J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 11
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Thus ^ is a potential function which is regular and bounded in the entire x, t/-plane. By Liou* This statementthis section.isnot valid in three dimensions as we shall see later on inWATER WAVES40ville'stheoremit istherefore a constant,andsincey=fory=0,the constant must be zero. Hence y vanishes identically.
From (3.1.14)it therefore follows that any solutions (p(x, y) of our boundaryvalue problem are also solutions of the differential equation(3.1.15)nup<pvThe most general=oo0,< y < 0.solution of this differential equation(3.1.16)<p=c(x)eisgiven bymvwith c(x) an arbitrary function of x alone. However,harmonic function and hence c(x) is a solution of<p(x,y)isa+ m*c =(8.1.17)which, in turn, has asits general solution the linear combinationscos mx. It follows, therefore, that the standing wavesolutions of our problem are indeed all of the form Ae mv cos (w#+oc),*of sinmx andAwith a andarbitrary constants fixing the "phase" and the amplitudeis a fixed constant which determines the waveof the wave, whileofthe given frequency a through (3.1.13).intermsAlengthmIn water of uniform finite depth h it is also quite easy to obtaintwo-dimensional standing wave solutions of our boundary valueproblem. One has, corresponding to the solutions (3.1.10) for water ofinfinite depth, the harmonic functions= cosh m(y + h) cos mx,= cosh m(y + h) sm mx,[<pf (p4(d.l.lS)as solutionswhich.satisfy thewhile the free surface conditionstantmsatisfiesboundary condition at the bottom,issatisfiedprovided that the con-the relationa2(3.1.19)= gm tanh mhinstead of the relation (3.1.11), as one readily sees.
Since tanh mh-+Ias h -> oo it is clear that the relation (3.1.19) yields (3.1.11) as limitrelation for water of infinite depth. The uniqueness of the solutions(3.1.18) for the two-dimensional case under the condition of boun-dedness at oo was* It canfirstproved by A. Weinstein [W.7] by a methodnow beseen that the negative sign in the free surface condition (3.1.7)our results: if this sign were reversed one would find that thesolution g? analogous to (8.1.16) would be bounded at oo only for c(x) = 0,because (3.1.16) would now be replaced by c(x)e~ mv with> 0.is decisive for,mSIMPLE HARMONIC OSCILLATIONS41from the method used above for water of infinite depthwhich can not be employed in this case (see [B.
12]).It is of interest to calculate the motion of the individual waterparticles. To this end let 6x and dy represent the displacements fromdifferentmean position (x, y) of a given particle. Our basic assumptionsmean that dx dy and their derivatives are small quantities; it followstherefore that we may writethe9=X= v(x, y) =y=u(x y)ydtddy-m A cos at cosh m(y + h) sin mx= mA coserfsinhm(y+ A) cos mxwithin the accuracy of our basic approximation. The constant A isan arbitrary factor fixing the amplitude of the wave. Hence wehave upon integrationdx=dy=sin at coshm(y+ h) sin mx,(3.1.20)7?? /-isin at sinham(y+/i)cos w#.The motionof each particle takes place in a straight line the directionof which varies from vertical under the wave crests (cos mx1) to==horizontal under the nodes (cos mx0).
The motion also naturallyh.becomes purely horizontal on approaching the bottom yThese consequences of the theory are verified in practice, as indicated=taken from a paper by Ruellan ami Wallet (cf. [R.12]).The photograph at the bottom makes the particle trajectories visible ina standing wave; this is the final specimen in a series of photographs ofparticle trajectories for a range of cases beginning with a pure progressing wave (ef. see. 3.2), and continuing with superpositions of progressing waves traveling in opposite directions and having the samewave length but not the same amplitudes, finally ending with ain Fig. 3.1.1,standingwave when the wave amplitudesof the two trains are equal.We proceed next to study the special class of three-dimensionalstanding waves that are simple harmonic in the time, arid whichdepend only on the distance r from the t/-axis.
In other words, weseek standing waves having cylindrical symmetry. Again we seeksolutions of (3.1.6) which satisfy (3.1.7). Only the case of water ofinfinitedepthwillbe treated here, and hence(3.1.8) is replacedby42WATER WAVESFig. 3.1.1. Particle trajectories in progressingand standing wavesthe condition that the solutions be bounded at oo in the negative//-direction as well as in the x- and ^-directions.
It is once more ofinterest to derive all possible standing wave solutions which areeverywhere regular and bounded at oo because of the fact that thesolutions in the present case behave quite differently from thoseobtained above for motions that are independent of the ^-coordinate.SIMPLE HARMONIC OSCILLATIONS43In particular, we shall see that all bounded standing waves withcylindrical symmetry die out at oo like the inverse square root ofthe distance, while in two dimensions we have seen that the assumption that the wave amplitude dies out at oo leads to waves of zeroamplitude everywhere.make useof cylindrical coordinates in derivingour uniqueness theorem. Thus we write (3.1.6) in the formItnatural tois*(3.1.21)_(r -?^dr Jr or \withr+ -? =dy^0,> -^r<oo,ooj/-axis.
The assumption that q> dependsand y and not on the angle 6 has already been used.only uponFor our purposesQ replacing rit isuseful to introduce aby means of the(3.1.22)gnew independentvariablerelation=logr,terms of which (3.1.21) becomes3V-Z + -Za> =2<?-*(3.1.23)OQ*This equation holds,j/,j/the distance from therin2g-plane.dywe0,2y< 0, -oo<<oo.observe, in the half-plane yto be satisfied at yThe boundary condition<=of theis(cf.(3.1.7)):(3.1.24)cp ymq)=0,m=a 2 /g.Wewish to find all regular solutions of (3.1.23) satisfying (3.1.24)which 99 and <p y are bounded at oo. To this end we proceed alongmuch the same lines as above (cf. (3.1.14) and the reasoning immediately following it) for the case of two dimensions, and introduceforthe function y(g, y) by the identityy(3.1.25)Sincey;=<p ym<p,2/<0,oo<p<oo.involves only a derivative of <p with respect to y and notit follows at once that y is a solution of (3.1.23).with respect to Q=from (3.1.24) it follows easily that it canvanishes at yintothe upper half-plane ybe continued analyticallyby settinglthatthean(functionwill be aresultingy)y)y(>y(>Sinceif)>=solution of (3.1.23) in the entirep,t/-plane.The function\pthusobtained will be bounded in the entire plane, since it was boundedin the lower half plane by virtue of the boundedness assumptions with respect to <p.
A theorem of S. Bernstein now yieldsWATER WAVES44the result that \p is everywhere constant* if it is a uniformly boundedsolution of (3.1.23) in the entire Q, t/-plane. Since ip vanishes on thet/-axis it follows that y vanishes identically. Consequently we conclude from (3.1.25) that(3.1.26)m<p(p yThe most general function= em(3.1.27)satisfies9?<p(g,vf($)(p=the relationy0,< 0.y) satisfying this equation= emvf(log r) =emis^g(r)with g(r) an arbitrary function. But (p(r, y) is also a solution of(3.1.21) and hence g(r) is a solution of the ordinary differentialequation(3.1.28)-:?rdr I',!\ drother words, g(r)or, in(3.1.29)weSincethatg(r)allAfactora Bessel function of= AJ<p(r,g(r, y)+(mr)restricted ourselves tosolutions(3.1.30)withisofrder zero:BY,(mr).boundedsolutions onlyy) of our problem are given=Ae^J^(mr),m=critfollowsby2/g,an arbitrary constant. Upon reintroduction of the timehave, therefore, as the only bounded velocity potentialswethe functions0(r, y;(3.1.31)Asiswellt)= Aeiate m vJ Q (mr).known, these functions behavefor large values of r asfollows:(3.1.32)0(r, y;t)~ AeiaicosI/nmre*'[mr-\\4/and thus they die out like l/Vr as stated above.In two dimensions we were able to find bounded standing waves>of arbitrary phase (in the space variable) at oo.
In the present caseof circular waves we have found bounded waves with only one phaseat oo. However,*Whatifwe wereto permit a logarithmic singularity at theis needed is evidently a generalization of Liouville's theorem to theequation (3.1.23) which has a variable coefficient. The theorem of Bernstein referred to is much more general than is required for this special purpose,but it is also not entirely easy to prove (cf. E. Hopf [H.17] for a proof of it).ellipticSIMPLE HARMONIC OSCILLATIONS45=and thus admit the singular Bessel function Y Q (mr) asa solution of (3.1.28), we would have as possible velocity potentialsthe functionsaxis r0(r, y;(3.1.33)which behaveBe iat em ^Y (mr)for large r as follows:0(r, y;(3.1.34)=t)t)~Be iatev]/*sinnmr(mr\--\.4/Admitting solutions with a logarithmic singularity on the i/-axisthus leads to standing waves which behave at oo in the same wayas those which are everywhere bounded, except that they differ by90 in phase at oo.
Thus waves having an arbitrary phase at oo canbe constructed, but not without allowing a singularity. It has, howevernot been shown that (3.1.31) and (3.1.33) yield all solutions with thisproperty.3.2.Simple harmonic progressing wavesSince our boundary problem is linear and homogeneous we canreintroduce the time factors cos at and sin at and take appropriatelinear combinations of the standing waves (3.1.5) to obtain simpleharmonic progressing wave solutionsthe form= A cosh m(y + h) cos(3.2.1)withmwater of uniform depth ofin(mxat+ a)and a satisfyinga2(3.2.2)=gm tanh mh,as before.The wave,or phase speed cc(3.2.3)or, inc=Xisbya/m,2n/m bytanh *!*.'==terms of the wave length A(3.2.3),Itof course givenis9%7l/useful to write the relation (3.2.2) in terms of the2n/m and then expand the function tanh mh in ato obtainwave lengthpower seriesWATER WAVES46Wesee therefore that2a 2 ->(3.2.5)(y)^=m*& h as\-**and hence thatVghc^=L(3.2.6)ifA/Aissmall.This last relation embodies the important fact that the wave speedbecomes independent of the wave length when the depth is small comparedwith the wave length, but varies as the square root of the depth.
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