Doppler Effect Calculator
Calculate the observed frequency due to the Doppler effect using f' = f₀ × (v ± v_o)/(v ∓ v_s). Enter the source frequency, source and observer velocities, and speed of sound to find the perceived frequency shift. See also our Frequency Calculator and Speed of Sound Calculator.
How to Calculate the Doppler Effect
The Doppler effect is the change in frequency (or wavelength) of a wave as perceived by an observer moving relative to the wave source. Named after Austrian physicist Christian Doppler who proposed it in 1842, this phenomenon is experienced daily — the pitch of an ambulance siren rises as it approaches and falls as it moves away. The effect applies to all waves: sound, light, water waves, and even gravitational waves.
For sound waves, the Doppler formula is: f' = f₀ × (v ± v_o)/(v ∓ v_s), where f₀ is the source frequency, v is the speed of sound, v_o is the observer's speed, and v_s is the source's speed. The signs depend on direction: use the upper signs (+v_o, -v_s) when source and observer approach each other, and lower signs (-v_o, +v_s) when they move apart. The key rule: motion that reduces the distance between source and observer increases the observed frequency.
The physical explanation is straightforward. When a source approaches, each successive wave crest is emitted from a position closer to the observer, so crests arrive more frequently (higher frequency, shorter wavelength). When the source recedes, each crest is emitted from farther away, so crests arrive less frequently (lower frequency, longer wavelength). The effect is asymmetric — approaching produces a larger frequency shift than receding at the same speed.
An important distinction: the Doppler effect for sound is different depending on whether the source or observer is moving (even at the same relative speed). This is because sound has a preferred reference frame — the medium (air). For light in vacuum, there is no preferred frame, and the relativistic Doppler effect depends only on relative velocity. This distinction was historically important in establishing special relativity.
Doppler Effect Formula
General Doppler formula (sound):
f' = f₀ × (v ± v_o) / (v ∓ v_s)
+ v_o when observer moves toward source
- v_s when source moves toward observer
Source approaching, observer stationary:
f' = f₀ × v / (v - v_s)
Source receding, observer stationary:
f' = f₀ × v / (v + v_s)
Observer approaching, source stationary:
f' = f₀ × (v + v_o) / v
Wavelength (source moving):
λ' = λ₀ × (v ∓ v_s) / v
Relativistic Doppler (light):
f' = f₀ × √((1+β)/(1-β))
β = v/c (relative velocity / speed of light)
Example Calculation
An ambulance with a 700 Hz siren approaches at 25 m/s. A stationary observer hears:
Given: f₀ = 700 Hz, v_s = 25 m/s, v_o = 0, v = 343 m/s
Approaching:
f' = 700 × 343/(343-25) = 700 × 343/318 = 755.0 Hz
Shift: +55 Hz (+7.9%) — pitch sounds higher
Receding:
f' = 700 × 343/(343+25) = 700 × 343/368 = 652.4 Hz
Shift: -47.6 Hz (-6.8%) — pitch sounds lower
Total perceived change as ambulance passes:
755.0 → 652.4 Hz (drop of 102.6 Hz = ~2 semitones)
Note: approaching shift (55 Hz) > receding shift (47.6 Hz)
The Doppler effect is asymmetric!
Doppler Effect Reference Table
| Scenario | Source (Hz) | Observed (Hz) |
|---|---|---|
| Ambulance approaching (vs=25m/s) | 700 | 755 |
| Ambulance receding (vs=25m/s) | 700 | 652 |
| Car horn approaching (vs=30m/s) | 440 | 483 |
| Car horn receding (vs=30m/s) | 440 | 405 |
| Train whistle (vs=40m/s, approaching) | 500 | 566 |
| Train whistle (vs=40m/s, receding) | 500 | 448 |
| Observer running toward (vo=5m/s) | 440 | 446 |
| Both approaching (vs=20, vo=10) | 440 | 481 |
| Race car (vs=80m/s, approaching) | 300 | 391 |
| Race car (vs=80m/s, receding) | 300 | 243 |
Calculated with v = 343 m/s (sound in air at 20°C)
Frequently Asked Questions
What is the Doppler effect?
The Doppler effect is the change in frequency of a wave relative to an observer moving relative to the wave source. When the source and observer approach each other, the observed frequency increases (higher pitch for sound, blueshift for light). When they move apart, the observed frequency decreases (lower pitch, redshift). The effect was first described by Christian Doppler in 1842 and confirmed experimentally by Buys Ballot in 1845 using musicians on a moving train. It applies to all types of waves.
Why does a siren change pitch as it passes?
As a siren approaches, it "catches up" to its own sound waves, compressing them together. The observer receives more wave crests per second, perceiving a higher frequency (higher pitch). As it recedes, it moves away from its emitted waves, stretching them apart. The observer receives fewer crests per second, perceiving a lower frequency (lower pitch). The actual siren frequency never changes — only the perceived frequency does. The sudden pitch drop as the vehicle passes is the most noticeable manifestation of the Doppler effect in daily life.
How is the Doppler effect used in medicine?
Doppler ultrasound measures blood flow velocity by bouncing ultrasound off moving red blood cells. The frequency shift of the reflected signal indicates how fast the blood is moving and in which direction. This is used to detect blood clots, assess heart valve function, monitor fetal heartbeat, and evaluate blood flow in arteries and veins. Color Doppler imaging displays flow direction and speed as colors overlaid on anatomical images. It is non-invasive, real-time, and uses no radiation.
What is redshift and blueshift?
Redshift and blueshift are the Doppler effect applied to light. When a light source moves away from an observer, its light is shifted toward longer wavelengths (red end of spectrum) — this is redshift. When approaching, light shifts toward shorter wavelengths (blue end) — blueshift. Edwin Hubble's 1929 observation that distant galaxies are all redshifted proved the universe is expanding. The amount of redshift indicates recession speed: z = Δλ/λ = v/c (for v << c). Cosmological redshift of distant galaxies reaches z > 10.
How do police radar guns use the Doppler effect?
Radar speed guns emit microwave radiation at a known frequency toward a moving vehicle. The reflected signal has a Doppler-shifted frequency proportional to the vehicle's speed. The gun measures the frequency difference (beat frequency) between transmitted and received signals to calculate speed: v = (Δf × c)/(2f₀). The factor of 2 accounts for the double Doppler shift (wave hits moving car, then reflects back). Modern guns use digital signal processing for accuracy within ±1 mph. Lidar guns use the same principle with laser light.
What happens at the speed of sound (Mach 1)?
When a source reaches the speed of sound (v_s = v), the denominator in the Doppler formula becomes zero, and the observed frequency theoretically becomes infinite. Physically, all the wave fronts pile up at the source, creating a shock wave (sonic boom). The Doppler formula is not valid at or above Mach 1. Beyond Mach 1, the source outruns its waves, creating a cone-shaped shock wave (Mach cone) with half-angle sin(θ) = v/v_s = 1/M. Objects inside the cone haven't yet "heard" the approaching source.
Doppler Effect Applications
The Doppler effect has remarkably diverse applications across science and technology. In astronomy, it reveals the radial velocities of stars and galaxies, detects exoplanets (via stellar wobble), and proved the expansion of the universe. In meteorology, Doppler radar measures wind speeds inside storms, enabling tornado warnings. In navigation, Doppler shift of satellite signals helps determine position and velocity. In acoustics, it affects the design of concert halls and the tuning of instruments on moving vehicles.
Modern technology exploits the Doppler effect in countless ways. Doppler lidar measures wind speeds for aviation safety and wind farm optimization. Doppler sonar tracks underwater objects and measures ocean currents. Automotive radar uses Doppler to detect approaching vehicles for collision avoidance systems. Even your smartphone uses Doppler-based techniques in some motion-sensing applications. The principle discovered in 1842 remains one of the most practically useful phenomena in all of physics.