= Acoustic waveguide cutoff
{title2=$\omega_{c,n}=n\pi c/H$}
For an <acoustic waveguide mode> with transverse index $n\geq1$, the <dispersion relation> $\omega^2=c^2[q^2+(n\pi/H)^2]$ permits a propagating axial <wavenumber> only above the displayed <angular frequency>. Below it the outgoing bounded response decays as $e^{-\alpha x}$, with $\alpha=\sqrt{(n\pi/H)^2-\omega^2/c^2}$. The $n=0$ mode has no positive cutoff. At cutoff, a nonzero imposed axial derivative requires $\Phi=A+Bx$, so no bounded periodic solution exists on a semi-infinite channel. Near cutoff, fixed wall-velocity forcing gives amplitudes proportional to $1/|q|$, which can invalidate <linearization>.
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