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).