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

Question

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?

Answer 1

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 $v-w$.