= Solution
Let $\widehat{\mathbf u}$ and $\widehat p$ be the complex <plane wave> amplitudes and set $k_x=k_\perp$, $k_z=k_\parallel$. The <linearized Euler equations> in the rotating frame are
$$
\begin{aligned}
-i\omega\widehat u_x&=-ik_\perp\widehat p+2\Omega\widehat u_y,\\
-i\omega\widehat u_y&=-2\Omega\widehat u_x,\\
-i\omega\widehat u_z&=-ik_\parallel\widehat p,\\
k_\perp\widehat u_x+k_\parallel\widehat u_z&=0.
\end{aligned}
$$
Here $p$ is <pressure> per unit <mass density>, including the centrifugal potential. The last line is <incompressibility>, not an additional equation of state.
Taking the <curl> eliminates <pressure>. For constant $\boldsymbol\Omega$ and a <divergence-free vector field>, $\nabla\times(2\mathbf u\times\boldsymbol\Omega)=2(\boldsymbol\Omega\cdot\nabla)\mathbf u$. Thus the <inertial-wave vorticity equation> gives
$$
-i\omega(i\mathbf k\times\widehat{\mathbf u})
=2i\Omega k_\parallel\widehat{\mathbf u},\qquad
\omega\mathbf k\times\widehat{\mathbf u}=2i\Omega k_\parallel\widehat{\mathbf u}.
$$
Apply $\mathbf k\times$ once more and use $\mathbf k\times(\mathbf k\times\widehat{\mathbf u})=-k^2\widehat{\mathbf u}$. For a nonzero oscillatory amplitude,
$$
-\omega k^2\widehat{\mathbf u}
=2i\Omega k_\parallel\frac{2i\Omega k_\parallel}{\omega}\widehat{\mathbf u},
$$
so the <inertial-wave dispersion relation> is
$$
\boxed{\omega^2=4\Omega^2\frac{k_\parallel^2}{k^2},\qquad
\omega_\sigma=\sigma\,2\Omega\frac{k_\parallel}{k},\quad\sigma=\pm1.}
$$
The restoring influence is the <Coriolis acceleration>. In particular $|\omega|\le2|\Omega|$. For $k_\parallel=0$, the projected linear dynamics have zero <frequency>; these are steady balanced disturbances rather than the nonzero-frequency branch used in the division above. The spatially uniform $k=0$ mode must be treated separately from this <dispersion relation>.
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