138 DYNAMICAL THEOKY OF SOUND 



have the same frequency, and the modes themselves accordingly 

 become to some extent indeterminate. The case would be 

 altered at once if the ring were not quite uniform, e.g. if it were 

 slightly thicker at one point. The normal modes in which 

 there is a node or a loop respectively, of radial vibration, at 

 this point would differ somewhat in character, and have slightly 

 different frequencies. Accordingly when both modes are excited 

 we should have beats between the corresponding tones. This 

 is a phenomenon often noticeable in the case of bells (and 

 finger-bowls), the inequality being due to a slight defect of 

 symmetry. 



The vibrations of a ring in its own plane were first investi- 

 gated by R. Hoppe (1871) ; a simplified treatment of the flexural 

 modes was subsequently given by Lord Rayleigh. The theory 

 of vibrations normal to the plane is more intricate, since torsion 

 is involved as well as flexure. The problem has been solved by 

 J. H. Michell (1889), who finds, in the case of circular cross- 

 section, 



s 2 + 1 + a- ' pa 4 ' 

 where a is Poisson's ratio. 



