expressed by means of this Green's function. This can be achieved by the 

 application of Green's theorem in similar fashion to the problem of steady 

 forward motion discussed in the former section. Because of the slender 

 body assumption, the first order solution takes the form in which the 

 singularities on the hull surface shrink to a line distribution of wave 

 sources along the x-axis. Then the far field potential is expressed by 



= - e 1Wt Im(x') G(x,y,z;x',0,0) dx' (124) 



where m(x) is the source density which is determined by the matching 

 procedure between the near field and far field solutions. Thanks to the 

 simple condition 3(f>/8z = at z = 0, the source density can be determined 

 in a simple way such as 



m ( x > " " h ( iaH " U Ix~) [B(x) ( z S-x^-C w )] (125) 



where B(x) is the width at the water plane at each transverse section of 

 the ship. The unknown function g, (x) is determined by the inner expansion 

 of the far field potential and the final result becomes 



-J' 



: ,| dm(x') sgn(x _ x .) £ n(2 |x-x'|) dx' 



- m(x') G'(x,0,0;x',0,0) dx' (126) 



where 



G'(x,y,z;x',y',z') = G(x,y,z;x' ,y' ,z' ) -^-~ (127) 



1 2 



We have assumed that m(x) vanishes at both ends of the ship. One can 

 formulate the forces and moments acting on the ship by the integration of 



48 



