= Acoustic waveguide mode
{title2=$\phi=\Phi(x)\cos(n\pi y/H)e^{-i\omega t}$}
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 $n\geq0$. The <velocity potential> satisfies the <wave equation>, and its axial factor obeys
$$
\Phi''+q_n^2\Phi=0,\qquad q_n^2=\omega^2/c^2-(n\pi/H)^2.
$$
Here $c$ 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.
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