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.



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.

You can also linearly combine these (effectively Fourier transforms) to get more solutions.

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!


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