Degenerate Mixing of Electrostatic Modes on a Finite-Length Nonneutral Plasma Column

Using cold fluid theory, we discuss the structure of standing electron plasma waves on a magnetized, nonneutral plasma column of finite length. Such eigenmodes can be surprisingly complex, involving a superposition of many component waves with different axial and transverse wavenumbers $k_z$ and $k_\perp$. The dispersion relation\footnote{A.W. Trivelpiece and R.W. Gould, J. Appl. Phys. {\bf 30}, 1784 (1959).} for the individual components [{\it i.e.}, $\omega = \omega_p k_z / \sqrt{k_z^2 + k_\perp^2}$] implies that waves with small $k_z$ and $k_\perp$ can be degenerate with waves with large $k_z$ and $k_\perp$. Reflection at the column ends mixes these degenerate components, yielding the complicated structure. We have in mind eigenmodes on a cryogenic plasma column, where cold fluid theory is valid even for waves with large $k_z$ and $k_\perp$. In a warmer plasma, kinetic effects ({\it e.g.}, Landau damping of the large wavenumber components) spoils the degeneracy and kills the mixing.