= Solution
At fixed <frequency>, the transformed equation has characteristic roots $\pm ik$, where
$$
\boxed{k^2=\omega^2-\mu^2.}
$$
For a consistent explicit branch choose
$$
k(\omega)=\begin{cases}
\sqrt{\omega^2-\mu^2},&|\omega|>\mu,\\
i\sqrt{\mu^2-\omega^2},&|\omega|<\mu.
\end{cases}
$$
This branch is even in $\omega$. Away from the two thresholds, the spatial solution is $\widetilde\sigma=A(\omega)e^{ikr}+B(\omega)e^{-ikr}$, so <Fourier inversion> gives
$$
\boxed{\sigma(t,r)=\frac1{2\pi}\int_{-\infty}^{\infty}
\left[A(\omega)e^{ik(\omega)r}+B(\omega)e^{-ik(\omega)r}\right]e^{i\omega t}\,d\omega.}
$$
At $|\omega|=\mu$, the spatial equation has a double zero characteristic root and its full solution is $C+D r$. The two exponentials at $k=0$ represent only its constant part. This isolated-frequency issue does not change an <integral> with ordinary integrable amplitudes; threshold-supported distributional modes require separate treatment.
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