J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 39
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One can also verify easily that q>&ef(z)satisfied all of the boundary and regularity conditions, except perhapsthe condition at oo on the upstream side. In Chapter 4, however, itiswas found(4.3.15)) that f(z) behaves at oo as follows:(cf.O/for Ste z\< 0,,Thusoo, but there are in general waves of/(s) dies out as x ->nonzero amplitude far downstream, i.e. at x = + oo. The uniquelydetermined harmonic function 9? = 9te f(z) is now seen to satisfy allconditions that were imposed.The waves at x = + oo are identical (within a term of order I/a)with the steady waves that we have found in the preceding section tobe possible when the stream is subject to no disturbance (cf. (7.1.9)),and the wave far downstream has the wave length A = 2nU 2 /g.However, we observe the curious and interesting fact (pointed out byLamb[L.3],=p.=wave mayalso vanish: clearly ifSte f(z) vanishes downstream as well404) that this., 99nn, n1, 2,as upstream, and this occurs whenever 2a/A is an integer, i.e.the length of the segment over which the disturbing pressurega/U2..wheneverisappliedan integral multiple of the wave length of a steady wave in water ofvelocity U (with no disturbance anywhere).
This in turn gives rise toisthe observation that there exist rigid bodies of such a shape that theycreate only a local disturbance when immersed in a running stream:one need only calculate the shape of the free surface which is, of2nn, take a rigid body having thecourse, a streamline for ga/U=WATER WAVES206shape of a segment of this surface and put it into the water. (Involvedis, as one sees, a uniqueness theorem for problems in which theshape of the upper surface of the liquid, rather than the pressure, isprescribed over a segment, but such a theorem could be proved along thelines of the uniqueness proof of the analogous theorem for simple harmonic waves given by F. John [J.5]. ) This fact has an interesting physical consequence, i.e., that such bodies are not subject to any wave resistance (by which we mean that the resultant of the pressure forceson the body has no horizontal component) while in general a resistancewould be felt.
This can be seen as follows: Observe the motion froma coordinate system moving with velocity U in the ^-direction. Allforces remain the same relative to this system, but the wave at + oowould now be a progressing wave simple harmonic in the time andhereU while at oo the wave amplihaving the propagation speedtude is zero. Thus if we consider two vertical planes extending fromthe free surface down into the water, one far upstream, the other fardownstream we know from the discussion in Chapter 3.3 that there isa net flow of energy into the water through these planes since energystreams in at the right, but no energy streams out at the left sincethe wave amplitude at the left is zero. Consequently, work must bedone on the water by the disturbance pressure and this work is doneat the rate RU = F, where R represents the horizontal resistanceand F the net energy flux into the water through two planes containwhich is theing the disturbing body between them.
Thus if .F =case if the wave amplitude dies out downstream as well as upstreamthen R = 0. This result might have practical applications. For example, pontoon bridges lead to motions which are approximately twodimensional, and hence it might pay to shape the bottoms of thepontoons in such a way as to decrease the wave resistance and hencethe required strength of the moorings. However, such a design wouldyield an optimum result, as we have seen, only at a definite velocity ofthe stream; in addition, the wave resistance is probably small compared with the resistance due to friction, etc., except in a streamflowing with high velocity.We conclude this section by giving the solution of the problem ofdetermining the waves created in a stream when the disturbance isconcentrated at a point, i.e. in the case in which the length 2a of thesegment over which the pressure p is applied tends to zero butlim 2p^a = P The desired solution is obtained at once from (7.2.7);9.it is:TWO-DIMENSIONAL WAVESig .= -/()(7.2.9)ig tf*LI &*e\J207ioodt.tThis solution behaves like I/* far upstream and like (2P /C7g)2exp {igz/U } far downstream.
Note that the amplitude downstreamdoes not vanish for any special values ofin this case. It is perhapsUalso of interest to observe that f(z) behaves near the origin like i log z,and hence the singularity at the point of disturbance has the characterof a vortex point;we recall that the singularity in the analogous caseby an oscillatory point source that were studied inof the waves createdChapter 4 had the character of a source point, since f(z) behaved likelog z rather than like ilogz (cf. 4.8.28)), with a strength factoroscillatory in the time. When one thinks of the physical circumstancesin these two different cases one sees that the present result fits thephysical intuition.7.3.Steady waves in water of constantfinitedepthIn water of constant finite depth the circumstances are more complicated, and in several respects more interesting, than in water ofinfinite depth.
This is already indicated in the simplest case, in whichthe free surface pressure is assumed to be everywhere zero and themotion is assumed to be steady. In this case we seek a function <p(x, y)satisfying the conditions (7.0.2) to (7.0.5), with <p and rj t both identtically zero.The boundaryconditions are thusU--2(7.3.1)<p yH(p xx=0,y= 0,y=0,gand(7.3.2)A<p,harmonic function which(p(x 9 y)(7.8.3)withA=Aand a arbitrary-h.satisfies these conditions is+ h) cos (mx + a)constants, and m a root ofgiven by:cosh m(ythe equationg^tanhmfe(7.8.4)ghThe conditionfree surface==ismh(7.8.4) ensures that thesatisfied, asfor the discussion in thisboundary condition on theone can easily verify.
Itisvery importantand the following section to study the rootsWATER WAVES208of the equation (7.8.4). The curvesplotted in Fig. (7.3.1).the intersections f==tanh f and f(U*/gh) f areroots of (7.3.4) are of course furnished byof these curves. One observes: 1)isThe= mhm=always a root; 2) there are two real roots different from zero if U /gh<l2Roots of the transcendental equation (U 2 /ghFig. 7.8.1.8) there are no real roots other than zerothe functiontan ifUm= i tanh2g tanhmhifU 2 /gh ^vanishes atm=<1; 4) iflikem;1)U3;=21/ghsince5)follows that (7.8.4) has infinitely many pure2/gh.imaginary roots no matter what value is assigned toOn the basis of this discussion of the roots of (7.3.4) we therefore,itUexpect that no motions other than the steady flow with no surfaceconst.
) will exist unless U 2 /gh < 1 Thesedisturbance (for which <pwaves are then seen to have the wave length appropriate for simple=.harmonic waves of propagation speed c = U in water of depth h, ascan be seen from (8.2.1), (3.2.2), and (3.2.8). It is possible to give arigorous proof of this uniqueness theorem which holds when no conditions at oo other than boundedness conditions are imposedbymaking use of an appropriate Green's function, or by making use ofthe method devised by Weinstein [W.7] for simple harmonic waves inwater of finite* depth, but we will not do so here.More interesting problems arise when we suppose that steady waves$re created by disturbances on the free surface, or perhaps also on thebottom.
Mathematically this means that a nonhomogeneous boundarycondition would replace one, or perhaps both, of the homogeneousTWO-DIMENSIONAL WAVESboundary conditions(7.3.1)and(7.3.2).209In addition, as we infer fromthe discussion of the preceding section, it is also necessary in generalto prescribe a condition of "radiation" type at oo in addition to boun-dedness conditions, and an appropriate such condition is that thedisturbance should die out upstream.
In the present problem, however, the additional parameter furnished by the depth of the waterleads to some peculiarities that are conditioned in part by the difference in behavior of the solutions of the homogeneous problem in theirdependence on the parameter U*/gh: Since the only solution of the20, one expectshomogeneous problem in the case U /gh S> 1 is q>that the solution of the nonhomogeneous problem will be uniquelydetermined in this case without the necessity of prescribing a radiation=<1 it is clear that the nonhomoU*/ghgeneous problem can not have a unique solution unless a conditionsuch as that requiring the disturbance to die out upstream iscondition at oo.
However,ifout the otherwise possible addition of the nonvanishing solution of the homogeneous problem. These cases have beenworked out (cf. Lamb [L.3], p. 407) with the expected results, asoutlined above, for U*/gh > 1 and U 2 /gh < 1, but the known representations of these solutions for the steady state make the waveimposed thatwill rule=1 and \x\ large.amplitudes large for U jghWe shall not solve these steady state problems directly here because the peculiarities not to say obscurities indicated above canall be clarified and understood by re-casting the formulation of theproblem in a way that has already been employed in the previouschapter (cf.
sec. 6.7)). The basic idea (cf. Stoker [S.22]) is to abandonthe formulation of the problem in terms of a steady motion in favor ofa formulation involving appropriate initial conditions at the time2t=and afterwards to make a passage to the limit in the solutionsunsteady motion by allowing the time to tend to oo. As was0,for theindicated in sec. 6.7, the advantage of such a procedure is that theinitial value problem, being the natural dynamical problem in New-tonian mechanics (while the steady state is an artificial problem), hasa unique solution when no conditions other than boundedness conditions are imposed at oo.
If a steady state exists at all, it should thenupon letting t -> oo, and the limit state would then automatihavethose properties at oo which satisfy what one calls radiacallytion conditions, and which one has to guess at if the steady stateproblem is taken as the starting point of the investigation.resultWe shall proceedalong these lines in the next section in attackingWATER WAVES210the problem of the waves created in a stream of uniform depth whena disturbance is created in the undisturbed uniform stream at the0. The subsequent unsteady motion will be determined whentime tboundednessconditions are imposed at oo. It will then be seenonlythat the behavior of the solutions as t -* oo is indeed as indicated=i.e.
the waves die out at infinity both upstream and downstream when U 2 /gh1, that they die out upstream but not downstream when U 2 /gh < 1. One might be inclined to say: "Well, what ofit, since one guessed the correct condition on the upstream side anyway?" However, we now get a further insight, which we did not21 there just simply is no steadypossess before, i.e. that for U /ghstate when t -> oo although a uniquely determined unsteady motionexists for every given value of the time t In fact it will be shown thatabove,>=.the disturbance potential becomes infinite like J 2/3 at all points of thefluid when t -> oo and U 2 /gh1, and that the velocity also becomes=infinite7.4.everywhere when-> oo.tUnsteady waves created by a disturbance on the surface of arunning streamThe boundary conditions on the disturbance potential <p(x, y\ t) atthe free surface (cf. Fig.
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