Since the boundary condition on the free surface, Equation (9) can be 

 written as 



„ 2 Y 3z 



dX 



z=0 



H 2 fe +V * ,v ) |v *' : 



z=C 



- fV-4*- 



J \3x 2 3z 



3 2 

 3 <b , 3d),, 



dz 



we can solve for the function $(x,y) as 



»(x,y) = 



i( 2 |_ + v+.v)r 



z=c 



J„ \3x 2 3z 



+y 



3 2 <j) 

 a 2 



dz 



dz 



(16) 



The first term on the right-hand side of Equation (15) defines the sources 

 and dipoles with their axes in the direction normal to the surface and 

 distributed over the hull surface S. The second term corresponds to a line 

 distribution along the water line of sources and x-directed dipoles, and 

 the third term means the source distribution over the horizontal plane or 

 the still water plane. 



Next let us show that the same potential can be expressed by a 

 distribution of sources only. The velocity potential given by Equation 

 (15) is valid outside the hull surface S. Here let us assume a fictitious 

 velocity potential cj) which is valid inside the surface S and satisfies the 

 linearized boundary condition at z = as 



2 * 



+ Y, 



1!_ = 



3x' 



3z 



= 



(17) 



Consider a closed surface composed of S and the portion of the plane z = 

 inside S denoted by £„. Apply Green's theorem, as before, to (J) and 



11 



