Let the surface of the ship be specified mathematically by the equation 



y = h(x,z,t) (1) 



and let the free surface be given by 



z = ?(x,y,t) (2) 



Then, the fluid motion can be expressed by the velocity potential ({)(x,y,z,t) 



cj)(x,y,z,t) = (t)^(x,y,z) + (^^ix,y,z)e (3) 



where (J), = -Ux + (|) (x,y,z), the potential due to steady forward motion, and (j) = un- 

 steady potential due to oscillation. 

 Equation (3) must satisfy the following conditions: 



1. The Laplace equation in the fluid domain: 



cf) + ({) + (j) = (4) 



XX yy zz 



2. The dynamic free-surface condition: 



4'^ + gC + ^ <^'t>l+^l+'i>l') = i U^ for z = C(x,y,t) (5) 



3. The kinematic free-surface condition: 



(J)C +({)? -(j) +^^ = forz= C(x,y,t) (6) 

 X X y y z t ■' 



4. The kinematic body condition: 



cf)h-c!)+cf)h+h^ = fory = h(x,z,t) (7) 



X X V Z Z t ^ V > J / 



