Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 2 / c

For the axisymmetric perturbation, defineLinearization about the uniform dust density and drifting equilibrium givesThe factors $2{\Omega }_K$ and ${\Omega }_K/2$ are the Coriolis acceleration and Keplerian-shear couplings. Applying ${\mathcal{D}}_{\tau }({\mathcal{D}}_{\tau }^2+{\Omega }_K^2)$ to the continuity equation and using the three momentum equations eliminates ${\mathbf{u}}^{\prime }$. All remaining forcing terms are proportional to $e^{-i\omega t}$, sowhere $F$ is independent of time.

On the forcing $e^{-i\omega t}$,The other three factors have roots with real decay rate $-1/\tau$, so a real-frequency gas wave can resonate only with the undamped factor $\mathcal{D}$. The resonance condition is thereforeAt resonance, $\mathcal{D}$ annihilates the forcing, so a particular solution acquires one power of time:This is a streaming instability resonance: the gas wave's phase velocity along $\mathbf{k}$ matches the dust drift projected along the same wavevector.