J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 45
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The wave amplitudes of bothinfinite according to these formulas for.systems of waves becomei.e. for points at the= 0*,boundary of the disturbed region, but the asymptotic formula(8.2.33)there. We shall consider thesenot valid at such points since <p" =Themoment.inadiverging system also has infinite amplitudepointsbutthisfor 2corresponds to the origin, and the infinite amn/2plitude there results from our assumption of a moving point impulseis=as amodel9for our ship.the amplitude of the waves along the boundary of thedisturbed region, we must calculate the value of ds(p/dt 3 at such pointsin order to evaluate the appropriate term in (6.5.2).
(The problem ofTo determineWATER WAVES242the character of the waves in this region has been treated by Hognerdifferentiating (8.2.29) after replacing dO/dt by c sin 6/roo), one finds readily(8.2.30) and (8.2.31) forBy[H.13].)(cf.R=- -(&MT)and from(8.2.28) in*5=-*.d zw(8.2.38)'V=4>gcregionwithresultis0:sin 2.of the waves along the boundary of the disturbedgiven by(6.5.2)):(cf.functions evaluated foralla cos 2\ct cos 0, ra 2 cos*0dt*The amplitude=combination with r=0*=arc sin l/\/f<*-is~(8.2.40)77-iexpWeobserve that the wave amA^ is a certain constant.1/ 21/3likeinsteadoflikeas they do in thedieoutI/aI/aplitudeswaveinterior of the disturbed region; i.e. theamplitudes are now of aThe quantitynow,and higher, order of magnitude.
As we have seen in all ofour illustrations of ship waves, the wave amplitudes are quite noticeably higher along the boundary of the disturbed region. The phasealso differs now by n/4> from the former values. On some of the photographs (cf. especially Fig. 8.2.4), there is some evidence of a ratherabrupt change of phase in the region of the boundary, though it maybe that one should interpret this effect as due rather to the finitedimensions of the actual ship, which then acts as though severalmoving point sources were acting simultaneously.In the treatment of the present problem by A.
S. Peters [P.4]mentioned in the preceding section, the complete asymptotic development of the solution was obtained.The above developments hold only for the case of a point impulsemoving on the surface of water of infinite depth. It has some interestto point out that there are considerable differences in the results ifthe depth of the water is finite. Havelock [H.8] has carried out theapproximation to the solution by the method of stationary phase fordifferent,WAVE PATTERN CREATED BY A MOVINGSHIP243the case of constant finite depth, with the following general results:1 ) If the speed c of the ship and the depth h satisfy the inequalityc 2 /gh< 1, the general patternof the wavesismuchthe same as forwater of infinite depth except that the angle of the sector withinwhich the main part of the disturbance is found is now larger thanfor water of infinite depth.
2) If c 2 /gk > I holds, the system of transverse waves no longer occurs, but the diverging system is found.Fig. 8.2.12. Speed boat in shallow waterFigure 8.2.12 is a photograph of a speed boat creating waves, presumably in shallow water, in view of the difference in the wave patternwhen compared with Fig. 8.2.4. Finally, if c 2 /gh 1 (i.e. for the caseof the critical speed), the method of stationary phase yields no reasonable results; that this should be so is perhaps to be understood in=the light of the discussion of the corresponding two-dimensionalprobleminChapter7.4.CHAPTERThe Motion of a9Ship, as a Floating Rigid Body, in aSeaway9.1.
Introductionand summaryThe purposeof this chapter is to develop a mathematical theory forthe motion of a ship, to be treated as a freely floating rigid body underthe action of given external forces (a propeller thrust, for example),under the most general conditions compatible with a linear theory andthe assumption of an infinite ocean.* This of course requires theamplitude of the surface waves to be small and, in general, that themotion of the water should be a small oscillation near its rest positionof equilibrium; it also requires the ship to have the shape of a thindisk so that it can have a translatory motion with finite velocity andstill create only small disturbances in the water. In addition, the mo-must be assumed to consist of small oscillationsmotion of translation with constant velocity.
Withintion of the ship itselfrelative to athese limitations, however, the theory presented is quite general inthe sense that no arbitrary assumptions about the interaction of theship with the water are made, nor about the character of the couplingbetween the different degrees of freedom of the ship, nor about thewaves present on the surface of the sea: the combined system of shipand sea is treated by using the basic mathematical theory of thehydrodynamics of a non- turbulent perfect fluid.
For example, thetheory presented here would make it possible in principle to determine the motion of a ship under given forces which is started witharbitrary initial conditions on a sea subjected to given surface pressures and initial conditions, or on a sea covered with waves of prescribed character coming from infinity.knownthat such a linear theory for the nonturbulent motion of a perfect fluid, complicated though it is, still doesnot contain all of the important elements needed for a thoroughgoingItisof course welldiscussion of the practical problems involved.
For example,*The presentation of the theory given heregiven in a report of Peters and Stoker [P.7].245isessentially theitignoressame as thatWATER WAVES246the boundary-layer effects, turbulence effects, the existence in generalof a wake, and other important effects of a non-linear character. Gooddiscussions of these matters can be found in papers ofLunde and Wig-ley [L.I 8], and Havelock [H.7], Nevertheless, it seems clear that anapproach to the problem of predicting mathematically the motion ofseaway under quite general conditions is a worthwhile enterand that the problem should be attacked even though it isships in aprise,recognized at the outset that all of the important physical factors cannot be taken into account.
In fact, the theory presented here leads atonce to a number of important qualitative statements without thenecessity of producing actual solutionsfor example,weshall seethat certain resonant frequencies appear quite naturally, and inaddition that they can be calculated solely with reference to the massdistributionand the given shape of thehull of the ship. Interestingobservations about the character of the coupling between the variousdegrees of freedom, and about the nature of the interaction betweenthe ship and the water, are also obtained simply by examining theequations which the theory yields.In order to describe the theory and results to be worked out inlater sections of this chapter,it isnecessary to introduce our notationand to go somewhat into details.
In Fig. 9.1.1 the disposition of two ofthe coordinate systems used is indicated. The system (X, Y, Z) is aAY-*.'2Fig. 9.1.1. Fixedsystem fixedX*and moving coordinate systemswith the X, Z-plane in the undisturbed freeand the F-axis vertically upward. A movingin spacesurface of the water(x, y, z) is introduced; in this system the #, ztoassumedcoincideplanealways with the X, Z-plane, and theisassumedtocontainthecenter of gravity (abbreviated to e.g.t/-axisin the following) of the ship.
The course of the ship is fixed by themotion of the origin of the moving system, and the #-axis is taken alongsystem of coordinatesisTHE MOTION OF A SHIP IN A SEAWAY247the tangent to the course. It is then convenient to introduce thespeed s(t) of the ship in its course: the speed s(t) is simply the magni-tude of the vector representing the instantaneous velocity of this point.At the same time we introduce the angular speed co(t) of the movingsystem relative to the fixed system: one quantity fixes this rotationbecause the vertical axes remain always parallel. The angle oc()indicated in Fig.
9.1.1defined byis(9.1.1)0)(t)(It,corresponds to an instant when the #-axis andInorder to deal with the motion of the ship as aJT-axis are parallel.rigid body it is convenient, as always, to introduce a system of coor-implying thattdinates fixed in the body. Such a systemFig.
D.I. 2.The#', t/'-planeisassumed(#',y\z') is(b)(a)Fig. 0.1. 2a, b.ofsymmetryindicated into be in the fore-and-aft planeAnother moving coordinate systemof the ship's hull, and the ?/'-axisisassumed to containthe e.g. of the ship. The moving system( r', j/', z') is assumed to coincide with the (iT, j/, z) system when the ship and the water are at resttin their equilibrium positions.Thee.g.of the ship will thus coincidewith the origin of the (#', j/', z') system only in case it is at the levelof the equilibrium water line on the ship; we therefore introduce theconstant y'c as the vertical coordinate of thedinate system.The motiontialof the water0(X, F, Z;t)whichisise.g. inthe primed coor-assumed to be given by a velocity poten-therefore to be determined as a solutionof Laplace's equation satisfying appropriate boundary conditions atthe free surface of the water, on the hull of the ship, at infinity, andalso initial conditions at the timet=0.The boundary conditions onthe hull of the ship clearly will depend on the motion of the ship,WATER WAVES248which in its turn is fixed, through the differential equations for themotion of a rigid body with six degrees of freedom, by the forces actingon it including the pressure of the water and its position and veloc= 0.
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