THE HYPOTHESIS:
Assuming that the equation for the Doppler frequency shift of a source
moving at constant velocity also applies under dynamically changing
velocity conditions, one would expect the propagating sound, that is
produced by a high-frequency source moving dynamically at a low
frequency around a fixed position, to be frequency modulated. One
would further expect that the instantaneous frequency of the
propagating sound would reflect the dynamic low-frequency velocity the
source. If so, the waveform of the fm-demodulated high-frequency
propagating sound, should follow on an instantaneous basis, the
dynamic velocity of the low-frequency velocity of the source.


THE SETUP:
A small circular piezoelectric bimorph, having a resonant frequency of
approximately 10KHz was attached to the 10-lb armature/shaft of a
linear motor. The displacement of the armature/shaft was monitored
by a linear displacement transducer attached to the opposite end of
the armature/shaft. The linear displacement transducer also provided
feedback for the servo amplifier which was driving the linear motor.
Because the linear motor was in a servo loop, the displacement of the
motor followed with reasonable accuracy both sinusoidal and
non-sinusoidal command signals that were applied to the amplifier.
The piezoelectric sound source was driven by a low-distortion
oscillator at 10KHz. The sound emitted by the source was measured by a
microphone at a distance of approximately one foot. The output of the
microphone was amplified, high-pass filtered and applied to a
frequency-to-voltage converter. The output of the
frequency-to-voltage converter was low-pass filtered to reduce the
level of the residual 10KHz carrier, amplified and applied to a signal
averager. The signal averager was triggered by the command signal
that was applied to the linear motor. Averaging was used in order to
remove non-coherent 60Hz that was present in the output of the
demodulator.


THE MOTION OF THE SOUND SOURCE
A triangular command signal having a 50-msec period was applied to the
servo amplifier. A triangular command signal was used in order to
simplify interpretation of the measurement result and to avoid the
phase shift vs time delay ambiguity that would otherwise exist with
fixed frequency sinusoidal excitation. The output of the displacement
transducer was monitored on an oscilloscope and found to be triangular
with rounded corners. The rounding of the corners is due to the
limited closed-bandwidth of the servo. The velocity of the linear
motor was therefore trapezoidal with relatively flat and relatively
long plateaus and relatively short transitions.


THE MEASUREMENT RESULT
The propagating 10KHz signal emitted by the piezo bimorph was applied
to an FFT analyzer in zoom-analysis mode with a resolution bandwidth
of 0.1Hz. When the piezo bimorph was stationary, the propagating
signal picked up by the microphone showed only a single spectral peak
at 10KHz. When the piezo transducer was moving back and forth with a
triangular displacement provided by the linear motor, the propagating
signal received by the microphone contained numerous sidebands which
were indicative of FM modulation. Additionally, the output of the FM
demodulator was observed to be trapezoidal and followed on an
instantaneous basis the velocity of the linear motor and the attached
piezo transducer.


THE CHALLENGE
In science, theory usually follows experimental results. In this case
the experimental result shows that a 10KHz signal applied to a small
piezoelectric source moving back and forth around a fixed position
becomes frequency modulated by the back and forth motion of the
source. The measurement further shows that the received,
FM-demodulated signal follows the instantaneous velocity of the
source. This result is exactly what is expected on the basis of
Doppler frequency shift extrapolated from constant velocity to dynamic
velocity conditions. While some might argue that the observed
FM-like sidebands and the trapezoidal demodulated waveform are the
result of IM distortion, and not Doppler FM, the ball is in their
court. It is now up to them to provide an explanation/analysis
involving an IM producing mechanism in the present experimental setup
that accounts for the present experimental result. Finally, it must
be noted that the purpose of the present measurement was to
demonstrate fundamental phenomenological behavior. The 10KHz carrier
and the 50-msec peridiocity for the displacement of the linear motor
were chosen solely to accommodate the hardware on hand. There is
presently no reason to believe that the outcome of the present
measurement would be different if other carrier frequencies or other
source displacement periodicities or waveshapes were used.