1. Whistler Waves
Whistler waves were first detected by Barkhausen in At that time, the equipment essentially consisted of a pair of earphones and an amplifier connected to a long wire. Barkhausen heard musical tones with continuously descending pitch (these were also heard by many field radio operators during WWI, ~ 10 kHz.) The early (and erroneous) explanation was that these were doppler-shifted noises from artillery shells. The correct explanation was made by Storey in He proposed that a lightning stroke at the earth's surface generates a pulse of electromagnetic waves. These are subsequently dispersed by propagation in the earth's ionized outer atmosphere. Storey proposed that the waves followed magnetic field lines from one hemisphere to the other at the conjugate point. The waves are confined to ducts (as we will rigorously demonstrate later). The basic geometry is shown below:

7. Field-line path followed by a ducted whistler
Field-line path followed by a ducted whistler. Inset diagrams show idealized spectra of whistler echo trains at conjugate points A and B.

16. We first need to show that the group velocity is a function of 
We first need to show that the group velocity is a function of  . We have
and, for
Differentiating the expression for k yields
the group velocity, vg , is then given by
We can then write the time of arrival of each component of a pulse as

17. Thus, the arrival time depends on the electron density along the path, the wave frequency, and the B-field strength. The arrival time is shorter for higher frequencies giving the descending pitch.
The signal time delay is often used to measure density at large distances in the earth's outer atmosphere (within the magnetosphere).
Idealized waveforms and spectra are shown in the following figures.

18. Idealized waveform and spectrum of a whistler (D 50)
Idealized waveform and spectrum of a whistler (D 50).  (a) Waveform with each cycle presenting 400 cycles on the original. (b) Curve of actual frequency with time.(c) Curve of with mc.

19. Typical Experimental Data
One-hop whistler of high amplitude with three-hop echo (a) Curve of versus t; (b) Dynamic spectrum; (c) Corresponding oscillogram of wide-band amplitude; (d) A section of (c) expanded in time by a factor of 20. In parts (c) and (d), filter passband was 600 cps to 15 kc/s.

20. Electron-Density Measurements Using Whistlers
The following figure shows the comparison between experimentally measured dispersion and theory.
Electron-Density Measurements Using Whistlers
Circles represent data scaled from a whistler recorded at Seattle, Feb. 5, 1958, 1336 UT. Curves were normalized to fit the whistler at its nose frequency of kc/s.

21. Parallel electromagnetic waves III
Resonances indicate a complex interaction of waves with plasma particles. Here k ->  means that the wavelength becomes at constant frequency very short, and the wave momentum large. This leads to violent effects on a particle‘s orbit, while resolving the microscopic scales. During this resonant interaction the waves may give or take energy from the particles leading to resonant absorption or amplification (growth) of wave energy.
/kc  1/2
At low frequencies,  << ge , the above dispersion simplifies to the electron Whistler mode, yielding the typical falling tone in a sonogram as shown above.

22. Whistlers in the magnetosphere of Uranus and Jupiter
Wideband electric field spectra obtained by Voyager at Uranus on January 24, 1986. fc
Wave measurements made by Voyager I near the moon Io at a distance of 5.8 RJ from Jupiter.
Whistlers

23. Whistler mode waves at an interplanetary shock
w = ge(kc/pe)2
Gurnett et al., JGR 84, 541, 1979

24. In the laboratory, there have been several observations of whistler waves. In one of the original experiments of the ZETA device, (toroidal high density pinch) whistler waves and whistler ducting were observed. The parameters were
In this experiment, they demonstrated that the waves were confined to channels (ducts)  1 mm in diameter.

26. The ZETA device at Harwell
The ZETA device at Harwell. The toroidal confinement tube is roughly centered, surrounded by a series of stabilizing magnets (silver rings). The much larger peanut shaped device is the magnet used to induce the pinch current in the tube.

28. Laboratory Studies of Whistler Wave Propagation
There have been many other laboratory experiments on whistler propagation.
However, the majority of these have been done in tiny nonuniform plasma columns,
where the boundary effects dominated. The exception is the whistler wave work done
by Stenzel and also by Sugai. Our introduction to Stenzel's work follows.[1]
Stenzel studied whistler wave propagation in the large magnetized plasma device
shown below. The chamber is 1m in diameter and 3.5m long, which is to be compared
to the whistler wavelength of  cm. Thus, infinite plasma assumptions appear
to be reasonably well satisfied. A large (50 cm diameter) oxide cathode provides high
energy ( 50 eV) to so-called “primary” ionizing electrons, which produce a plasma
via impact ionization of the neutral background (typically krypton at  210-4 Torr).
The experiment is operated in the afterglow of the discharge, providing a very
quiescent Maxwellian plasma. Parameters are typically

29. Schematic view of the plasma device

30. Typical time sequence of the pulsed plasma experiment
The system is repetitively pulsed with a timing sequence, shown below.
Typical time sequence of the pulsed plasma experiment

31. The whistler waves are launched and detected with magnetic and electric dipole probes, as shown in the following figure. Note that because of the small size of the exciters (as compared to a wavelength), one expects the wavefronts to be diverging and not plane (in the absence of ducting). Also note that by rotating the probes, one can determine the wave polarization.

32. Schematic view of antennas used to excite and detect whistler waves

33. The whistler wavelength is determined interferometrically

34. An example of the raw data from such an interferometer is shown below.
Axial Distance from Exciter (2 cm)  Raw data of axial whistler wave interferometer traces. In a uniform plasma (a) the amplitude decays due to the divergence of the energy flux; in a density trough (b) the wave is ducted and propagates at a constant amplitude in a collisionless plasma.

35. From such data, the dispersion relation is determined by varying  /c , etc. An example is shown below.
Whistler wave dispersion relation. The solid line with data points is measured, the dashed line is the theoretical dispersion.

36. The dashed line in the figure above is a plot of
To determine the group velocity, one launches phase coherent tone bursts.
The system employed by Stenzel is shown below, along with actual experimental data.

37. Block diagram for exciting and detecting phase coherent whistler wave bursts in a slowly pulsed afterglow plasma

38. (a) Phase-coherent whistler wave bursts vs
(a) Phase-coherent whistler wave bursts vs. time at different positions from the exciter antenna. (b) Time-of-flight diagram of the burst envelope [outlined by thin lines in (a)] which gives a direct measure of the group velocity .

39. The theoretical group velocity is given by
Evaluating this for his experimental parameters of  / c = 0.6,  p / c = 25, Stenzel found which is in good agreement with his experimental measurements.
The strong dispersive nature of this wave is illustrated below for two cases:
(a) and (b)

40. (a) Applied (top trace) and dispersed (bottom trace) wave packet near cyclotron resonance; (b) A short low frequency wave burst applied to the exciter (top trace) disperses into a nose whistler (bottom trace).

41. In (a), the lower frequency, less damped components appear first, and the higher frequency components near  c are more strongly damped. In (b) we see the results of propagation near the so-called “nose” frequency, which we will study in much more detail later when we examine whistler wave ducting. Here, it will suffice to note that the group velocity maximizes for  / c  0.25, yielding the term “nose” frequency. We see that this frequency component appears first, followed by both a higher frequency component and a lower frequency one of descending tone.
Finally, the wave polarization measurement technique is illustrated in the axial interferometer traces displayed in the following figures.

42. (a) Axial interferometer traces at different dipole polarization angles indicating right-hand circular polarization. (b) Schematic view of measurement principle for determining the wave polarization