= Variable-Coriolis shallow-water height equation
{title2=$\mathcal F_y=\partial_t^2+f(y)^2$}
On a <beta plane> with $f=f_0+\beta y$, set $\mathcal F_y=\partial_t^2+f(y)^2$. Exact elimination of the <depth-integrated shallow-water transports> gives
$$
\mathcal F_y\left[\nabla_h^2\eta-\frac{\mathcal F_y\eta}{c^2}\right]_t=\beta(\eta_{xtt}+2f\eta_{yt}-f^2\eta_x).
$$
The essential commutator is $\partial_y(\mathcal F_yV)=\mathcal F_yV_y+2f\beta V$. Freezing undifferentiated $f$ factors to $f_0$ is a local approximation. The slow-wave limit gives the <shallow-water Rossby-wave dispersion relation> with a negative zonal phase velocity for $\beta>0$.
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