All-101 



2h 



^ . u^ ^^ . v- i^ . ,,,. .,. (J,= 



ay 

 9n^ 



2 



R 



9>1. 



- g[b — ^ + a --il ] + 1 (V . A.V)u' 



ax ax ^ „ 1 ' 2 



(8) 



— ^ + u, -^ 



et 



2 a 



, av^ 



X 



+ Vp — ^ + 2Qao sin(-^) = 

 2 ay 



I 



^2 



.y, 



■R' 



an^ arin n , 



- g[b —.Li + a ~li ] + 1 (V • A.V)v' 



ay 



I 



ay 

 t 



I 



auo avo av/o 

 ax ay az 



(9) 



(10) 



where a = p 2_/p 2 , b =( p 2 - P]). /p 2 = A p /p . 



The problem defined by equations ( ^) - (10) v;ith appro- 

 priate boundary conditions is quite general in that no assumption 

 has been made concerning the vertical distribution of velocity. 

 As we shall see later, when the equations are integrated over z 

 and linearized, the simplified problem is still too difficult 

 to solve. For this reason we formulate a second problem v/hich 

 allows a more general density distribution but which is more 

 restricted in other respects. 



In this prololem we retain, for the time being, the gen- 

 eral form p =: p[z - T(x,y,t)], Then the pressure terms in 

 equations (1) and (2) are* 



1 M = E. 



p ax p 



1 _ap ^ g 



p ay P 



i£ dz +1 In po 

 2 ax p ax *^° 



nT 



9£ dz + ^ an 

 ay p ay t'o 



(11. a) 

 (ll.b) 



* For the present problem the plane z = lies on the undisturb- 

 ed equilibriiim free surface. 



