= Sommerfeld half-plane diffraction
{c}
For a Neumann screen on the negative horizontal axis, the <Helmholtz equation> has an exact solution in <Fresnel integrals>. With time dependence $e^{i\omega t}$, a wave incident from angle $0<\theta_0<\pi$ has total field $e^{ikr\cos(\theta-\theta_0)}\mathcal F(s_-)+e^{ikr\cos(\theta+\theta_0)}\mathcal F(-s_+)$, where $s_\pm=\sqrt{2kr}\cos[(\theta\pm\theta_0)/2]$ and $\mathcal F(s)=\tfrac12\operatorname{erfc}(-e^{i\pi/4}s)$. The angular derivatives cancel on both faces of the screen. Away from the two shadow boundaries, the <Fresnel tail expansion> splits this into incident/reflected <geometrical optics> fields and an outgoing edge wave proportional to $r^{-1/2}e^{-ikr}$.
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