Kotik and Lurye 



y„(t) ^ ^y.(O) 



(3.29) 



g^t^ 



A comparison of this expression with the corresponding one for cylindrical 

 bodies, Eq. (3.26), shows that: 



1. the approach to buoyant equilibrium in three dimensions is asymptotically 

 faster than in two dimensions by a factor proportional to l/t^, and 



2. the approach to equilibrium in three dimensions is asymptotically from 

 the side of the equilibrium position defined by the initial displacement; in two 

 dimensions the approach is from the side opposite the initial displacement. 



It is our intention to present, in a future publication, calculations of tran- 

 sient forces and displacements using Hi-Fi approximations-^' to P^Co-). 



4. NUMERICAL DETERMINATION OF HYDRODYNAMIC 

 COUPLING COEFFICIENTS FROM VOLUMETRIC 

 SINGULARITY DISTRIBUTIONS 



In this section we outline briefly a numerical scheme for calculating the 

 hydrodynamic coupling coefficients H. ^ (already defined in Sections 2, 3, and 4) 

 for a fully or partially submerged body engaging in small time harmonic oscil- 

 lations. Our computer program so far covers only the zero speed case, but its 

 extension to forward speed should present no difficulty in principle; the chief 

 additional complication would center on the calculation of the time-harmonic 

 Green's function for a point source in a steady stream below a free surface. 



The idea of the method is to approximate the velocity potential exterior to 

 the oscillating body by the potential of a time-harmonic finite set of singularities 

 contained in the interior of the body surface. These singularities will usually 

 be either sources or dipoles although higher order multipoles can also be used. 

 The strengths of the singularities are determined by the requirement that the 

 normal velocity they induce on the submerged portion of the undisplaced body 

 surface, s^, should best approximate the actual normal velocity of s^ in a cer- 

 tain mean square sense. t Specifically, let P^^Cm = l, . . . M) be the points where 

 the M singularities of complex strength q^ are located interior to S^, and let 

 P"(n = 1, . . . N) be a set of points on S^ with N >M. Let a^^ be the complex 

 normal velocity at P" due to a singularity of unit strength at P^, the singularity 

 potential satisfying the linearized free surface condition. Finally let V" by the 

 actual complex normal velocity of s^ at P" due to the oscillation. Then we 

 seek to determine the q so as to minimize the mean square expression 



IN 



V" 



E 



(4.1) 



'■'Examples are given in [5]. 



THowever, we plan to consider other types of approximation as well. 



420 



