1889.] Vibrations of an infinitely long Cylindrical Shell. 443 



IV. ' Note on the Free Vibrations of an infinitely long Cylin- 

 drical Shell." By LORD RAYLEIGH, Sec. RS. Received 

 February 26, 1889. 



In a recent memoir* Mr. Love has considered this question among 

 others ; but he has not discussed his result [equation (95)], except in 

 its application to a rather special case involving the existence of a 

 free edge. When the cylinder is regarded as infinitely long, the pro- 

 blem is naturally of a simpler character ; and I have thought that it 

 might be worth while to express more fully the frequency equation, as 

 applicable to all vibrations, independent of the thickness of the shell, 

 which are periodic with respect both to the length and the circum- 

 ference of the cylinder. 



In order to prevent misunderstanding, it may be well to premise 

 that the vibrations, whose frequency is to be determined, do not 

 include the gravest of which a thin shell is capable. If the middle 

 surface be simply bent, the potential energy of deformation is of a 

 higher order of magnitude than in the contrary case, and according 

 to the present method of treatment the frequency of vibration will 

 appear to be zero. It is known, however, that the only possible modes 

 of bending of a cylindrical shell are such as are not periodic along the 

 length, or rather have the wave-length in this direction infinitely long.f 

 When the middle surface is stretched, as well as bent, the potential 

 energy of bending may be neglected, except in. certain very special 

 cases. 



Taking cylindrical co-ordinates (r, 0, z), and denoting the displace- 

 ments parallel to z, 0, r by u, v, w respectively, we have for the 

 principal elongations and shear at any point (a, 0, z) J 



du " _ w 1 dv _ 1 du dv ,-,*. 



ffl = Tz' ff *-a+adj>' "ad^dz ..... C 



and the energy per unit of area is expressed by 



... ....... (2), 



where 2h denotes the thickness of the shell, and m, n are the elastic 

 constants of Thomson and Tait's notation. 



* " On the small Free Vibrations and Deformation of a thin Elastic Shell," 

 ' Phil. Trans.,' A, vol. 179 (1888), p. 491. 



f " On the Bending and Vibration of thin Elastic Shells, especially of Cylindrical 

 Form," ' Eoy. Soc. Proc.,' supra, p. 105. 



See a paper on the Infinitesimal Bending of Surfaces of Eevolution (' London 

 Math. Soc. Proc.,' vol. 13, p. 4, Nov. 1881), and those already cited. 



