BARS 



137 



It follows from (1) that the extension is negligible, and the 

 energy mainly flexural. The frequencies are in fact comparable 

 with those of transverse vibration of a bar. In the mode of 

 order s there are 2s nodes, or places of vanishing radial motion, 

 but these are not points of rest, the tangential motion being 

 there a maximum*. In the case 5=1 the circle is merely 

 displaced as a whole, without deformation, and the period is 



Fig. 48. 



accordingly infinite. The most important case is that of s = 2, 

 where the ring oscillates between two slightly elliptical extreme 

 forms. The arrows in the annexed figure shew the directions of 

 motion at various parts of the circumference at two epochs, 

 separated by half a period, when the ring passes through its 

 equilibrium position. The dotted lines pass through the nodes 

 of the radial vibration. 



One farther point is to be noticed. Owing to the assumed 

 uniformity of the ring the origin of 6 is arbitrary, and other 

 modes, with the same frequencies, are obtained by adding a 

 constant to 0. In particular we have the flexural mode 



u = A sin sO . cos (nt + e), v = cos sd . cos (nt + e), (15) 



s 



with the same value of n 2 as in (14). We have here an instance 

 of the kind referred to in 16, where two distinct normal modes 



* This point is illustrated by the vibrations of a finger-bowl when excited by 

 drawing a wetted finger along the edge. The point of rubbing is a node as 

 regards the radial vibration, and the crispations on the contained water are 

 accordingly most conspicuous at distances of 45 on either side, where the radial 

 motion is a maximum. 



