= Solution
The acoustic pressure satisfies $p_{tt}=c_0^2\Delta p$. Rigid walls impose vanishing normal velocity, equivalently the <Neumann boundary condition> on pressure. Separation of variables gives transverse modes $\cos(m\pi x/a)\cos(n\pi y/a)$ with $m,n\ge0$. With axial factor $e^{ikz-i\omega t}$, their <dispersion relation> is
$$
\boxed{\omega^2=c_0^2\left[k^2+\frac{\pi^2}{a^2}(m^2+n^2)\right].}
$$
The $(0,0)$ mode is nondispersive. Every other mode propagates only above its <waveguide cutoff frequency> $\omega_{mn}=\pi c_0\sqrt{m^2+n^2}/a$. Thus $\omega_{\min}=\pi c_0/a$.
Expand the prescribed entrance pressure in this cosine basis, with coefficients $A_{mn}$. Impose outgoing propagation and decay at infinity, with no incoming disturbance. At $\omega=\omega_{\min}/2$, only the constant transverse mode propagates, with $k_{00}=\pi/(2a)$. All other modes have $k=i\chi_{mn}$, where $\chi_{mn}=\pi\sqrt{m^2+n^2-1/4}/a$. The exact modal field is the outgoing constant mode plus $\sum_{m+n>0}A_{mn}\cos(m\pi x/a)\cos(n\pi y/a)e^{-\chi_{mn}z-i\omega t}$.
Since the slowest evanescent decay rate is $\sqrt3\pi/(2a)$, the far field is
$$
\boxed{\widetilde p(x,y,z,t)\sim\left[\frac1{a^2}\int_0^a\int_0^a\widetilde P(x,y)\,dx\,dy\right]e^{i\pi z/(2a)-i\omega t},\qquad z\gg a.}
$$
If that area average is zero, there is no propagating far-field component, and the field decays exponentially instead. For sufficiently regular entrance data the omitted terms are $O(e^{-\sqrt3\pi z/(2a)})$.
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