 Boundary work and mean-flow energy of an internal wave

For a traveling sinusoidal boundary forcing with horizontal phase speed $c_p$, the pressure and horizontal velocity polarization gives the displayed relation between upward energy input and vertical wave momentum flux. A slowly established outgoing wave envelope deposits horizontal mean momentum as its front advances. In the frame moving at $c_p$, the order-amplitude-squared change of mean kinetic energy is $-{\rho }_0{c}_p\int \overline{u}dz$. Its loss rate equals the boundary work in the laboratory frame. The comparison uses a finite perturbation of the infinite uniform background energy.