Low-Frequency Waves and Instabilities on the Positive Column in a Magnetic Field. I. Analysis and Study of Axisymmetric Modes

An analysis is presented for the normal modes of low-frequency wave propagation on the positive column in an axial magnetic field. Dispersion characteristics are obtained from a hydrodynamic treatment by perturbing a steady-state discharge model which includes generation and ion inertia, and is valid for arbitrary pressure. The resulting complex eigenvalue problem is solved numerically by use of a Fourier-Bessel expansion. In general, two types of waves are found for a given frequency: ion-acoustic waves and electron waves. For the regions of parameter space appropriate to laboratory discharges, the ion-acoustic wave branches are predicted to be heavily damped, expect at low pressures where the computer solutions are in general agreement with available experimental data. The electron wave branches do not show instability for axisymmetric modes (m = 0), but can be unstable for asymmetric modes (m ≠ 0). Stable modes only are considered and it is shown that some hitherto unexplained damped waves, observed experimentally, are identifiable as symmetric electron-wave modes (m = 0).