J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 20
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Hence y is a solution of (5.3.2) in the case Aby use of the aboveB(x).and g(y) =satisfied by <psatisfies (5.3.14). Since (5.3.12) holdsthat the free surface condition (5.3.3)the fact that dH V (ikr)/dyfor y(=is=^itinfollowsview of0.We have still to show that a solution B(x) of (5.3.14)can be chosenWAVES ON SLOPING BEACHES AND PAST OBSTACLESso thatr.(p=Aie* fJof the integralsince tx=in (5.3.13) yields the following forV#>isl)(ikVx*+t=should be recalled that the upper limit ynegative; thus the integrand has a singularity foris included in the interval of integration and(ikr) is singular for r--Aie vfJq>:0.
ItV* |-2A^[HM (ikVx*OX00We0.=show that lim dyjdxshallx->0=provided that #,(0)(jW0,QCprovided that x11^xby partsintegration5.3.17)and that g? has the desired behavior forActually, these two things go hand in hand. Anfor(p xlarge values of910.We>have, for xand y<0:+ t*)]dtdxThe second term on thezero as x ->right hand side is readily seen to approachterm can be written as the product of x andbounded for y < 0. For the same reason it is clearsince thisa factor whichisthat the only contribution furnished by the integral in the limitsince the factor x mayas x - >arises from a neighborhood of tbe taken outside of the integrallim fV< Aa->oJeOXThe function ill^r=<sign.We therefore[i//[i//>(ikVx*+72consider the limite)]d*,>0.(ikr)has the following development valid near(ikr)J=0:HI^ov-n+[JQ (ikr)/ logp(r)~\Lg r -r FV;jovin which p(r) represents a convergent power series containing onlyeven powers of r, and J is the regular Bessel function with thefollowing developmentJ(ikr)=i+.-L- +....It follows thatox[iHM(ikr)]= -|- J Q (ikr)n\r 2+J'o(ikr)12-logr +xg(r) 1rJWATER WAVES92=which g(r)(l/r)dp/dr is bounded as o?->0 since j/<0.
The contribution of our integral in the limit is therefore easily seen to beingiven by2 f~*-'i-limByuintroducingthe limit=we maylim-*-new</# asxB(o?)equation (5.3.14)SinceH^ (ikr)<p- f-=-= -----k-by=A[*<pi(x,y)2ie*(ikVx*with^f 2an arbitrary-2.and thedifferentialsinVl -2itfollows that theB(x) defined by (5.3.19) behaves(5.3.17) withtherefore givenand passing to__o AA(5.3.20)dttdies out exponentially as r -> oogiven byis-dt.2A.lfat oo like e* sin [(1k*) *x].solution 9? 2 of our problem which(5.3.15))00satisfies this conditionVla?---provided thatisB(x)(5.3.19)solutionwhich2= -,dt^(0)(5.3.18)fiintegration variableIt therefore follows that lim d(p/dxThe function-limwrite2 f- e-f~2=d*+real constant.).^.=Vl-sin(cf.q) l___n^y-2i/out of phase withisVl -k2Standing wave solutions"I__k2 x,J2arethen given by(5.3.20)'<Z>x ii\y)( V*2\>fcos kz])1\..,f[sin kz\taking appropriate values of k progressing waves tending at ooany arbitrary plane wave solution for water of infinite depth canbe obtained by forming proper linear combinations of solutions of theBytotype (5.3.15)' and (5.3.20)'.
For a progressing wave traveling towardshore, for example, we may writeWAVES ON SLOPING BEACHES AND PAST OBSTACLES(5.3.210(x,)y, z; t)-A/1=Ax, y) cos kzk293"1q> 2 (x,-\y) sin kzcos at['Jfcs-A\pinwhichAlandA2"199 2 (#,in (5.3.15)andy)cos kz sin atboth taken equallike Ae* cos (Vlk 2x+kz+at)(5.3.20) areto ^. The solution (5.3.21) behaves at ooas one can readily verify by making use of the asymptotic behaviorof q>i(x, y) and q> 2 (x, y)* and it is the only such solution since <p^andare uniquely determined.I has a certain interest. It corresponds tospecial case kwaves which at oo have their crests at right angles to the shore.9? 2=TheOnereadily sees from (5.3.15) and (5.3.20) that as k ->gressing wave solution (5.3.21) tends to0(y,(5.3.22)thatis,the progressingz; t)=waveAe vcos (z1the pro-+ at)solution for this caseisindependentfree of a singularity at the origin, and the curves of constanta?,phase are straight lines at right angles to the shore line all propertiesofisthat are to be expected.Theprogressingwavesolution (5.3.21)was studied numericallyKO-22KFig.
5.8.1. Standingwave.)Jsolution for a vertical cliff (with crests at an angleof 30 to shore)* We remark once more that the original space and time variables can bereintroduced simply by replacingy, z by ma?, my, mz and k by kjm.,WATER WAVES94for k=1/2,which the wavefor the case ini.e.to a straight line inclined at 30crests tend at ooThe functionto the shore line.is plotted in Figure 5.3.1.
With the aid of these values the<p%(%9 0)were calculated and are given in Figure 5.3.2. Thesecontours forare also essentially contour lines for the free surface elevationin accordance with theformularj=<P t v .The water.\77,surfacegispair of successive "nodes" of <Z>, that is, curves0.
These curves go into the 2-axis (the shore line)shown between a=for whichunder zero angle, as do all other contour lines. Thisfrom their equation (cf. (5.3.21) with at = n/2)(5.3.23)Sincesin kz<p 20) cos kz(pi(x,-> oo as+integer,isall=kzwhile 9^ remains bounded,# ->const.it isclear thaton any such curve. That theat the points z = 27W, n anthes-axistotangentmust approach zerocontours are#, 0) sinseen at onceisas x ->also readily seen.Itisinteresting to observe that the121086^4-0912.-09l'2\m--4IFig.
5.3.2. Level lines for a23wave approaching a40.0xvertical cliff atan angleheight of the wave crest is lower at some points near to the cliff thanit is at oo. It may be that the wave crest is a ridge with a numberof saddle points.WAVES ON SLOPING BEACHES AND PAST OBSTACLESIt should95be pointed out that we are no more able to decide in thepresent case than we were in the two-dimensional cases whetherthe waves are reflected back to infinity from the shore, and if so towhat extent. Our numerical solution was obtained on the assumptionthat no reflection takes place, which is probably not well justifiedfor the case of a vertical cliff, but would be for a beach of small slope.5.4.Waves onsloping beaches.
General caseWe discuss here the most general case of periodic waves on slopingbeaches which behave at oo like an arbitrary progressing wave inparticular, a wave with crests at an arbitrary angle to the shore lineandthisbeach sloping at any angle. As has been mentioned earlier,problem was first solved by Peters and Roseau (cf. the remarksfor ain section 5.1).We seek a harmonic function 0(x, y, z; t) of the form exp'9>(#9y) n the region indicatediFig. 5.4.1. Sloping{i(at+kz)}in cross section in Figure 5.4.1.Atbeach of arbitrary angleshould behave like exp {i(at+kz+vix)} exp {a 2y/g}with k and a arbitrary. The function <p(x, y) is not a harmonic function, but satisfies, as one readily sees, the differential equationoo the function9^*(5.4.1)+ p - *V =>the free surface condition(5.4.2)cp ym<p=0,y=0,m=a2,WATER WAVES96and the condition at the bottom*=<p n(5.4.3)y0,=x tanco.we have done before) the new dimensionless=xmx, y l = my, a x = a/w, A?! = k/m the conditions ofquantities lthe problem for <p(x, y) can be put in the formByintroducing (as<p 9X(5.4.1)!+<p yy(5.4.2)!y>=^0,- = 0,v= 0,ny =<p(5.4.3)!k*<p9?y=fc^1,0,x tano>dropping subscripts.
Since we require 99(0?, t/) to behave likeexp {IK^XI + t/J at oo, it follows from (5.4.1) thatm 2 & 2 = and hence that a? A;* 1. Thus fc in (5.4.1)!a2^ k ^ 1,(really it is A^) is, as indicated, restricted to the rangeaftere i&x emy++=and this fact is of importance in what follows.** Finally, we knowfrom past experience that a singularity must be permitted at theorigin.
(In the problems treated earlier in this chapter we haveprescribed only boundedness conditions at oo in a way which led toa statement concerning the uniqueness of the solution. In the presentcase we do not obtain a similar uniqueness theorem in fact, as hasbeen pointed out by Ursell [U.7, 8], Stokes showed that there existmotions different from the state of rest and which die out at oo.For these motions, however, the quantity k is larger than unity).We seek functions (p(x, y) satisfying the above conditions as thereal or the imaginary part of a complex function f(z, z) which isanalytic in each of the variables z = x + iy and its conjugatez = xiy.
In the two-dimensional cases, it was sufficient to considerfunctionsf(z) of one complex variable, but in the presentanalyticcase it is necessary to take more general functions since q>(x y) isnot a harmonic function. Note that we now use the variable z in adifferent sense than above, where it is one of the space variables;no confusion should result since the space variable z hardly occursagain in the discussion to follow. It is useful to calculate some of thederivatives of such functions with respect to x and y; we have,9clearly:* Peters[P.6] solves the problem when the condition (5.4.3) is replaced bythe more general mixed boundary condition <p n + a<p0, a = const.** Involved in this remark is thethat the derivatives of the solutionassumptionsame as the derivatives of its asymptotic development;indeed the case, as we could verify on the basis of our final represen-btehave asymptotically thebut thisistation of the solution.WAVES ON SLOPING BEACHES AND PAST OBSTACLES=/**x +/!fx97=/, +/z,= i(h - /i),fy= 4/ZABB + fyyConsequently our differential equation(5.4.1)!can be replaced bythe differential equationsince the real or the imaginary part of any solution of it is clearlya solution of (5.4.1)!.Among the solutions of the last equation are the following simpleby separating thespecial solutions (obtained, for example,= f^z)in writing //,()):f(z, z)whenwhich,KkI+iz\I4/!kz 2C& + T=f,ix11|)4/\{=const.,of the formisk2\([exp]cfor example,\yvariables\> ,andthisisa solution ofjwhich has the proper behavior at oo, at least.
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