Ogilvie 



term may be considered as a correction for two effects: (1) The steady velocity 

 potential satisfies a condition on the wrong surface, viz., on the undisplaced hull 

 surface. (2) Rotational displacements of the ship interact with the steady flow 

 to produce an additional cross-flow. Both of these effects yield contributions of 

 the same order of magnitude as the desired perturbation effects. 



Now we state that a solution can be written in the form (cf. Eq. (7)): 



6 6 



(D(x,t) = -Vxi + cp„(x) + ]2 °-k(^) ^ik^^'^) + E ''kCt) ^2k(^) 



6 /-t 6 /-t 



E YikCx, t-T) i^^Cr) dr + ^ VjkCx, t-T) a^(r)dT, (11) 



k = l-'-a> k=l''-oo 



where the new unknown functions, ^j^Cx), v ^^(x, t) satisfy: 



>Ajk = , on X3 = ; 



301^ fa • ik • k = 1, 2, 3 , 



3n 



■ ik-3 '^ " • k = 4, 5, 6 , 



90,. f^-^^ [i '^^o^'')] • k = 1, 2. 3. 



Bn 



n • Vx [(1^.3 XX) X v^(x)] , k = 4, 5, 6 , 



on S ; 



on S„ ; 



^^-2V^^^V^-^^g^ = 0. onx3 = 0; 



^— = , on S ; 



an o 



= 0. 



This solution is quite analogous to the zero-speed solution. In fact, if v = 0, the 

 functions si'n.Cx) and Vi^(x,t) reduce to the corresponding fvinctions introduced 

 previously and the functions ^jk^^'S^ ^^^ ><'2k(^'i' ^^ become identically zero. It has 

 been necessary here to introduce an extra double set of functions to take care of 

 the Timman-Newman boundary condition correction. It can be checked straight- 

 forwardly that this solution does satisfy (10a) and (10b). (In addition, there 

 shoxild be further conditions at infinity — which the solution is assumed to 

 satisfy.) 



30 



