where G„ n is given by Equation (15). The two-dimensional source strength a in 

 Equation (24) is distributed on the ship's contour (Figure 2), and 0-. is a concen- 

 trated source strength at the origin. In other words, a, is the solution of the 

 Frank close-fit method and o.. is the solution of the multipole expansion method. 

 The relation between O, and a 7 can be expressed by 



-.-I 



| o d e K? cos Kn d£ (36) 



c 



The same relation between a. and a is given by 



J- 



a Q = 2 I a e K? cos Kri d£ (37) 



The total unsteady potential <j) for heave and pitch motion is given by Equations 

 (23), (25), and (32): 



<j> 2 = <P + <P 7 + ? 3 <fi 3 + 5 5 V 5 (38) 



where 5n is the heave amplitude and £,. is the pitch amplitude. In the frame of the 

 unified slender-body theory, only heave and pitch motions are considered. The 

 pressure p in the fluid is given by Bernoulli's equation 



p = pUox^-VcjyVc^) e" ia)t - pgC (39) 



By substituting Equation (38) into Equation (39) and integrating on the body sur- 

 face, the hydrodynamic forces are expressed by 



f. = - plj (icon. 4m.) (^ 4^ 7 +C 3 ^ 3 +C 5 ^ 5 ) dS e~ i6Jt (j = 3,5) (40) 



where n, and m s are given by 



11 



