Caption: An animation of a sinusoidal wave propagating to the right at an unspecified phase velocity relative to an unspecified medium.

Features:

1. The phase velocity is the velocity at which the waveform propagates. It is NOT the velocity of the oscillation.

2. Because the wave in this case is a sinusoidal wave, the oscillation at any one point in space, is simple harmonic motion. So the blue dot is executing simple harmonic motion.

   For the cognoscenti, the wave could have the formula

   y(x,t)=A*cos(kx-ωt)+B*sin(kx-ωt)    .

3. The Doppler effect for mechanical waves in the classical limit is easily understood from this animation.

   Let's assume the observer (i.e., the blue dot) as shown is at rest in the medium reference frame (which is an inertial frame), and thus the phase velocity is relative to the medium.

   Note the phase velocity relative to the medium is what one always means if one says phase velocity without qualification: e.g., this is what one means by sound speed which is a phase velocity without qualification.

   The phase velocity (without qualification) is set just by the properties of the medium.

   If one qualifies phase velocity by saying the OBSERVER-FRAME phase velocity, then that phase velocity also depends on the observer's velocity in the direction of wave propagation. The formula for the cognoscenti is

     v_phase_observer = v_phase - v_observer*cos(θ)                               in general
                      = v_phase - v_observer*cos(θ=0°)   = v_phase - v_observer   decrease
                      = v_phase - v_observer*cos(θ=90°)  = v_phase                no change
                      = v_phase - v_observer*cos(θ=180°) = v_phase + v_observer   increase, 

   where θ is the angle between the wave propagation direction and the observer propagation direction, v_observer is the observer speed (i.e., magnitude of the observer velocity), v_phase_observer is the OBSERVER-FRAME phase velocity, and v_phase is the medium-frame phase velocity (i.e., the phase velocity unqualified).

   Note, v_phase_observer DECREASES if the observer is trying to catch up the waves and INCREASES if the observer is plowing head-on into the waves.

4. If the observer (i.e., the blue dot) moves left/right, the observer (plows into)/(runs away from) the waves and observes increased/decreased frequency relative to what the observer observes in the medium reference frame. In astro-jargon, the frequency is blueshifted/redshifted.

   For reference note:

     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, in astro jargon a blueshift is usually 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.

5. The classical Doppler effect formulae are derived and somewhat explicated in Waves file: doppler_effect_classical_derivation.html.