An inviscid fluid with sound speed $c_{0}$ occupies the region $0<y<\pi \alpha, 0<z<\pi \beta$ enclosed by the rigid boundaries of a rectangular waveguide. Starting with the acoustic wave equation, find the dispersion relation $\omega(k)$ for the propagation of sound waves in the $x$-direction.

Hence find the phase speed $c(k)$ and the group velocity $c_{g}(k)$ of both the dispersive modes and the nondispersive mode, and sketch the form of the results for $k, \omega>0$.

Define the time and cross-sectional average appropriate for a mode with frequency $\omega$. For each dispersive mode, show that the average kinetic energy is equal to the average compressive energy.

A general multimode acoustic disturbance is created within the waveguide at $t=0$ in a region around $x=0$. Explain briefly how the amplitude of the disturbance varies with time as $t \rightarrow \infty$ at the moving position $x=V t$ for each of the cases $0<V<c_{0}$, $V=c_{0}$ and $V>c_{0}$. [You may quote without proof any generic results from the method of stationary phase.]