= Solution
Use the conservative <continuity equation> and momentum balance, with $\sigma_{ij}$ the viscous stress:
$$
\rho_t+\partial_i(\rho u_i)=0,\qquad \partial_t(\rho u_i)+\partial_j(\rho u_iu_j)=-\partial_i p+\partial_j\sigma_{ij}.
$$
Differentiate the first equation in time and take the divergence of the second. Eliminating momentum gives
$$
\rho_{tt}=\partial_i\partial_j(\rho u_iu_j+p\delta_{ij}-\sigma_{ij}).
$$
Subtract $c_0^2\Delta\rho$, and define the <Lighthill stress tensor> by
$$
\boxed{T_{ij}=\rho u_iu_j+[(p-p_0)-c_0^2(\rho-\rho_0)]\delta_{ij}-\sigma_{ij}.}
$$
The constant reference subtraction has no double divergence. The exact <Lighthill acoustic analogy> is
$$
\boxed{(\partial_t^2-c_0^2\Delta)\rho'=\partial_i\partial_jT_{ij},\qquad \rho'=\rho-\rho_0.}
$$
This is an exact rearrangement; linearization is not used to eliminate the nonlinear source tensor.
Convolving with the <retarded acoustic Green function> and transferring the source derivatives by <integration by parts> gives
$$
\rho'(\mathbf x,t)=\frac1{4\pi c_0^2}\partial_{x_i}\partial_{x_j}\int\frac{T_{ij}(\mathbf y,t-|\mathbf x-\mathbf y|/c_0)}{|\mathbf x-\mathbf y|}\,d^3y.
$$
The <acoustic compact-source approximation> requires source size $\ell$ to satisfy $\Omega\ell/c_0\ll1$ for its characteristic frequency $\Omega$; the <acoustic far field> additionally requires $r\gg\ell$ and $\Omega r/c_0\gg1$. Internal retardation is then negligible at leading order, and the radiating parts of the two observer derivatives act on the common <retarded time>. With $n_i=x_i/r$ and $S_{ij}=\int T_{ij}\,d^3y$, the <far-field acoustic force and stress moments> give
$$
\boxed{\rho'(\mathbf x,t)\sim\frac{n_in_j}{4\pi c_0^4r}\ddot S_{ij}(t-r/c_0).}
$$
The two spatial derivatives supply two factors of $1/c_0$, in addition to the $1/c_0^2$ in the Green function.
For a subsonic fluctuating source of velocity scale $V$, size $\ell$ and turnover time $\ell/V$, suppose its stress scale is $T_{ij}=O(\rho_0V^2)$. Then $S_{ij}=O(\rho_0V^2\ell^3)$ and $\ddot S_{ij}=O(\rho_0V^4\ell)$. The <compact acoustic quadrupole Mach-number scaling> is therefore
$$
\boxed{\frac{\rho'}{\rho_0}=O\left(\frac\ell r\left(\frac V{c_0}\right)^4\right)=O((\ell/r)m^4).}
$$
This fourth-power statement concerns the acoustic density amplitude under the specified eddy-time and stress scaling. With an independently imposed frequency, the general estimate instead retains $V^2\Omega^2\ell^3/c_0^4r$.
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