The classical vibrational modes are calculated for a thin-walled hollow cylinder of isotropic material. An analytical solution is obtained for all of the modes. The results are applied to calculate the low-frequency vibrational modes of a single-wall carbon nanotube. Analytical formulas are obtained for the Raman frequencies, and for the velocities of acoustical modes.

• Received 1 February 2002

DOI:https://doi.org/10.1103/PhysRevB.65.235402