= Pressure-based acoustic analogy with a potential body force
{title2=$Q,W_{ij}$}
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,
$$
(\widehat c_0^{-2}\partial_t^2-\nabla^2)q=\partial_i\partial_jW_{ij}+Q_{tt},\qquad
W_{ij}=\rho u_i u_j-\sigma_{ij},\qquad Q=q/\widehat c_0^2-(\rho-\widehat\rho_0).
$$
No energy equation is needed for this identity. The arbitrary reference fields should match the stationary medium where the outgoing sound is evaluated.
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