3^ = (%-^-^) l^- " U(Z -X*) ^ 



'(£) 



(103) 



while the latter is the diffraction potential (J) which has to satisfy the 

 boundary condition 



3n' 



3n' 



(104) 



Let us consider next the boundary condition at the free surface. If 

 the form of the free surface is given by the equation 



z = C(x,y,t) 



(105) 



the kinematical condition is 



K ± (^ M\ 1? /+ |i |£ _ M = 



? + ( u+ I 4 ) 



)t \ 9x/ 



3x 9y 3y 3z 



(106) 



while the condition of constant pressure is 



St 



+ »!M((-) 2+ (t) 2+ (i) 2 ) + ^° 



(107) 



Eliminating L, between the above equations, we obtain 



"i_ + ( u+ J£\ i_ + M i_ + li i_ 



3t \ 3x/ 3x 3y 3y 3z 3z 



f«^4((i) 2+ (f) 2 + (f)>- 



(108) 



This is the exact nonlinear form of the free surface condition. Let us 

 consider first the near field and examine the order of magnitude of each 

 term. We have written the velocity potential in the form 



43 



