= Solution
Let $\widetilde\rho$, $\widetilde p$, and $u$ be first-order perturbations. Linearized mass and <momentum> conservation and the homentropic relation are
$$
\partial_t\widetilde\rho+\rho_0\nabla\cdot u=0,
\qquad
\rho_0\partial_tu=-\nabla\widetilde p,
\qquad
\widetilde p=c_0^2\widetilde\rho.
$$
Differentiate the first equation in time and use the divergence of the second to obtain
$$
\boxed{\partial_t^2\widetilde p-c_0^2\nabla^2\widetilde p=0.}
$$
The perturbation is irrotational, so write $u=\nabla\phi$. Integrating the <momentum> equation spatially, with a purely time-dependent constant absorbed into $\phi$, gives
$$
\boxed{\widetilde p=-\rho_0\partial_t\phi.}
$$
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