= Kirchhoff–Helmholtz representation
{c}
With the normal directed out of an obstacle $D$ into the fluid, a field solving $(\Delta+k^2)\psi=-Q$ outside $D$ has the representation
$$
\psi(\mathbf r)=\int Q(\mathbf r')G_k(\mathbf r,\mathbf r')\,d\mathbf r'+\int_{\partial D}[\psi\partial_{n'}G_k-G_k\partial_{n'}\psi]\,dS'.
$$
Here $G_k$ is the <outgoing Green function for the three-dimensional Helmholtz equation> and the observation point lies outside the obstacle. The formula follows from <Green second identity>; the normal to the exterior fluid on its inner boundary is the negative of the stated obstacle normal. The surface term represents the scattered field.
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