Consider an isotropic wave propagating outward from a central point. The wave equation is given by
where v is the speed of the wave, but in spherical coordinates with no - or -dependence (i.e., having angular symmetry), the Laplacian Eric Weisstein's World of Math simplifies, giving
However, this can be rewritten
which, from the wave equation, has the general solution
The harmonic solution is then given by
where is the angular frequency, k is the wavenumber, and is the phase, which can be rewritten as
References
Bekefi, G. and Barrett, A. H. Electromagnetic Vibrations, Waves, and Radiation. Cambridge, MA: MIT Press, pp. 158-161, 1987.
