Pressure-based acoustic analogy with a potential body force ($Q,W_{ij}$)

Eliminating momentum between the continuity equation and conservation of momentum gives ${\rho }_{tt}={\partial }_i{\partial }_j(\rho u_i{u}_j-{\sigma }_{ij})+{\nabla }^2(p+\chi )$ when the force density is $-\nabla \chi$. For $q=p-(P-\chi )$ and time-independent reference fields,No energy equation is needed for this identity. The arbitrary reference fields should match the stationary medium where the outgoing sound is evaluated.