J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 27
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(It is nothard to see in a general way how this latter nonlinear effect arisesmathematically. In the Bernoulli law, the nonlinear term of the form0y + 0% would lead, through an iteration process starting with= Aemy cos mx cos at, to terms involving cos 2 at and thus toharmonics with the double frequency.) It happens that seismic wavesin the earth of very small amplitudescalled microseismsand offrom310oftoseismosecondsareobservedsensitiveperiodsbygraphs; these waves seem unlikely to be the result of earthquakes orlocal causes; rather, a close connection between microseisms and disturbed weather conditions over the ocean was noticed.
However, sinceit was thought that surface waves in the ocean lead to pressurevariations which die out so rapidly in the depth that they could notbe expected to generate observable waves in the earth, it was thoughtunlikely that storms at sea could be a cause for microseisms. The resultof Miche stated above was invoked by Longuet-Higgins and Ursell[L. 14] in 1948 to revive the idea that storms at sea can be the originof microseisms. (See also the paper of 1950 by Longuet-Higgins[L.13].) In addition, Bernard [B.8] had collected evidence in 1941theindicating that the frequency of microseisms near Casablanca wasjust double that of sea waves reaching the coast nearby; the sameratio of frequencies was noticed by Deacon [D. 6] with respect tomicroseisms recorded at Kew and waves recorded on the north coastof Cornwall.
Further confirmation of the correlation between seawaves and microseisms is given in the paper of Darby shire [D. 4].WATER WAVES138A reasonable explanation for the origin of microseisms thus seems to beOfthat standingwaves are generated, but Longuet-Higgins has shown that the neededeffects are present any time that two trains of progressing wavesavailable.course,this explanation presupposesin opposite directions are superimposed, and itimagine that such things would occur in a storm areamovingisnot hard tofor example,through the superposition of waves generated in different portions ofa given storm area. It might be added that Cooper and LonguetHiggins [C.3] have carried out experiments which confirm quantitatively the validity of the Miche theory of nonlinear standing waves.
Itis perhaps also of interest to refer to a paper by Danel [D.2] in whichstanding waves of large amplitude with sharp crests are discussed.In Chapter 6 some references will be made to interesting studiesconcerning the location of storms at sea as determined by observations on shore of the long waves which travel at relatively high speedsoutward from the storm area (cf. the paper by Deacon [D.6]).of predicting the character of wave conditionsofshoreacourse, interesting for a variety of reasons,is,givenalongreasons(see Bates [B.6], for example). Methodsincluding militaryfor the forecasting of waves and swell, and of breakers and surf aretreated in two pamphlets [U.I, 2] issued by the U.S. HydrographicThe general problemOffice.Anecessary preliminary to forecasting studies, in general, is aninvestigation of ways and means of recording, analyzing, and representing mathematically the surface of the ocean as it actually occursin nature.
Among those who have studied such questions we mentionhere Seiwell [S.9, 10] and Pierson [P.IO'J. The latter author concernshimself particularly with the problem of obtaining mathematical representations of the sea surface which are on the one hand sufficientlyaccurate, and on the other hand not so complicated as to be practicallyunusable. The surface of the open sea is, in fact, usually extraordinarily complicated. Figure 5.0.4 is a photograph of the sea (taken fromthe paper by Pierson) which bears this out.
Pierson first tries representations employing the Fourier integral and comes to the con-clusion that such representationstheir use. (In Chapter 6 we shallwould be so awkward as to precludehave an opportunity to see that it isindeed not easy to discuss the results of such representations even formotions generated in the simplest conceivable fashion by applyingan impulse at a point of the surface when the water is initially at rest,for example.) Pierson then goes on to advocate a statistical approachWAVES ON SLOPING BEACHES AND PAST OBSTACLES139to the problem in which various of the important parameters areassumed to be distributed according to a Gaussian law.
These developments are far too extensive for inclusion in this book -besides, theFig. 5.6.4. Surfacewaves on the open seaauthor is, by temperament, more interested in deterministic theoriesin mechanics than in those employing arguments from probabilityand statistics, while knowing at the same time that such methods arevery often the best and most appropriate for dealing with the complex problems which arise concretely in practice. It would, however,seem to the author to be likely that any mathematical representationsof the surface of the sea whether by the Fourier integral or any otherwould of necessity be complex and cumbersome in proporintegralstion to the complexity of that surface and the degree to which detailsare desired.Before leaving this subject, it is of interest to examine another photograph of waves given by Pierson [P.10], and shown in Fig. 5.6.5.Near the right hand edge of the picture the wave crests of the pre-dominant system are turned at about 45 to the coastline,and theyare broken rather than continuous; such wave systems are said to beshort-crested.
About half-way toward shore it is seen that thesewaves have arranged themselves more nearly parallel to the coast(indicating, of course, that the water has become shallower) and atthe same time the crests are longer and less broken in appearance,140WATER WAVESFig. 5.6.5. Aerialphotograph over OracokeWAVES ON SLOPING BEACHES AND PAST OBSTACLES141though no single one of them can be identified for any great distance.Near the shore, the wave crests are relatively long and nearly parallelto it.
On the photograph a second train of waves having a shorterwave length and smaller amplitude can be detected; these waves aretraveling almost at right angles to the shore (they are probably causedby a breeze blowing along the shore) and they are practically notdiffracted. Each of the two wave trains appears to move as thoughthe other were not present: the case of a linear superposition wouldthus seem to be realized here. One observes also that there is a shoal,as evidenced by the crossed wave trains and the white-water due tobreaking over the shoals.Wepass next to a brief discussion of a few problems in which ouremphasis is on the methods of solution, which are different from thoseemployedin the preceding chapters ofPartII.Thefirstsuch problemto be discussed employs what is called the Wicner-Hopf method ofsolving certain types of boundary problems by means of an ingenious,though somewhat complicated, procedure which utilizes an integralequation of a special form.
This method has been used, as was mentioned in the introduction to this chapter, by Heins [11.12, 13] and byKeller and Weitz [K.9] to solve the dock problem and other problemshaving a similar character with respect to the geometry of the domainsin which the solution is sought. However, it is simpler to explain theunderlying ideas of the method by treating a different problem, i.e.the problem of diffraction of waves around a vertical half-plane inother words, Sommerfeld's diffraction problem, which was treated bya different method in the preceding section. We outline the method,following the presentation of Karp [K.3].
The mathematical formulation of the problem is as follows. A solution q>(a\ y) of the reducedwave equationVV + &V =(5.6.1)isto be found subject to the(5.6.2)<p y=boundary conditionfory=x0,in the domain excluding this raya solution in the formand regulartion,(jp(5.6.4)(cf.Fig. 5.6.6). In addi-<P=<PQ+Vi(5.6.3)with>Qdefined by<p=elk <* cos e+ v sin V,<<27r,WATER WAVES142and with <p prescribed to die out at oo is wanted.
In other words, aplane wave comes from infinity in a direction determined by the angleand the scattered wave caused by the presence of the screen, and,Fig. 5.6.6. Diffractionaround a screenv is to be found. It is a peculiarity of the Wienor-IIopfnot only in the present problem but in other applications todiffraction problems as wellthat the constant k is assumed to be anumberthana real number, as in the preceding(rathercomplexsection ) given, say, by kkik^ with & 2 small and positive. Withthis stipulation it is possible to dispense with conditions on (p of theradiation type at oo, and to replace them by boundedness conditions.We employ a Green's function in order to obtain a representationof the solution in the form of an integral equation of the type to whichthe Wiener-Hopf technique applies.
In the present case the Green'sgiven by<pmethod=function G(x y\ #+y ) is defined as that solution of (5.6.1) in thewhole plane which has a logarithmic singularity at the point (# y Q )and dies out at oo (here the fact that k is complex plays a role). This9,,functioniswell-known;it is,in fact, theHankel functionof the first kind:(5.6.5)G(x, y;x,y)=4(k[(x-#2)+(y-The next step is to apply Green's formula to the functions 9? and Gthe domain bounded by the circle C 2 and the curves marked C xininBecause of the fact that G is symmetric, has a logarithmicsingularity, and that 99 and G both satisfy (5.6.1), it follows by arguments that proceed exactly as in potential theory in similar cases thaty) can be represented in the formFig.
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