The analytical model of a uniform cavity (normal modes)

The simplest approach assumes a uniform (homogeneous, ideally symmetric) spherical cavity and seeks its normal modes — the natural oscillations that the cavity supports. For such an idealization an analytical solution exists.

The eigenfrequencies of the modes are given by the relation

f_n = \frac{c}{2\pi a}\sqrt{n(n+1)},

where c is the speed of light, a is the radius of the Earth and n = 1, 2, 3, \dots is the mode number. This relation gives a series of resonances whose spacing increases with n. The spectra of the individual modes have a Lorentzian shape in this model (resonance peaks with a characteristic width given by the losses in the cavity).

The main advantage is speed: the formula provides an immediate order-of-magnitude estimate of the frequencies without the need for numerical simulation. The price for the simplicity is that the model does not capture the inhomogeneities of the real cavity — the day-night asymmetry, geographic differences in ionospheric conductivity, or line splitting of modes. It therefore serves as a reference starting point and a sanity check for more demanding methods.