# Doppler measurement of velocity

Our labs do not have written instructions. In keeping with this spirit, the description given here will be brief and general. The intent is that each performance of the lab will be unique; in each nature will reveal a slightly different face to the observer.

In this lab a sound transmitter is driven by a single signal generator with a fixed frequency (about 40kHz). The transmitter is attached to a mass which moves in simple harmonic motion. A sound detector converts the radiated sound from the moving source to an electric signal The frequency of the detected sound is Doppler shifted higher (if the transmitter is approaching) and lower (if the transmitter is departing) than the signal generator frequency.

This experiment was developed by senior physics major Steven A. Kirk for his senior seminar research project. Besides the ultrasonic transducers, he used an operational amplifier and an analog-to-digital converter (ADC) which can make at least 100 measurements per second. (Steve used the MultiPurpose Lab Interface from Vernier Software set for 500 points per second.) A sketch of his set-up is shown below.

The Doppler shifted signal is mixed with a signal at the original frequency in a heterodyne circuit. Our ADC does not respond to the 40kHz signal, so at the heterodyne output, the only observed signal was at the difference (or "beat") frequency. The circuit adjusted the two signals to have (roughly) the same amplitude, added the two signals, amplified the sum, and sent the sum signal through a diode. A small DC signal was also added to the sum signal so that the diode was biased to operate in its most non-linear fashion. A sketch of Steve's circuit is shown below:

In the sketch, the resistor labeled 1 is to draw current through the diode. The resistor and capacitor labeled 2 form a low pass filter. The capacitor labeled 3 blocks the DC component of the output.

After removing background variations caused by the varying distance that the sound had to travel, the difference signal showed a variation in frequency as sketched in the figure below.

As marked on the graph, high difference frequency corresponds to large Doppler shift, and therefore high velocity of the oscillating mass. The frequency variation of the difference signal equals the variation in Doppler shift of the detected signal. The observed variation correspond to a velocity of the oscillating mass having a maximum value of about 0.8 m/sec and a sinusoidal time dependence with a period of about 1 sec.

Steve analyzed the data in two ways: by hand calculation of the period of each cycle of the difference signal, and by spreadsheet curve fitting of the data.