= Wiener-Hopf solution of rigid half-plane diffraction
{c}
{title2=$\widehat\phi^-=-i\sqrt{2k_0}\sin(\theta_0/2)/[(\kappa+k_0\cos\theta_0)\sqrt{\kappa+k_0}]$}
For a rigid screen $x<0$, use time dependence $e^{i\omega t}$ and inverse <Fourier transform> $e^{-i\kappa x}$. The scattered field is odd in $y$. Its upper-face trace is supported on $x<0$, while its normal derivative on $x>0$ is unknown. With the outgoing factor $\gamma=\sqrt{\kappa^2-k_0^2}$, these traces satisfy $\gamma\widehat\phi^-+G^+=b/(\kappa+a)$, where $a=k_0\cos\theta_0$ and $b=k_0\sin\theta_0$. Factor $\gamma=\sqrt{\kappa-k_0}\sqrt{\kappa+k_0}$ into upper/lower analytic factors. <Pole subtraction in a Wiener-Hopf equation> and the finite-energy edge condition give the displayed trace. The forcing-pole <residue> yields <geometrical optics>, while <stationary phase> yields the outgoing edge wave
$$
\phi_d\sim\sqrt{\frac{2}{\pi k_0r}}\frac{\sin(\theta_0/2)\sin(\theta/2)}{\cos\theta+\cos\theta_0}e^{-ik_0r-i\pi/4}.
$$
The angular formula excludes the shadow-boundary transition regions; the exact integral supplies their uniform continuation.
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