= Rossby-wave equation for a sheared zonal current
{c}
{title2=$(\partial_t+U\partial_x)(\nabla_h^2-R_D^{-2})\phi+Q_y\phi_x=0$}
Linearizing <shallow-water quasi-geostrophic potential vorticity> about a zonal current $U(y)$ gives $Q_y=\beta-U^{\prime\prime}+U/R_D^2$. A normal mode $\widehat\phi(y)e^{i(kx-\omega t)}$ obeys $(U-\omega/k)[\widehat\phi^{\prime\prime}-(k^2+R_D^{-2})\widehat\phi]+Q_y\widehat\phi=0$. A global transverse <plane wave> is generally unavailable for an arbitrary nonuniform jet; boundary conditions or a local short-wavelength approximation are required.
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