The potential iJj(C) is known since 9(})/8n is given on the ship surface h. The poten- 

 tial L (^;(f)), on the other hand, is evidently not known. 



An approximate solution of the integro-dif f erential Equation (5.1) may be 

 obtained by seeking a solution of Equation (5.1) of the form (t)(^) = k(t)ijj(t), where 



->- . -^ 



the function k(C) - 't>(.0/^(.0 is assumed to be slowly varying. Specifically, by 

 adding the term k(OL^(?;ip) to both sides of Equation (5.1) and multiplying the 

 resulting equation by ii(^) , we may obtain 



tt)(t){[l-w(|)]4;(|)+L^(t;iP)} = ilj^it) + (J)(t)L'(|;i|j) - i>(X)l^' (t'A) (5.5) 



If the potential (p were actually proportional to the potential ^, the term 

 <)>(S)L (C;'!') - ^KOl (C;<)>) would vanish, and the modified integro-dif f erential Equa- 

 tion (5.5) would yield the solution 



*(!) = 4^M)/{[l-w(t)]t|;(|) + L^it-A)} (5.6) 



More generally, the above expression for the potential can be regarded as the 



-. ..., _. . ..J, (n) . , 



first approximation in the sequence of iterative approximations <p associated 



with the recurrence relation 



<,,(n+l)(|) ^ ^(X)&\t)/ill-^(t)]&\t) + h^it-A'-''^)} for n > (5.7) 



and the initial (zeroth) approximation (}) (?) = i)(K) ■ An approach to the numerical 

 evaluation of these iterative approximations is presented in the following section. 



21 



