= Solution
Write the density and pressure as $\rho=\rho_0+\rho'$ and $p=p_0+p'$, and let $\mathbf u$ be the small fluid velocity. The <linear homentropic acoustic equations> are
$$
\boxed{
\rho'_t+\rho_0\nabla\mathbin\cdot\mathbf u=0,
\qquad
\rho_0\mathbf u_t=-\nabla p',
\qquad
p'=c_0^2\rho',
}
$$
where the <adiabatic sound speed> is
$$
c_0^2=\left(\frac{\partial p}{\partial\rho}\right)_s.
$$
If the flow is <irrotational flow>[irrotational], write $\mathbf u=\nabla\phi$. The momentum equation integrates spatially to
$$
p'=-\rho_0\phi_t,
$$
a purely time-dependent integration term having been absorbed into $\phi$. Substitution into continuity gives the <wave equation>
$$
\boxed{\phi_{tt}-c_0^2\nabla^2\phi=0,}
$$
so the wave speed is $c_0$.
To obtain energy conservation, dot the momentum equation with $\mathbf u$ and use the <product rule>:
$$
\frac{\partial}{\partial t}\left(\frac12\rho_0|\mathbf u|^2\right)
=-\nabla\mathbin\cdot(p'\mathbf u)+p'\nabla\mathbin\cdot\mathbf u.
$$
The continuity and pressure-density relations give
$$
p'\nabla\mathbin\cdot\mathbf u
=-\frac{p'p'_t}{\rho_0c_0^2}
=-\frac\partial{\partial t}\left(\frac{p'^2}{2\rho_0c_0^2}\right).
$$
Therefore the <acoustic energy conservation> law is
$$
\boxed{\frac{\partial E}{\partial t}+\nabla\mathbin\cdot\mathbf I=0,}
$$
with <acoustic energy density> and <acoustic energy flux>[intensity]
$$
\boxed{
E=\frac12\rho_0|\mathbf u|^2+\frac{p'^2}{2\rho_0c_0^2},
\qquad
\mathbf I=p'\mathbf u.
}
$$
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