= Finite-amplitude stationary channel internal waves
{title2=$\nabla^2\psi'=-N^2\psi'/U^2,\quad\sigma'=N^2\psi'/U$}
In a nonrotating ideal <Boussinesq> layer with uniform current $U>0$, constant <buoyancy frequency> $N$ and rigid walls, any stationary superposition of channel wave modes with $k_n^2+(n\pi/H)^2=N^2/U^2$ obeys the displayed relations. With $(u,w)=(\psi_z,-\psi_x)$, the total streamfunction is $\Psi=Uz+\psi'$ and total buoyancy is $\sigma=(N^2/U)\Psi$. Its material derivative vanishes. The perturbation <vorticity> is proportional to $\psi'$, so its nonlinear self-advection vanishes as well. The modes are therefore exact finite-amplitude solutions, although large amplitudes may violate physical approximation assumptions or become unstable.
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