All-101 



90 



If wo Integrate Cl)-(3) from a depth z = - h(x,y,t) 

 where the motion is assumed negligible to the free surface 

 z = T] (x,y,t) , then 



P,v. 



.PyU = 



'^ |J2 dz + AAU + A. ^y' 



8x J az 



-h 



P n 



-h 



-|E dz + AAV + A. 1^' 

 9y J Qz 



-h 



1 

 -h 



(^) 



(5) 



^ + -^ = 

 9x Qy 



(6) 



where the non-llncar terms resulting from the interchange of 

 derivatives and into.'7;rals in the viscous terms have boon 

 neglected. 



The terms A^ -^J' 



i az 



-h 



'^x-"4 A3 II' 



= 1^.. 



-h 



provide the v/ind-stress components at the free surface (see 

 sec. 3 of report). The depth z = -h has been chosen as that 

 depth where the velocities arc negligible so that the contribu- 

 tions of the above terms at the lower limit are negligible. 



When the ^ term in the continuity equation is integrated, 

 8z -107 



it provides a contribution Involving a time-derivative, viz. 5 

 — ^ riP'^ , so that it vanishes in the present problem. 



The pressure terms are 



r ^ ^. dz 

 ] -h ^^- 



^ ^^ dz 



ey 



9 Pn p dz - -gii p^ - -gi p 



3x 



-h 



ax 



ax ^-h 



U 



-h 



A.\'^ pdz-lip -.,9hp 

 9y 1 9y n qy -h 



