Kotik and Lurye 



the coupling coefficients are determined as a function of the frequency, the 

 imaginary parts can be calculated from the Kramers- Kronig relations. 



We quote the result for a distribution of sources: 



Here the qin,(m= i, . . . N-) are the strengths of the sources at the points P-^, 

 these sources generating the approximate motion in the ith mode, while the 

 Qjj^C j = 1, ■ • • Nj) have the same significance for the jth mode. Some of the points 

 P.^ and P.^ may coincide. The function of position '^(^i^^P^O is the regular 

 part of the Green's function G(P.^, P.j^) satisfying the free surface condition. 

 Finally, p is the fluid density and cr the angular frequency of the oscillation. 



Pressure Integrals 



The most obvious way to arrive at the forces and moments on the body is to 

 use the singularity strengths to obtain the pressure distribution on the sub- 

 merged body surface and then form the appropriate pressure integrals over that 

 surface. From a computational standpoint, it is extremely important to note 

 that the integrations need not be carried out over the actual surface of the body. 

 Rather, one can express each component of force or moment as an integral or 

 combination of integrals over the plane domains defined by projecting the sub- 

 merged part of the body surface onto each of the three coordinate planes. Thus 

 only ordinary double integrals over plane regions need be computed. 



We conclude with the results of a preliminary numerical test. These re- 

 sults were obtained by applying our procedure to the case of a prolate spheroid 

 in an infinite fluid. The assumed motion of the spheroid was a small time- 

 harmonic translation in the direction of its axis (surge). The thickness-to- 

 length ratio was 1/8. For the singularity distribution, we chose a set of 45 axi- 

 ally directed dipoles located on the axis of the spheroid. Having determined the 

 dipole strengths in the manner already described, we then calculated the ampli- 

 tude of the linearized time-harmonic pressure on the surface of the spheroid. 



Our results are shown in Figs. 1 and 2. Figure 1 is a plot of the normal- 

 ized real amplitude of the time-harmonic dipole moment vs normalized axial 

 distance. The normalized real amplitude is defined as m/Mq? where a^ is the 

 real amplitude of the unnormalized dipole moment, and Mq is the amplitude of 

 the dipole moment at the center of the spheroid. The normalized axial distance 

 is x/a , where x is the distance from the center of the spheroid measured along 

 its axis and a is the half-length of the spheroid. The solid curve represents 

 the exact continuous distribution of dipole strength — this is known to be para- 

 bolic for surge in an infinite fluid — while the two broken curves represent ap- 

 proximations computed by our procedure. In both of the latter, a discrete dis- 

 tribution of 45 equally spaced axial dipoles was assumed to lie between the foci. 

 The two approximations differ in that the mean- square boundary condition in- 

 volved 48 points on the spheroid surface in the one case and 96 points in the 



422 



