= Solution
For the single-frequency response, use <separation of variables> with $u=R(r)\sin\omega t$. Set $k=\omega/c$ and $V=rR$; then $V''+k^2V=0$, so $V=C\sin kr+D\cos kr$. A finite origin value forces $D=0$, since $\cos kr/r$ is singular. Since $\sin kr/r\to k$, the prescribed origin amplitude gives $C=A/k$. The <regular time-harmonic spherical wave> is therefore
$$
\boxed{u(r,t)=A\frac{\sin(\omega r/c)}{\omega r/c}\sin\omega t},
$$
with the quotient defined by its limit at zero.
This answers the literal finite-origin condition in the class of regular single-frequency responses. It is a standing combination of inward and outward waves: the numerator is proportional to $\cos(kr-\omega t)-\cos(kr+\omega t)$. A purely outgoing wave from an ideal point source would instead be proportional to $r^{-1}\sin[\omega(t-r/c)]$, which has no finite prescribed value $u(0,t)$. Such a source must be specified by a singularity strength, for example $\lim_{r\downarrow0}ru$. The source's wording supplies neither an initial field nor an outgoing singular-source normalization; without restriction to the stated harmonic response, a general initial-value problem is not uniquely determined by the one origin condition.
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