 Stationary mountain-wave group velocity

Define intrinsic positive frequency $\omega$ and ground frequency $\hat{\omega }=Uk+\omega$ for a nonrotating internal gravity wave. Stationarity with $U>0$ selects $k<0$ and $\sqrt{k^2+m^2}=N/U$. Differentiating the full dispersion before imposing that condition gives the displayed ground-frame group velocity. Upward energy propagation requires $m<0$ and $|k|<N/U$. Its ray inclination is ${\cos }^{-1}(U|k|/N)$. The ground-frame group velocity is opposite the normal intrinsic phase velocity; the intrinsic group velocity is perpendicular to that phase velocity. A strictly stationary ground-frame phase pattern has zero phase velocity, so mixing frames can produce a false angle claim.