Kotik and Lurye 



where y^Ct) is the heave displacement measured with respect to the position of 

 buoyant equilibrium. For three-dimensional bodies, M is the mass of the body, 

 and A^ its cross- section area in the free surface, while for two-dimensional 

 bodies M is the mass per unit length of the body and A^ its width in the free 

 surface. 



Taking the Laplace transform of Eq. (3.14), introducing the initial conditions 

 that YoCt) = y^CO) at t = 0, y^CO) = 0, and converting Fourier transforms, we 

 find for Y^(cr), the Fourier transform of y^Ct), 



- iyo(O) o-[tpp'(o-) +M] 



Y^(a) = (3.15) 



PgA^ - cr2 [rpp'(a) + M] 



Separating real and imaginary parts, we can write Eq. (3.15) in the form 



V^) = - iyo(0)[Yo''(^°'^ + iY^^a)] (3.16) 



where the primes mean that -iy^CO) has been factored out as shown. Y^'^ and 

 Y^^ have the following forms: 



cr(r/:p; + M) [pgA^ - a2(Tpp;+ M)] - cr^rV' Pd' f^.n^ 



Y'R(cr) = ■ W-i-T) 



[pgAe-o-2 (Tpp^+ M)2J +a'*T2p^Pd^ 



CTTO'^ p J gA„ 



Y'H^) -- " ^ ■ (3.18) 



[pgA^ - 0-2 (rpp; + M) 2] + a'^rV^Pd 



Since p^(a-) is an even function and PdCo") an odd function of cr, one sees 

 from Eqs. (s'.l?) and (3.18) that y;"^ is odd and y;^ is even in a. It follows that 

 upon taking the inverse Fourier transform of Eq. (3.15) we can write 



y.CO) 



y^(t) = — Y^'C'^) COS at - Y^^(cy) sin at da. 



(3.19) 



We now use Eq. (3.19) together with Eqs. (3.17) and (3.18) to infer the as- 

 ymptotic form of y^Ct) as t -co. This form depends on the behaviour of Y^V^) 

 and Y^^(a) in the neighborhood of a = o. We treat the cases of two- and three- 

 dimensional bodies separately. 



Two-Dimensional Bodies 

 In this case [5] 



2A' (3.20) 



418 



