Solution
= Solution
The homogeneous equation can be written
$$
v''+\left[
\frac{S^2k_y^2x^2}{v_s^2}-k_y^2-
\frac{\kappa_r^2}{v_s^2}
\right]v=0.
$$
Its solutions are locally oscillatory where
$$
S^2k_y^2x^2>\kappa_r^2+v_s^2k_y^2,
$$
equivalently
$$
\boxed{|Sx|>\sqrt{v_s^2+\frac{\kappa_r^2}{k_y^2}}>v_s}.
$$
The relative background orbital speed is $|\mathbf u_0|=|Sx|$, so propagating wave zones occur only where the relative motion of the satellite and disk material is <supersonic>.