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XX. On a Type of Vibration of an Elastic Spherical Shell. 



By NlKHlLRAJS T JAN SEN *. 



IT has been pointed out by Lovej that Stokes in his treat- 

 ment of the vibration of a spherical shell in a gaseous 

 medium neglects one important condition of motion, and his 

 solution, though it is remarkably successful in explaining 

 Leslie's experiment, does not bring into prominence the 

 decaying factors of vibration. The damping factors are a 

 necessary consequence of the circumstances of the motion, 

 which, as Love has shown, is generally damped harmonic and 

 to the firstforder of approximation is simple harmonic for the 

 sphere, and partly simple harmonic and partly exponential for 

 the medium. Also the waves of rapidly damped harmonic type 

 which are almost insensible a short distance from the sphere 

 come into prominence again near the surface of discontinuity 

 of the advancing disturbance, and are responsible for the 

 fulfilment of the necessary condition at this boundary. In 

 the present paper the shell is taken to be elastic, which 

 requires the satisfaction of the additional condition of equality 

 of traction and pressure at the spherical surface. The 

 resulting motion has been found to consist of two parts : one 

 damped harmonic, and the other of the purely exponential 

 type. The additional condition does not, however, alter the 

 general nature of motion of the medium. The period of 

 vibration has been shown to depend on the ratio of the 

 thickness of the shell to the radius, and there are two critical 

 thicknesses within which the motion is found to be entirely 

 aperiodic. In this respect this purely theoretical problem 

 appears to be of some interest. 



The kinematical condition to be satisfied at the wave front 

 is obtained by considering the motion of a cylindrical element 

 of the medium standing on the surface of discontinuity. 

 The momentum of this element of the medium is changed in 

 a short interval of time from zero to that attained by it at 

 the end of that interval by pressures on the base of the 



* Communicated by Prof. D. N. Mallik, Sc.l)., F.R.S.E. 

 t " Some Illustrations of Modes of Decay of Vibratory Motions," Proc. 

 Lond. Math. Soc. (2) vol. ii. p. 88 (1904). 



\ Rayleigh's ' Theory of Sound,' vol. ii. p. 239. 



