= Solution
With buoyancy perturbation $b=-g\rho'/\rho_0$, the inviscid <Boussinesq equations> are
$$
\frac{D\mathbf u}{Dt}=-\frac1{\rho_0}\nabla p+b\widehat{\mathbf z},
\qquad
\nabla\mathbin{\cdot}\mathbf u=0,
\qquad
\frac{Db}{Dt}+N^2w=0.
$$
For
$$
\mathbf u=\nabla\times(\psi\widehat{\mathbf y})
=(-\psi_z,0,\psi_x),
$$
incompressibility is automatic. Define the <Jacobian determinant>
$$
J(A,B)=A_xB_z-A_zB_x.
$$
The material derivative is $D/Dt=\partial_t+J(\psi,\mathord\cdot)$. Taking the $y$ component of the curl of momentum gives the exact finite-amplitude system
$$
\boxed{
\partial_t\nabla^2\psi+J(\psi,\nabla^2\psi)=b_x,
}
$$
$$
\boxed{
\partial_tb+J(\psi,b)+N^2\psi_x=0.
}
$$
If the disturbance amplitude is small enough that each Jacobian is asymptotically smaller than its corresponding time derivative, the system can be linearized. Eliminating $b$ then gives
$$
\boxed{
\partial_t^2\nabla^2\psi+N^2\partial_x^2\psi=0
}.
$$
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