Internal-wave work in a phase-speed frame (${\dot{K}}_{\mathrm{mean}}^{\mathrm{moving}}=-\overline{W}$)

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 $-{\rho }_{0}c\int \overline{u}dz$ 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.