= Solution
Use complex amplitudes with time dependence $e^{i\omega t}$, so the outgoing two-dimensional <Helmholtz equation> kernel has phase $e^{-ik_0r}$. Dividing the harmonic form of <Lighthill acoustic analogy> by $c_0^2$ and integrating the two source derivatives by parts gives an amplitude proportional to
$$
\widetilde\rho'\sim\frac{k_0^2}{c_0^2}\frac{e^{-ik_0r}}{\sqrt{k_0r}}\,n_in_j\int\widetilde T_{ij}(\mathbf y)\,d^2y.
$$
Only the magnitude is needed here; the specified kernel omits its constant phase and normalization. In the <acoustic compact-source approximation>, $\int\widetilde T_{ij}d^2y=O(\rho_0U^2\ell^2)$. With $\omega=O(U/\ell)$, hence $k_0\ell=O(m)$, the <two-dimensional compact quadrupole scaling> is
$$
\frac{|\widetilde\rho'|}{\rho_0}=O\left[\frac{U^2}{c_0^2}(k_0\ell)^2(k_0r)^{-1/2}\right]=\boxed{O\left(m^{7/2}\sqrt{\ell/r}\right).}
$$
The 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; $k_0r\gg1$ is still required.
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