J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 26
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In addition, the next terms in the asymptotic expansion, oforder 1/V0 are a ' so determined, as follows:inin,0<0<ft,fta^2-1 a 2V^2?-__O__a9cos--aoccos(5.5.56)'in~/2 V 2ftgftt2V2ftg+ft-a<0<ft+a2a2cos0-a/2V2ft0 cos2We observe,i/2v 2ft/)+ai0+a2that these expansions do not hold uniformly in becausena, i.e. at the lines ofof zeros in the denominators for= idiscontinuity of the function g(p, 0).With the aid of the function g(p, 0) defined in (5.5.56)a function(5.5.57)Thus h/>(p,/({),0)define= g({>, 0) + h(Q, 0).of necessity a discontinuous function sincewhile g has jump discontinuities along the linesfunction h is given by (cf. (5.5.50)):iswe0) by the equation/ is=continuousfta.TheWATER WAVES130(5.5.58)with the proviso that the integrals should be deformed into theupper half-plane in the vicinity of the origin in case either A_ or k +vanishes:i.e.,has one ofin caseThat the sum g+hreallyour solutionisand thatlight of our discussion above;which just compensate those of gWealso be easily verified.twoitscriticalvaluesna.rather clear in the/ ishasitjump discontinuitiesmake / continuous canin order toshall not carry out the calculation here.view of (5.5.56) and (5.5.57) thus yields whatmight be called the "scattered" part of the wave.In order to show that our solution / satisfies the conditions of theThe functionA(g, 0), inuniqueness theorem proved above, we proceed to show that h asdefined by (5.5.58) satisfies the radiation condition (5.5.9); afterwardswe will prove that / behaves at theOur solution / will thus be provedThat the functionconditionisnot atA(p, 6) definedallorigin as prescribedto be unique.byby(5.5.11).(5.5.58) satisfies the radiationobvious: one sees, for example(cf.
(5.5.50)'),from being uniform in the angle 0.In fact, the transformation of h to be introduced below is motivatedby the desire to obtain an estimate for the quantity dh/dQ + ihthatitsbehavior at ooisfarl>|which figures in the radiation condition, that is independent of 0;and this in turn means an estimate independent of the quantitiesA- and A + defined by (5.5.49). The function A(g, 0) canbe put in the form~ *e Q d*fc/Me*m=m6)= -~M hLWe-A2Jo(as one readily verifies.,r-*~*fjM---.j2A_+LVr3Joproceed as follows: FirstwewritefV^ f Y<^>'21("V* fV^'J'dfatt +A,JoJoJofirst of allJothen carry out the integrations with respect to A to obtain*~i(?r,-7--U--2VmFromLthis representation of3h5p+ tA =tr*__**-**r ---i+V,Jo (pA_-4V7ri L+0*00r^**-iJo((?+02h we obtain00r,5fJo/-dt---3+^'.,+t)i00f/'A------I,Jo+)iJWAVES ON SLOPING BEACHES AND PAST OBSTACLES1312important to observe that A _ and Al have pure imaginary values,as we know from (5.5.49). We also observe that the exceptionalItisnwhich correspond to A^ =0, simply have the effectthat one of the two terms in the brackets in the last equation vanishes.From the Schwarz inequality we havelinesa,+fih2*1IX-eIi___!_&nweright;''/*'f<dt2IJofind:002rf<2rX,Iterm on thefirst(Itif2!Jo (gIAJo (o +t)lConsider thertAe^- l diIQ\d~2GO/*\dh+ <)zJo(eSince the same estimate holds for the second term,dhanditfollows that.this estimate holds for all values of 0, since it holds for theexceptional values 6rangefgrgj2n.=na as well as foralltwoother values in theWe have thus verified that the radiation conditionwe have shownit holds in thestrong form.behavesatthe origin.properly/(p, 6)To this end, we start with the solution in the form (cf.
(5.5.48) andholdsin fact,Weproceed tothatshow that(5.5.45)):=Lthe path of Fig. (5.5.7). The transformation A\/'z is thenis like themade, so that the new path of integrationpath C incircularnowhasaradiusthethat5.5.8large enoughpartFig.exceptwithDWeto include the singularities of the integrands in its interior.mayto have the value 1/p, sincetake the radius of the circular part ofDwecare only for small values of Q in the present consideration.Theu then leads to the following formula for /(g, Q):transformation QZ=WATER WAVES132Dwith Dj a path of the same type asofDislexcept that the circular partnow the circle of unit radius. For small values of Q the integralsin the last expressioncan be expressed..u./*thisform\duftW+ e*^Fromin the6^wdu+....expansion we see clearly thatas0.This completes the verification of the conditions needed for theapplication of the uniqueness theorem to our solution /.It has been shown by Putnam and Arthur [P.18] (see also CarrandStelzriede [C.I]) that the theory of diffraction of waterFig.
5.5.9.around a vertical barrierwavesWaves behind a breakwaterisingood accord with the physicalfacts,the accuracy being particularly high in the shadow created by theWAVES ON SLOPING BEACHES AND PAST OBSTACLES133breakwater. Figure 5.5.9 is a photograph (given to the author byH. Carr of the Hydrodynamics Laboratory at the CaliforniaInstitute of Technology) of a model of a breakwater which gives someindication of the wave pattern which results.J.5.6. Brief discussions of additional applicationsand of other methodsof solutionThe object of the presentsectionisto point out a few furtherproblems and methods of dealing with problems concerned, for themost part, with simple harmonic waves of small amplitude.The first group of problems to be mentioned belongs, generallyspeaking, to the field of oceanography.
For general treatments ofthis subject the book of Sverdrup, Johnson, and Fleming [S.32]should be consulted. One type of problem of this category whichwas investigated vigorously during World Warthe problem ofwave refraction along a coast, or, in other terms, the problem ofthe modification in the shape of the wave crests and in the amplitudeIIisof ocean waves as they move from deep water into shallow water.have seen in the preceding sections that it is not entirely easyWeto give exact solutions in terms of the theory ofwaves of smallamplitude even in relatively simple cases, such, for example, as thecase of a uniformly sloping bottom. As a consequence, approximatemethods modeled after those of geometrical optics were devised,beginning with the work of Sverdrup and Munk [S.35]. Basically,these methods boildownto the assumption that the local propagationspeed of a wave of given length is known at any point from the formulas derived in Chapter 4 for water of constant depth once thedepth of the water at that point is known; and that Iluygens' principle,or variants of it, can be used to locate wave fronts or to constructthe rays orthogonal to them.
The errors resulting from such an assumption should not be very great in practice since the depth variations are usually rather gradual. Various schemes of a graphicalcharacter have been devised to exploit this idea, for example byJohnson, O'Brien, and Isaacs [J.7], Arthur [A.3], Munk and Traylor[M.16], Suquet [S.30], and Pierson [P.8].
Figure 5.6.1 is a refractiondiagram for waves passing over a shoal in an otherwise level bottomin the form of a flat circular hump, and Fig. 5.6.2 is a picture of theactual waves. Both figures were taken from a paper by Pierson [P.8],and they refer to waves in an experimental tank. As one sees, thereWATER WAVES134Fig. 5.6.1. Theoreticalwavecrest-orthogonal pattern for waves passing over aclock glass. No phase shifteven good agreementis fair general agreement in the wave patternsin detail over a good part of the area.
However, near the center ofthe figures there are considerable discrepancies, since the theoreticaldiagram shows, for instance, a sharp point in one of the wavecrestsWAVES ON SLOPING BEACHES AND PAST OBSTACLESwhichislacking in the photograph.The135that thereis a caustictheby geometrical optics (i.e.orthogonalsto the wave crests have an envelope), and in thevicinity of such aregion the approximation by geometrical optics is not good.
One offactisin the rays constructedFig. 5.6.2.Shadowgraphforwaves of moderate length passing over a clockglassthe interesting features of Fig. 5.6.2 is that the shoal in the bottomresults in wave crests which cross each other on the lee side of theshoal, although thewaves. Figure 5.6.3oncoming waves form a single train of planean aerial photograph (again taken from theisWATER WAVES130paper by Pierson) showing the same effect in the ocean at a pointoff the coast of New Jersey; the arrow points to a region where therewould appear to be three wave trains intersecting, but all of themappear to arise from a single train coming in from deep water.Fig.
5.6.3. Aerialphotograph at Great EggInlet,NewJerseyIn the case of sufficiently shallow water Lowell [L.I 6] has studiedthe conditions under which the approximation by geometrical opticsis valid; his starting point is the linear shallow water theory (forwhich see Ch. 10.13) in which the propagation speed of waves isVgh, with h the depth of the water, and it is thus independent ofthe wave length. Eckart [E.2, 3] has devised an approximate theoryWAVES ON SLOPING BEACHES AND PAST OBSTACLES187which makes it possible to deal with waves in both deep and shallowwater, as well as in the transition region between the two.There is an interesting application of the theory of water waves toa problem in seismology which will be explained here even though itis necessary to go somewhat beyond the lineartheory on which thisofthebookisbased.WehaveseeninpartChapter 3 above that thedisplacements, velocities, and pressure variations in a simple harmonicstanding wave die out exponentially in water of infinite depth.However, it was pointed out by Miche [M.
8] that this is true only offirst order terms in the development of the basic nonlinear theorywith respect to the wave amplitude; if the development is carried outformally to second order it turns out that the pressure fluctuateswith an amplitude that does not die out with the depth, but dependson the square of the amplitude. (For progressing waves, this is nottrue. ) In addition, the second order pressure variation has a frequencywhich is double the frequency of the linear standing wave.
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