Q:

Is it possible to explicitly solve the inhomogeneous Helmholtz equation in a rectangle?

Consider the following Helmholtz equation in a rectangle $\Omega$ and Neumann boundary conditions:
$$
\begin{align}
\Delta u + k^2 u = \delta_y, \quad \quad x \in \Omega, \\
\frac{\partial u}{\partial \nu} = 0, \quad \quad x \in \partial \Omega.
\end{align}
$$
Here $\delta_y$ is some point source emitted from the point $y\in \Omega$. Can an explicit solution be found for this equation?

A:

If you know a free-space solution of
$$
        \Delta v+k^2v=\delta_y
$$
then you can solve for $w$ such that
$$
                   \Delta w+k^2w=0 \\
                   \frac{\partial w}{\partial n}=\frac{\partial v}{\partial n}
$$
and the solution you want will be $v-w$.

