Mass-injection term in the acoustic analogy ($({\partial }_t^2-c_0^2\Delta ){\rho }^{\prime }={\partial }_i{\partial }_j{T}_{ij}+{\partial }_{t}M-{\partial }_i{S}_i$)

If the continuity equation has mass source $M$ and the conservative momentum equation has momentum source $S_i$, elimination of momentum gives the displayed wave equation. With no imposed momentum source, its extra term is ${\partial }_{t}M$. A compact source of total mass rate $\mathcal{M}=\int MdV$ radiates the acoustic monopole ${\rho }_M^{\prime }=\dot{\mathcal{M}}(t-r/c_0)/(4\pi c_0^{2}r)$. The momentum source supplies an additional acoustic dipole; specifying mass injection alone does not fix that dipole.