 Finite-amplitude stationary channel internal waves

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 $\Psi =Uz+{\psi }^{\prime }$ and total buoyancy is $\sigma =(N^2/U)\Psi$. Its material derivative vanishes. The perturbation vorticity is proportional to ${\psi }^{\prime }$, 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.