Exact 3D Doppler Effect Explorer

Analytic formulae and simulations of exact Doppler effects for arbitrary motions in 3D.

Descriptions of the Doppler effect are usually restricted to one-dimensional uniform motions of a detector and a sound source. This educational Live Script provides general exact expressions for the non-relativistic Doppler effect for a monopole source that is not necessarily harmonic and an isotropic detector, both undergoing arbitrary motions in three dimensions, properly accounting for time delay using retarded and advanced time functions. Exact general analytic formulae are obtained for the detected sound for uniform motion of either the source or detector and arbitrary motion of the detector or source respectively. These formulae may be evaluated using parametric representations of these motions. For a harmonic source, expressions are derived for the detected instantaneous frequency over time for a source/detector in free fall near a stationary detector/source. The script computes numerically the retarded and advanced time delay functions, the detected sound, and the instantaneous frequency for these and other exemplary motions and compares the results to the theory .

This script will interest students and instructors of physics and related fields. "Try this" suggestions, coding "Challenges", hyperlinks, and references are supplied for further exploration. Additional educational Live Scripts by the author are here.