J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 16
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As we know from Chapter 3, such a train ofwaves is accompanied by a flow of energy in the direction toward theshore. If we assume that there is little or no reflection of the wavesfrom the shore which observations show to be largely the case for ait follows that there must exist some mechagently sloping beach*nism which absorbs the incoming energy; and that mechanism is ofcourse the breaking of the waves which converts the incoming waveenergy partially into heat through turbulence and partially into theenergy of a different type of flow, i.e.
the undertow. In terms of thelinear theory about the only expedient which we have at our disposalto take account of such an effect in a rough general way is to permitthat the wave amplitude may become very large at the shore line, or,in mathematical terms, that the velocity potential should be permitted to have an appropriate singularity at the shore line. As wehave already hinted at the end of the preceding chapter, the appropriate singularity for a two-dimensional motion seems to belogarithmic, and hence the wave amplitude would be logarithmicallyinfinite at the shore line. Indeed, it turns out that no progressingwave solutions without reflection from the shore line exist at allwithin the framework of the linear theory unless a singularity atleast as strong as a logarithmic singularity is admitted at the shoreline.Once the frequencybeendifferenttwohasof the wave motionfixed,types of standingTheactual procedure works out as follows:* This fact is also used inlaboratory experiments with water waves: theexperimental tanks are often equipped with a sloping ''beach" at one or moreof the ends in order to absorb the energy of the incoming waves through breaking,and thus prevent reflection from the ends of the tank.
This makes it possible toperform successive experiments without long waits for the motions to subside.WATER WAVES72waves are obtained, one of which has finite amplitude, the otherinfinite amplitude, at the shore line. These two different types ofstanding waves behave at oo like the simple standing wave solutions for water of infinite depth obtained in Chapter 3; i.e. one ofthem behaves like emv sina) while the other behaves like(mx +hencethetwo may be combined with appropriatea-});{time factors to yield arbitrary simple harmonic progressing wavesat oo. If the amplitude at oo is prescribed, and also the condition(cf. the last two sections of the preceding chapter) requiring that thewave at oo be a progressing wave moving toward shore, then thee my cosmxuniquely determined; in particular, the strength of thelogarithmic singularity at the shore line is uniquely fixed once theamplitude of the incoming wave is prescribed at oo.The fact that progressing waves over uniformly sloping beachessolutioniscan be uniquely characterized in the simple way just stated is nota thing which has been known for a long time, but represents ratheran insight gained in relatively recent years (cf.
the author's paper[S.18] of 1947 and the other references given there). The methodemployed in the author's paper makes essential use of an idea dueto H. Lewy to obtain the actual solutions for the case of two-dimensional waves over beaches sloping at the angles Tt/Vn, with n aninteger; H. Lewy [L.8] extended his method also to solve the problemfor slope angles (p/2n)n with p an odd integer and n any integersuch that p < 2n. For the special slope angles 7t/2n the progressingwave solutions were obtained first by Miche [M.8] (unknown to theauthor at the time because of lack of communications during WorldWar II), and somewhat later by Bondi [B.14], but without uniqueness9statements.
Actually, the special standing wave solutions for thesesame slope angles which are finite at the shore line had alreadybeen obtained by Hanson [H.3].All of these solutions for the slope angles eo == jr/2n, become morecomplicated and cumbersome as n becomes larger, that is, as thebeach slope becomes smaller. In fact, the solutions consist of finitesums of complex exponentials and exponential integrals, and thenumber of the terms in these sums increases with n. Actual oceanbeaches usually slope rather gently, so that many of the interestingcases are just those in which the slope angle is small of the orderof a few degrees, say.
It is therefore important to give at least anapproximate representation of the solution of the problem valid forsmall angles eo independent of the integer n. Such a representationWAVES ON SLOPING BEACHES AND PAST OBSTACLES73has been given by Friedrichs [F.I 4], To derive it the exact solutionis first obtained for integer n in the form of a singlecomplex integral,which can in turn be treated by the saddle point method to yieldasymptotic solutions valid for large n, that is, for beaches with smallslopes. The resulting asymptotic representation turns out to be veryaccurate. A comparison with the exact numerical solution for co = 6shows the asymptotic solution to be practically identical with theexact solution all the way from infinity to within a distance of lessthan a wave length from the shore line.
Eckart [E.2, 3] has devisedan approximate theory which gives good results in both deep andshallow water.For slope angles which are rational multiples of a right angle ofthe special form co = pji/2n with p any odd integer smaller than 2n,the problem of progressing waves has been treated by Lewy, as wasmentioned above.
Thus the theory is available for cases in which cois greater than jr/2, so that the ''beach" becomes an overhangingcliff. The solution for a special case of this kind, i.e. for co = 135or p = 3, n = 2, has been carried out numerically by E. Isaacson[1.2]. It turns out that there is at least one interesting contrast withthe solutions for waves over beaches in which co < n/2. In the latterit has been found that as a progressing wave moves in towardshore the amplitude first decreases to a value below the value at oo,before it increases and becomes very large at the shore line.
(Thiscasefact has also often been verified experimentally in wave tanks).The same thing holds for standing waves: at a certain distance fromshore there exists always a crest which is lower than the crests at oo.In the case of the overhanging cliff with co135, however, the=maximumgoing outward fromthe shore line is about 1higher than the height of the crests at oo.Still another fact regarding the behavior of the solutions near theshore line is interesting. In all cases there exists just one standingwave solution which has a finite amplitude at the shore line; Lewy[L.8] has shown that the ratio of the amplitude there to the amreverseisfound to be true: the first%11given in terms of the angle co by the formula (n/2co) *.Thus for angles co less than n/2 the amplitude of the standing wavewith finite amplitude is greater on shore than it is at infinity (becomingplitude at ooisvery large as co becomes small) while for angles co greater than jr/2the amplitude on shore is less than it is at oo.
Since the observationsindicate that the standing wave of finite amplitude is likely to be thewave which actually occurs in nature for angles co greater thanWATER WAVES74about 40, the above results can be used to give a rational explanationfor what might be called the "wine glass" effect: wine is much moreapt to spill over the edge of a glass with an edge which is flared outward than from a glass with an edge turned over slightly toward theinside of the glass.limit case of theAproblem of the overhanging cliff has a specialco approaches the value n and theproblem becomes what might be called the "dock problem": thewater surface is free up to a certain point but from there on it isinterest, namely the case in whichcovered by a rigid horizontal plane.
The solutions given by Lewyare so complicated as p and n become large that it seems hopelessto consider the limit of his solutions as co -> n. Friedrichs and Lewy[F.I 2] have, however, attacked and solved the dock problem directlyfor two-dimensional waves. For three-dimensional waves in water ofconstant finite depth the problem has been solved by Hcins [H.I 3](also see [H.12]).It would be somewhat unsatisfying to have solutions for the slopingbeach problem only for slope angles which are rational multiplesis imposed by the methods usedand not by any inherent characteristics of theproblem itself. The two-dimensional problem has, in fact, been solvedfor all slope angles by Isaacson [I.I].
Isaacson obtained an integralrepresentation of Lewy's solutions for the angles pn/2n analogousto the representation obtained by Friedrichs for the angles n/2n,and then observed that his representation depended only upon theratio of p to n and not on these quantities separately. Thus theof n:it isclear that this limitationto solve the problemsolutions for all angles are givenbythis representation. Peters [P.5]has solved the same problem by an entirely different method, whichmakes no use of solutions for the special slope angles pyi/2n.The problem of two-dimensional progressing waves over slopingbeaches thus has been completely solved as far as the theory ofwaves of small amplitude is concerned.
Only one solution for threedimensional motion has been mentioned so far, i.e. the solution byHeins for three-dimensional motion in the case of the dock problem.For certain slope angles co = n/2n the method used by the author[S.18] can be extended in such a way as to solve the problem ofthree-dimensional waves on sloping beaches; in the paper cited thesolution is carried out for the case co = n/2 i.e. for the case of wavesapproaching at an angle and breaking on a vertical cliff. Roseau[R.9] has used the same method for the case co = jr/4. Subsequently9WAVES ON SLOPING BEACHES AND PAST OBSTACLES75the problem of three-dimensional waves on sloping beaches has beensolved by Peters [P.6] and Roseau [R.9], who make use of a certainfunctional equation derived from a representation of the solutionby a Laplace integral.
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