J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 29
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Such problems are discussed in Lamb[L.3], p. 370.The caseof internal waves inmedia with a continuousvariation in density has considerable importance also for tidal motionsin both the atmosphere (cf. Wilkes [W.2]) and the oceans (cf.Fjeldstad [F.4]).SUBDIVISION BMOTIONS STARTING FROM REST.
TRANSIENTS.CHAPTER6Unsteady Motions6.1.General formulation of the problem of unsteady motionsIn the region occupied by the water we seek, as usual, a harmonicfunction 0(x, y, z; t) which satisfies appropriate boundary conditionsand, in addition, appropriate conditions prescribed at the initial instant /0.
At the free surface we have theboundary conditions-#i+i? =(6.1.1)^(0.1.2)in\for1y=t0,>*t+gn=--v\terms of the vertical elevationrj(<x, z; t)of the free surface and thepressure p(x, z; t) prescribed on the surface. As always in mechanics,a specific motion is determined only when initial conditions at thetime t are given which furnish the position and velocity of allparticles in the system. This would mean prescribing appropriateconditions on0, but since wethroughout the fluid at the time tas well as for tshall assumeto be a harmonic function at tit is fairly clear that conditions prescribed at the boundaries of theis then determined uniquely throughoutfluid only will suffice sinceits domain of definition in terms of appropriate boundary conditions.*As initial conditions at the free surface, for example, we might therefore take==(6.1.3)r,(x,z;0)=MX,z)\a=>Q(6.1.4)andarbitrary functions characterizing the initial elevationwith/!andvertical velocity of the free surface./2In water wave problemsit isof particular interest to consider cases* We shall see later on(sections 6.2 and 6.9) that the solutions are indeeduniquely determined when the initial conditions are prescribed only for theparticles at the boundary of the fluid.149WATER WAVES150inwhich the motion of the watersive pressure to the surfaceisgenerated by applying an impul-when the waterinitially at rest.isToobtain the condition appropriate for an initial impulse we start from<S tr.
The(6.1.2) and integrate it over the small time interval^resultispdt(6.1.5)=Q0(x,eg0, *; r)\r)dt,JoJosince 0(x, y, z; 0) can be assumed to vanish. One now imagines thatr -> 0+ while p -> oo in such a way that the integral on the left tendsthe impulse / per unit area. Since it is natural toassume that 77 is finite it follows that the integral on the right vanishesas r -> + and we have the formulato a finite value,/(6.1.6)=Q0(x,0, z;+)for the initial impulse per unit area at the free surface in terms of thethere. If/ is prescribed on the free surface (together withvalue ofappropriate conditions at other boundaries), it follows that0(x, y, z; 0^ ) can be determined, or, in other words, the initial velocityof all particles is known.at the freeIt is also useful to formulate the initial condition onsurface appropriate to the case in which the water is initially at restunder zero pressure, but has an initial elevation rj(x z; 0).
The condi9tionisobtained at once from (6.1.2);(6.1.7)sincept=its firstt6.2.It+ ),>instead of.+initialconditionsinsteadweshall simply.Uniqueness of the unsteady motions in bounded domainsisofsomemotions, forinterest to consider the uniqueness of the unsteadyone thing because of the unusual feature pointed outin the preceding section: itissufficient to prescribe the initial positionbut only of those on the boundary.basedonlaw of conservation of energy willtheuniqueness proofandA- - grj(x, z;isnot be used in formulatingwrite+)0. Prescribing the initial position and velocity ofthus equivalent to prescribing the initial values oftime derivative t From now on the notation + willforthe free surfaceand(x, 0, z;it isVelocity, not of all particles,be given.To this end, consider the motion of a bounded volume of water confined to a vessel with fixed sides but having a free surface (cf.
Fig.UNSTEADY MOTIONS6.2.1).andEits151In Chapter 1 we have already discussed the notion of energytime rate of change with the following results. For the energyitselfwehave, obviously:h(x,z)Fig. 6.2.1.E(t)(6.2.1)HereTheRx,-z-plancis,in++Q JJJrefers to thefree surface,Water containeda vessely]dx dydz.volume occupied by the water at anyinstant.as usual, taken in the plane of equilibrium of theh(x z) are assumed to be therj(x z; t) and yand y==99equations of the free surface and of the containing vessel, respectively.can now be written in the formThe expression forE(6.2.2)E(t)1tf+<2Z)dx dy dzBy Sis meant the projection on the #, s-plane of the free surface andthe containing vessel. In Chapter 1 the following expression for therate of change of the energy E was derived:dE(6.2.3)=ffJJ[Q&-vn)- pv n]dS.WATER WAVES152By Rmeant the boundary surface of R, while v n means the normalcomponent of the velocity of R.
It is essential for our uniquenesson the freeproof to observe that in the special case in which p =surface we haveisdE(6.2.4)=E=0,const.===on the fixed partThis follows at once from the fact that v nnandonfree surface.theof the boundary, while v nnpSo far no use has been made of the fact that we consider only a=on the assumption of small oscillations about theequilibrium position. Suppose now that the initial position and velocity of the water particles has been prescribed, or, as we have seen inthe preceding section that rj(x z; 0) and 0(x y, z; 0) are given func-linear theory based99tions:^(*> *:)(0(x,Iy, z; 0)= /i(*= f (x2*)y, z).9We proceed next in the customary way that one uses to prove uniqueness theorems in linear problems.
Suppose that rj l9are two solutions of the initial value problem. Thenandr\=r] 2ij larc functions which satisfyall19and=r\^^22of the conditionsimposed on t and 17 f except that fl and / 2 in (6.2.5) wouldvanish, and the free surface pressure would also vanish(cf. (6.1.2) and (6.1.7)). (Here the linearity of our problem is used inan essential way. ) It follows therefore that dE/dt = 0, and E = const,when applied toand 77, as we have seen.
But at the initial instant==and0, so thatrforiginallynow bothE= -(0.2.6)htdxdz,5fromly(6.2.2) as appliedwe have=<t>ljjjand 17=rj lr] 2.Consequent-^ dx dz == and =grad+ 01 + 01) dx dy dz + gRand2the resultfff (0*(6.2.7)to0jjff0,Ssuminobviously vanishes only ifrjfollows thatanaddiforunessentiall2 (excepttive constant), rj land the uniqueness of the solution of therj 29thisother wordsinitial=it=value problemisproved.UNSTEADY MOTIONS153The proof given here applies only to a mass of water occupying abounded region. Nevertheless, it seems clear that the uniqueness ofinitial value problem is to be expected if the wateran unbounded region, provided that appropriate assumptionsconcerning the behavior of the solution at oo are made.
In the follow-the solution of thefillsa variety of such cases will be treated by making use of the technique of the Fourier transform and, although no explicit discussionof the uniqueness question will be carried out, it is well-known thating,uniqueness theorems (of a somewhat restricted character, it is true)hold in such cases provided only that appropriate conditions at ooare prescribed. Recently these uniqueness questions have been treatedby Kotik [K.I 7] and Finkelstein[F.
3].The latter,for example, provesthe uniqueness of unsteady motions in unbounded domains inwhich rigid obstacles occur, and both writers obtain their uniquenesstheorems by imposing relatively weak conditions at infinity. In sec. 9of this chapter the theory devised6.3.by Finkelsteinwillbe discussed.Outline of the Fourier transform techniqueAs indicated above, the solutions of a series of problems of unsteadymotions in unbounded regions as determined through appropriateinitial conditions will be carried out by using the method of theFourier transform.
The basis for the use of this method is the fact thatof our free surface problems are given in thecase of two-dimensional motion in water of infinite depth, for examplespecial solutions-b(6.3.1)0(a: y\9t)= e* sin (at + a) cos m(x -r)witha2(6.3.2)= gwfor arbitrary values of a and r. From these solutions it is possibleto build up others by superposition, for example, in the formand(6.3.3)0(x, y\t)=h(a)evsin (at+ <x.)dmJoo 00f(r) cosm(xT) dr,f(r) are arbitrary functions. This in turn suggeststhat the Fourier integral theorem could be used in order to satisfyinwhich h(a) andgiveninitial conditions, since thistheorem states that an arbitraryWATER WAVES154function f(x) defined foroo<x <oo canbe representedin theformf(x)(6.3.4)=i rr-nJda/(^) cos a(^x)dr\J ~oooprovided only that f(x) is sufficiently regular (for example, that J(x)is piecewise continuous with a piecewise continous derivative is morethan sufficient) and that f(x) is absolutely integrable:r(6.3.5)Indeed,Jwewould have|00/(a?)do?|<oo.=weset h(a)1/n andexactly the integral in (6.3.4) forsee thatifat= yt/2in (6.3.3)and y=0,weandhence 0(x 0; 0) would reduce to the arbitrarily given function f(x).Thus a solution would be obtained for an arbitrarily prescribed initialcondition on 0.It would be perfectly possible to solve the problems treated belowby a direct application of (6.3.4), and this is the course followed byLamb [L.3] in his Chapter IX.
Actually the problems were solvedfirst by Cauchy and Poisson (in the early part of the nineteenth cen9who derived solutions given by integral representations beforethe technique of the Fourier integral was known. It might be addedthat these problems were considered so difficult that they formed thetury),subject of a prize problem of the Academic in Paris.We prefer, in treating these problems, to make use of the techniqueof the Fourier transform (following somewhat the presentation givenby Sneddon[S.ll ], Chapter 7) since the building up of the solution tothe prescribed conditions then takes place quite automatically.However, the method is based entirely upon (6.3.4) and thus alsofulfillits validity that the functions f(x) to which the techniquebe represeritable by the Fourier integral.
This is ashouldappliedrestriction of a non-trivial character: for example, the basically important solutions given by (6.3.1) are not representable by the Fou-requires forisrier integral.Itis(6.3.4).useful to express the Fourier integral in a form different fromWewritef( x)=1- limnff->oo J -QOf*i(n)driJcoss(r]x)ds.UNSTEADY MOTIONSButf*Jsin5sinceS(TIcosIx)ds%=x)dss(rjwe may0,J/(a?)We nowcoswritesexp[is(x=/*GOf^-s(rjcosJOrj)} ds,x)ds ands(rjx)ds=and hence/*00s2^J~00 e*ds\J-00f(r))e^dr\.set(6.3.7)and2f *T(6.3.6)=155t (s)the Fourier transform of /(#). It follows at once fromthatthe(6.3.6)original function f(x) is obtained from its transforml( s ) by the inversion formulacall J(s)In our differential equation problems it will be essential to expressthe Fourier transform of the derivatives of a function in terms of thetransform of the function itself.
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