I wanted to investigate the doppler effect on pendulums and set up an experiment where:
- The bob of a pendulum is a speaker which emits a specific sound frequency.
- Directly underneath the lowest point of the pendulum sits a microphone that observes the frequency being emitted by the pendulum.
In the experiment, I would release the speaker from various amplitudes and record the frequency that the microphone perceives as the speaker passes the equilibrium position.
I know that the standard doppler equation is:
where $f$ is the observed frequency
$f_0$ is the initial frequency
$c$ is the speed of sound
and $v$ is the speed of the moving source
However this equation only applies to non-accelerating systems; even though the speed of the speaker at the bottom of the pendulum can be predicted with the equation
substituting this velocity into the doppler equation will not give accurate results because of the acceleration of the pendulum.
So, my question is, how can one calculate the frequency observed by the microphone at the bottom of the pendulum given the various release heights that the speaker is released from?