Doppler Effect Calculatorf′ = f (v + vo) ÷ (v − vs)
The Doppler effect is the change in frequency you hear when a sound source or listener moves: f′ = f (v + vo) ÷ (v − vs) when they approach each other. That’s why a passing ambulance siren drops in pitch. Get the frequency heard while a source approaches and while it recedes, work out a vehicle’s speed from the two pitches, or see how a police radar gun turns a frequency shift into a speed.
Formula verified against OpenStax — University Physics, Volume 1 · HyperPhysics, Georgia State University
By Manzoor Ahmad · Pharmacist & calculator developerReviewed & updated
Source & listener
Speeds are towards each other while approaching. Use 0 for a stationary listener.
Step-by-step working
What is doppler effect?
The Doppler effect is the apparent change in frequency of a wave caused by motion of the source, the observer, or both. Waves bunch up in front of a moving source and spread out behind it.
Christian Doppler described it in 1842. Today it is used in radar speed guns, medical ultrasound, weather radar and astronomy, where the redshift of distant galaxies revealed that the universe is expanding.
Quick answers
Why does a siren change pitch as it passes?
While it approaches, each wave is emitted closer to you, so crests arrive more often and the pitch is higher. Once it passes, the crests are stretched out and the pitch drops.
How do I calculate the frequency heard from a moving source?
f′ = f × v ÷ (v − vs) while approaching. A 500 Hz horn on a car at 25 m/s sounds like 539.3 Hz coming towards you and 466.0 Hz going away.
How does a radar gun measure speed?
It measures the frequency shift of microwaves bounced off the car: v = cΔf ÷ 2f₀. On K-band (24.125 GHz), every 1 km/h adds about 44.7 Hz of shift.
Can you work out speed from a recording?
Yes. Measure the pitch before and after the vehicle passes, then vs = v (f₁ − f₂) ÷ (f₁ + f₂). The local speed of sound matters, so correct it for temperature.
Doppler effect formula
Valid for speeds well below the speed of sound. Light needs the relativistic version.
| Symbol | Meaning | SI unit |
|---|---|---|
| f′ | Observed frequency | Hz |
| f | Emitted frequency | Hz |
| v | Speed of sound in the medium | m/s |
| vs | Source speed | m/s |
| vo | Observer speed | m/s |
| Δf | Radar frequency shift | Hz |
How to calculate the Doppler shift
- Find the speed of sound: 343 m/s at 20 °C.
- Approaching: add the listener speed on top and subtract the source speed underneath.
- Receding: flip both signs.
- Multiply the ratio by the source frequency.
Worked example: an ambulance at 30 m/s
An ambulance siren at 700 Hz passes you at 30 m/s (108 km/h). What do you hear?
Receding: f′ = 700 × 343 ÷ (343 + 30) = 643.7 Hz
Drop as it passes ≈ 3.04 semitones
Working backwards from 767.1 and 643.7 Hz gives vs = 343 × 123.4 ÷ 1,410.8 ≈ 30 m/s.
Where the Doppler effect is used
| Application | What is measured |
|---|---|
| Police radar and lidar | Vehicle speed from the reflected frequency shift |
| Doppler ultrasound | Blood flow speed in arteries and the fetal heart |
| Weather radar | Wind speed inside storms (rotation, tornadoes) |
| Astronomy (redshift) | How fast stars and galaxies are receding |
| Sports radar | Pitch and serve speeds |
Common mistakes to avoid
- Mixing up the signs. Approaching always raises the pitch; check your answer is above f.
- Assuming source and observer motion are symmetric. At the same speed, a moving source shifts the pitch more than a moving listener.
- Using the sound formula for light. Electromagnetic Doppler shifts need special relativity.
Manzoor Ahmad
Pharmacist turned Web Developer & Calculator Engineer · 13+ years in web developmentI’m a pharmacist by training who turned into a full-time tech geek, and for 13+ years I’ve designed and coded web tools. My pharmacy degree included physical pharmacy and pharmaceutics — states of matter, heat, flow and reaction kinetics — so the physics behind these tools is familiar ground, not a new hobby.
I’m not a physicist, so I don’t ask you to take my word for it: every formula on this page is traced to an open university textbook or NIST reference data, and the calculator is tested against published worked examples before it goes live. Spot an error? Tell me and it will be fixed and dated on the next update.
Doppler Effect Calculator: frequently asked questions
The change in observed frequency of a wave when the source and observer move relative to each other: higher when approaching, lower when receding.
f′ = f (v ± vo) ÷ (v ∓ vs), using the upper signs when source and observer approach each other.
Not directly. It changes pitch. Loudness changes because the source gets closer and then farther away.
The wavefronts pile up into a shock wave, heard as a sonic boom, and the formula breaks down.
Yes, as redshift and blueshift, but it must be calculated with special relativity.
References & sources
- OpenStax — University Physics, Volume 1 — Ch. 17.7, The Doppler Effect
- HyperPhysics, Georgia State University — Doppler effect for sound
- NOAA National Weather Service — Doppler radar — weather applications
Disclaimer: This calculator is an educational tool built on idealised textbook models. It is not a substitute for professional engineering or safety calculations.