= Stratified acoustic pressure equation
{title2=$a=p_0^{1/\gamma}/\rho_0$}
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 give
$$
\rho_0\mathbf u_t=-\nabla p',\qquad
p'_t=-\mathbf u\cdot\nabla p_0-\gamma p_0\nabla\cdot\mathbf u.
$$
Differentiating the second equation and substituting the first yields
$$
c_0^{-2}p'_{tt}-a^{-1}\nabla\cdot(a\nabla p')=0,\qquad
c_0^2=\gamma p_0/\rho_0,\quad a=p_0^{1/\gamma}/\rho_0.
$$
The 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>.
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