Degenerate Mixing of Trivelpiece-Gould Waves on a Cold, Finite-Length Plasma Cylinder

In the cold-fluid dispersion relation $\omega = \omega_p / [1 + ( k_\perp / k_z)^2]^{1/2}$ for Trivelpiece-Gould waves on an infinitely-long magnetized plasma cylinder, the transverse and axial wavenumbers appear only in the combination $k_\perp / k_z$. As a result, for any frequency $\omega < \omega_p$, there are infinitely many degenerate waves, all having the same ratio $k_\perp / k_z$. On a cold finite-length plasma cylinder, each longitudinal normal mode is a mixed superposition of these degenerate waves. Here several such modes are calculated for a single-species plasma cylinder with rounded ends. A striking feature of these modes is that the short-wavelength waves add constructively along cones\footnote{R.K. Fisher and R.W. Gould, Phys. Rev. Lett. {\bf 22}, 1093 (1969).} given by $dz / dr = \pm (\omega_p^2 / \omega^2 - 1)^{1/2}$. Thus, the mode structure of even a low order mode is substantially more complicated than the single sine wave approximation typically assumed. Also, the admixture of short wave lengths substantially enhances the viscous damping of the mode.