Langmuir Waves: When electrons in a plasma are displaced from their equilibrium positions,
the restoring electrostatic force causes oscillations at the plasma frequency
ωₚ = √(ne²/ε₀mₑ). The Bohm-Gross dispersion relation for Langmuir waves is
ω² = ωₚ² + 3k²v_th² where v_th = √(k_B T/mₑ) is the thermal velocity. At long wavelengths
(k→0), all modes oscillate at ωₚ — the plasma cannot support frequencies below ωₚ (cutoff).
The Debye length λ_D = v_th/ωₚ = √(ε₀k_BT/ne²) sets the screening length; electrostatic
perturbations are screened exponentially beyond λ_D. The wave plot shows the electron density
perturbation propagating; the dispersion curve shows ω(k) vs the light-cone and plasma cutoff.