Doppler Effect¶
The Story Behind the Math¶
In 1842, Austrian physicist Christian Doppler proposed a bold idea: the color of light from a star depends on whether it's moving toward or away from us. He published his theory in a paper titled "On the coloured light of the double stars and certain other stars of the heavens."
The problem that fascinated him: Why do some binary stars appear to change color as they orbit each other? Doppler reasoned that just as the pitch of a sound changes when a source moves, the frequency of light must shift too.
The first experimental confirmation came in 1845, when Dutch meteorologist Christoph Buys Ballot hired a group of musicians to play a calibrated note on an open train car while trained musicians on the platform listened. As the train approached and receded, the listeners confirmed the pitch shift — exactly as Doppler predicted.
The key insight: When a wave source moves toward you, the waves get "bunched up" — the wavelength shortens and the frequency increases. When the source moves away, the waves get "stretched out" — the wavelength lengthens and the frequency decreases. This happens because each successive wave crest is emitted from a slightly different position.
In 1848, Hippolyte Fizeau independently extended the idea to light, explaining how spectral lines would shift toward the blue (higher frequency) or red (lower frequency) end of the spectrum. This is why the effect for light is sometimes called the Doppler-Fizeau effect.
Why It Matters¶
The Doppler effect is one of the most widely used phenomena in science and technology:
- Astronomy — Edwin Hubble used redshift to discover that the universe is expanding, leading to the Big Bang theory
- Radar and speed guns — Police radar measures the Doppler shift of reflected microwaves to determine vehicle speed
- Medical ultrasound — Doppler ultrasound measures blood flow velocity by detecting frequency shifts
- Weather forecasting — Doppler radar tracks storm movement and wind speed
- Satellite communication — GPS systems must correct for Doppler shifts due to satellite motion
- Music and acoustics — The familiar pitch change of a passing siren or train horn
The Formula¶
For a source emitting waves at frequency \(f_0\), the observed frequency \(f\) depends on the relative velocity between source and observer.
For sound (source and observer moving along the line connecting them):
Where: - \(f\) = observed frequency - \(f_0\) = emitted (source) frequency - \(v\) = speed of the wave in the medium (e.g., speed of sound) - \(v_o\) = speed of the observer (positive if moving toward the source) - \(v_s\) = speed of the source (positive if moving toward the observer)
For light (relativistic Doppler effect):
Where \(\beta = v/c\) is the relative velocity as a fraction of the speed of light, with \(v > 0\) for approach.
Derivation¶
Step 1: A Stationary Source¶
Consider a source emitting sound waves with frequency \(f_0\) (period \(T_0 = 1/f_0\)) in a medium where the wave speed is \(v\). When the source is stationary, the wavelength is:
The wave crests are evenly spaced in all directions, separated by \(\lambda_0\).
Step 2: Moving Source¶
Now suppose the source moves toward the observer at speed \(v_s\). Between emitting one crest and the next (a time interval \(T_0\)), the source moves a distance \(v_s T_0\) closer to the observer.
The second crest is emitted from a position \(v_s T_0\) closer than the first crest was. But the first crest has already traveled \(v T_0\) in the same time.
So the distance between successive crests (the wavelength) in the direction of motion is:
Step 3: The Observed Frequency (Moving Source)¶
Since the wave still travels at speed \(v\) in the medium, the observed frequency is:
When the source moves toward the observer (\(v_s > 0\)), the denominator decreases, so \(f > f_0\) — the pitch goes up.
When the source moves away (\(v_s < 0\), or equivalently replace \(v_s\) with \(-v_s\)), \(f < f_0\) — the pitch goes down.
Step 4: Moving Observer¶
Now consider a stationary source and a moving observer. The wavelength \(\lambda_0\) is unchanged because the source is stationary. But the observer moves through the wave pattern at speed \(v_o\).
The wave crests pass the observer at a rate determined by the relative speed between the observer and the wave:
When the observer moves toward the source (\(v_o > 0\)), crests arrive more frequently and \(f > f_0\).
Step 5: The General Formula¶
Combining both effects — moving source and moving observer:
This is the general Doppler formula for sound. The numerator accounts for the observer's motion; the denominator accounts for the source's motion.
Step 6: The Relativistic Case (Light)¶
For light, there is no medium — light always travels at speed \(c\) regardless of reference frame (Einstein's postulate). The classical formula breaks the symmetry between source and observer, but relativity demands they be equivalent.
Starting from the classical approach and applying time dilation (a moving clock runs slow by factor \(\gamma = 1/\sqrt{1 - \beta^2}\)):
The source emits with period \(T_0\) in its own frame, but the observer measures the source's clock as running slow:
Using the classical moving-source result with \(v = c\):
Since \(\gamma = \frac{1}{\sqrt{1 - \beta^2}} = \frac{1}{\sqrt{(1-\beta)(1+\beta)}}\):
This is the relativistic Doppler formula. It depends only on relative velocity, as required by special relativity.
Variables Explained¶
| Symbol | Name | Unit | Description |
|---|---|---|---|
| \(f\) | Observed frequency | Hz | Frequency detected by the observer |
| \(f_0\) | Source frequency | Hz | Frequency emitted by the source |
| \(v\) | Wave speed | m/s | Speed of the wave in the medium |
| \(v_s\) | Source speed | m/s | Speed of source along the line to observer |
| \(v_o\) | Observer speed | m/s | Speed of observer along the line to source |
| \(\lambda\) | Wavelength | m | Distance between successive wave crests |
| \(\beta\) | Velocity ratio | dimensionless | \(v/c\), fraction of the speed of light |
| \(c\) | Speed of light | m/s | \(\approx 3 \times 10^8\) m/s |
Worked Examples¶
Example 1: Ambulance Siren¶
An ambulance emitting a siren at \(f_0 = 700\) Hz approaches you at \(v_s = 30\) m/s. The speed of sound is \(v = 343\) m/s. You are standing still (\(v_o = 0\)).
Approaching:
Receding (after it passes, \(v_s = -30\) m/s):
The pitch drops from 767 Hz to 644 Hz as the ambulance passes — a drop of about 123 Hz, or roughly a minor third in musical terms.
Example 2: Receding Galaxy¶
A galaxy is moving away from Earth at 10% the speed of light (\(\beta = -0.1\), or equivalently \(v/c = 0.1\) recession). A hydrogen spectral line is emitted at \(f_0 = 4.57 \times 10^{14}\) Hz (656 nm, red light).
Using the relativistic formula (with \(\beta = -0.1\) for recession):
The corresponding wavelength:
The line has shifted from 656 nm (red) to 726 nm (deeper red / near infrared) — a redshift.
Example 3: Police Radar¶
A police radar gun emits microwaves at \(f_0 = 10.5\) GHz. A car approaches at 36 m/s (about 130 km/h). Since \(v \ll c\), we can use the low-velocity approximation:
The factor of 2 appears because the wave travels to the car and back (double Doppler shift). The radar measures this 2.5 kHz shift to calculate the car's speed.
Common Mistakes¶
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Mixing up signs: The sign convention must be consistent. Decide whether "toward" is positive or negative and stick with it. In our formula, velocities toward each other are positive.
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Using the sound formula for light: Sound requires a medium and the formula is asymmetric (moving source ≠ moving observer). Light has no preferred frame — use the relativistic formula.
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Forgetting the medium matters for sound: The Doppler shift for sound depends on motion relative to the air, not just relative motion between source and observer. Wind affects the result.
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Assuming linear frequency change: The pitch doesn't gradually slide from high to low as a car passes. It stays approximately constant (but shifted) during approach, changes rapidly at the closest point, then stays approximately constant (shifted the other way) during recession.
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Confusing Doppler shift with sonic boom: The Doppler effect occurs at all speeds. A sonic boom occurs only when the source exceeds the wave speed (\(v_s > v\)), creating a shock wave.
Related Formulas¶
- Simple Harmonic Motion — Wave frequency and period fundamentals
- Fourier Transform — Frequency analysis of signals
History¶
- 1842 — Christian Doppler publishes his theory on the frequency shift of waves from moving sources
- 1845 — Buys Ballot confirms the effect experimentally using musicians on a train
- 1848 — Hippolyte Fizeau independently extends the theory to light (spectral line shifts)
- 1868 — William Huggins uses Doppler shifts to measure the radial velocity of Sirius
- 1901 — Aristarkh Belopolsky confirms the optical Doppler effect in the laboratory
- 1905 — Einstein derives the relativistic Doppler effect from special relativity
- 1929 — Edwin Hubble uses galactic redshifts to discover the expansion of the universe
- 1950s — Doppler radar developed for weather tracking
- 1960s — Medical Doppler ultrasound introduced for blood flow measurement
References¶
- Doppler, C. (1842). Über das farbige Licht der Doppelsterne und einiger anderer Gestirne des Himmels. Abhandlungen der Königlichen Böhmischen Gesellschaft der Wissenschaften.
- Einstein, A. (1905). Zur Elektrodynamik bewegter Körper. Annalen der Physik, 322(10), 891–921.
- Hubble, E. (1929). A relation between distance and radial velocity among extra-galactic nebulae. Proceedings of the National Academy of Sciences, 15(3), 168–173.
- Rosen, J., & Gothard, L. Q. (2009). Encyclopedia of Physical Science. Facts on File.