= Acoustic equipartition in a rigid waveguide
{title2=$\langle E_{\rm kin}\rangle=\langle E_{\rm comp}\rangle$}
For a separated acoustic potential with transverse Neumann eigenvalue $\kappa^2$, integration by parts gives $\int|\nabla_\perp F|^2=\kappa^2\int F^2$. The time-averaged kinetic energy is proportional to $(k^2+\kappa^2)\int F^2$, while compression energy is proportional to $\omega^2\int F^2/c_0^2$. The dispersion relation $\omega^2=c_0^2(k^2+\kappa^2)$ proves equality. At a ray speed below $c_0$, nondegenerate stationary phase gives dispersive amplitudes of order $t^{-1/2}$; the zero transverse mode instead preserves an undispersed pulse on the sound-speed ray.
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