J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 41
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In fact, this function behaves like l/^/t since one can easilywhole of the semicircles.two points of stationary phase, i.e. two=whereand /"( /3 ) ^ 0. (The point s =/-(pointsA>)j8where /_($) vanishes.)lies between the origin and the point s =Thus the steady state is again given by <p^. However, unlike the preceding case, the steady state does not furnish a motion which diesout both upstream and downstream. This can be seen as follows.Consider first the behavior upstream, i.e. for x < 0. On the semicircuverify that /_(s) has exactlyPin the lower half-plane we see that the expolar parts of the pathe i8X in (7.4.17) has a negative real part, and therefore by thenent insame argumentvanish as x-P make contributions whichP also make contributionsas above, these parts ofoo. The straight parts ofWATER WAVES216which vanish for large x (either positive or negative), by the RiemannLebesgue theorem.
Thus the disturbance vanishes upstream. On thedownstream side, i.e. for x > 0, we cannot conclude that the semicircular parts of P make vanishing contributions for large x since thei8Xnow has a positive real part. We therefore make useexponent in ePof the standard procedure of deforming the paththrough the polestheresiduesattheseandpoles. It is clear that/?subtractingat s=the semicircles in the upper half-plane yield vanishing contributionsto <pM when x -><x>: the argument is the same as was used above.+This leads to the following asymptotic representation (obtained fromthe contributions at the poles), valid for x large and positive:(7.4.19,Heres=..^.r.W'(ft)/?,00)=Q coshW^is the value of the derivative of(cf.
(7.4.15)) atfact that W(/J) is an odd function has been used. Inthe surface pressure p(x) were given by the delta func-and theparticular, iftion p(x)=d(x),i.e. ifthe disturbance were causedby a concentratedpressure point at the origin, (7.4.19) would yield,~,^/x?.(*,;(7.4.19),,).=since the transform of d(x) is l/\/2n. Another interesting specialcase is that in which p(x) is a constant p over the intervalax^and zero over the rest of the free surface. In this case p =(2p / \/2n) ( sin sa )/s and q> x behaves for large positive x and t as follows^a:x.(, y; oo)(7A19),cosh-fi(y+ h).Dn ^asm /to..This yields the curious result (mentioned above) that under the proper circumstances the disturbance may die out downstream as well as=nn, i.e.
if the length 2a of theupstream; it will in fact do so if (iasegment over which the disturbing pressure is applied is an integralmultiple of thevelocityUwave lengthand the depthat oowhichis,in turn, fixedby theh.=1, and begin by disFinally we consider the critical case gh/U*in <p as t -> oo. Forthebehaviorofthetimetermscussingdependentthispurposeit isthis function:convenient to dealfirstwith the time derivative ofTWO-DIMENSIONAL WAVES(7.4.80)*,=217P cosh $A \/g$ tanhThe integrand has nosingularities on the real axis and consequentlybe deformed into the real axis. Thus the principle ofstationary phase can be employed once more. Since the derivative of= sU \/gs tanh sh evidently does not vanish for any real sf + (s)while the derivative of /_(s) has one zero at s0, it follows that theintermtheofleadingasymptotic development9?^ for large t arisesfrom the term exp { itf__(s)}. Since, in addition, '(0) = but /'"(O)^ we have (cf.
sec. 6.8):the pathP can+(7.4.21)0f=- Ap(0).-,A =const.^0.ffrom zero, it follows that <p ^ behaveslike rand hence that q> becomes infinite everywhere (for all xand y, that is) like t*/3 as t -> oo.* Thus a steady state does not existif one considers it to be the limit as t -> oo. It might be thought thatSince p(0)isin general different(t)1/3the existence in practice of dissipative forces could lead to the vanishing of the transients and thus still leave the steady state (p^ as given(7.4.17) as a representation of the final motion. That is, however,also not satisfactory since<p^ becomes unbounded for x large when21: at the origin there is a pole of order two since W($) begh/Uby=and consequently the term isx in the power series for e isxleads to a contribution from this pole which is linear in x.
In lineartheories based on assuming small disturbances one is reconciled tosingularities and infinities at isolated points, but hardly to arbitrarilyhaveslike s 4large disturbances in whole regions. All of this suggests that theis that the linearreasonable attitude to take in these circumstancestheory, which assumes small disturbances, fails altogether for flowsat the critical speed U*/ghI and that one should go over to a non-=* Itmight seem odd that we have chosen to discuss the function <p^ ratherthan the function <p(aswedid in the other cases).The reason is that the asymp-not easily obtained directly by the method of stationarysince the coefficient of the leading term in this developincasethephasepresentment would be zero.
However, one could show (by using Watson's lemma, forexample, which yields the complete asymptotic expansion of the integral) thatbehaves like *^3behaves like f~^3 and hence that q>q>totic behavior of q>^is,.218WATER WAVESlinear theory in order to obtain reasonable resultsfrom the physicalpoint of view. In Chapter 10.9, which deals with the solitary wave (anessentially nonlinear phenomenon), we shall see that such a steadywave exists for flows with velocities in the neighborhood of thecritical value.CHAPTERWaves Caused by a MovingTheory of the8.1.An idealizedWave8Pressure Point. Kelvin'sPattern Created by a Moving Shipversion of the shipwave problem.
Treatment by themethod of stationary phaseThepeculiar pattern of the waves created by objects moving overthe surface of the water on a straight course has been noticed byeveryone: the disturbance follows the moving object unchanged inform and it is confined to a region behind the object that has the samev-shape whether the moving object is a duck or a battleship.
An explanation and treatment of the phenomenon was first given byKelvin [K.ll], and this work deserves high rank among the manyimaginative things created by him. As was mentioned earlier, Kelvininvented his method of stationary phase as a tool for approximatingthe solution of this particular problem, and it is indeed a beautifulandstrikingly successful example of its usefulness.should be stated at once that there is no notion in thisand thenext following section of solving the problem of the waves created byan actual ship in the sense that the shape of the ship's hull is to betaken into account; such problems will be considered in the nextchapter.
For practical purposes an analysis of the waves in such caseseven a large fraction if theis very much desired, since a fractionto the forward motion of aisoftheoftheresistancespeedshiplargeisduetotheusedinmaintaining the system of gravityenergyshipupwaves which accompanies it. The problem has of course been studied,in particular, in a long series of notable papers by Havelock,* but thedifficulties in carrying out the discussion in terms of parameters whichfix the shape of the ship are very great. Indeed, a more or less complete discussion of the solution to all orders of approximation even inthe very much idealized case to be studied in the present chapter, isby no means an easy task in fact, such a complete discussion, alongItReferences to some of these papers will be given in the next chapter.219WATER WAVES220lines quite different from those of Kelvin, has been carried out onlyrather recently by A.
S. Peters [P. 4] (cf. also the earlier paper byHogner [H.16]). However, we shall follow Kelvin's procedure here ina general way, but withThe problem we havedifferences in detail.manyinmindto discuss as a primitive substitutefor the case of an actual ship is the problem of the surface wavescreated by a point impulse which moves over the surface of the water(assumed to be infinite in depth). We shall take the solution of section6.5 for the wave motion due to a point impulse and integrate it alongthe course of the "ship" in effect, the surface waves caused by theship are considered to be the cumulative result of impulses deliveredat each point along its course.
The result will be an integral representation for the solution, in the form of a triple integral, which can beby the method of stationary phase. However, it is necessarymethod of stationary phase three times in succession, andif this is not done with some care it is not clear that the approximationis valid at all; or what is perhaps equally bad from the physical pointdiscussedto apply theof view, it may not be clear where the approximation can be expectedto be accurate.
Thus it seems worth while to consider the problem withsome attention to the mathematical details; this will be done in thepresent section, and the interpretation of the results of the approximation will be carried out in the next section (which, it should be said,can be read pretty much independently of the present section).From section6.4 the vertical displacement*r)(x, y, z; t)of the waterparticles due to a point impulse applied on the surface at thecan be put in the formand at the time t =(8.1.1)in77(0?, t/,which a 2functionJz;t)= gm(mr)byJfor reasons=whichI-r2andits/*ooasmat'emvmdmI= x + z We22.origin/w/2cos(mrcos/J)d/Jhave replaced the Besselintegral representationf*/ 2(mr)=2willbecomecos (mr cosft) df}clear in amoment. As we haveindicated, our intention is to sum up the effect of such impulsesas the "ship" moves along its course C.
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