J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 37
Текст из файла (страница 37)
In addition, we know that 0y+(l/g)0= ( l/g)P for * > from (6.9.25). It follows that (6.9.31 ) can be writtenin the formtt*One wayto use the Poisson integral formula expressing <&on the surface of a sphere centered at the pointin question. Differentiation of this formula yields for any first derivative of <P tan estimate of the form M/6 wheredepends only on the bound for 3> t on thesphere and b is the radius of the sphere. Finally, since our domain for 6 > (;r/2)-h<5contains spheres of arbitrarily large radius at points arbitrarily far from theorigin, the result we need follows.** Since<P(x t/, z; 0) is harmonic, it is uniquely determined by its boundaryvalues on y'and the boundedness conditions at oo.atany pointto proveit istin terms of its valuesMtUNSTEADY MOTIONS0(x,y,z;r)(6.9.32)-11954>(x, y, z; 0)-/ 2 G,)S+I|-o("-GJogtPdt\di-dl*S\an explicitJ;=0We nowsec thatifthere are no immersed surfaces0(x y, z\ r) is given at once by (6.9.32) in terms of the initialconditions, which fixy f=0 and / 2 and the condition on the freesolution9,|Psurface pressure fixingin fact, our general argument shows thatevery solution having the required properties is representable in thisform.
Consequently, the uniqueness theorem is proved for these cases.In particular, the Green's function constructed above is thereforeuniquely determined since its regular part, B, satisfies the conditionsimposed above on 0.In case there are immersed surfaces present the equation (6.9.32)does not yield the solution 0, but it docs yield an integral equationfor it in the following way (which is the standard way of obtainingan integral equation for a harmonic function satisfying variousboundary conditions): One goes back to the derivation of(6.9.30),but considers that the singularity is at a point (x, y, z) of *S\.
If S t issufficiently smooth (and we assume that it is) the equation (6.9.30)isstill holds, except that the factor l/4jt is replaced by I/2n, andto r isthen of course given only on S The integration on t fromonce more performed, and an equation analogous to (6.9.32) is obtained; it can be written in the formt0(x,(6.9.33)i/,*;r)-F(x,.y, z; r)-T~dSJJ J["Ffunction obtained by adding together what corresponds to the first two integrals on the right hand side of (6.9.32).As we see, this is an integral equation for the determination ofwithaknownon S t could0(x, y, z- r) on S { If it were once solved, the value ofbe used in (6.9.32) to furnish the values ofeverywhere.We may make use of (6.9.32) to obtain our uniqueness theorem in9.the following fashion.
Suppose there were two solutionsl and $ 2ofsatisfies allthe conditions imposed onThenSetl2that the nonhomogeneous boundary conditions and1 and2 except.=.WATER WAVES196now replaced by homogeneous conditions, i.e.= since 0(x y, z; 0) is aandhence0; f ly toQ/2= sinceharmonic function which vanishes for y = 0, andtn= on S Thus for we would have the integral representation:ninitialconditions are==P=t(6.9.34)9\.0(x,y, z; r)= -i- ff ff**^5,SinceGnboundedof0 onlike I/or 3 (cf. (6.9.23)) and valuessurfaces Si are alone in question, it follows thatbehaves at oolike I/a 3 at oo fortthealso be-since the surfaces Si are bounded.beshown to die out at oo at least ascould alsoThe derivatives of3rapidly as I/a since the derivatives of G could be shown to have thisproperty for example, by proceeding in the fashion used to obtainhavesany fixed r(6,9.23).As a consequence the following functionoft(essentially the energyintegral) exists:*(6.9.35)E(t)=-[IT[01+01+ 0\] dxdydz + -Ldxdz.jT#fs,Differentiation of both sides with respect to(6.9.36)E'(t)=+[(*).*.(**),(*)+tyields(*),(*).] dxdydzR-&t&ttgsS,of Green'sby applicationboundary of R.
Butfirstn =formula,on S^ and= andIt follows therefore that E'(t)t=and henceidentically zero,JB=andfromwithB= y=WE = const.S3(t+ Sf/g)<Pand our uniqueness theoremis=ButX9(6.9.85). It follows thatythus also vanishes identically. Hencetheon,S/.atZ=are2proved.* It shouldperhaps be noted that the energy integral for the original motionsneed not, and in general will not exist, since the velocity potential and its derivatives are required only to be bounded at oo.SUBDIVISION CWaves on a Running Stream.ShipWavesIn this concluding section of Part II made up of Chapters 7, 8, and9, we treat problems which involve small disturbances on a runningstream with a free surface; that is, motions which take place in theneighborhood of a uniform flow, rather than in the neighborhood ofthe state of rest, as has been the case in all of the preceding chaptersof Part II. In Chapter 7 the classical problems concerning steadytwo-dimensional motions in water of uniform (finite or infinite) depthare treatedfirst.
It isof considerable interest, however, to consideralso unsteady motions (which seem to have been neglected hitherto)both because of their intrinsic interest and because such a studythrows some light on various aspects of the problems concerningsteady motions. In Chapter 8 the classical ship wave problem, inwhich the ship is idealized as a disturbance concentrated at a pointon the surface of a running stream, is studied in considerable detail.In particular, a method of justifying the asymptotic treatment of thesolution through the repeated use of the method of stationary phase isgiven, and the description of the character of the waves for bothand curved courses is carried out at length. Finally, in Chapter1) theproblem of the motion of a ship of given hull shape is treatedunder very general conditions: the ship is assumed to be a rigid bodyhaving six degrees of freedom and to move in the water subject onlyto the propeller thrust, gravity, and the pressure of the water, whilethe motion of the water is not restricted in any way.straight197CHAPTER7Two-dimensional Waves on a Running Streamin Water of Uniform DepthAs indicatedin Fig.
7.0.1weconsider waves created in a channel'yy= -hFig. 7.0.1.Waves on a running streamUof constant depth h, when the stream has uniform velocityin thepositive ^-direction in the undisturbed state. Such a uniform flow canreadily be seen to fulfill the conditions derived in Chapter 1 for aas a free surface under constant pressure.potential flow with y=We assume that the motions arising fromdisturbances created in theuniform stream have a velocity potential 0(x,(7.0.1)0(x9Since<p(x 9y; t)=Ux+ <p(x, y;t) 9-ooy\ t)<x <oo,a harmonic function of x and y</>(#, y; t)isalsoharmonic:y\ t)isW=(7.0.2)9and we- h<ityset<r).follows that<>The function<p(x, y;t) is assumed to yield a small disturbance on therunning stream, and we interpret this to mean that <p and its derivatives are all small quantities and that quadratic and higher orderterms in them can be neglected in comparison with linear terips.
Weassumealso that the vertical displacement y198= rj(x; t)of the freeTWO-DIMENSIONAL WAVES199measured from the undisturbed position y0, is also asmall quantity of the same order as <p(x, y; t). Under these circumstances the dynamic free surface condition as given by Bernoulli'slaw (cf. (1.4.6)) and the kinematic free surface condition (cf.( 1.4.5))surface, astake the forms(7.0.3)L+O +gr,I1V<pI tQ(7.0.4)+ Up x + -2 I/* =IM/If-at y=0,r, twhen quadratic termsadditive constantand consistentisin<pand77are neglected and an unessentialAt the same time, it is properignored in (7.0.3).*such an approximation to satisfy the free surfaceconditions at yinstead of at the displaced position yr\.
(Thereason for this is explained in Chapter 2 actually only for the case[7 = 0, but the discussion would be the same in the present case.)At the bottom yh we have the conditionin==(7.0.5)(p v=at y=In case the channel has infinite depth(7.0.5)'y andy=itsderivativesweh.replace (7.0.5) withup to second order are bounded atoo.In addition to the conditions (7.0.2) to (7.0.5) it is necessary also to<x> and, unless the motion to be studiedpostulate conditions at xis a steady** motion withof t, it is also necessary to<p independent=The cases to be treated in0.impose initial conditions at the time tthe remainder of this chapter differ with respect to these varioustypes of conditions, and we shall formulate them as they are needed.=7.1.Steady motions in water of infinite depth withp=on thefree surfaceIf the disturbance potential <pthe free surface it follows that* It isperhapsisindependent of99(0?,y) satisfiest,andifp=onthe conditionsexplicitly that it would be inappropriate toassume that U, the velocity of the stream, is a small quantity of the same orderas r] and (p: to do so would lead to the elimination of the terms in U and theresulting theory would not differ from that of the preceding chapters.** In thischapter the term "steady motion" is used in the customary way todescribe a flow which is the same at each point in space for all times.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
