plus a constant, where (j)(x,y,z,t) is the velocity potential, whose 

 gradient is the fluid velocity vector. This potential must satisfy the 



Laplace equation 



2 2 2 



3 (fi 9 (j) 3 (f) 



V(|) = + ^+-— =0 [10] 



2 2^2 ■■ -" 



dx^ 3y 3z 



throughout the fluid; the linearized free surface of the boundary condition 



d^(i> 3<t) 



■~;+g:^=0 onz = h [11] 



3t^ ^^ 



where z = h is the plane of the free surface, and the kinematic boundary 

 condition 



3(|> 



— = V [12] 



3n n 



on the body, where V is the normal velocity of the body surface. For a 

 slender body of revolution, the surge contribution to the normal velocity 

 will be small, compared to the pitch contribution, and 



V = - e X cos(n,z) [13] 



n 



The potential i> will consist of an incident wave potential ()>. ; and a body 

 potential <\>,, due to the presence of the body. For plane progressive 

 waves of circular frequency w. 



g A 



exp iK(z-h) + i K X cos 3 + i Ky sin g - iwt 1 [14] 



2 

 where the real part is understood. Here A is the wave amplitude; K = w /g 



is the wave number; and g is the angle of incidence (3=0 for head waves) . 



Combining Equations [12] to [14] , the body potential must satisfy 



^— = - e X cos(n,z) - — - [is; 



9n 9n 



