= Boundary work and mean-flow energy of an internal wave
{title2=$\mathcal W=c_p\rho_0\overline{u'w'}$}
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 $-\rho_0c_p\int\overline u\,dz$. Its loss rate equals the boundary work in the laboratory frame. The comparison uses a finite perturbation of the infinite uniform background <energy>.
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