A satellite on a circular orbit is stationary in the corotating sheet. Write its tidal forcing and the response as
$$
\Psi=\psi(x)e^{ik_yy},
\qquad
v_y=v(x)e^{ik_yy}.
$$
Since $D=-iSk_yx$, the forced wave equation becomes
$$
\boxed{
-v_s^2v''+
[\kappa_r^2+v_s^2k_y^2-S^2k_y^2x^2]v
=(2\Omega-S)\psi'-Sxk_y^2\psi}.
$$
This ordinary differential equation describes a <satellite-forced density wave in an astrophysical disk>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 2 / d / Solution

A satellite-forced density wave is a compressive disk response to the satellite's periodic tidal potential. In a local corotating sheet, propagating zones lie where the relative orbital motion is supersonic.


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