An acoustic waveguide consists of a long straight tube $z>0$ with square crosssection $0<x<a, 0<y<a$ bounded by rigid walls. The sound speed of the gas in the tube is $c_{0}$. Find the dispersion relation for the propagation of sound waves along the tube. Show that for every dispersive mode there is a cut-off frequency, and determine the lowest cut-off frequency $\omega_{\min }$.

An acoustic disturbance is excited at $z=0$ with a prescribed pressure perturbation $\tilde{p}(x, y, 0, t)=\tilde{P}(x, y) \exp (-i \omega t)$ with $\omega=\frac{1}{2} \omega_{\mathrm{min}}$. Find the pressure perturbation $\tilde{p}(x, y, z, t)$ at distances $z \gg a$ along the tube.