= Equatorial shallow-water dispersion relation
With $c=\sqrt{gH}$, <beta plane> parameter $\beta>0$, and <Hermite polynomial> index $n\geq0$, meridional trapping of the nonzero meridional-velocity branch requires
$$
\omega^2-c^2k^2-\frac{\beta kc^2}{\omega}=(2n+1)\beta c.
$$
Its meridional velocity is proportional to $H_n(Y)e^{-Y^2/2}$, with $Y=y\sqrt{\beta/c}$. Exceptional roots with $\omega^2=c^2k^2$ must be checked in the original equations, since eliminating velocity and height divides by this factor.
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