As discussed in context of plasma theories, the thermal kinetic energy of plasma particles is ignored in cold plasmas. In practice this means neglection of the pressure terms in the equations. In the theory of waves, this approximation is justified if the particle thermal velocity is small when compared with the wave phase velocity. Accordingly, for waves with very small phase velocities, warm plasma theory must be used.
Another often used approximation is to consider only electron motion. The theory of wave propagation in a cold homogenous electron gas immersed in a magnetic field (B) is known as magnetoionic theory. The neglection of ions is valid only for high frequency waves, i.e., for frequencies large compared to the ion cyclotron frequency. To go on to the next level of approximation, the theory of high-frequency small-amplitude plane waves propagating in an arbitrary direction with respect to B is called the Appleton-Hartree theory. This was first developed to study wave propagation in the Earth's ionosphere. The restriction of small-amplitude waves makes it possible to uselinear perturbation theory, where the wave related variations in the plasma parameters are small when compared to the undisturbed parameters. The use of plane waves makes the equations easier, but implyes no loss of generality, since any wave motion can always be synthesized in terms of plane waves. The hydromagnetic extension of magnetoionic theory deals with multicomponent cold plasma.
In the cold electron gas, important variables are the electron number density and the average electron velocity. They must satisfy
The wave modes are found from the dispersion relation calculated using the above equations. We start investigating the results from a non-magnetized (isotropic) plasma, before including the external magnetic field so important for space physics. Ion motion is not included (but some remarks about ion effects are given).
There are one oscillation mode and one real wave mode in cold isotropic electron plasma:
Figure: Frequency dependence of the phase and group velocities for the transverse waves in a collisionless isotropic cold electron plasma.
Things get quite a bit more complicated when external magnetic field B is taken into account (even with the Appleton-Hartree approximation):
Figure: Frequency dependence of the phase and group velocities for the transverse RCP and LCP waves propagating along magnetostatic field in a collisionless cold electron plasma.
An important point is that, in the very low frequency limit, the phase velocities of the RCP and LCP waves (if ion motion is included) go to zero. This means that in order to get Alfven waves (see MHD waves) one has to use warm plasma theory, where the low frequency phase velocity is, approximately, Alfven velocity.
We are not going into details here (perhaps later)!
A phenomenon known as Faraday rotation occurs in the frequency range of RCP and LCP waves, and can be used as a plasma diagnostic tool.