1182 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



With the boundary condition that A = s — .t = at ?/ = 0, we thus 

 obtain 



A(l) = S - X = tan ^ f (1 - [To/iTo + ^)]^)* d^, (68) 



Z Jo 



where S and X are the dimensionless values of s and x at the ship. 



Next we let 



CO — s -\- X, 0} = o:/h. 



Then we have 



do} _ I dA 



dy I dy' 



and, as can be seen from (66) through (68), 



J^d) =&^X = ctn ^ f (1 - [n/(n + ^)]')~^ dl (69) 



Z Jo 



For convenience we define u and R by 



u = ^ 



R = 



To + r 



To 



(70) 

 ci(l) =foctn" r ,,/^ .. . (71) 



2 Jfl M^l — WT)^ 



1 + To' 

 In terms of m and R (68) and (69) become 



A(l) = To tan ^ f ' ^^^^^^ di^, 



2 Jr v? 



du 



\i - u^y^ 



Further, integration by parts gives 



r (1 - u')' ,^. _ (1 - m' , 7 rOj::^ , _T r__^!^ 



i« ^^ "^'^ R "^ 2 i« ^^ '^'^ 2 i« 2^H1 - ^*^)^- 



Combining the above three ecjuations and making the approximation 

 (1 - R'^f X 1, we find 



(l - fj A(l) + I tan^ ^ (1,(1) = (1 + To) tan ^ . (72) 



Thus A(l) and d)(l) are related, and we need evaluate only one of the 

 quantities numerically by means of equation (70) or (71) in order to 

 compute both A(l) and <I)(1), and hence S and A". 



