J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 64
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The second equation leads through (10.9.23) to theconditionfirstequationisr["(10.9.25)= 9^,one readily verifies. Once r l has been determined, one seesalso immediately fixed by the first equation in (10.9.23).Boundary conditions are needed for the third order nonlinear differential equation given by (10.9.25); we assume these conditions to befor TV asthatisX(10.9.26)TJ,(o)-o,(oo)=1,These conditions result from our assumed physical situation: the firstis taken since a symmetrical form of the wave about its crest is expected and hence X (0) = 0, the second arises from (10.9.18), whilethe third is a reasonable condition that is taken in place of what looks=the more natural condition ri(oo)automatically satisfied, in view of thelikesince the latter conditionequation of (10.9.23)and 0i(oo)0, and thus docs not help in fixing r uniquely.An integral of (10.9.25) is readily found; it is:isfirst==l r + const.,Tiiand the boundary conditions yieldFromthis(10.9.27)one obtains,finally,T!(<P)the solution:= 1-3 sech2(3<p/2),then fixed by (10.9.23).
From these one finds for the shape ofthe wave that is, the value of y corresponding to y = 1and forthe horizontal component u of the velocity the equationsand 6 lis,(10.9.28)y=(10.9.29)u=11+ 3*2sech 2,23* 2 sech 2 -.2LONG WAVES IN SHALLOW WATERIncalculating351thesequantities,higher order terms in x have beenneglected. The expression for thewaveandisprofilefoundidentical with thoseby Boussinesq,Rayleigh,For the velocity u,Ithe two former authors give uwhile Keller gives the same expression as above except that the factor3# 2 is replaced by another whichdiffers from it by terms of orderKeller.=tt4or higher.Thus asolitarywaveofsym-metrical form has been found withan amplitude which increases withits speed 17. Careful experimentsto determine the wave profile andspeed of the solitary wave havebeen carried out by Daily andStephan [D.I], who find the waveprofile and velocity to be closelyapproximated by the above formulas with a maximum error in the%at the highest amplitude-depth ratio tested.
Fig. 10.9.2is a picture of a solitary wave takenby Daily and Stephan; three frameslatter of 2.5Fig. 10.U.2.10.10.Asolitaryfrom a motion picture filmshown.waveThe breaking of wavesin shallow water.In sections 10.4 and 10.6 aboveshallow water theory, whichisareDevelopment of boreshas already been seen that themathematically analogous to theitin a gas, leads to a highly interestingincasesresultinvolving the propagation of disturbsignificantances into still water that are the exact counterparts of the corre-theory of compressible flowsandsponding cases in gas dynamics involving the motions due to the actionof a piston in a tube filled with gas. These cases, which are very easilyWATER WAVES352described in terms of the concept of a simple wave (cf.
sec. 10.3),lead, in fact to the following qualitative results (cf. sec. 10.4): there isa great difference in the mode of propagation of a depression waveand of a hump with an elevation above the undisturbed water line;in the first case the depression wave gradually smooths out, but inthe second case the front of the wave becomes progressively steeperuntil finally its slope becomes infinite.
In the latter case, the mathematical theory ceases to be valid for times larger than those at whichthe discontinuity first appears, but one expects in such a case that thewave will continue to steepen in front and will eventually break. Thisis the correct qualitative explanation, from the point of view ofhydrodynamical theory, for the breaking of waves on shallow beaches.It was advanced by Jeffreys in an appendix to a book by Cornish[C.7] published in 1934.
Jeffreys based his discussion on the fact thatthe propagation speed of a wave increases with increase in the heightwave above the undisturbed level. Consequently, if a wave isof acreated in such awayas to cause a rise in the water surfaceitfollowsthat the higher points on the wave surface will propagate at higherspeed than the lower points in front of them in other words there isa tendency for the higher portions of the wave to overtake and tocrowd the lower portions in front so that the front of the wave becomes steep and eventually curls over and breaks; the same argumentindicates that a depression wave tends to flatten out and becomesmoother as it advances.It is of interest to recall how waves break on a shallow beach.Figures 10.10.1, 10.10.2, and 10.10.3 are photographs* of waves onthe California coast.
Figure 10.10.1 is a photograph from the air,taken by the Bureau of Aeronautics of the U.S. Navy, which showshow the waves coming from deep water are modified as they movetoward shore. The waves are so smooth some distance off shore thatthey can be seen only vaguely in the photograph, but as theymoveinshore the front of the waves steepens noticeably until, finally,breaking occurs. Figures 10.10.2 and 10.10.3 are pictures of the samewave, with the picture of Figure 10.10.3 taken at a slightly later timethan the previous picture. The steepening and curling over of thewave are very strikingly shown.At this point it is useful to refer back to the beginning of section10.6 and especially to Fig.
10.6.1. This figure, which is repeated here* Thesephotographs were very kindly given to the authorof the Sctipps Institution of Oceanography.by Dr. Walter MunkLONG WAVES IN SHALLOW WATERFig. 10.10.1.Waves ona beachindicates in terms of the theory ofhappens when a wave of elevation is created byfor the sake of convenience,characteristicswhat358Fig. 10.10.2.Wavebeginning to break354WATER WAVESFig. 10.10.3.Wavebreakingpushing the moveable end of a tank of water into it so that a disturbance propagates into still water of constant depth: the straightcharacteristics issuing from the "piston curve" AD, along each ofFig.
10.10.4. Initial point of breakingLONG WAVES IN SHALLOW WATER355c = Vg(h + rj) are constant,The point E is a cusp on the enve-which the velocity u and the quantityeventually intersect at the point E.lope of the characteristics, and represents also the point at which theslope of the wave surface first becomes infinite. The pointmightEsomewhat arbitrarily,thusing point (x b9-be taken as defining the breakoneof the wave, sinceexpects the wave to starttb )it istruecurling over after this point is reached. It is possible to fix the valuesof x b and t b without difficulty once the surface elevation r\77(0, t) is==0; we carry out the calculation for the interestingprescribed at xcase of a pulse in the form of a sine wave:(10.10.1)Forandit ist=17(0, t)0,x>its=Asincot.we assume the elevation r\ of the waterU Q to be constant (though not necessarilyto be zerozero, sincevelocityof interest to consider the effect of a current on the time and placeof breaking).As we know, the resulting motion is easily described in terms of thecharacteristics in the x, f-plane, which arc straight lines emanatingfrom theJ-axis, asThe values of u and cThe slope dx/dt of anyindicated in Figure 10.10.6.are constant along each such straight line.straight characteristic through the point (0, r)dx(10.10.2)whichiswhile cthe same as (10.3.6).=Vg(Acharacteristicxj(t9c-2ris+MOgiven by>The quantity C Q has the valuec=Vgh,+17), as always.
On the other hand, the slope of thisclearly also given in terms of a point (x, t) on it byr) so that (10.10.2)(10.10.3)in=atisx=(tcan be writtenr)[8c(r)-+M2<?which we have indicated explicitly thatin thecformO]depends only on r since(as well as all other quantities) is constant along any straightcharacteristic. Thus (10.10.3) furnishes the solution of our problem,itonce c(r ) is given, throughout a region of the x, -plane which is covered by the straight lines (10.10.3) without overlapping. However, theinteresting cases for us are just those in which overlapping occurs,i.e. those for which the characteristics converge and eventually cuteach other, and this always happens if an elevation is created atx0.
In fact, if c is an increasing function of r, then dx/dt as givenby (10.10.2) increases with r and hence the characteristics for x >WATER WAVES856mustIn this case, furthermore, the family of straight chaan envelope beginning at a point (xb9 tb ), which wehave defined to be the point of breaking.We proceed to determine the envelope of the straight lines (10.10.3).As is well known, the envelope can be obtained as the locus resultingfrom (10.10.3) and the relationintersect.racteristics has(10.10.4)=--[8c(r)2c+ u + 3(t ]T)C'(T)obtained from it by differentiation with respect to T.
For the points(x C9 t c ) on the envelope we then obtain the parametric equationsMAinKt(10.10.5)*c==r- [3c(r)-2C+ utf,andte(10.10.6)Weare interested mainly in the "first" point on the envelope, thattb ) for which t e has its smallest value since we iden-the point (x b9is,tify this point as the point of breaking. To do so really requiresproof that the water surface has infinite slope at this point. Suchaaproof could be easily given, but we omit it here with the observationthat an infinite slope is to be expected since the characteristics whichintersect in the neighborhood of the first point on the envelope allcarry different values for c.We have assumed that 77(0,t) isgiven by (10.10.1 ) and consequentlythe quantity c(r) in (10.10.5) and (10.10.6)(10.10.7)Ifwe assume=c(r)Vg(hisgiven by+ A sin COT).A > we see that C'(T) is a positive decreasing functionof T for small positive values of T.
Since c(r) increases for small positive values of r it follows that both x c and t c in (10.10.5) and (10.10.6)=Aare increasing functions of T near T0.minimum value of x c andt e must therefore occur for T0, so that the breaking point is given=bynoini(10.10.8)*x,= 2<?o(c + "o)2andas one can readily verify.Wenote that the point (x b,tb )lieson theLONG WAVES IN SHALLOW WATERinitial characteristic=x(C Q+w)f,asitshould since r357=for thisFrom the formulas we can draw a number of interestingSince c = Vgh we see that breaking occurs earlier incharacteristic.conclusions.shallower water for a pulse of given amplitude A and frequency a).Breaking also occurs earlier when the amplitude and frequency arewaveslarger. It follows that shortwillbreak sooner than long waves,since longer waves are correlated with lower frequencies.
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