J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 19
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as soon as A is fixed) sincethe only standing wave solutions of ourand<pi(x, y)(p 2 (x, y) yieldalso within a real factor. As wearedeterminedandtheyproblemas the amplitudeishave already stated in the preceding section, the progressing wavesolutions (5.2.24) have been determined numerically (cf.
[S.18]) for90, 45, and 6, with results whose general featuresslope angles to=were already discussed in that section.5.3.Three-dimensional waves against a verticalcliffsome three-dimensional problems of wavesbeachesapver slopingby method similar to the method used in thepreceding section for two-dimensional waves, in spite of the factthat it is now no longer possible to make use of the theory of analyticItispossible to treatWAVES ON SLOPING BEACHES AND PAST OBSTACLES85functions of a complex variable. In this section we illustrate themethod by treating the problem of progressing waves in an infiniteocean bounded on one side by a vertical cliff when the wave crestsmay make anyangle with the shore line (cf.
[S.18]).seek solutions 0(x, y, z; t) of V 2 x>1/ ^)0in the regionwiththeoo2ootakennormalto thex *^Q, y0,j/-axisundisturbed free surface of the water and the z-axis* taken alongat ooWe=(< <^i.e. at the water line on the vertical cliffs = 0.Progressingwaves moving toward shore are to be found such that the wavecrests (or other curves of constant phase) at large distances fromshore tend to a straight line which makes an arbitrary angle withthe shore line. For this purpose we seek solutions of the formthe "shore",0(x,(5.3.1)thatAsis,=y, z; t)exp+ kz + ^)}(p(x{i(at9y)solutions in which periodic factors in both z and t are split off.preceding section, we introduce new variables and para-in the=meters through the relations x lMX, y l ~- my, z lmz,2mo jg and cbtain for <p the differential equation=Vf,^ -(5.3.2)and the^=-k/m,=-Artyfree surface condition(5.3.3)(p y(p==for y0,The conditionat(i.e. at the shore line on the cliff)origin x ~~ 0, yrequire, as in former cases, that (p should be of the formweafter dropping the subscriptthe cliff is, of course,on=0--^(5.3.4)1allforquantities.x=0.oxAt the(5.3.5)+small values of r =(pIplog rr <C 1,<p,2+2with <p and ^ certainbounded functions with bounded first and second derivatives in aneighborhood of the origin.
The functions q> and !p should be consideredfor sufficiently(xf/1/2f).at present as certain given functions; later on, they will be chosenspecifically.For large values ofrwe wishto have 0(x, y,z; t)behavelike* It hasalready been pointed out that functions of a complex variable arenot used in this section, so that the reintroduction of the letter 2 to represent aspace coordinate should cause no confusion with the use of the letter z as a complexvariable in earlier sections.WATER WAVES86evexp{i(at+ kz + cue +/?)}with & 2+oc2=1but k and a otherwisearbitrary constants, so that progressing waves tending to an arbitraryplane wave at oo can be obtained.
This requires that <p(x, y) shouldbehave at oo like e v exp {i(onx +^2)} because of (5.3.1). However,it is no more necessary here than it was in our former cases to requirethat (p should behave in this specific way at oo; it suffices in factto require that(5.8.6)i.e.\<P\that(p++\<Px\\<Pxy\<Mforr> R,and the two derivatives of <p occurring in (5.3.6) shouldoo. As we shall see, this requirement leadsbe uniformly bounded atto solutions of the desired type.proceed to solve the boundary value problem formulated inequations (5.3.2) to (5.3.6).
The procedure we follow is analogousWeto that used in the two-dimensional cases in every respect.with, we observe that/}(5.3.7)/(/)_dx \dy\i\<p=oboth xfor=and y=Tobegin0,Jbecause of the special form of the linear operator on the left handside together with the fact that (5.3.3) and (5.3.4) are to be satisfied.A function yj(x, y) is introduced by the relation(5 - 8 - 8)The"essential point of our-T*(TVmethodisthat the function\pisdetermineduniquely within an arbitrary factor if our function 9?, having theproperties postulated, exists.
Furthermore, y> can then be given explicitly without difficulty. The properties of \p are as follows.1. \p satisfies the same differential equation as y>, i.e. equation(5.3.2), as one sees from the definition (5.3.8) of \p.><and vanishes, in view2. y) is regular in the quadrant x0, yof (5.3.7), on xand0. Hence y can bex0, y0,ycontinued over the boundaries by the reflection process to yield a=<=>continuous and single-valued function having continuous secondk 2 \pderivatives y xx and y>yv (as one can readily see since V 2y>0,withtheexontheintheentireand \px, t/-plancboundaries)==ception of the origin. (Hereof angle rc/2.)we usethe fact that our domainisa sectorWAVES ON SLOPING BEACHES AND PAST OBSTACLES3.At theorigin,\phas a possible singularity whichof the formisandas one can see from (5.3.5)99 regular,(jp(x, t/)//*This statement clearly holds for the function y when2with,87it(5.3.8).has beenextended by reflection to a full neighborhood of the origin.4.
The condition (5.3.6) on q> clearly yields for \p the conditionthat y is uniformly bounded at oo after \p has been extended to thewhole plane.Thusisis\puniformly bounded atk\ =V 2y>a solution ofoo.At thecertain regular functions (q>on the entire x and y axes.^Wein the entire plane+2originq>/rip<pwithwhichand9?=not excluded).
In addition, \pshall show, following Weinstein[W.5],* that the function(5.3.9)\p(x,y)= AiH= Vx + y2(ikr) sin 20, r(2,^k^1the unique solution for \p in polar coordinates (r, 0) with A anthe Hankel function of order twoarbitrary real constant, and->Theroo.function ip has real values for rwhich tends to zero asisH^real.(The notation given in Jahnke-Emde, Tables of Functions,isused.)Theobtained by Weinstein in the following way.In polar coordinates (r, 6) the differential equation for ^ issolutionyis3r 2^For any fixed value of_r drr*r the function\p*can be developed in thefollowing sine series:n=lsinceyvanishes forare given=Cn rJo/2y(r, O) sin*3n/2;and thecoefficients c n (r)2nOdd,n=1, 2,...,C n a normalizing factor.
From this formula one finds by differen-tiations with respect to rthat0, rc/2, n,byc n (r)with=c n (r) satisfiesand use of thedifferential equation for^the equationIn the author's paper the solutionuniqueness statement.\pwas obtained, but with alessgeneralWATER WAVES88Thehandcan be seen byandmaking use of theintegrating the first term twice by partsboundary conditions \p = for = and 6 = n/2.
Thus the functionsrightside of this equation vanishes, asc n (r) are Bessel functions, as follows:c n (r)= A wi*^H<(ikr) + B ZaI 2n (kr),A 2n and B2narbitrary real constants. The functions I 2n arebehave like r~ 2n fortheHankel functionsunbounded at oo;->rand tend to zero exponentially at oo. It follows therefore thatthe Fourier series for ip in our case reduces to the single term givenby (5.3.9) because of the boundedness assumptions on \p.For our purposes it is of advantage to write the solution \p in thefollowing form:withH^(5.3.10)ip=-- H32Ai(1)r(ikr),= Vx + y22,oxoyany real constant and H (l) is the Hankel function oforder zero which is bounded as r -> oo.
It is readily verified that thissolution differs from that given by (5.3.9) only by a constant multiplier: for example, by using the well-known identities involving theinwhichAisderivatives of Bessel functions of different orders.Once(5.3.11)y>isdetermined we(may-- I\w = Aiox \oywrite (5.3.8) in the form- HM(ikr),Aarbitrary.oxoy/This means that our function<p,if it exists,mustsatisfy (5.3.11) aswell as (5.3.2). By integration of (5.3.11) it turns out that we areable to determine q> explicitly without great difficulty on accountof the simple form of the leftto do.handside of (5.3.11).
Thiswe proceedIntegration of both sides of (5.3.11) with respect to x leads to(5.3.12)--iv= Ai- H((ikr)+ g(y),in which g(y) is an arbitrary function. But g(y) must satisfy (5.3.2),0.k 2gsince all other terms in (5.3.12) satisfy it. Hence d 2g/dy 2==,In addition g(0)0, since the other terms in (5.3.12) vanish for=0ofbecause(5.3.3) and the fact that dHyl9y=(ik)'^(ylr)dH^ /dr.isbounded as y ->oo because of condition (5.3.6)Finally, g(y)and the fact that dH V /By tends to zero as r -+ oo. The function(WAVES ON SLOPING BEACHES AND PAST OBSTACLESg(y) is therefore readily seen to be identically zero.of (5.3.12) we obtain (after setting g(y)0):=(5.8.13)<p=TAieye~*~[H(ikVx2+t+2)]dtBy89integrationB(x)c*.utJ+ooThe function B(x) and the real constant A are arbitrary. The integralVconverges, since d(H)/dt dies out exponentially as t -* oo.We shall see that two solutions (x, y) and <p 2 (x, y) satisfying all(^conditions of our problem can be obtained from (5.3.13) by takingin the other case, and that thesein one case andA=A ^solutions will be 90"out of phase" at oo.
(This is exactly analogousto the behavior of the solutions in our previous two-dimensional cases. )Consider first the case A = 0. The function 9? given by (5.3.13)satisfies (5.3.2)ifonly+-(5.3.14)-(1k 2 )B(x)=0.2important to recall that k < 1. The boundary condition= o for x = requires that B x (0) = 0. The condition q> y <p =(p xfor y =0.is automatically satisfied because of (5.3.12) and g(y)2=withcoskHence B(x)AlA l arbitrary, and the solutionVlx,Itis=^(cr, y)(5.3.15)This leads to solutions(5.3.15)'0^,j/,2; t)= A^v cos Vlin thelk*x.form of standing waves,* as= Af^e* cosVl~^~k*xfollows:C S((sink < 1.be valid.for2Ifk=1,the solutionAs we have alreadystated,Xgiven by (5.3.15)' continues towe obtainsolutions <p%(x,y) fromwhich behave for large x like sin Vlk 2 x rather(5.3.13) for A ^k 2 x and with these two types of solutionsthan like cos Vlprogressing waves approaching an arbitrary plane wave at oo canbe constructed by superposition.We begin by showing that (5.3.2) is satisfied for all x > 0, y <by (p as given in (5.3.13) with A ^ 0, provided only that B(x)9*solutions of this type (but not of the type with a singubeaches sloping at angles n/2n were obtained by Hanson [H.3] by aquite different method.The standing wavelarity) forWATER WAVES90>satisfies (5.3.14).
Since x0, it is permissible to differentiate underthe integral sign in (5.3.13), even though t takes on the value zero(since the upper limit y is negative). By differentiating we obtain(5.3.16)V 2 <p-=topAif{*V I fj^ +(v(1-\-2A; ) oca solution of (5.3.2) the operator (9 2 /9a? 2229 /9j/ and hencecurring under the integral sign can be replaced bythe integral can be written in the formSinceWeffWisintroduce the following notationand obtain through two integrations by parts theresulte*inwhich we have made use of the fact that the boundary terms arczero at the lower limitto zero as r ->by I I+/ 3 andoo.+Thesince all derivatives ofoo,integral of interest to usthis in turnisisH^(ikr)tendgiven obviouslygiven byU)_3dt2a*/a?/irelations for I m Hence the quantity in the firstbracket in (5.3.16) is identically zeroin other words the termcontaining the integral on the right hand side of (5.3.13) is a solutionifof (5.3.2).
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