= Solution
For a <normal mode> proportional to $e^{ikx-i\omega t}$, the linear equations give
$$
-i\omega U_x=2\Omega U_y-ik\frac{c_s^2}{\Sigma_0}\hat\sigma,\qquad -i\omega U_y=-\frac\Omega2U_x,\qquad -i\omega\hat\sigma=-ik\Sigma_0U_x.
$$
For $\omega\ne0$, eliminate $U_y=-i\Omega U_x/(2\omega)$ and $\hat\sigma=k\Sigma_0U_x/\omega$. The <density-wave dispersion relation in a non-self-gravitating disk> is
$$
\boxed{\omega^2=\Omega^2+c_s^2k^2.}
$$
The $\Omega^2$ term is the <radial epicyclic frequency> squared for <Keplerian rotation>: <Coriolis force> and background shear restore an orbital displacement. The $c_s^2k^2$ term is acoustic restoration by a <pressure> <gradient>. Long wavelengths are mainly <epicyclic motion>; short wavelengths are mainly <acoustic waves>. The given model contains no disk self-gravity term.
For completeness, the full three-variable determinant is proportional to $\omega(\omega^2-\Omega^2-c_s^2k^2)$. Its additional $\omega=0$ branch has $U_x=0$ and $2\Omega U_y=ikc_s^2\hat\sigma/\Sigma_0$. This is a stationary <pressure>/Coriolis balance, the <zero-frequency balanced mode of an axisymmetric disk>, rather than a propagating density wave. At $k=0$ it reduces to a uniform surface-density change. Dividing by $\omega$ is legitimate only for the wave branches just derived.
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