J.J. Stoker - Water waves. The mathematical theory with applications, страница 9

The differential equations (2.3.5)and (2.3.6), together with the "adiabatic" lawp = gg*/2g given by(2.3.4), are identical in form with the equations of compressible gasdynamics for a one-dimensional flow except for the term gQhx onthe right hand side of (2.3.5), and this term vanishes if the originalundisturbed depth h of the water is constant.

The "sound speed" ccorresponding to our equations (2.3.5) and (2.3.6) is, in analogy withingas dynamics, given by chas the value(2.8.7)C==VdpldQ, and=Vg(rjthisfrom(2.3.4)and(2.3.1)+ h).It will be seen later that c(x, t) represents the local speed at whicha small disturbance advances relative to the water.THE TWO BASIC APPROXIMATE THEORIES2.4. Systematic derivation of the shallow27water theoryIt is of course a matter of importance to know under what circumstances the shallow water theory can be expected to furnishsufficiently accurate results. The only assumption made above inaddition to the customary assumptions of hydrodynamics was thatis given as in hydrostaticsby (2.2.9), but no assumptionwas made regarding the magnitude of the surface elevation or thethe pressurevelocity components. Consequently the shallow water theorymaybe accurate for waves whose amplitude is not necessarily small,provided that the hydrostatic pressure relation is not invalidated.The above derivation of the shallow water theory is, however, opento the objection that the role played by the undisturbed depth of thewater in determining the accuracy of the approximation is not putIn fact, since we shall see later on that all motions dieout rather rapidly in the depth, it would at first sight seem reasonableto expect that the hydrostatic law for the pressure would be, on thein evidence.whole, more accurate the deeper the water.

That this is not thecase in general is well known, since the solutions for steady progressingwaves of small amplitude (i.e. for solutions obtained by the linearizedtheory) in water of uniform but finite depth are approximatedaccurately by the solutions of the shallow water theory (when it alsois linearized) only when the depth of the water is small comparedwith the wave length (cf. Lamb [L.3], p. 368).

It is possible to givea quite different derivation of the shallow water theory in which theequations (2.2.11) and (2.2.12) result from the exact hydrodynamicalequations as the approximation of lowest order in a perturbationprocedure involving a formal development of all quantities in powersof the ratio of the original depth of the water to some othercharacteristic length associated with the horizontal direction.* Therelation (2.2.9) is then found to be correct within quadratic termsin this ratio. In this section we give such a systematic derivationof the shallow water theory, following K. O.

Friedrichs (see theappendix to [S.19]), which, unlike the derivation given in section* In this book theparameter of the shallow water theory is defined in twodifferent ways: in dealing with the breaking of waves in Chapter 10.10 it isthe ratio of the depth to a significant radius of curvature of the free surface;in dealing with the solitary wave, however, it is essentially the ratio of the depthzthe propagation speed of the wave, and in thisto the quantityjg, withacase the development is carried out for/gh near to one.

In still otherproblems it might well be defined differently in terms of parameters that arecharacteristic for such problems.UUUWATER WAVES28capable of yielding higher order approximations.The disposition of the coordinate axes is taken in the usual manner,with the x, 3-plane the undisturbed water surface and the i/-axis2.2above,isThe free surface elevationand the bottom surface by y =h(x, z).positive upward.isgiven by yrj(x 9 z, t)Werecapitulate for thesake of convenience the differential equations and boundary conditions in terms of the Euler variables, that is, the equations ofcontinuity and motion, the vanishing of the rot at ion, and the boundaryconditions:wz =(2.4.1)0,vu(2.4.2)wt+ uwxpxvvwv z= --- pyvwwwz----= wx vx = u= v at y =nt + uri x + wr]at y =p =at y =uhx -\-v-\~ wh =(2.4.3)pz,(2.4.4)z(2.4.5)(2.4.6)77,77,/i.zWe nowintroduce dimensionless variables through the use of twoanddk, with d intended to represent a typical depth and klengthsa typical length in the horizontal direction it is characteristic ofthe procedure followed here that the horizontal and vertical directions are not treated in the same way.

The new independent variablesare as follows:x(2.4.7)while the(2.4.8)x/k,y=y/d, z=z/k,new dimensionless dependentu=p=^= 7^/d,In addition,(2.4.9)=(Vgd)~lrv=h/d.tVgd/k,variables are= (kVgd/d)-u,=1w = Vgd)~ wlv,(p,Awe introducethe important parametera=THE TWO BASIC APPROXIMATE THEORIESinterms of whichissmall the water29quantities will be developed; when this parameterconsidered to be shallow. This means, of course,allissmall compared with fc, and hence that the x and z coordinates (cf. (2.4.7)) are stretched differently from the y coordinatethat dandisin a fashionSinceitiswhich depends upon the development parameter.the horizontal coordinate whichstrongly stretchedseems reasonable to refer to theisrelative to the depth coordinate, itresulting theory as a shallow water theory.The stretching processcombined with a development with respect to a is the characteristicfeature of what we call the shallow water theory throughout thisbook.

The dimensionless development parameter a has a physicalbut its interpretation will vary dependingon the circumstances in individual cases, as has already been notedabove. For example, consider a problem in which the motion is to bepredicted starting from rest with initial elevation y77 (#, j/, z)wehavefrom(2.4.8)prescribed;significance, of course,y^=-r)dfj Q (x 9 y, z)i- i^'/I?0xyU'"<*'from which we obtainIt isnatural to assume that the dimensionless second derivativebe at least bounded and consequently one sees that theassumption that a is small might be interpreted in this case asmeaning that the product of the curvature of the free surface of thetfficxwillwater and a typical depthis a smallquantity.to consider a sequence of problems dependingon the small parameter a and then develop in powers of or. Introduc-The object nowtion of thenewisvariables in the equations (2.4.1) to (2.4.6) yields+ v y + aw = 0,a[u + uux + wu + px + vuy = 0,a[v + uvx + wv + p y + 1] + TO, =a[w + uwx + ww + p + vwy = 0,w y = V u = wx vx = u y= v at y =a[ijt + ur) x + wr]aux(2.4.1)'zzt(2.4.2)'(2.4.3)'(2.4.4)'zt.]ztZ9zg],z],77,0,WATER WAVES80p(2.4.5)'when bars over allThe next step isw,whz ]a[uhx(2.4.6)'=at y== Oa-j-t;=h,quantities are dropped and r is replaced byto assume power series developments for u,t.v,and p:rj,==wv(2.4.10)-f...,4-...,= p(0)andinsertandt.themin the equations (2.4.1)' to (2.4.6)' to obtain, byequating coefficients of like powers of a, equations for the successivecoefficients in the series, which are of course functions of x, y, z,The terms ofzero order yield the equationsnT .(0)(2.4.1/(2-4.4);a%<> =vv_7 ,(0) 7 ,()(2.4.2);(2.4.3);'y)\vv_(0)MUy==u (0>(2-4.5);p(o)(2.4.6);rj(o)==o=oo,ou=y = ^y =at yatat_(0)^(0)Wy.

, V X(0)Uy,(O),T?(0),A.These equations yield the following:i><>(2.4.11)(2.4.12)(2.4.13)(2.4.14)wu<)p<>(0,=0,= w<>(ff,=u(0)i^W),z,<)(o;, 2, i),a,*)=0,which contain the important results that the vertical velocity component is zero and the horizontal velocity components are independentof the vertical coordinate y in lowest order.The first order terms arising from (2.4.1)' to (2.4.6)' in their turnTHE TWO BASIC APPROXIMATE THEORIES81yield the equationsu+=-*W,(2.4.1);+ pf = 0,p<,> +1=0,+ u^wx + a;a>< + p<> - 0,=at y = ?,rjf + u^rif + wr)W= at y = - h,u^hx + ojtWA, +t tt<(2.4.2);0)+ w %f + w(<0).((0) tt< 0)0)}K>!<(2.4.4);<(2.4.6)iupon making use of(2.4.11), (2.4.12),andcan be integrated at once since u w and(2.4.13).w(0)Equation(2.4.1)Jare independent of yto yieldyd)(2.4.15)Fwith- - (> + o,<>+) z,F(<c, z, t),an arbitrary function which can be determined by using(1 >(2.4.6);; the result for u<(2.4.16)= -(M<>+isu,<then0))y-[(<>/*),+()*).],._..To secondin theorder the vertical component of the velocity is thus lineardepth coordinate.