= Density-wave dispersion relation in a non-self-gravitating disk
{title2=$\omega^2=\kappa_r^2+c_s^2k^2$}
Let $c_s^2=(dP/d\Sigma)_0$ be the <barotropic closure of a razor-thin disk> sound-speed squared and $\kappa_r$ the <radial epicyclic frequency>, with $\kappa_r^2=2(2-q_r)\Omega^2$ for <orbital shear parameter> $q_r$. For a radial axisymmetric <normal mode> in a uniform barotropic <shearing sheet>, let $U_x,U_y$ be velocity amplitudes and $\sigma$ the surface-density amplitude. Linear momentum gives $-i\omega U_x=2\Omega U_y-ikc_s^2\sigma/\Sigma_0$ and $-i\omega U_y=-\kappa_r^2U_x/(2\Omega)$; mass conservation gives $\omega\sigma=k\Sigma_0U_x$. Eliminating the amplitudes for $\omega\ne0$ proves the displayed dispersion relation. The rotational term is epicyclic restoration; the <pressure> term is acoustic restoration. Without a perturbed gravitational potential, no self-gravity term belongs in the formula.
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