= Solution
Use an <acoustic velocity potential> $\varphi$ with $\mathbf u=\nabla\varphi$ and $p'=-\rho_0\varphi_t$. It obeys the <wave equation> $\varphi_{tt}=c_0^2\nabla^2\varphi$. The rigid walls require zero normal velocity, hence <Neumann boundary conditions> on the transverse variables. Separation gives
$$
\varphi=\operatorname{Re}\left\{A e^{i(kx-\omega t)}\cos(my/\alpha)\cos(nz/\beta)\right\},\qquad m,n\in\{0,1,2,\ldots\}.
$$
Define $\kappa_{mn}^2=m^2/\alpha^2+n^2/\beta^2$. The <dispersion relation>, <phase velocity> and <group velocity> for positive $k,\omega$ are
$$
\boxed{\omega=c_0\sqrt{k^2+\kappa_{mn}^2},\qquad c=\frac\omega k=c_0\sqrt{1+\frac{\kappa_{mn}^2}{k^2}},\qquad c_g=\frac{c_0k}{\sqrt{k^2+\kappa_{mn}^2}}=\frac{c_0^2}{c}.}
$$
For a dispersive mode, $\omega(0)=c_0\kappa_{mn}>0$, $c$ decreases from infinity to $c_0$, and $c_g$ increases from zero to $c_0$. The $(0,0)$ mode has $\omega=c_0k$ and $c=c_g=c_0$: it propagates without dispersion.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2011/ii/paper-1-waveguide-dispersion.png]
{title=Acoustic waveguide dispersion and velocities; a nonzero transverse wavenumber is compared with the nondispersive mode}
Let $\mathcal A=\pi^2\alpha\beta$ be the cross-sectional area. The appropriate time and cross-sectional average at fixed $x$ is
$$
\langle h\rangle=\frac{\omega}{2\pi\mathcal A}\int_0^{2\pi/\omega}\int_0^{\pi\alpha}\int_0^{\pi\beta}h\,dz\,dy\,dt.
$$
Write $F_{mn}(y,z)$ for the product of cosines and $Q=\mathcal A^{-1}\int F_{mn}^2\,dy\,dz$. Its transverse eigenvalue and wall conditions give $\int|\nabla_\perp F_{mn}|^2=\kappa_{mn}^2\int F_{mn}^2$ by integration by parts. The <kinetic energy density> and compressive acoustic energy density are $\rho_0|\mathbf u|^2/2$ and $(p')^2/(2\rho_0c_0^2)$. Time averaging each squared real oscillation contributes a factor $1/2$, so
$$
\boxed{\langle E_{\rm kin}\rangle=\frac{\rho_0|A|^2Q}{4}(k^2+\kappa_{mn}^2)=\frac{\rho_0|A|^2Q\omega^2}{4c_0^2}=\langle E_{\rm comp}\rangle.}
$$
This holds for every dispersive mode, and also for the zero mode.
For localized smooth initial data, a dispersive mode contains integrals of the form $\int a(k)e^{it[kV-\omega(k)]}\,dk$ along $x=Vt$. When $0<V<c_0$, a stationary point exists at $k_* =\kappa_{mn}V/\sqrt{c_0^2-V^2}$. It is nondegenerate because $\omega''(k)=c_0\kappa_{mn}^2/(k^2+\kappa_{mn}^2)^{3/2}>0$. The <stationary phase method> therefore gives an oscillatory amplitude $O(t^{-1/2})$ for each excited dispersive mode, and for their sum under sufficient mode summability.
At $V=c_0$, no dispersive mode has a finite stationary wavenumber; with smooth localized data these contributions decay, while the nondispersive right-moving pulse retains its original amplitude. Thus a general disturbance with nonzero zero-mode component has \b[an undiminished $O(1)$ pulse on the sound-speed ray]. If that component is absent, this surviving contribution is absent too. For $V>c_0$, <finite propagation speed> puts a compactly supported initial disturbance outside this ray for all sufficiently large $t$, so \b[the field is eventually zero]. Rapidly localized, rather than strictly compact, data give a correspondingly negligible tail instead.
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