= Solution
Transform in time, leaving $r$ as a parameter. Applying the <derivative> identity twice gives $\widetilde{\partial_t^2\sigma}=-\omega^2\widetilde\sigma$. Therefore
$$
-\omega^2\widetilde\sigma-\partial_r^2\widetilde\sigma+\mu^2\widetilde\sigma=0,
\qquad
\boxed{\partial_r^2\widetilde\sigma=(\mu^2-\omega^2)\widetilde\sigma.}
$$
These manipulations can be justified for sufficiently decaying smooth fields, or with distributional <Fourier transforms> when ordinary integrability is unavailable.
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