All -1^1 



27 



Let us next integrate equations (12) and (13) from 

 z = - h to z = r). The pressure terms become 



-h 



p ax ^ ^ ax ^ Ap ^ ax 



(I8aa) 



r n 



, 1 9P dz =: - gD On - g i=h n i^. 

 J_^ P ay ^ ay ^ Ap ' ay 



where D = C + d/2, and the complete equations are 



! 

 -h i-^-h 



9U ., -I '^u' 9ui dz + p i 'v' P-H! dz - 2f.V sin (|) 

 at ^ J ax ^ I dy R 



(I8.b) 



= . gD 9m - g L-i . iaa +AAU + (A, 1^) 



^ ax ^ Ad ^ ax ' 3 az ' 



at 



Ap ' a 



+ P 



u 



' 9V' (J2 + p 



ax 



J -h 



J_ 



h 



-h 



v' iLl dz + 2QU sin (I) 

 ay R 



(19) 



gD 



jnp 

 8y 



-g.^h^|^.AAV.(A3|^) 



avS 



-h 



where 



U = 



n 



pu'dz, V = 



-h 



pv 'dz, 



J-h 



(20) 



p is a constant 5 average density, 



and X(x,y,z,t) ^ = \(x,y,ri,t) - X(x,y,-h,t). 



l-h 



The non-linear terms, u' (x,y , rj^t) ari/at, etc., from 



* See Appendix U(b) for the details, 



** Since the viscous terms are, in any case, only approximations 

 to the actual shear stresses, v;e have made the further approx- 

 imation 



J 



f ^ 1 A (A, M-l)dz ~ 1 f "^ -1(A, lu^l = lA, 9u 



J -h P 9z -^ sz ^ J -h az^ 3 -aFidz ^""B q- 



n 



-h 



