J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 42
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The notations to be used for*Actually, we have considered only the displacement of the free surface inthat section, but it is readily seen that (8.1.1) furnishes the vertical displacementof any points in the water.WAVE PATTERN CREATED BY A MOVINGSHIP221which is to be considered asa vertical projection of the free surface on any plane y = const. Thecourse of the ship is given in terms of a parameter t by the relationsthispurpose are indicated in Figure8.1.1,^ ^t(8.1.2)T,and t is assumed to mean the time required for the ship to travelfrom any point Q(xv z^ on its course to its present position at thekP(x,z)wave problemFig. 8.1.1. Notation for the shiporigin.shipis(8.1.3)Weseek the displacement of the water atat the origin;it istherefore determined(x, y, z)when theby the integralr](x,y,z)I=/r/ak(t) dt%ngQ JoIn this formulasin at e mym dmpjr/2cos (mr cosJoJok(t) represents the strength of the impulse,might reasonably assume to be constantwhich wethe speed of the ship istheisthereforeconstantthisconstant;only parameter at our disposalwhich might serve to represent the effect of the volume, shape, etc.of a ship's hull.Weifwrite the last relation in the formrooji/2(8.1.4)mv iri(x,y>z)~K [[(ame [e(at~mrcos+ei(at+ mrcos]d@dm dtooowith the understanding that the imaginary part of the integral is to betaken.
(K is a constant the value of which is not important for theWATER WAVES222=+discussion to follow.) It should be noted that r 2(xx^ 23 X ) 2 Since y0, the integral converges strongly because of the(z<.exponential factor.One of the puzzling features (to the author, at least) of existingtreatments of the problem by the method of stationary phase is thatmadewhat parameter is large in the exponentials as thetoeach of the three integrals in turn, so that one isappliednot quite sure whether there might not be an inconsistency.
Thentatter is easily clarified by introduction of appropriate dimensionlessit isnotmethodclearisquantities, as follows(cf.Figure 8.1.1):(8.1.5)x=rR cos a,xl == R V(A cos ar=xR=z,= R sin a,+ (Asinaxcos a) 2R,/Rct/R,cos a xl*A,=sin a) 2^,4c 2Rz=Rlsina l5/,m = ^-l2.4r 2Here the quantityc represents the speed of the ship in its course. Itshould be noted that x, y, and z are held fixed they represent thepoint at which the displacement is to be observed, but that x l9 z l(and hence R l and o^), and r all depend on t. We have also introduceda new variable of integration f replacing m, which depends on t.
TheJacobian 9(ra, t)/d(, r) has the value gt 2 Rg/(2cr 2 ) and hence vanishesonly for t = 0. In terms of the new quantities the integral (8.1.4) isfound to take the form:,r)(x,y,z)(8.1.6)TO oojz/2w^T 54xr^yfm-,3000=where T OcT//e.<Again we remark0.that the integral converges uniformly for yHowever, the integrand has a singularity if the point (x, y, z) happensto be vertically under a point on the course of the ship: in such a casewe havevalue rR=RTinl (i.e.A ==1),the interval=a^ TOand afSr,factor, the integral continues to exist,from(8.1.4) that taking r=does notfor a certainso that I =Because of the exponentialhowever. Indeed, one seeslfmakethe integrand singular;WAVE PATTERN CREATED BY A MOVING228SHIPthe fact that a singularity crops up in (8.1.6) arises from our choiceof the variable | which replaces ra.
This disadvantage caused by introduction of the new variables is much more than outweighed by thefact thatwe now cansee that the approximationby the method ofstationary phase depends only on one parameter, i.e. the parameterx = gjR/4r 2 in the exponentials. We can expect the use of the methodof stationary phase to yield an accurate result if this parameter islarge, and that in turn is certainly the case if R is large, i.e.
for pointsnot too near the vertical axis through the present location of the ship.The application of the method of stationary phase to the integralin (8.1.6)can now be justified by an appeal to the arguments used insection 6.8. In doing so, the multiple integral is evaluated by integrating with respect to each variable in turn; at the same time, theintegrands are replaced by their asymptotic representations as fur-nished by the method of stationary phase.
One need only observe, inverifying the correctness of such a procedure, that the integrandsremain, after each integration, in a form such that the arguments ofthat section apply in particular that they remain analytic functionsof their arguments provided only that points (x, y, z) on or under theand that an asymptotic series can be inship's course are avoided*tegrated termwise. It is not difficult to see that the contributions toYI(X, y, z) of lowest order in \\x are made by arbitrarily small domainscontaining in their interiors a point where the derivatives 9?^, <p^ <p r of2cos ^)r 2 /l(r) vanish simultaneously.the phase 9?(2Even for points on the ship's course the argument of section 6.8will still hold provided that no stationary point of the phase <p occurs9== 0: the reason for this is that thein section 6.8 only to treat ausedassumption of analyticity wasneighborhood of a point of stationary phase, while for other segmentsof the field of integration only the assumptions of integrability andthe possibility of integration by parts are needed.
It happens that thecases to be treated later on are such that l(r) does not vanish at anypoints of stationary phase, and hence for them the asymptoticapproximation is valid also for points on the ship's course.There is one further mathematical point to be mentioned. Thefor a value of rsuch thatl(r)/* In section 6.8 theintegrals studied&were of the formy(x) exp (ikq>(x)} dx,Jo/bonewhile here the integral is of the formy>(x, k) exp (ifop(x)} dx. However,Jacan verify that the argument used in section 6.8 can easily be generalized toinclude the present case.WATER WAVES224above discussion requires that we take y < 0, and it is not entirelyis legitimate in the approxiclear that the passage to the limit y ->mate formulas, so that the validity of the discussion might be thoughtopen to question for points on the free surface.
Indeed, it would appearto be difficult to justify such a limit procedure for the integral in(8.1.1), for instance, since it certainly=does not convergeifwesetsince the integrand then does not even approach zero as m -> oo.yHowever, this is a consequence of dealing with a point impulse.
If wehad assumed as model for our ship a moving circular disk of radius aover which a constant distribution of impulse is taken, the result forthe vertical displacement due to such a distributed impulse appliedat t =could be shown to be given byrj(x, y, z\ t)=K/*00lasin at^ mvJrjo(mr)J 1 (ma)dmKwith Ji(ma) the Bessel function of order one and l a certain constant.This integral converges uniformly for y0, as one can see from theasymptotic behavior of JQ (mr) and J^(ma).
Consequently rj(x, y, z; t)is continuous for y0. On the other hand, if the radius a of the diskis small the result cannot be much different from that for the pointimpulse. Thus we might think of the results obtained in the nextsection, which start with the formula (8.1.1 ) for a point impulse, as anapproximation on the free surface to the case of an impulse distributedover a disk of small radius.It has already been mentioned that the problem under discussionhere has been treated by A.
S. Peters [P.4] by a different method.Peters obtains a representation for the solution based on contourintegrals in the complex plane, which can then be treated by thesaddle point method to obtain the complete asymptotic developmentof the solution with respect to the parameter x defined above, whilewe obtain here only the term of lowest order in such a development.However, the methods used by Peters lead to rather intricate deve-^~lopments.8.2.Theclassical shipwave problem.In the preceding sectionwe haveDetails of the solutionjustified the repeated applicationmethod of stationary phase to obtain an approximate solutionfor the problem of the waves created when a point impulse moves overthe surface of water of infinite depth. In particular, it was seen thatthe approximation obtained in that way is valid at all points on theof theWAVE PATTERN CREATED BY A MOVING225SHIPsurface of the water not too near to the position of the "ship" at theinstant when the motion is to be determined (provided only that acertain condition is satisfied at points on the ship's course).
In thissection we carry out the calculations and discuss the results, returninghowever to the original variables since no gain in simplicity would beachieved from the use of the dimensionless variables of the precedingsection.Kelvin carried out his solution of the ship wave problem for thecase of a straight line course traversed at constant speed. Up to acertain point there is no difficulty in considering more general courses\P(x,z)Fig.
8.2.1. Notation for the shipwave problemfor the ship. In Figure 8.2.1 we indicate the coursegiven in terms of a parameter t by the equationsx=for(8.2.1)<,t<Casany curveT.The parameter t is taken to represent the time required for the shipto pass from any point (# 15 2 X ) to its present position at the origin O,=to correspond to the origin so thatconvenient to take tthe point (x v y x ) moves backward along the ship's course as t increases.The shape of the waves on the free surface is to be determined at themoment when the ship is at the origin.
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