J.J. Stoker - Water waves. The mathematical theory with applications, страница 8

areoriginally undisturbed surface of the water y,from=Scondition onon S(cf.arisessection 1.4);poweri711*j= 0,= 0.=(2.1.5) it followsall to be satisfied0.byontheThe other boundaryfrom the fact that the water particles stayit is expressed in the form#ifcInsertion of the+ 0]e+ #.%+*]t=andseries fory77onS.in this expression leads tothe conditions- 0,fjW(2.1.8)(2.1.9)(2.1.10)which are also to be satisfiedIn view of the fact that iy (0)be put in the form(2.1.12)gqW+Qp^O,+ 0W = _OT(2.1.13)g^<(2.1.11)Cw>+ 0| =w)for=y0,J[(0W)=0.the free surface conditions can+(0^)2+((PW )],<i)0W,jP^-D,which the symbol pi n -u refers to a certain combination of the(k) with fcfunctions rj (k) andnj1, and all conditions are to beinsatisfied forbecomesy=0. Similarly,the other set of free surface conditionsTUP:(2.1.14)fi(2.1.15),(2.1.16)T?<TWOBASIC APPROXIMATE THEORIES- >,< -=**,>21+W)whichk ^ niny_= o.G (n ~(fc)(k) withanddepends only upon functions ?yand once more all conditions are to be satisfied forThis theory therefore is a development in the neighborhoodl)I9of the rest position of equilibrium of the water.The relations (2.1.11) to (2.1.16) thus, in principle, furnish ameansof calculating successively the coefficients of the series (2.1.1) and(2.1.2), assuming that such series exist: The conditions (2.1.11) and(2.1.14) at the free surface together with appropriate conditions atother boundaries, and initial conditions for /0, would in conjunc2(1)lead to unique solutions r/ (l and <2> (1) Oncetion with V=).andare determined, they can be inserted in the conditionsrf(2(2) andwhichand(2.1.12)(2.1.15) to yield two conditions for rj(2)serve towith the subsidiary boundary and other conditions onl)(l)>One could interpret the work of Levi-Civita [L.7]referredto in section 1.7 as a method of proving[S.29]the existence of progressing waves which are periodic in x by showingdetermine them,etc.and Struikandthat the functionsrjcan indeed be represented as convergentpower series in e for e sufficiently small.In what follows in Part II of this book we shallcontent ourselvesin the main with the degree of approximation implied in breakingoff the perturbation series after the terms e0 (l} and erj (l) in the series(2.1.1)and(2.1.2),i.e.we= e0set(l)andrj=(l)er].Withthisstipulation the free surface conditions (2.1.11) and 2.1.14) yield(2.1.17)gl?+0 = o) for =ytI(2.1.18)Byelimination ofritr\-0 =tf0.oJbetween these two relations the single conditionon 0:(2.1.19)tt+ g0y =fory=obtained; this condition is the one which will be used mainly infrom V 2 = 0, after which the freePart II in order to determinesurface elevation r\ can be determined from (2.1.17).

The usualmethod of obtaining the last three conditions is to reject all butisWATER WAVES22and77and dynamicandthe linear terms intheir derivatives in the kinematic(cf. (1.4.6)) free surface boundary con(1.4.5))ditions.proceeding in this way we can obtain a first approximation(cf.Byto the pressurep (which was not consideredin theabove generalperturbation scheme) in the form:-= -gy-0(2.1.20)t.*QWe can now see the great simplifications which result throughthe linearization of the free surface conditions: not only does theproblem become linear, but also the domain in which its solutionisto be determined becomes fixed a prioriand consequently thewave problemsin this formulation belong, from the mathematical point of view, to the classical boundary problems of potentialsurfacetheory.2.2.Shallow water theory to lowest order.

Tidal theoryAdifferent kind of approximation from the foregoing linear theoryof waves of small amplitude results when it is assumed that thedepth of the water is sufficiently small compared with some othersignificant length, such as, for example, the radius of curvature ofthe water surface.

In this theory it is not necessary to assume thatthe displacement and slope of the water surface are small, and theresulting theory is as a consequence not a linear theory. There arecircumstances in nature under which such a theory leads toa good approximation to the actual occurrences, as has already beenmentioned in the introduction. Among such occurrences are the tidesin the oceans, the "solitary wave" in sufficiently shallow water, andmanythe breaking of waves on shallow beaches. In addition, many phenomena met with in hydraulics concerning flows in open channels suchas roll waves, flood waves in rivers, surges in channels due to suddeninflux of water, and other kindred phenomena, belong in the nonlinearshallow water theory.

Chapters 10 and 11 are devoted to the workingout of consequences of the shallow water theory.The shallow water theory is, in its lowest approximation, the basictheory used in hydraulics by engineers in dealing with flows in open* In case the surfacepressure pa (x t z;isreplaced(2.1.20)!t) isnot zero one finds readily that (2.1.17)bygriwhile (2.1.18) remains unaltered.+ & = - Po/Q,tTHE TWO BASIC APPROXIMATE THEORIES23channels, and also the theory commonly referred to in the standardtreatises on hydrodynamics as the theory of long waves.begin bygiving first a derivation of the theory for two-dimensional motionWealong essentially the lines followed by Lamb [L.3], p. 254.

As usual,the undisturbed free surface of the water is taken as the o?-axis andthe t/-axis is taken vertically upwards. The bottom is given by=h represents the variable depth of the undisturbedsurface displacement is given by yr](x, t). The velocityaredenotedandv(x, y, t).componentsby u(x, y, t)yh(x), so thatwater.=TheThe equationof continuityUx(2.2.1)The conditionsis+ Vy = 0.to be satisfied at the free surface are the kinematicalcondition:(2.2.2)(r) t+ ur, x -v) y=ri=\0;and the dynamical condition on the pressure:p(2.2.3)At the bottom the condition(uhx(2.2.4)=!,_0.is+ v)|V= _A=0.Integration of (2.2.1) with respect to y yieldsfh(2.2.5)Use of the condition+v\\ = Q.and (2.2.4)y =Ux )dy(J(2.2.2) ath yieldsat y77the relationr(2.2.6)JWehux dy+r) t+u\^ +u\_ hhx=0.introduce the relationg(2.2.7)^vp()J _ fc(jBand combineitu dy=rnurj x\+u)\hxy =_ h+J -h\uxdy.with (2.2.6) to obtain(2.2.8)Upto this point no approximations have been introduced.an approximate theory which resultsfrom the assumption that the y- component of the acceleration ofThe shallow water theoryisWATER WAVES24the water particles has a negligible effect on the pressure p, or,what amounts to the same thing, that the pressure p is given as inhydrostatics*by= gety - y).p(2.2.9)Q is the density of the water.

A number of consequencesof (2.2.9) are useful for our purposes. To begin with, we observe thatThe quantitypx(2.2.10)= ggjfc,pindependent of y. It follows that the ^-component ofthe acceleration of the water particles is also independent of y\so thatx isand henceu, the^-component of thevelocity,is=also independentWeof y for all t if it was at any time, say at t0.shall assumethis to be true in all casesisittrue for example in the importantwhich the water was at rest at t =so that u= u(x, t)onfromandtnowon.Asxdepends onlyequation of motion in the^-direction we may write, therefore, in view of (2.2.10):special case inu(2.2.11)t+ uux =-gife.This is simply the usual equation of motion in the Eulerian form,use having been made of u y0.

In addition, (2.2.8) may now bewritten=(2.2.12)[u(riprju dysince= u\pr\hJhJ+ h)] x =-ij i9dy on account of the fact that uisindependentof y. The two first order differential equations (2.2.11) and (2.2.12)for the functions u(x, t) and r\(x, t) are the differential equations ofthe nonlinear shallow water theory. Once the initial state of the fluidisprescribed,i.e.once the values of u andgiven, the equationsmotion.and(2.2.11)77(2.2.12)at the timeyieldt=arethe subsequentassumption of the shallow water theorywe assume that u and 77, the particle velocityIf in addition to the basicexpressed by (2.2.9)and free surface elevation, and their derivatives are small quantitieswhose squares and products can be neglected in comparison withlinear terms, it follows at once that equations (2.2.11) and (2.2.12)simplify tou(2.2.13)(2.2.14)Wehave p vt(uh) x== = -gjfcifc,,gg and (2.2.9) results through the use of p=fory=r).THE TWO BASIC APPROXIMATE THEORIESfrom which7725can be eliminated to yield for u the equation(uh) xx(2.2.15)~u tt=0.SIf,in addition, thesatisfiesdepth hisconstantitfollows readily thatuthe linear wave equation-uuxx(2.2.16)tt=0.In this case 77 satisfies the same equation.

One observes thereforethe important result that the propagation speed of a disturbance isgiven by Vgh. In principle, this linearized version of the shallow watertheory is the one which has always been used as the basis for thetheory of the tides. Of course, the tidal theory for the oceans requirescomplete formulation the introduction of the external forcesacting on the water due to the gravitational attraction of the moonand the sun, and also the Coriolis forces due to the rotation of theearth, but nevertheless the basic fact about the tidal theory fromthe standpoint of mathematics is that it belongs to the linear shallowwater theory.

The actual oceans do not from most points of viewimpress one as being shallow; in the present connection, however,the depth is actually very small compared with the curvature ofthe tidal wave surface so that the shallow water approximation is anexcellent one. That the tidal phenomena should be linear to a goodapproximation would also seem rather obvious on account of thesmall amplitudes of the tides compared with the dimensions of theoceans. A few additional remarks about tidal theory and some otherapplications of the linearized version of the shallow water theory tofor itsconcrete problems (seiches in lakes, and floating breakwaters, forexample) are given in Chapter 10.13.2.3.ItinGas dynamics analogyispossible to introduce a different set of dependent variablesway that the equations of the shallow water theory becomesuch ato thefundamental differential equations of gas dynamics fora compressible flow involving only one space variable x.(This seems to have been noticed first by Riabouchinsky [R.8].)To this end we introduce the mass per unit area given byanalogousthe case of(2.3.1)Q=Q(rj+h).WATER WAVES26Since h depends only on x==e^.ft(2.3.2)Wewe havenext define the force p per unit width:p(2.3.3)=\\pdy,nJwhich, in view of (2.2.9) and (2.8.1), leads toP(2.3.4)The=^(n+h}*=-j^Q*.=thus of the form pAQY withy==2, that is, the "pressure" p and the "density" Q are connectedby an "adiabatic" relation with the fixed exponent 2.Equation (2.2.11) may now be writtenrelationbetween p and Qq(r)andthis,in turn,is+ h)(u + uux = - gQfa + h)qx)tbe expressed through use ofmay(2.3.1)and(2.3.4) as follows:(2.3.5)+ uu9 = -Q(u t)px+ gQh x,as one can readily verify.The equation(2.2.12)maybe written as(2.3.6)view of (2.3.2) as well as (2.3.1).