows atonce from (5.5.22) that the Fourier coefficients behavefor large Q as follows:(5.5.26)lim e *A nf2(e)=0.>00YSince the Bessel functions / n / 2 (o) andn j 2 ((>)1/V(? i* follows that all of the coefficientsall^4behave atn/2 (g)oo likemust vanish.Consequently % vanishes identically outside a sufficiently large circle,hence it vanishes* throughout its domain of definition, and the* This could beis now seen to satisfyof Fig. 5.5.5.proved in standard fashion since #homogeneous boundary conditions in the domainD*WATER WAVES120uniqueness theoremproofismuchisproved.
As was stated above, this uniquenesslike that of Rellich[R.7].easily in such aThe above proof can be modifiedwayas to applyto a region with a sector, rather than a ray, cut out at oo. The onlydifference is that the Fourier series for #(g, 0) would then not havethe period 4jr and that the Bessel functions involved would not beof index n/2.Once it has become clear that the decomposition of the solutioninto the sum of the two discontinuous functions g(p, 6) and A(g, 6)a procedure that is really natural and suitable forthis problem, one is then led to the idea that such a decompositionmight be explicitly used in such a way as to determine the solutionof the original problem ((cf. (5.5.8)) in a direct and straightforwardway. Our next purpose is to carry out such a procedure.defined earlierWeisset (cf.
Figure (5.5.4)(5.5.27)and equation(5.5.13)):/(0,with g(p, 0) defined bycos (0-ooie Q*<?_|_cos(0+oc)^(5.5.28)9n0,In addition< <tta+ < < 2n.ocwe have(5.5.29)fand wealso require(5.5.30)o7t-QL<0-*)lim0-^00<\/Q=(^-we=for0,+ <*) === 2rcuniformly in0,/since the validity of the radiation condition in this strong form canbe verified in the end.Theby developing /(p, 0) into aand determining the coefficients of theseries through use of the radiation condition in the strong form(5.5.30); afterwards, the series can easily be summed to yield aconvenient integral representation of the solution.
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