This is the disturbance velocity potential for a double body in the 

 uniform flow. Now we write 



>0 + *l 



(30) 



and assume $. = 0(Fn ) . This assumption is not self-evident but seems to 



be legitimate when one considers the case of a vertical cylinder for which 



/3z is zero everywhere. Then, the first approximation for the free 



ur 



surface elevation is 



3x + 2 i Ux / + \ 3y 



(31) 



z=0 



The first approximation for the generalized Kochin function is given by 

 the substitution of (j> for <f> in Equation (27) 



H(k,9) = - 



1_ 



471 JJ 



3n v 3n 



exp[kz+ik(x cos + y sin 0)] dS 







AfTY„ I I 7T- - -ik cos 9*(b_ / exp[ik(x cos + v sin 0)] dy 



3 L V8X 0/ z=0 



4fTY n J J 



L 



(x,y) exp[ik(x cos + y sin 0)] dxdy 



(32) 



where 



* (x,y) 



\3x 



k + t v V v 



2 



1 3z 



z=0 



18 



