= Resonant dust entrainment by an epicyclic gas wave
{title2=$\mathbf v'-\mathbf u'\propto e^{-\epsilon\Omega t}$}
If the prescribed gas velocity is itself a free <epicyclic motion> solution, subtract its two momentum equations from the <forced dust epicycle with aerodynamic drag> equations. The velocity difference obeys a damped epicycle and decays as $e^{-\epsilon\Omega t}$. Therefore dust approaches the gas velocity with identical phase and amplitude for every positive drag rate. At zero drag the difference does not decay, so the late-time limit is nonuniform as drag vanishes. A finite-wavelength <inertial-acoustic wave> is detuned from the free epicycle; matching its phase and amplitude in the weak-drag limit additionally requires $|\omega-\Omega|\ll\epsilon\Omega$. For a gas density wave of frequency $\omega$, the exact radial transfer amplitude is
$$
\frac{V_x}{U_x}=\frac{\epsilon\Omega[\epsilon\Omega-i\omega-i\Omega^2/\omega]}{(\epsilon\Omega-i\omega)^2+\Omega^2},
$$
obtained by solving the two dust equations and using $U_y=-i\Omega U_x/(2\omega)$. At $\omega=\Omega$ it is exactly one.
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