A theory is presented for a two-stream free-electron laser (FEL) with a background plasma. A dispersion relation (DR) for the unstable couplings of wave modes is derived using fluid formulation. This DR is solved numerically to find the unstable modes and their growth rate. The effect of the velocity difference of the two electron beams as well as the background plasma on the FEL resonance and the two-stream instability is studied. It is shown that their separate as well as combined effects can increase the growth rates.
