J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 36
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The free surface conditionnow be satisfied, as one can easily sec, if T(t) satisfies the(6.9.9) willdifferentialequationT +g*T =(6.9.12)ttThe functionTisinitialconditionsresultis(6.9.13)now uniquely determined fromT=T =tT(t, s)Thus we have2g.forGfort=(6.9.12)and theT derived from (6.9.4); the= 2 LTthe function(6.9.14)+2 f "VtH-*)[i_cosJoandclearly satisfies all of the conditions prescribed above, exceptpossibly the conditions at oo, which we shall presently investigate insome detail because of later requirements.
Before doing so, however,itwe observe/,i.e.thatG is symmetrical not only in thebut also in the time variables r andthe important fact thatspace variables f,77,and#, y, z 9WATER WAVES190G(,(6.9.15)f; rr),= G(x, y,z\t\y, *; t)a?,|f,??,f r) and;We turn next to the discussion of the behavior of G at oo.
Considerfirst the function A == l/R1/i?'. This function evidently willbehave atitoo likea dipole; hencefollows thatoriginfor large a:Aanda represents distance from theA a behave as followsifradial derivativeitsA ~1(TOnwe havethe free surface where yA=3To determinethe behavior ofexpandcosB=(6.9.18)Vgs= 0,andy =for yforl/a[12(rBi.e.in at)]large a.of the integral in (6.9.14)powerseries in ry negative.
The=4R Jo(*(6.9.19)and fromitwe obtain==//and write2It is clearly legitimate to integrate term-wise forformula (6.9.8) can be expressed in the formwithtwecos0,It follows, sincer$nby a well-known formula for spherical harmonics.are bounded functions, that the leading termP n (//)asymptotic expansion of B arises from the first term in thesquare bracket. Hence the behavior of B is seen from (6.9.20) for thecase- nI to be given byin the=l?~l/(r(6.9.21)2,a large and any fixed values of r and t. The derivative B y is seen,also from (6.9.20), to behave like I/a 3 and the derivative B r also canbe seen to behave like I/a 3 thus the radial derivative Ba behaves inthe same way and we havefor;(6.9.22)Ba ~I/a3,Bv ~I/a3.UNSTEADY MOTIONS191Summing up, we have for the Green's function G the following behaviorat oo:G ~Ga ~Gy ~(6.9.23)I/aI/aI/a233.All of these conditions hold uniformly forthe values of r and t.any fixedfiniteranges inWe turn next to the consideration of a water wave problem of very<is filled with water andgeneral character, as follows.
The space yin addition there are immersed surfaces S t of finite dimensions havinga prescribed motion (which, of course, must of necessity be a motion ofsmall amplitude near to a rest position of equilibrium). The pressureon the free surface S f is prescribed for all time, and the initial positionand velocity of the particles on the free surface and the immersedsurfaces are given at the time t = 0. At infinity the displacement andvelocity of all particles are assumed to be bounded.
The resultingin terms of a velocitymotion can be described for all times t >whichsatisfiesof the kind studied inconditionspotential 0(x y, z; t)9the first section of this chapter; these conditions are:(6.9.24)V2,in the regionRV(Z=<exterior to the imconsisting of the half space yOn the free surface the conditionmersed surfaces S^(6.9.25)tt+g0 v = ~-Pt=:P(xz;t)999t>0,y=Qis prescribed, with p the given surface pressure (cf. (6.1.1 ) and (6.1.2)).At the equilibrium position of the immersed surfaces the condition(6.9.26)withn=Von S i9V the normal velocity of S=St at totherwiseis,of course, assumed(cf. 6.1) thatwe knowthe free surface at(6.9.27)i9tt^0,prescribed.
The initial position ofknown, and for the initial conditionsisitsuffices to prescribeandton0:j^O^O)^*,*)#(*,0,*;0) = /,(*,*).IAtoowe assume that 0,bounded.tandtheir first derivatives are uniformlyWATER WAVES192Webyproceed now to set up a representation for the functionInobtainedabove.caseGreen'sfunctiontherearenoimtheusingmersed surfaces this representation furnishes an explicit solution ofthe problem, and in the other cases it leads to an integral equation forit.
In all cases, however, a uniqueness theorem can be obtained.To carry out this program we begin, in the usual fashion, by applyingGreen's formula to the Green's function G and to t (rather than 0)in a sphere centered at the origin of radius a large enough to includethe immersed surfaces and the singular point (|, 77, ) of the Green'sfunction minus a small sphere of radius e centered at the singular point.are both harmonic functions and G behaves like l/RSince G andtat the singular point,ittheory thatt) ist(x 9 y> z;followsby the usual argumentsin potentialobtained in the form of a surface integral,as follows:(6.9.28)tThe symmetryof(x, y,=-?- (T4rc JJ(G0tn~ &tG n)dS.G has been used at this point. The integration varia-Even though6? depends on the difference tr thedepends only on t; that is, only the singular part ofthe behavior of G matters in applying Green's formula, and the reon the time at which t andt depends onlysulting expression fortn are measured.
The surface integral is taken over the boundaryof the region just described (cf. Fig. 6.9.1), and n is the normal takenbles are f,77,.integral in (6.9.28)JkysFig.6.9.1.Domainflfor application of Green's formulaoutward from the region. The boundaryiscomposed of threedifferentparts: the portion of the sphere S a of radius a lying below the planecut out by the sphere S a9 and0, the part S f of the plane yy==UNSTEADY MOTIONSthe immersed surfacesplane y=St193(which might possibly cut out portions of the0).important to show first of all that the contribution to thesurface integral provided by S a tends to zero as a -> oo, and that theintegral over S f exists as a -> oo.
The second part is readily shown:The integrand to be studied is G0 tyG y From the symmetryof G and (6.9.23) we see that the above integrand behaves like I/a 2for large a sinceandare assumed to be uniformly bounded attyoo hence the integral converges uniformly in t and r for any fixedItist.t;Garanges of these variables. To show that the integral of G0 totover S a tends to zero for a -> oo requires a lengthier argument. ConGaGalike I/a 3 for large a whileis bounded, it is clear that the integral of this term behaves liketI/a and hence tends to zero as a -> oo. The integral over the remainingsider first thetermis(6.9.29)termt.Sincebehavesbroken up into two parts, as follows:!(0 ia GdSJJ=fJo"f*,Ga*sinOded(oJinl:(a 12) +6+f2n f(nl2)+6* iaGa>tsindddo).JO Jnj2ITheintegrations arc carried out in polar coordinates,and disa smallangle (cf.
Fig. 6.9.2); the second integral represents the contributionfrom a thin strip of the sphere S a adjacent to the free surface. SinceFig. 6.9.2.GThe sphere S abehaves like I/a 2 for large a, it is clear that theabsolute value of the second contribution (i.e. that from the thin strip)can be made less than e/2, say, if d is chosen small enough. Once d hasbeen fixed, it can be seen that the contribution of the remaining partta isbounded andWATER WAVES194of S a can also bemade less than e/2 inonce shownabsolute valueifaistaken largethen clear that the integral in ques->oo.
The proof of this fact is, however,tion vanishes in the limit as anot difficult: we need only observe that t is by assumption boundedenough. If thisisit isandit is a well-known fact* thatta then tends to zero uniformlya along any ray from the origin which makes an angle ^ 6 withthe plane y = 0. Thus the integrand in the first term of (6.9.29)at oolike11behavesandlike I/ait(6.9.30)tmade arbitrarily small bywe now have the representationtherefore can betaking a sufficiently large.
Thus for=(x, y, z; t)ti- [((G0tr)0,Siwhichof course, understood thatany parts of the plane yareomittedinthefirsttintegral. The next step is tosidesofwithboth(6.9.30)respect to t from to r. The resultintegrateincut outit is,by Sis0(x(6.9.31)*9y, z; r)- 0(xy, z; 0)9If [J7=*0=1ffto JJ [LS,(CO*,+TltG-t )gf^G, +JoolttGt )ddt\Sd'C+IJ7=0+ gGy =when G ttas asymbolwhile fort=fory=for the integralwe havetisused(cf. (6.9.2))and /over S^ We have G = G ==/t2,andy\<=s0isintendedfort=r;uniquely determined by /j**from the conditions (6.9.27).
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