^^"^ = (D. + 



1 1 



(47) 



On the body surface, (}). and <p . satisfy the following conditions 



p , = - lOJn . 



(48) 



and 



4>. = m. 



(49) 



In Equation (46) (}) is the conjugate of (j) . . The solution of (})„ for heave motion can 

 be expressed by 



^3 = ^3Sd + 



m=l 



co s 2m9 

 ^m 



K cos (2m-l)e 



2m- 1 



2m- 1 



(50) 



where G is defined by Equation (29), O- is the two-dimensional source strength at 

 the origin, and the terms under the summation are the higher order multipoles which 

 form the wave-free potentials. Equation (50) was first introduced by Ursell. A 

 similar solution for c})„ can be given by 



S = ^sSd + 



m=l 



cos 2m9 

 2m 



K cos (2m-l)( 



2m- 1 



2m- 1 



(51) 



For pitch motion we can have similar expressions to Equations (50) and (51) by 



substituting a„, a , O^, and a for -xa„, -xa , -xa„, and -xa , respectively. From 

 ^ 3' m 3' m 3' m 3 m' '^ ^ 



Equation (50) the outer approximation of the potential (}) . is 



^3 = "j Sd^^'^) 



(52) 



12 



