= Cold-plasma radio dispersion
{title2=$n^2=1-\omega_p^2/\omega^2$}
Take a cold, unmagnetized, collisionless <plasma> with uniform equilibrium <electron density> $n_e$, fixed neutralizing <ions>, and a small <electromagnetic wave> that is a <transverse wave>. Cold means that thermal-motion corrections are neglected. For <electron mass> $m_e$, <elementary charge> magnitude $e$ and <electric field> $\mathbf E=\operatorname{Re}(\widehat{\mathbf E}e^{-i\omega t})$, the linear <Newton's second law> equation is $m_e\dot{\mathbf v}=-e\mathbf E$. Thus $\widehat{\mathbf v}=-ie\widehat{\mathbf E}/(m_e\omega)$ and the <current density> is $\widehat{\mathbf J}=in_ee^2\widehat{\mathbf E}/(m_e\omega)$. Writing the induced <electric polarization> as $\widehat{\mathbf J}=-i\omega\widehat{\mathbf P}$ gives a negative <electric susceptibility>. With <vacuum permittivity> $\epsilon_0$,
$$
\epsilon_r=1-\frac{\omega_p^2}{\omega^2},\qquad
\omega_p^2=\frac{n_ee^2}{\epsilon_0m_e},\qquad
c^2k^2=\omega^2-\omega_p^2,\qquad
n^2=\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2}{\omega^2}.
$$
The <plasma frequency> $\omega_p$ controls the response. For $\omega>\omega_p$, the propagating branch has real <refractive index> $0<n<1$. For $\omega<\omega_p$, this homogeneous model has an <evanescent wave>, not an ordinary propagating wave with a negative real <refractive index>. In a slowly varying vertical profile, a normally incident ray approaches a turning region where $\omega_p(z)=\omega$; the geometrical-ray approximation fails right at that region. At oblique incidence the conserved horizontal <wave vector> changes the turning condition. A smooth density gradient can therefore return a <skywave> without a mirror or a layer of metal.
For a homogeneous propagating section, direct <differentiation> of the <dispersion relation> gives
$$
v_{\rm ph}=\frac{\omega}{k}=\frac c n>c,\qquad
v_g=\frac{d\omega}{dk}=cn<c,\qquad
n_g=n+\omega\frac{dn}{d\omega}=\frac1n.
$$
A <phase velocity> greater than $c$ does not send information faster than $c$. Here a narrow <wave packet> has <group velocity> $v_g$, while the full causal response must still respect <special relativity>. Even this stripped-down <plasma> is strongly dispersive near its <plasma frequency>. In the actual <ionosphere>, collisions produce absorption, and the <magnetic field> of <Earth> changes the response with direction and <wave polarization>; a single real scalar $n$ cannot capture all of that behavior.
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