There are two driving forces responsible for second order internal waves: 

 (i) The body force fi in the fluid due to the nonlinear terms in the 

 equations of motion and the incompressibility condition. It is 

 given by the inhomogeneous part of (36) 



f (T,Z,t)=- - V;/ 



p 



,„u,.,^,„_^^^.,_,,„ 





p dzdt h Y dt ^ ^ 



(39) 



(ii) The surface force fg due to the deviations of the sea surface from 

 its mean level, given by the inhomogeneous part of (37) 



f„ (x,z = 0,0 



dt 



h-\^ 



p ">% *(i).v77r(i 



p dt 



grC 



u'^^'-Vu, 



dw' 



-Vi 



n) ^„,(i) 



-gVi^^'V-^^*'''^^^''* 



(40) 



Comparing these equations with the simple example given at the beginning, 

 the problem (9) to (14) corresponds to equation (5), and (32) to (34) corresponds 

 to (6), and (36) to (38) corresponds to (8). 



THE PRIMARY WAVE FIELD AND THE FORCES f 



AND fo 



Suppose that the primary waves_are progressive waves traveling in a direc- 

 tion given by the wave number vector k s (k. t]). We will use the notation 

 k = |T|=^2 + ^2 . A set of solutions of (32) to (34) is 



T;j<l> (7,z,t) = 



k (M 



- 1^'^V-^ ^smC.x-.„.i 



(41) 



w^^'(x,z,t) 



Saw 



(42) 



p*^' (x,z,t) 



dz 





(43) 



15 



