Doppler effect

The Doppler effect is the change in the observed frequency or wavelength of a wave when the source and the observer move toward or away from each other. It occurs for any wave, including sound, light, and water waves, and it is named for the Austrian physicist and mathematician Christian Johann Doppler (1803–1853), who proposed it in 1842.¹ The everyday version is the siren of a passing ambulance, which sounds higher in pitch as it approaches and lower after it passes, even though the siren emits a constant frequency.¹

Mechanism

For a wave traveling through a medium such as air, the wave speed is fixed by the medium and is the same whether the source moves or not. If the source moves toward an observer, each successive wave crest is emitted from a point closer to the observer than the last, so the crests arrive bunched together: the wavelength is shortened and the frequency, related to the wavelength by $v = f \lambda$, is raised. A receding source stretches the crests apart and lowers the frequency. The same result follows when the observer moves, because a moving observer meets successive crests sooner or later than a stationary one would.¹

The two cases are not equivalent for sound. Because the speeds are measured relative to the medium, the equations differ depending on whether the source, the observer, or the air itself is moving. For a moving source and stationary observer,

$$f_{\text{o}} = f_{\text{s}} \left( \frac{v}{v \mp v_{\text{s}}} \right),$$

where $v$ is the speed of sound, $v_{\text{s}}$ the source speed, and the minus sign applies to approach. For a moving observer and stationary source,

$$f_{\text{o}} = f_{\text{s}} \left( \frac{v \pm v_{\text{o}}}{v} \right).$$

Both cases combine into the general expression $f_{\text{o}} = f_{\text{s}} \left( \frac{v \pm v_{\text{o}}}{v \mp v_{\text{s}}} \right)$, with the top sign for approaching motion.¹

The asymmetry shows in numbers. A train with a 150 Hz horn moving at 35.0 m/s on a day when sound travels at 340 m/s is heard by a trackside observer at 167 Hz while approaching and 136 Hz after passing; the shifts are 17.0 Hz toward and 14.0 Hz away, so the shift is not symmetric for equal approach and recession speeds. An engineer riding on the train hears exactly 150 Hz, because source and observer share the same motion relative to the air.¹

When a source reaches the speed of the waves themselves, the wavefronts ahead of it bunch up and arrive together; for sound this is the mechanism of the sonic boom produced when an airplane reaches and surpasses the speed of sound.⁶

Doppler effect for light

Because light needs no medium, its Doppler shift in vacuum is determined solely by how fast the source and observer move relative to one another. At speeds comparable to the speed of light $c$, the classical formula must be replaced by the relativistic Doppler shift, which combines the classical effect with time dilation:

$$\lambda_{\text{obs}} = \lambda_{\text{s}} \sqrt{\frac{1 + v/c}{1 - v/c}}, \qquad f_{\text{obs}} = f_{\text{s}} \sqrt{\frac{1 - v/c}{1 + v/c}},$$

where $v$ is positive for recession and negative for approach.³ As a worked example, a galaxy receding at $0.825c$ that emits radio waves of wavelength 0.525 m is observed on Earth at 1.70 m.³

Time dilation adds a transverse shift. In classical physics a shift appears only when the motion has a component toward or away from the observer. In special relativity a source moving exactly sideways still shows a lowered frequency, because all processes on the moving source, including the emission of successive wave crests, are slowed by time dilation. This is the transverse Doppler effect.⁶

Redshift, blueshift, and astronomy

When a light source moves away from the observer, the observed wavelength is longer, a shift called a redshift; when it approaches, the wavelength is shorter, a blueshift. The terms became established because the Doppler effect was first used with visible light in astronomy, and astronomers now apply them to radio waves and X-rays as well.⁴

For speeds small compared with $c$, the shift along the line of sight follows the simple relation

$$\frac{\Delta\lambda}{\lambda} = \frac{v}{c},$$

so the recession or approach speed is $v = c \cdot \Delta\lambda/\lambda$. A hydrogen emission line emitted at 656.3 nm and observed at 656.6 nm therefore indicates a gas cloud receding at 140 km/s.⁴ The shift is hard to detect in a continuous spectrum but easy to measure in absorption and emission lines, whose laboratory wavelengths are known precisely; the whole pattern of lines shifts together, so the pattern remains recognizable while its displacement gives the speed.⁴

Astronomers use these measurements to determine how fast galaxies move away from us, estimate their distances, and measure the expansion rate of the universe.⁷ A related but distinct phenomenon is cosmological redshift: the redshift of most galaxies is caused not by motion through space but by the stretching of space itself as the universe expands, and the farther a galaxy is, the more its light is redshifted.⁷

Doppler broadening of spectral lines

The Doppler effect also broadens spectral lines rather than only shifting them. Thermal motion of emitting atoms or ions gives each emitter a small random velocity along the line of sight, so the emitted frequencies spread around the rest value. For a Maxwellian velocity distribution the line shape is Gaussian, with a full width at half maximum of

$$\Delta\lambda_{1/2}^{D} = (7.16 \times 10^{-7})\, \lambda \left( \frac{T}{M} \right)^{1/2},$$

where $T$ is the temperature of the emitters in kelvin and $M$ the atomic weight in atomic mass units. Doppler and pressure broadening are the principal physical causes of spectral line broadening.⁸

Applications

Radar and weather. A Doppler radar determines target velocity from the phase shift of a reflected pulse, the electromagnetic analog of the pitch change of a passing siren. The WSR-88D weather radar transmits pulses lasting 1.57 microseconds, repeated about 1300 times per second, and spends the great majority of each hour listening for returns; the measured phase shift of the returned wave gives the velocity of rain, snow, and other targets toward or away from the radar.⁵ Doppler radar is also used in meteorology to track the motion of storm clouds.¹

Medicine and speed measurement. Doppler shifts in ultrasound reflected from blood are used to measure blood velocity in medical diagnostics, and police radar uses the Doppler shift of microwaves to measure vehicle speeds.¹

History

Doppler presented the principle on 25 May 1842 in a lecture to the Royal Bohemian Scientific Society in Prague titled "Concerning the coloured light of double stars and of some other heavenly bodies" (Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels), arguing by analogy with sound and water waves that the frequency, and hence the color, of light changes when the source or observer moves.² The effect was tested with musicians playing on a moving open railroad car and beside the tracks, with the frequency changes measured on and off the train.^{1 • 10} His theory was challenged by the Dutch scientist Christophorus Buys Ballot, who set out to disprove it using sound; the effect was first verified for sound waves three years later, in 1845.⁹

References

1. The Doppler Effect, University Physics Volume 1, OpenStax
2. The Doppler effect (Ch. 6), J. B. Hearnshaw, The Analysis of Starlight, Cambridge University Press
3. Doppler Effect for Light, University Physics Volume 3, OpenStax
4. The Doppler Effect, Astronomy, OpenStax
5. How Radar Works, NOAA JetStream
6. Waves, motion and frequency: the Doppler effect, M. Pössel, Einstein Online
7. Spectroscopy 101 – Beyond Temperature and Composition, NASA Science
8. Spectral Line Shapes, Widths, and Shifts, NIST Atomic Spectroscopy
9. The Oxford Handbook of the History of Modern Cosmology (preview)
10. The fall and rise of the Doppler effect

Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Waves and optics › Wave phenomena and acoustics › Doppler effect › Doppler effect (overview and general theory)

Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026

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