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Your muddying the water. Your first answer is correct as
far as it goes.
I demonstrate the Doppler effect with an apparatus I build using one of
those obnoxious solid state beepers from Radio Shack. (As my musical
friends tell me, they are obnoxious because they produce a pure sine wave
and not a musical note, but this makes them ideal for this demonstration.)
It is mounted on a small piece of wood with an nine volt battery, The wood
is fastened to a stout cord so that it can be sung around my head. The
students in the class hear the Doppler effect as it changes speed in their
direction. I, at the center, hear only a small effect since my ears are not
exactly at the center of rotation.
One of the real powers of the method of analysis we use in
physics is that
we are always making simplifications, when you start relaxing those the
answers can quickly become murky. Since I have two ears, I probably can
hear some phase variation as well, but I have always attributed most of the
change to the small velocity component because the ear is not at the
center. If I make the string truly at the center of my head will I also
hear a beat do to the opposite Doppler effects in each ear? Into the
mud we go!
Gary
Gary Karshner