There is a condition of constant pressure which is expressed by the 

 Bernoulli equation. 



\ (u 2 +v 2 +w 2 -l) + Yn ? = (6) 



Then the free surface elevation is given by 



1 2 2 2 

 C = ^ (1-u -v -w Z ) (7) 



Keeping in mind the fact that the boundary conditions are satisfied at the 

 curved surface z = Z, and that ? is a function of x and y, the explicit £ 

 in Equations (5) and (6) is eliminated by the substitution of partial 

 derivatives of Equation (7) with respect to x and y in Equation (5). The 

 result is 



H u h + v i + w h) (- 2+ v 2+ - 2 ) + y w ■ ° ^ 



If we substitute the velocities by the expression of Equation (3) , the 

 above equation becomes 



l( 2 |_ +v ,. v )|V„2 + f| +Yo |i =0 (9) 



dx 



This relation holds on the unknown surface z = £, so that the boundary 

 condition is quite nonlinear. The usual way of solution is the perturba- 

 tion expansion of the boundary condition assuming a small parameter which 

 relates to the shape of the hull surface. The first term of the expansion 

 is the linearized solution. In order to make such a linearized solution 

 valid, a condition of small disturbance is necessary which imposes a 

 restriction on the hull shape. Instead of the application of the perturba- 

 tion expansion to the boundary condition, let us seek a general expression 



