Wave propagation through a magnetized plasma

For a plasma ($\omega_0=0$) the dispersion relation (4.51) reduces to

$$n_\pm^{~2}(\omega) = 1 - \frac{\omega_p^{~2}}{\omega(\omega\mp{\mit \Omega})}.$$ (700)

The upper sign corresponds to a left-handed circularly polarized wave and the lower sign to a right-handed polarized wave. Of course, Eq. (4.65) is only valid for wave propagation along the direction of the magnetic field. Wave propagation through the Earth's ionosphere is well described by the above dispersion relation. There are wide frequency intervals where one of $n_+^{~2}$ or $n_-^{~2}$ is positive and the other negative. At such frequencies one state of circular polarization cannot propagate through the plasma. Consequently, a wave of that polarization incident on the plasma is totally reflected. The other state of polarization is partially transmitted.

The behaviour of $n^{~2}_-(\omega)$ at low frequencies is responsible for a strange phenomenon known to radio hams as ``whistlers.'' As the frequency tends to zero, Eq. (4.65) yields

$$n_-^{~2} \simeq \frac{\omega_p^{~2}}{\omega\,{\mit \Omega}}.$$ (701)

At this sort of frequency $n_+^{~2}$ is negative, so only right-hand polarized waves can propagate. The wave-number of such waves is given by

$$k_- = n_-\,\frac{\omega}{c} \simeq\frac{\omega_p}{c} \sqrt{\frac{\omega} {\mit \Omega}}.$$ (702)

Energy transport is governed by the group velocity (see later)

$$v_g(\omega) = \frac{d\omega}{dk_-} \simeq 2c\, \frac{\sqrt{\omega\,{\mit\Omega}}} {\omega_p}.$$ (703)

Thus, low frequency waves transmit energy slower than high frequency waves. A lightning strike in one hemisphere of the Earth generates a wide spectrum of radiation, some of which propagates along the dipolar field lines of the Earth's magnetic field in a manner described approximately by the dispersion relation (4.68). The high frequency components of the signal return to the surface of the Earth before the low frequency components (since they travel faster along the magnetic field). This gives rise to a radio signal which begins at a high frequency and then ``whistles'' down to lower frequencies.