Electron plasma oscillations, Bohm–Gross dispersion, and Landau damping
Langmuir waves are longitudinal electrostatic oscillations of the electron fluid in a plasma.
The dispersion relation (Bohm–Gross): ω² = ωₚ² + 3k²vₜ² where ωₚ = √(ne²/mε₀) is the plasma frequency.
Key features: (1) plasma frequency cutoff — waves can't propagate below ωₚ;
(2) the wave is longitudinal (E∥k), unlike light;
(3) Landau damping (1946) — even in collisionless plasma, waves damp when vₚhₐₛₑ ≈ vₜ,
because electrons near the phase velocity resonantly absorb energy (no collisions needed!).
The left panel shows the oscillating electron density; right shows the dispersion curve ω(k)
with the forbidden gap ω < ωₚ shaded.