J.J. Stoker - Water waves. The mathematical theory with applications, страница 10

In similar fashion the second of the equations(2.4.2)jx, z,tcan be integrated and the additive arbitrary function ofdetermined from (2.4.14); the result isp(0(2.4.17)whichfy, z, t)- ijW(x z,t)-y9obviously the hydrostatic pressure relation (in dimensionform).In the derivation of the shallow water theory given in the precedingislesssection this relationwas taken as the starting point;here,itisderived as the lowest order approximation in a formal perturbationscheme. However, it is of course not true that we have proved thatinsome sense an appropriate assumption:instead, itshould be admitted frankly that our dimensionless variables were(2.4.17)isintroduced in just such a way that (2.4.17) would result.

If it couldbe shown that our perturbation procedure really does yield a correctasymptotic development (that the development converges seemsunlikely since the equations (2.4.1)' to (2.4.6)' degenerate in orderso greatly for a0) then the hydrostatic pressure assumption couldbe considered as having been justified mathematically.proof that=Awould be of great interest, since it would give amathematical justification for the shallow water theory; to do so inthis is the caseWATER WAVES82way would seemto be a very difficult task, but Friedrichsshownthat the development does yieldhave[F.18]inthe important special case of thethe existence of the solutionwave(cf.

Chapter 10.9). (Keller [K.6] had shown earliersolitaryformalthethatprocedure yields the solitary wave.) The problemis of considerable mathematical interest also because of the followinga generaland Hyersintriguing circumstance: the approximation of lowest order to thesolution of a problem in potential theory is sought in the form of asolution of a nonlinear wave equation, and this means that thesolution of a problem of elliptic type is approximated (at least in thelowest order) by the solution of a problem of hyperbolic type.and p (0) given by (2.4.16) and (2.4.17) are nowthe first and third equations of (2.4.2)J and in (2.4.4)!'The valuesinserted inof v (l)to yield finallytt<>t*Ww>< 0)(2.4.19)(2.4.20)+ w^uf + rif = 0,+ i*<> H>W + w^wf + i?<> - 0,+ [ttWfoW + h)] x + [w<>fo<o) + h)], =uf +(2.4.18)T?<>as definitive equations for t^ 0),w (Q \ and(0)iyall0,of which,werepeat,{Q)isdepend only upon #, 2, and /.

If the superscript is dropped, wtaken to be zero, and it is assumed that all quantities are independentof z, one finds readily that these equations become identical withequations (2.2.11) and (2.2.12) except for the factor g in (2.2.11)which is missing here because of our introduction of a dimensionlesspressure.It is clear that theabove process can be continued to obtain theorderhigherapproximations. An example of such a calculation willbe given later in Chapter 10.9, where we shall see that the first non-trivialtermin thesecond order.development which yields the solitary waveisofPARTIISummaryIn Part IIwhichariseswetreat a variety of problems in terms of the theorythrough linearization of the free surface condition (cf.the preceding chapter); thus the problems refer to waves of smallamplitude.

To this theory the names of Cauchy and Poisson areusually attached. The material falls into three different types, orclasses, of problems, as follows: A) Waves that are simple harmonicin the time. These problems are treated in Chapters 3, 4, and 5 andthey include a study of the classical standing and progressing wavesolutions in water of uniform depth, and waves over sloping beachesand past obstacles of one kind or another. The mathematical toolsemployed here comprise, aside from classical methods in potentialtheory, a thorough-going use of integrals in the complex domain.B) Waves created by disturbances initiated at an instant when the wateris at rest.

These problems, which are treated in Chapter 6, comprisea variety of unsteady motions, including the propagation of wavesfrom a point impulse and from an oscillatory source. Uniquenesstheorems for the unsteady motions are derived. The principle mathematical tool used in solving these problems is the Fourier transform.The methodof stationary phase is justified and used. C) Wavesimmersed in a running stream. This categoryarising fromof problems differs from the first two in that the motion to be inobstaclesvestigated is a small oscillation in the neighborhood of a uniformflow, while the former cases concern small oscillations near the stateof rest. This difference is in one respect rather significant since thefirst two types require no restriction on the shape ofimmersed bodies, or obstacles, while the third type of problemrequires that the immersed bodies should be in the form of thin disks,since otherwise the flow velocity would be changed by a finite amount,and a linearization of the free surface condition would not then beproblems of thejustified.In other words, the problems of this third type require3536WATER WAVESa linearization based on assuming a small thickness for any immersedbodies, as well as a linearization with respect to the amplitude of thesurface waves.

These problems are treated in Chapters 7, 8, and 9.The classical case of the waves created by a small obstacle in a runningstream of uniform depth is first treated. This includes the classicalshipwave problem, discussed in Chapter 8, in which the "ship" istreated as though it could be replaced by a point singularity. Atreatment is given in Chapter 9 of the problem of the waves createdby a ship moving through a sea of arbitrary waves, assuming theship to be a floating rigid body with six degrees of freedom and withits motion determined by the propeller thrust and the pressure of thewater on its hull.Finally, in an Appendix to Part II a brief summary of some ofthe more recent literature concerned with the above types of problemsis given, since the cases selected for detailed treatment here do notby any means exhaust the interesting problems which have beensolved.SUBDIVISION AWAVES SIMPLE HARMONICCHAPTERINTHE TIME3Simple Harmonic Oscillations in Waterof Constant Depth3.1.Standing wavesIn Chapter 2 we have derived the basic theory of irrotationalwaves of small amplitude with the following results (in the lowestorder, that is).

Assuming the #, 2-plane to coincide with the freesurface in its undisturbed position, with the t/-axis positive upward,the velocity potential 0(x,(8.1.1)V2<Z>y, z; t) satisfies=XX+the following conditions:vybounded above by the plane y =and elsewhere byThesurfaces.freecondition undersurfaceothergiven boundaryanyin the regionthe assumption of zero pressure there(8.1.2)tt+ g0y =isfor=y0.=Thecondition at fixed boundary surfaces is that d0/dn0; forconst, we have therefore the conditionwater of uniform depth h(8.1.3)y==forY)(x,(3.1.4)7?==-h.has been determined the elevationOnce thevelocity potentialz;t) of the free surfaceyisgiven by--#,(*,0, *; f).%Conditions at oo as well as appropriatemustinitialconditions att=also be prescribed.In this section we are interested in those special types of standingwaves which are simple harmonic in the time; we therefore write37WATER WAVES880(x,(3.1.5)y, z; t)= e M<p(x, y, z) *a real function, and with the understanding that either thereal or the imaginary part of the right hand side is to be taken.The problems to be treated here thus belong to the theory of smalloscillations of dynamical systems in the neighborhood of an equilib-withrium<pposition.Thegiven above translate into the followingconditions onconditions on<p:V 2<p =(3.1.6)(3.1.7)<y <Q-<p = Q,h0,<poo9y=< #,*<oo,Q,S(3.1.8)=<p yAs conditions atArbitraryinitialooy=-h.0,we assume that y and (p y are uniformly bounded.**now be prescribed, of course,conditions cannotwe have assumed the behavior of our system to be simpleharmonic in the time.

The free surface elevation is given bysince(3.1.9)ri=--tcseM.<p(x, 0, z).SWe look first for standing wave motions which are two-dimensional,so that(jpdepends only upon x andy:<pthe case of water of infinite depth,readily that the functionsfirst(3.1.10)\(p=e mv siny(x, y),i.e.hand=- oo.also considerOneverifiesmxare harmonic functions which satisfy the free surface conditionsatisfies the relation(3.1.7) provided that the constantm(3.1.11)m=o*/g.In addition, the conditions at oo are satisfied. In particular, it is ofinterest to observe that the oscillations die out exponentially in thedepth. The free surface elevation is then given by= f(t)(p(x, y, z).

ThisThe most general standing wave would be given bymeans, of course, that the shape of the wave in space is fixed within a multiplyingfactor depending only on the time. Thus nodes, maxima and minima, etc. occurat the same points independent of the time.** This means that the verticalcomponents of the displacement and velocityare bounded at oo. One could prescribe more general conditions at oo withoutimpairing the uniqueness of the solutions of our boundary value problems, butit does not seem worth while to do so in this case.*SIMPLE HARMONIC OSCILLATIONS^_^(3.1.12)g\sin39mxbe pointed out specifically that our boundary problem,though it is linear and homogeneous, has in addition to the solutiona two-parameter set of "non trivial" solutions obtained by<p ==taking linear combinations of the two solutions given in (3.1.10).The surface waves given by (3.1.10) are thus simple harmonic inx as well as in t.

The relation (3.1.11) furthermore states the veryIt shouldimportant fact that the wave length A given byA ==(3.1.13)is2n/m=2<2ng/onot independent of the frequency of the oscillation, but variesinversely asitssquare.discussion yields standing wave solutions of physicallyreasonable type, but one nevertheless wonders whether there mightThe abovewaves which are not simplesine or cosine functions of x, but rather waves with amplitudes which,for example, die out as x tends to infinity.

Such waves do not occurin two dimensions,* however, in the sense that all solutions for waternot be othersfor example, standing= 0,of thehomogeneous boundary problemformulated in (3.1.6) and (3.1.7) together with the condition that <pand (py are uniformly bounded at oo are given by (3.1.10) withsatisfying (3.1.11).

This is a point worth pausing to prove, especiallysince the method of proof foreshadows a mode of attack on ourexceptof infinite depth,cpmproblems whichfirstwillbe usedina more essentialstep in the uniqueness proofdefinediswaylater on.Theto introduce the function y(x, y)by(3.1.14)\p=(jpymy,m>0.a harmonic function, obviously yj is also a harmonic funcon account of its definitiontion. In addition, y) vanishes for yand (3.1.7). Hence \p can be continued by reflection over the iT-axisSinceq>is=into a potential function which is regular and defined as a singlevalued function in the entire x, j/-plane. Since <p and q>y were assumedto be uniformlyis\pboundedboundedin the entire lower half planeitfollows thatin the entire x, i/-plane since reflection in the #-axisdoes not destroy boundedness properties.