= Solution
At rest, all time derivatives and advective terms vanish. The <continuity equation> and the stated energy equation are then identities, while the momentum equation reduces to $\boxed{\nabla(p_0+\chi)=0}$. The reference state need not have uniform <mass density> or <entropy>. Assume smooth positive reference <pressure> and <mass density>, and retain only first-order perturbations.
The linearized <continuity equation>, momentum balance and adiabatic energy relation are
$$
\rho'_t+\mathbf u'\cdot\nabla\rho_0+\rho_0\nabla\cdot\mathbf u'=0,
\qquad \rho_0\mathbf u'_t=-\nabla p',
$$
$$
p'_t+\mathbf u'\cdot\nabla p_0
=c_0^2\left(\rho'_t+\mathbf u'\cdot\nabla\rho_0\right)
=-\rho_0c_0^2\nabla\cdot\mathbf u'.
$$
The perturbation of $c^2$ multiplies a vanishing reference material derivative, so it does not enter at first order. Differentiate the last equation in time, then substitute the linearized momentum equation:
$$
p'_{tt}=\frac{\nabla p_0}{\rho_0}\cdot\nabla p'
+\rho_0c_0^2\nabla\cdot\left(\frac{\nabla p'}{\rho_0}\right).
$$
Expanding the divergence yields the <stratified acoustic pressure equation>
$$
\boxed{\frac{p'_{tt}}{c_0^2}-\nabla^2p'
=\left(\frac{\nabla p_0}{c_0^2\rho_0}-\frac{\nabla\rho_0}{\rho_0}\right)\cdot\nabla p'.}
$$
For a <perfect gas>, $c_0^2\rho_0=\gamma p_0$. Put $a(\mathbf x)=p_0^{1/\gamma}/\rho_0$. Then
$$
\nabla\log a=\frac{\nabla p_0}{\gamma p_0}-\frac{\nabla\rho_0}{\rho_0},\qquad
\frac1a\nabla\cdot(a\nabla p')=\nabla^2p'+\nabla\log a\cdot\nabla p'.
$$
Thus the divergence-form <stratified acoustic pressure equation> is
$$
\boxed{c_0^{-2}p'_{tt}-a^{-1}\nabla\cdot(a\nabla p')=0.}
$$
This is homogeneous linear propagation through the static medium, including its inhomogeneity. In the <acoustic analogy> of part (a), $W$ has no first-order contribution when <viscosity> is neglected and the reference velocity is zero. Nevertheless the chosen left-hand operator lacks the gradient terms in the genuine propagation equation. Those effects must consequently appear in $Q_{tt}$. They describe propagation of an existing disturbance, rather than an independent source. Since reference-field choices also change the division between the operator and forcing, \b[$Q_{tt}$ cannot be identified unambiguously with newly generated noise]. Its presence in the right-hand side is a consequence of the chosen analogy, not a source-classification theorem.
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