J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 75
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Under any immersed bodies the value of fj will be fixed by thestatic equilibrium position of the given bodies. Thus fj is in all casesa given function of x and z; for a floating rigid body, for example, ittime.)surface,would be determined by hydrostatics.The condition to be satisfied at the upper surfaceconditionisthe kinematic:(10.13.3)XYX +ZY Zy+Y =t0,which states that a particle once on the surface remains onsurface, the condition to be satisfied isit.At thebottom(10.13.3)!xh x+zhz+V=at yBernoulli's law for determining the pressure at(10.13.4)-+t+ gy -W== -h(x, z).any point in the water is0.QHere we have assumed that theremaybe other external forces besideWATER WAVES416and theseassumed to be determined by a workwhoseW(x j/, z; t)space derivatives furnish the force components; in this case it is known that the motion can be irrotationaland that Bernoulli's law holds in the above form (cf.
the derivationsgravity,functionininforces are9Chapter1).We now write the equation of the moving upper surfacethe formy(10.13.5)and assume=Y(x,z; t)=ij(x 9 z)+ *?(#, z;in accordance with our statementt)above thatr)(x, z; t) re-presents a small vertical displacement from the equilibrium positiongiven by yq. Upon insertion in (10.13.3) and (10.13.4) we find=after ignoring quadratic terms in(10.13.6)xfj x+e fjz-y+r, tr/andandtheir derivatives:=at y=fj(x, z)(10.13.7)boundary conditions to be satisfied at the equilibrium position ofthe upper surface of the water. At points corresponding to a free surand hencewe would have, for example, fj =face where p =as_0 y+ ^ == o(10.13.8)'(10.13.9)A special case might be that in which the motion of a portion of theupper surfaceissumed known;suffice asprescribed, i.e. rj(x, z; t) as well as fj would be prein such a case the condition (10.13.6) alone woulda boundary condition for the harmonic function 0.
In someof the problems to be treated here, however, we do not wish to assumethat the motion of some immersed body, for example, is known inadvance; rather, it is to be determined by the interaction with thewater which exerts a pressure p(x, z; t) on it in accordance with (10.13.7). Thus the exact formulation of our problems would requirethe determination of a harmonic function 0(x, t/, z; t) in the spacebetween y =h(x, z) and y = fj which satisfies the conditions (10.and13.6)(10.13.7) at the upper surface (in particular the conditionsand(10.13.8)(10.13.9) on the free surface) and (10.13.3)! at thebottom. Additional conditions where immersed bodies occur (to beobtained from the appropriate dynamical conditions for such bodies)would be necessary to determine the pressure p, which provides the"coupling" between the water on the one hand and the immersedbodies on the other.
Finally, appropriate initial conditions for theLONG WAVES IN SHALLOW WATER417water and the immersed bodies at the initial instant would be neededif one were to study non-steady motions, oras will be the case hereconditions at oo of the radiation type would be needed if simpleharmonic motions (that is, steady vibrations instead of transients)are studied.
It need hardly be said that the difficulties of carryingout the solutions of such problems are very great indeed (cf. Chapterso much so that we turn to an approximate theory9, for example)which is based on the assumption that the depth of the water issufficiently small and that the immersed bodies are rather flat.*In the derivation of the shallow water theory we start from theand integrate it with respect to yLaplace equation (10.13.1) forfrom the bottom to the equilibrium position** of the top surfacey=fj(x, z)(10.13.10)to obtain, after integration"_& yv dy=y-yby= -parts:(0 XX+zz)dymeans that it isHere, and in what follows, a bar over the quantityto be evaluated at the equilibrium position of the upper surface of the=water, i.e. for y?/(#, 2), and a bar under the quantity means that itis to be evaluated at the bottom yh(x, z).
From the kinematic=surface condition (10.13.6) and the condition (10.13.3 ) t at the bottom,we have therefore (due regard being paid to the fact that a bar shouldnow be put over(10.13.11)ingSincein (10.13.3)!):expresses the factthenextresult of integratConsiderincompressible.really a continuity conditionisby parts the(10.13.12)and under=-ntThis conditionthat the waterin (10.13.6)rightXdyhand=side of (10.13.11); in particular:fj0 x+ h0 x -y0 xydy.we have* In the course of the derivation the termsneglected are given explicitly sothat a precise statement about them can be made.** One seesreadily that carrying out the integration to y ~ 7] rather thanto yyields the same results within terms of second order in smallrj + //=quantities.WATER WAVES418(10.13.18)we mayhJeliminate(10.13.14)Xh$xv dyfrom (10.13.12) to obtain:+ h)0 x -dy=(rjX= h$x - h&x(h+ y)0 xy dy.Indeed, we have quite generally for any function F(x, y,z;t) theformula:(*(10.13.14)!J/iFdy=(fj+ h)F - JP n (h + y)F y dy.Making use of the analogous expressionfor the integral ofZweobtain from (10.13.11) the relation(10.13.15)in= -r, t+ h)0 x x -[(fj][(fj+ h)0z] z+ IX + Jzwhich(10.13.16)=/*"+ y)0 xy dy, J =(hIn addition, we have from (10.13.10)in+ y)0zy dy.(hcombination with (10.13.14)the condition:(10.13.17)y= -+ h)[0 xx +(fjas one can readily verify.Up to this point we havearisingfrommate theoryturnlinearizing.isjustifiedtives ofare boundedx-hzmade no approximationsThein the rightz+I X + J29other than thoseessential step in obtaining our approxi-in neglecting the terms I x and Jz This inassumed that certain second and third deriva-when htives are small* of theand Jzhxnow takenif it isisZZ ]hand.issmall and that77andits firstderiva-same orderas h: one sees that the terms I xsides of (10.13.15) and (10.13.17) are then of2order h while the remaining terms are of order h.
Under the free surface in the case of a simple harmonic oscillation one can show that thisapproximation requires the depth to be smallwaveincomparison with thelength.differentiating the relation (10.13.7) for the pressure at thesurfaceof the water with respect to t (again noting that a barupperUponshould be placed over the termwe find the equation* Thiswhich aretin (10.13.7))means that the theory developed hereflat.and using (10.13.15)applies toimmersed bodiesLONG WAVES IN SHALLOW WATER&(10.13.18)+L-wt=tt[(jj+ h )0 x ] xand Jz Thisafter dropping the terms I x.is419+the basic differential equa-tion for the function 0(x, z; t) which holds everywhere on the uppersurface of the water.
In particular, we have at the free surface whereandpfj~the equation+(h$ x ) x(10.13.19)- -$ = -(A0,),ttWt.ooWerecall that W(x, y, z; t) represents the work function for anyexternal forces in addition to gravity (tide generating forces, forits value on the free surface, would be given byexample), so thatW9W(x, 0, 2; t).
If, in addition, it is assumed that h is a constant, i.e. thatthe depth of the water is uniform, and that gravity is the only externalforce, we would have the equation&xx +(10.13.19)!thatis,the linearthe time/.$-wave equationAs a consequence,itin theallcase with the constant speed c$ =0,two space variables#, zanddisturbances propagate in such a=Vgh, asiswellknownfor thisequation.If thereis an immersed object in the water, the equation (10.13.19)holds everywhere in the x, z-plane exterior to the curve C whichdefines the water line on the immersed body in its equilibrium position.
The curve C is supposed given by the equationsx(10.13.20)=zx(s\= z(s)Wemust have boundary, or perhaps it isbetter to say, transition conditions at the curve C which connect thesolutions of (10.13.19) in the exterior of C in an appropriate mannerwith the motion of the water under the immersed body. Reasonableconditions for this purpose can be obtained from the laws of conservation of mass and energy. In deriving these conditions we assume= 0, since we wish to deal only with gravity as the external forcewhen considering problems involving immersed bodies.
Consider aninterms of a parameters.Welement of length ds of the curveimmersed body(cf.CFig. 10.13.2).representing the water line of theThe expressionWATER WAVES420represents the mass flux through a vertical strip having the normalnC(x(s),y(s))Boundary at waterFig. 10.18.2.an immersed bodyline ofand extending from the bottom to the top of the water. Fromfor F =n we havex applied(10.13.21)edyn=e (ij^Q(fj+ h)0 n + h)0 n(hQ(10.13.+ y)0 nv dywhere the second term is ignored because it is of order h 2 Thus itwould be reasonable to require that (77 + h)0 n should be continuouson C since this is the same as requiring that the mass of the water is conserved within terms of the order retained otherwise in our theory.
Forthe flux of energy across a vertical strip with the normal n we have.(10.13.22)(J^ p0 n dyds=)(-QJ^0#n dy- gQJ^ y0 n dy) dsupon making use of the Bernoulli law (10.13.4) for the pressure p= 0). Once more we may ignore the second term in the(whenWbrackets sinceF=tn(10.13.23)Sinceitisof order h 2.Uponapplying (10.13.14)! withand again ignoring a term of order h 2 we findj\F n p0 n dy)ds = {-Jwe have already requiredthat(fj+ h)0 n should be continuous,wesee that the additional requirement,continuity of the energy flux.As reasonabletransition conditionsimmersed body at(10.13.24)ds.t}i(fjitswater+ h)0 n,linettcontinuous, ensures theon the curvewe haveCthereforecontinuous on C.delimiting theLONG WAVES IN SHALLOW WATER421Ofcourse, if fj is continuous (e.g.
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