J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 14
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Again one observes an odd kindof "resonance" phenomenon: large amplitudes are conditioned bythe wave length in space of the applied pressure once the frequencyhas been fixed.4.3.Thevariable surface pressuresurfaceIn this sectionweisconfined to a segment of theconsider the case in which the surface pressuresin at.Ix\<pa{Pwith P a constant. Some of the motions which can arise under suchcircumstances are discussed by Lamb [L.2] in the paper quoted above.However, here as elsewhere, Lamb assumes fictitious dampingforces* in order to be rid of the free oscillations and thus achieve aunique solution, and he also makes use of the Fourier integral technique which we prefer to replace by a different procedure.
In fact,the present problem is a key problem for this Part II and a peg uponwhich a variety of observations important for other discussions in laterchapters will be hung. As we shall see, the present problem is alsodecidedly interesting forenough made no attemptitsownsake,althoughLambstrangelyin his paper to point out the really strikingresults.In addition to prescribing the pressure p through (4.3.1) it isnecessary to add to the conditions imposed in section 4.1 appropriateconditions at the points (it #, 0) where p has discontinuities. Inview of (4.1.3) it is clear that a finite discontinuity in t or r\ orboth must be admitted and it seems also likely that the derivativeswould be unbounded near these points.
We shall makeX andy of*Lambassumes resistances which are proportional to the velocity. Inthisway the irrotational character of the flow is preserved, but it is difficult to seehow such resistances can be justified mechanically. It would seem preferableto secure the uniqueness of the solution in unbounded domains by imposingphysically reasonable conditions on the behavior of the waves at infinity.SIMPLE HARMONIC SURFACE PRESSURE59the following requirements(4.3.2)tbounded;y=O(Q-I^e),>a neighborhood of the points (^ a, 0) with Q the distance fromthese points. This means, in particular, that the surface elevation isbounded near these points and that the singularity admitted is notas strong as that of a source or sink.
We recall thatand y weretinrequired to be uniformly bounded atoo.madeTheso far do not ensure the uniqueness of thestipulationssolutionof our problem any more than similar conditions ensureduniqueness of the solution of the problem treated in the precedingsection. However, we have in mind now a physical situation in whichwe expect the solution to be unique: We imagine the motion resultingfrom the applied surface pressure p given by (4.3.1) to be the limitapproached after a long time subsequent to the application of p tothe water when initially at rest.
Under these circumstances one feelsinstinctively that the motion of the water far away from the sourceof the disturbance should have the character of a progressing wavemoving away from the source of the disturbance, since at no timeis there any reason why waves should initiate at infinity. (We shallshow (of. (6.7)) that the motion of the water arising from such initialconditions actually does approach, as the time increases withoutlimit, the motion to be obtained here.) Consequently we add to ourthe condition often called the Sommerfeld conditionconditions onproblems concerning electromagnetic wave propagation that thezvares should behave at oo like progressing waves moving away frominAs we shall see, this qualitative conditionleads to a unique solution of our problem.In solving our problem there are some advantages to be gained bythe source of the disturbance.not stipulating at the outset that the Sommerfeld condition shouldbe satisfied, but to obtain first all possible solutions of the form(4.1.5), and only afterwards impose the condition.
We have thereforeto find the harmonic functions (p which satisfy the condition (cf.(4.1.6)(4.3.3)and(4.3.1))rv<p- my =(c,|\(0,|xx\|^>aay,with(4.3.4)ra=a 2 /g,c=Pa68=WATER WAVES60on the free surface, and the boundedness conditions which followfrom those imposed on 0:andicpvI Vbounded atbounded and{cp<pThe functions yin (4.1.5),=<p yoo,70(e~"1+e),e>at x0,=a.those which yield the waves of phasei.e.<Z>,satisfy the same conditions as in section 4.1 and aretherefore given by (4.1.9).
We have therefore only to determine thesin at infunctions<p.Tothisendit isconvenient to introducenew dimen-sionless quantities= mx,= c/m soxl(4.3.6)together with c^takes the form_v=V(4.3.7)In what follows(= my,yla= mathat the free surface condition (4.3.3)cl,|xl\^!,ft-0.weuse the condition in this form but drop the subscripts for the sake of convenience.In most of the two-dimensional problems treated in the remainderwe make use of the fact that any harmonic functioncanbetaken as the real part of an analytic function f(z) of(p(x, y)the complex variable z = x -\- iy and writeof Part II(4.3.8)f(z)= <p(x, y) + iy(x, y) = f(x + iy).In our present problem f(z) is defined and analytic in the lower halfplane. To express the surface condition (4.3.7) in terms of f(z) wewritedi-~%in-=/)die (if-f),which the symbol 8&e means that thereal part ofwhatfollowsisto be taken.
Consequently the free surface condition has the form:(4.3.9)V.-VSIMPLE HARMONIC SURFACE PRESSUREWe nowintroduce anewanalytic function F(z)F(z)(4.3.10)=if'(z)61by the equation*- /()and seek to determine F(z) uniquely through the conditions imposedon 9?Ste J(z). We observe to begin with that F(z) satisfies theconditionfrc9\|TxIi^<.nain view of (4.3.9). We show now that F(z) is uniquely determinedwithin an additive pure imaginary constant, as follows: Supposethat G(z) - Fi(z)F 2 (z) is the difference of two functions F(z)from (4.3.10) through those onon the entire real axis, exceptfrom (4.3.11). Hence 3&eG(z) is asatisfying the conditions resultingf(z).
Then die G(z) would vanishia, as one seespossibly at x -functionwhichcan be continued analytically by reflectionpotentialover the real axis into the entire upper half plane; it will then bedefined and single-valued in the whole plane except for the points(i fl, 0). At oo, 8&e G(z) is bounded in the lower half plane, while1+eJle G(z)ea in view of the regularity0, at x),0(g~~==>conditions and the definition of G(z). These boundedness propertiesare evidently preserved in the analytic continuation into the upperhalf plane.
Consequently Sfce G(z) has a removable singularity at thea on the real axis since the singularity is weaker thanpoints xiorder and the function is single-valued in the neighborhood of these points. Thus 3te G(z) is a potential function whichis regular and bounded in the entire plane, and is zero on the realaxis; by Liouville's theorem it is therefore zero everywhere. Con-a pole offirstsequently the analytic function G(z) is a pure imaginary constant,and the result we want is obtained.
On the other hand it is rathereasy to find a function F(z) which has the prescribed properties forexample by first finding its real part from (4.3.11) through use ofthe Poisson integral formula. We simply give it:(4.3.12)F(z)ic=__logr:za;verifies readily that it has all of the required properties.take that branch of the logarithm which is real for (z - a)/(zoneWe+ a)* This device has been usedby Kotschin [K.I 4], and it was exploitedinandtheauthor[S.18]studying waves on sloping beaches.[L.8]by LewyWATER WAVES62realandpositive.Ojice F(z) has been uniquely determined, the complex velocitypotential f(z) is restricted to the solutions of the first order ordinarywhich means that the solutions dependwhich multiplies the non-vanishingconstantonthearbitraryonly= 0. Butsolution e~ iz of the homogeneous equation if'(z)fiz =vxe (A cos9te (A + iB)e~+ B sin x) and these are the standing0.
The mostwave solutions for the case of surface pressure pdifferential equation (4.3.10),=general solution of (4.3.10), with F(z) given by (4.3.12), can bewritten, as one can readily verify, in the formCzcelog/(*)=-*n(4.3.13)Jzntat+adt,with the initial point * and the path of integration any arbitrarypath in the slit plane. Changing # obviously would have the effectof changing the additive solution of the homogeneous equation. Itis convenient to replace (4.3.13) by the following expression, obtainedthrough an integration by parts:= ci f (4.3.14)y/v;/(*)vrcz-alogfo/I--(\t-aLand at the same time toC*fix the1t+apath of integration as indicatedint- plane(a)Fig.
4.3.1a,b.(b)Path of integration in f-planeFigure 4.3.1. This path comes from oo along the positive imaginaryaxis and encircles the origin, leaving it and the point (-a, 0) toSIMPLE HARMONIC SURFACE PRESSUREthe63Use has been made of thefact that log (z -a)/(*+a) ->0also that the integrals converge on accountof the exponential factor.That 95(3?, y)&te f(z) as given through (4.3.14) satisfies theleft.whenz -> oo;we observe=free surface and the regularitycondition at the points (i a, 0) is easy to verify. We proceed todiscuss the behavior of f(z) at oo (always for z in the lower half plane).boundary conditions imposed at theForthispurposeitsuffices to discuss the integralsP iiatOO tdt since the function logbehaves like l/z at oo (as one readilyonce by parts to obtainCzi"rJ( fy\2(z)sees).*i*Toe i(t,.J\*00i"end we integrate~z}I,flthisIf,\2fitatm#)Wesuppose that the curved part of the path of integration in Figure4.
3. la is an arc of a circle. It follows at once that the complex numbertz has a positive imaginary part on the path of integration aslong as the real part of z is negative, and hence we havedt1#fJfoodtf\ta\JConsequently I(z) behavesz is negative,andlike l/z at infinityf(z) likewise. The situation\ta\when theisreal part ofdifferent,however,the real part of z is positive. To study this case, we add and subtractcircular arcs, as indicated in Figure 4.3.1b, in order to have anifaintegral over the entire circle enclosing the singularities atas well as over a path symmetrical to the path in Figure 4.3.
la.By the same argument as above, the contribution of the integralover the latter path behaves like 1/2 at oo, and hence the non#vanishing contribution arises as a sum of residues at the pointsThese contributions to I(z) are at once seen to have the values2nie^ iz e^ ia Thus we may describe the behavior of f(z) as given.by(4.3.14) at oo as follows:(4.3.15)/(*)= '(7)4ci (sin a) e~ iz+OIIfor Ste z<for die z>0.WATER WAVESFromone sees that f(z) has the same behaviora factorand with iti. Hence f(z)=thebehavioratItisconvenientoo.3te /(*), haspostulated(jp(x, y)to write down explicitly the behavior of <p(x, y) at oo:at(4.3.10) and (4.3.12)oo as f(z) 9 except for(4.3.16) <p(x y)99= 9te f(z) =Vy; t)9=< 0,forx>}+OIt follows that all simple harmonic solutions ofgiven by linear combinations of0(xx.4c sin a e v sin x(4.3.17)forr '(die f(z)+Ae*sinx+I10.\rjour problem arcBe* cos x) cosatand<P(x y\(4.3.18)9t)=y(Ce sin x+ De v cos x) sin atwhich A, B, C, and 1) are arbitrary constants, and f(z) is givenby (4.3.14).
In other words, the standing waves <p(x y) cos at justfound above, together with the standing waves which exist forin9vanishing free surface pressure, constitute all possible standing waves.We now impose the condition that the wave 0(x y\ t) we seek9behaveslikean outgoing progressing wave atoo, i.e.+ at) + K cosat xthatitbehaveslikeS_:andev(Hsin (x(x+erf))=oolike+ M cos (x at)) at x = + oo.In view of the behavior of (p(x, y) = <%ef(z) at x =oo (of.
(4.3.16)),S+i.e.:e v (L sin (xthe fact thatat)dies out there,itit iswe may combineway asclear thatthe standing wave solutions (4.3.17) and (4.3.18) in such ato obtain a progressing wave solution(4.3.19)0(x9y;t)= e*(H sin (x + at) + K cos+ <p(x y) cos at(x+ at))9valid everywhereconstantsHandand whichKstillsatisfiesarbitrary.the conditionAt x= +_,oo thiswith the twowave has thebehavior(4.3.20)&(x, y;t)= e*[(H sin (x + at) + K cos(x+ at))4c sin a sin x cos at]+O11SIMPLE HARMONIC SURFACE PRESSURE65view of (4.3.16).
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