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An acoustic quadrupole has forcing ${\partial }_i{\partial }_j{T}_{ij}$. The acoustic compact-source approximation reduces its leading far-field density to $n_i{n}_j{\ddot{S}}_{ij}/(4\pi c_0^{4}r)$ in three dimensions, with $S_{ij}=\int T_{ij}{d}^{3}y$ evaluated at retarded time.

Use complex amplitudes with time dependence $e^{i\omega t}$, so the outgoing two-dimensional Helmholtz equation kernel has phase $e^{-i{k}_{0}r}$. Dividing the harmonic form of Lighthill acoustic analogy by $c_0^2$ and integrating the two source derivatives by parts gives an amplitude proportional toOnly the magnitude is needed here; the specified kernel omits its constant phase and normalization. In the acoustic compact-source approximation, $\int {\tilde{T}}_{ij}{d}^{2}y=O({\rho }_0{U}^2{\ell }^2)$. With $\omega =O(U/\ell )$, hence $k_0\ell =O(m)$, the two-dimensional compact quadrupole scaling isThe half-power difference from three dimensions comes from cylindrical spreading, including its $k_0^{-1/2}$ factor. As in part (i), the Mach number power is stated with the geometrical range factor separated; $k_{0}r\gg 1$ is still required.

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.