= Outgoing modes of an open rigid acoustic waveguide
{title2=$k_n^2=k_0^2-(n\pi/b)^2$}
For an even field between rigid plates at $y=\pm b$, the transverse modes are $\cos(n\pi y/b)$, with longitudinal wave numbers $k_n=\sqrt{k_0^2-(n\pi/b)^2}$. With time dependence $e^{i\omega t}$, outgoing modes into $x>0$ use $e^{-ik_nx}$: choose positive real $k_n$ for cut-on modes and negative imaginary $k_n$ for cut-off modes. A lower-half-plane <Fourier inversion> extracts these modes by <residues>. Their decaying components are <evanescent waves>, not backward-propagating waves. At an exact cutoff one takes the radiation continuation before its limit.
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