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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?

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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 $v-w$.

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  • $\begingroup$ Why does $u=v-w$? How do you know that? $\endgroup$
    – user522521
    Feb 21, 2018 at 22:50
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    $\begingroup$ @majormaki : $\Delta(v-w)+k^2(v-w)=(\Delta v+k^2v)-(\Delta w+k^2w)=\delta_y$ and $\frac{\partial}{\partial n}(v-w)=0$, which are the equations for $u$. $\endgroup$ Feb 21, 2018 at 23:17

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