Acoustic equipartition in a rigid waveguide ($⟨E_{\mathrm{kin}}⟩=⟨E_{\mathrm{comp}}⟩$)

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.