The spherical resonator and eigenvalues (eigenmodes)

The Earth-ionosphere cavity can be understood as a spherical resonator: waves with a wavelength comparable to the Earth's circumference (\approx 40\,000 km) combine around the circumference into a standing wave. Resonance occurs when an integer number of wavelengths "goes around" the planet — hence the discrete spectrum of natural modes (eigenmodes) numbered n = 1, 2, 3, \dots

Mathematically the angular number n arises from the requirement that the solution of the Legendre equation be regular over the whole sphere; only integer values of n are admissible, to which belong the Legendre polynomials P_n(\cos\theta). The mode n has n nodal circles on the sphere, so n=1 is the lowest (dipole) configuration, n=2 the quadrupole one, etc. Each mode is, in the ideal (spherically symmetric) cavity, (2n+1)-fold degenerate in the azimuthal number m — this degeneracy is lifted in the real, inhomogeneous cavity and leads to the splitting of modes (see #24).

The eigenvalue determines the resonance frequency f_n of the given mode. For the ideal cavity it has a closed form (see #17), for the real cavity it is obtained numerically.