= Stationary mountain-wave group velocity
{title2=$\widehat{\mathbf c_g}=Uk(k,m)/(k^2+m^2)$}
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.
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