The pulsation of a rotary speaker and our electronically ORBITED speaker produce the same acoustic spatial effect.
The sound source has to physically move (or be displaced) towards and away from you for the pitch shift of the Doppler Effect to work.
The rotary speaker achieves this displacement (movement) of sound by using a rotary mechanism that spins the horn where the mouth of the horn moves in a circle making an orbit.
The sound source is the mouth of the horn, so as the mouth moves toward and away, the pitch is shifted
Rotation movement does not cause displacement (movement) of sound and, therefore, cannot create the Doppler Effect
The ORBITED speaker accomplishes this sound movement (displacement) by electronically, physically displacing the sound source, moving it from one face of the cabinet to the next around in a circle making an orbit.
When the sound source moves from the front, to a side and to the back face, the pitch is shifted down. The pitch shifts up as the sound source returns from the back to the front.
If the movement of the sound source is only side-to-side, there is no change in distance to the listener and no pitch shift occurs.
Similarly, if a speaker driver simply rotates or has a short arm (as in the many other older competing designs) as opposed to a long arm moving in a circle, there is no change in distance to the listener and again little or no pitch shift occurs.
The reality is that most speakers are omni-directional with the sound going out in all directions.
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