J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 33
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0.6/2. (Continued)173WATER WAVES174which correlate the occurrence ofstorms in the Atlantic with the long waves which move out from thestorm areas and reach the coast of Cornwall in a relatively short time.By analyzing the periods of the swell, as determined from actualwave records, it has been possible to identify the swell as having beencaused by storms whose location is known from meteorological observations.
Aside from the interest of researches of this kind from thepurely scientific point of view, it is clear that such hindcasts could, inprinciple, be turned into methods of forecasting the course of stormsassociates have carried out studiesat sea in areas lacking meteorological observations.6.7.to a periodic impulse applied to the water wheninitially at rest. Derivation of the radiation condition for purelyWaves dueperiodicwavesIn section 3 of Chapter 4 we have solved the problem of two-dimensional waves in an infinite ocean when the motion was a simplein the time that was maintained by an applicationof a pressure at the surface which was also simple harmonic in thetime. In doing so, we were forced to prescribe radiation conditionsat ooeffectively, conditions requiring the waves to behave like out-harmonic motiongoing progressing waves at oo in order to have a complete formulation of the problem with a uniquely determined solution.
It wasremarked at the time that a different approach to the problem wouldbe discussed later on which would require the imposition of boundedness conditions alone at oo, rather than the much more specific radiation condition. In this sectionweshall obtain the solutionworked outwithout imposing a radiation condition by considering it as thean unsteady motion as the time tends to infinity.
However, ithas a certain interest to make a few remarks about the question ofradiation conditions in unbounded domains from a more general pointof view (cf. [S. 21]).In wave propagation problems for what will be called here, exceptionally, the steady state, i.e., a motion that is simple harmonic inthe time, it is in general not possible to characterize uniquely thein 4.3limit ofsolutions having the desired physical characteristics by imposing onlyboundedness conditions at infinity.
It is, in fact, as we have seen inspecial cases, necessary tocase in which the mediumimpose sharper conditions. In the simplestis such as to include a full neighborhood ofthe point at infinity that is in addition made up of homogeneous matter,UNSTEADY MOTIONSthe correct radiation condition175not difficult to guess. It is simplyat infinity behaves like an outgoing spherical wavefrom an oscillatory point source, and such a condition is what isthat theiswavecalled the radiation, or Sommerfeld, condition. Amongother things this condition precludes the possibility that there mightbe an incoming wave generated at infinity which, if not ruled out,commonlywould manifestly make a unique solution of the problem impossible.If the refracting or reflecting obstacles to the propagation of waveshappen to extend to infinity for example, if a rigid reflecting wallshould happen to go to infinity it is by no means clear a priori whatconditions should be imposed at infinity in order to ensure the uniqueness of a simple harmonic solution having appropriate propertiesotherwise.* A point of view which seems to the author reasonable isthat the difficulty arises because the problem of determining simpleharmonic motions is an unnatural problem in mechanics.
One should inprinciple rather formulate and solve an initial value problem byassuming the medium to be originally at rest everywhere outside asay, and also assume that the periodicdisturbances are applied at the initial instant and then maintainedwith a fixed frequency. As the time goes to infinity the solution of thesufficiently large sphere,initial value problem will tend to the desired steady state solutionwithout the necessity to impose any but boundedness conditions atinfinity.**The steadyin the author's view, atstate problem is unnatural-because a hypothesis is made about the motion that holdsfor all time, while Newtonian mechanics is basically concerned withleasta unique way, furthermore of the motion of amechanical system from given initial conditions. Of course, in mechanics of continua that are unbounded it is necessary to impose conditions at oo not derivable directly from Newton's laws, but for theinitial value problem it should suffice to impose only boundednessconditions at infinity.
In sec. 6.9. the relevant uniqueness theorem forthe special case to be considered later is proved.the predictionin* For a treatment of the radiation condition in such cases see Rellich[R.7],John [J.5], and Chapter 5.5.** The formulation of the usual radiation condition is doubtlessly motivatedby an instinctive consideration of the same sort of hypothesis combined with thefeeling that a homogeneous medium at infinity will have no power to reflectanything back to the finite region. Evidently, we also have in mind here onlyif such modes ofcases in which no free oscillations having finite energy occuroscillation existed, clearly no uniqueness theorems of the type we have in mindcould be derived.WATER WAVES176If one wished to be daring one might, on the basis of these remarks,formulate the following general method of obtaining the appropriateradiation condition: Consider any convenient problem in which thepart of the domain outside a large sphere is maintained intactinitially at rest.
(In other words, one might feel free to modify inconvenient way any bounded part of the medium.) Next solveinitial value problem for an oscillatory point source placed atandanytheanyconvenient point. Afterwards a passage to the limit should be made inallowing the time t to approach oo, and after that the space variablesshould be allowed to approach infinity. The behavior at the far distantportions of the domain should then furnish the appropriate radiationconditions independent of the constitution of the finite part of thedomain.
It might be worth pointing out specifically that this is a caseinwhich the order of the two limit processes cannot be interchanged:obviously, if the time / is first held fixed while the space variables tendto infinity the result would be that the motion would vanish at oo,and no radiation conditions could be obtained.The writer would not have set down these remarks which are of acharacter so obvious that they must also have occurred to manyothers if it were not for two considerations.
Every reader will doubtlessly have said to himself: "That is all very well in principle, but willnot be prohibitively difficult to carry out the solution of the initialvalue problem and to make the subsequent passages to the limit?"In general, such misgivings are probably all too well founded. How-itproblem concerning water waves to be treated here happensto be an interesting special case in which (1) the indicated programcan be carried out in all detail, and (2) it is slightly easier to solve theinitial value problem than it is to solve the steady state problem withthe Sommerfeld condition imposed.We restrict ourselves to two-dimensional motion in an x, z/-plane,with the y- axis taken vertically upward and the #-axis in the originallyever, theundisturbed horizontal free surface.
The velocity potentialis a harmonic function in the lower half-plane:(6.7.1)The(p xxfree surface(6.7.2)+q> yv=0,y< 0,boundary conditions aret(cf.>(p(x, y\ t)0.(6.1.1),(6.1.2)):UNSTEADY MOTIONS177=As usual, r\r\(x\ t) represents the vertical displacement of the freesurface measured from the #-axis, and pp(x; t) represents thepressure applied on the free surface.suppose that 9? and its firstWeand second derivatives tend to zero atoo for any given time tin factthat they tend to zero in such a way that Fourier transforms existbut we do not, in accordance with our discussion above, make anymore specific assumptions about the behavior of our functions as-> oo./At the timewet<p(x, 0;(6.7.4)prescribe the following initial conditions0) ===0; 0)<p t (x,0,which state (cf.
(6.1.6), (6.1.7)) that the free surface is initially atrest in its horizontal equilibrium position.In what follows we consider only the special case in which the surface pressure p(x;(6.7.5)int)isgiven byp(x;.which d(x)ist)=imtd(x)ethe Dirac d-function.>t,Wehave not made explicit usewe have used it implicitly in section6.1 in dealing with concentrated impulses. It is to be interpreted inthe same way here, i.e.
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