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The vertical relative vorticity isDifferentiate the $v$-momentum equation with respect to $x$ and the $u$-momentum equation with respect to $y$, then subtract. The mixed pressure derivatives cancel, givingThe mass-conservation equation says ${\eta }_t=-h(u_x+v_y)$. Therefore the linearized shallow-water potential vorticitysatisfiesThus

The basic state has velocity fieldand buoyancy $B=N^{2}z$. For disturbances independent of $y$, the linearized equations areSubstituting a plane wave proportional to $\exp [i(kx+mz-\omega t)]$ and eliminating $p$, $v$, and $b$ gives the dispersion relationThus the requested coefficients are

An instability exists precisely when some wavenumber pair makes ${\omega }^2<0$. Since the Brunt–Väisälä frequency satisfies $N^2>0$, this is possible exactly whenThe basic relative vorticity is $\Lambda {\mathbf{e}}_z$, so its absolute vorticity is $(f+\Lambda ){\mathbf{e}}_z$. Its Ertel potential vorticity is thereforeThe instability criterion can consequently be written as $fQ<0$: the vertical absolute vorticity has the opposite sign to the planetary vorticity. This is inertial instability, approached most directly by disturbances with $|m/k|\gg 1$.

Expand both the forcing and response in the orthogonal vertical normal modes. Projection onto $P_n$ givesIf the horizontal forcing scale obeys $L_h\gg c_n/f_0$, the relative vorticity term ${\nabla }_h^2{\varphi }_n$ is small compared with the stretching term. The resulting long-wave equation isThus each mode communicates the forcing westward at its long Rossby wave speed $C_n$, with the barotropic mode fastest because $c_0$ is largest.