All-101 29 



Where w ' | is negligible by definition of h(x,y,t). Substitut- 



-h 

 ing the free surface condition, we have 



9U . qV ^ _ ln£ . (21) 



ax ay at 



Equations (19) and (20) are now further simplified by 

 neglecting the non-linear terras. The reader is referred to 

 Appendix 2 for a detailed plausibility argument concerning this 

 step * 



Two final simplifications will be made in equations (19) 



and (20). The Corlolis parameter 2 Q sin(^) will be linearized 



R 



by writing 2 ."2sin(--) ~ By where p = 2Q/R. 



R 



In addition, if the velocities are found in some manner, 

 then the free surface shape can be obtained by integrating the 

 equations (19) and (20) (neglecting the integrals of the non- 

 linear terms) v/ith respect to x and y respectivelyo This yields 



Where X denotes a known function. The solution of this quad- 

 ratic equation in r\ is 



- D + D Jl + — -^^ i JL 

 Ap 



* It must be emphasized that the argument presented in Appendix 

 2 is one of plausibility and not one of justification. In 

 view of the desirability of obtaining an analytic solution, 

 we neglect the non-linear terms in the hope that the results 

 will agree qualitatively with observation and will so furnish 

 a mathematical description of the ocean circulation. 



