= Internal-wave work in a phase-speed frame
{title2=$\dot K_{\rm mean}^{\rm moving}=-\overline W$}
A right-going, boundary-forced <internal gravity wave> has horizontal momentum per unit inertial density $P=(k/\omega)\overline E$. Its envelope-induced <Eulerian mean> acceleration is $\overline u_t=P_t$ to leading slow order. In the frame moving at horizontal <phase velocity> $c=\omega/k$, the change of mean kinetic energy relative to the initial current $-c$ is $-\rho_0c\int\overline u\,dz$ at quadratic wave order. Its time derivative is minus the boundary work $\overline W=\rho_0\,d\int\overline E\,dz/dt$. The lab-frame mean kinetic energy alone is fourth order in wave amplitude; retaining the moving-frame cross term is essential.
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