J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 92
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Instead, we proceed to formulate boundary conditionsfor the special problem of breaking of a dam, which is in any casetypical for the type of problems for which the present procedureisTHE BREAKING OF A DAM517We assume therefore that the region occupied initiallywaterthe(or rather, a vertical plane section of that region) is thebybaoo,h, as indicated in Fig. 12.1.1. Thehalf-striprecommended.<^^ ^b=hb=0Fig. 12.1.1.damr isisThe breaking of a damof course located at ainitially at restwhen=0.Sincewe assume that the waterwe have the conditionsfilling the half-strip=a,Y(a,b;0)X (a, b; 0) =0.YX(a.b;0)(12.1.9)o=b,and(12.1.10)Whenthetdamist(a, b;0)=broken, the pressure along0.itwillbe changedsuddenly from hydrostatic pressure to zero; it will of course be prescribed to be zero on the free surface.
This leads to the followingboundary conditions(p(a, h;(12.1.11)\t)for the pressure:==0,a0,b< GO,< h,> 0,t=results from thebFinally the boundary condition at the bottombottom remaintheatwaterthethatparticles originallyassumptionthehavewein contact with it; as a resultboundary condition(12.1.12)Y(a,a0;*)=0.<oo,t> 0.satisfied because of the(12.1.9) are automaticallyform (cf. (12.1.5)) chosen for the scries expansion. The conditions(1)(l)0.(a, b)(a, b)(12.1.10) are satisfied by takingThe conditions=FX=X(2)(2)(a, 6), it is(a, b) and FIn order to determine the functionsthedifferentialtoinadditionnecessary to obtain boundary conditionsandSuchboundary(12.1.7).equations given for them by (12.1.6)in conandobtainedbe(12.1.12)conditions canby using (12.1.11)fromThusseriesdevelopments.junction with (12.1.1) and the powerWATER WAVES518^we<oo (indeed, F<">(a, 0)aforfind F< 2) (0, 0) =would be zero for all n).
Insertion of the series (12.1.5) and use of theh yieldsboundary conditions for 6(12.1.12)=XW(a,(12.1.13)= 0,h)of the equations in (12.1.1).of (12.1.1) leads to the conditionupon using thefirstF< 2 >(0,&)=:(12.1.14)We knowThe second equation-12+is an analytic function of theiXthat Z(z) == F< 2)ib in the half-strip, and we now haveacomplex variable zits real or its imaginary part on each of theeitherforvaluesprescribedthree sides of the strip; it follows that the function Z can be deter-=+mined by standard methods for example by mapping conformallyon a halfplane. In fact, the solution can be given in closed form, asfollows: Since X(a, h) = 0, we see that X^(a h) = 0, and hencethat F 2) (a, h) = Q since X (2) and F (2) arc harmonic conjugates.9Therefore the harmonic function F (2) (a, b) can be continued over theh by reflection into a strip of width 2/i, as indicated in Fig.line b(2)are also shown.
Thus a complete12.1.2; the boundary values for F=YFig. 12.1.2.(2)-bBoundary value problemforFb:2hb=0(2)(a,b)formulated boundary value problem for F (2) (0, b) in a half-striphas been derived. To solve this problem we map the half-strip on thecosh (nz/2h)upper half of a w-plane by means of the function wly=eitherformulaby inspection or by using the Schwarz-Christoffel mappingand observe that the vertices z = and z = 2ih of the half-w=of the wj-plane, as indicated in(2}(w) on the realFig.
12.1.3. The appropriate boundary values for(2)axis of the w-plane are indicated. The solution for(w) understripmapinto the points1YYTHE BREAKING OF A DAM519w- plane-HFig. 12.1.3.Mapping on the w-planeknown; it is the function Y (2) (P) ~0J, with X and 2 the angles marked in Fig. 12.1.3.(g/2n)(6 2The analytic function of which this is the real part is well known; it istheseconditionsiswelly<2>as can inany case be+1easily verified. Transferringback to the z-planewe haveucoshZ(z)=Y(2)+ iX=-12hlog,coshand upon separationinto realand imaginary parts we have+cos 2X<*>(a b)9=-finally:sinh 2-*-snsinh 2(12.1.15)nb.sinF< 2 >(a, 6)One checks2h=n.,smhnaX(2that the boundary conditions0,>(a, h)Theinitialare satisfied, and that F (2) (0, 6)F< 2 >(a, 0)g/2.(2)(0)(a.
b)pressure distribution p (a, 6) can be calculated, now that=easily=Xisknown, by using thefirstequation of (12.1.1), which yields520WATER WAVES(12.1 16)pj-In the present case there are advantages in working first with thepressure p(a, b; t) and determining the coefficient of the series for itdirectly by solving appropriate boundary value problems; afterwardsXand Y are easily found.
The mainreason for basing the calculation on the pressure in the first instanceis that the boundary conditions at bh and aare very simple,and hence p<*>for all indices i. The boundary conditionsi.e. pthe coefficients of the series for=bottom 6at the====involve the displacements Y. For instance, onesame general way as above that p^ =gg,_ __ QgY^ as boundary conditions at 6 = 0.finds readily in theSincep2)pj,and=. 0,pW<0)isharmonic,placementsanF(2)found at once without reference toit isinteresting fact in itself.Once p (o)isfound,X(2)dis-andcan be calculated without integrationsp^ == 0.(a, b)Since=for b0,and p (1)is(cf. (12.1.16), for example).also harmonic, it follows thatSince F (2) is now known,it follows that a complete setp (1)of boundary conditions for p (2) (a, b) is known, and p (2) (a, 6) is thendetermined by solving the differential equation(12.1.17)=rd(x (z) F (2 M^ST-'-MeLL9 /h*- V2 {(Jf(2))2 +(F(2))2d(a, b)IIcoshnCLcosh\n ^\Ihjobtained after a certain amount of manipulation.
This process can be continued. One would find next that^ and F (4) can be found once (2) isF<3)o, and thatpknown. However, the boundary condition at the bottom, and the righthand sides in the Poisson equations for the functions p (i) (a,b) becomewhose right handsideis=XW =X(more and more complicated.The initial pressure p (o) (a, b) can be discussed more easily on thebasis of a Fourier series representation than from the solution inclosed form obtainable from (12.1.16); this representation is(12.1.18)=p<>(a, 6)^.,oglhb)}---V-e--zhcosTHE BREAKING OF A DAM521We note that the first term represents the hydrostatic pressure, andthat the deviation from hydrostatic pressure dies out exponentiallyas a -> oo and also as h -> 0, i.e.
on going far away from the dam andon considering the water behind the dam to be shallow (or, better,considering a/h to be large). This is at least some slight evidence ofthe validity of the shallow water theory used in Chapter 10 to discussalsosame problemof the breaking of atoo close to the site of the dam.thisdamat least at points notThe shape of the free surface of the water can be obtained for smalltimes from the equationsX=(12.1.19)=abevaluated for a(for the particles at the face of the dam) and forbh on the upper free surface. The results of such a calculation forthe specific case of a dam 200 feet high are shown in Fig. 12.1.4.50 40 30 2010010Fig. 12.1.4.
Free water surface after the breaking of aOnea=xmilesa singularity at the originby the discontinuity in thehas a logarithmic singularity for a0,of the peculiarities of the solutionwhich is brought about0, 6Xdamis=pressure there. In fact,is negative infinite for allb0, as one sees from (12.1.15) and0. This, of course, indicates that the approximation is not goodtat this point; in fact, there would be turbulence and continuousbreaking at the front of the wave anyway so that any solution=^(2}Xignoring these factors would be unrealistic for that part of the flow.In the thesis by Pohle [P.ll], the solution of the problem of thecollapse of a liquid half-cylinder and of a hemisphere on a rigid planeare treatedbyessentially thesame methodas has been explained forWATER WAVES522the problem of the breaking of a dam.
These problems have also beentreated by Penney and Thornhill [P.2], who also use power series inthe time but work with the Eulerian rather than the Lagrangian representation, which leads to what seem to the author to be more complicated calculations than are needed when the Lagrangian representation12.2.isused.The existence of periodic waves of finite amplitudeIn this section a proof, indetail, of the existence oftwo-dimensionalperiodic progressing waves of finite amplitude in water of infinitedepth will be given.
This problem was first solved by Nekrassov[N.I, la] and later independently by Levi-Civita [L.7]; Struik[S.29] extended the proof of Levi-Civita to the same problem forwater of finite constant depth. A generalization of the same theoryto liquids of variable density has been givenby Dubreuil-Jacotinhas given a different method of[D.I 5, 15a], Lichtenstein [L.ll]solution based on E. Schmidt's theory of nonlinear integral equations.Davies [D.5] has considered the problem from still a different point ofview. Gerber [G.5] has recently derived theorems on steady flows inwater of variable depth by making use of the Schauder-Leray theory.We shall start from the formulation of the problem given by LeviCivita (and already derived in 10.9 above), but, instead of provingdirectly, as he does, the convergence of a power series in the amplitude to the solution of the problem, an iteration procedure devisedby W.
Littman and L. Nirenberg will be used to establish the existenceof the solution. The two procedures are not, however, essentiallydifferent.Itisconvenient to break up this rather long section into sub-sec-tions as ameans of focusing attention on separate phases of theexistence proof.12.2a. Formulation of the problemAs in sec. 10.9, the problem of treating a progressing wave whichmoves unchanged in form and with constant velocity is reduced to aproblem of steady flow by observing the motion from a coordinatesystem which moves with the wave.
A complex velocity potential (seesec. 10.9 for details) %(z) is therefore to be found in the #, t/-plane(cf.Fig. 12.2.1):LEVI-C1 VITA'SFig. 12.2.1. Periodicx(12.2.1)The==+<pi\p=THEORYwaves offinitezjf(z),523=Theamplitudex+iy.harmonic functions<p(x,y) and y(x,y) represent the velocity potential and the streamfunction. The complex velocity w is given byvelocity at yoo shouldbew =(12.2/2)[7.urealivdzwith u, v the velocity components.
This follows at once from theCauchy-Riemann equations:(12.2.3)(p xwsince993.= yy =w,<p v= - yx = ^-f iy^.Weproceed to formulate the boundary conditions at the free surfree surface condition can be expressed easily because the free surface is a stream line, and we may choose y(x> y)along it. The dynamic condition expressed in Bernoulli's law is givenface.The kinematic=by|(12.2.4)|w2|+ gy =The problemas one can readily verify.conditionisatconst.y=0,of satisfying this nonlinearof course the source of the difficulties in deriving anAt oo the boundary condition isexistence proof.w(12.2.5)and wbounded.isthuswe(12.2.6)->Uuniformly as y ->Weoo,supposed to be nowhere zero and to be uniformlyseek waves which are periodic in the ^-coordinate andin additionrequire ^ to satisfy the conditionX (zwith h a real constant.+ h) -x(z)=0,WATER WAVES524Following Levi-Civita, we assume that the region of flow in thes-plane is mapped into the 99, ^-plane by means of %(z).
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