surface is not convenient to finding the solution, let us perform the 

 following transformation to the vertical coordinate. 



z = C Q + z' (55) 



After the transformation of the Laplace equation to new coordinates and 

 taking only the first order terms, we find that the Laplace equation does 

 not change, namely 



2 2 2 

 9 <j> 3 <j) 3 <J> 



+ ^e + k = (56) 



2 2 2 

 9x 9y 9z 



while the boundary condition at the free surface is 



2 2 2 



2 d *1 8 *1 2 S *1 9 *1 



l o 7T + 2 u o v o ^97 + v o tt + Y o ir- = Y o D(x *y ) (57) 



9x J 9y 



at z' = 0. The homogeneous equation, which is obtained by setting the 

 right-hand side equal to zero, is the boundary condition of free waves on 

 a nonuniform flow field. 



2 2 2 



9 (j) 1 9 <|>, „ 9 <j>, 3c|>, 



+ 2 u n v n -^ + < —^ + Y„ ^T = (58) 



„ 2 v 9x9y v . 2 *0 3z' 

 3x J 9y 



Now let us assume the wave potential 



^ = A exp Y [k(x,y)z'+i S(x,y)] (59) 



On substituting in the Laplace equation and taking terms of the lowest 



-] 



( 



order with respect to y n , we have the relation 



27 



