Upstream decay of an open parallel-plate waveguide (${\varphi }_{\mathrm{tot}}(x,y)\sim b\sqrt{2k_0/(\pi |x|)}{e}^{i{k}_{0}x+i\pi /4}$)

For fixed $|y|<b$ and $x\to -\infty$, the incident pole in the scattered transform cancels the analytic continuation of the incident plane wave. Near $k=-k_0$, the remaining transform is proportional to $(k+k_0{)}^{-1/2}$, so the total outgoing field has the displayed inverse-square-root cylindrical decay. A claim of square-root growth at negative infinity is incompatible with this outgoing solution.