= Solution
Linearize about a uniform horizontal flow $U$ and a hydrostatic, vertically stratified basic state. The basic-flow <material derivative> is $D_0=\partial_t+U\partial_x$. The first two equations are the horizontal and vertical perturbation momentum balances: <pressure> gradients accelerate the disturbance, while the vertical equation retains the <buoyancy> force $-g\rho'/\rho_0$ after the basic <hydrostatic balance> is subtracted.
The <Boussinesq approximation> neglects fractional <density> variations in <inertia> and enforces <incompressibility>, $u_x+w_z=0$, but retains them in <buoyancy>. Advecting the stable background <density> gradient gives $D_0\rho'+w\,d\bar\rho/dz=0$. With $N^2=-(g/\rho_0)d\bar\rho/dz>0$, this is the fourth supplied equation. A displaced parcel then has a restoring <buoyancy> tendency.
Small amplitude allows products of perturbations to be discarded; two-dimensional motion removes $y$ dependence and transverse flow. Rotation, viscosity, <diffusion>, diabatic heating and mean shear are neglected. $\rho_0$ is the reference inertial <density>, whereas the small background <density> gradient still sets $N$. In a compressible real atmosphere the appropriate stability measure involves <potential temperature>; this Boussinesq system is a local idealization that filters acoustic waves rather than a full compressible atmospheric model.
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