= Solution
Use a locally uniform eastward barotropic current on a beta plane. For horizontal wavevector $(k,l)$, the <Barotropic Rossby wave> dispersion relation is
$$
\omega=Uk-\frac{\beta k}{k^2+l^2},\qquad
\beta=\frac{2\Omega\cos\phi}{a},\qquad k_n=\frac{n}{a\cos\phi}.
$$
The <stationary zonal-wavenumber-two Rossby-wave condition> is therefore $U=\beta/(k_2^2+l^2)>0$. For the usual assumption $l=0$,
$$
\boxed{U=\frac{\beta}{k_2^2}=\frac{\Omega a\cos^3\phi}{2}\simeq82.5\,\mathrm{m\,s^{-1}},\quad\text{eastward}.}
$$
Here $a=6.4\times10^6\,\mathrm m$ is supplied; $\Omega=7.292115\times10^{-5}\,\mathrm{s^{-1}}$ is an explicitly identified external nominal rotation constant, not an invented historical handbook value. <IERS Conventions (2010) by International Earth Rotation and Reference Systems Service>[Table1.2,printedp18/physicalPDFp19, nominal angular velocity]{ref} The number assumes no meridional structure; a nonzero l or finite deformation radius changes the required speed. Zonal wavenumber alone does not fix those additional parameters.
For a shorter zonal disturbance with integer $n>2$ and $l=0$, the <shorter Rossby-wave propagation inside a stationary long wave> has
$$
\boxed{c_{\rm phase}=U\left(1-\frac4{n^2}\right)>0,\qquad
c_{{\rm group},x}=U\left(1+\frac4{n^2}\right)>U.}
$$
Its intrinsic phase motion is westward relative to the air, but both the weather pattern and its packet move eastward relative to the ground. Packet energy propagates downstream faster than the phase, giving <Rossby-wave downstream development>: downstream systems can amplify as upstream systems decay. Disturbances also superpose and interfere. In the strictly linear uniform-flow model they do not directly scatter from one another; finite-amplitude eddies can exchange momentum and heat and modify the background. Their detailed nonlinear interaction requires amplitudes and geometry, not just a wavenumber. <Fundamentals of Atmosphere-Ocean Dynamics by Michael Edgeworth McIntyre>[Rossby-wave phase/group propagation and wave–mean-flow interaction]{ref}
\Image[/past-exam-of-the-atmospheric-physics-course-of-the-university-of-cambridge/2012/paper-1-ap2012-rossby-speeds.png]
{title=Rossby phase and packet speeds}
{source=paper-1-ap2012-rossby-speeds.py}
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