Paldoussis 



Re{«) = O 



4 6 



Re(a>) 



Fig. 10 The dimensionless complex frequencies of the zeroth and 

 first modes of a flexible cylinder with f i = 1 - C| = 1 , 

 f2=l-c2=l, ecM=ec^=l, A=l, X| = X2 = O'O^' ^s 



a function of the dimensionless towing speed 

 theory). 



(New 



curve corresponding to that relating to the first mode in Fig, 6 will 

 now exhibit a maximum at f2< 1; i.e. the system is least stable 

 in its first mode, not for a perfectly streamlined tail as predicted 

 by the old theory, but for a somewhat less perfectly streamlined 

 tail. The second and third modes, on the other hand, are both un- 

 conditionally stabilized as the tail is made blunter; thus, the 

 threshold of instability of the second mode is at u = 2,83 for f g = 1 , 

 at u= 3.85 for f2=0.6,andat u= 4.38 for f2=0.4. 



Figures 12 and 13 show the dynamical behavior of a system 

 with €CN=ec^=l, A=l, Xi=X2 = ^»^^' f, = 0.7, c, = and 

 f2= 1 - C2 = 0.7. We note that the effect of a less than perfectly 

 streamlined nose is to destabilize the system in all its ocillatory 

 modes. Thus the first mode is unstable for < u < 2.80, and the 

 second and third modes lose stability at, respectively u = 3.21 and 

 u = 5.40 (cf. values given above). 



Also shown in Figs. 12 and 13 (dashed line) is the behavior 

 of a system with ec,,j = ecy =0.5 and all other parameters the same, 



998 



