J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 52
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The center of buoyancyEquation (9.3.27)plane of symmetry, and equation (9.8.28) is anexpression of the second law of equilibrium of a floating body; namelythat the center of buoyancy for the equilibrium position is on thesame vertical line, the t/'-axis, as the center of gravity of the ship.The function <p l must satisfyof the shipin theis=+ <Plw +A where D the half space y< 0,<Plxxin thedomainD<PizzisandAistheplane disk defined by the projection of the submerged hull on theassume thatx, y-plane when the ship is in the equilibrium position.intersects the x, 2-plane. The boundary conditions at eacli side ofWeAAare(9.3.36)= + sofa +The boundary condition(9.3.10)and2 i)-=co =at y(9.3.11). SinceK+is#21)*found by eliminatingthese equations are<Pit<Piv+~ Mix + ^i<rj lfrom==and they yield(9.3.37)sfo lxx-2^9?!^+ g<fi y + <Pw =y = 0.
The boundary conditions (9.8.36) and (9.8.87) show thatPi depends on co^t), O n (t) and 21 (0- Th e problem in potential theoryfor (p l can in principle be solved in the formforwithout using (9.3.29), (9.3.30), (9.3.32). The significance of this hasWATER WAVES278already been discussed inThe general proceduresec. 9.1 in relation toto be followed in solvingequations (9.1.14).allproblems was alsodiscussed there.The remainder of this chapter is concerned with the special case ofa ship which moves along a straight course into waves whose crests areat right angles to the course. In this case there are surging, heavingand pitching motions, but we have d l=0,2=0, co=0; inadditionwe notethat the potential function <p can be assumed to be an evenfunction of z.
Under these conditions the equations of motion aremuchsimpler.Theyare(9.8.38)Mj^glJA h(9.3.39)M^y l =-2Qgy 1 \hdx-2 e g6^ \xhdx+2 ex (s<fpi x ~<Pit)dA+TJL(9.3.40)/3i03i= -2eg03if(y-y'e }hdA-2 Q gyA xhdxJLJAfih y (s Q<p lx -<p lt )dAJAJLx*hdx+lTJL[xh y -(y-y'c )hx }(s<fp lx -(p lt )dA.+2QJAIt will be shown in the next section that an explicit integral representation can be found for the corresponding potential function and thatthis leads to integral representations for the surge s v the heave y land the pitching oscillation 31.9.4.Method of solution of the problem of pitching and heaving ofa ship in a seaway having normal incidenceIn this section we derive a method of solution of the problem ofcalculating the pitching, surging, and heaving motions in a seawayconsisting of a train of waves with crests at right angles to the courseof the ship, which is assumed to be a straight line (i.e.,propeller thrust is assumed to be a constant vector.The harmonic functionco= 0).
The9^ and the surface elevation iy t thereforesatisfy the following free surface conditions (cf. (9.8.10) and (9.8.11),withco= 0):THE MOTION OF A SHIP IN A SEAWAY(9.4.1)279=The kinematic condition(9.3.14) with21=arisingfrom the hull of the shipis(cf.= o>! = 0):= - V**-n(9.4.2)<p lzBefore writing down other conditions, including conditions atexpress <p l as a sum of two harmonic functions, as followsoo,we9^ (x, y,(9.4.3)Here Xois=^z; t)(a?,y, 2)+ Xl (x, y, z;t).a harmonic function independent of t which is also an evenWe now suppose that the motion of the ship is a steadyfunction of z.simple harmonic motion in the time when observed from the movingcoordinate system ox, y, z. (Presumably such a state would resultafter a long timethrust.)upon starting fromConsequently werestunder a constant propellerinterpret X Q(X, J/ z ) as the disturbancetherefore dies out at oo; while %i(x t/, z; t)caused by the ship, whichrepresents a train of simple harmonic plane waves covering the wholesurface of the water.
Thus fa is given, with respect to the fixed coordinate system O X, Y, Z by the well-known formula (cf. Chapter 8):9Xl=~YCe"/a+ -X2sin lat\+yl,with a the frequency of the waves. In the ox, y, zsystem we have,therefore:o(9.4.4)Xl (x, y, z; t)= Ce^r"sin22\/x+la+^J*n<+YWe observe that the frequency, relative to the ship, is increased aboveis positivei.e. if the ship is heading into thetobeofis,course,expected. With this choice ofthatsatisfiesthe following conditions:easy to verify%Qthe value aand(9.4.5)if $thisS$XQXX+ gXov =obtained after eliminating(9.4.6)fa,r/ l=froms^h xat y(9.4.1),=waveslfit is0,andon A,with A, as above, the projection of the ship's hull (for z > 0) onatvertical mid-section. In addition, we require that XQ ->itsoo.WATER WAVES280remarked at this point that the classical problem concerning the waves created by the hull of a ship, first treated by Michell[M.9], Havelock [H.7], and many others, is exactly the problem ofdetermining # from the conditions (9.4.5) and (9.4.6).
Afterwards,It should bethe insertion of^ =%Q in (9.3.38),withix=0,<p lt= 0, leads to theformula for the wave resistance of the ship i.e. the propeller thrust Tis determined. Since y l and 3 are independent of the time in this case,one sees that the other dynamical equations, (9.3.39) and (9.3.40),yield the displacement of the e.g. relative to the rest position ofequilibrium (the heave), and the longitudinal tilt angle (the pitchingangle). However, in the literature cited, the latter two quantities aretaken to be zero, which implies that appropriate constraints wouldbe needed to hold the ship in such a position relative to the water.interest, though, is the wave resistance, and itthefirstorder theory, at least) by the heave and pitch.(inThe main quantity ofisnot affectedWe proceed to the determination of # using a method differentfrom the classical method and following, rather, a course which it ishoped can be generalized in such a way as to yield solutions in more,difficult cases.Suppose that we know the Green's function G*(f 77, x,y,z) such thatis a harmonic function for rj < 0, f >except at (#, y, z) whereit has the singularity 1/r; and G* satisfies the boundary conditions,;G*(9.4.7)G+ kG* =G* =on77on C==where k = g/s We shall obtain this function explicitly in a moment,and will proceed here to indicate how it is used.
Let 27 denote the half> 0; and let Q denote the half plane f 0, r] < 0.plane r\ = 0,From Green's formula and the classical argument involving thesingularity 1/r we haveThen, since=0,THE MOTION OF A SHIP IN A SEAWAYwe have anexplicit representation of the solution in the(*>(9.4.8)281)= -2/>,(*, y,JorXo*G*ddr),jA(,z)=formq, 0; *, y, z^ )<?*(,^upon using(9.4.6).In order to determine G* consider the Green's functionx,y,z) for the half spaceon=770.77<whichG(,r],C;satisfiesThis function can be written aswhereand gisa potential function in.,onr)_.d=0.77<whichasatisfiesiThe formula1-^^-2A;(obtained from the well-known analogous representationwhich the Besscl function J can be expressed as.___2-)]=-2 fn/2^Jooos [p(f-#) cos 0] cos [p(C-s) sin 0] d0,allows us to write4A;4&^forr\=rfJandrf00/2~~ a?coscoscosJj/< 0.It isfor l/r) innoweasy to see that~~WATER WAVES2824Jfegtf +*&,=,=isff*/2pc'Wrt cos [p(f-a?) cosn Jo Jo<a potential function in 77An6] cos[p(C-a) sindB dp9]which satisfies the boundary condition.interchange of the order of integration gives=4JfcM*f*dQ 9teJoftcos [p(f-*) sinp&e denotes the real part.wheredpJoIf we think of p as acomplex variable,to oo in the last result can be replaced by anythe path fromequivalent path L, to be chosen later:4fk,=f ^2fdOMei p cosn Jo\[p(t-*) sindpJLSince the right hand side of this differential equation for gas a superposition of exponentials in |a solution ofcos o]0]e[(m)+-)andrjisexpressedto be expected thatit iscan be found in the formit&p-pL2cos 2provided the path L can be properly chosen.
The path L, which willbe fixed by a condition given below, must, of course, avoid the polep= A/cosItG(f,26.can now be seen that the functionrj,;+ G(|,x, y, z);?j,a?,imposed on the Green's function employedright has the proper singularity inboundary condition (9.4.7) andGc (f,iszero at=0.??,C; x, y, z)Thus we have-rf=o8ftf*^"JoThethe77;>x, y,G* the0,it#, y, z)conditionssum on- z)i-a;/2-f ensJL(<nzsin 0^ ^(v+i?)+<(e-*)ftpcos2thesatisfies therepresentation:substitution of this in (9.4.8) gives finallyA,)=^ff2*JJ;in (9.4.8): the< 0,c (, ??,for17,all'LA/++GG*(f,satisfies2)y,cos a],THE MOTION OF A SHIP IN A SEAWAYA conditionimposed on % Q (x,This conditionissatisfied ify, z) isthat #283-*(#, y, z)asx->we take L to be the path shown in Fig.+oo.9.4.1 .(P)c/cos>Fig.
9.4.1.The function g^isi2*iThe path^Lin the p-plane,Wu -Vltc=kgiven byand therefore the important quantity(9.4.9)with- -CceIf this is substituted in the9s<fp lxra 2jccos<p ltsI+Iaisgiven bya2\1+ --\t + y + StfQxequation (9.3.38) for the surgeT.we have+|(T+The last equation shows thatmust take for T the value(9.4.10)in order tokeepT = -whereigp coss1boundedfor alltweWATER WAVES284T is determined by the other time-independent term in theofmotion. Equation (9.4.10) gives the thrust necessary toequationmaintain the speed s or inversely it gives the speed s which corresIneffect,,ponds to a given thrust.
The integral in (9.4.10) is called the waveresistance integral. As one sees, it does not depend on the seaway. Theform as follows.integral can be expressed in a simplerThe function fax (x,t/,is0)asumof integrals of the type?;*,an integral of this typewe haveIfwave resistanceintegralA4say. Thissubstituted in theisthe same asis(IIJAAand we see that /=if/(,??;#,=t/)f(x,y;,Therefore,AAwhere/1"I2=d0f&eJoSince 3teI(igP C^^SOS [p( *-~ xg-spcosJLisr,)/,">cos 0] dp2zero except for the residue from the integration alongJLthe semi-circular path centered at the point_g2*Jwecos 6&cos 2 6'find from the evaluation of this residue that/ 1==f*!r*Q Jsec 8 6e^+rt "^ e cos [k(-x)cos 6] d6.THE MOTION OF A SHIP IN A SEAWAYWe285introduce MichelPs notation:Q(0)=((h x (x, y)e kv ***=h x (x, y)e kv^cos (kx sec 6) dxdyesin (kx sec 0)dxdyAand can then write-*-\This(P2+Q23)sec 0d0.the familiar formula of Michcll for the wave resistance.isThe surgeisgiven bygAHereafterwewillisno couplingThesubstitutionsuppose for simplicity that therebetween (9.3.39) and(9.3.40), so thatxhdxJL=0.of (9.4.9) in (9.3.39) therefore gives the following equation for theheave:2eJ\\AThe time independent part of yr the heave component of thewe denote by r/f; it is given by(9.4.11)gtrim,hAHere y* is the vertical displacement of the center of gravity of theship from its rest position when moving in calm water.
The integralon the right hand side of (9.4.11) is even more difficult to evaluatethan the wave resistance integral.The response to the seaway in the heave component is given byWATER WAVES286/y2gCa \\h y e[~acos2To2\I-1h+y-f|cr+s,** ~~For the case under consideration, the theory predicts that resonancein the heave occurs whengThe equationI"fLJAfor the pitching angleis(y-y'e )hdA+ \ a*hfa\JLJcosgThe time independent part of2fig[ fLJA81 ,+ L + *-t+y\dxdyewhich we denote by0*j_isgiven by(y-y'c )hdA+ [x^hdx\Q^JLJf [xh.-JAThe angle 0*!iscalled the angle of trim;of a ship whichTheit ismoves with the speedoscillatory part of theheave31the angular displacementcalm water.S Q into the seais9ff [xh y -(y-y' )h xeJJ-]cos(I8+ lo+ ^]t+y\g /268and we see that the theory predicts resonance when\)dxdyTHE MOTION OF A SHIP IN A SEAWAY287Ofcourse, the differential equations for y l and 31 permit alsosolutions of the type of free undamped oscillations of a definite fre-having the resonant frequencies just discussed) butwith arbitrary amplitudes which could be fixed by appropriate initialconditions.
This point has been discussed at length in the introductionquency(in fact,to this chapter.PARTIIICHAPTERLong Waves10.1. Introductory10Waterin ShallowRemarks and Recapitulation of the Basic EquationsThebasic theory for waves in shallow water has already been derived at length in Chapter 2 in two different ways: one derivation,along conventional lines, proceeded on the basis of assuming thepressure to be determined by the hydrostatic pressure law py) ( see Fig. 10.1.1), the other by making a formal develop&Q(ninmentpowers of a parameter a; the two theories are the same in=y-.Free Surfaceh(x)>BottomLong wavesFig.
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