Two-dimensional compact quadrupole scaling
= Two-dimensional compact quadrupole scaling
{title2=$\rho'/\rho_0=O(M^{7/2}\sqrt{\ell/r})$}
In two dimensions, the outgoing harmonic <Helmholtz equation> kernel has magnitude proportional to $(k_0r)^{-1/2}$. 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.