1890.] vibrations of a revolving cylinder or bell. 



107 



This may be interpreted as representing oscillations of the ring or 

 cylinder of period 27r/s3- s having 2s nodes or nodal meridians (where 

 dv/dcp = 0) which rotate about the axis with angular velocity 



s 2 -l 



-5 -, CO. 



s 2 + l 



The nodal angular velocity is thus in every case less than co, but 

 in the higher tones its difference from co becomes less and less. 



As the nodes are carried round in succession past the direction 

 of the observer's ear, beats will be heard, and their number per 

 revolution of the material will be 



2.9 



s 2 -l 



sr+r 



The numerical values of these results are tabulated below for a 

 few of the smaller values of s : 



The pitch of the intermittent sound is determined by -sr, 

 when co = 0, fx = 0, we have 



2 /3 *V-1) 2 



Now 



era 



+ 1 



(13), 



as found by Hoppe* and Lord Rayleigh-f*. Denoting this value 

 of -or, by H s , equation (11) shows that if /x = co 2 , so that the 

 attraction counterbalances the purely statical effects of centrifugal 

 force, we shall have w g 2 > IT/, or the pitch will be somewhat raised 

 by the rotation. 



If however fi = 0, and the frequency of rotation is small com- 

 pared with the frequency of the vibrations in a non-rotating 

 cylinder, so that co is a small quantity of the first order compared 

 with II s , equation (11) shows that ot s differs from 1T S by small 

 * Crclle, Bd. 63, 1871. t Theory of Sound. 



