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Write the perturbation velocity as $(u,v)$, the density amplitude as ${\sigma }_1$, and ${\gamma }_d=ϵ\Omega$. Axisymmetry removes advection by the background azimuthal flow, but the radial perturbation advects the Keplerian shear: $u{\partial }_x{U}_{g,y}=-3\Omega u/2$. Combining this with the Coriolis force gives the linearized shearing sheet equationsThe razor-thin disk Poisson kernel supplies the self-gravity term. The $\Omega /2$ coefficient, rather than $2\Omega$, is essential: it includes the perturbed advection of the background velocity.

Put $A=c^2{k}^2-2\pi G{\sigma }_0|k|$ and ${\omega }^2={\Omega }^2+A$. The determinant of the three amplitude equations isExpanding it gives the dust gravitational dispersion relation with gas dragUsing a determinant avoids division by $s$ and retains the neutral/secular branch. The radial epicyclic frequency of this Keplerian shearing sheet is $\Omega$, so the three terms in ${\omega }^2$ represent rotational support, self-gravity and dust pressure.

For any nonzero positive drag, the cubic dust gravitational dispersion relation with gas drag is negative at $s=0$ if ${\omega }^2<{\Omega }^2$ and positive at sufficiently large positive $s$. The intermediate value theorem therefore guarantees a growing real root. This existence argument is independent of the weak-drag expansion and includes the dynamically unstable ${\omega }^2<0$ regime.

For fixed ${\omega }^2={\Omega }^2-2\pi G{\sigma }_0|k|+c^2{k}^2>0$, the two oscillatory roots of the dust gravitational dispersion relation with gas drag acquire negative real parts at first order in $ϵ$. Their frequencies have no first-order shift. The expansion is not uniform near ${\omega }^2=0$, where the oscillation and damping scales can become comparable.