with 



(2)^^^Ai)^ _-;:(l).V^<l>at<=0 (29) 



dt dz 



,(2)_^(2)^^^(l)^_i_l_|ate = (30) 



dz dz 2 dz" 



w^2> = at z = H (31) 



The 0th order equations characterize the mean stable conditions; the 1st 

 order ones describe linear internal and surface waves in that fluid; and the 2nd 

 order equations give nonlinear effects of these waves, including energy transfers 

 to other frequency bands. It is this set of equations which leads to energy flux 

 from surface waves to internal waves. 



THE EQUATIONS FOR w[Zz,\\ 



We intend to eliminate u, v, p, and p in the left-hand sides of the 1st and 

 2nd order equations in order to get equations for w. The procedure is well- 

 known. 13 For the 1st order equations we get 



where 



1 dp 



riz) = - — 



p dz 



Combining the boundary conditions (23) and (24) together with the horizontal 

 components of (20) and the continuity equation (22), we find 



3,„(1) 



d^w 



gV 2a'<l> = 0at2 = (33) 



dt-dz 



and (25) remains the same 



«'<^' = Oatz = H (34) 



Internal and suiface waves of the first order are governed by equation (32) 

 with boundary conditions (33) and (34). The problem has been solved analytically 



13 



