Doppler Effect

The Doppler effect describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of that wave.

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How to measure it in class

FizziQ’s spectrogram and frequency meter make it possible to track in real time the frequency of a moving sound source and deduce its speed, without ever needing to know the frequency actually emitted.

Steps:

• Attach a smartphone emitting a stable pure tone (frequency synthesizer set to around 1,000 Hz) to a support that will pass in front of the class: a bicycle, a cart, or a long pendulum.
• Place a second, stationary smartphone in spectrogram mode along the path, and start recording before the source begins to move.
• Have the source pass at a roughly constant speed in front of the receiver; on the spectrogram, the trace forms a high plateau, a rapid transition at the moment of passage, then a low plateau.
• Read the approach plateau frequency f₊ and the receding plateau frequency f₋ on the portions where the trace is clearly horizontal, far from the transition.
• Calculate the speed using v = c(f₊ − f₋)/(f₊ + f₋), with c ≈ 343 m/s: the emitted frequency cancels out, so there is no need to measure it.
• Check the result by timing the passage of the source between two markers about ten meters apart.

Scientific activities on this topic

Many experiments that can be carried out with a smartphone demonstrate the Doppler effect. Here are two experiments that are easy to perform in class or in remote learning to demonstrate and measure the Doppler effect:

• Doppler effect: Measure the speed of a sound source using the Doppler effect
• Sound pendulum: Analysis of the variations in the frequency emitted by a rotating pendulum

Our article Five Doppler Effect Experiments with a Smartphone details other protocols.

Learn more

The Doppler effect was named after the Austrian physicist Christian Doppler (1803-1853), who formulated the principles of this effect in 1842 in a paper entitled “Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels” (On the colored light of binary stars and some other stars of the heavens). Doppler noticed that the frequency of the light emitted by a star moving relative to the Earth would be modified according to the speed of that star relative to us. He thought this would explain the color of binary stars, which was a mistake: stellar speeds are far too small to produce a perceptible color change. The principle itself, however, was correct. It was confirmed for sound as early as 1845 by Buys Ballot, who had trumpet players perform on a railway wagon while musicians on the platform noted the perceived pitch.

This phenomenon is observed when the source of a wave (such as sound or light) and an observer move relative to each other. This effect causes a perceived change in the frequency of the wave:

• If the wave source approaches the observer, the waves are compressed, which increases the perceived frequency (and decreases the wavelength). For sound, this results in a higher-pitched sound.

• If the wave source moves away from the observer, the waves are stretched, which decreases the perceived frequency (and increases the wavelength). For sound, this results in a lower-pitched sound.

The formula for calculating the Doppler effect depends on the context (sound or light) and on the relative speed between the source and the observer. For sound waves in a medium such as air, the formula is as follows:

f’ = f*(v+vo)/(v+vs)

where f’ is the frequency perceived by the observer, f the frequency of the source, v the propagation speed of the wave, vo the speed of the observer relative to the medium (positive if approaching the source) and vs the speed of the source relative to the medium (positive if moving away from the observer).

Note that this expression involves separately the speed of the source and that of the observer, both measured relative to the air. This is not a minor detail: a source approaching at 30 m/s and an observer approaching at 30 m/s do not give exactly the same shift. In acoustics, the propagation medium constitutes a privileged reference frame, and the two situations are physically distinct.

For light, things are quite different. There is no propagation medium, hence no privileged reference frame: only the relative speed of the source and the observer can play a role. The relativistic Doppler effect is written f’ = f·√((1 − β)/(1 + β)) for a receding source, with β = v/c. It is therefore not the acoustic formula “slightly corrected”: it is a formula of a different form, symmetric by construction. At low speeds (β ≪ 1) it reduces to f’ ≈ f(1 − v/c), which explains why the classical result appears to be recovered in everyday astronomy. It also predicts a transverse Doppler effect, with no acoustic equivalent: a source passing perpendicularly, neither approaching nor receding, still appears redshifted, purely through time dilation.

The Doppler effect has a wide and fascinating range of applications that spans various scientific and practical fields. In astronomy, it is essential for studying the motion and properties of stars and galaxies; observing the redshift or blueshift of the light emitted by these celestial bodies tells us about their speed and direction relative to Earth. This method is crucial for understanding the expansion of the universe and helped in the formulation of cosmological theories such as the Big Bang.

In the medical field, the Doppler effect is used in Doppler ultrasound to visualize blood flow through veins and arteries, giving physicians a detailed view of blood circulation and enabling early detection of conditions such as venous thrombosis.

In meteorology, Doppler radars analyze precipitation and wind patterns, playing a crucial role in forecasting and tracking storms and extreme weather events.

Finally, in everyday life, the Doppler effect can be observed in the change in pitch of an ambulance siren as it approaches and then moves away, providing an audible illustration of this fascinating principle of physics.

Formula

For a sound wave in air, the perceived frequency is written:

f’ = f·(c + v_o)/(c + v_s)

where:

• f’: frequency perceived by the observer (Hz)
• f: frequency emitted by the source (Hz)
• c: speed of sound in the medium, about 343 m/s in air at 20 °C
• v_o: speed of the observer relative to the medium, counted positive when approaching the source (m/s)
• v_s: speed of the source relative to the medium, counted positive when moving away from the observer (m/s)

Case of a moving source passing in front of a stationary observer. Let f₊ be the frequency perceived during the approach and f₋ during the recession:

f₊ = f·c/(c − v) and f₋ = f·c/(c + v)

By forming the ratio of the difference to the sum, the emitted frequency f disappears:

v = c·(f₊ − f₋)/(f₊ + f₋)

where:

• v: speed of the source along the line of sight (m/s)
• f₊: frequency measured during the approach (Hz)
• f₋: frequency measured during the recession (Hz)

This relation is exact and it is the one used in class, because it requires no prior knowledge of the source. The relative gap between the two plateaus is:

(f₊ − f₋)/f ≈ 2v/c

the factor 2 coming from the fact that the shift occurs once in one direction on approach, and once in the other on recession.

For light, in the absence of a propagation medium:

f’ = f·√((1 − β)/(1 + β)), with β = v/c

where:

• v: relative recession speed of the source and the observer (m/s)
• c: speed of light in vacuum, 3.00 × 10⁸ m/s

Application examples

• A car traveling at 50 km/h (13.9 m/s) emits a 1,000 Hz sound. One perceives 1,042 Hz on approach and 961 Hz on recession, that is a jump of 81 Hz at the passage: about 8%, which is 2v/c and not v/c.

• A traffic radar sends a wave onto a vehicle and analyzes the echo. Here the shift is doubled a second time, because the vehicle successively plays the role of observer and then of source.

• Doppler ultrasound measures the speed of red blood cells. At 5 MHz and for a blood flow of 0.5 m/s, the shift is about 3 kHz, an audible frequency that the device renders as sound.

• The redshift of distant galaxies allowed Hubble to establish in 1929 that the Universe is expanding.

• The radial velocity method detects exoplanets: the planet makes its star oscillate by a few meters per second, which periodically shifts its spectral lines.

• Doppler weather radars distinguish rain that is approaching from rain that is receding, and can thus detect the characteristic rotation of tornadoes.

FAQ

Q: How can a speed be measured with the Doppler effect without knowing the emitted frequency? A: By reading the two plateaus f₊ and f₋ on either side of the passage, then applying v = c(f₊ − f₋)/(f₊ + f₋). The emitted frequency cancels out in the calculation. This is the great advantage of the “source passing by” configuration over the “source only approaching” configuration.

Q: Is the gap between the two frequencies v/c or 2v/c? A: 2v/c. This mistake is common. The shift on approach is about +v/c and the one on recession about −v/c; the gap between the two therefore accumulates both contributions. Using v/c leads to reporting a speed twice too large.

Q: Is a source approaching the same as an observer approaching? A: For sound, no. The formalism involves speeds relative to the air, which serves as a privileged reference frame: the two cases give different expressions, even though they coincide at first order in v/c. For light, there is no medium, so only the relative speed matters and the question has no meaning.

Q: Does the wind change the result? A: Yes, indirectly. The wind does not create a Doppler effect by itself, but it moves the medium, which changes the propagation speed relative to the ground. An outdoor measurement in strong wind should be interpreted with caution; indoors, the problem does not arise.

Q: Why does the sound not change abruptly but glide as the source passes? A: Because only the component of the velocity directed toward the observer plays a role. When the source reaches its closest point, this component vanishes and then changes sign. The transition is all the faster as the source passes close to the observer.

Related concepts

Frequency - Speed of Sound - Wavelength - Spectrogram