J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 22
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ForequationA() we have from(5.4.14)and(5.4.15) the-WAVES ON SLOPING BEACHES AND PAST OBSTACLESv,. 41ft(5.4.16)MOThis equationis=2,nm(L).k'(f-friXf-fr,)solved by introducing the function /() bylog A(C)(5.4.17)and=103=l(),one finds at once that /() satisfies the difference equation(5.4.18)l(e**t)(-/(C)-log m(C)=w(f).In solving this equation we shall begin by producing a solutionfree of singularities in the sectorcoco, after which theargfunction /&() which is (cf. (5.4.17)) then also regular in the same^^can be continued analytically into the whole -plane slitalong the negative real axis (or, if desired, into a Riemann surfacehaving the origin as its only branch point) by using (5.4.16). As ansectoraid in solving equation (5.4.18)\co(5.4.19)V1C== Ta< arg f < coitsinset< a ^ 1,*(*") - L(r),OCTT,and operate nowwe,Mr")=W(r)One observes that thein a r-plane.9sectorthe C-plane corresponds to the r-planecoslitalongnegative real axis.
For L(r) one then finds at once from (5.4.18)the equationL(re(5.4.20)2ni)-L(r)=W(r).Ourobject in putting the functional equation into this form(following Peters) is that a solution is now readily found by makinguse of the Cauchy integral formula. Let us assume for this purposethat L(r) is an analytic function in the closed r-plane slit along itsnegative real axis* (which would imply that l() is regular in^^the sectoro>co, as we see from (5.4.19)); in such aarg fcase L(r) can be represented by the Cauchy integral formula:L(r)= -L(5.4.21)Cthe path in the {-plane indicated by Figure 5.4.4. If wesuppose in addition that L() dies out at least as rapidly as, say,l/{ at oo, it is clear that we can let the radius R of the circularwithpart of C tend to infinity, draw the path of integration into the twoedges of the slit and, in the limit, find for L(r) the representationWeshall actuallyproduce such a regular solution shortly.WATER WAVES104(-planeFig.
5.4.4.L(r)(5-4.22)CPath=in the {-plane2m J2m>in readilyOn makingunderstandable notation.drawing the two integrals together,it isuse of (5.4.20), andreadily seen that L(r) isgiven by(5.4.23)L(r)=d.-.r2jwJ_aofthe negative real {-axis, and W() is ton. Since W() has no singularities (of.The path of integrationbe evaluated for arg{is(5.4.18)), it follows thatL(r) as given by (5.4.23)in the slit r-plane. L(r) also hasisindeed regularno singularity on theslitexceptat the origin, where it has a logarithmic singularity. Since thenumerator in the integrand behaves like l/{ a a0, at oo (cf.(5.4.19), (5.4.18), (5.4.16)), it is clear that the function L(r) dies,out>like I /r at oo in the r-plane. This function therefore has all ofthe properties postulated in deriving (5.4.23) from (5.4.21), andhence is a solution of the difference equation (5.4.20) in the slitplane including the lower edge of the slit.Asolution of (5.4.18) can(5.4.19); the resultnow bewrittendown throughuse ofis:=with w( a ) to be evaluated for arg {so far only for f in the sectora>arg^n.
This solutionis^regular,co,whereit isvalidWAVES ON SLOPING BEACHES AND PAST OBSTACLESaswe know fromthe discussion above. However,define the functionA()=e l( ^(cf.it is105necessary to(5.4.17)) in the entire slit-can be done by analytic continuation with the aidplane, andof the functional equation (5.4.16). In the process of analytic conthistinuation, starting with the original sector in which /(), and henceA(), is free of singularities, one sees that the only singularities whichcould occur in continuing into the upper half-plane, say, would arisefrom the function on the right hand side of the equation (5.4.16).=The onlyir l 2singularities of this function occur obviously atnoConsequentlysingularity of A() appears in the analytic con-.tinuation into the upper half-plane, through widening of the sectorin which &() is defined, until the pointsir l andir 2 havebeen covered, and one sees readily from (5.4.16) that the first suchsingularities of /i()poles of first order -appear at the points=ri,2ex P{*(2co+ 7T/2)},thenextatr2exp=(i(4co+ ?r/2)},etc.,though some, or all, of these poles may not appear on the first sheetof the slit f -plane, depending on the value of the angle co.
The continuation into the lower half-plane is accomplished by writing(5.4.16) in the equivalentform(5.4.16)'Again we see that poleswilloccur in the lower half-plane in thecourse of the analytic continuation, this time at r l%2 exp {-i(2co +JT/2}rCX P {i(4a) +yt/2)} etc. The situation is indicated in Figurei,29g() at the points?>! andThus the function A() is defined in the slit -5.4.5; /i() lacks the singularities of?V 2(cf.(5.4.15)).can alsobe continued analytically over the slit whichofrotationathe path of integration.) We see that /()permitsinthehavepolesopen left* half-plane, on two circles of radiimayplane.and(Itbut the poles closest to the imaginary axis are at theangular distance 2co from it.
There is also a simple pole of A() atthe origin, but g() (cf. (5.4.15)) is regular there.The behavior of /i() at oo in the slit plane is now easily discussed:In the original sector we know from (5.4.24) that /() dies out at oo1/a Hence= e l( ^ is bounded in the sector, and since theliker1l/r2,.A()right hand side of (5.4.16) is clearly bounded atA() is bounded at oo in the -pUme.all++itfollows thattr t ) (cf.
(5.4.15)) can nowfViHf*()/(of the properties needed to identify the functionsThe function g()be seen to have=ooWATER WAVES106{-planeFig. 5.4.5.Thesingularities of /i()andwhose real part andwave solutionsthedesiredstandingimaginary part, respectively, yieldof our problem. To this end we write down the integrals/(*) in (5.4.4)(5.4.25)and f^z)in (5.4.10) as functions-i*L').(Cover the paths indicated in Figure 5.4.6, where the direction is indicated only on the part of the path in the lower half-plane, sincethe paths P Pl differ only in the direction in which the remainderof the path is traversed.Since /&() is bounded at oo, and 0te z > 0, the integrals clearlyconverge. One sees also that the paths of integration can be rotated9co about the origin without passing over singularitiesof the integrand, and also without changing the value of the inprove next thattegrals. (This was needed in deriving (5.4.8)'.)through the angleWeg()boundary condition (5.4.11).
To begin with, we shallshow that J() as defined by (5.4.24) is real when is real and positive.Once this is admitted to be true, then h() as given by (5.4.17) wouldhave the same property, and the function g() defined by (5.4.15)satisfies thewouldeasilybe seen toonly to show that /()Wehave, then,satisfy the condition (5.4.11).is real for real , and this can be seen as follows:WAVES ON SLOPING BEACHES AND PAST OBSTACLES-Fig. 5.4.0.The paths P, P^plonein the C-plane=ais to be evaluated for argjr.)(5.4.21) log w(case one sees easily from the equation (5.4.16) defining111a=ro/jr,ef.a(5.4.19)) that m(^ ) haswhen argrr,and hencethe path of integration;it107itsButin thisaw(f ) (withvalues on the unit eirclelogarithm is pure imaginary onfollows at once from (5.4.24) that /()itsandpositive.
Since g(C) was constructed in such atoassatisfy (5.4.13) we know that (5.4.12) is satisfied autowayour standing wave solutions satisfy the boundaryThusmatically.isreal forrealconditions.Finally, we observe that the behavior of / and f l for &e z -> oowhat was prescribed. To this end we deform the path of integrationinto a path running along the two banks of the slitted negative= ir^ 2 contribute terms alreadyreal axis. The residues atis+oo.discussed above which furnish the desired behavior for 3te z ->suretheresiduesatmakethatthetoWe have, then, onlyremainingpoles and the integrals along the slit make contributions which dieout as 3te z ->+oo.Asfor the residues at the poles at the pointsWATER WAVES108w=r1>2exp{i(2ncon = 1, 2,Ae z ^n but+ rc/2)},contributions are of the form9...,we observe thatsinceco^arg zthese^itis clear that these contributions die out exponentially when z tendsCDto infinity in the sectorarg 2 fg 0.
As for the integrals along^they are known to die out like 1/2, as we have seen in similarcases before, or as one can verify by integration by parts. Thus allof the conditions imposed on f(z) and f^z) are seen to be satisfied.theslit,We observe, however, that the integrals in (5.4.25) over the pathsP and P converge only if 9&e z ^ 0, and hence this representationlthe bottom slopes down at an anglen/2 the solution can bejr/2. For an overhanging cliff, when coobtained by first swinging the path of integration clockwise throughof our solutionisvalid onlyif>^990(and swinging the slit also, of course); the resulting integralsand the solutionswould then be valid for all z such that ^m zwould hold forn.coIt is perhaps of interest to bring the final formulas together for^<^the simplest special case, i.e. the dock problem for two-dimensionalmotion (first solved by Friedrichs and Lewy [F.12]), in which theyFig.
5.4.7.The dock problemangle co has the value n, as indicated in Figure 5.4.7.* In this casethe function /() is given by(5.4.26)integral defines it at once in the entire slit -plane. The3le f(z) and y 1 (#, y)f^z)standing wave solutions <p(x, y)and the=Smare determined through*As was mentioned in section 5.1, the dock problem in water of uniformdepth and for the three-dimensional case was first solved by Heins [H.I 3]finitewith the aid of the Wiener-Hopf technique.WAVES ON SLOPING BEACHES AND PAST OBSTACLES(5.4.27)f(z)=~|i(5.4.28)109e^r./ t (z)with A(C) defined by(5.4.29)As was remarked above, the integrals in (5.4.27) and (5.4.28) converge only if &e z ^ 0.
However, the analytic continuation into theentire lower half-plane is achieved simply by swinging the pathsP and P! into the positive imaginary axis (which can be done sinceA(C) is bounded at oo), while staying on the Riemann surface of A(),and theseintegrals are then valid for all z in the lower half-plane.Finally, it is also of interest to remark that the functions f(z)and f^z) do not behave in the same way at the origin: the first isboundedthere,and the secondonly for the spcciaj case of theunder consideration here.5.5.
Diffraction ofnot, and this behavior holds notdock problem, but also in all casesiswaves around averticalwedge.Sommerfeld's diffraction problemwe are primarily concerned with the problem ofof a barrier in the form of a vertical rigideffectthedetermininginasindicatedFig. 5.5.1, on a plane simple harmonic wavewedge,In this sectionyiFig. 5.5.1. Diffraction of a planecoming frominfinity.cylindrical coordinatesIn this case(r, 0,y).wave by awedgeconvenient to make use ofseek a harmonic functionit isWeverticalWATER WAVES110<&(r> 0> y\ t) in the regionexterior to the wedge of anglewhenat rest.variables(r,<v,2nvThe problem is reducedby settingh<yandinto one<Q,in thein the regioni.e.water ofdepth htwo independentfinite6)<2>(r, 0,(5.5.1)yi t)=/(r, 0)+ h)e= 0, 6 = viotcosh m(y= forThe boundary conditionseto the rigid walls of the wedge yield for.correspondingthe0)boundary con-/(r,ditions(5.5.2)Thef=free surface conditionQ=v.0-0,0,g0 y+tt=at y =-yields the con-ditionm tanh mh= at they(5.5.3)while the conditionautomatically.
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