= Radiation condition for a waveguide mode
{title2=$\Phi_x-iq\Phi\to0$}
For a rightward <acoustic waveguide mode> with time convention $e^{-i\omega t}$, $\omega>0$ and real $q>0$, the displayed condition selects $e^{iqx}$ and excludes waves incident from positive infinity. If $q=i\alpha$ with $\alpha>0$, the corresponding selection is the decaying factor $e^{-\alpha x}$. Boundedness alone cannot distinguish outgoing and incoming propagating waves, while requiring decay for every frequency would incorrectly remove the propagating response. The condition must match the chosen temporal phase convention.
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