slender and wall-sided at the water plane. For the vertical force, we take 

 the z-component of the first equation, while for the pitching moment, the 

 y^-component of the second equation is taken. In the present case, we put 



VMU, U^,U^) (147) 



Next we define vector functions ){, |, )( , and ip , which are two-dimensional 



harmonic functions in the lower half space outside the ship and satisfy the 

 boundary conditions on the hull surface, such as 



3X W 



= m 



(148) 



tt— = n, U -r— = m 

 9n 3n - 



ax* * a$* , 



-7T— = n , U -r— = m 

 3n - 3n - 



Furthermore, these functions are assumed to satisfy the free surface 

 condition 



|i _ v <j> = atz=0 (149) 



dz 



We have expressed the near-field expression of the radiation potential for 

 a slender ship by 



^N = <j)(2D) + (1+VZ) g l (x) (150) 



Then, the vertical force is written in the form 

 (2D) 3z 



p dx i 



J C(x) 



. T dc 

 3n' 



(2D) 3<J>< 2D) (2D) 3^ 2D) (2D)) 

 C(x) I ' (cont -> 



56 



