J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 78
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(It isnecessary also to permit <p(x) and v(x) to be complex-valued functions of the real variable x.) The conditions (10.13.32) and (10.13.33)now becomed 2cp(10.13.37)I +(10.13.38)Tl* +The equationa2=.1*1>a1*1<aT^'(10.13.37) has as general solution(10.13.39)=y(x)Ae~ ikx+Be ik*with k given byFor 0(x,=k(10.13.40)we havet)&(x,(10.13.41)a/Vgh.therefore= Ae-Wx - a +t)Be'l***"*,the first term representing a progressing wave moving to the right,the second a wave moving to the left. In our problems we prescribethe incoming wave from the right, and hence for(10.13.42)in<p(x)Bwhichis=Be ikx|R(10.13.43)ToRe~ tkxx,wewrite> a,the amplitude of the reflectedamplitude) is to be determined.prescribed, while .Kwave (more precisely,At the left we writewith+cp(x)\is its<p(x)=Te ikx,Tthe amplitude of the transmitted wavecomplete the formulation of the problem itto be determined.isnecessary to con-dynamics of our floating rigid body.
We shall treat two cases:theis held rigidly fixed in a horizontal position, b) the boardboarda)sider thefloats freely in the water.a) RigidlyFixed Board.=0, and hence (cf.rigidly fixed we have 77(0?, t)(10.13.36)) v(x) =0. It follows from (10.13.88) that <p xx vanishesIf theboardisLONG WAVES IN SHALLOW WATERidenticallyunder the board and hence that(10.13.44)Since<p(x)= yx + d,a433a linear function:<p(x) is< x < + a.furnishes the horizontal velocity component of thefollowsfrom (10.13.44) and (10.13.35) that the velocity underwater,the board is given by ye iat , i.e.
it is constant everywhere under theboard at each instant a not unexpected result.x (z, t)itWe now(10.13.34),downwritemaking useand (10.13.43) at xthe transition conditions at x=of (10.13.35) and of (10.13.42) at xa; the result is:= -Be tkaBe zka= ya + 6= y/ffc= _ ya=Re~ lkaRe~ ika+-(10.13.45)from= +aTe -ika_|Te -ikaOnce the real number B which fixes the amplitude of the incomingwave has been prescribed, these four equations serve to fix the constants R, T, y, and 6 and hence the functions 0(x, t) and rj(x, t). Thepressure under the board can then be determined(10.13.46)p(x,/.)= --Q&t=(10.13.47)(cf..in the present case.)2a/A,the ratio of the length of the board to thebythe expressioniatQia(p(x)e(Observe that the quantity y in (10.13.4) is zeroIn terms of the dimensionless parametersurface, given(cf.from(10.13.4) for Bernoulli's law)wave length A on thefree(10.13.41))A(10.13.48)the solution of (10.13.45)=2^/fc,isR=Oni+1(10.13.49)7dThereflection_dni07i-la=Beeni.and transmissioncoefficients are obtained at once:WATER WAVES434CrQn=(10.13.50)ViThey depend only upon theratio=+2a/A, asone would expect.
They+also satisfy the relation C*1, as they should: this is an exCfpression of the fact that the incoming and outgoing energies are thesame. The following table gives a few specific values for these coefficients:0.51.02.0Thus a fixed board whose length is half the incoming wave length hasthe effect of reducing the amplitude behind it by about 50 percent andof reflecting about 72 percent of the incoming energy. One should,however, remember that the theory is only for long waves in shallowit seems likely that the length of the boardaroleindetermining the accuracy of the approximation.playThis question has been investigated by Wells [W.10] by deriving theshallow water theory in such a way as to include all terms of third orderin the depth h and studying the magnitude of the neglected terms inwater, and, in addition,will alsospecial cases; in particular, the present case of a floating rigid bodyisinvestigated.
Wells finds that if A/A is small and if a/A (the ratio ofthe half-length of the board to the wave length) is not smaller than 1,the neglected higher order terms are indeed negligible, but if a/A is lessthe higher order terms need not be small. In other words,floating obstacles ought to have lengths of the order of the wavelength of the incoming waves if the shallow water theory to lowestthan1,order in hItisThisisisexpected to furnish a good approximation.of interest to study the pressure variation under the board.given in the present case (cf. (10.13.46)) by(10.13.51)(x, t)=dotd)ethe real part only to be taken.
Thus the pressure varies linearly in x>but it is a different linear function at different times since y and d areLONG WAVES IN SHALLOW WATERcomplex constants. Theresult of taking the real part of the rightside of (10.13.51) can be readily put in the(10.13.52)p(x,t)435= Pi(x) cos athandformp 2 (x)sin atwithPi(x)(10.13.53)p 2 (x)= agBfiifa) sin r + b (x) cos r)= agB(b (x) sin r b^x) cos r)22and(10.13.54)-i-55board.Fig. 10.13.11. Pressure variations for a stationary* /0i=1,pin poundsl(ft) 2WATER WAVES436We have assumed inmaking thesecalculations, as stated above, thatB.
which represents the amplitude of the incoming wave,isa realnumber.In Fig. 10.13.11 the results of computations for the pressure distribution for time intervals of 1/4 cycle over the full period are given fora special numerical case in which the parameter 6 has the value = 1,i.e. the length of the board is the same as the wave length. One ob-serves that the pressure variation is greater at the right end than atthe left, which is not surprising since the board has a damping effect onthe waves.One observesatmospheric(i.e. it isalso that the pressurenegative at times, whileissometimespislessthanthe assumedpressure at the free surface).b) Freely Floating Board.In Fig. 10.13.12 the notation for the present case is indicated:u(t), v(t) represent the coordinates of the center of gravity of theboard in the displaced position, andthe angular displacement.As before, we consider only simple harmonic oscillations and thustake u v, and CD in the formco(t)9(10.13.55)u= xeiQv\Fig.
10.13.12.A= yeiat,CDfreely floating= we*,boardin which x, y, and w arc constants representing the complex amplitudes of these components of the oscillation. For rj(x, t) we have,therefore(10.13.56)when termsri(x 9 1)==[y+ (x - x)w]e+ xw)eiatiat(yof first order in x, y, andw only are considered.(The hori-component of the oscillation is thus seen to yield only a secondorder effect.) The relation (10.13.56) now yields (cf. (10.13.38)):zontal(10.13.57)<p xx=-^(y +LONG WAVES IN SHALLOW WATEEinwhich (piat(p(x)e.is487the complex amplitude of thevelocity potential 0(x,is the<pfollowing cubic polynomial:t)=Hence(10.13.58)<p(x)Since the pressurepresent case(10.13.59)=isp(x)!given by p=[-icre0>(0)=-g0we havegg^f+ xw)]ex =nowiatQg(y.The transition conditions (10.13.34) atflsame way as above from (10.13.42) and (10.13.43),Be ika+in thelead, in theto the equationsRe~ n(10.13.60)oiTe~ tkatffei=*\ya\9(*A\ 2J4kV.YThese four equations are not sufficient to determine the six constantsff, T, w, y< y, and 6.
We must make use of the dynamical equations ofmotion of the floatingrigidbodyfor this purpose.Wehave theequations of motion:F-(10.13.61)at our disposal. In the first equationforceon the board andacceleration ofitsitsLandAlv,Fand= /aM are the totalmass, per unit width, and vcenter of gravity, / themomentisverticalthe verticalof inertia,Ltheand a the angular acceleration. These dynamical conditionsthen yield the following relations:torque,p dx(10.13.62)J-oand these=px dxMv,= lib,J-ain turn lead to the equations1"J_a0.13.63)--!.WATER WAVES488In the first equation we have ignored the weight of the board, sinceit is balanced by the hydrostatic pressure. The equations (10.13.60)and (10.13.63) now determine all of the unknown complex amplitudes.Weomit the details of the calculations, which can be found in thepaper by Fleishman [F.5], In Fig.
10.13.13 the results of calculationsfor the pressure distribution in a numerical case are given. The parameters were chosen as follows:= 1,M = 18.72Ith=1B=ft,apounds/ft,=ft*/sec,a4.46 rad/sec,A1might be added that the value chosenture sinksdown0.0375 feetwhenforMis=4=8ft,ft.such that the struc-in equilibrium.A few observations should be made. First of all, we note that in both= +cases the pressure variation at the right end (xa), where theincoming wave is incident, is greater than at the left end.
This is tobe expected, since the barrier exercises a damping effect on the wavegoing underThe pressuredistribution in the case of the floatingboard is quadratic in #, in contrast with the case of the fixed board inwhich the distribution of pressure was linear in x. Next, we note thatit.the pressure variation near the right end of the stationary board isgreater than at the same end of the floating one; this too might beexpected since the fixed board receives the full impact of the incidentwave, while the floating one yields somewhat. Finally, we see that atend the oppositeeffect occurs: there the pressure variationunder the stationary barrier is less than that under the floating barrier.This is not surprising either, since the fixed board should damp thetheleftwave more successfully than the movable board.Finally, we take up the case of a floating elastic beam (cf.
[F.5] ).The beam is assumed to extend from x ~ltox = Q and, as in theabovecases, tobe in simple harmonic motion due to an incomingwave from x = + oo. The basic relations for <(#, t) on the freeface, and for rf(x, t) under the beam are the same as before:(10.13.64)XX=-1xtt9>0,x<-sur-I,gh(10.13.65)ij t= - h0 xx,- < x < 0./We assume once more that the beam sinks very little below the watersurfacewhenin equilibrium(i.e.verylittlein relation to thedepth ofLONG WAVES IN SHALLOW WATER439p-200<T 1=IT/4-200500ix/o-5-500P4002002005.5-1-55x/aP400200200Fig. 10.13.13. Pressure variations for floating board.*'J6=1,pinpounds I (ft)*WATER WAVES440the water), so that the coefficient ofh rather than(h+r))(cf.(10.13.66)Xandand(10.13.15)),beamtions at the ends of the<& xx in (10.13.65)arecontinuous at x<P tcan be taken asalso the transition condi-=x0,=/.After writing(10.13.67)we<P(x, t)= (p(x)eiat17(0, /)9=V(X)CMfind, as before:2a-= 0,<p xx H(10.13.68)x<p> 0,x<~Igh(10.13.69)<p xx+The conditions at(10.13.70)A:oo<p(x)(10.13.71)withturn^h v =<p(x)Ihave the=-Be ik*effect that (cf.
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