= Slowly modulated internal-wave boundary forcing
{title2=$A_T+c_{gz}A_Z=0$}
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 $\omega^2=N^2k^2/(k^2+m^2)$; 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.
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