The other two solutions to Equation (5.88) are obtained by setting the determinant involving the - and -components of the electric field to zero. The first wave has the dispersion relation
For the right-handed wave, because is negative, it is evident that as . This resonance, which corresponds to , is termed the electron cyclotron resonance. At the electron cyclotron resonance, the transverse electric field associated with a right-handed wave rotates at the same velocity, and in the same direction, as electrons gyrating around the equilibrium magnetic field. Thus, the electrons experience a continuous acceleration from the electric field, which tends to increase their perpendicular energy. It is, therefore, not surprising that right-handed waves, propagating parallel to the equilibrium magnetic field, and oscillating at the frequency , are absorbed by electrons.
When lies just above , we find that is negative, and so there is no wave propagation. However, for frequencies much greater than the electron cyclotron or plasma frequencies, the solution to Equation (5.90) is approximately . In other words, , which is the dispersion relation of a right-handed vacuum electromagnetic wave. Evidently, at some frequency above , the solution for must pass through zero, and become positive again. Putting in Equation (5.90), we find that the equation reduces to
The dispersion curve for a right-handed wave propagating parallel to the equilibrium magnetic field is sketched in Figure 5.1. The continuation of the Alfvén wave above the ion cyclotron frequency is called the electron cyclotron wave, or, sometimes, the whistler wave. The latter terminology is prevalent in ionospheric and space plasma physics contexts. The wave that propagates above the cutoff frequency, , is a standard right-handed circularly polarized electromagnetic wave, somewhat modified by the presence of the plasma. The low-frequency branch of the dispersion curve differs fundamentally from the high-frequency branch, because the former branch corresponds to a wave that can only propagate through the plasma in the presence of an equilibrium magnetic field, whereas the latter branch corresponds to a wave that can propagate in the absence of an equilibrium field.
The curious name ``whistler wave'' for the branch of the dispersion relation lying between the ion and electron cyclotron frequencies is originally derived from ionospheric physics. Whistler waves are a very characteristic type of audio-frequency radio interference, most commonly encountered at high latitudes, which take the form of brief, intermittent pulses, starting at high frequencies, and rapidly descending in pitch.
Whistlers were discovered in the early days of radio communication, but were not explained until much later (Storey 1953). Whistler waves start off as ``instantaneous'' radio pulses, generated by lightning flashes at high latitudes. The pulses are channeled along the Earth's dipolar magnetic field, and eventually return to ground level in the opposite hemisphere. Now, in the frequency range , the dispersion relation (5.90) reduces to
The shape of whistler pulses, and the way in which the pulse frequency varies in time, can yield a considerable amount of information about the regions of the Earth's magnetosphere through which the pulses have passed. For this reason, many countries maintain observatories in polar regions--especially Antarctica--which monitor and collect whistler data.
For a left-handed circularly polarized wave, similar considerations to those described previously yield a dispersion curve of the form sketched in Figure 5.2. In this case, goes to infinity at the ion cyclotron frequency, , corresponding to the so-called ion cyclotron resonance (at ). At this resonance, the rotating electric field associated with a left-handed wave resonates with the gyromotion of the ions, allowing wave energy to be converted into perpendicular kinetic energy of the ions. There is a band of frequencies, lying above the ion cyclotron frequency, in which the left-handed wave does not propagate. At very high frequencies, a propagating mode exists, which is basically a standard left-handed circularly polarized electromagnetic wave, somewhat modified by the presence of the plasma. The cutoff frequency for this wave is