= Solution
Put $v(r,t)=ru(r,t)$. Multiplication of the radial <wave equation> by $r$ gives $v_{tt}=c^2v_{rr}$ for $r>0$. In the <characteristic coordinates> $\xi=r+ct$, $\eta=r-ct$, this becomes $v_{\xi\eta}=0$. Integrating first in $\xi$ and then in $\eta$ gives $v=f(\xi)+g(\eta)$. Consequently
$$
\boxed{u(r,t)=\frac{f(r+ct)+g(r-ct)}r}\qquad(r>0).
$$
The two terms are incoming and outgoing radial disturbances. If the field is finite at the origin, their numerators must cancel there; arbitrary independent choices generally produce a $1/r$ singularity.
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