= Solution
Use <mass conservation> and conservative <momentum conservation>, with <viscous stress> $\sigma_{ij}$ defined so that its divergence is the viscous force:
$$
\rho_t+\partial_i(\rho u_i)=0,\qquad
(\rho u_i)_t+\partial_j(\rho u_i u_j)=-\partial_i p+\partial_j\sigma_{ij}.
$$
Differentiate the first equation in time and eliminate the momentum time derivative using the second. This gives
$$
\rho_{tt}=\partial_i\partial_j(\rho u_i u_j)+\nabla^2p-\partial_i\partial_j\sigma_{ij}.
$$
Set $\rho'=\rho-\rho_0$ and $p'=p-p_0$, where the reference values and $c_0$ are constant. Subtract $c_0^2\nabla^2\rho'$ to obtain the exact <Lighthill acoustic analogy>
$$
\boxed{(\partial_t^2-c_0^2\nabla^2)\rho'=\partial_i\partial_jT_{ij},\qquad
T_{ij}=\rho u_i u_j+(p'-c_0^2\rho')\delta_{ij}-\sigma_{ij}}.
$$
The <Lighthill stress tensor> includes nonlinear momentum transport, departure from the reference linear pressure-density relation, and <viscous stress>. For an inviscid fluid the last term is absent. The derivation is an exact rearrangement of the conservation equations; treating the tensor as a known localized source is a subsequent approximation.
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