Linear Theories — Viscous 89 



Cross-differentiation of the first two equations and use of the third result 

 in the following equation: 



^ , 8f Fn . ny ^/8v 8u\ 



In the actual oceans, h is so much smaller than D that to a first degree of 

 approximation this may be rewritten as 



D ^ . ny 8v du ^ 



_^„+ysinJ^ + ---=0, (5) 



where the following definitions have been made: fi = df 1 8y; a,nd y = Fn/Rb. 

 This equation is called the vorticity equation. To the same degree of 

 approximation, the equation of continuity may be replaced by 



P+l^ = 0. (6) 



8x 8y 



A stream function yjr is introduced now by the relations: u = 8ijrl8y; and 

 v= —8ijrl8x. The vorticity equation is now rewritten in terms of the stream 

 function: 



The boundary conditions are that the shore of the ocean be a streamline 



if{0, y) = fir, y) = ir{x, 0) = ir{x, b)=0. (8) 



If / is a linear function of y, then iDjR)j3 is a constant. The general 

 solution is 



f=xy-rg)%inf. (9, 



where 



Y = ^{CjSinnjy + djCosnjy), (10) 



X = 'Z{pje^J^ + qje^J''). (11) 



The constants A, and Bj have been defined thus: 



The quantities c„ dj, Pj, q, are undetermined constants. This solution i^ very 

 general, but reduces to a simple closed form when the boundary conditions 

 are imposed. Fkst of all, the d^ and c, vanish, except Cj corresponding to 



