384 FOFONOFF [sect. 3 



the angular velocity of the wave motion. The real parts of k and I are the wave 

 numbers. 



Substituting for the variables in (226) and (227), we obtain for the baro- 

 tropic mode 



iojtiB^-fvB^ = -ik{gor,^ + g'-ni^) (232) 



ftiB^ + icoVB^' = -iligov^' + G'Vi^), (^33) 



where the zero superscript denotes the coefficient of e*'<'"'+*-^+'^) for each variable. 



We have assumed that rj^ and rji^ are independent of y and x. Solving for the 



velocity components, wc obtain 



UB^ = ^^^^A9ov' + U'Vi') (234) 



^BO = ^^^^(^o^o + f7'^»0). (235) 



Similar expressions can be derived for the baroclinic velocity components. 



From (234) and (235), we can see that the motion in a wave is completely 

 specified by the momentum equations except for a relation between angular 

 velocity and the wave numbers. If we assume that rji^ is zero and cd is real and 

 represent the vertical velocity, ivb^, at the surface by ioj-q^, we note that the 

 coefficients satisfy the relation 



luB^-kvB^ + ^^^P^ ojb' = 0. (236) 



Assuming further that k and I are real, we see that the particle motion lies in 

 a plane perpendicular to the vector [I, - k, gof{k^ + l")laj{f" - co^)].! This vector 

 is perpendicular to the direction of travel [parallel to { — k, — I, 0)] of the wave 

 and inclined at an angle, y, to the horizontal plane, where 



tan y = — -p; — ■ (237) 



At high frequencies, co^f, the angle y approaches zero and the particle motion 

 is in a nearly vertical plane. At low frequencies, co<tf, and at co=f, the motion 

 is almost entirely in a horizontal plane. The particle motion in the plane forms 

 an elliptical circuit in the general case. 



A relation between the angular velocity, to, and the wave numbers, k and /, 

 is obtained by substituting (234) and (235) for the velocity components in the 

 continuity equations. The substitution yields the simultaneous equations 



\aj + {goH + g'Hi)Z]r^o + g'Ho^Z7^tO = (238) 



goH2Z-qO + l^^ + g'HoZ]rjiO = 0, (239) 



where 



Z 



k-^ + r^ mf~ + ^~)^ -W/ 



P-CO-^ ^(/2_^2)2' (/2_^2)2_ 



(240) 



1 This is seen more readily if we interpret {'I'MS) as tlie scalar product of the vectoi- 

 with the velocity vector at the ocean surface. 



