J.J. Stoker - Water waves. The mathematical theory with applications (796980), страница 79
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(10.13.41) et seq.):+isxIfe-"*2V**= cr/Vgfe- All of thisnow- < x < 0.0,xthe<-same> 0,Z,as for the previous cases.Weto the conditions which result from the assumption that thebody is a beam.The differential equation governing small transversea beam isfloatingof(10.13.72)EEIr, xxxxoscillations+ mr, = p,ttthe modulus of elasticity, / the moment of inertia of across section of unit breadth (or, perhaps better, El is the bendinginwhichismthe mass per unit area, and p is the pressure.
Wetheofthe beam and at the same time disregard theignoreweightcontribution of the hydrostatic pressure term in p corresponding tothe equilibrium position of the beam i.e. the pressure here is that duestiffness factor),entirely to the(10.13.73)dynamics of thep= ~situation.Q&ThustInsertion of this relation in (10.13.72)and use of (10.13.69) leads atonce to the differential equation for<p(cc):(10.18.T4)Eldx*ElhLONG WAVESthatvalid under the beam.isINTTheSHALLOW WATER441case of greatest importance for usthat of a floating beam used as a breakwater leads obviously to theboundary conditions for the beam which correspond to free ends, i.e.and bending moments should vanishmean that rj xx andshouldvanishattheendsoftheandfrombeam,(10.13.67) andr] xxx(10.13.69) we thus have for q> the boundary conditionsto the conditions that the shearat the ends of the beam.
These conditions in turn-J?ax*(10.13.75)The==-atxax 5=0,x=-I.transition conditions (10.13.66) require that <p and q> x be conti0, xI, and this, in view of (10.18.70) and (10.13.nuous at x==71), requires that(10 ' 13 76)'*'f\<p(-1)=Ter*,<p m(-l)=ikTe~ ikl.remark once more that the constant B is assumed to be real, butR and T will in general be complex constants, and that the realand rj as given by (10.13.67) are to be taken at the end.parts ofIn order to solve our problem we must solve the differential equation (10.13.74) subject to the conditions (10.13.75) and (10.13.76).A count of the relations available to determine the solution should bemade: The general solution of (10.13.74) contains six arbitrary constants, and we wish to determine the constants R and T (the amplitudes of the reflected and transmitted waves) occurring in (10.13.76) once the constant B (the amplitude of the incoming wave) has beenfixed.
In all there are thus eight constants to be found, and we have in(10.18.75) and (10.13.76) eight relations to determine them. Oncethese constants have been determined, the reflection and transmissioncoefficients are known, and the deflection of the beam can be foundfrom (10.13.69). The maximum bending stresses in the beam can then= Elrf xxbe calculated from the usual formula: s = Me//, withand c the distance from the neutral axis to the extreme outer fibres ofthe beam.WethatMIn principle, therefore, the solution of the problem is straightforward.
However, the carrying out of the details in the case of the beamof finite length is very tedious, involving as it does a system of eightlinear equations for eight unknowns with complex coefficients. Inaddition, one must determine the roots of a sixth degree algebraicequation in order to find the general solution of (10.13.74). TheseWATER WAVES442roots are in general complex numbers and they involve the essentialparameters of the mechanical system. Thus it is clear that a discussion of the behavior of the system in general terms with respect toarbitrary values of the parameters of the system is not feasible, andone must turn rather to concrete cases in which most of the parametershave been given specific numerical values.
The results of some calculations of this kind, for a case proposed as a practical possibility, will begiven a little later.The case of a semi-infinite beam i.e. a beam extending from x =oois simpler to deal with in that the conditions in theto x =second line of (10.13.76) fall away, and the conditions (10.13.75) atxoo can be replaced by the requirement that q> be bounded atxoo. The number of constants to be fixed then reduces to fourinstead of eight, but the determination of the deflection of the beamstill remains a formidable problem; we shall consider this case as wellas the case of abeamof finite length.Webegin the program indicated with a discussion of the generalsolution of the differential equation (10.13.74). Since it is a lineardifferential equation with constant coefficients we proceed in thestandard fashion by settingfor K the equationx(10'.13.77)q>=*,inserting in (10.13.74), to find+ ax + b2--=witha cubic equation in x 2 = /?, which happens to be in the standardform to which the Cardan formula for the roots of a cubic appliesdirectly.
For the roots /^ of this equation one has thereforeThisis=u+vs 2v(10.13.79)H CTwith u and v defined by(10.18.80)/u=l-\andeb2+T/b 2(\4+a s \i\*I|,27//'the following cube root of unity:=/b1\2/b 23a \i\i_+_))\427//LONG WAVES IN SHALLOW WATER443The constant awave,isis positive, since or, thefrequency of the incomingso small in the cases of interest in practice thatqg is muchlarger than^ma 2 The.constant bisobviously positive. Consequentlyand negative since \u\ <\v\ and v is negative.2Thus the roots x = + /?J' 2 x 2 =are pure imaginary.
TheJ/andarequantities f} 2/? 3complex conjugates, and their square rootsoftwoyieldpairscomplex conjugatesthe rootisreal,For/9 2and/? 3we have(10.13.82)&- -(10.13.83)/?,-+ v) + i(it2= -1+(u&-)i-2(u-(u-~v),v).and ^ 3 both have positive real parts. We suppose the rootsf} 2#Hx& 4 5 x 6 to be numbered to that x 3 and x 5 are taken to have positive real parts, while x 4 and x 6 have negative real parts.
The generalThus,,solution of (10.13.74) thus(10.13.84)<p(x)is- a^i* +a<** x+ atf** x + a 4^4* + a^x + a^ x.beam covering the whole surface of the water, i.e.fromoo to + oo, the condition that <p be bounded atextending= 4 = 5 = 6 = since the exr = ^ oo would require that3ponentials in the corresponding terms have non-vanishing real parts.The remaining terms yield progressing waves traveling in opposite= 2n/\ x 2directions; their wave lengths are given by A = 2n/\ Kand thus byIn the case of ac:I\=A(10.13.85)2n/V\ u+v|,+with uv defined by (10.13.80).
The wave length and frequency arethus connected by a rather complicated relation, and, unlike the caseof waves in shallow water with no immersed bodies or constraints onthe free surface, the wave length is not independent of the frequencyand the wave phenomena are subject to dispersion.In the case of a beam extending from the origin towater surfaceisfree forx>0,oo while thethe boundedness conditions for==<patsince x 4 and x 6 have negativeaeoo requires that we take a4**would yield exponentiallyande**real parts and consequentlyunbounded contributionsto<patx=oo.We knowthatjand x 2WATER WAVES444are pure imaginary with opposite signs, with x 2 say, negative imaginary.
Since no progressing wave is assumed to come from the left, we,xmust then take a 2 = 0. Thus the termyields the transmittedwave and the terms involving a 3 and a 5 yield disturbances which dieout exponentially at oo. The conditions (10.13.70) and (10.13.71 atnow yield the following four linear equations:x =a^)=(10.13.86)for the constants a v a 3wave one- B+Rx 6 at~ik(B-R)+"a,#5,RR. For the amplitudeof the reflectedfinds(10.13.87)-1-1-1K!*3*5135-f1,/*Even in this relatively simple case of the semi-infinite beam the reflection coefficient is so complicated a function of the parameters(even though it is algebraic in them) that it seems not worthwhile towrite it down explicitly.
The results of numerical calculations basedon (10.13.87)willbe given shortly.beam/of finite length extending from <rto xthe eight conditions given by (10.13.75) and (10.13.76) mustbe satisfied by the solution (10.13.84) of the differential equationIn the case of the=and these conditions serve to determine the six constantsof integration and the amplitudes R and T of the reflected and trans(10.13.74),mitted waves.
The problem thus posedis quite straightforward butitinvolvesastedioussolving eight linear eq nations forextremelydetailsconstants.Forreference is again made to theeight complexwork of Wells [F.5].This case of a floatingbeam wassuggested to the author by J. H.LONG WAVESINSHALLOW WATER445Carr of the Hydraulics Structures Laboratory at the California Institute of Technology as one having practical possibilities; at hissuggestion calculations in specific numerical cases were carried outin order to determine the effectiveness of such a breakwater. Thereason for considering such a structure for a breakwater as ameans of creating relatively calm water between it and the shoreis the following: a structure which floats on the surface without sinking far into the water need not be subjected to large horizontal forcesand hence would not necessarily require a massive anchorage.
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