Examples of 2D wave equations with analytic solutions I need to numerically solve the following wave equation$$\nabla^2\psi(\vec{r},t) - \frac{1}{c(\vec{r})^2}\frac{\partial^2}{\partial t^2}\psi(\vec{r},t) = -s(\vec{r},t)$$ subject to zero initial conditions $$\psi(\vec{r},0)=0, \quad \left.\frac{\partial}{\partial t}\psi(\vec{r},t)\right|_{t=0}=0$$ where $\vec{r} \in \mathbb{R}^2$ and $t \in \mathbb{R}$.
The problem is that I don't know if my numerical solution is right or not, so I wonder if there are some simple cases where the analytic solution can be calculated (besides the green's function, i.e. the solution when $c(\vec{r}) \equiv $ const and $s(\vec{r},t) = \delta(\vec{r},t)$), so I can compare it with my numerical solution. 
Thanks!
 A: In my former life, I wrote finite-difference time-domain solvers for the wave equation (really Maxwell's equations, but we'll keep the geometry simple so we stay scalar and hence valid for acoustics).  We assume a monochromatic wave so the time dependence is $e^{i \omega t}$.  I will also set $s = 0$ and have a nonzero initial condition, but perhaps this is equivalent to there being a source of my plane waves.
So the equation is 
$$\nabla^2 u(\vec{r}) + k^2 u(\vec{r}) = 0$$
If you can deal with this scenario, I have some favorite test cases that are highly nontrivial, that I used to verify my own code.  One is a rigorous solution to the wave equation (in the optics case, a rigorous solution to Maxwell's equations in a particular polarization state), corresponding to diffraction of an incident plane wave by a perfectly reflecting (i.e., perfectly conducting in optics) half-plane.  Thus, on the half-plane, the field is zero.
The solution is due to Sommerfeld and may be found in Born & Wolf, 6th Ed., p. 569.  (The derivation is a thing of beauty which I highly recommend reading in detail.)  The expression is, for a normally incident  plane wave:
$$u(r,\theta) = \frac{e^{-i \pi/4}}{\sqrt{\pi}} \left [ e^{-i k r \sin{\theta}} F(-\sqrt{2 k r} \cos{(\theta/2 - \pi/4)}) - e^{i k r \sin{\theta}} F(-\sqrt{2 k r} \cos{(\theta/2 + \pi/4)}) \right ]$$
where
$$F(y) = \int_y^{\infty} ds \: e^{i s^2}$$
Note that the origin is the edge of the half-plane and $\theta=0$ coincides with the half-plane.  Here is a picture of $|u|^2$:

Note the reflections off the plane, and the diffraction effects around the plane.  I hope this helps.
A: One easy trick is to take $\psi(r,t)$ as your favorite function as long as it satisfies the initial conditions, and then compute the inhomogeneous term $s(r,t)$ from the equation itself.
A: Solution using Green functions and using Sommerfeld radiation condition, in cylindrical coordinates. 
\begin{eqnarray}
u_s(\rho, \phi, t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \{ -i \pi H_0^{(2)}\left(k | \sigma - \sigma_s| \right) F(\omega) e^{i\omega t} d\omega\}
\end{eqnarray}
Where $s$ denotes the source position.
More details and implementation can be found in this post here from Computational Science.It uses the following reference: Morse and Feshbach, 1953, p. 891 - Methods of theoretical physics: New York, MacGraw-Hill Book Co., Inc. 
Here are two snapshots $\mathbb{R}^2$ in [200, 200] lattice with space increment of 100 meters (is not in the axis), velocity 8000 m/s.

The source function $ s(\vec{r},t) = \delta(\rho, \phi)f(t) $ bellow sampled with $\Delta t = 0.05 $ seconds, obviously placed in the origin $(\rho=0,\phi=0)$. In fact can be anything you want as long you can have its Fourier Transform.
 
