THE 

 LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



\b b a r p; 





[FIFTH SERIES.] 



APRIL 1899. 



XXIX. Longitudinal Vibrations in Solid and Hollow 

 Cylinders. By C. Chree, Sc.D., LL.D., F.B.S* 



Preliminary. 



§ 1. TT1HE frequency kfiir of longitudinal vibrations in the 

 A ideal isotropic bar of infinitely small cross section 

 has long been known to be given by 



k=p K/Efp, (1) 



where p is the density, E Young's modulus ; p is given by 

 <p = i7r/l when the ends are both free or both fixed 

 p=(2i + l)7r/(2l) when one end is free, the other fixed, 

 I being the length of the rod and i a positive integer. 



For the fundamental or lowest note 2 = 1. 

 For a circular bar whose radius a, though small compared 

 to I, is not wholly negligible, the closer approximation 



k=p(E/p)i{l-}pWa 2 \, (2) 



where tj is Poisson's ratio, was obtained independently by Prof. 

 Pochhammer j" and Lord RayleighJ fully 20 years ago. 



§ 2. The subject has been treated by myself in three 

 papers in the ' Quarterly Journal of . . . Mathematics ' (A) 

 (p. 287, 1886), (B) (p. 317, 1889), (C) (p. 340, 1890). 



In (A) I arrived at (2) describing it (/. c. p. 296) as 



* Communicated by the Physical Society : read December 9, 1898. 



t Crelle, vol. lxxxi^. (1876). 



\ i Theory of Sound,' vol. i. art. 157. 



Phil. Mag. S. 5. Vol. 17. No. 287. April 1899. 2 A 



