Acoustic waveguide mode ($\varphi =\Phi (x)\cos (n\pi y/H)e^{-i\omega t}$)

A separated acoustic wave in a straight rigid channel $0<y<H$ has transverse Neumann boundary conditions, giving factors $\cos (n\pi y/H)$ for integers $n\ge 0$. The velocity potential satisfies the wave equation, and its axial factor obeysHere $c$ is the sound speed. Real $q_n$ permits propagation; positive imaginary $q_n$ produces an evanescent wave. A prescribed wall velocity proportional to one transverse factor excites that mode alone in the linear straight-channel model. The physical field is the real part of the complex expression.

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.