Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 3 / c

Let $\delta ={\Sigma }^{\prime }/{\Sigma }_0$ and define the derivative following the background Keplerian shear byThe linearized continuity and momentum equations areThe coefficient $\Omega /2$ combines the background shear with the Coriolis acceleration.

Applying ${\mathcal{D}}_0$ to the shearing-wave ansatz produces an extra phase termThe perturbation equations have spatially uniform amplitude coefficients only when this term vanishes. HenceFor $k_y\ne 0$, the radial wavenumber changes linearly. A leading disturbance with $k_x/k_y<0$ first opens until $k_x=0$, then becomes an increasingly tightly wound trailing disturbance with $k_x/k_y>0$. This is the geometric swing of a shearing wave.

The linearized vortensity amplitude isSince the exact vortensity obeys $D_t\zeta =0$ and the time-dependent $k_x$ cancels the background shear in the phase, linearization gives

On the zero-vortensity branch,Substitution in the two momentum equations yields the linear system of differential equationswithwhere $k_x=k_x(t)$ is the result from part (ii).

When $k_y=0$, $k_x$ is constant and the system becomesThusFor time dependence $e^{-i\omega t}$, the dispersion relation isExplicitly,and $\tilde{\delta }=2i{k}_x{\tilde{u}}_y/\Omega$. This is an axisymmetric inertial-acoustic wave: pressure supplies the $c_s^2{k}_x^2$ restoring term and epicyclic motion supplies the ${\Omega }^2$ term.