Solution
= Solution
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>.