= Solution
For a single phase $\phi=kx+mz-\omega t$, every disturbance field is a function of $\phi$ alone. Its velocity is tangent to the phase surfaces:
$$
\mathbf k\mathbin{\cdot}\mathbf u
=k(-\psi_z)+m\psi_x=0.
$$
Consequently both nonlinear Jacobians vanish exactly, irrespective of amplitude. Substitution gives
$$
i\omega(k^2+m^2)\widetilde\psi=ik\widetilde b,
\qquad
-i\omega\widetilde b+iN^2k\widetilde\psi=0.
$$
A nonzero amplitude therefore requires the <internal gravity wave> dispersion relation
$$
\boxed{
\omega^2=\frac{N^2k^2}{k^2+m^2}
}.
$$
The wave is an exact nonlinear solution of the inviscid Boussinesq equations. The parameter $\epsilon$ merely sets its amplitude; it does not change its frequency or waveform. Very large $\epsilon$ can nevertheless invalidate the Boussinesq model or make the total stratification locally overturn.
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