J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 93
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The free surface in the physical plane corresponds to the real axis \pof thewethattheandassumeentireoftheflowintheregion2-plane^-plane,way on the lower half of the ^-plane. (Weafunctionshall prove shortly that%(z) satisfying the conditions givenabove would have this property.) In this case the inverse mappingz(%) exists, and we may regard the complex velocity w(z) as an analytic function of # defined in the lower half of the ^-plane. In thisway we are enabled to work with a domain in the <p, y-plane that isfixed in advance instead of with an unknown domain of the x, j/-planc.Levi-Civita goes a step further by introducing a new dependent variaismappedbleco,in a one-to-onereplacingw,\by theso that(cf.=w(12.2.7)(joisrelationUe~l= 6 + ir;co><an analytic function of-(-cpiy.Consequently we have(12.2.2))(12.2.8)IThus rw = Uer|=,argw.is the inclination of the velocity vector.log (\w\/U), whileway as in sec.
10.9 (cf. the equations following (10.9.11))In the samethe boundary condition (12.2.4) can be put in the form(12.2.9)withA'V= AV 3T sin 0,fory>=0,defined byA'(12.2.10)Our problem nowis=3g/f/.to determine an analytic function a>(#)<+ if(<p,in the lower halfplane yand a constant A' inty)suchthatisforcontinuousfor \po>0,0,a)(12.2.9)analytic\piscontinuousfortheOandnonlinearconditionb) vboundaryyas(12.2.9) is satisfied, c) co has the period Uh in 99, d) co(%) ->0(<p 9 ip)<^^yj->oo, c)|a)(%)|fg.Thelasttwo conditions are motivated bythe conditions imposed on w at oo and the condition w 3= 0: the condition d) from (12.2.5) and (12.2.7), while the condition e) is imposedin order to ensure that w is uniformly bounded away from both zeroandinfinity.As weshall see, the condition c) leads to the periodicitycondition (12.2.6) on #.Weproceed to show briefly (again following Levi-Civita) that asolution of the problem we have formulated for co would lead through(12.2.7) to a function w(%) and then to a function %(z) through theLEVI-CI VITA'STHEORY525=differential equation d%(z)/dzw(%) which satisfies all of the conditions formulated above.
The essential items requiring verificationare the periodicity condition and the one-to-one character of thez(%) definedmappingover the halfplaneWe^by< 0.Fromproceed to investigate the second property.we(12.2.7)have+e~ r (cosl/w(%)isin 0).and hence thatfM = i-*WSinceco\\^wfollows that^, itrx ci,the integralt/isbounded away fromconverges. Since both|r|^zero, so thatandJ||^J,itJofollows that &e(I/w)ispositive (webounded away from and oo.so that yv = ^(l/w)and x 99^(7 tobe positive) and= l/woc^iy^9&te(\\w}\ since <#e(I/w)+>9itfollowsa strictly monotonic increasing function of y, andthe mapping z(%) is one-to-one, and(p.
Consequentlyoo whenaddition y ->oo when y ->oo, while # ->->since &e(I/w) is positive and bounded away from zero; thustherefore that yx similarlyinassumeWe have ^ + iyv = i/wisiniithe flowWeismapped ontothe entire halfplaneconsider the periodicity condition next.\p< 0.We have&dz+since a) has the period Uh by assumption.
This implies that z(%Uh)const. This constant is easily seen to have the value h by%(%)=letting-formulaoo in theipdx=IJsincew->Uuniformly when*(*y(Xz( xy>+Uh)oo.Consequently we have+ UA) - x( X = h,+ Uh) - y(x) = 0.)WATER WAVES526We know from=^const,(12.2.8) andJ that the stream lines \pvertical tangents, hence they can be represented in the formy(v), and the last two equations show that they are periodic in||have noy :=x of periodtions a)1h.e)isThe problem of determiningco(%) subject to the conditherefore equivalent to the problem formulated for %(z).2.2b.
Outline of the procedure to be followed in proving the existenceof the functionThe proofof the existence of the analytic function co(%) whichproblem will be carried out as follows. First of all, we ob-solves ourserve that the problem has always the solution co(%) =0, correspondwith undisturbed free surface.ing to the uniform flow w=UWe^exists, and will thenby assuming that a solution a>(%)usethetheoffor co, to derive aassumedpropertiesproceed, throughfunctional equation for the values 0(99, 0) of co on the boundary ip0,shall beginoo < <p < oo. It will then be shown that the functional equationhas a solution co(<p 0) ^= in the form of a complex- valued continuousfunction &>(<p), and this function will be used to determine an analytic9functionco(<p, y})boundaryvalues,inoowhichis<y>< 0,oo<(pthen shown to satisfy<alloo,withco(<p)asof the conditionsa)-e).It will occasionno surprise to remark atthis point that the solutionwe obtainwithwill give a motion in a neighborhood of thehorizontal free surface, i.e.
with an amplitude in auniform flowneighborhoodshould be remarked that the problemin perturbation theory which thus arises involves a bifurcation phenomenon, since the desired solution of the nonlinear problem, once thewave length is fixed, requires that the perturbations take place in theneighborhood of a definite value of the velocity U. In other words, thedesired solution bifurcates from a definite one of the infinitely manypossible flows with uniform velocity which are exact solutions of theof the zero amplitude.
Also,itnonlinear problem.Thedecisive relation in the process just outlinedisthe nonlinearboundary condition (12.2.9). It is convenient to introduce at this pointsome notations which refer to it, to recast it in a different and moreconvenient form, and also to derive a number of consequences whichflow out of it. At the same time, some factors which motivate all thatfollows will be put in evidence.LEVI-CIVITA'SSinceco (<p,we wish0) ofa>,it=>(<p)=and then to introduce the operatorwithe defined-boundary valuesuseful to introduce the notationeo(p, 0)/[]527to concentrate attention on theis(12.2.11)(12.2.12)THEORY-)i(e~*~ sin0)+6(<p)f[co]ii(<p),defined by+ ee~^ sin =F(<p)bye(12.2.13)-V-LThe constant Awill be given an arbitrary but fixed value; the quantitythen be the period of the function co(%). The constant e, andwith it A' through (12.2.13), will be fixed by the solution a)(%) in amanner to be indicated below, and the propagation speed U is thendetermined by the formula (12.2.10).
As can be seen at once, theboundary condition (12.2.9) now takes the form27T/A will(12.2.14)V- A0 -F(<p),y=0.The reasonsfor writing the free surface condition in the form(12.2.14) are as follows: As remarked above, we seek a motion in theneighborhood of a uniform flow, so that co as defined by (12.2.7) shouldsome sense. It would seem reasonable to set up an iterationwhichstarts with that solution (o^y, y) of the problemprocedurewhich results when F(<p), which contains the nonlinear terms in thebe smallinfree surface condition, vanishes identically.
Afterwards the successiveapproximations will be inserted in F(cp) to obtain a sequence of linearproblems whose solutions co k converge to the desired solution of ourproblem.The problem of determining co^y, ^>), when F0, is exactly thediscussedat length inwhichwaslinearthetheoryproblem posed byChapter 3; in fact, if F(q>) vanishes, we know from the discussion inChapter 3 that the only bounded conjugate harmonic functionsther thanT I = 0, in the lower half plane10i (^ V>) r i(V> V0>thewhichsatisfyhomogeneous free surface conditionip <=QI VA0!=are the functions1 (^,y>)r i(<P) V)=a^ sinA<p,a i e^ cos ^Ponce (p is taken to be zero at a crest or trough of the wave.
Thus theboundedness condition at oo and the homogeneous free surface condition lead automatically to waves which are sines or cosines of <p.WATER WAVES528'The 'amplitude" a tof course, arbitrary on account of theis,The corresponding functiongeneity of the problem.We suppose,that|ct^co 1 (^)homothenisnaturally, that the "amplitude" a x is small and hencesame order. The basic parameter in the||also small of theis|iteration procedure will be the quantity a v and the procedure willbe so arranged that the quantity s in (12.2.13) as well as the iterateso x |. It is then easily seen that F((p) will alwaysa) k will be of order|be of order|2ax|Wewhich indicates that such a scheme of iteration,show thatisdoes indeed lead to a sequence CO Awhich converges to the desired solution co for all sufficiently smalland that the solution co fixes a value of e, and hence ofvalues ofreasonable.|A'%shallit|,once for all fixed, in a manner to be explained in a moment.It might be mentioned that it is not difficult to verify that thesince Aiscorresponding motion furnished in the physical plane by Xi( z ) would,be given byup to terms of first order in a x|,|andthis coincideswith what was found in Chapter 3 by a more directprocedure.Iterations, asthe solution a)^we have=ia l e*(indicated, are to be performed, starting withv~ i(p] of the linearizedproblem, with a x regardedas a small parameter.
Thisisthen inserted in the right hand side ofin the half plane(12.2.14); a bounded harmonic function2 (g9, y)is then determined through this nonhomogeneous boundaryipcondition and the corresponding analytic function co 2 (%)7T^22<+with'In order to solve the boundary problem for 2 (and through ithowever, it is necessary to dispose of the parameter e in (12.2.13)appropriately. This comes about because, as we have just seen, thehomogeneous linear boundary value problem for 1 has a non-trivialit.o> 2 ),solution, 6 lF(<p)isa^ sinneeded whichh<p,nonhomogeneous problemthat the integraland hence an orthogonality condition onwill ensure the existence of the solution of theforF(<p) sin2.This conditionishydy should vanish.wellknownIt turnsto beout thatthe value of e so determined really is of the same order as a v Continuing the iterations in this fashion, the result is a sequence offunctionseo n (#),and a sequence of corresponding valuese n of ssuehLE VI -CI VITA'S THEORYthatandare529is to be shown that bothand a number e which solvethe problem, the quantity A' in (12.2.10) being fixed by e = lim e n andthe arbitrarily chosen value of A through (12.2.13).We observe that this whole procedure is in marked contrast with|a) n||en\allof order|axIt|.sequences converge to yield a function CD (/)the method of solution of the problem of the solitary wave givenby Friedrichs and Hyers [F.13] and explained in sec.
10.9; in the lattercase the iteration procedure is quite different and it is carried outwith respect to a parameter which has an entirely different significance from the parameter a x which is used here.The procedure outlined here also differs from the procedure followedby Liechtenstein [L.I 1 J in solving the same problem. Lichtenstcin appliesE. Schmidt's bifurcation theory to an appropriate nonlinear integralequation (essentially the counterpart of the functional equation to beused here).
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