J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 65
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Finally wenotice that early breaking of a wave is favored by small values foru Q theuniform velocity of the quiet water. In fact, if u$ isthe water is flowing initially toward the point wherenegative,the pulse originates, the breaking can be made to occur more quickly.Everyone has observed this phenomenon at the beach, where the breaking of an incoming wave is often observed to be hastened by waterrushing down the beach from the breaking of a preceding wave.It is of some importance to draw another conclusion from our theoryfor waves moving into water of constant depth: an inescapable consequence of our theory is that the maxima and minima of the surfaceelevation propagate into quiet water unchanged in magnitude with respect to both distance and time.
This follows immediately from the fact,initiali.e. ifthat the values of the surface elevation are constant along the straightcharacteristics so thatifsay, then this value of7?rj==has a relative maximum for x0, tr,be a relative maximum all along thewillfrom x0, t = r. The waves changetheir form and break, but they do so without changes in amplitude.In a report of the Hydrographic Office by Sverdrup and Munk[S.36] some results of observations of breakers on sloping beaches aregiven in the form of graphs showing the ratio of breaker height todeep water amplitude and the ratio of undisturbed depth at the breaking point to the deep water amplitude as functions of the "initialcharacteristicwhichissuessteepness" in deep water, the latter being defined as the ratio ofamplitude to wave length in deep water.
The "initial steepness" is thusessentially the quantity Ao> in our above discussion, and our resultsindicate that it is a reasonable parameter to choose for discussion ofbreaking phenomena. The graphs given in the report reproduced hereshow very considerable scatteringin Figures 10.10.5a and 10.10.5bof the observational data, and this is attributed in the report to errorswhich are apparently difficult to make withof our above conclusion that the breaking ofbasistheaccuracy.a wave in water of uniform depth occurs no matter what the amplitudein the observations,Onof thewave may bein relation to theundisturbed depthwecouldWATER WAVES358offeranother explanation for the scatter of the points in Figuresand 10.10.5b, i.e.
that the amplitude ratios are relatively10.10.5aindependent of theinitial steepness.Ofcourse, the curves of FiguresW.H.0.1.AB.E.B~1:6.3A B.E.B1:20.4or.33.3B.E.BTHEORY0.0030.0050.01INITIALFig. 10.10.5a.0.020.03STEEPNESS0.050.15H'Ratio of breaker height to wave height in deep water, ///assuming no refractionLONG WAVES IN SHALLOW WATER85910.10.5a and 10.10.5b refer to sloping beaches and hence to cases inwhich the wave amplitudes increase as the wave moves toward shore;but still it would seem rather likely that the amplitude ratios would berelatively independent of the initial steepness in these cases also sincethe beach slopes are small.
The detailed investigation of breaking ofwaves byHamadain character, shouldwhichboth theoretical and experimentalbe consulted for still further analysis of this and[H.2],isother related questions. The papers by Iversen [1.6] and Suquet[S.31 ] also give experimental results concerning the breaking of waves.We continue by giving the results of numerical computations forthree cases of propagation of sine pulses into still water of constantdepth.
The cases calculated are indicated in the following table:CaseType of pulseCase 1 is a half-sine pulse in the form of a positive elevation, case 2is a full sine wave which starts with a depression phase, and case 3consists of several full sine waves.Figure 10.10.6 shows the straight characteristics in the x, f-planefor case 1. (In all of these cases, the quantities x and y are now certaindimensionless quantities, the definitions of which are given in [S.19].)We observe that the envelope begins on the initial characteristic inwith earlier developments in this section. Theenvelope has two distinct branches which meet in a cusp at thebreaking point (xb tb ).
Figure 10.10.7 gives the shape of the wave forthis case, in accord,twodifferent times.As wesee,the front of the=tc bfinally becomes vertical for xflattens out. The solution given,t=-tb ,by thewave steepensuntilwhile the back of theitwavecharacteristics in FigureWATER WAVES360Region ofConstant StateRegion of Constant StateFig. 10.10.6. Characteristicdiagramin the x, /-planeFig. 10.10.7. Wave height versus distance for a half-sine wave of amplitudeh Q /5 in water of constant depth at two instants, where h n is the height of thestillwater levelLONG WAVES IN SHALLOW WATERFig.
10.1O.8.Wave361profile after breakingFig. 1O.1O.9. Characteristicdiagram in thea?,-planeWATER WAVES36210.10.6not valid for xis> x& >tt^and we expect breaking toensue. However, we observe that the region between the two branchesof the envelope is quite narrow, so that the influence of the developingFig. 10.10.10.amplitudebreaker/*Wave/5 inmayheight versus distance for awater of constant depth attfull=negative sine wave witht = 5.0, and t = 6.283.0,not seriously affect the motion of the water behindit.Thus we might feel justified in considering the solution by characteristics given by Figure 10.10.6 as being approximately valid for values ofLONG WAVES IN SHALLOW WATDRslightly greater than863t b (This also seems to the writer to be intuitiveratherfromthe mechanical point of view.) Figure 10.10.8lyplausibletwas drawn on.a time considerably greater than tb Theportion of the curve was obtained from the characteristics outsidethe region between the branches of the envelope, while the dottedportion which is of doubtful validity was obtained by using thecharacteristics between the branches of the envelope in an obviousthis basis for.fullmanner.
In this way onecurling over of a wave.isable to approximate the early stages of theFigures 10.10.9, 10.10.10, and 10.10.11 refer to case 2, in which adepression phase precedes a positive elevation. In this case the envelope of the characteristics does not begin, of course, on the initialcharacteristic but rather in the interior of the simple wave region, asindicated in Figure 10.10.9. Figure 10.10.10 shows three stages in theprogress of the pulse into still water. The steepening of the wave frontvery marked by the time the breaking point is reached much moremarked than in the preceding case for which no depression phase occurs in front.
Figure 10.10.11 shows the shape of the wave a short timeafter passing the braking point. This curve was obtained, as in theispreceding case, by using the characteristics between the branches ofthe envelope. Although this can yield only a rough approximation, still=7 for non-sloping bottom where the pulse is anFig. 10.10.11. r\ versus x at tentire negative sine-wave.
The dotted part of the curve represents r\ in the regionbetween the branches of the envelopeonewave really would break very soonsomewhatthe point we havearbitrarily defined as the breakingisafterpoint.rather convinced that theWATER WAVES364Figure 10.10.12 shows the water surface in case 3 for a time wellbeyond the breaking point.Fig. 10.10.12.Waterprofile after breakingIn gas dynamics where u andc represent the velocity and soundof a tube containing the gas,crossanentiresectionspeed throughoutnot possible to give a physical interpretation to the regionbetween the two branches of the envelope in the cases analogous tothat shown in Figure 10.10.6, since the velocity and propagationspeed must of necessity be single-valued functions of x.
However, in ouritclearlyisu and c refer to values on the water surface so thatno reason a priori to reject solutions for u and c which are notsingle- valued in x. Thus we might be tempted to think that the dottedpart of the curve in Figure 10.10.8 is valid within the general framework of our theory, but this is, unfortunately, not the case: our fundamental differential equations are not valid in the "overhanging"part of the wave, simply because that part is not resting on a rigidbottom.
It may be that one could pursue the solutions beyond thepoint where the breaking begins by using the appropriate differentialequations in the overhanging part of the wave and then piecing tocase of water wavesthereisgether solutions of the two sets of differential equations so that continuity is preserved, but this would be a problem of considerableIn this connection, however, it is of interest to report thea calculation by Biesel [B.10] for the change of form ofprogressing waves over a beach of small slope. Not the least interestingdifficulty.results ofaspect of Biesel's results is the fact that they are based essentially onthe theory of waves of small amplitude, i.e. on the type of theorywhich forms the basis for the discussions in Part II of this book.However, in Part II only the so-called Eulerian representation wasused, in which the dependent quantities such as velocity, pressure,LONG WAVES IN SHALLOW WATER365etc., are all obtained at fixed points in space. As a result, when linearizations are introduced the free surface elevation 77, for example, isand must of necessity be single-valued.
Biesel,however, observes that one can also use the Lagrangian representation* just about as conveniently as the Eulerian when a developmenta function of x andtwith respect to amplitude is contemplated. In this approach, all quantities are fixed in terms of the initial positions of the water particles(and the time, of course). In particular, the displacements (, 77) ofthe water particles on the free surface would be given as functions ofa parameter, i.e. = f (a, t), r\ = r\(a, t), and there would be no necessitya priori to require that rj should be a single-valued function of x.Biesel has carried out this program with the results shown in Figs.Asinusoidal progressing wave infirst two figures refer to the theory when10.10.13 to 10.10.16 inclusive.deep water is assumed. Thecarried out only to first order terms in the displacements relative to therest position of equilibrium.
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