J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 96
Текст из файла (страница 96)
The function r(<p) is introduced:r(<p)=^-&-<*,and a new transformationTondefined(12.2.30)(12.2.31)Trris= 1 SRfafr + t>byLEVI-CIVITA'STHEORY539In other words, T is applied only to those functions r such thata i( rie****) is in the domain of definition of R. The functionalequation (12.2.29) is now seen to be equivalent to the equation+r(12.2.32)and we seek a solution ofit=Tr,in the space B.Weshall solve (12.2.32) by an iteration process which starts withsuch that the corresponding function /?ft> 1 is in za function i\ inBB=(i.e. such that R is applicable to o^), inserts it in r 2Tr^ etc., thus=rrwithInaTrordertomakesure that_kk vobtainingsequence k(12.2.28) holds for the solution we stipulate that the parameter e in(12.2.12) be fixed at each stage of the iteration process so that (12.2.28)done by settingsatisfied; this isisnsk(12.2.33)At the sameX( e~~k=sn asnaIf 2 *MQ*-.e~*p * sin<x.,k sin Acpckptime, this ensures that the transformationTisreallyapplicable to the members of the sequence r k Of course, it will benecessary to show that the denominators in the equations (12.2.33)are bounded away from zero and that the sequence e k converges.
The.existence of a sequencerkconverging to a solutionr of (12.2.32) will%be shown by disposing properly of the arbitrary constant(thetheinoftheofthewavelinearizedsolutionproblem),amplitudecan be chosen small enough so that thei.e. by. showing thatt 7^sequences r k and e k each of whichnote in passing that if d) k<x-f,=fca function of a r converges. Weis small of order %, then e k asijj kisgiven by (12.2.33) is also of order a l later on, we give an explicitestimate for it so that the quantities e k should not turn out to be ofthe wrong order.The convergence of the sequence of iterates r k to a solution r ofr=Trwillbe shown by proving thatallof the functions r k , forvalues of a x less than a certain fixed constant, satisfy the following1conditions: for some real positive constant r] and real positive<I)\\r\\^r,implies \\Tr\\ ^r],II)llrJMIr.H^impliesfor||JVj-2V.Hx||1^-r,||,any pair of functions r v r 2 Condition I) says that the transformaT carries any function in the closed "sphere" of radius 77 intotion.WATER WAVES540another function in the same "sphere", and Condition II)isa Lip-schitz condition.TheiterationschemeBTr = rTake any function r Q (<p)and consider the iteratesfor solving the functional equationproceeds in the following standard fashion.inis applicablewith || r ||to whichr\<rw =TTr n _.
From I) we see thatbyhave a bounded norm. We have, evidently:r n definedr n+ iwe maySince II) holdsIIfVu-r n =-rw||=Tr n-such functions rallTr n _ vwrite||Tr n-Tr n ^^*||-rn\\r n _,\\.and henceWeIIconsider next the- rnrm||=||(rmnormof rm- r^) +rn(rm _^-m ^ n:,rm _ 2 )+...-r1+(r n^-rn)\\-(The triangle inequality is of course used here.) Since x < 1 it is clearthat the sequence r n is a Cauchy sequence and hence it converges toa unique limit function r in B with norm less than r).
(The uniquenessstatement holds of course only for functions r with norm less than rj. )That the limit function r satisfies Tr = r is clear, since the sequencer n is identical with the sequence Tr n _ and hence both converge tothe same limitr.In order to establish conditions I) and II) for the functions in thesequence r k9 and hence to complete our existence proof, it is convenient to introduce certain continuous functions F l (N) F 2 (N),.,= 0, and increasing2> 0, bounded nearwhich are defined for realwith N..9Suppose that rand(cf.CBissuch that(12.2.30)) recall that r l+== N. In what follows,||lrl^rj.\\We+ lTl - ie -*<pset OJ 1so thatOt||+o>,||ir l^<*ihowever, we omit the circumflexand r, and we shall also omit the subscript on a.The following inequalities hold when a is sufficiently small:aioverIl.NNI(11?)|\LEVI-CIVITA'S1.2|<r**i sin|3Tll*- ' si'(12284),|THEORY541|n0i-\\e-^sme -e-^sinem \\^\a\\\8.i4.11e~3T sin-Ze- 3rsin d m-(B l-^\a\\\6 m )\\-rmri\\NFt (N).00These inequalities arebased on the fact thatallan absolutely convergent powerif= J & nf n isothen f ^ NA(f )series for all real f,1|00implies|A(|)^ ^\o\hnN n Wederive the second and third of the.\above inequalities as typical cases the others are derived in a similarway.
Consider the second inequality; we write= 0(*- 3T - 1) + e- 3T (sin - 0)e-* sin ||||||||..Fwith2(Ar)=3^ 3N+ Ate4Ar.Consider next the third inequality.--3einwhichFrom,*-*. sinwith0,this-e-**m sinF 3 (A^) =snT^rm +^Tsome values on the segmentwe have0*, T* are(0 m rm ).||Frommean value theorem we havethem^||t||rcosjoining,- m(0j,.r t ) and-0 m4e3JV in view of the definition of r l9,rmgiven in (12.2.30).Itis also essential togive an estimate for e k in (12.2.33). First weobtain a lower bound for the denominator. We have, from the defini-and the second inequality above:tion of rrsinsin/2/2^1/Ar(e~^sin0) sin Ag? dgpsinJoJo-f|Jo~+[|a|(1-27?)-a|(sin Arc+ ^?* ^)sinWATER WAVES542N=+a \(lSubscripts have been dropped in the above.
Sinceq)andaclear that for r/a (1)Jsufficiently small, say athe resulting expression is greater than k a |, with k a positive con-^it is|\\|\^,\stant depending on a (1) Use of this fact together with the second inequality above in the definition (12.2.33) of e leads at once to the.inequality:5.Thus|.|aordere is of|ifTJ\^aJ,sinceNisaof order|Thus the\.defined by (12.2.33), are in fact of the correct order.In the same fashion, by using all four of the above inequalities, onequantities e k , asobtains6.(12.2.34)We\s lnoware-m \^\\r-rmlaa position to show that the conditions I) and II)inhold once proper choices of 17 and a have been made. We suppose thatraa (1) any value xJ and choose a such thatr\||<^^||<in thea)K<1 israngedefined by r satisfies||taken.o>||^Asa||\before, the|(1+77)^;normof the function= N ^ ||a\.Our nextto give an estimate for Tr as defined by (12.2.31).objectiveviewof (12.2.31), and (12.2.27) and (12.2.25):inhave,is|andaaIIWea|Ithis in turn yields:A<Tr|\a\\a\K a fixed constant, upon using the first, second, and fifth of ourof order a.
Thusinequalities, together with the fact that Nwithisa (2)^a(1)isifKa F 7 (^aa positive constant such that(2)(2})^r\itfollows that||Tr||^r\if||rThis establishes the conditionout inII^2IImuch^il-\\<^I).r\^Janda|The proof that\^a (2).II) holdsisthe same way. Suppose that r l9 r 2 are such thathave, upon using the inequalities 1. to 6.:Wecarried||rl||,LEVI-CIVITA'STHEORY543a-fsin e i9r ,sin02-0! +0 2 )\a\+e 1 (tf~ 3Ti<r 3T 2 sinsin d l2)-f e~*T26^e^)sinKJV 3XIwithal42^ a fixed positive constant. Ifstant such that 3?a >F e (a<(aand the conditionII)is8))^*,(3)^3<z(2)is5Tal8la fixed positive con-thenverified.that an iteration process starting with an arbitraryfunction r in B, such that Ra) lies in B 2 with \\rQwill^rj(3)r ifaaThe functionconverge to a solution r of TrIt follows,=a>a(r+solutioniste""*^)(12.2.29) whichinlies\\<|\^^.then a solution of the functional equation/?, and which is furthermore not the "trivial"=^^cbr \\)(which always exists), sincea(lThis concludes the proof for the existence of acontinuous solution co(^) of the nonlinear functional equation.
Oncefa sincethis hasa)\\r||\\\\||fg J.been done we have seen at the end of the preceding sub-section thatd>(<f>)isactually analytic in <.eachIt is also clear that the quantities e k assigned tocb kandrkand that they converge since the e k form a Cauchy sequence in-> 0.rnview of the sixth inequality above and the fact that rmIf we set elim e k it is clear that the resulting value of A' obtainedexist,11\\,from (12.2.13), in conjunction with the arbitrarily prescribed value ofA, yields the propagation speed U through (12.2.10) as a function ofthe amplitude parameter a.
Since co(#) has the period 2yr/A, it followsfrom the discussion at the beginning of this section that the motionin the physical plane has the period, orwavelength, 2jr/AC7.BibliographyArthur, R. S.Revised wave forecasting graphs and procedure. Scripps Institution of[A.I]Oceanography of the University of California. Wave Report 78, 1947.i.A.2]Variability in direction of wave travel. Annals of the New York Academyof Sciences, Vol. 51, Art. 3, 1949, pp. 511-522.[A. 3]Refraction of shallow water .... Transactions of the American Geophysical Union, Vol.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
