= Upstream decay of an open parallel-plate waveguide
{title2=$\phi_{\rm tot}(x,y)\sim b\sqrt{2k_0/(\pi|x|)}\,e^{ik_0x+i\pi/4}$}
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.
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