UNCLASSIFIED F/'G 20/3 NL

- ' Nonneutral plasmas confined by conducting walls exhibit, among other things, steady oscilla-" 0 tioms of the type referred to as virtual cathode oscillations. In previous work we studied the instability of steady plasma flow between two conducting grids in one dimension (Sullivan and Coutsis, 1981; Coutias and Sullivan, 1q83). In the later " reference we used ideas of bifurcation theory to numerically demonstrate the existence of an oscillatory state for neutral beams which was not previously established. What was still lacking was a method of dealing with the analytical problems created when a cold beam develops regions of multivalued flow. The main theoretical difficulties are present in a simplii. I one-dimensional model in which monoenergetic electrons are injected through a grid in a semi-infinite space and return to the grid due to the image-charge attraction. For the dynamical description of the evolution of a monoenergetic streim wt used a simplified version of the Vlaaov equation for the electron distribution function. Our equation is essentially the Hamilton-Jacobi equation for the action of an electron, subject to the self-consistent electric field, i.e., the field caused by the other electrons plus any externally imposed field