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The Doppler Effect
Waves & Optics
A siren sounds higher as an ambulance approaches and lower as it drives away. That pitch change is the Doppler effect, and it happens for light as well as sound.
What the Doppler effect is
The Doppler effect is the change in observed frequency when a wave source and an observer move relative to each other. For sound, when a source moves toward you, each new wave is emitted from a point closer to you, so the waves are squeezed together: the wavelength shortens and the frequency rises. When the source moves away, the waves are stretched out and the frequency falls. The effect depends only on the relative motion along the line joining source and observer; motion at right angles has no first-order effect. The same idea applies to any wave, including water waves and light, although light at high speeds needs a relativistic treatment.
The formula
For a source moving toward a stationary observer, the observed frequency is the emitted frequency multiplied by the speed of sound divided by the speed of sound minus the source speed. The formula shows why the effect grows as the source speed approaches the speed of sound: the denominator shrinks and the frequency climbs sharply. If the observer moves instead, the formula changes: a moving observer meets more wavefronts per second when approaching the source. When both move, the two effects combine. At the speed of sound itself the waves pile up into a shock wave, heard as a sonic boom.
Where you see it
The falling pitch of a passing siren or racing car is the everyday example, and radar guns use the Doppler shift of reflected radio waves to measure a car's speed. Weather radar tracks the motion of rain, and medical ultrasound measures blood flow by the Doppler shift of sound reflected from moving blood cells. In astronomy, the redshift of light from distant galaxies showed that the universe is expanding: the faster a galaxy recedes, the more its light is shifted toward longer wavelengths. Police, doctors and astronomers all rely on the same simple effect.
Worked example: pitch of an approaching siren
An ambulance siren emits f = 1000 Hz and approaches at vs = 20 m/s. Speed of sound v = 343 m/s. For a source moving toward a stationary observer, f' = f x v / (v - vs). Substitute: f' = 1000 x 343 / (343 - 20) = 1000 x 343 / 323. 343 / 323 = 1.062, so f' = 1062 Hz. The observer hears 1062 Hz, higher than the 1000 Hz emitted. After it passes, the frequency drops below 1000 Hz.
Common mistakes
- Saying the source changes its frequency: the source emits the same frequency; the observer receives waves squeezed together or stretched out.
- Using the wrong sign: approaching means a smaller denominator and higher frequency; receding means a larger denominator and lower frequency.
- Forgetting that light needs the relativistic formula at high speeds: the sound formulas above are for sound in a medium, not for light near light speed.
Practice questions
- A 500 Hz source approaches at 30 m/s. What frequency is heard? (v = 343 m/s.)
Show answer
f' = 500 x 343 / (343 - 30) = 500 x 343 / 313 = 548 Hz.
- What frequency is heard after the same source passes and recedes at 30 m/s?
Show answer
f' = 500 x 343 / (343 + 30) = 500 x 343 / 373 = 460 Hz.
- What happens to light from a galaxy moving away from us?
Show answer
The wavelength is stretched, so the light shifts toward the red end of the spectrum.
Frequently asked questions
Does the formula change when the source moves away?
Yes, for a source moving away the denominator becomes v + vs, so the observed frequency is lower.
What is the Doppler effect called for light from distant galaxies?
Redshift, where light from distant galaxies is shifted to longer wavelengths because the galaxies are moving away and space is expanding.