The Doppler effect: why a passing siren drops in pitch
The Doppler effect is the change in frequency you hear when the source of a sound and the listener are moving towards or away from each other. As an ambulance approaches, each sound wave is emitted a little closer to you than the last, so the waves arrive bunched up and the pitch is higher; as it drives away they are stretched out and the pitch is lower. The sudden drop as it passes is the switch from one to the other. The Doppler effect calculator works out the frequencies for sound and for light.
How it works
The pitch you hear is the number of pressure peaks reaching your ear each second. If the source is still, the peaks spread out evenly at the speed of sound, about 343 m/s in air at 20 Β°C. If the source moves towards you, it chases its own waves: each new peak starts from a point closer to you, so the wavelength is shorter in front of the source and longer behind it. A 700 Hz siren has a wavelength of 343 Γ· 700 = 0.49 m when standing still; moving at 30 m/s, the waves ahead of it are squeezed to (343 β 30) Γ· 700 β 0.447 m and those behind stretched to about 0.533 m. The siren itself never changes its note. The Austrian physicist Christian Doppler proposed the effect in 1842, and C. H. D. Buys Ballot tested it for sound in 1845 in experiments on a Dutch railway.
The formula
For sound in still air, with c the speed of sound, f the emitted frequency, vβ the source's speed and vβ the observer's speed, both counted as positive when they are moving towards each other:
fβ² = f Γ (c + vβ) Γ· (c β vβ)
When they move apart, the signs flip. Worked example: a 700 Hz siren approaching at 30 m/s (108 km/h) is heard at 700 Γ 343 Γ· 313 β 767 Hz, and as it recedes at 700 Γ 343 Γ· 373 β 644 Hz. If instead you drove at 30 m/s towards a stationary 700 Hz siren, you would hear 700 Γ 373 Γ· 343 β 761 Hz: close to 767 Hz, but not the same. The two cases differ because for sound what matters is motion relative to the air, not just relative to each other. The speed of sound itself changes with temperature, which the speed of sound calculator works out.
Why the pitch drops as it passes
From 767 Hz down to 644 Hz is a frequency ratio of about 1.19, a drop of three semitones, which musicians call a minor third (Music intervals explained: names, semitones and inversions explains the names). A slower vehicle at 20 m/s gives a drop of about two semitones. The change is not instant. The formula applies to motion straight along the line between you, and as a vehicle passes to one side, the part of its velocity that points towards you falls through zero, so the pitch glides down over a second or two. The closer the road runs to you, the more abrupt the drop. To hear which musical notes those frequencies are, the note frequency calculator finds the nearest note and how many cents sharp or flat it is.
Sonic booms
As a source approaches the speed of sound, the bottom of the formula, c β vβ, heads towards zero and the waves in front of it pile up on top of each other. At and above the speed of sound the source outruns its own sound, and the overlapping waves merge into a cone-shaped shock wave trailing behind it. When that cone sweeps over the ground, people hear a sonic boom, often as a sharp double bang from a supersonic aircraft. Contrary to a common belief, the boom is not a one-off made at the moment the aircraft passes the speed of sound: it is produced continuously for as long as the aircraft flies supersonically, and is heard by everyone along its track.
Light, radar and redshift
Light shows the Doppler effect too, but there is no air to move through, so the relativistic formula is used, which depends only on the relative speed of source and observer. Light from a source receding at 10% of light speed is stretched by about 10.6%, a redshift of z β 0.106; approaching sources are shifted towards the blue. Astronomers use this to measure how fast stars and galaxies are moving, although the redshift of very distant galaxies is better understood as the expansion of space itself. Closer to home, police speed guns, weather radar and medical ultrasound all measure speed from the Doppler shift of a reflected wave. For converting between the wavelengths and frequencies involved, the wavelength and frequency calculator is useful, and How far away is lightning? Counting the seconds to thunder puts the speed of sound to work in a different way.
Sources and further reading
The claims in this guide rest on these references, which were checked when the guide was last updated. Spotted an error? The contact page says how to report it.