The Doppler Effect

In the real world, a Doppler shift is an apparent change in acoustic frequency content of a sound source due to motion of the source relative to the listener. You have probably heard the pitch of a horn drop lower as it passes by (e.g., from a moving train). While a pitched sound-source is moving toward the listener, the heard pitch is higher than it is at rest; while the source is moving away from the listener, its pitch is lowered.

As derived in elementary physics texts, the Doppler shift is given by

$$\omega_l = \omega_s \frac{1+\frac{v_{ls}}{c}}{1-\frac{v_{s,l}}{c}} \protect$$ (1)

where $\omega_s$ is the radian frequency emitted by the source at rest, $\omega_l$ is the frequency received by the listener, $v_{ls}$ denotes the speed of the listener relative to the propagation medium in the direction of the source, $v_{s,l}$ denotes the speed of the source relative to the propagation medium in the direction of the listener, and $c$ denotes sound speed. Note that all quantities in this formula are scalars.