The plane wave solutions we’ve explained in great detail are a useful set of solutions to the wave equation, because they form a complete set of functions, and therefore sums (or integrals) over a set of plane waves can be used to represent any arbitrary wave. However, there are other useful solutions to the wave equation. One of these is a spherical wave:
where is the spherical radius. This is a wave whose constant phase surfaces are spheres. It emanates out in all directions from a spherical or point source. For distances very far from the source, the constant-phase spheres are very large, and for a small observer the spherical wave resembles a plane wave.
There’s a lot that can be said about spherical wave solutions to Maxwell’s Equations (see Jackson’s Classical Electromagnetics text) but we’re not going to pursue this further for now