Circumferential normal modes in an empty elastic cylinder

The so-called circumferential normal modes propagating in an empty elastic cylinder are considered. A dispersion equation for the wave numbers of these waves, an equation for the critical frequencies, and expressions for the eigenfunctions of such a waveguide are derived. Solutions to these equations are obtained by numerical methods for different values of the parameter d representing the relative thickness of the cylinder. An analysis of the solutions is performed, and the main properties of the dispersion curves are described, including those for the low-frequency waves of the new type, which correspond to the branches in the form of open loops. Individual normal modes are identified on the basis of the calculations and subsequent analysis of eigenfunctions.