Laser Photonics Other wave equation solutions: spherical waves

Other wave equation solutions: spherical waves

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:

\displaystyle \frac{B}{r} \cos{(k r - \omega t)}

where r 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