Radiation condition for a waveguide mode (${\Phi }_x-iq\Phi \to 0$)

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.