= Solution
{c}
<Coastal Kelvin-wave radiation of long Rossby waves> requires a common <phase> convention to match the two <waves>. The supplied <Rossby wave> formula conventionally multiplies $e^{i(k_Rx+\ell_Ry-\omega_Rt)}$, while the coastal expression uses $e^{i(\omega_Kt-\ell_Ky)}$. Since physical fields are real, matching may use the complex-conjugate representative; in either convention the boundary forcing fixes the absolute <frequency> and alongshore <wavenumber>. Thus $|\omega_R|=|\omega_K|$ and $|\ell_R|=|\ell_K|$.
For $k_R^2+\ell_R^2\ll L_d^{-2}$, the <shallow-water Rossby-wave dispersion relation> reduces to
$$
\omega_R\simeq-\beta L_d^2k_R,
\qquad
\boxed{|\omega_K|\simeq\beta|k_R|L_d^2
=\frac{\beta|k_R|g'h_0}{f_0^2}.}
$$
Choosing $k_R>0$ recovers the printed magnitude relation. Both the zonal <phase velocity> and zonal <group velocity> are approximately $-\beta L_d^2$, so the long <Rossby waves> carry the coastal signal westwards into the basin.
The coastal condition $|\ell_K|L_d\ll1$ alone does not guarantee a long zonal <wavelength>. Since $|\omega_K|=c|\ell_K|$, matching requires $|k_R|=c|\ell_K|/(\beta L_d^2)$ and hence the stronger consistency condition
$$
|\ell_K|\ll\frac{\beta L_d}{c},\qquad\text{equivalently}\qquad |\omega_K|\ll\beta L_d.
$$
For example, at $10^\circ$ a coastal <wavenumber> with $|\ell_K|L_d=0.05$ gives $|k_R|L_d\simeq0.56$, which is not asymptotically small even though the meridional <wavelength> is long. Thus the source's proposed excitation is possible at sufficiently low <frequency>; it is not an automatic consequence of its meridional long-wave assumption.
At $10^\circ$ N or S the <beta plane> parameter is
$$
\beta=\frac{2\Omega\cos10^\circ}{R_E}
=2.25\times10^{-11}\,\mathrm{m}^{-1}\mathrm{s}^{-1},\qquad
\beta L_d^2\simeq0.225\,\mathrm{m\,s}^{-1}.
$$
Taking a representative Pacific basin width $W\simeq1.5\times10^7$ m gives
$$
\boxed{t_{\rm cross}\simeq\frac W{\beta L_d^2}
=6.65\times10^7\,\mathrm s\simeq2.1\,\mathrm{yr}.}
$$
A width of $1.4$–$1.7\times10^7$ m gives roughly $2.0$–$2.4$ years. This is propagation near $10^\circ$ latitude, not an equatorial $f=0$ calculation. Slow <Rossby wave> adjustment supplies a basin-scale delay relevant to <El Niño–Southern Oscillation>: changing <thermocline> depth can affect later <sea surface temperature> and coupled wind forcing. Reflection and coupling may help produce interannual memory, but this travel-time estimate alone does not establish a climate oscillation or its period.
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