= Solution
For the proposed <plane wave>, define the positive squared effective <wavenumber>
$$
\mathcal K^2=k^2+l^2+\frac{f_0^2}{N^2}m^2.
$$
The horizontal <Laplacian> contributes $-(k^2+l^2)\psi'$ and the vertical stretching operator contributes $-(f_0^2/N^2)m^2\psi'$. Substitution in the <Linear Rossby-wave equation> therefore gives
$$
[-i(\omega-Uk)](-\mathcal K^2)+i\beta k=0,
\qquad
\boxed{\omega=Uk-\frac{\beta k}{k^2+l^2+(f_0^2/N^2)m^2}.}
$$
The intrinsic <phase velocity> is westward for positive $\beta$. A stationary disturbance with $\omega=0$ and $k\ne0$ must have its westward intrinsic <phase velocity> exactly cancelled by the eastward background current. Solving for the squared vertical <wavenumber> yields
$$
m^2=\frac{N^2}{f_0^2}\left(\frac\beta U-k^2-l^2\right).
$$
For <Earth>, with northward $y$, $\beta=2\Omega\cos\phi/a>0$ in both hemispheres away from the poles. Thus an oscillatory vertical structure requires
$$
\boxed{0<U<\frac\beta{k^2+l^2},\qquad N^2>0,\quad f_0\ne0,\quad k\ne0.}
$$
The background flow must be eastward but not too strong for the chosen horizontal scale. Long horizontal <wavelengths> have the largest allowed speed. An easterly current, $U<0$, gives $m^2<0$ and an <evanescent wave>; so does an eastward current exceeding the upper bound. At equality $m=0$ and the vertical <group velocity> vanishes. A zonally uniform disturbance, $k=0$, is a degenerate stationary pattern with no vertical <Rossby wave> energy propagation, so it cannot be obtained by cancelling $k$ in the preceding calculation.
A real $m$ alone does not select \i[upward] propagation. Differentiate the <dispersion relation> at fixed $k,l$ to obtain the vertical <group velocity>:
$$
c_{g,z}=\frac{\partial\omega}{\partial m}
=\frac{2\beta k(f_0^2/N^2)m}
{[k^2+l^2+(f_0^2/N^2)m^2]^2}.
$$
The outgoing <radiation condition> for a surface source requires
$$
\boxed{c_{g,z}>0\quad\Longleftrightarrow\quad km>0\quad(\beta>0).}
$$
For the convention $k>0$, take $m>0$. The stationary <phase> lines then obey $dx/dz=-m/k<0$: they tilt westward with height while <wave energy> travels upward. The real conjugate component $(-k,-l,-m)$ describes the same physical upward-propagating wave.
The figure suggests a winter extratropical path from the lower <terrestrial atmosphere> into the <stratosphere>, where the mean <zonal flow> is eastward. The summer <stratosphere> has westward flow and blocks such stationary <Rossby waves> in this idealization. A zero-wind surface is a stationary-wave <critical layer in a shear flow>; it also bounds the winter westerlies aloft near the upper part of the diagram. For a slowly varying real wind profile, local propagation needs $m^2(z)>0$ along the path, sufficient surface forcing, and no intervening absorbing <critical layer in a shear flow>. A layer where the eastward wind becomes too strong is a turning barrier for the given horizontal <wavenumbers>. The uniform-current result is a local guide, not a complete prediction through a sheared spherical <terrestrial atmosphere>.
For scale, at $60^\circ$ latitude $\beta\simeq1.14\times10^{-11}\ \mathrm{m^{-1}\,s^{-1}}$. A planetary <zonal wavenumber> $s$ has local $k=s/(a\cos\phi)$. Taking $l=0$ gives the most permissive limit
$$
U_{\max}=\frac{\beta a^2\cos^2\phi}{s^2}
\simeq\frac{116}{s^2}\ \mathrm{m\,s^{-1}}.
$$
The resulting limits are approximately $116$, $29$ and $13\ \mathrm{m\,s^{-1}}$ for $s=1,2,3$. Therefore the long planetary waves are favored; the strong core of the winter vortex can exclude shorter ones even though it is westerly. Nonzero meridional <wavenumber> lowers these limits further. A compressible <log-pressure coordinate> model would also contain a density-scale-height correction; it is absent from the equation supplied here and should not be silently inserted.
Interchanging the Northern and Southern Hemispheres changes the sign of $f_0$, but only $f_0^2$ enters the <dispersion relation> and vertical <group velocity>. With the same eastward $x$ and northward $y$, $\beta$ stays positive. Thus the \b[propagation criterion is the same in both hemispheres]: it selects the winter westerlies, irrespective of which hemisphere is winter. Actual amplitudes can differ because mountains, land-sea contrasts, forcing spectra and mean winds differ between hemispheres; those source differences do not reverse this propagation criterion.
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