= Solution
Take $U_0>0$, $k=2\pi/L>0$, and represent the stationary <topographic internal gravity wave> by phase $kx+mz$. Its laboratory <angular frequency> is zero, and its <intrinsic frequency> is $\widehat\omega=-kU_0$. The <internal gravity wave> <dispersion relation> gives
$$
m^2=\frac{N_0^2}{U_0^2}-k^2.
$$
The <radiation condition> selects $m>0$, since this makes the vertical <group velocity> upward. The lower <kinematic boundary condition> is $w=U_0h_x$, so a corresponding linear wave field is
$$
w=kU_0h_0\cos(kx+mz),\qquad
\zeta=h_0\sin(kx+mz),
$$
where $\zeta$ is the vertical <fluid displacement>. The <constant-phase lines of an internal gravity wave> slope down to the right, with $dz/dx=-k/m$.
Let $K^2=k^2+m^2$. In the frame fixed to the hills the phase pattern is stationary: its <phase velocity> is zero, while the <group velocity> is
$$
\boxed{\mathbf c_g=\frac{U_0}{K^2}(k^2,km).}
$$
Energy travels up and downwind, with ray slope $dz/dx=m/k$. In coordinates $x'=x-U_0t$ moving with the wind, the phase is $kx'+mz+kU_0t$. Thus
$$
\boxed{\mathbf c_p'=-\frac{kU_0}{K^2}(k,m),\qquad
\mathbf c_g'=\frac{U_0}{K^2}(-m^2,km).}
$$
The intrinsic <phase velocity> is upstream and downward; the intrinsic <group velocity> is upstream and upward, perpendicular to the <wave vector>. Adding $(U_0,0)$ to the group velocity recovers the laboratory energy ray.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/iii/paper-71-mountain-frames.png]
{title=Stationary mountain-wave crests and energy directions in the ground and wind frames}
Vertically propagating waves require $kU_0<N_0$, equivalently $\boxed{L>2\pi U_0/N_0}$. At equality the vertical <group velocity> vanishes; above the <mountain-wave cutoff> the bounded response is an <evanescent wave>. The <linearization> requires small terrain slope $kh_0\ll1$ and small wave-induced displacement gradients $|m|h_0\ll1$, often summarized by $N_0h_0/U_0\ll1$. The model also neglects rotation, viscosity and compressibility; its scales must permit those approximations.
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