Caption: An animation of a sinusoidal wave propagating to the right with some phase velocity.
Features:
So the blue dot is executing simple harmonic motion.
Let's assume the observer (i.e., the blue dot) as shown is at rest in the medium reference frame, and thus the phase velocity is the medium phase velocity.
And usually one means medium phase velocity when one says phase velocity without qualification---e.g., when one says sound speed.
If the observer (i.e., the blue dot) moves left/right, the observer (ploughs into)/(runs away from) the wave and observe increased/decreased frequency relative to what the observer observes in the medium reference frame.
In astro-jargon, the frequency is blueshifted/redshifted:
Plow in Run away from Run transversely blueshift redshift no shift classically f_obs > f_medium f_obs < f_medium f_obs = f_medium
Note that in astro jargon a blueshift is often considered a NEGATIVE redshift.
The observed frequency for moving in direction of the waves can go to zero if the observer is moving at the medium phase velocity. It can even be NEGATIVE if the observer is plowing into the waves from their back ends. However, we seldom think of frequency as being NEGATIVE.
There is NO Doppler effect for transverse motion: i.e., motion perpendicular to the line-of-sight.
The relativistic Doppler effect for electromagnetic radiation (EMR) does have a transverse Doppler effect, but it's usually very small and NOT detectable.