 Acoustic waveguide cutoff

For an acoustic waveguide mode with transverse index $n\ge 1$, 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.