Far-field pattern

The far-field pattern is the directional coefficient $f_{\infty }(\hat{\mathbf{r}})$ in an outgoing asymptotic field $u_s(\mathbf{r})=e^{ikr}{f}_{\infty }(\hat{\mathbf{r}})/r+O(r^{-2})$.

The Sommerfeld radiation condition selects outgoing solutions of the exterior Helmholtz equation. In three dimensions it requires $r({\partial }_{r}u-iku)\to 0$ as $r\to \infty$ uniformly in direction.