= Solution
Convolution with the <retarded acoustic Green function>, followed by two integrations by parts in the source coordinate, yields
$$
\rho'(x,t)=\partial_{x_i}\partial_{x_j}\int\frac{T_{ij}(y,t-|x-y|/c_0)}{4\pi c_0^2|x-y|}\,dV_y.
$$
For the <acoustic compact-source approximation>, replace the distance and the source <retarded time> by $r=|x|$ and $\tau=t-r/c_0$ in the leading moment. Thus the integral is $S_{ij}(\tau)/(4\pi c_0^2r)$, where $S_{ij}=\int T_{ij}\,dV$. In the <acoustic far field>, each spatial derivative acts predominantly on retarded time: $\partial_{x_i}\tau=-n_i/c_0$, where $n_i=x_i/r$. Derivatives of $1/r$ and of $n_i$ give lower powers of $r$. The two retarded derivatives therefore give
$$
\boxed{\rho'_Q(x,t)\sim\frac{n_i n_j\ddot S_{ij}(\tau)}{4\pi c_0^4r}
=\frac{x_i x_j\ddot S_{ij}(\tau)}{4\pi c_0^4r^3}}.
$$
This is the compact <acoustic quadrupole> field, retaining its angular stress projection.
For the <compact acoustic quadrupole Mach-number scaling>, let the source length be $\ell$, typical fluctuation speed be $U$, and typical time be $\ell/U$. The low-<Mach number> stress is $T=O(\rho_0U^2)$, so $S=O(\rho_0U^2\ell^3)$ and $\ddot S=O(\rho_0U^4\ell)$. Consequently
$$
\boxed{\rho'_Q/\rho_0=O\left(\frac{\ell}{r}m^4\right),\qquad m=U/c_0}.
$$
With the geometric ratio separated, this is the stated fourth-power dependence. It assumes an advective source time scale and compactness $\omega\ell/c_0=O(m)\ll1$. An externally driven source with an independent <frequency> does not acquire the same Mach power automatically.
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