J.J. Stoker - Water waves. The mathematical theory with applications, страница 7

The resultisWATER WAVES14/ITT=P P PQ(J J J~dtsThe integrandin the first integral&x(t)xand hence thecan be expressed in the form+ ^y(^t)y + &z(t)z =gra d&gradtbe written as the following surface integral:integral can\^dS,sview of Green's formula and the fact thatinexpression for dE/dt, the rate ofput into the following form:J Tjl.(1.6.3)We/*-f-JJs=0.change of the energy in/=V2[Q0 t (0 n-vn )-Thus theR9can bepv n ]dS.means the normal velocity component of S, andcomponent of the fluid taken in the directionof the normal to S which points outward from R.recall that v nn refers to the velocityhappens frequently that the boundary surface S of R is madenumber of different pieces which have different propertiesor for which various different conditions are prescribed.

Supposefirst that a portion S P of S is a "physical" boundary containingn and v n are identical (cf.always the same fluid particles. Thenand(1.4.2))Itupof adEdS.(1.6.4.)dtSPSP=fixed in space, i.e. v n0, thecontribution of S P to dE/dt evidently vanishes, as it should, since noenergy flows through a fixed boundary containing always the sameIf, in addition, the surfaceisfluid particles. Similarly, the contribution to the energy flux alsovanishes in the important special case in which S F is a free surfaceon which the pressure p vanishes; this result also accords with whatone expects on physical grounds.Suppose now that SG is a "geometric" surface fixed in space, butnot necessarily consisting of the same particles of water.

In thiscasewe havevn=and the flow of energy through SGisgiven byBASIC HYDRODYNAMICS15dE,,.6.5)An-//<important special case for us0(x,(1.6.6)y, z; t)isthat in whichwavepotential for a plane progressingisthe velocitygiven, for example,by= <p(x-ct, y, z),which represents a wave moving with constant velocity c in thedirection of the #-axis.

The flux through a fixed plane surface Sorthogonal to the #-axis is easily seen from (1.6.5) to be given by(iv(1.6.7)-^=SThe negativesign results since our stipulations amount to sayingthat the regionoccupied by the fluid lies on the negative sideof S (i.e. on the side away from the positive normal, the #-axis);Rand consequently the energy flux through S due to a progressingwave moving in the positive direction of the normal (so that c ispositive) is such as to decrease the energy in jR, as one would expect.Itisto be noted that theresurfacen E^=isalways a flow of energy through aS orthogonal to the direction of a progressing wave ifeven though the motion of the individual particles of thefluid shouldhappen, for example, to be such that the particleswave.movein a direction opposite to that of the progressing1.7.Formulation of a surface wave problemperhaps useful although somewhat discouraging, it must beadmitted to sum up the above discussion concerning the fundamental mathematical basis for our later developments by formulating a rather general, but typical, problem in the hydrodynamics ofsurface waves.

The physical situation is indicated in Figure 1.7.1;what is intended is a situation like that on any ocean beach. Thewater is assumed to be initially at rest and to fill the space R defined byItish((r,+2)^2/^0,oo<z <oo,=oo in the ^-direction. At the time t0, a givencreated on the surface of the water over a region(by the wind, perhaps), and one wishes to determine mathematicallythe subsequent motion of the water; in particular, the form of theand extending todisturbanceisDWATER WAVES16Fig. 1.7.1.Avery general surface wave problem=basis of theser\(x, z; t) is to be determined.

On theshouldbesatisfied:theconditionsFirst ofassumptionsfollowingis, of course, theall, the differential equation to be satisfied byfree surfaceyLaplace equationV*0(I.T.I)(x s (z=XX+^Qy-fort)9h(x9ooz)^y^ rj(^ z;<00[to be noted that x s (z;^x<Z<t)00t) the abscissae of the water line on shore,and rj(x z; t), the free surface elevation, are not known in advancebut are rather to be determined as an integral part of the solution.As boundary condition to be satisfied at the bottom of the sea wehaveItis99-(1-7.2)=for yon=h(x z)99while the free surface conditions are the kinematic condition(cf.(1.4.5))(1.T.3)xrj x-0 + & + =yzr] zrj tfory= rj(x9z; t) 9and the dynamic condition(1.T.4)gqwith F(x z;disturbance+t+ \(0l +01+ 0%) = F(x9z; t)- rj(x z;D where theon y=9t),everywhere except over the regionAt oo, i.e.

for x -> oo and z -> oo, wethatandmight prescribey remain bounded, or perhaps even thatthey and certain of their derivatives tend to zero. Next we have the9initial(1.T.5)(1.T.6)t)iscreated.|conditionsqfa=Xz; t)==Zyfors=t=fort0,=0,|BASIC HYDRODYNAMICS17appropriate to the condition of rest in an equilibrium position.Finally we must prescribe conditions fixing the disturbance; thiscould be done, for example, by giving the pressure p over thedisturbed regionof the surface, in other words by prescribing thefunction F in (1.7.4) appropriately there.DOne has only to write down the above formulation of our problemto realize how difficult it is to solve it.

In the first place the problemis nonlinear, but what makes forperhaps even greater difficulties isthe fact that the free surface is not known a priori and hence thedomainknownininwhich the velocity potential is to be determined is notadvance aside from the fact that its* boundary varieswith the time.These are, however, not the onlyabove problem.If we assume that the functionis regular throughout the interiorof R and uniformly bounded (together with some of its derivatives,perhaps) in R, the formulation of the problem given above wouldseem to be reasonable from the point of view of mechanics.

However,the solution would probably not exist for all t >for the followingreason: everyone who has visited an ocean beach is well aware thatthe waves do not come in smoothly all the way to the shore (exceptpossibly in very calm weather), but, rather, they steepen in front,curl over, and eventually break. In other words, any mathematicalformulation of the problem which would fit the commonly observedfacts even for a limited time would necessitate postulating theexistence of singularities of unknown location in both space and time.Because of the difficulty of the general nonlinear theory very littleprogress has been made in solving concrete problems which employdifficulties in theAnexception is the problem of proving the existence of twodimensional periodic progressing waves in water of uniform depth.This was done first by Nckrassov [N.I], [N.I a] and by Levi-Civitait.water of infinite depth, and later by Struik [S.29] for water ofconstant finite depth.

In Chapter 12 an account of Levi-Civita's theoryis given. In both cases the authors prove rigorously the existence ofwaves having amplitudes near to zero by showing that perturbationseries in the amplitude converge. Another exception to the above[L.7] forstatement is the problem of the solitary wave, the existence of which,from the mathematical point of view, has been proved recently byLavrentieff [L.4] and by Friedrichs and Hyers [F.13]; an account ofthe work of the latter two authors is given in Chapter 10.9.It seems likely that solutions of problems in the full nonlinearWATER WAVES18version of the theory will, for a long time to come, continue to beof the nature of existence theorems for motions of a rather specialnature.In order tomakeprogress with the theory of surface wavesit isin general necessary to simplify the theory by making special hypotheses of one kind or another which suggest themselves on the basisof the general physical circumstances contemplated in a given classof problems.

As we have already explained in the introduction, upnow attention has been concentrated almost exclusively upon thetwo approximate theories which result when either a) the amplitudeof the surface waves is considered small (with respect to wave length,todepth of the water is considered small (againwith respect, say, to wave length). The first hypothesis leads to alinear theory and to boundary value problems more or less of classicaltype; while the second leads to a nonlinear theory for initial valuefor example), or b) theproblems, which in lowest orderisof the type employed inwavepropagation in compressible gases.

If both hypotheses are made, theresult is a linear theory involving essentially the classical linearwave equation; the present theoryof the tides belongs in this classof problems.In the next chapter we derive the approximate theories arisingfrom the two hypotheses by starting from the general theory andthen developing formally with respect to an appropriate parameteressentially the surface wave amplitude in one case and the depthof the water in the other and in subsequent chapters we continuebytreating a variety of special problems in each of thetwoclasses.CHAPTER2The Two Basic Approximate Theories2.1.Theory of waves of small amplitudeIt has already been stated that the theory of waves of smallamplitude can be derived as an approximation to the general theorypresented in Chapter I on the basis of the assumption that thevelocity of the water particles, the free surface elevation yr)(x, z; t),Weandtheir derivatives, are all small quantities.assume, in fact,that the velocity potentialand the surface elevation rj possess thefollowing power series expansions with respect to a parameter e:=(2.1.1)e0 (l)+e2(2)+e3(3)+...,It follows first of all that each of the functionsa solution of the Laplace equation,V 2 0><*>(2.1.3)Weand0< k) (x,y, z; t) isi.e.0.turn next to the discussion of the boundary conditions.

At a(cf. section 1.4) of the fluid we have clearlyfixed physical boundarythe conditions- -- =(2.1.4)a0,which d/dn represents differentiation along the normal to theboundary surface.is zero weAt a free surface S: yr](x> z; t) on which the pressurehave two boundary conditions. One of them arises from the Bernoullilaw and has the formingrj+t+ i(<Z>* + *I + 01) =on S.insertion of (2.1.1) and (2.1.2) in this condition and developing&t> &x> etc systematically in powers of e (due regard being paid toetc. are to be evaluated forthe fact that the functions 0^ k) 9,Upon'(19WATER WAVES20yit)= ri(x, z\ t) and= 77<> erjM-)--\-that...)in its turn is given in terms of77one finds readily the conditions(2.1.5)?7gqM(2.1.6)(2.1.7)W +><P+ i[(0> + (0<21})=2)+ (#> )] +=(O)and since ?y (0)to be satisfied for yT?that the conditions (2.1.6), (2.1.7), etc.