Maxwell's equations in a cavity resonator

The Schumann resonance is the natural electromagnetic oscillation of the spherical cavity between the Earth's surface and the lower ionosphere. The fields are derived from Maxwell's equations written on a spherical shell with inner radius a (the Earth's surface) and a conductive upper boundary (the ionosphere) Sentman 1995Nickolaenko Hayakawa 2002.

The cavity is dominated by TM modes (transverse magnetic): they have a vertical (radial) electric field E_r and a horizontal magnetic field; the component of E in the direction of propagation is present, whereas H is purely transverse. This corresponds to the physics of the ELF source — lightning as vertical current channels — and to the geometry of the thin cavity, whose thickness (tens of km) is much smaller than the wavelength (thousands of km) and the Earth's circumference.

From the source-free Maxwell's equations, for the harmonic dependence e^{-i\omega t}, one obtains the wave (Helmholtz) equation \nabla^2 \psi + k^2 \psi = 0 with k = \omega/c. Thanks to spherical symmetry the solution separates into a radial and an angular part; the angular part leads to spherical harmonics Y_n^m(\theta,\varphi), respectively Legendre functions P_n(\cos\theta). The indices n number the natural modes of the cavity.