Acoustic dipole Mach-number scaling (${\rho }^{\prime }/{\rho }_0=O((l/r)M^3)$)

For a small body of size $l$ in flow of speed $u$, force of order ${\rho }_0{u}^2{l}^2$ varying on time $l/u$ produces an acoustic dipole density amplitude of the displayed order, where $M=u/c_0$ is the Mach number. Its sound power scales as ${\rho }_0{u}^6{l}^2/c_0^3$. The corresponding compact acoustic quadrupole Mach-number scaling has density amplitude of order $(l/r)M^4$ and power ${\rho }_0{u}^8{l}^2/c_0^5$.