The recent Plum Brook—NASA experiments on counterstreaming plasma instabilities in which electric field emissions at frequencies [(n + ½) ±α]ωe (n = 1, 2, 3,…, 9, α < 0·5) have been observed, raised the theoretical problem of the non-symmetric two-stream instability. This is the case in which the counterstreaming beams have different velocities in the directions parallel and perpendicular to the static magnetic fields as well as different particle density.

We investigate theoretically this problem. The instability of quasi-electrostatic waves at shifted half odd integer multiples of the cyclotron frequency, due to two non-symmetric counterstreaming electron beams, is considered. The beam velocities parallel and perpendicular to the static magnetic field are represented by different double Dirac delta functions; no parameter limitation is imposed. A systematic investigation of (i) the coupling between plasma modes and cyclotron modes and (ii) the coupling between cyclotron modes (beam 1) and cyclotron modes (beam 2) resulting in shifted half odd integer multiples of the cyclotron frequency is carried out.

The results include approximate but simple analytical expressions for maximum growth rates and for marginal stability as well as exact numerical solutions for the detailed unstable spectra (1 ≤ m ≤ 20) in both ωτ- and kr-space. The relative weakening or suppression of certain modes is also predicted.