J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 40
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7.0.1 and equations (7.0.3) and (7.0.4)) arev+ gr, +11(7.4.1)<p t+U<p x+U2 -(7.4.2)ty+Ur, x-<p y=0,=0,*Q=>for all times t0. The quantity pp(x;ythe pressure prescribed on the free surface. At the bottom ywe have, of course, the conditionto be satisfied at=is(7.4.3)At the<p yuniform flow,(7.4.4)Fromt^0.=q>(x, y;(7.4.1),t0,which we= ^(x; 0) = p(x; 0) = 0.assume to hold at = 0, we0)tthus have thecondition(7.4.5)Finally,(7.4.6)(f> twehwe suppose the flow to be the undisturbedand hence we prescribe the initial conditions:instantinitial=t)(x, y;0)=0.prescribe the surface pressurep= p( x)9t>0.pfort>0:TWO-DIMENSIONAL WAVES211(The surface pressure is thus constant in time.) At oo we make noassumptions other than boundedness assumptions.
We shall notformulate these boundedness conditions explicitly: instead, they areused implicitly in what follows because of the fact that Fourier transforms in x foroo < x < oo are applied to q> and p and theirOfderivatives.course, this means that these quantities must not onlybe bounded but also must tend to zero at oo, and this seems reasonablesince the initial conditions leave the water undisturbed at oo.Wehave, therefore, the problem of finding the surface elevationhand the velocity potential <p(x, y; t) in the stripy 5^ 0,^r)(x; t)satisfy the conditions (7.4.1) to (7.4.6).
Weoursolutionoftheproblem by eliminating the surface elevationbeginfrom the first two boundary conditions to obtain:77oo<x <(7.4.7)<p ttoo,which+ t/Vxx + W<pxt+g<p y=-- pat yx,=0.QThe Fourier transform with respect to x is now appliedto yield-(7.4.8)s*y=<tp(7.4.9)= A (*>v(*> y>')=(p(s,inview of+Acosh s (yy; t) ofa function to be determined. The transformwith the result:__+ 2isU<p + g<p v -Viit_UWcp=<p.From(7.4.8)must)>t)_(7.4.10)+<p vv =0,where the bar over refers to the transform ip= for y = h; hence ip,(7.4.3) we have q> ybe of the formto (7.4.7)(p xxsec. 6.3):(cf.with A(s;tois-- _p,next appliedat y=0,eandthis yields,(7.4.11)Here p(s)t=from(7.4.9) forAU + 2isUA +tisfe*y== 0,tanh shthe differential equation-sU2of course the transform of p(x).for A($;t)in conjunctionwe have fromwith(7.4.12)The function A(s;t)(7.4.4)and2]A=Asinitialis=At(si0)conditions at(7.4.5) the conditions (again(7.4.9)):A(a\ 0)isUpQ cosh sh=then easily found;0.it isWATER WAVES212s2(7.4.13)A(s;t)U2tanh shgseUVpjQ cosh sh2 Vgs tanh sh=2 A/gs tanhThesolution y(x> y;inverting<p(s,t)of our problemw(x, y;The path of5/i-Vgs tanh(sU-shVVgs tanhst7of courseis+itnowobtained byy; t):I(7.4.14)sUe1+ Vgs tanh sh)-it (sUt)f00==V%nintegrationisA(s;t)cosh s(y+ h) elsxds.J -oothe real axis.Onefinds easily that thevintegrand behaves for large s like e^ /s since the denominators ofthe terms in the square brackets in (7.4.13) behave like s 2 the ratiovcosh s(yA)/cosh sh behaves like 0W for large s, and p(s) tends to9,+zero at oo in general.
Since y is negative (cf. Fig. 7.0.1) it is clear thatthe integral converges uniformly. (We omit a discussion of the be-=havior on the free surface corresponding to y0, although such adiscussion would not present any real difficulties.) Upon examiningthe function A(s; t) in (7.4.13) it might seem that it has singularitiesat zeros of the denominators (and such zeros can occur, as we shall see)but in reality one can easily verify that the function has no singulariwhen the three terms in the square brackets are taken togetherasone might also put it, any singularities in the individual termsor,cancel each other.
Thus the solution given by (7.4.14) is a regularh ^ y <for all time t, or, inharmonic function in the stripother words, a motion exists no matter what values are given to theparameters. In addition, the fact that the integral exists ensures thatalso its derivatives) tends to zero for any given time when9? (andties-> oothis is the content of the so-called Riemann-Lebesgue\x\theorem. This means that the amplitude of the disturbance dies outat infinity at any given timenot unexpected result since a certain/atime must elapse before any appreciable effects of a disturbance arefelt at a distance from the seat of the disturbance.*However, we know from our earlier discussion (and from everyday** It should bepointed out once more that disturbances propagate at infinitespeed since our medium is incompressible. Each Fourier component, however,propagates with a finite speed.TWO-DIMENSIONAL WAVES213observation of streams, for that matter) that as t -> oo it may happenthat a disturbance also propagates downstream as a wave with non-vanishing amplitude.
Our main interest here is to study such a passageto the limit. It is clear that one cannot accomplish such a purposesimply by letting t -> oo in (7.4.14), since, for one thing, the transformof <p cannot exist if <p does not tend to zero at oo. What we wish to doy>is to consider the contributions of the separate items in the bracketsand to avoid any singularities caused by zeros in theirdenominators by regarding A (s; t) as an analytic function in the neighborhood of the real axis of a complex $-plane and deforming the pathof integration in (7.4.14) by Cauchy's integral theorem in such a wayas to avoid such singularities.
One can then study the limit situationin (7.4.13),ast-> oo.In carrying out this program it is essential to study the separateterms defining the function A(s; t) given by (7.4.13). To begin with,we observe that the function Vgs tanh sh can be defined as an analyticand single- valued function in a neighborhood of the real axis sincethe function s tanh sh has a power series development at sthat2is valid for all s and begins with a term in sand, in addition, the=,=0. Once the function Vgs tanh shfunction has no real zero except shas been so defined, it follows that each of the terms in (7.4.13) is ananalytic function in a strip containing the real axis except at real zerosof the denominators.
It is important to take account of these zeros,as we have already done in sec. 7.3. For our present purposes it isuseful to consider the function(7.4.15)W(s)= gs/U 2_ _\.[\ghshtanh sh=sU2)2gs tanh sh/= (sU + Vgs tanh sh)(sU= /+(')/-(*)Vgs tanhsh)With reference to Fig. 7.3.1 above and the accompanying discussion,one sees that there are at most three real zeros of the function W(s):s=isin all cases a root,and thereexist in addition2two otherrealroots if the dimensionless parameter gh/Ugreater than unity. Also,2 =1 the origin is a double root of W(s), but isit is clear that if gh/Ua quadruple root if gh/U 2 = 1. In case gh/U 2 > I the real rootsftof W(s) are simple roots. (It might be noted in passing that W(s) hasinfinitely manyIt follows atis=n1, 2,..)i/? npure imaginary zerosonce that if we deform the path of integration,..inWATER WAVES214(7.4.14)from thereal axis to thepathP shown in Figure 7.4.1 we canconsider separately the contributions to the integral furnished by eachof the three items in the square brackets in (7.4.13), since the separate+ 13The pathFig.
7.4.1.integralswould thenrather than9?itself.Forin the s-planewe proceedto do, except that we preof the disturbancecomponents q> 9<p yThisexist.fer to consider the velocityPandweq> xwrite*= ?i + V,f)(7.4.16)9>with <pW anddefined (in accordance with (7.4.13) and (7.4.14)) as<pfollows:2p(s)s cosh s(y(7.4.17)7.4.18)+ h)>t'x,W(s) cosh sh?W --u+ h)p(s)s* cosh s(y[ettf+S__eltf ~ (S]cosh sh \/gs tanh shThe functions W(s),f-(s),and f + (s) have been definedin (7.4.15).Evidently the notation <p^\ (p has been chosen to point to the factshouldthat <p^ should yield the steady part of the motion while q>furnish "transients" which die out asaswe now show,unity,at leastits criticalwhenindeed the case,the parameter gh/U is not equal tot-> oo.
Thisis2value.<the case gh/U*1. In this case there are no singueven at the origin (cf. (7.4.18)), since / + and/_ vanish to the first power and i/gs tanh sh vanishes to the first0. Since p(s) is regular at sand $ 2 occurs in thepower also at sConsiderlaritiesfirston thereal axis,==numerator of the integrand our statement follows. Consequently the'path P can be deformed back again into the real axis. In this case the*Thediscussionwoulddiffer innoessentialwayfor<p vinstead of<p x.TWO-DIMENSIONAL WAVESbehavior of <pary phase(215can be obtained by the principle of stationIn the present case the functions f+(s) andfor larget(cf. sec.
6.8)./_(s) have non-vanishing first derivatives for all s, and consequently->at least like l/t since there are no points where the phase is<p(no attempt is made to give theasymptotic behavior with any more precision than is necessary forstationary. (Hereandinwhatfollowsthe purposes in view. ) As t -> oo therefore we obtain the steady statesolution cpW. The behavior of q>W for \x\ large is also obtained atonce: one sees that the integrand in (7.4.17) has no singularities inthis case also, and it follows at once from the Riemann-Lebesgueoo. Thus a steady state exists, andhas the property that the disturbances die out both upstream andtheorem that 9?^ ->itas-\x\downstream.We turn next to the more complicated case in which gh/U 2 > 1.The integrand for q>^ has no singularity at the origin, but it has=db P (cf.
Figure 7.4.1) furnished by simple zerossimple poles at sas t -> oo.of f_(s) at these points. Again we show that (pW ->Consider first the contribution of the semicircles at s/?. (Sinceis not a singularity, we deform the path back into the real axiss==there.) In the lower half-plane near sf-(s)expaxis{=P onesees readily thatthus the exponent inhas a negative imaginary part, andsince /_($)itf_(s)} has a negative real part,andits firsthaves like c(s ^f/?)isrealon therealpositive there (so that /_($) bewith c a positive constant). Thus for any closedderivative f_(s)isportion of the semicircles which excludes the end-points the contribution to the integral tends to zero as t -> oo, and hence also for theOnthe straight parts of the path the prinasciple of stationary phase can be used again to show that <p$ ->t -> oo.
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