J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 66
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The second figure is a detail of the motionin a neighborhood of the location shown by the dotted circle in thefirst figure. Fig. 10.10.15 and Fig. 10.10.16 treat the same problem,but the solution is carried to second order terms. In both cases thedevelopment of a breaker is strikingly shown. A comparison of theresults of the first order and second order theories is of interest; themain conclusions are: if second order corrections are made the breaking is seen to occur earlier (i.e.
in deeper water), the height of the waveat breakingismuchgreater,and the tendency of the wave to plungedownwardafter curling over at the top is considerably lessened.Actually, our shallow water theory cannot be expected to yield agood approximation near the breaking point since the curvature ofbe large there. However, since the motionshould be given with good accuracy at points outside the immediatevicinity of the breaking point it might be possible to refine the treatment of the breaking problem along the following lines: consider themotion of a fixed portion of the water between a pair of planes locatedthe water surfaceislikely toin front and in back of the breaking point. The motionof the water particles outside the bounding planes can be considered asmight then seek to determinegiven by our shallow water theory.some distanceWethe motion of the water between these two planes by making use of arefinement of the shallow water theory or by reverting to the originalexact formulation of the problem in terms of a potential function with* InChapter 12.1 this representationproblems involving unsteady motions.isexplained and used to solve other366Fig.
10.10.13. ProgressionWATER WAVESand breaking of a wave on a beach of1 in10 slope.First-order theoryFig. 10.10.14. Details of breaking ofwave shown,theoryin Fig. 10.10.18. First-orderLONG WAVES IN SHALLOW WATERFig. 10.10.15. Progressionof a wave on a beach ofSecond-order theoryand breakingFig. 10.10.16. Details of breaking ofwave showntheory3671 in10 slope.in Fig. 10.10.15. Second-orderWATER WAVES368the nonlinear free surface condition and determineitby usingfinitea bounded region.It is of interest now to return to the problem with which we openedthe discussion of the present section, i.e. to the problem of a tank witha moveable end which is pushed into the water. As we have seen, thewave which arises will eventually break. Suppose now we assumethat the end of the tank continues to move into the water with a uniform velocity.
The end result after the initial curling over and breaking will be the creation of a steady progressing wave front which issteep and turbulent, behind which the water level is constant and thedifferencemethodsinFig. 10.10.17.The borein the TsicnTang Riverwater has everywhere the constant velocity imparted to it by the endof the tank. Such a steady progressing wave with a steep front iscalled a bore. It is the exact analogue of a steady progressing shockwave in a gas. In Figure 10.10.17 we show a photograph, taken fromthe book by Thorade [T.4], of the bore which occurs in the TsienTang River as a result of the rising tide, which pushes the water intoa narrowing estuary at the mouth of the river.
The height of this boreapparently is as much as 20 to 30 feet. According to the theory presented above, this bore should have been preceded by an unsteadyphase during which the smooth tidal wave entering the estuary firstcurled over and broke. Methods for the treatment of problems involving the gradual development of a bore in an otherwise smoothflow have been worked out by A. Lax [L.5] (see also Whitham [W.I 2] ).We have, so far, used our basic theory the nonlinear shallowLONG WAVES IN SHALLOW WATER369water theory to interpret the solutions of only one type of problem,i.e. the problem of the change of form of a pulse moving into stillwater of constant depth. The theory, however, can be used to studythe propagation of a wave over a beach with decreasing depth justas well (cf. the author's paper [S.19]), but the calculations are mademuch more difficult because of the fact that no family of straightcharacteristics exists unless the depth is constant.
This problem, into the fore the difficulties of a computational naturefact, bringswhich occur in important problems involving the propagation of floodwaves and other surges in rivers and open channels in general. Suchproblems will be discussed in the next chapter.On an actual beach on which waves are breaking, the motion ofthe water, of course, does not consist in the propagation of a singlepulse into still water, but rather in the occurrence of an approximatelyperiodic train of waves. However, experiments indicate that only aslight reflection of the wave motion from the shore occurs. The in-coming wave energy seems to be destroyedin turbulencedue to break-ing or to be converted into the energy of flow of the undertow.
Inother words, each wave propagates, to a considerable degree, in amanner unaffected by the waves which precededit.Consequently theabove treatment of breaking, in which propagation of a wave intowater was assumed, should be at least qualitatively reasonable.stillAn-other objection to our theory has already been mentioned, i.e. thatlarge curvatures of the water surface near the breaking point seem suretomakethe results inaccurate. Nevertheless, the theory should bemany cases of waves on sloping bea-valid, except near this point, inwave lengths arc usually at least 10 to 20 or more timesthewater in the breaker zone, hence the theory presentedthe depth ofabove should certainly yield correct qualitative results and perhapsches, since thealso reasonably accurate quantitative results.Waves do not by any means always break in the manner describedup to this point.
In Fig. 10.10.1 8a, b we show photographs (given tothe authorby Dr. Walter Munk of waves breaking)ina fashion con-Wesiderably at variance with the results of the theory presented here.observe that the waves break, in this instance, by curling over slightly atthe crest, but that the wave remains, as a whole, symmetrical in shape,while the theory presented here yields a marked steepening of the wavefront and a very unsymmetrical shape for the wave at breaking.Observation of caseslikethat shown in Figure 10.10.18 doubtlesslyled to the formulation of the theory of breakingdue to Sverdrup andWATER WAVES370(a)(b)Fig.
l().l().18a, b.MunkWaves breakingat crestsbased on results taken from the study ofthe solitary wave, which has been discussed in the preceding section.*The solitary wave is, by definition, a wave of finite amplitude con*An[S.33]; their theoryisinteresting mathematical treatment of breaking phenomena from thisofview was given some time ago by Kculegan and Patterson K.lllj.pointLONG WAVES IN SHALLOW WATERsisting of a single elevation of suchchanged in form. At first sight, thiswave forma shape thatit871can propagate un-would seem to be a rather curiousto take as a basis for a discussion of thephenomenaofbreaking, since it is precisely the change in form resulting in breakingthat is in question.
On the other hand, the waves often look as inFigure 10.10.18 and do retain, on the whole, a symmetrical shape,*with some breaking at the crest. Actually, the situation regarding thetwo different theories of breaking from the mathematical point ofview is the following, as we can infer from the discussion of the preceding section: Both theories are shallow water theories. In fact, asKeller [K.6], and Friedrichs and Hyers [F.13], have shown, the theoryof the solitary wave can be obtained from the approximation of next higherorder above that used in the present section, if the assumption is made thata steady motion. In other words, the theory used bySverdrup and Munk is a shallow water theory of higher order thanthe theory used in this section, which furnishes in principle the constant state as the only continuous wave which can propagate un-themotionisaltered in form.Onthe other hand, the theory presented hereFig.
10.10.10.itmakesSymmetrical waves breaking at crestspossible to deal directly with the unsteady motions, while Sverdrupand Munk are forced to approximate these motions by a series ofdifferent steady motions. One could perhaps sum up the whole matterby saying that waves break in different ways depending upon theindividual circumstances (in particular, the depth of the water compared with the wave length is very important), and the theory whichshould be used to describe the phenomena should be chosen accordingly.
In fact, Figures 10.10.17 and 10.10.18 depicting a bore and*Sverdrup and Munk, like the author, assume that, when considering breakingphenomena, each wave in a train can be treated with reasonable accuracy asthough it were uninfluenced by the presence of the others.WATER WAVES872crests of otherwise symmetrical wavesa whole series of cases which includeinextremesperhaps representand 10.10.3 as an intermediatein10.10.2the breaker shownFigurescase.
Some pertinent observations on this point have been made byMason [M.4]. A theory has been developed by Ursell [U.ll] whichdiffers from the theories discussed here and which may well be appro-waves breaking only at thepriate in cases notamenable to treatment by the shallow water theory.referred to above should also be men-The paper by Hamada [H.2]tioned again in this connection.
In particular, Fig. 10.10.19, takenfrom that paper, shows waves created in a tank which break by curlingat the crest but still preserving a symmetrical form. It is interestingwave length in this case is almost the same as theIt is also interesting to add that in this case atheofwater.depthcurrent of air was blown over the water in the direction of travel ofto observe that thesomewhatThe two waves were both generated by a wavethe waves. Fig. 10.10.20 shows a similar case, but withgreaterwavelength.Fig.
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