= Forced dust epicycle with aerodynamic drag
{title2=$D_\epsilon=\partial_t+\epsilon\Omega$}
Let $v_x,v_y$ and $u_x,u_y$ be the dust and gas velocity perturbations after subtracting the common background shear, suppressing primes. Put $t_s>0$ for the <aerodynamic stopping time>. Linear axisymmetric dust perturbations about a <Keplerian shearing sheet> obey $D_\epsilon v_x-2\Omega v_y=\epsilon\Omega u_x$ and $D_\epsilon v_y+\Omega v_x/2=\epsilon\Omega u_y$, with $\epsilon\Omega=t_s^{-1}>0$. Eliminating $v_y$ gives
$$
[\partial_t^2+2\epsilon\Omega\partial_t+(1+\epsilon^2)\Omega^2]v_x=\epsilon\Omega\partial_tu_x+\epsilon^2\Omega^2u_x+2\epsilon\Omega^2u_y.
$$
Thus a gas epicycle $u_x=u\cos(kx-\Omega t)$, $u_y=u\sin(kx-\Omega t)/2$ supplies both $2\epsilon\Omega^2u\sin(kx-\Omega t)$ and $\epsilon^2\Omega^2u\cos(kx-\Omega t)$. Keeping the latter is necessary for an exact linear equation beyond first order in drag. The homogeneous roots are $-\epsilon\Omega\pm i\Omega$, separating damping from epicyclic frequency.
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