A numerical solution of the electrostatic plasma-wave mode equation for cold, nonneutral plasmas

The low-frequency mode equation of a nonneutral plasma is a hyperbolic partial differential equation inside the plasma and an elliptic equation outside the plasma. This equation can now be solved numerically by directly solving the set of finite difference equations using a band-matrix inverter than gives highly accurate results for this particular problem. When analyzing plasmas with sharp boundaries, special care must be taken in implementing the matching conditions at the plasma boundary. The results of this solution are compared to the results of a particle simulation and to analytic solutions of special cases discussed by Prasad and O'Neil and by Dubin. The author has studied the effect of the wall on the mode frequency of short plasmas and have also looked at the indirect effect of the wall by way of its effect on the shape of the plasma. The direct effect on the frequency is small for radii less than a certain percentage of the wall radius. This percentage is mode dependent and can be estimated by a Gould-Trivelpiece analysis. For large plasma radii, however, the Gould-Trivelpiece analysis overestimates the frequency shift. The effect as the plasma radius approaches the wall radius. Both effects seem to suggest that the effect of image charge in the trap wall is to lower the oscillation frequency