= Solution
Hold $l,m,f_0,N$ fixed when differentiating the <dispersion relation> of <Rossby waves>, so that $\alpha^2\geq0$ is independent of $k$. For $k\ne0$, stationary zonal phase requires
$$
0=\frac\omega k=U-\frac\beta{k^2+\alpha^2},
\qquad U=\frac\beta{k^2+\alpha^2}.
$$
The zonal <group velocity> measured relative to the ground is
$$
c_{g,x}=\frac{\partial\omega}{\partial k}
=U+\frac{\beta(k^2-\alpha^2)}{(k^2+\alpha^2)^2}.
$$
Using the stationarity condition,
$$
\boxed{c_{g,x}=\frac{2\beta k^2}{(k^2+\alpha^2)^2}
=\frac{2Uk^2}{k^2+\alpha^2}>0.}
$$
Here $\beta=2\Omega\cos\phi/a>0$ on the ordinary midlatitude <beta plane>, so the stationary <Rossby wave> requires an eastward background current and carries its <wave packet> eastward. If $\alpha>0$, a real nonzero stationary $k$ exists only for $0<U<\beta/\alpha^2$. The restriction $k\ne0$ matters: a zonally uniform disturbance with $k=0$ has $\omega=0$ identically, but its zonal <phase velocity> is undefined, so it is not a counterexample within the stated meaning of stationary waves.
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