Formula for the eigenfrequencies f_n = (c/2πa)·√(n(n+1))

For an ideal cavity bounded by perfect conductors (infinite conductivity of both the surface and the ionosphere, lossless), the solution of the wave equation on the sphere has a closed form:

f_n = \frac{c}{2\pi a}\sqrt{n(n+1)}, \qquad n = 1, 2, 3, \dots

where c is the speed of light, a \approx 6\,370 km is the Earth's radius and the term \sqrt{n(n+1)} follows from the eigenvalues of the Legendre equation on the sphere. The prefactor comes out as

\frac{c}{2\pi a} \approx 7{,}5\ \text{Hz}.

Substituting for n we obtain the ideal modes: approximately 10.6; 18.4; 26.0; 33.5; 41.1 Hz for n = 1 to 5. These values are systematically higher than the actually observed frequencies — the difference is a consequence of neglecting losses and the finite conductivity of the ionosphere (see #18). The formula therefore serves as a reference "zeroth" estimate, not as a prediction of the measured values.