Acoustic multipole source

Acoustic sources may be organized by the number of spatial derivatives in their effective wave equation forcing. A direct scalar source is an acoustic monopole, a divergence of a force density an acoustic dipole, and a double divergence of a stress tensor an acoustic quadrupole.

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.

For a low-Mach number source with stress $O({\rho }_0{U}^2)$ and advective time $\ell /U$, the three-dimensional compact acoustic quadrupole has density amplitude $O({\rho }_0{M}^4\ell /r)$, where $M=U/c_0$. The power depends on these source-time and stress hypotheses, not only on the quadrupole label.

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.

An acoustic dipole represents an applied force density. A surface force defect contributes $-{\partial }_i(L_i{\delta }_s)$ to the density wave equation.

An acoustic monopole represents unsteady mass or volume injection. A moving-interface mass-flux defect contributes ${\partial }_t(Q{\delta }_s)$ to the density wave equation.