Two-stream electromagnetic instability

The characteristics of the instability of an electromagnetic wave propagating obliquely to a double stream in a plasma are investigated. Instability is shown to exist for all directions of the stream. The direction normal to the wave vector is a singular case for which the unstable wave has a zero frequency of oscillation and a growth rate which increases from zero as the wavenumber is increased and asymptotically approaches a maximum value for very large wavenumbers. For all other directions unstable waves exist for wavenumbers below a critical value. The frequency of oscillation increases monotonically from zero as the wavenumber is increased and reaches a maximum value at the cutoff wavenumber but the growth rate increases from zero as the wavenumber is increased, reaches a maximum value, and then decreases to zero at the critical wavenumber. The dependence of the frequency of oscillation and the growth rate of the unstable wave on the direction of the stream with respect to the wave vector is also examined. Many of the features of this instability, particularly for other than the singular direction, are predicted correctly by the quasi-static approximation.