Slowly modulated internal-wave boundary forcing ($A_T+c_{gz}{A}_Z=0$)

For a plane internal gravity wave in uniform buoyancy frequency, substitute $\zeta =A(\mu t,\mu z)e^{i(kx+mz-\omega t)}$ into the displacement equation. The leading order gives ${\omega }^2=N^2{k}^2/(k^2+m^2)$; the next order gives the displayed transport equation with $c_{gz}=-\omega m/(k^2+m^2)$. A boundary amplitude $a(T)$ radiates as $A=a(T-Z/c_{gz})$, choosing the sign of $m$ so the group velocity points into the fluid. The wavelength-averaged energy is $N^2|A{|}^2/2$ and its vertical flux is $c_{gz}$ times that energy. A causal ramp therefore creates a translating front, not an instantaneous monochromatic field throughout the half-space.

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.