J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 63
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This procedure ofLavrentieff thus also starts with a nonlinear first approximation.The problem thus furnishes another good example of the well-knownfact that it is not always easy to guess how to set up an approximationscheme for solving nonlinear boundary value problems, since thesolution may behave in quite unexpected ways for particular values ofthe parameters. Hindsight, however, can help to make the necessityfor procedures like those of Friedrichs and Hycrs and of Lavrentieffin the present case more apparent: we have seen in Chapter 7.4 that=a steady flow with the critical speed UVgA is in a certain sensesincetheunstabledisturbancewould lead, in termshighlyslightestof the linear theory for waves of small amplitude, to a motion in whichinfinite elevations of the free surface would occur everywhere; thusthe linear theory of waves of small amplitude seems quite inappropriate as the starting point for a development which begins with a'* L.Nirenberg [N.2] has recently proved the existence of steady waves ofamplitude caused by flows over obstacles in the bed of a stream.finiteLONG WAVES IN SHALLOW WATER345uniform flow at the critical speed, and one should consequently usea basically nonlinear treatment from the outset.Wenowturnto the discussion of the solution of the solitarywaveproblem.
The theory of Friedrichs and Hyers begins with a formulation of the general problem that is the same as that devised by LeviCivita for treating the problem of existence of periodic waves of finiteamplitude, and which was motivated by the desire to reformulate theproblem in terms of the velocity potential <p and stream function \p asindependent variables in order to work in the fixed domain betweenthe two stream lines \pconst, corresponding to the bottom and the=free surface instead of in the partially unknown domain in the physical plane.therefore begin with the general theory of irrotationalWewater when a free surface exists.
The wave is assumed tobe observed from a coordinate system which moves with the samevelocity as the wave, and hence the flow can be regarded as a steadywavesinflow in this coordinate system.Fig. 10.9.1.Acomplex velocity potential(10.9.1x')sought in anisTheis=?'%'(x', y')+ *V>#', i/'-plane (cf.solitarywave=%'(z'):=x +'*'{y'Fig. 10.9.1) such that at infinity theU and the depth of the water is h. %' is of course an analyticvelocityfunction ofTheharmonic functions y' and \p' represent thevelocity potential and the stream function. The complex velocitywf=z'.d%'/dz'isrealgiven byw'(10.9.2)inwhichof theu'andv'=u'iv' 9are the velocity components. This followsCauchy-Riemann equations:(10.9.3)<p' x<= v>VVV = ~V>'*''by virtueWATER WAVES846since w'=xq>'>+i\p'x>.Itconvenient to introduce new dimensionlessisvariables:(10.9.4)zw== z'/h,and a parameter= V + *V = X'/( hUxw'/U,)>y:= gWy(10.9.5)2-1In terms of these quantities the free surface corresponds to \psincethetotalflowaoveris assumed given by y>0,curve extending from the bottom to the free surface is Uh.
Theboundary conditions are now formulated as follows:if=the bottomv(10.9.6)=\ \w\(10.9.7)Jmw =2+ yy =The second conditionaty>atconst.=\p0,=1.from Bernoulli's law on taking thepressure to be constant at the free surface and the density to be unity,as one sees from equation (1.3.4) of Ch. 1. At oo we have the conditionresultsw(10.9.8)-> 1as|xWe assume now that the physical planeby means of %(z)is\pmapped=into the99,in a one-to-one-* oo.|(i.e.they-plane in such a#, i/-plane) iswaymappedthat the entire flowandthe strip bounded by \pmapping function z(%) exists, and weway on1.* In this case the inversecould regard the complex velocity=was a function of# defined in thethe ^-plane. We then determine the0, \pstrip\panalytic function w(%) in that strip from the boundary conditions(10.9.6), (10.9.7), (10.9.8), after which %(z) can be found by an inte-bounded by1 in=Jm # =1.gration and the free surface results as the curve given by \pIt is convenient, however, again following Levi-Civita to replacethe dependent variable w by another (essentially its logarithm)through the equationw(10.9.9)=<r l( *+ tA).It follows that(10.9.10)and thus A\w\=log|io|,with=e\w\x,=argiei,the magnitude of the velocity vector,* Ourassumption that the mapping of the flow on the /-plane is one-to-onecan be shown rather easily to follow from the other assumptions and Levi-Civitacarries it out.
The equivalence of the various formulations of the problem is thenreadily seen. In Chapter 12.2 these facts are proved.LONG WAVES IN SHALLOW WATERandWethe inclination relative to the #-axis of the velocity vector.proceed to formulate the conditions for the determination of 6isand A6847=in theattiate\p=y plane.<p,(10.9.6) becomes, of course,To transform the0.with respect to(10.9.11)The condition|wcondition (10.9.7) we first differenline ipthe1 to obtainalong=q>d \w\^dy+ y-^ =-|=on wrfSince x and y are conjugate harmonic functions ofwritedocdy~dv(10.9.12)<pand\pwe mayu(p x~~~~<pl + cplVx-rVvdwr7{1.dq>dcpdyoxvd<pd\p\wjw2|2|in accordance with well-known rules for calculating the derivativesof functions determined implicitly, or fromdzud%d>Xu11uiv2+ iv+v'2dzAs a consequence we have fromor,since|w~exandv=(10.9.11):J>w e~ i(P+ iK}e*sinO:\A6^=^--w~2A sin 0,t/9?and since ^A/S^it=dO/dy because A and 6 are harmonic conjugatesfollows finally thatdO(10.9.13)= w- 3A sin0atw=1.oy==and (10.9.13) at \p = 1 areLcvi-Civita's conditions, but the condition at oo imposed here is replaced in Levi-Civita's and Struik's work by a periodicity condition inand this makes a great difference.
Levi-Civita and Struik proceedx,on the assumption that a disturbance of small amplitude is createdrelative to the uniform flow in which w = const.; this is interpreted toThe boundary conditionsfor\fWATER WAVES848+meanthat 6iX is a quantity which can be developed in powers ofa small parameter e, and the convergence of the series for sufficientlysmall values of e is then proved. In Chapter 12.2 we shall give a proofof the convergence of this expansion. (In lowest order, we note thatthe condition (10.9.13) leads for small A and 6 to the condition dO/dy1in agreement with what we have seen in Part II.)at \fyd=In the case of the solitary wave such a procedure will not succeed, asit would not yield anything but auniform flow. The procedure to be adopted here consists in developing,1; but,roughly speaking, with respect to the parameter y near yas in the shallow water theory in general in the version presented insection 4 of Chapter 2, we introduce a stretching of the horizontalcoordinate <p which depends on y while leaving the vertical coordinateunaltered (see equation (10.9.19)).
This stretching of only one of thecoordinates is the characteristic feature of the shallow water theory.(The approximating functions are then no longer harmonic in the newindependent variables.) Specifically, we introduce the real parameterx by means of the equationwas explained above, or rathere~**(10.9.14)= y = gh/U*.<This implies that gh/U 21, but that seems reasonable since all ofthe approximate theories for the solitary wave lead to such aninequality.
We also introduce a new function r, replacing A, by therelationr(10.9.15)For6((f> 9 \p)andr(y,y>)90(10.9.17)For<p->goowe have2.we then have the boundary6(10.9.16)=A +x==0,e~ 3x sin\pconditions=0,y>=1.the conditions imposed by the physical pro-blem:(10.9.18)0->0,r->* 2,=Kthe latter resulting since A ->at oo from we andw -> 1at oo. As we have already indicated, the development we use requiresstretching the variable <p so that it grows large relative to \p when x|small; this is done in the present casedependent variablesis\|\by introducing the newin-LONG WAVES IN SHALLOW WATER= xy,y(10.9.19)349= y.ijpThe dependent variables 6 and r are now regarded as functionsand ip and they are then expanded in powers of x:ofy(109 20}'''{(We have omitted writing down a number of terms which in thecourse of the calculation would turn out to have zero coefficients.)Friedrichsand Hyers have proved that the lowest order termsinthese series, as obtained formally through the use of the boundaryconditions, are the lowest order terms in a convergent iteration schemeusing x as small parameter.
Their convergence proof also involves theexplicit use of the stretching process. However, the proof of thistheorem is quite complicated, and consequently we content ourselveshere with the determination of the lowest order terms: we remark,however, that higher order terms could also be obtained explicitlyfrom the formal expansion.The series in (10.9.20) are now inserted in all of the equations whichserve to determine 6 and r and relations for the coefficient functionsr i(*P> yO and Oiiy* V) are obtained.
The Cauchy-Riemann equations9and r lead to the equationsfor(10.9.21)in0^=terms of the variables-XT-,<pand\p,Ty= xOjand theseries (10.9.20)then yieldthe equations(10.9.22)ThusTJ=Tl5-TI(IJ>) is0,0<-= -T^,T 2^= 0*.independent of y, and integration of the remainingequations gives the following results:(10.9.23)T,= -4yTj'+refer to differentiation with respect to <p. An additivefunctionof (p in the first of these equations was taken to bearbitrary0.for \pzero because of the boundary condition 6 lcondition(10.9.Upon substitution of (10.9.20) into the boundaryThe primes=17)wefind=WATER WAVES350and consequently we have the equations(10.9.24)Theautomatically satisfied because of the first equation of (10.9.23).
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