J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 35
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Thus / x is indeed of order I/A:, as statedabove. It might be noted that this argument really does not requirethe analyticity of <p and \p but only that the integrands be integrable9and that integration by parts may be performed. Infinite limits for theintegrals could also be permitted if cp(x) and \p(x) behave appropriatelyat oo.UNSTEADY MOTIONS183Suppose now that <p(x) has one stationary value at x = a in thee l ^ x ^ a + i> s l >i.e., <p'(# ) vanishes only atsegment a=xa in this interval. Suppose, in addition, that the second derivative y" (x) does not vanish at xa, and indeed is positive there:incasewhich0.>(The<p"(<x.)9/'(a) is negative and the more critical=will be discussed later.) We shall show thatcase in which 9?"(a)a positive number e ^ s l exists such that9/,(*)(6.8.2)=dx(*)'*=oe-,In other words, we shall show that a fixed segment of length %e containing a exists such that its contribution to / is independent of e andisof order l/\/k, with an error of order I/A;.these statements we begin by introducingTo provenewvariablesas follows:x(6.8.3)Consider(6.8.4)first/ a (fc,=a+ u,=<p(x)y(a)+ w(u).the integral / 2 (&, e^:cx )- ^^W^fcM)(tt)+ M) du = ^^^(a(a)J.J-iItisJconvenient to write the integral(6.8.5)Jas therff^(i) y ( a==+ MI)duj+Since99(0?)has a+Jiinw(t4 2 ) in the intervalat x52 =--t isand J 2 become,(6.8.7)+ ^2) ^2y( a^i/2^isx.e1that w(u l )^% ^0,isa positiveand likewiseHence we may introduce a newfurthermore^%w(u l )ine1w(u 2 )in^ureal, in5^ 0,each of thein-and.\t/!ie lkw(u *>a, it followsthe intervalintegration variable t, whichtegrals, defined as follows:In each intervalof two terms:t/ 2 .minimummonotonic function(6,8.6)sumJoJ-ej-e2<^taken as the positive square root.as one readily sees:TheintegralsWATER WAVES184with==tfjVw(solutions ofand t 2 = Vwfa).
The functions %() w 2 (tf) are2For w(u) we have the power series develop-fii),w(ui )=t.ment(6.8.8)w(u)since w?(0)==<p(ocw'(Q)+ u)=9?(a)<fii-3.3.=2a 2(6.8.3)); in addition(cf.contains the entire intervalSince+aw +a 9u 2.g/'()>>We suppose that this series converges in a circle whichby assumption.e2=J2=w>(w 2 )2we may^u ^in2itsinterior,withwritefor<Su2^and=2Since a 2 7^we may express the square roots as power series in u iand then invert the series to obtain u^ and u 2 as power series in t, asfollows:(6.8.10)u,withCj= +=+cc,2<+2V/ 2/9?"(a). Hence we may....writePwhich P^(t) and 2 (0 are convergent power series. It may be thatthese series do not converge up to the values ^ and t 2 of the upper limits of the above integrals J l and J 2 in (6.8.7).
In that case we simplyinassume the length of the segmentistaken to bestill lessthan 2e 2 sopermissible and the seriesandPtoPi(t)appropriate values ?x and 2 It is2 (0 converge upclear that numbers ?x and 12 with these properties exist. The integralsJ t and J 2 may now be written in the form*that the inversion of the series (6.8.8)is.=fjhe* {^(a) + tP^t)}anddt,_(6.8.11)J2-t8c lV (a)pV*'jo<tt+ jo( *e*tPt (t)dt=J3+J4.* Therequirement of analyticity for (p and y is used to permit this simpleintroduction of t as variable of integration. However, the existence of a finitenumber of derivatives would clearlv have sufficed.UNSTEADY MOTIONSWe proceed to study the integrals J 3 andasnewJ 3 wevariable inpiQpi.(6.8.12)rci^(a)2 \/k J od8de=e__dQ\/0/*oopiQdB/*oo-Joby a known formula.
The lastorder l/\/& by integration bypiQ\\V^Jofirstkt 2introducing 6write(*JctaTheUpon.obtain_But we may/4185J kt 2 \/v\/^integral can now beparts, as follows:shownto be ofcontribution on the right hand side is obviously of order12 is a fixed number; as for the second, we havesincepvL-dBand hence the second contributionwe have the resultJ(6.8.13)TheintegralThusalso of order l/\/k.forJ3=is./ 4isfirstintegratedbyparts to obtaintJo-P,(0)and hence(6.8.14)|J4|^and the right hand1sideviously be treated in the|isP,(0)thus of orderP't (t)|1/fc.Thedi|integralJ l canob-same way as J 2 and with an exactly analogousWATER WAVES186result;consequentlywe have from (6.8.13), and (6.8.14) for the integralgiven in (6.8.4) the result(6.8.15)!,<*, e 3 )'** w+ i= /,(*) = y(a)+once 3 has been chosen small enough.
One observes how it comesabout that the lowest order term is independent of the values of ?x and?2 and hence of the length of the segment: the entire argument requires only that ?x and 2 be any fixed positive numbers since one needsonly the fact that the products kl\ and kl\ grow large with k.If (p (x) had been assumed to have a maximum at xa, withdifferencewouldbeandthethat0,rc/4onlyk<p"(u.)9/'(<x)would appear in the final formula instead ofk(p"(<x.) and + vt/4.,<+Consequently, in all cases in(6.8.16)I,(k)which<p" (a)^we have= y() (^LU\<p(a)and the sign of the term w/4 should agree with the sign of=<p"(oc).but 9?'"(oc) ^it is not difficult toderive the appropriate asymptotic formula for /(ft).
In fact, the stepsare nearly identical with those taken just now for the case <p"(oc) ^ 0.One introduces x = a + u, <p(x) = y(a) + ^( w ) as before and thenFinally, in case <p"(oc)0,makes use of power series in the variable t defined by 2 3same way as above. The result is, for e sufficiently(6.8.17)I 2 (k)==w(u )in thesmall:a+fV**<*) y(x) dxwhere F(\) refers to the gamma function. Hence the contributionarising from the stationary point is now of a different order of magni1 31 2tude, i.e., of order 1/& / instead of I/ft / This fact is of significancein the case of the ship wave problem which will be treated later..Naturally the lowest order terms in I(k) consist of a sum of termsfurnished by the contributions of all of the points of stationary phasein the interval S.
It is important enough to bear repetition that if nosuch points exist, then I(k) is in general of order 1/fc.In case a stationary point falls at an end point xb ofa or xthe interval of integration, one sees readily that the contribution=furnishedby such a pointto I(k)is=the same as that given above inUNSTEADY MOTIONS^187case <p"except that a factor 1/2 would appear in the final result.On the other hand, if g/'at an end point, then thebut <p'"contribution differs in phase as well as in the numerical factor from=^the contribution given above in (6.8.17).Atime-dependent Green's function. Uniqueness of unsteadymotions in unbounded domains when obstacles are present6.9.In sec.
6.2 above the uniqueness of unsteady wave motions forwater confined to a vessel of finite dimensions was proved. More general results have been obtained by Kotik [K.17], Kamp de Feriet andKotik [K.l], and Finkelstein [F.3] with regard to such uniquenessquestions. In the present section a rather general uniqueness theoremwill be proved, following the methods of Finkelstein, who, unlike theother authors mentioned, obtains uniqueness theorems when obstaclesarc present in the water. The essential tool for this purpose is a timedependent Green's function, which is in itself of interest and worthwhile discussing for its own sake quite apart from its use in derivinguniqueness theorems.
With the aid of such a function, for example, allof the problems solved in the preceding sections can be solved oncemore in a different fashion, and still other and more complicated unsolved problems can be reduced to solving an integral equation, asweshall see.Wetime-dependent Green's function in questionmotion in water of infinite depth,nowouldbedifficulty to obtain it in other cases asalthough therewell. The Green's function G in question is required to be a harmonicfunction in the variables (x, y, z) with a singularity of appropriatecharacter at a certain point (, 77, ) which is introduced at the timer andt = r and maintained thereafter; thus G depends upon f 77, fx, y, z; t: GG( r], f r x, y, z; t).
In fact, G is the velocity potential which yields the solution of the following water wave problem:A certain disturbance is initiated at the point (, 77, ) at the timetr. The pressure on the free surface of the water is assumed to ber the water is assumed to have beenzero always, and at the time tat rest in its equilibrium position. Since G is a harmonic function inshall derive thefor the case of three-dimensional=,,;;\reasonable to expect that the correct singularity to imposeat the point (, rj, ) in order that it should have the properties onelikes a Green's function to have is that it behaves there like I/R, withx, y, z it isR^V^-r-(i?)* -Hfa).ThusGshould satisfy theWATER WAVES188following conditions: It should be a solution of the Laplace equation+G =G xx + G yy(6.9.1)-forZZoo< y < 0,t^T,satisfying the free surface conditionGG, G(6.9.2)tt+ gGy ==y0,0.and their first derivatives to be uniformlyAt oo we requiretbounded at any given time t.
(Actually, they will be seen to tend towe requirezero at oo.) At the point77, f,G(6.9.3)Asinitialconditions at the timeG=G =(6.9.4)wewe havertfortAs weWeto be bounded.Rt=shall see later on, these conditionsGproceed to construct the function=yr,sec. 6.1)(cf.0.Gdetermineexplicitly.uniquely.As afirststepset(6.9.5)G(f,;17,T|v, y, *; t)= A(,f17,|*, y, a)+5(1,with^f(6.9.6)*?,C;T|a?, /y,z;A=1 /t-L with'=V(f/v-V)Tfo "+2/F+ G ^^-contains the prescribed singularity, and we may requireto be regular. Sinceis a harmonic function, it follows that BThus^4Aharmonic; in addition,(6.9.7))defined byB +gBy =BsatisfiesBisthe free surface conditionttHex)--t- r)'z)"]"-t- (t,di= 0'r at+ + (C ~ *)as one can readily verify. To determine B from this and the other con9*7 [(Iditions arising-*)from those imposed on2in preceding sections.]G it would be possible to employthe Hankel transform in exactly the sameform was used2 1/22*?wayHowever,as the Fourier transitseems betterin thepresent case to proceed directly by using the special, but well-known,Hankel transform for the function e- b */s (cf., for example, Sneddon[S.ll], p.
528); this yields theformulaUNSTEADY MOTIONSir=-(6.9.8)Va +2e~ bsJ189(as) ds,Job*> 0. By means of this formula the right hand side of (6.9.7)can be written in a different form to yieldvalid for bB + gB y =(6.9.9)tt2g JL fV< J=(sr) dsfyJo*>*2g fJoJQ (sr)dsat yvalid forr\<and with=r(6.9.10)V(S-x)*+B isSinceseek9B=(6.9.11)-(Ca harmonic function in x y,among functions of the formit=*)z, itI sT(t, s)e *+ti'Jo(J2.would seem reasonable to(sr) ds,which are harmonic functions.
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