= Solution
For an inviscid <Boussinesq approximation> fluid in a nonrotating frame, write buoyancy as $\sigma$ and kinematic pressure as $p$. The governing equations are
$$
\frac{D\mathbf u}{Dt}=-\nabla p+\sigma\widehat{\mathbf z},
\qquad
\frac{D\sigma}{Dt}=0,
\qquad
\nabla\mathbin\cdot\mathbf u=0.
$$
They require density variations to be small compared with a constant reference density, while retaining those variations in <buoyancy>; the flow scale must be small compared with the background density scale height. Ideal flow additionally neglects viscosity and scalar diffusion.
Linearize about
$$
\overline{\mathbf u}=\overline u(z)\widehat{\mathbf x},
\qquad
\frac{d\overline\sigma}{dz}=N^2(z)>0,
$$
where $N$ is the <buoyancy frequency>, and define
$$
D_t=\partial_t+\overline u\,\partial_x.
$$
For two-dimensional disturbances, the linear equations are
$$
\boxed{D_tu'+\overline u_z w'=-p'_x},
$$
$$
\boxed{D_tw'=-p'_z+\sigma'},
$$
$$
\boxed{D_t\sigma'+N^2w'=0},
$$
$$
\boxed{u'_x+w'_z=0}.
$$
The condition $N^2>0$ expresses <stable density stratification>.
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