J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 31
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Indeed, the asymptotic representation yields all of the qualitative features contained in the exactsolution (6.4.24), and is also accurate even for rather small values ofthe quantity gt 2 /2x (cf. Sneddon [S.ll], p. 287). For this purpose ithappens to be rather easy to work out an asymptotic development ofvalid for large values of gt 2 /2x and thiswrite (6.4.19) in the formto do, following Lamb.the solution thatis9we proceedWe0(x,y;t)(6 .4.25)-Iff= --n200/a 2 *ye*sinUo\\+Iat] daSrI*y.egsin.\gJo==\(a*xIat]gd#. New quantities | andVgs, 2odaintroduced in (6.4.25) by the relationsmaking use of adaI\}\o>arefrom which--cogThe expression0(x, 0;(6.4.26)where f(6.4.25)isto vanish.t)isthus readily found to take the form-2gl/2/o>-^ Jon%'sin (| 2- o>2)dnew variable of integration and y is assumedThe corresponding free surface elevation is given byintroduced asWATER WAVES1621(6.4.27)g'as one readily verifies.
In order to study the last expressionsider the integral(6.4.28)dccI==::It is welldccIIJcu*/0knowndc*6IiiJOwe con-thatf(6.4.29)Jowhile the second contribution can be treated as follows:*f(6.4.30)c*-"d!;=I(2 Jco 2 V*Jo)<-" 2 dt>r=_^iirH ^<'*> L + -2 J^La<"i^^- |2) ^~iJ2as new variable, and an integrationthrough introduction of tnextthatthefinal integral is of the order cw" 1showWeby parts..as follows:3|\tZ6[ut^tIJw 2Jet) 2rdt=Jo) 2considering the real parts of (6.4.28), (6.4.29), (6.4.30), andinserting in (6.4.27) one findsUpon(6.4.81),)=-^ (g)[cos(gJwhich the function O(co~ l ) refers, as one readily verifies, to a termwhich behaves like (g/ 2 /4#)~ 1/2 Consequently, if o> 2 = g^ 2 /4<r is suffi'ciently large, we may assume for the free surface elevation due to aconcentrated surface elevation at x =and t = the approximatein.expressionUNSTEADY MOTIONS12U1632n\,<.-,_ (!-)*(--_).(6.4.32)/g*/g*continuing the integration by parts, as in (6.4.30), it would bepossible to obtain approximations valid up to any order in the quanti1 '212but such an expansion would not be convergent;ty co"(gJ /^)-*it is rather an asymptotic expansion correct within a certain order inco" 1 when an appropriate finite number of terms in the expansion isBy=,taken.
Expansions of this type are as in other branches of mathematical physics very useful in many of our problems and we shallhave many other occasions to employ them.The casetimetof a concentrated point impulse applied at xat thebeinasthecasecantreatedexactly the same mannerjust considered: one has only to begin with the solution (6.4.17) instead of (6.4.19), and proceed along similar lines.
In particular, thelapproximate solution valid (to the same order in a)~ ) for large values2of g/ /4a? can be obtained; the result for the free surface elevation is-(6.4.33)t),j(x;'~ --22/g* \-23/2.siny-\AtxJ/gt^\x- x\.4/The method usedto derive these asymptotic formulas is ratherbecannotspecial:very easily used to study the cylindrical wavesforgiven by (6.4.23),example. We turn, therefore, in the next sectionto the derivation of asymptotic approximations in all of these casesitby the application of Kelvin's method of stationary phase.
Afterwards, the motions themselves will be discussed in section 6.6 on thebasis of theapproximate formulas.6.5. ApplicationTheof Kelvin's method of stationary phase.integrals of section 6.4 can all be put into theI(k)(6.5.1)much= f \(,k)ejikform*W d(ia form peculiarly suited to anapproximate treatment valid for large values of the real constant k.In fact, Kelvin seems to have been led to the approximate methodwithoutknownas thedifficulty,andthisismethod of stationary phase throughhis interest inin particular the shipproblems concerning gravity waves,problem. The general idea of the method of approximationiswaveas fol-WATER WAVES164Whenk is large the function exp (ik<p(!;)} oscillates rapidly asunlesschanges,(p() is nearly constant, so that the positive and negative contributions to the value of I(k) largely cancel out, providedlows.thatisk)y>(f ,when knot a rapidly oscillating function ofislarge.Hence one might expect thelargest contributions to the integral tofromtheariseneighborhoods of those points in the interval from ato b at whichmostvaries(p'()=o.the phase of the oscillatory part of the integral,slowly, i.e., from neighborhoods of the points where<p(),This indeed turns out to be the case.
In section 6.8itwillbe shown that/(*)(6.5.2)By(1we mean a function which tends to zero like l/& 2/3 as k -> oo.expressions the sums are taken over all the zeros oc r of q/()/&In these= 2 V(r. *)2/3)in the interior of the intervala^ ^ b at which 9p"(a ^=^r)and overbut <p'"(ocj0. The signq>'($) at which <p"(ocjsumbetakentoshouldinfirstthejr/4agree with theThe relation (6.5.2) is valid if y(f, k) andg?() arc ana-the zeros a s ofof the termsign of 9/'(ar ).lytic functions of<6 and if the only stationary points ofandsuchthat9?"()<p'"(l;) do not vanish simultaneously.*9?(|)areobtaintheWe proceed toapproximate solution (6.4.32) obtainedin the previous section once more by this method.
The motion of thewater was to be determined for the case of an elevation of the watersurface concentrated at a point; the formula for the velocity potentialwas put(6 .5.3)in theinform0(x, y;t)a ?g {(cf.=9(6.4.25)):-1fn]00*Lfeo/o*xsinI00fJo* If a zero ofS\[Jo\+ at]*1e^doJ/o*xsin\\at\1g/da]\Jhigher order should occur, then terms of other typesless rapidly in A:. It should also benoted that the coefficient function y of section 6.8 is assumed to be independentof k, which is not true in some of the examples to follow.
However, it is notdifficult to see that the proof of section 6.8 can be modified quite easily in sucha way as to include all of our cases.<p'()ofstillwould appear, and the error would die outUNSTEADY MOTIONS165This can in turn be put in the form0(x(6.5.4)withm9= --Iffy; t)a 2 fg. It00e m * e i(mx +n Uoistaken at the end.Itt}da-f*e my e i(mx~ at}\daJo)understood that the imaginary part only is to beis convenient to introduce a new dimensionlessvariable of integration as follows:interms of which (6.5.4)0(x, yi(6.5.6)t)--=- -'(Ior,readily found to take the formis'-XX2x=f(6.5.5)<"** *'* <**+*> rff-|fee'*&~**>JoJod\)withopA:--(6.5.7)v4cTmis of course also aas a dimensionless parameter. The quantityfunction of f, and exp {m(g)y} plays the role of the function ^() inWhen(6.5.1).solutionthe parameterby usingisA:(6.5.2).?()-(0.5.8)large,we may approximate the</;() we haveFor the phases22with stationary points given by?/()(0.5.0)=2f2=0,and we see that | ~ 1 is the only such point in the interval < f < ooover which the integrals are taken.
Consequently only the second integral in (6.5.6) possesses a point of stationary phase, and at this pointwe have(6.5.10)We? /'(l)=?(!)=2,-1-obtain therefore from (6.5.2) the approximate formula(6.5.11)<t>(x,as one readily verifies,large values of ky\ t) c^andgt*/&x."formula is a good approximation forcan also calculate the free surface eleva-thisWenxWATER WAVES166tion77same way fromin therj=(l/g)& tlv =o*the resu ltiseasilyfound to be^.^(6.5.12)just as before(cf.(6.4.32)).For the case of a concentrated impulse the method of stationaryphase as applied to (6.4.17) or (6.4.18) leads to the following approximation valid once again for large values of gt 2 /4>x:^^^(6.5.13)andthis coincides with the result given in (6.4.33).In the case of an impulse distributed over a segment one obtainsfrom (6.4.16) the result(6.5.14)2P=-(,;.gt*a./{>t*--^ ^ ^nsm-n\-J,valid for large values of gt 2 /4tx.*For the ring waves furnished by(6.5.15)To obtainJ,(r;thisformula(sr) in (6.4.23)byI)= -it isits(6.4.23) the asymptotic formulagt*jffisgt*.necessary to replace the Bessel functionintegral representation2ftt/2cos (sr cos/?)dpand then apply the method of stationary phase twice in succession.Since such a procedure is discussed later on in dealing with thesimplified ship wave problem (cf.
Chapter 8.1), we omit a discussionof it here, except to remark that the approximate formula (6.5.15) isvalid foranyr=and2g* /4r sufficiently large.=* Itis a singularmay seem strange that this formula indicates that xis not singular in the exact formula (6.4.16). This comespoint for r), while x=about through the introduction of the new variable (6.5.5) and the parameter kin (6.5.7) which were used to convert the original integral to the form (0.5.1).However, the validity of the formula (6.5.2) is assured, as one can see fromsection 6.8, only if x 96 0.UNSTEADY MOTIONS6.6. Discussioninitiated167of the motion of the free surface due to disturbanceswhenthe waterisat restWe proceed to discuss the motions of the water surface in accordance with the results given in the preceding section. The generalcharacter of the motion is well given by the approximate formulas,and we shall therefore confine our discussion to them.
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