Disturbances in a multi-velocity plasma

The single-velocity treatment of ac disturbances in electron flow is often accurate enough even when the flow actually has an infinitely broad but peaked velocity distribution, such as a Maxwellian or a bell-shaped distribution. In seeking a better approximation, one is tempted to expand in terms of the moments of the velocity distribution and to disregard terms beyond that involving the second moment. This leads to a dispersion equation describing waves which strictly do not exist. A linearized analysis by transform methods predicts a field which oscillates in an exponentially damped manner. This is not the oscillation of a normal mode, however, for various velocity classes of the charge distribution have ac densities which grow with time. This indicates that the linear expression will hold over a finite interval only. A simple example which compares a multi-stream analysis with an analysis based on the charge density in phase space (the distribution function) indicates the same sort of failure of both of the linearized theories at large times. Thus, it appears that the failure is characteristic of the problem rather than of the method of solution. The method of solution is a matter of choice.