= Stratified acoustic pressure equation
{title2=$a=p_0^{1/\gamma}/\rho_0$}
For adiabatic perturbations of a static perfect gas, pressure obeys
$$
c_0^{-2}p'_{tt}-a^{-1}\nabla\cdot(a\nabla p')=0,\qquad a=p_0^{1/\gamma}/\rho_0.
$$
The stationary balance is $\nabla(p_0+\chi)=0$. Density and entropy need not be spatially uniform. Expanding the divergence exposes the gradient terms that distinguish this equation from the homogeneous <acoustic pressure wave equation>. Its divergence form yields <weighted acoustic Green-function reciprocity>.
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