Pendulum Doppler Effect Theory and Simulation

Live Script simulation of the Doppler effect for a source on the end of a nonlinear pendulum.

An at-home experiment that integrates mechanics and acoustics is to let your mobile phone swing from its charging cable while it plays a tone and record the sound with a stationary laptop or another phone at some location. The received frequency will slightly warble due to the Doppler effect. This educational Live Script models this experiment. It assumes planar pendulum motion. The pendulum rod can swing "up and over the top" and the receiver location is arbitrary. It derives a symbolic expression for the frequency shift over time based on the approximation that the time delay due to the speed of sound is negligible. In this case, the instantaneous received frequency is determined by the instantaneous component of source velocity in the direction of the receiver. An explicit expression for the received frequency over time is derived for the case of small-angle pendulum motion. It calculates the numerically large-angle motion and the associated frequency shift and compares the small-angle approximation and large angle numerical results. Optionally, the script will display and save the simulated audio signal.

This script may interest students and instructors of physics. It illustrates symbolic methods and the numerical solution of a first order ordinary differential equation. 'Try this' suggestions are available to adjust parameters. Coding 'Challenges' are provided for further exploration. Additional educational live scripts are available here.