Surface representation of an obstacle far-field pattern
= Surface representation of an obstacle far-field pattern
For a scattered field satisfying $\psi_s=e^{ikr}f_\infty(\widehat{\mathbf r})/r+O(r^{-2})$, the <Kirchhoff–Helmholtz representation> gives
$$
f_\infty(\widehat{\mathbf r})=-\frac1{4\pi}\int_{\partial D}e^{-ik\widehat{\mathbf r}\cdot\mathbf r'}[\partial_{n'}\psi_s+ik(\widehat{\mathbf r}\cdot\mathbf n')\psi_s]\,dS'.
$$
The sign assumes the normal points out of the obstacle.