Plasma-Wave Dispersion

The $\omega$ - $k$ diagram of the basic plasma waves. The electron plasma frequency $\omega_p=\sqrt{n e^2/\varepsilon_0 m_e}$ sets the scale. The ordinary (O) mode $\omega^2=\omega_p^2+c^2k^2$ has a cutoff at $\omega=\omega_p$ (no propagation below it) and a superluminal phase speed with $v_\phi v_g=c^2$ . The extraordinary (X) mode has cutoffs $\omega_{L,R}$ and an upper-hybrid resonance $\omega_{UH}=\sqrt{\omega_p^2+\omega_c^2}$ , with a stop-band between the resonance and the right cutoff. The Bohm-Gross Langmuir branch is $\omega^2=\omega_p^2+3k^2v_{th}^2$ ; the ion-acoustic branch $\omega=kc_s/\sqrt{1+k^2\lambda_D^2}$ saturates; the Alfven wave $\omega=kv_A$ is non-dispersive. A marker sweeps the branch reporting the phase and group speed; the inset is a wave at the marked $(k,\omega)$ travelling at the phase speed. All curves are closed form (gate-tested).

Figure 1. Log-log dispersion diagram of the plasma-wave branches with the light line, the omega = omega_p reference, and (X-mode) the cutoffs and the upper-hybrid resonance; the sweeping marker reports the phase and group speed. Method: closed-form cold/warm fluid dispersion relations (gate-tested sim.js), Canvas2D log-log plot; the live readouts are omega_p, the cutoffs, and v_phase / v_group at the marker.

mode

omega_p (rel.)1.00

omega_c (rel.)0.60

WHAT TO TRY

Switch the mode (O, X, Langmuir, ion-acoustic, Alfven): the omega-k curve reshapes and the cutoffs move. The ordinary mode has a clean omega_p cutoff, the X-mode adds resonances from the cyclotron frequency.
Tune omega_p and omega_c: the cutoff and resonance frequencies track them, and the curve asymptotes to the light line omega = ck at high k. The gap below the cutoff is where the wave cannot propagate.
Read the phase and group velocities: where the curve is steep the group velocity is high, and the traveling wave at the bottom moves at the phase speed omega/k for the selected point.