If a harmonic solution of the equations of motion, Equations [U6a,b], 

 exists when the stream speed is S, its circular frequency being o), then u) 



and S must solve Equations [53c, d], or, if D ^ 0, Equations [53a,b]. 



2 

 Conversely, any solution of these equations m which w > leads to a 



harmonic motion. 



The problem of finding such solutions is complicated by the fact that 



the parameters a , a , a , a , and D all vary with oj. In numerical 

 11 12 21 22 



computation, the only feasible procedure seems to be to assume successive 



2 

 values of 0) , to calculate the paramaters for each value and then to 



attempt to find a value of S that satisfies both equations. Perhaps the 



work may be facilitated by rewriting the equations as quadratics in S; 



perhaps for Equations [53a, b], with all coefficients reversed in sign for 



convenience, the following will result: 



bS^+bS+b =0,eS +eS + e =0 [5ha,^] 



12 3 12 3 



where 



b = B [d"""" (a k k + Ls k ) + co^ (hm - Lm + LC + E^ ) ] 

 1 21 1 2 2 1 L 



2 



b2 = u (^i-^L '*' ^2^^ 



b3 = 



"-'■k,kp + uj |d [ms,k2 + loS^k-, 



+ hm (a^2 "^ ^21^ ^2,^2'^'^ '^1^2f "^o" 



e = c LB 

 1 1 



e- = - BD kp (s-, + La-ipk-, ) + CD ap-^k-j^kp 



- E D'-'-s k +(/[{! - hLm) B + hmC + mE, ] 

 L 2 1 e I. 



52 



