The other two solutions to Eq. (536) 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, it is evident, since is negative, 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 is just above , we find that is negative,
and so there is no wave propagation in this frequency range. However, for frequencies
much greater than the electron cyclotron or plasma frequencies, the solution
to Eq. (538) is approximately . In other words,
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 Eq. (538), 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 Fig. 10. 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 which propagates above the cutoff frequency, , is a standard right-handed circularly polarized electromagnetic wave, somewhat modified by the presence of the plasma. Note that the low-frequency branch of the dispersion curve differs fundamentally from the high-frequency branch, because the former branch corresponds to a wave which can only propagate through the plasma in the presence of an equilibrium magnetic field, whereas the high-frequency branch corresponds to a wave which 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. Figure 11 shows the power spectra of some typical whistler waves.
Whistlers were discovered in the early days of radio communication, but
were not explained until much later. Whistler waves start off as ``instantaneous''
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. Fig. 12 illustrates the typical
path of a whistler wave. Now, in the frequency
, the dispersion
relation (538) 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 they have passed. For this reason, many countries maintain observatories in polar regions, especially Antarctica, which monitor and collect whistler data: e.g., the Halley research station, maintained by the British Antarctic Survey, which is located on the edge of the Antarctic mainland.
For a left-handed circularly polarized wave, similar considerations
to the above give a dispersion
curve of the form sketched in Fig. 13. In this case, goes to
infinity at the ion cyclotron frequency, , corresponding to
the so-called ion cyclotron resonance (at
). At this resonance, the
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