In all seriousness though I'm going through your post to see if you've found a gaping problem =). I don't think there can be different density velocity ratios at any point of displacement in the wave's reference frame because the same net number of particles has to pass through each point or you start making a permanent buildup on one side or the other of that point, which you can't do because in the reference frame of the still air that point is moving at the speed of sound.
A moving point passes by the same number of unique particles at the speed of sound as it would if it were going the speed of turtle (if you ignore thermal dispersion).
"the same net number of particles has to pass through each point"
Agreed: the corollary to the "the sparse particles just in front of the dense ones" is the "sparse ones just behind the next wave of dense ones", which are, in fact, decelerating, or moving backward even closer to the speed of sound than the dense ones relative to the reference frame. So individual particles oscillate back and forth, but net effect is no movement (to the reference frame of the average speed of the particles, which is a third reference frame :-) )
Surely that animation exaggerates how little net molecule displacement occurs. I'm not sure what would cause many of the molecules to reverse direction while in the low-pressure part of the wave.
http://www.acs.psu.edu/drussell/Demos/waves/Lwave-v8.gif
In all seriousness though I'm going through your post to see if you've found a gaping problem =). I don't think there can be different density velocity ratios at any point of displacement in the wave's reference frame because the same net number of particles has to pass through each point or you start making a permanent buildup on one side or the other of that point, which you can't do because in the reference frame of the still air that point is moving at the speed of sound.
A moving point passes by the same number of unique particles at the speed of sound as it would if it were going the speed of turtle (if you ignore thermal dispersion).