= Solution
Write $\zeta=(\nabla\times\mathbf u)\cdot\mathbf e_z$ and $q=\zeta-f\eta/h_0$, so the PDF's vector <potential vorticity> is $\mathbf Q=q\mathbf e_z$. The local TeX loses some of this boldface. Taking the vertical <curl> of the momentum equation gives $\zeta_t=-f\nabla\cdot\mathbf u$, while <mass conservation> gives $\eta_t=-h_0\nabla\cdot\mathbf u$. Thus
$$
\boxed{q_t=\zeta_t-\frac f{h_0}\eta_t=0,\qquad \mathbf Q_t=0}.
$$
This is the linear perturbation of <shallow-water potential vorticity>, scaled by $h_0$; it should not be confused with the full nonlinear conserved ratio $(f+\zeta)/h$.
Since $\nabla\cdot(\mathbf e_z\times\mathbf u)=-\zeta$, taking the <divergence> of momentum gives $(\nabla\cdot\mathbf u)_t=f\zeta-g\nabla^2\eta$. Differentiate <mass conservation> and use $\zeta=q+f\eta/h_0$ to obtain
$$
\boxed{\eta_{tt}-gh_0\nabla^2\eta+f^2\eta=-h_0 f q=-h_0\mathbf f\cdot\mathbf Q}.
$$
For $\mathbf Q=0$, substitution of the stated <plane wave> gives the <linear rotating shallow-water dispersion relation>, with $c=\sqrt{gh_0}$:
$$
\boxed{\omega^2=f^2+c^2k^2,\qquad\omega_\pm(k)=\pm\sqrt{f^2+c^2k^2}}.
$$
The two even branches approach $\pm c|k|$ at large $|k|$ and have values $\pm|f|$ at $k=0$. The original <dispersion diagram> below also compares the speed magnitudes.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/ib/paper-4-dispersion.png]
{title=A <dispersion relation> and its velocity branches}
{description=Rotating shallow-water frequency branches and magnitudes of <phase velocities> and <group velocities>, with dimensionless <wavenumber>.}
For $k\ne0$, the <phase velocity> and <group velocity> magnitudes are
$$
\boxed{|c_p|=\frac{|\omega|}{|k|}=\sqrt{c^2+\frac{f^2}{k^2}},\qquad
|c_g|=\left|\frac{d\omega}{dk}\right|=\frac{c^2|k|}{\sqrt{f^2+c^2k^2}},\qquad |c_p||c_g|=c^2}.
$$
The printed convention $e^{i(kx+\omega t)}$ means the signed <phase velocity> is $-\omega/k$ and the signed <group velocity> is $-d\omega/dk$. The speed magnitudes above are independent of that sign convention.
For $f\ne0$, longer <wavelengths> have greater <phase velocity> but smaller <group velocity>: individual crests and <wave packets> therefore answer “faster or slower” differently. At long <wavelength>, $|k|\ll|f|/c$, the <frequency> is nearly the inertial <frequency> $|f|$, $|c_p|\sim|f|/|k|$, and $|c_g|\sim c^2|k|/|f|$. Rotation has its largest relative effect here, and the waves are strongly <dispersive>. At short <wavelength>, $|k|\gg|f|/c$, both speeds tend to $c$ and rotation gives only a small, weakly <dispersive> correction. If $f=0$, the gravity waves are <nondispersive>, with speed $c$; at $k=0$ the <phase velocity> formula is undefined.
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