Spherically Symmetric Solutions

The general spherically symmetric solution can be written

$$\Psi(r,t) = \frac{1}{r}\left(f(r+t)+g(r-t)\right)$$ (part1071)

where the functions $f$ and $g$ can be freely chosen.

Making the additional requirement of time symmetry at $t=0$, forces

$$f(r)=g(r)$$ (part1072)

Thus if the solution at t=0 is given by $\phi(r)$, the general solution will be

$$\Psi(r,t) = \frac{1}{2r}\left( (r+t)\phi(r+t)+(r-t)\phi(r-t) \right)$$ (part1073)