Regular time-harmonic spherical wave ($u(r,t)=A\sin (\omega t)\sin (\omega r/c)/(\omega r/c)$)

For a spherically symmetric function, $v=ru$ reduces the radial wave equation to $v_{tt}=c^2{v}_{rr}$. A time-harmonic separated solution has radial numerator $C\sin (kr)+D\cos (kr)$, with $k=\omega /c$. Finiteness at the origin excludes $D$, and the prescribed finite origin value fixes $C$. This is a standing combination of incoming and outgoing waves. A purely outgoing point-source wave instead has a $1/r$ singularity and must be specified by its source strength, not by a finite value at the origin.