= Solution
A mountain width is not an exactly specified sinusoidal wavelength. Taking its characteristic horizontal inverse length as $|k|\sim1/a$ with $a=10\,\mathrm{km}$ gives $|k|\sim10^{-4}\,\mathrm{m^{-1}}$. Since $N/U=10^{-3}\,\mathrm{m^{-1}}$, the stationary vertical <wavenumber> is
$$
m=-\sqrt{(N/U)^2-k^2}\simeq-9.95\times10^{-4}\,\mathrm{m^{-1}},\qquad
\boxed{\lambda_z=\frac{2\pi}{|m|}\simeq6.3\,\mathrm{km}.}
$$
This is close to the long-horizontal-scale/hydrostatic estimate $2\pi U/N=6.28\,\mathrm{km}$. If instead the quoted width is identified with a sinusoidal wavelength $\lambda_x=10\,\mathrm{km}$, then $|k|=2\pi/\lambda_x$ and $\lambda_z\simeq8.1\,\mathrm{km}$. Thus several kilometres, roughly $6$–$8\,\mathrm{km}$ under these two common width conventions, is the justified estimate; the terrain profile would fix its actual forcing spectrum.
For the inverse-width choice, both $k$ and $m$ are negative for upward <energy> radiation. The stationary phase lines $kx+mz=\text{constant}$ tilt upstream with increasing height, while <energy> propagates upward and slightly downstream. Numerically $\widehat{\mathbf c_g}\simeq(0.10,0.995)\,\mathrm{m\,s^{-1}}$, with ray inclination about $84.3^\circ$. Intrinsic <phase velocity>[phase propagation] points upstream and downwards; the ground pattern itself remains fixed. The original sketch below uses this explicit convention.
A ground observer may see stationary lenticular or wave <clouds> where ascent cools sufficiently moist air, with continuing air flow through a <cloud> whose position remains almost fixed. Descending regions can clear or warm. An aircraft crossing the pattern encounters alternating updraughts and downdraughts; gliders can exploit the lift. Strong amplitudes, shear or rotors may cause <turbulence>, but a linear <gravity wave> can also be smooth, and clear-air waves need not produce any visible <cloud>. Mountain height, humidity and nonlinear stability are not supplied, so neither a vertical-wind amplitude nor a definite <atmospheric precipitation> forecast follows.
\Image[/past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2018-stationary-mountain-waves.png]
{title=Stationary mountain-wave phase lines, intrinsic phase direction, energy rays and an upper-layer propagation cutoff}
{description=Original schematic and dispersion calculation. The ridge height is illustrative; the phase sketch uses $|k|=1/(10\,\mathrm{km})$, while the upper-layer panel shows the cutoff for the reduced Brunt–Väisälä frequency. No cloud or turbulence amplitude is inferred.}
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