where L is the line integral in Equation (325). If the waterline of the 



ij 

 ship is slender, L.. becomes small. By neglecting L we have the reverse 



-*-J •* 



flow theorem 



T.. = T.* (331) 



This reciprocal relation was first shown by Hanaoka in the case of a thin 

 ship. Timman and Newman gave a proof to the general validity for non- 

 slender ships, with a somewhat intuitive hypothesis. 



When a train of regular waves is superimposed on the uniform flow, 

 the periodical potential is composed of the incident wave potential cj> 

 and the diffraction potential cj> if the ship is fixed in the stream. The 

 wave excitation forces and moments are given by Equation (318) if <j>. is 

 substituted by <J) + A, or 



■ ■ $ 



F 7j -p <w ir ds (332) 



If we apply Green's theorem as before and make use of the boundary 

 condition 



v-£ + -5-^ = ° (333) 



dn dn 



on S n , we can write 







F 



l--pJl(*wa^ L -35 E *J ds + L 73 (334) 



7j 



If the line integral is omitted, the result is similar to the relation 

 which was given by Haskind in the case of zero forward speed. The 

 integral on the large cylindrical surface £ is related to the Kochin 

 function for the radiation problem. Therefore, the exciting forces and 

 moments are derived from the solution of the radiation problem in still 

 water. It should be noted that the above formula does not give a con- 

 sistent expression for the wave excitation if the finite forward speed 

 is present as mentioned in the section on wave pressure on slender ships. 



117 



