= Solution
Let $K=(k^2+l^2)^{1/2}$ and $c=\sqrt{g'h_0}$. The three branches are
$$
\boxed{\omega=0,\qquad \omega_\pm=\pm\sqrt{f^2+c^2K^2}.}
$$
For nonzero $K$ and $f$, the zero-frequency mode is stationary <geostrophic flow>, with $f\mathbf k\times\mathbf u=-g'\nabla\eta$ and zero horizontal divergence. It carries the linear potential-vorticity disturbance. There is no Rossby-wave propagation on a constant-$f$ plane; a varying <Coriolis parameter> would be needed for that.
The other two branches are oppositely phased <inertia-gravity waves>. Their phase-speed magnitude and radial <group velocity> are $|\omega|/K=\sqrt{c^2+f^2/K^2}$ and $|\partial\omega/\partial K|=c^2K/\sqrt{f^2+c^2K^2}$. When $KL_R\gg1$, gravity dominates: $|\omega|\simeq cK$ and the waves are approximately nondispersive, with both speeds approaching $c$. When $KL_R\ll1$, rotation dominates: $|\omega|\simeq|f|+c^2K^2/(2|f|)$, the <frequency> approaches the <Coriolis parameter>[inertial frequency] and the <group velocity> tends to zero. The large long-wave <Phase velocity>[phase speed] is not a large signal speed.
Without rotation the propagating branches become nondispersive <Gravity wave>[gravity waves], $\omega=\pm cK$, and the stationary branch is a nondivergent vortical velocity disturbance rather than a finite-pressure <geostrophic balance>. At $K=0$, the modes reduce to constant interface height and spatially uniform <Inertial oscillation>[inertial oscillations]. The wave branches have zero linear potential-vorticity perturbation; it is the stationary branch that retains the initial <potential vorticity>.
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