1890.] Mr Bryan, On a revolving cylinder or bell. 101 



The following Communications were made to the Society : 



(1) On the beats in the vibrations of a revolving cylinder or bell. 

 By G. H. Bryan, M.A., St Peter's College. 



In this paper I propose to investigate the nature of the beats 

 which may be heard when a vibrating shell in the form of a 

 cylinder or other surface of revolution has imparted to it a rotatory 

 motion about its axis of figure. 



It might at first appear that, unless such a body were revolving 

 with angular velocity comparable with the frequencies of the 

 vibrations, the latter would not be affected in any sensible manner, 

 and that the only important effect of rotation would be in per- 

 manently straining the body, owing to centrifugal force. This is, 

 for example, the point of view taken by Mr Love in his paper on 

 " The free and forced vibrations of an elastic spherical shell con- 

 taining a given mass of liquid,"* in which the author uses the 

 expressions for the accelerations referred to moving axes when 

 dealing with the oscillations of the liquid, but not when dealing 

 with those of the elastic envelope, although both are supposed to 

 rotate together. But, while we may be justified in neglecting the 

 effect on any single period of such small changes in the system as 

 those due to rotation, yet the slight opposed changes produced in 

 the periods of two similar vibration forms which travel in different 

 directions round the shell may produce phenomena of beats, which 

 in the case of very rapid vibrations like those of sound, are among 

 the most noticeable effects of the rotation. 



If a straight wire of circular section, clamped symmetrically at 

 one end, be made to rotate slowly about its axis while executing 

 transverse vibrations, it is well known that the plane of vibration 

 will remain fixed in space instead of turning with the wire. If the 

 vibrations are audible we shall, therefore, hear a continuous sound. 

 In the case of a tuning-fork the plane of vibration must necessarily 

 turn with the fork, so that beats are heard if it be rotated. 



When however the vibrating body is such as a bell, rotation 

 about its axis will produce an intermediate effect by causing the 

 nodal meridians-{- to revolve with angular velocity less than that of 

 the body, and depending in each case on the mode of vibration 

 considered. This phenomenon, which forms the subject of the 

 present paper, appears to be new, yet nothing is easier than to 

 verify it experimentally. If we select a wine-glass which when 

 struck gives, under ordinary circumstances, a pure and continuous 

 tone, we shall on twisting it round hear beats, thus showing that 

 the nodal meridians do not remain fixed in space. And if the 

 observer will turn himself rapidly round, holding the vibrating 



* Proc. Lond. Math. Soc. xix. 



t That is meridians along which the vibration has no radial component. 



