J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 25
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That such aprocess will be successful can be seen very easily: The Fourier seriesfor /(g, 0) will, on account of the boundary condition (5.5.29) and thefact that / is a solution of the reduced wave equation, be of the formdesired solution will be foundFourier series in*for fixed p,WAVES ON SLOPING BEACHES AND PAST OBSTACLES121]c nJ n/2 (0)cos n0/2; the Fourier coefficients for g(g, 6) as defined(5.5.28) are given in terms of integrals of the formby&2*12fln=\Jo==forsince this function also satisfies the condition ge0, 2n.Since / n / 2 ((?)> and its derivatives as well, behave like l/\/(? f r largevalues of Q and the integrals I nby a straightforward applicationof the method of stationary phase, for example,also behave inthis way, it is clear that the limit relation (5.5.30) when used inconnection with (5.5.27) will serve to determine the coefficients c nWe proceed to carry out this program. The finite Fourier transform./of/ isintroduced by the formula(5.5.31)n)J(Q,2*i/*=Aj/5f(o, Q)cosdO.Sincefor 6/9we2jt0,find for fm the transformJ^-^-^J,4(5.5.32)by using two integrations by(5.5.33)Q*feefollows thatit.2Jo/ isQQand solutions ofefQis=0,1=+ foe +Q2fa solution ofa solution of&(5.5.34)+parts.
Since /this+QfQ+g2~0,equation are given byJ(e.n)(5.5.35)=a nJ nl2 ( Q ).(The Bcsscl functions Y n 2 (()) of the second kind are not introducedbecause they are singular at the origin; the solution we want is inany case obtained without their use.)jThe transformof g(p, 0)g(0, n)(5.5.36)is,=of course, given/* 2*ig( e , 6) cosJoand we have,inview ofC 2nJoor,also:(5.5.8),nQh(Q, 6) cos(5.5.37)dO2byA|/5=d0,2the relation:a nJ nl2 (Q)-f 2*n6g(g 6) cosdO9Jo2WATER WAVES122We=h( e , n)(5.5.88)-a nJ nl2 (g)g( e , n).+0to both sidesmust next apply the operation yp (9/^eand then make the passage to the limit, with the result*of (5.5.87)=(5.5.89)+lim ^/Q(|*->\OQSince the functionsi) \a n/ LJ nl2(e)-*^( *g( e 6) cos2 d6\.JOJ,behave asymptotically as follows:t/ n 2 (g)/2and since these asymptotic expansions can beIL +(5.5.40)i)J n/2(Q)differentiated,we have~ I/Aas an easy calculation shows.
The behavior of the integral over gcan be found easily by the well-known method of stationary phase,which (cf. Ch. 6.8) states thatfJawhich aQ<Pa simple zero of the derivative <p'(Q) in the rangea < 6 < ft, and the ambiguous sign in the exponential is to be takenthe same as the sign of ^"(a). In the present case, in which g(p, 0)is defined by (5.5.28) one sees at once that there are three pointsinis= a,of stationary phase, i.e.
at 6the three contributions only the=nfirst, i.e.+ a. Ofthe contribution at 6 = a,**a,furnishes a non-vanishing contribution for Q -> ooVp(9/9p(5.5.41)Use officients+ i)isapplied tor+A(I/Jowe(5.5.40)anandit;one finds, ing(e 6) cos,nandwhen the operatorfact:^ dQ ~ 2 V^ cos2fe2X3.(5.5.41) in (5.5.39) furnishes, finally, the coef-:(5.5.42)The Fourieran=2n cose?*T.2series for /(g, 0) is* It should be noted that theargument goes through if the radiation conditionused in the weak form.** This hasphysical significance, since it says that only the incoming wave iseffective in determining the Fourier coefficients of the solution.isWAVES ON SLOPING BEACHES AND PAST OBSTACLES/(<?>or,from(5.5.35)0)=andaw/(g, o)+/( e ,TTn) cos~i1232(5.5.42),cos-.It is not difficult to sum the series for /(p, 0).
If we use the representation (for a derivation, see Courant-Hilbert [C.10, p. 413])(5.5.43)wherewe,( e )= J! f *-lH)2^i Jpcidc,Pis the path in the complex-plane shown in Figure 5.5.6,find that /(p, 0) can be expressed as the integral of the sum of-planeFig. 5.5.6.The pathPin the f-planea constant plus four geometric series. The summation of the geometricseries and a little algebra yields, finally, a solution in the form(5.5.44)ftp* 0)=1fe~Wt)--Sjti J'3wp37i"^Weproceed to analyze the solution (5.5.44) of our problem withrespect to its behavior at oo and the origin, and we will show thatWATER WAVES124the conditions needed for the validity of the uniqueness theoremproved above are satisfied.
We will also transform it into the solutiongiven by Sommerfeld (cf. equation (5.5.47)). Not all of the detailsof these calculations will be given: they can be found in the paperby Peters and Stoker [P.19].Ifweset/(g, 0)(5.5.45)and definesee)+ /( e0+ot),,0-<x)by- 1L=/(g,x)(5.5.46)we/(g,- J( ei3;ie 2on comparison with(5.5.44) thatwe wishcorrectly as the solution(5.5.45) defines /(p, 0)to investigate.obtain the solution in the form given by Sommerfeld.denominators of the fractions in square bracketsin (5.5.46) are rationalized, and the fractions combined to yieldLet usTofirstthis end, theQf1J(e>*)= 7-;3*3rre~2\ ~C/C2+22*C114+2tS/4cos-fLCIf----(c--)r.2mdl2^....2V c/*.(f2+{,Jpsee ^^ a * tta last equation+cos(2ivK\!;I/IT=^r^eisis~M^4 ^-^ cos 2readily found:inv=Vtoioo ,,-<A(l4-cosx)/oo/fcoijf2Jeforequivalent to./# of the non-homogeneous equation whichvanishes as Q -> oo23-and the well-known trigonometric formulas(5.5.43)J-HZ(Q) wesolutionLJ>J-? i-.-^:--^^A1xt'~2 f^i/2(?)>+;2^008-we usecos-y1verify readily that / satisfies the differential equationdlIf-,.3nFi~r^+ 2zC cos xi4m JpOne can thenl\Cin general1WAVES ON SLOPING BEACHES AND PAST OBSTACLESThus for / the appropriatemust besolution of the differential equation1=6+*#newvariable of integration z in the expression for2z 2 and use of the formulathrough the relation 2A cos x/2Introduction of aIN125==e~ iz2 dz/,_4\/7t e-00leads with no difficulty to the expressionin/I(Q, x)(5.5.47)and=this leads, in conjunction6*flcos xe *x/V 2c cos 2-ae~ iz dzwith (5.5.45), to Sommerfeld's solution.derive the asymptotic behavior of 7(p, x) as Q -> oo we proceeda little differently.
The fractions in the square brackets in (5.5.46)Toare combined, and some algebraic manipulation(QtA new'~isapplied, to yield-uri4^ J P _1_(C, /2integration variable Anow introduced by the equationis2v/2V/2with the result-_V2 ^AP4cos 2transformed into the path L shown inFig. 5.5.7, as one readily can see. The path L leaves the circle ofradius \/2 centered at the origin on its left. This representation ofthe function 7(g, K) is obviously a good deal simpler than thatfurnished by (5.5.46), and it is quite advantageous in studying theproperties of the solution: for one thing, the plane waves at ooThe path(cf.Fig.
5.5.6)iscan be obtained as the residues at the poles3*<(5.5.49)A= V2 c *-=-aQcosiWATER WAVES126In fact,mayiforis a pole in the upper half of the A-plane (and therenot be, depending on the values of both 6 and a) onetheremayX-planeFig. 5.5.7.The path Lhas, after deformation of thepathLin the A-planeoveritand into thereal axis,for I(Q, K) the result:(5.5.50)I(o,9c)=*w- e".,e(*\iQVv 2 e4cos ~2piq cos x"*Ifx=7iAthe only case in which thereA/ 2 eis4cos 2a singularity on the realz-planeFig. 5.5.8.in the z-planea pole at Awe assume that the path of integrationdeformed near the origin into the upper half-plane. It is convenientaxis,is=The path Ci.e.WAVES ON SLOPING BEACHES AND PAST OBSTACLES=to introduce the variable z'-~-gA2in the integral,"'127with the resultC ~* dZ/($,*)=(5.5.51)=cos x/2, and C the path of integration shownin Fig.
5.5.8 For large values of g, and assuming A x0, the squarebracket in the integrand can be developed in powers of (Z/Q ) 1/2 , andwith A x\/ 23 4e' *'^we maywritee-*dze~ z dzCIwe may allow e -> (see Fig. 5.5.8) and hence thecandeformedinto the two banks of the slit along thebepathreal axis; each of the terms in the square brackets then can beevaluated in terms of the /^-function (cf., for example, MacRobert[M.2], p. 143). It is thus clear that for A x ^ 0, the leading term inthe asymptotic expansion of the integral in (5.5.51) behaves liken fact, we have for /(g, ):l/\/PIt is clear thatC*(5.5.52)Since /'(^)= V^ an d7(p, x)(5.5.53)V^ ^^x~e**cosx4cos ~wc have- --_-4-A.zVzno^ cos -2Of course,this holds only ifHlies inthe rangeoccurs in the upper half of the A-plane onlyto A x=0.WemustSince K =(cf.
(5.5.50)).^ ^ < n since a polewhencos */2ispositivealso exclude the value KT a,wen, correspondingsee that the valuesn a cor-=n, and these values of 0, inrespond to the exceptional value Kinthe physical plane acrosslinesturn, are those which yield thewhich our solution / behaves discontinuously at oo. (Cf. Fig. 5.5.4).WATER WAVES128The discussion of the last paragraph yields the result, in conjunctionwith equation (5.5.45) which defines our solution in terms of /(p, H):/(0, 6 )(5.5.54 )~ e*cos2V2no*andfor large Q<rangea<-cos-2V2no*2<cos-~2<for angles 6 such thatna, and a in then\ only in this case are there poles of both of theintegrals in (5.5.45) in theTheupper halfplane.discussion of the behavior of the solution in other sectors ofthe physical plane and along the exceptional lines can be carried outi371 / 4cos ( +oc/2 ) 0,in the same way as above.
For example, if A+\/2nand hencea, it follows that there is only one pole in the===upper halfplane and our solution/(g,na)isgiven by5.5.48),(cf.(5.5.49)):e -iQ f e -Q&+ ?_ML_27riJ Lor also(cf.f(g9(5.5.51)and Fig.A5.5.8) by:e~ z dzn-*) =e -iQ/e -,dz.AnijcThe asymptotic behavior ofwayas above; the result(5.5.55)f(Q,n- a)/zcan now be determinedin the2/-_-_if!2 V2jrp* cos2ocJtthe second term resulting from the pole at the origin.In this fashion the behavior of /(g, 6) for large values of Qmined, and leads tosameis2isdeter-WAVES ON SLOPING BEACHES AND PAST OBSTACLESe iQ(5.5.56)cos (<>-a)tg cos (0-a)/(ftThisis,129,ft,fta<<ft+a < <+a2ft.of course, a verification of one of the conditions imposedat oo.
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