given by Equation (5.6), actually is exact, as is proved in Reference 11; this first 

 approximation was also shown to provide a good approximation to the exact potential 

 for arbitrary translatory motions in an unbounded fluid of a cylinder in the shape 

 of an ogive with arbitrary thickness ratio. 



The plan of the study is as follows. The basic potential-flow problem of the 

 three-dimensional theory of flow about a ship moving with constant speed in regular 

 waves is briefly formulated in Section 2; a more detailed formulation of the problem 

 may be found elsewhere, for instance in Reference 1. The basic equations satisfied 

 by the Green function associated with the free-surface boundary condition (2.5) are 

 given in Section 3. Specifically, the Green function, G, satisfies Equations (3.3a 

 and b) or (3.4a and b) , depending upon whether the singularity is fully submerged 

 (5<0) or exactly at the mean sea surface (C=0), respectively. Equations (3.3a and b) 

 for a fully submerged source are well known. However, Equations (3.4a and b) , cor- 

 responding to a flux across the mean sea surface, are proper in the limiting case 

 when the singularity is exactly at the mean sea surface. Equations (3,3a and b) and 



(3.4a and b) generalize the corresponding equations obtained previously for the par- 



12 

 ticular cases of ship wave resistance and of wave radiation and diffraction at 



13 

 zero forward speed. Equations (3.3a and b) and (3.4a and b) are used in Section 4 



for obtaining basic integral identities satisfied by the velocity potential. The 



three classical identities (4.10a, b, and c) — valid strictly outside, inside, and on 



the ship surface, respectively — are obtained first. However, the main new result of 



Section 4 is identity (4.13). This identity is valid outside, inside, and exactly on 



the hull surface, and indeed is equivalent to the set of the three usual identities 



(4.10a, b, and c) . The integral identity (4.13) yields an integro-dif f erential 



equation for determining the velocity potential on the surface of a ship moving at 



constant speed in regular waves. This equation is examined in Section 5. Finally, 



an approach to the numerical evaluation of the iterative approximations defined in 



Section 5 is presented in Section 6. 



