In the case of Equation (15), on the other hand, the singularities are the 

 surface distribution of sources and dipoles on S, the line distribution of 

 sources and dipoles along L, and the distribution of sources on the hori- 

 zontal plane £_. Then, the corresponding Kochin function is 



H(k,6) = - 4^f (|f " ♦ 1^) exp[kz+ik(x cos + y sin 6)] dS 



+ -7 ( ^ -ik cos 9 • (J) j exp[ik(x cos 9 + y sin 9)] dy 



z=0 



4TTY r 



>(x,y) exp[ik(x cos + y sin 9)] dxdy 



(27) 



In order to distinguish the above function from the ordinary Kochin 

 function such as Equation (26) , it may be called the generalized Kochin 

 function. Keeping in mind the fact that $(x,y) decays out much faster than 

 the fluid velocity on going away from the ship, one can express the wave 

 resistance by Havelock's formula 



2 2 2 

 R = 8TTPU £ Yn 



tt/2 



-it/ 2 



|H(y n sec 2 9, 9) | sec 3 9 d9 



(28) 



Now, let us consider the first approximation for low Froude numbers. It is 



easily understood by the condition of Equation (5) that the vertical 



2 

 velocity w at the free surface is of the order of Fn , since the free 



2 

 surface elevation is 0(Fn ) by Equation (7). Therefore, the zeroth 



approximation for the velocity potential at low Froude numbers, designated 



by <j) , is obtained from the condition 



/9z =0 at z = 



(29) 



17 



