J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 62
Текст из файла (страница 62)
10.8.5. Thethe bore,i.e.,=water surface at the front of the wave on the downstream side istangent to the bottom and moves with the speed 2clB On the otherhand, when h^h^ approaches the other extreme value, i.e. unity, it isclear that the height h 2h of the bore must again approach zero.Hence the height of the bore must attain a maximum for a certain-ParabolaxFig.
10.8.5.=x =-c,t2c,txMotion down the dry bed of a streamvalue of h Q /h v In Fig. 10.8.6 we give the result of our calculations forA as a function of h^h^. The curve rises very steeply to itsA2-A = .32^ for h^h^ = .176 and then falls to zeroh v It is rather remarkable that the bore can attaina height which is nearly 1/3 as great as the original depth of the waterbehind the dam.maximumA2again when h Q=0.110.2Fig. 10.8.6.04Maximum10.60.8IX)height of the boreLONG WAVES IN SHALLOW WATER341motion by means of the #, -planetwolimitvaluesAO/AJunity and zero. These two casesare schematically shown in Fig.
10.8.7. When hhv we note thatthe zone (3) is very narrow and that the shock speed f approaches cl9Itiswheninstructive to describe theisnearits~the propagation speed of small disturbances in water of depth h l9corresponding to the fact that the height of the shock wave tends toi.e.zero as A -> h^more(cf.Fig.
10.8.6).The otherlimit situation,i.e.hc0,wetacitly consider h to remain fixed in ourandhencethat h 2 is also fixed since we are in thepresent discussion,supercritical case, it follows (for example from (10.6.23) with A 2 inisinteresting. Sinceplace of hi) that f ->oo as A -> 0.Free surface for(I)(3)(2)Onthe other hand asweseefromttQ(0)Fig. 10.8.7. Limit casesA of the shock wave tends to zero ratherFig. 10.8.6, the height h 2becomes the front of the waveslowly as A -> 0. In the limit, pointPwith the motion indicated by Fig. 10.8.5.
Thus asA -> the shock wave becomes very small in height but moves downstream with great speed; or, as we could also say, in the limit the waterin front of the point P is pinched out and P is the front of the wave.in accordanceWATER WAVES34210.9.Thesolitarywavehas long been a matter of observation that wave forms of a permanent type other than the uniform flows with an undeformed freeItsurface occur in nature; for example, Scott Russell [R.14] reportedin 1844 his observations on what has since been called the solitarywave, which is a wave having a symmetrical form with a single humpand which propagates at uniform velocity without change of form.Later on, Boussinesq [B.16] and Rayleigh [R.3] studied this problemmathematically and found approximations for the form and speed ofsuch a solitary wave.
Korteweg and de Vries [K.15] modified themethod of Rayleigh in such a way as to obtain waves that arc periodicand which tend to the soliin formcalled cnoidal waves by theminthe limiting case of long wave lengths.tary wave found by RayleighAsystematic procedure for determining the velocity of the solitarywave has been developed by Weinstein [W.6],At the beginning of section 10.7 we have shown that the only continuous waves furnished by the theory used so far in this chapterform are of a very special and ratheruninteresting character, i.e., they are the motions with uniform velocity and horizontal free surface.* This would seem to be in crass contradiction with our intention to discuss the solitary wave in terms ofthe shallow water theory, and it has been regarded by some writers as aparadox. **The author's view is that this paradox like most othersbecomes not at all paradoxical when properly examined.
What isinvolved is a matter of the range of accuracy of a given approximatetheory, and also the fact that a perturbation or iteration scheme ofuniversal applicability does not exist: one must always modify suchschemes in accordance with the character of the problem. In the present case, the salient fact is that the theory used so far in the presentwhich progress unchangedinchapter represents the result of taking only the lowest order terms inthe shallow water theory as developed in section 4 of Chapter 2, andit isnecessary to carry out the theory to include terms of higher order* If motions witha discontinuity are included in the discussion, then the motionthe only other possibility up to now in this chapter with regard towaves propagating unchanged in form.** Birkhoff[B.ll, p. 23], is concerned more about the fact that the shallowwater theory predicts that all disturbances eventually lead to a wave whichbreaks when on the other hand Struik [S.29] has proved that periodic progressingwaves of finite amplitude exist in shallow water.
In the next section the problemof the breaking of waves is discussed. Ursell [U.ll] casts doubt on the validityof the shallow water theory in general because it supposedly does not give riseto the solitary wave.of a boreisLONG WAVES IN SHALLOW WATERif348one wishes to obtain an approximation to the solution of the problemof the solitary wave.
This has been done by Keller [K.6],that the theory of Friedrichs [F.ll] presented in Chapter 2,who findswhen car-wave and cnoidalwaves of the type found by Korteweg and de Vries [K.15] (thus theshallow water theory is capable of yielding periodic progressing wavesof finite amplitude). As lowest order approximation to the solutionof the problem, Keller finds (as he must in view of the remarks above),that the only possibility is the uniform flow with undeformed freeried out to second order,* yields both the solitarysurface,butifthe speedUof the flowistaken at thecriticalvalueU=\/gk with h the undisturbed depth, then a bifurcation phenomenon occurs (that is, among the set of uniform flows of all depths andvelocities, the solitary wave occurs as a bifurcation from the specialflow with the critical velocity) and the second order terms in the de-velopment of Friedrichs lead to solitary and cnoidal waves withspeeds in the neighborhood of this value.
To clinch the matter, it hasbeen found by Friedrichs and Hycrs [F.13] that the existence of thesolitary wave can be proved rigorously by a scheme which starts withthe solution of Keller as the term of lowest order and proceeds byiterations with respect to a parameter in essentially the same manneras in the general shallow water theory.** In the following, we shallderive the approximation to the solution of the solitary Wave problemmethod of Friedrichs and Hyers rather than the generalschemewhich was used by Keller, and we can then stateexpansionthe connection between the two in more detail.following theThe author thus regards the nonlinear shallow water theoryto beand not at all paradoxical. Indeed, the linear theory ofwaves of small amplitude treated at such length in Part II of thisbook is in essentially the same position as regards rigorous justification as is the shallow water theory: we have only one or two cases sofar in which the linear theory of waves of small amplitude is shown tobe the lowest order term in a convergent development with respect towell foundedamplitude.
We refer, in particular, to the theory of Levi-Civita [L.7]and Struik [S.29] in which the former shows the existence of periodicprogressing waves in water of infinite depth and the latter the samething (and by the same method) for waves in water of finite constant* In order to fix all terms of second order, Keller found it necessary to makeuse of certain relations which result from carrying the development of some ofthe equations up to terms of third order.** W. Littman, in a thesis to appear in Communs. Pure and AppJ.
Math.,has proved rigorously in the samewaythe existence also of cnoidal waves.844WATER WAVESdepth.* This theory will be developed in detail in Chapter 12. It mightbe added that those who find the nonlinear shallow water theoryparadoxical in relation to the solitary wave phenomenon should bythe same type of reasoning also find the linear theory paradoxical,since it too fails to yield any approximation to the solitary wave, evenwhencarried out to terms of arbitrarily high order in the amplitude,except the uniform flow with undisturbed free surface.
In fact, if onewere to assume that a development exists for the solitary wave whichproceeds in powers of the amplitude as in the theory discussed in thefirst part of Chapter 2, it is easily proved that the terms of all ordersin the amplitude are identically zero. There is no paradox here, however; rather, the problem of the solitary wave is one in which thesolution is not analytic in the amplitude in the neighborhood of itszero value, but rather has a singularity possibly of the type of abranch point there.
Thus a different kind of development is needed,and, as we have seen, one such possibility is a development of the typeof the shallow water theory starting with a nonlinear approximation.Another possibility has been exploited by Lavrentieff [L.4] in adifficult paper; Lavrentieffproves the existence of the solitary waveby starting from the solutions of the type found by Struik for periodicwaves of finite amplitude and then making a passage to the limit byallowing the wave length to become large and, presumably, in such a2way that the parameter gh/U tends to unity.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
