Far-field acoustic force and stress moments (${\rho }^{\prime }=\frac{n_i{n}_j{\ddot{S}}_{ij}}{4\pi c_0^{4}r}+\frac{n_i{\dot{F}}_i}{4\pi c_0^{3}r}$)

For compact Lighthill stress tensor and applied-force distributions, define $S_{ij}=\int T_{ij}{d}^{3}y$ and $F_i=\int f_i{d}^{3}y$. Differentiating the retarded acoustic Green function at far distance gives the displayed leading radiation field, evaluated at $t-r/c_0$. Source compactness requires its size to be small compared with the acoustic wavelength, not merely with the observer distance.