where a„ is the two-dimensional source strength located at the coordinate origin. 

 The terms under the summation are the higher order multipoles which form the wave- 



free potentials with multipole strengths a 



(0) 



2D 



is the two-dimensional Green 

 10 



function due to a source at the origin and is given by Wehausen and Laitone as 



(0) 



2D 



1 r>™ e Z cos(ky) . Kz ,„ . 

 = - PV — R _ k — *— dk - x e cos(Ky) 



(15) 



where K = co /g, and PV in Equation (15) denotes a principal value integral. The 

 solution of <J>„ by the Frank close-fit method is 



G 2D dl 



(16) 



where a is the source strength distributed on the ship's contour and G„ is a two- 

 dimensional Green function due to a unit source on the ship's contour. G_ , in this 

 case, is slightly different from Equation (15) and is given by 



G 2D =Re 2^ 



log(y+iz-n-iO-log(y+iz-n+i<;) 



+ 2 PV 



C -ik(y+iz-n+iO 

 e 



K-k 



dk 



- i Re(e- iK( ^ +iz - ri+U) ) 



+ Re 



2tt 



log(y+iz+n-i£)-log(y+iz+ri+ic;) 



+ 2 PV 



a 



s 



-iMy+iz+n+i?) 

 K-k 



dk 



i Re( e- iK( y +iz+ri+i ^) 



(17) 



