Two cold electron beams drifting at +v0 and -v0 over a neutralising ion background obey the dispersion relation 1 = (wp^2/2) [ 1/(w - k v0)^2 + 1/(w + k v0)^2 ], where wp is the total plasma frequency. Solving for w^2 gives w^2 = k^2 v0^2 + wp^2/2 - sqrt(wp^4/4 + 2 wp^2 k^2 v0^2), which is negative (growing waves) for k v0 < wp. The fastest mode sits at k v0 = sqrt(3/8) wp with gamma_max = wp / (2 sqrt 2) = 0.354 wp.

The simulation is a 1D electrostatic particle-in-cell code: 4000 super-electrons, cloud-in-cell weighting on 64 cells, a spectral Poisson solve and leapfrog time stepping (dt = 0.05/wp). In the phase-space plot the beams roll up into vortices (particle trapping) once the wave amplitude is large enough, which saturates the growth. Thermal spread weakens and eventually removes the instability.

Simplifications: 1D, electrostatic, immobile ions, periodic box, normalised units (wp = 1). The check fits the growth of mode 1 in a box tuned to its fastest mode and compares it with gamma_max, and monitors total-energy drift.