Stratified acoustic pressure equation ($a=p_0^{1/\gamma }/{\rho }_0$)

Consider adiabatic perturbations of a static ideal gas, with background pressure $p_0(\mathbf{x})$, mass density ${\rho }_0(\mathbf{x})$ and constant heat-capacity ratio $\gamma$. Assume a prescribed force density $-\nabla \chi$ that does not change when the fluid is perturbed. Static balance is $\nabla (p_0+\chi )=0$. Linearized Euler equations and adiabatic compression then giveDifferentiating the second equation and substituting the first yieldsThe divergence form gives weighted acoustic Green-function reciprocity. Neither uniform mass density nor uniform entropy is required. A prescribed gravitational acceleration instead produces a force density proportional to the perturbed mass density; its extra buoyancy term cannot be omitted under this derivation. The homogeneous limit is the acoustic pressure wave equation.