Mixed Dirichlet-Neumann half-plane diffraction ($\beta (\alpha )=\sqrt{k^2-{\alpha }^2}$)

For a scalar wave above a line with a Dirichlet half and a Neumann half, the boundary transform factors into ${\beta }_{+}(\alpha )=\sqrt{k+\alpha }$ and ${\beta }_{-}(\alpha )=\sqrt{k-\alpha }$. For $e^{-i\omega t}$, the limiting absorption principle puts $-k$ below and $k$ above the inversion contour. With Dirichlet conditions on the negative half-line and incident tangential wavenumber $a$, pole subtraction gives the diffracted trace $U_{+}=2i\sqrt{k-a}/[(\alpha +a)\sqrt{k+\alpha }]$. Its large-transform decay gives a bounded field with a square-root edge trace. The normal derivative has an inverse-square-root singularity on the Dirichlet half; it is identically zero on the Neumann half.