The complex refractive index and the phase velocity of ELF waves

The lossiness of the cavity is formally summarized in the complex refractive index. For a harmonic wave e^{i(kz-\omega t)} one introduces a complex wave number k = \omega\, \tilde{n}/c, where \tilde{n} = n' + i n'' is the complex refractive index: the real part n' determines the phase velocity, the imaginary part n'' the attenuation. The phase velocity of ELF waves is then

v = \frac{c}{n'}.

Because the cavity is lossy and the ionosphere has finite conductivity, $n' > 1$, and therefore the phase velocity $v < c$ — on the order of v \approx 0{,}7-0{,}8\,c. This is another, equivalent way of explaining the lower observed frequencies relative to the ideal (see #18): the wave circles the Earth more slowly than light in vacuum, so the resonance condition "an integer number of wavelengths per circumference" is satisfied at a lower frequency.

The complex refractive index (and thus also v and the attenuation) depends on the conductivity profile — that is, on the two characteristic heights (#21) and the "knee" model (#22), which determine its value across the SR band.