J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 47
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9.1. 2b), in its equilibrium position,9^ are9r/-plane.The boundary conditions on(9 1 12)2 i)K+-+ Ou(y ~= 0+ and 2 =02i)<*2/c)A-onA + and .^__ refer to the two sides z0_ of theTheonarediskA.conditionsthefreesurfaceboundaryplaneinwhich(9.1.13)The first of thesefrom the condition that the pressure vanisheson the free surface, the second arises from the kinematic free surfacecondition. If s Q9 co l9 21 an d 0n were known functions of t, these boundary conditions in conjunction with (9.1.11) and appropriate initialconditions would serve to determine the functions cp l and rj l uniquely;i.e.
the velocity potential and the free surface elevation would beknown. Of course, the really interesting problems for us here are thosein which the quantities s co l9 21 an d O ll9 referring to the motion ofthe ship, are not given in advance but are rather unknown functionsof the time to be determined as part of the solution of the boundaryproblem. In principle, one method of approach would be to apply theLaplace transform with respect to the time t to (9.1 .11), (9.1.12), andof course taking account of initial conditions at the time(9.1.13)/ =and then to solve the resulting boundary value problemfor the transform q>i(x y z; a) regarding s and the transforms oi^cr),02i( (T )> a]Q d fliifa) as parameters.
However, for the purposes of thisintroduction it is better to concentrate on the most important specialcase (already mentioned above) in which the ship has a motion oftranslation with uniform speed combined with small simple harmonicoscillations of the ship and the sea having the same frequency.* Inresults,,9this caseelevationwerj l9,9write the velocity potential ^(tf,z; t) 9j/,and the other dependent quantitiesy9z; t)^(a?, z; t)== \pt(x- Q^\y;andy9z)+ ^(x9yformiat9z)e+ //!<#, z)e M- en e M- e^e M//(*, z)(<?!(#,a*!The functions9the surfacein the^n9ai.are of course both harmonic functions.expect the functions g^ andrj lto have time-independentWecomponents* It can beseen, however, that the discussion which follows would takethe same course if more general motions were to be assumed.muchTHE MOTION OF A SHIP IN A SEAWAY253due to the forward motion of the ship; certainly they would appearin the absence of any oscillatory components due, say, to a wave trainin the sea.Uponand (9.1.13) weinsertion of these expressions in equations (9.1.12),find for y> the conditions:Aon9at yandfor=0,the conditionsy>isy) lz21-\-(Q l -\-io0 2l )x.g//!+ *oVi.r*aVi-'-Qio&n(yy') on AI!We observe,in passing, that y> satisfies the same boundary conditionsas in the classical Michcll-Havelock theory.little later we shall see,Ain fact, that thewaveindeed independent of all components of the motion of the ship (to lowest order in /3, that is) except itsuniform forward motion with speed s and that the wave resistanceresistanceis,isdeterminedin exactly thesame wayas in the Michell-HavelockWe continue the description of the equations which determinetheory.the motion of the ship, and which arise from developing the equationsof motion with respect to ft and retaining only the terms of order /?and /ft 2 (We observe that it is necessary to consider terms of both.MM = M^of the ship is given byorders.) In doing so the massa constant, since we assume the average density of the shipwithlto be finite and its volume is of course of order 0.
The moments ofM9Thepropeller thrust is assumed to beacting in the ^'-direction and in the x' y'vertical distance from the e.g. is/; the thrustinertia are then also of order (3.a force of magnitude Tplane at a point whoseT is assumed to be of order /? 2 since the mass is of order /? and accelerations are also of order /?.* The propeller thrust could also, of course,be called the wave resistance.9,The termsof order/?yield the following conditions:We have in mind problems in which the motion of the ship is a small deviationfrom a translatory motion with uniform finite speed.
If it were desired to studyas would be necessary, for example, if themotions with finite accelerationsit would clearly be necessaryship were to be considered as starting from restto suppose the development of the propeller thrust T to begin with a term of firstorder in /?, since the mass of the ship is of this order. In that case, the motionof the ship at finite speed and acceleration would be determined independentlyof the motion of the water: in other words, it would be conditioned solely by theinert mass of the ship and the thrust of order 0.*WATER WAVES254=(9.1.15)*(9.1.16)2 eg f0,fihdA- MJg,J Af(9.1.17)=xfihdA0,J A- Wlc )]+ <L4 -(9.1.18)fJ x(9.1.19)f[a( VlfJ Af(9.1.20)[( Vli[i/^,-*tf> lx )]*+8<ff lx )-]0,cW- 0,dA= 0.J Aoccurring here means that the jump in the quantity in brackets on going from the positive to the negative side of theprojected area A of the ship's hull is to be taken.
The variables of in-The symbol]*[tegration are x and y. The equation (9.1.15) states that the term oforder zero in the speed is a constant, and hence the motion in thea small oscillation relative to a motion with uniformvelocity. (This really comes about because we assume the propellerthrust T to be of order /? 2 .) Equation (9.1.16) is an expression of the^-directionislaw of Archimedes: the rest position of equilibrium must be such thatthe weight of the ship just equals the weight of the water it displaces.Equation (9.1.17) expresses another law of equilibrium of a floatingbody, i.e. that the center of buoyancy should be on the same verticalline as the center of gravity of the ship.
The remaining three equations(9.1.18), (9.1.19), and (9.1.20) in the group serve to determine the dis-Uplacements21 ,,andco vwhich occurin theboundary condition(9.1.12) for the velocity potential 9^. In the special caseweconsiderwe observethat these three equations would determinethe values of the constants19n and zl (the complex amplitudesof certain displacements of the ship) which occur as parameters in the(cf.(9.1.14))Qboundary conditionsfor the,harmonic functiontpi(x, y, z)given in(9.1.14)!.Wespeednowaresisfixed, itmonic function(9.1.14),(jp 1(9.1.14),draw someinteresting conclusions. Once thefollows that the problem of determining the har-able toiscompletely formulated through the equations(9.1.14)!,and(9.1.18) to (9.1.20) inclusive (to-gether with appropriate conditions at oo).
In other words, the motionof the water, which is fixed solely by <p v is entirely independent of theTHE MOTION OF A SHIP IN A SEAWAY255pitching displacement 31 (0 the heave yi(t) and the surge s^t), i.e.of all displacements in the vertical plane except the constant forward9speedSQ.A little reflection, however, makes this result quite plausible:Our theoryisbased on the assumption that the shipdisposed vertically in the water,Hence onlywhose thicknessisisa thin diska quantity ofdisplacements of the disk parallel tothis vertical plane could create oscillations in the water that are offirst order. On the other hand, displacements of first order of the diskfirst order.finiteat right angles to itself will create motions in the water that are alsoof first order.
One might seek to describe the situation crudely in thefollowing fashion. Imagine a knife blade held vertically in the water.Up-and-down motions of the knife evidently produce motions of thewater which arc of a quite different order of magnitude from motionsproduced by displacements of the knife perpendicular to the plane ofits blade. Stress is laid on this phenomenon here because it helps topromote understanding of other occurrences to be described later.The terms of second order in /? yield, finally, the following conditions:(9.1.21)A/!*!-h x [(<Pitp-s<p lxY+(<p ltJ A(9.1.22)^i^~%f JL (2/i+^31h y [(q'it)Mr-pJA(9.1.23)'si^ai^-2(y-Vc) h *A-lQSyA xhdxeg3i|-2gg0 31f,r2Mr+/TJ Lf-Q\J\^h v-(y~yc)h x ][((p lt~s^ lx + +)((p lt-AWenote that integrals over the projected water-line L of the ship onthe vertical plane when in its equilibrium position occur in additionto integrals over the vertical projection A of the entire hull.
The/?/31 for the moment ofquantity / 31 arises from the relation /=an axis through its e.g. parallelthe surge s v and alsodeterminesto the s'-axis. The equation (9.1.21 )the speed s (or, if one wishes, the thrust T is determined if S Q isinertia / of the ship with respect toWATER WAVES256assumed to be given). Furthermore, the speed s is fixed solely by Tand the geometry of the ship's hull. This can be seen, with referenceto (9.1.14) and the discussion that accompanies it, in the followingway: The term y (#, y, z) in (9.1.14) is the term in q>^ that is indepenin (9.1.21).dent of t. It therefore determines T upon insertion ofThis term, however, is obtained by finding the harmonic functiony as a solution of the boundary problem for \p Q formulated in (9.1.14)In fact, the relation between s and T is now seen to be exactly thesame relation as was obtained by Michell.
(It will be written down ina later section. ) In other words, the wave resistance depends only onthe basic translatory motion with uniform speed of the ship, and notat all on its small oscillations relative to that motion. If, then, effects^.on the wave resistance due to the oscillation of the ship are to beobtained from the theory, it will be necessary to take account of higherorder terms. Once the thrust T has been determined the equations(9.1.22) and (9.1.23) form a coupled system for the determination ofyl and 31 since 9^ and n have presumably been determined previously.
Thus our system is one in which there is a considerable amount of,cross-coupling. Itmightalsobe noted that the trim,i.e.the constantvalues of y l and 31 about which the oscillations in these degrees offreedom occur are determined from (9.1.22) and (9.1.23) by the time-independent terms in these equations including, for example, themoment IT of the thrust about the e.g.We proceed to the discussion of other conclusions arising from ourdevelopments and concerning two questions which recur again andagain in the literature. These issues center around the question: whatis the correct manner of satisfying the boundary conditions on thecurved hull of the ship? Michell employed the condition (9.1.12),naturally with n = 21 = coj = 0, on the basis of the physical argument that s h x represents the component of the velocity of the waternormal to the hull, and since the hull is slender, a good approximationwouldresult by using as boundary condition the jump conditionfurnished by (9.1.12).
Havelock and others have usually followed thesame practice. However, one finds constant criticism of the resultingtheory in the literature (particularly in the engineering literature)because of the fact that the boundary condition is not satisfied at theactual position of the ship's hull, and various proposals have beento improve the approximation. This criticism would seemmadeto be beside the point, since the condition (9.1.12) is simply the consequence of a reasonable linearization of the problem. To take accountTHE MOTION OF A SHIP IN A SEAWAY257of the boundary condition at the actual position of the hull would, ofbut then, it would be necessary to dealcourse, be more accuratewith the full nonlinear problem and make sure that all of the essentialcorrection terms of a given order were obtained.
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