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Let $r=|\mathbf{x}|$, $n_i=x_i/r$, and let the source have size $\ell$. The retarded acoustic Green function gives the outgoing solution of Lighthill acoustic analogy asThis follows by integrating the source derivatives by parts in the retarded convolution. Assume the source is localized and the boundary terms vanish. In the acoustic compact-source approximation, $k_0\ell \ll 1$, the retardation across the source can be neglected, while $r\gg \ell$ permits replacement of the denominator by $r$. The integral becomes $S_{ij}(t-r/c_0)/r$.

In the radiation region $k_{0}r\gg 1$, derivatives of the retarded argument dominate derivatives of the spreading factor. Since ${\partial }_{x_i}(t-r/c_0)=-n_i/c_0$, the leading acoustic quadrupole field isThe two negative retardation derivatives give a positive sign. This is a far-field approximation, with smaller near-field terms omitted.

For low-Mach number aerodynamic fluctuations of speed $U$ and advective time $\ell /U$, take $T_{ij}=O({\rho }_0{U}^2)$, $S_{ij}=O({\rho }_0{U}^2{\ell }^3)$, and two time derivatives of order $(U/\ell {)}^2$. ThereforeThe quoted fourth power is the compact acoustic quadrupole Mach-number scaling, with geometric spreading shown explicitly. It assumes the source strength and time scale just stated; the source must be acoustically compact, and the observation point must remain in the radiation region.