solve a “wave equation” with an extra term

I want to solve the following "wave equation" $$\nabla^2\psi(\vec{r},t) - \frac{1}{c(\vec{r})^2}\frac{\partial^2}{\partial t^2}\psi(\vec{r},t) = R(r)\psi(\vec{r},t)$$ subject to initial conditions $$\psi(\vec{r},0)=\psi_0(\vec{r}), \quad \left.\frac{\partial}{\partial t}\psi(\vec{r},t)\right|_{t=0}=0$$

Note that the source term $R(r)\psi(\vec{r},t)$ depends on the wavefield $\psi(\vec{r},t)$.

I plan to solve it numerically, and treat the computed wavefiled $\psi(\vec{r},m\Delta t)$ at time step $m\Delta t$ as the source term (after multiplying $R(r)$), but I am not sure if this approach is correct, so I wonder if there is any simple case where I can calculate the analytic solution so that I can compare it with my numerical solution to verify my method.

Thanks.

If you set $R,c$ to constants, you obtain a massive scalar field equation (kinda like a time dependent Helmholtz equation).
This has solutions of the form $$\psi = \exp (i\omega t - i \mathbf k \cdot\mathbf x)$$ where you can figure out the relationship needed between the parameters.
Separating variables will also give you many more more complicated solutions if you want them, if you choose $R,c$ nicely, I should think.
That is, you can write $$\psi = f(r) \exp(i\omega t)$$ and obtain an ODE for $f$. Choosing $R,c$ nicely will result in an ODE you can solve!