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

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.