Doppler effect

When a source of waves and an observer are in relative motion, the perceived frequency is no longer the emitted one: this is the Doppler effect. If source and receiver approach each other, the wavefronts arrive more closely spaced and the perceived frequency rises; if they move apart, the wavefronts spread out and the frequency falls. You hear it every time an ambulance passes: the siren is higher-pitched as it approaches, lower as soon as it has passed.

Doppler effect: when the source moves, the wavefronts bunch up ahead (higher frequency) and spread out behind (lower frequency).

Key formula — Doppler effect law

$\ev{f' = f\,\frac{c - v_R}{c - v_E}}$ where $c$ is the speed of the wave in the medium, $v_E$ the speed of the source (emitter) and $v_R$ the speed of the receiver. Velocities are positive in the source $\to$ receiver direction.

Sign convention

If the source approaches the receiver: $v_E > 0$, the denominator decreases and $f'$ increases. If the receiver moves away: $v_R > 0$, the numerator decreases and $f'$ falls. Getting a sign wrong reverses the effect: it’s always worth checking the result has the right sense (approaching $\to$ higher-pitched sound).

Curiosity — Sirens and stars

The acoustic Doppler effect explains why the sound of an ambulance is higher-pitched as it approaches and lower as it moves away. In astrophysics, the redshift of light from distant galaxies — discovered by Hubble in 1929 — is the optical analogue of the same effect: it shows that those galaxies are moving away from us and that the universe is expanding.

Links

Topics: Onde Concepts: Effetto Doppler

Related exercises: Esercizio svolto — Il ciclista e lo specchio acustico · Problema — Doppler radar e velocità dell’auto · Problema — Doppler ambulanza in avvicinamento