So far we have superposed waves of the same frequency. What if the two frequencies differ slightly? The answer is one of the most useful effects in acoustics: beats.
Beats are the periodic rise and fall in the loudness of sound heard when two notes of slightly different frequencies are sounded together. At some instants the two waves are in phase and reinforce (loud); a moment later they slip out of phase and cancel (soft). One loud–soft cycle is one beat.
- Beat frequency $=|f_1-f_2|$ — the number of beats heard per second equals the difference of the two frequencies.
- Beats are clearly audible only when the difference is small (roughly up to 6–7 Hz); beyond that the ear cannot resolve the separate loud–soft cycles.
- Beats are the basis of tuning: a musician adjusts an instrument against a reference until the beats slow down and vanish, meaning the frequencies match exactly.
For example, two tuning forks of 256 Hz and 260 Hz sounded together produce $|260-256|=4$ beats per second.
The Doppler effect is the apparent change in the frequency (pitch) of a sound when there is relative motion between the source and the observer. You hear it every day: the pitch of an approaching ambulance siren is higher than its true pitch, and it suddenly drops to a lower pitch as the ambulance passes and recedes.
The general formula for the observed frequency $f'$ when the source emits frequency $f$ in still air of sound speed $v$ is:
- $f'=f\left(\frac{v\pm v_o}{v\mp v_s}\right)$, where $v_o$ is the observer's speed and $v_s$ is the source's speed.
- Sign convention: choose the signs so that motion of either body towards the other raises $f'$, and motion away lowers it. Use the upper sign in the numerator ($+v_o$) when the observer moves towards the source; use the upper sign in the denominator ($-v_s$) when the source moves towards the observer.
Some standard cases (sound speed $v$):
- Source approaching, observer at rest: $f'=f\left(\frac{v}{v-v_s}\right)$ — pitch rises.
- Source receding, observer at rest: $f'=f\left(\frac{v}{v+v_s}\right)$ — pitch falls.
- Observer approaching, source at rest: $f'=f\left(\frac{v+v_o}{v}\right)$ — pitch rises.
Note that the Doppler effect for sound is not symmetric in $v_o$ and $v_s$ — a moving source and a moving observer at the same speed give slightly different shifts, because sound travels in a medium (air) which provides a frame of reference. Applications include speed-detecting radar guns used by traffic police, weather radar for storms, medical ultrasound to measure blood flow, and sonar for detecting submarines. Astronomers use the Doppler shift of starlight (red shift) as evidence that the universe is expanding.