Two-dimensional compact quadrupole scaling (${\rho }^{\prime }/{\rho }_0=O(M^{7/2}\sqrt{\ell /r})$)

In two dimensions, the outgoing harmonic Helmholtz equation kernel has magnitude proportional to $(k_{0}r{)}^{-1/2}$. Combining this spreading with the compact acoustic quadrupole source and $k_0\ell =O(M)$ changes the density-amplitude Mach number power to $7/2$ when geometric range is separated.