Forgive me for interfering in the detailed analysis of the doppler and related satellite engineering parameters, but there is a comment from Robin Clark to the effect that at apogee the satellite will have no "vertical motion".
Satellites orbit around the mass of the earth but the earth is not a sphere and does not have uniform density. This means that any movement with respect to a fixed point on the surface of the earth will result in a change of gravity and thus a change in altitude.
Radio waves are not affected by gravity [1] so a change of absolute vertical distance caused by the earth's "lumpy" gravitational field will result in a doppler shift.
Apogee is the point in the orbit of maximum altitude so
at this point there will be no vertical motion.
[1] Actually radio waves are affected by gravity due to relativistic effects but the change is rather small.