J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 15
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In order that S+ should hold for this solutiont one seesmust satisfyreadily that the constants // andinKfor allthe linear equations(4.3.21HLL4e sin a= -H\M = K, M = - K,)from which we conclude that<>fL-2c sin a,//=2c sin a{*-*_..Thus the solution is now uniquely determined through impositionof the Sommerfcld condition, and can be expressed as follows:(4.3.23)0(x, y;t)~---- sinmaemv/ 1 \sin(mx ~ot)+ O\\,\ rx>/original variables and parameters (cf.with(4.3.6)),0(l/r) representing a function which dies out atlikeof course yields a wave with syminfinity1/r. The functionmetrical properties with respect to the i/-axis.
We observe thatthe wave length A = 'Infm of these waves at oo is the same as that offree oscillations of the same frequency, as one would expect.The most striking thing about the solution is the fact that forcertain frequencies and certain lengths of the segment over whichthe periodic pressure differs from zero, the amplitude of the progressingupon rcintroduction of thewavezero at oo; this occurs obviously for sin maSince..1, 2, 3,kn, k2n/h with A theis=0,i.e.formwave lengthmaof a free oscillation of frequency a, it follows that the amplitude ofthe progressing wave at oo vanishes when-(4.3.24)..2a -frA,k=1,2 ____,when the length of the segment on which the pressure is appliedan integral multiple of the wave length of the free oscillation havingi.e.isthe same frequency as the periodic pressure.
This does not of coursemean that the entire disturbance vanishes, but only that the motionin this case is a standing wave given by(4.3.25)&(x, y;t)K= y(x, y) cos at,in (4.3.19) are now both zero. Sincesince the quantities // andnow behaves like 1/r at both infinities, the amplitude of the standing<pWATER WAVES66Awave generating device based on theto zero at infinity.physical situation considered here would thus be ineffective at certainfrequencies. It is clear that no energy is carried off to infinity inwave tendsand hence that the surface pressure p on the segmenta 5* x ^ + a can do no net work on the water on the average.Since r/ t =V it follows that the rate at which work is done by thethis case,pressure p (per unit width at right angles to the #, t/-plane)yarpq>cos atdx,andsincep has the phasesin at itisisindeed clearthat the average rate of doing work is zero in this case.There is a limit case of the present problem which has considerableinterest for us.
It is the limit case in which the length of the segmentover which p is applied shrinks to zero while the amplitude P of pincreases without limit in such a way that the product 2aP approachesway weobtain the solution for an oscillatingpressure point. One sees easily that the function f(z) given by (4.3.13),which yields the forced oscillation in our problem, takes the followinga finiteformlimit.In thisin the limit:(4.3.26)withCi(z)=^e-^dt,the real constant 2aPa/gg. At oo this function behaves asfollows((4.3.27)/()='for die z< 0,for S&e z>V * 7)2Ci e~ iz+(I0.In this limit case of an oscillating pressure point we see that there areno exceptional frequencies: application of the external force alwaysleads to transmission of energy through progressing waves at oo.Thesingularity of f(z) at the origin is clearly a logarithmic singularitysince f(z) behaves near the origin like(4.3.28)We/(*)C= -*-"nC*dt_+....Jtsee that a logarithmic singularity is appropriate at a source orsink of energy when the motion is periodic in the time.SIMPLE HARMONIC SURFACE PRESSURE4.4.
Periodic progressingwaves against a67vertical cliftWith theaid of the complex velocity potential defined by (4.3.13)discuss a problem which is different from the one treated inthe preceding section. The problem in question is that of the deter-we canmination of two-dimensional progressing waves moving toward avertical cliff, as indicated in Figure 4.4.1. The cliff is the verticalFig. 4.4.1.Wavesagainst a verticalcliffplane containing the //-axis.
As in the preceding section, we assumealso that a periodic pressure (cf. (4.3.1)) is applied over the segmenta at the free surface. To solve the problem we need only<j| x^combine the standing waves given by (4.3.17) and (4.3.18) in sucha way as to obtain progressing waves which move inward from thetwo infinities, and this can be done in the same way as in section 4.3.result will be again a wave symmetrical with respect to theXalong the j/-axis; thus sucht/-axis, and hence one for whicha wave satisfies the boundary condition appropriate to the verticalThe=cliff.forWe would find for the velocity potentialx(4.4.1>)the expression, valid0:0(x,ij;t)=2Pasinmae m ^[sm(nix+ at)] +to/ 1 \II'with 0(l/r) a function behaving like 1/r at oo but with a singularity(a, 0). In order to obtain a system of waves which are not reflectedback to oo by the vertical cliff it was necessary to employ a mechanismxa on the freethe oscillating pressure over the segmentat^^WATER WAVES68which absorbs the energy brought toward shore by the inwave.However, the particular mechanism chosen here, i.e.comingone involving an oscillatory pressure having the same frequency asthe wave, will not always serve the purpose since the amplitude Asurfaceof the surface elevation of the progressing(4.4.1)and(4.1.4),waveat ooisgiven,frombyA =(4.4.2)2Psinma.68Thus thePapplied on the watersurface near shore to the amplitude of the wave at oo would obviously0.
In other words, such a mechanism wouldbecome oo when sin maratio of the pressure amplitude=waves whose wave length A at oo satisfiesk A/2, with k any integer, only if infinite pressurethe relation afluctuations at the shore occur. Presumably this should be interpretedas meaning that for these wave lengths the mechanism at shore isnot capable of absorbing all of the incoming energy, or in other words,some reflection back to oo would occur.
This remark has a certainpractical aspect: a device to obtain power from waves coming towarda shore based on the mechanism considered here would functionachieveitspurpose fordifferently at different wave lengths.It is of interest in the present connection to consider the same limitcase as was discussed at the end of the preceding section, in whichthe segment of length a shrinks to zero whilePa remainsfinite.Inno exceptional wave lengths or frequencies occur. However,the limit complex potential now has a logarithmic singularity at theshore line, as we noticed in the preceding section, and the amplitudethis caseof the surface would therefore also be infinite at the shoreline.Whatreally happen, of course, is that the waves would break alongthe shore line if no reflection of wave energy back to oo occurred,wouldand theamplitude obtained with our theory represents thebest approximation to such an essentially nonlinear phenomenonthat the linear theory can furnish.This limit case represents the simplest special case of the problemof progressing waves over uniformly sloping beaches which will betreated more generally in the next chapter.
However, the presentcase has furnished one important insight: a singularity of the complexvelocity potential is to be expected at the shore line if the conditionat oo forbids reflection of the waves back to oo, and the singularityshould be at least logarithmic in character.infiniteCHAPTERWaves on5.1. Introduction5Sloping Beaches and Past Obstaclesand summaryPerhaps the most strikingand perhapsalso themost fascinating-among all water wave phenomena encountered inthe breaking of ocean waves on a gently sloping beach.The purpose of the present chapter is to analyze mathematically thebehavior of progressing waves over a uniformly sloping beach insofaras that is possible within the accuracy of the linearized theory forsingle occurrencenatureiswaves of small amplitude; that is, within the accuracy of the theorywith which we are concerned in the present Part II.
Later, in Chapter10.10, we shall discuss the breaking of waves from the point of viewof the nonlinear shallow water theory.To begin with, it is well to recall the main features ofwhatisoftenobserved on almost any ocean beach in not too stormy weather.Some distance out from the shore line a train of nearly uniformprogressing waves exists having wave lengths of the order of sayfifty to several hundred feet. These waves can be considered as simplesine or cosine waves of small amplitude. As the waves move towardshore, the line of the wave crests and troughs becomes more andmore nearly parallel to the shore line (no matter whether this wasthe case in deep water or not), and the distance between successivewave crests shortens slightly.
At the same time the height of thewaves increases somewhat and their shape deviates more and morefrom that given by a sine or cosine in fact the water in the vicinityof the crests tends to steepen and in the troughs to flatten out untilfinally the front of the wave becomes nearly vertical and eventuallythe water curls over at the crest and the wave breaks. These observations are all clearly borne out in Figures 5.1.1, 5.1.2, whichare photographs, given to the author by Walter Munk of the ScrippsInstitution of Oceanography, of waves on actual beaches. At thesame time, it should be stated here that the breaking of waves alsooccurs in a manner different from this a fact which will be discussed695.1,1.WavesFig. 5.1.2. Breakingandon,adiffraction ofwaves at aninletWAVES ON SLOPING BEACHKS AND PAST OBSTACLES71Chapter 10.10 on the basis of other photographs of actual wavesand a nonlinear treatment of the problem.inclear that the linear theory we apply here can not in principleyield large departures from the sine or cosine form of the waves inItisdeep water, andstill lesscanityield the actual breakingphenomena:obviously these are nonlinear in character.
On the other hand thelinear theory is to be applied and should yield a good approximationfordeep water and for the intermediate zone between deep water andthe actual surf region. However, the fact that breakers do in generaloccur in nature cannot by any means be neglected even in formulatingthe problems in terms of the linear theory, for the following reasons.Suppose we consider a train of progressing waves coming from deepwater in toward shore.
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