Figure 1 - Coordinate System Fixed with Respect to 



the Ship. This System Moves with a Speed U 



Relative to a System Fixed in Space 



region, the three-dimensional Laplace equation is solved under a three-dimensional 

 free-surface condition and an imposed radiation condition. The inner solution is 

 governed by the two-dimensional Laplace equation with two-dimensional boundary 

 conditions. These two solutions are matched in the overlapping domain or inter- 

 mediate region to determine the solution of the unified slender-body theory. 



The governing equations of velocity potentials, boundary conditions, and their 

 solutions will now be given. The details of the derivations are given in References 

 4 and 6 . 



POTENTIAL OF STEADY FORWARD MOTION 



The velocity potential of steady forward motion can be expressed as 



(x,y,z) = - U x + <j) (x,y,z) 



(1) 



where (J) (x,y,z) satisfies Laplace's equation and the free surface and body boundary 

 conditions given by 



|> + <J> + <J> =0 z<0 



oxx oyy ozz 



(2) 



(U /g) cj) +0 =0 z = 



oxx oz 



(3) 



