Effects of warm streaming electrons on electrostatic shock solutions

A plasma model consisting of warm downward streaming electrons, cool Boltzmann electrons, cold upward streaming ions, and hot Boltzmann ions is considered. It is shown that there is little difference between having a finite temperature in the electron beam and leaving the beam cold. This implies a greater utility for a cold electron beam model, and adds weight to the idea that the smaller amplitude electrostatic shocks seen in conjunction with ion beams are nonlinear ion acoustic modes.