= Solution
The rest state is $\rho=\rho_0$, $p=p_0$, $\mathbf u=\mathbf0$. At <linear order> in the perturbations, the <continuity equation>, <Euler momentum equation> and the given <equation of state> become
$$
\rho'_t+\rho_0\nabla\cdot\mathbf u=0,\qquad
\rho_0\mathbf u_t=-\nabla p'=-c^2\nabla\rho'.
$$
For the acoustic component take $\mathbf u=\nabla\phi$. This is consistent with initially zero <vorticity>, since taking the <curl> of the linear momentum equation gives $\partial_t(\nabla\times\mathbf u)=0$. Any pre-existing linear vortical component is time independent and is separate from the potential oscillations being sought.
Spatial integration of $\rho_0\nabla\phi_t=-c^2\nabla\rho'$ gives $\rho_0\phi_t+c^2\rho'=K(t)$. Adding a function of time to the <velocity potential> removes $K(t)$ without altering the <velocity field>. Hence
$$
\boxed{\rho'=-\frac{\rho_0}{c^2}\phi_t,\qquad p'=-\rho_0\phi_t.}
$$
Substitute the first relation into <mass conservation>:
$$
-\frac{\rho_0}{c^2}\phi_{tt}+\rho_0\nabla^2\phi=0.
$$
Thus the <velocity potential> satisfies the <wave equation>
$$
\boxed{\phi_{tt}-c^2\nabla^2\phi=0.}
$$
Differentiating that equation in time and using the same density-potential relation gives
$$
\boxed{\rho'_{tt}-c^2\nabla^2\rho'=0.}
$$
Alternatively, differentiate the linear <continuity equation> in time and take the <divergence> of the linear momentum equation. The coefficient $c$ is the <sound speed>, since $dp/d\rho=c^2$.
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