J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 53
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10.1.1.in shallowwaterWith one exception, the present chapter will make useofthetheory to lowest order and consequently the derivation ofonlyit given in sections 2 and 3 of Chapter 2 suffices for all sections oi thislowest order.chapter except section 9.We recapitulate the basic equations. In terms of the horizontalu(x, t), and the free surface elevationvelocity component u=rj=rj(x(10.1.1)(10.1.2)9t)the differential equations(2.2.11),(cf.+uu x = -gr) x=[wfo + h)] xut,fi t291.(2.2.12)) areWATER WAVES292sometimes useful and interesting to make reference to the gasdynamics analogy, by introducing the "density" Q throughItis=Q(10.1.3)and the "pressure" p by p=Q(r)+ A),pdy, which in view of the hydrostatic\pressure law yields the relationP(10.1.4)8r=='/-an "adiabatic law" with "adiabatic exponent" 2 connectingpressure and density.
As one sees, it is the depth of the water, essentially, which plays the role of the density in a gas. In terms of thesequantities, the equations (10.1.1) and (10.1.2) take the formThisis(10.1.5)g(u t(10.1.6)+ uu x = - p x + gQh= -Q(QU) X)tX9.These equations, together with (10.1.4), correspond exactly to theequations of compressible gas dynamics for a one-dimensional flow ifh x = 0, i.e. if the depth of the undisturbed stream is constant.
Itfollows that a "sound speed" or propagation speed c for the pheno-mena governed by=Vdefined by cdp/dp, as inacoustics, and this quantity in our case has the valuethese equationsc(10.1.7)aswesee from (10.1.4)=and=isVg(7?(10.1.3).+h)9Later on, we shall see thatit isindeed justified to call the quantity c the propagation speed since itrepresents the local speed of propagation of "small disturbances"relative to the moving stream.
We observe the important fact that c(which obviously is a function ofx and t) is proportional to the squareroot of the depth of the water.The propagation speed c(x, t) is a quantity of such importance thatworthwhile to reformulate the basic equations (10.1.1 ) and (10.1.2)with c in place of 77. Since c xghx )/2c and c tgr) t j2c one(grj xit is==+finds readily(10.1.8)(10.1.9)ut+ uux + 2ccx -Hx =2c + 2uc x + cu x = 0,twith(10.1.10)H = gh.0,LONG WAVES IN SHALLOW WATER.293The verification in the general case that the quantity c representsa wave propagation speed requires a rather thorough study of certainbasic properties of the differential equations.
However, if we restrictourselves to motions which depart only slightly from the rest position==of equilibrium (i.e. the state with rj0, u0) it is easy to verifythecthatquantity then is indeed the propagation speed. From (10.1.7)we would havein this case c=+ e(x,ct),withc=Vgh andsasmall quantity of first order.
We assume u and its derivatives also tobe small of first order and, in addition, take the case in which thedepth handisconstant.Underthese circumstances the equations (10.1.8)(10.1.9) yield+ 2^ = 0,2e +c u x =u(10.1.11)(10.1.12)Qtorder terms only are retained.if firstu thetByeliminating sweobtain fordifferential equationu tt(10.1.13)-c*u xx-0.wave equation all solutions of which areu(x i c t) and this means that the motionsarc superpositions of waves with constant propagation speed c = Vgfe.The role of the quantity c as a propagation speed (together withmany other pertinent facts) can be understood most readily by discussing the underlying integration theory of equations (10.1.8) andThisisthe classical linearfunctions of the formu(10.1.9) by using what is called the method of characteristics; weturn therefore to a discussion of this method in the next section.10.2.
Integration of the Differential EquationsCharacteristicsby the Method ofThe theory of our basic differential equations (10.1.8) and (10.1.9),which are of the same form as those in compressible gas dynamics,has been very extensively developed because of the practical necessityfor dealing with the flow of compressible gases. The purpose of thepresent section is to summarize those features of this theory whichcan be made useful for discussing the propagation of surface wavesin shallow water. In doing so, extensive use has been made of thepresentation given in the book by Courant and Friedrichs [C.9]; infact, a good deal of the material in sections 10.2 to 10.7, inclusive,follows the presentation given there.WATER WAVES294The essential point is that the partial differential equations (10.1.8)and (10.1.9) are of such a form that the initial value problems associated with them admit of a rather simple discussion in terms of apair of ordinary differential equations called the characteristic differential equations.proceed to derive the characteristic equationsWewhichfor the special case in[cf.(10.1.10)]H x = m = const.(10.2.1)the case in which the bottom slope is constant.
In fact, this is theonly case we consider in this chapter. If we add equations (10.1.8)and (10.1.9) it is readily seen that the result can be written in thei.e.form:9(10.2.2)The^-+ u+c.a.)(^.(uexpression in brackets+ 2c-mt)=0.of course, to be understood as a dif-is,ferential operator. Similarly, a subtraction of (10.1.9)from (10.1.8)yieldsid_1(10.2.3)[ot+(u_cd\1.()tt_2c-ox]mt)=0.But theinterpretation of the operations defined in (10.2.2) and(10.2.3) is well known (cf. (1.1.3)): the relation (10.2.2), for example,states that the function (u2cmt) is constant for a point moving++c), or, as we may also put it,through the fluid with the velocity (ufor a point whose motion is characterized by the ordinary differential=u + c.
Equation (10.2.3) can be similarly interpreThat is, we have the following situation in the x, /-plane: There aretwo sets of curves, Cl and C 2 called characteristics, which are theequation dx/dtted.,solution curves of the ordinary differential equationsCxdx:(10.2.4)dx.=u=U+c,andCand we have the relations1u+ 2cmtu2cmt==Aj&2==const, along a curveconst, along a curveC l andC2.Of course the constants k^ and k 2 will be different on different curvesin general.
It should also be observed that the two families of charac-LONG WAVES IN SHALLOW WATERteristicsdetermined by (10.2.4) are really distinct because of the fact=^+since we suppose thath)Vg(r)the water surface never touches the bottom.that c295rj>thath, i.e.By reversing the above procedure it can be seen rather easily thatthe system of relations (10.2.4) and (10.2.5) is completely equivalentto the system of equations (10.1.8) and (10.1.9) for the case of constant bottom slope, so that a solution of either system yields a solution of the other.
In fact, if we set f(x, t)2cumt and ob-== &! = const, along any curve=citfollowsthat along such curvesu+dxfdtserve that f(x,t)(10.2.6)/,In the same+way+ /.dx-h+(uthe function g(x,+ c)f, =t)=ux-=x(t) forwhich0.2cmtsatisfies therelation(10.2.7)ft+(u-c)g x=uc. Thus wherever the curvealong the curves for which dx/dtfamilies C 1 and C 2 cover the r, J-plane in such a way as to form a nonsingular curvilinear coordinate system the relations (10.2.6) andnow(10.2.7) hold. Ifthe definitions ofequation/(#,(10.1.8)equations (10.2.6) and (10.2.7) are added andand g(x, t) are recalled it is readily seen thatt)subtracting (10.2.7) from (10.2.6)obtained.
In other words, any functions u and cresults.Byequation (10.1.9) iswhich satisfy the relations (10.2.4) and (10.2.5) will also satisfy(10.1.8) and (10.1.9) and the two systems of equations are thereforenow seen to be completely equivalent.As we would expect on physical grounds, a solution of the originaldynamical equations (10.1.8) and (10.1.9) could be shown to be=0,uniquely determined when appropriate initial conditions (for tsay) are prescribed; it follows that a solution of (10.2.4) and (10.2.5) is also uniquely determined when initial conditions are prescribedsince we know that the two systems of equations are equivalent.At first sight one might be inclined to regard the relations (10.2.4)(10.2.5) as more complicated to deal withferential equations, particularly since the rightandknownthan the originalhanddif-sides of (10.2.4)advance and hence the characteristic curves are alsonot known: they must, in fact, be determined in the course of determining the unknown functions u and c which constitute the desiredsolution.
Nevertheless, the formulation of our problems in terms ofare notinWATER WAVES296the characteristic formisquite useful in studying properties of theandalso in studying questions referring to the appropriateness of various boundary and initial conditions. It is useful to beginsolutionsbriefly a method of determining the characteristics andthus the solution of a given problem by a method of successive approximation which at the same time makes possible a number of usefulby describingobservations and interpretations regarding the role played by thecharacteristics in general.
Let us for this purpose consider a problemwhich the values of the velocity u and the surface elevation rj(or, what amounts to the same thing, the propagation or wave speedinct= Vg(?7 + h)) are prescribed for all values of at the initial instant= 0. We wish to calculate the solution for > by determining ua?tthrough use of (10.2.4) and (10.2.5) and the givenwe assume thattions. At tandcinitialcondi-=u(x, 0)(10.2.8)(x,0)= u(x)= c(x)which u(x) and ~c(x) are given functions.
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