J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 18
Текст из файла (страница 18)
It turns out that the resulting solution behaves at oo like* Far lessstringent conditions at the boundaries could be prescribed, sinceanalytic continuations over the boundaries can easily be obtained explicitly inthe present ease.WATER WAVES80knownwavesin water having infinite depth everydeterminedby prescribing the amplitudeuniquelytheof the wave at oo together withassumption that it should be,thesolutions forwhere and thatit isan incoming wave.say,WeallToproceed to carry out this program, without however givingof the details (which can be found in the author's paper [S.18]).begin with, the ordinary differential equation for f(z) and theoperators(5.2.10)Ltare givenby= L! LL(D)fL82...L 2nfnwith the complex constants(5.2.11)a*=e~ indefined byv.
k(L + 1}fc,One observes that L(D) and L 2n (D)given in (5.2.8) and (5.2.9). It=1, 2,...,2n.coincide with the definitionsin fact, not difficult to verify thatis,&eF(z)(5.2.12)D - I)/=on both boundaries of the sector, by making use of the propertiesof the numbers a* and of the fact that Ste L^D) and Ste L 2n (D)fvanish on the bottom and the free surface, respectively, by virtueof the boundary conditions (5.2.7) and (5.2.6).So far we have not prescribed conditions on f(z) at oo and at theorigin, and we now proceed to do so.
At the origin we assume, inaccordance with the remarks made in section 5.1 and the discussionin the last section of the preceding chapter, that f(z) has at most a<kklogarithmic singularity; we interpret this to mean that d f(z)/dzkwithin a neighborhood of the origin for k., 2/i,1, 2,k /\ zMM\|\kcertain constants.withdkAt=koowerequire that==<p.ffle f(z)for k., 2n be uniformly1, 2,f(z)/dzz -> oo in the sector.
(These conditions could be||...togetherbounded whenweakened con-siderably, but they are convenient and are satisfied by the solutionswe obtain. ) In other words, although we expect the solutions of ourproblem to behave atoo in accordance with (5.2.4) it is not necessaryto prescribe the behavior at oo so precisely since the boundednessconditions yield solutions having this property automatically. Oncethese conditions on f(z) have been prescribed we see that the functionF(z) defined by (5.2.10) has the following properties: 1)|F(z)\isWAVES ON SLOPING BEACHES AND PAST OBSTACLESuniformly bounded in the sector, and 2)neighborhood of the origin.|=F(z)\0(1 jz812n)in thehave already observed that 3te F(z) =on both boundariesof the sector and that F(z) can therefore be continued as a singlevalued function into the whole plane, except the origin, by thereflection process.
Here we make decisive use of the assumptionthat co, the angle of the sector, is n/2n with n an integer. Since theboundedness properties of F(z) at oo and the origin are preservedin the reflection process, it is clear from well-known results concerninganalytic functions that F(z) is an analytic function over the wholeplane having a pole of order at most 2n at the origin. Since in addition the real part of F(z) vanishes on all rays z = r exp{ikn/2n},Wek1, 2,...,4tt,itfollows that F(z)F(z)(5.2.13)withA 2nan arbitraryisgiven uniquely by=Z 2nreal constantwhichmaythe value zero.
Thus the complex potential f(z)in particularwehaveseek satisfies thedifferential equation(5.2.14)faDfoJ) -Our problemis1) ... (a 2n . 1 D)(a 2n Z)-reduced to finding a solutionI)/-^.f(z) of this differentialequation which satisfies all of the conditions imposed on f(z). Fromthe discussion of section 5.1 we expect to find two solutions f^z)of our problem which behave differently at the origin andat oo; at the origin, in particular, we expect to find one solution,say fi(z), to be bounded and the other, / 2 (s), to have a logarithmicandf 2 (z)singularity.the solution of (5.2.14) which oneobtains by taking for the real constant A 2n the value zero, while0. In other words the solution of the nonf 2 (z) results for A 2nTheregular solution f^z)is^homogeneous equation contains the desired singularity at theOneinorigin.finds for f^z) the solutionwhich the constantsckandft kare the following complex numbers:WATER WAVES82n + l--k\\1} cot= exp {ml ---n.(5.2.16)*\=The constants4\2/J(k-l-l)n--2ncot2n...cot-2nA:cn= 2, 3, ...,n.c k areobtained by adjusting the arbitrary constantsin the solution of (5.2.14) so that theat the free surface,2nand the bottom areboundary conditions onsatisfied; thatf(z)such a resultcan be achieved by choosing a finite number of constants is at firstsight rather startling, but it must be possible if it is true that a function f(z) having the postulated properties exists since such a functionmust satisfy the differential equation (5.2.14).
The calculation of theconstants c k is straightforward, but not entirely trivial. The functionuniquely given by (5.2.15) within a real multiplying factor.-> oo in the sector, all terms clearly die out exponentiallysince all (i k sn, which is c n exp {iz}except the term for kEventermn diestheforkrealhaveparts.except f} nnegativeout exponentially except along lines parallel to the real axis. (Thef^z)isAsz||=value of c n,by the way,9isexp-{each other for kof fi(z):behaviorasymptoticin (5.2.16) cancelThel)/4} since the cotangentsin(n=n.)This term thus yields thesolution / 2 (*) of the nonhomogeneous equation (5.2.14) whichthe boundary conditions is as follows:satisfies(5.2.18)f 2 (z)=nra**-lwhich thefor the case in^kLr**Pk **'J00dt-nme****Jtreal constantA 2nisset equal to one.Theconstants {i k are defined in (5.2.16); and the constants a k are defined byak(5.2.19)=c k /{(n-I)\Vn}>is, they are a fixed multiple (for given n) of the constants c kdefined in (5.2.16).
The constants a k9 like the c k are uniquely deter-that,real multiplying factor. The path of integration forall integrals in (5.2.18) is indicated in Figure 5.2.2. That the pointsin theizft k lie in the lower half of the complex plane (as indicatedmined within afigure) canfact that zbe seen from our definition of the constantsisrestricted to the sectorjr/2n(i k^ arg 2^0.and theWAVES ON SLOPING BEACHES AND PAST OBSTACLESPath of integration in f-planeFig.
5.2.2.The behavior83depends on the behaviornot hard to show for example,by the procedure used in arriving at the result given by (4.3.15)that these functions behave asymptoticallyin the preceding chapterofatf 2 (z)oo of courseof the functions in (5.2.18).
Itisas follows:fi*(* ke(5.2.20)>*i*k\IJ*%/tooOncethatthis factf 2 (z)it(()(VI/}dt~\[2mIestablishedis0,9/1\.o[~]itisJm (iz0 k ^0,,clear)0.from (5.2.19) and (5.2.18)behaves asymptotically as follows:n(5.2.21),(nsince the term for k>=n dominatesallothers(cf.(5.2.20))andComparison of (5.2.21) with (5.2.17) showsthat the real parts of f^z) and f 2 (z) would be 90 out of phase at oo.That the derivatives of / 2 (z) behave asymptotically in the sameSte(izp k )in this case.fashion as f 2 (z) itself is easily seen, since the only terms in the derivatives of (5.2.18) of a type different from those in (5.2.18) itself areof the form b k /z k9k an integer^1.Finally,itisclear that f 2 (z)Hence f^z) andhas a logarithmic singularity at the origin.f 2 (z)satisfy all requirements.
Just as in the 90 case (cf. the last sectionWATER WAVES84=+of the preceding chapter) it is now clear that f(z)b^^z)b%f 2 (z),with bi and 6 2 any real constants, yields all standing wave solutionsof our problem.The relations(5.2.17)and(5.2.21of the real potential functionsp^x, y)(5.2.22)= 9keji~~(p^~\T~rt-(n(5.2.23)<p 2 (x,y)=3tef 2~it isobserved that c n=e " cosx\exp-+/-.7-7-** sinIJiynn74\7T-(nwhenyield for the asymptotic behaviorthe relations:q> 2)and*W(a?+l)/4). It_14is\n\-\\//nowpossibleto construct either standing wave or progressing wave solutions whichbehave at oo like the known solutions for steady progressing wavesCT(/&{water which is everywhere infinite in depth.
In particular weobserve that it makes sense to speak of the wave length at oo in ourcases and that the relation between wave length and frequencysatisfies asymptotically the relation which holds everywhere in waterinof infinite depth. For this, it is only necessary to reintroduce theoriginal space variables by replacing x and y by mx and my, witho 2 lg (cf. (5.2.3)), and to take note of (5.2.22) and (5.2.23).m=Finally,welike e y cos (xwritedown a+ + a),ti.e.t) which behaves at ooa steady progressing wave movingsolution 0(x, y\liketoward shore:(5.2.24) 0(x, y;t)=A[cpi(x, y) cosAs our discussion shows,(tthis solution+ a)is<p 2 (x,y) sin(t+ a)].uniquely determined as soonprescribed at oo (i.e.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
