Transfer Function of Earth-ionosphere Waveguide
Sferics can be simulated approximately by the electromagnetic radiation field of a vertical Herz dipole. The maximum spectral amplitude of the sferic typically is near 5 kHz. Beyond this maximum, the spectral amplitude decreases as 1/f if the Earth's surface were perfectively conducting. The effect of the real ground is to attenuate the higher frequencies more strongly than the lower frequencies (Sommerfeld's ground wave).
R strokes emit most of their energy within the ELF/VLF range (ELF = extremely low frequencies, < 3 kHz; VLF = very low frequencies, 3–30 kHz). These waves are reflected and attenuated on the ground as well as within the ionospheric D layer, near 70 km altitude during day time conditions, and near 90 km height during the night. Reflection and attenuation on the ground depends on frequency, distance, and orography. In the case of the ionospheric D-layer, it depends, in addition, on time of day, season, latitude, and the geomagnetic field in a complicated manner. VLF propagation within the Earth-ionosphere waveguide can be described by ray theory and by wave theory.
When distances are less than about 500 km (depending on frequency), then ray theory is appropriate. The ground wave and the first hop (or sky) wave reflected at the ionospheric D layer interfere with each other.
At distances greater than about 500 km, sky waves reflected several times at the ionosphere must be added. Therefore, mode theory is here more appropriate. The first mode is least attenuated within the earth-ionosphere waveguide, and thus dominates at distances greater than about 1000 km.
The Earth-ionosphere waveguide is dispersive. Its propagation characteristics are described by a transfer function T(ρ, f) depending mainly on distance ρ and frequency f. In the VLF range, only mode one is important at distances larger than about 1000 km. Least attenuation of this mode occurs at about 15 kHz. Therefore, the Earth-ionosphere waveguide behaves like a bandpass filter, selecting this band out of a broadband signal. The 15 kHz signal dominates at distances greater than about 5000 km. For ELF waves (< 3 kHz), ray theory becomes invalid, and only mode theory is appropriate. Here, the zeroth mode begins to dominate and is responsible for the second window at greater distances.
Resonant waves of this zeroth mode can be excited in the Earth-ionosphere waveguide cavity, mainly by the continuing current components of lightning flowing between two return strokes. Their wavelengths are integral fractions of the Earth's circumference, and their resonance frequencies can thus be approximately determined by fm ≃ mc/(2πa) ≃ 7.5 m Hz (with m = 1, 2, ...; a the Earth's radius and c the speed of light). These resonant modes with their fundamental frequency of f1 ≃ 7.5 Hz are known as Schumann resonances.
Read more about this topic: Radio Atmospheric
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