Linearized Potential Flow Theory for ACVs in a Seaway 



G(x,y, z,S ,7i , f ) = G(x,y, z; £,' b , £"') + ( S- O 



3G 



+ ( v -b ) 



ao 



a» 



+ (f -r) 



(€.b f r) 



^G 



u,b,n 





It is assumed that this expansion is permissible even for the singular 

 part of G . A similar expansion has been carried out for the poten- 

 tial in Appendix III for the evaluation of the pressure in the (x 1 , y', z') 

 system. As the potential is expressed in terms of G , this implied 

 an expansion of G . It may be hoped that the singular terms arising 

 from the two expansions will either cancel with each other or become 

 of a higher order than that we are concerned with at the moment. 



Now, 



. 2 



S - £ = x + ( r' + h ) e + o(e ) 



V -V' = 5 h 



(£,£"') in the case 



of S 



and 



f _ r= i - Z 6+ 0(0 ) 



£6 



and as — — has ^ as a factor and we are only concerned with poten- 

 tials of (5 , /3, 8a, /3a) we may use only the (a) term in the expansion 

 for G 



G(x, y, z; £, rj , f ) = G(x, y, z; £; b, f ) + ae 

 so that 



lot 



dG 



>+ r' 



ac 



ooi a $ ooi a f 



G^ (x,y, z; S, i; , f) = G^ (x, y, z; *', b ,f') + 



+ a e 



icrt 



r ooi 3£'cV 



where we have written 



a 2 G 



tj =b 



2 35 



