J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 38
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In thepreceding chapters we have sometimes used this term (in conformity with established custom in the literature dealing with wave propagation) in a differentsense.worth notingWATER WAVES200(7.1.1)VV =(7.1.2)<p v+_0,J7 2=<pxxoo<yyo,^0,=-sIn addition,werequire that(7.1.3)andits9?thoughderivativesthis conditionisupto second order aremore restrictive thanisof these conditions was obtained from (7.0.3)bounded atnecessary.and(7.0.4)oo,The secondbydifferen-and eliminatingtiating (7.0.3)77.It is interesting to find all functions q>(x, y) satisfying these conditions, and it is easy to do so following the same arguments as wereused in Chapter3.1.Using(7.1.1)we mayU*v*-v,i, =(7.1.4)re-write (7.1.2) in the formy>=o-o(This of coursemakes use of the fact that <p is harmonic for y = 0,true.
One could easily show, in fact, that thewhich we assume to bean analytic continuation of 9?actually harmonic in a domain includingobserve that (7.1.4) is the same conditionfree surface condition (7.1,2) permitsover yy==0,so that9?in its interior.)isWewas imposed on the function called <p in Chapter 3, and weproceed as we did there by introducing a harmonic function y(x y)onas<p v9throughV =<Pv(7.1.5)-U2,<f>yy2/^0-eand can therefore be continuedanalytically by reflection into the upper half plane.
Since (p and itsderivatives were assumed to be bounded in the lower half plane, itfollows that \p is bounded in the entire plane and hence by Liouville'stheorem it is a constant; hence \p vanishes identically since \p =for y = 0. Thus we have for y> y a differential equation given by (7.1.5)with \f = 0, and it has as its only solutions the functionsThis function vanishes on y(7.1.6)Since<p y9> visalso a0,=c(harmonic function,itfollows that c(x)of the differential equation/a\ 2c= 0.isa solutionTWO-DIMENSIONAL WAVESHence<pis(7.1.8)201given by<p(x,y)=vAeu* cos /-L *+ a + c^x)jwith A and a constants and c^x) an arbitrary function of x.
By makingd 2cuse of (7.1.2), however, one finds that0, and hence thatdx 2c == const, since cp is bounded at oo. There is no loss of generality in0. The only solutions of our problem are therefore giventaking c l==by(7.1.9)y)<p(x,=vAeu* cosUL + AThus the only steady motions satisfying our conditions, aside froma uniform flow, are periodic in x with the fixed wave length A given byA(7.1.10)=7722n.gThe amplitude and phase of the motions are arbitrary.
If we were toobserve these waves from a system of coordinates moving in the xdirection with the constant velocity E7, we would see a train of progressing waves givenby<p=Ae mvcosm=gm (x-\~Ut)withC/=22n.AThese waves are identical with those already studied in Chapter 3(cf. sec. 3.2). The phase speed of these waves would of course be thevelocity U and the wave length A would, as it should, satisfy the relation (3.2.8) for waves having this propagation speed. In other words,the only waves we find are identical (when observed from a coordinatesystem moving with velocity U ) with the progressing waves that aresimple harmonic in the time and which have such a wave length thatthey would travel at velocity U in still water.7.2.Steady motions in water of infinite depth with a disturbingpressure on the free surfaceThe same hypotheses are made as in the previous section, exceptwe assume the pressure on the free surface to be a functionthatWATER WAVES202over the segmented in Fig.7.2.1.Thea^ x ^ a and zero otherwise, as indicat-free surface condition, as obtainedfrom(7.0.8)p0-o-i-aUFig.
7.2.1. Pressure disturbanceandof(7.0.4)t,ison a running streamby eliminating rj and assuminggiven byr\and9?to be independentnow4.^(7.2.1)JL m vU^=J^lon,z/=0,UQas one readily verifies. We prescribe in addition that <p and its firsttwo derivatives are bounded at oo.The solutions <p of our problems are conveniently derived by introducing the analytic function f(z) of the complex variable z = x + iywhose real part is 9?:(7.2.2)Sinceyy(7.2.3)/(a)=-yx,= <p(x, y) + iy(x9y).the condition (7.2.1) can be put in the form~i V=p~r-+ const.,on yand the constant can be taken as zero without=0,loss of generality, sincep can not affect the motion.We consider now only the case in which the surface pressure p is aconstant p = p over the segment \x\ ^ a, and zero otherwise.
Sincethis surface pressure is discontinuous at xa, it is necessary toadmit a singularity at these points; we shall see that a unique solutionof our problem is obtained if we require that q> is bounded at these10, with r the distancepoints while <p x and <p y behave like 1/r "*, eadding a constant to=>TWO-DIMENSIONAL WAVES203a on the free surface. (This singularity isfrom the points x =weaker than the logarithmic singularity of <p appropriate at a sourceor sink.)In terms of f(z), the free surface condition (7.2.3) clearly can be putin theform(7.2.4)z-(if^/A)=for',\x\>Jm a =0.aThe device of applying the boundary condition in this form seems tohave been used first by Keldysh [K.21].
We now introduce the analytic function F(z) defined in the lower half plane by the equationF(*)(7.2.5)= y.- A/.This function has the following properties: 1) Its imaginary part isprescribed on the real axis. 2) The first derivatives of its imaginarypart are bounded attwo derivatives of 99 are assumedand fz are bounded in view of theCauchy-Riemann equations. 3) Near z = o its imaginary part be~haves like l/\z ^r a\ l e> 0, as one readily sees. It is now easy toshow that F(z) is uniquely determined,* within an additive real conJP XF 2 be the difference of two functionsstant, as follows: Let GthesethreeJm G then vanishes on the entireconditions.satisfyingreal axis, except possibly at the points (#> 0), and G can thereforebe continued as a single-valued function into the whole plane exceptat the points (a, 0).
However, the singularity prescribed at theisweakerthan that of a pole of first order, and hencea,0)points (tohavethisoo, sincetheproperty and hencefirstfzz,the singularities at these points are removable. Since the first derivaG are bounded at oo, it follows from the Cauchy-Riemanntives ofJmequations that Gz is bounded at oo. Hence Gz is constant, by Liouville'sGd. Sincecztheorem, and G is the linear function: Gon the real axis, it follows that c and d are real constants. However,d on the left hand side of (7.2.5) leads to aa term of the form cz=term of the formOLZ++ 0,awitha=c,=Jm+in the solution of this* InChapter 4, the function F(z) given by (4.3.10)satisfied identical conditions except that the condition 2)tive in the present case.had aiswhichreal partslightlymorerestric-WATER WAVES204equation for /(*), and since f(z) is assumed to be bounded at oo.
itfollows that c0.We have here the identical situation that has been dealt with insec. 3 of Chapter 4, except that it was the real part of the functionF(z), rather than the imaginary part, that was prescribed on the real=and we can take over for our present purposes a number of theresults obtained there. The function F(z), now known to be uniquelydetermined within an additive real constant, is given byaxis,*(,(7.2.6)UQTtZ+a,from F(z) as given by (4.3.12) essentially only in theshould.
In any case, one can readily verify that F(z)satisfies the conditions imposed above. We take that branch of thelogarithm that is real for z real and \z\ > a, and specify a branch cuta and going to oo along the positive real axis. Thestarting at z =isnow an ordinary differential equation for the funcequation (7.2.5)tion f(z) which we are seeking.whichdiffersfactoriasitThe differential equation (7.2.5) has, of course, many solutions,and this means that the free surface condition and the boundcdnessconditions at oo and at the points (a, 0) are not sufficient to ensurethat a unique solution exists.
In fact, it is clear that the non-vanishingsolution of the homogeneous problem found in the preceding sectioncould always be added to the solution of the problem formulated upto now. A condition at oo is needed similar to the radiation conditionChapter 4. In the presentcan be made unique by requiring that the disturbance created by the pressure over the segment x fS a shoulddie out on the upstream side of the channel, i.e. at xoo.
The onlyaside from the fact that itjustification for such an assumptionmakes the solution unique is based on the observation that one neversees anything else in nature.* In sec. 7.4 we shall give a more satisfactory discussion of this point which is based on studying the unsteady flow that arises when the motion is created by a disturbanceinitiated at the time t = 0, and the steady state is obtained in thelimit as t -> oo. In this formulation, the condition that the motionimposedin the analogous circumstances incase, the solution|*\Lamb [L.3], p. 399, makes use, once more, of the device of introducingdissipative forces of a very artificial character which then lead to a steady stateproblem with a unique solution when only boundedness conditions are prescribedat oo.TWO-DIMENSIONAL WAVES205should die out on the upstream side is not imposed; instead, it turnsout to be satisfied automatically.A solution of the differential equation (7.2.5) (in dimensionlessform) has been obtained in Chapter 4 (cf.
(4.3.13)) which has exactlythe properties desired in the present case;(7.2.7) /(*)-2-=it is:& -^+te-&['logdt,atThe pathof integration (cf. Fig. 4.3.1) comes from iao along the positive imaginary axis and encircles the origin in such a way as to leaveitand the pointa, 0) to(theleft.That(7.2.7) yields a solution of=(7.2.5)easily checked.
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