Physics
How To Calculate The Doppler Effect
The Doppler effect is the change in observed frequency caused by motion of the source, the observer, or both. Approaching motion raises the pitch, receding motion lowers it, and one formula covers every case.
Quick Answer
f' = f x (c + vo) / (c - vs)
- f'
- The observed frequency, in hertz
- f
- The frequency emitted by the source, in hertz
- c
- The speed of sound, about 343 metres per second in air
- vo
- Observer speed toward the source, in metres per second
- vs
- Source speed toward the observer, in metres per second
Multiply the source frequency by the speed of sound plus the observer's speed toward the source, then divide by the speed of sound minus the source's speed toward the observer. With both at rest a 440 hertz siren stays at 440 hertz.
What Is The Doppler Effect?
The Doppler effect is the shift in frequency that appears when a wave source and an observer move relative to each other. The source still emits at its own frequency; what changes is the rate at which wave crests arrive at the listener.
A source moving toward you compresses the waves in front of it. The crests are closer together, so more of them arrive each second and the pitch rises. A source moving away stretches the waves behind it and the pitch falls.
An observer moving toward the source meets crests more often than a stationary one, which raises the observed frequency without changing the wavelength in the air at all. This is why the two motions enter the formula in different places.
The formula multiplies the emitted frequency by the speed of sound plus the observer's speed toward the source, all divided by the speed of sound minus the source's speed toward the observer. Signs are handled by choosing the direction.
Direction conventions are the main source of error. Speeds toward each other are positive, so a source approaching at 30 metres per second makes the denominator smaller and the pitch higher. Reversing the sign reverses the shift.
The speed of sound is about 343 metres per second in air at 20 degrees Celsius. It rises with temperature, roughly 0.6 metres per second per degree, so the same siren shifts slightly differently on a cold day.
Light shows the same effect but needs relativity, because the speed of light is the same for every observer. The result is a redshift for receding galaxies, which is how astronomers measure how fast the universe is expanding.
There is a hard limit. When the source reaches the speed of sound the crests pile up into a shock front, and beyond it the simple formula stops applying because the source outruns its own waves. This is the sonic boom.
Everyday examples are everywhere: a passing ambulance, a train horn, a police siren. The pitch drops noticeably as the vehicle passes, because the motion switches from approaching to receding in an instant.
Radar uses the effect to measure speed. The reflected wave is shifted by twice the relative motion, and the shift is converted directly into a speed reading, which is how speed guns work.
Medical ultrasound uses it to measure blood flow. Sound reflected from moving blood cells is shifted, and the shift reveals both the speed and the direction of the flow, all without breaking the skin.
The shift is asymmetric in an interesting way. For the same relative speed, a moving source produces a slightly different shift than a moving observer, because the medium carries the sound and only the source changes the wavelength itself.
Formula
f' = f x (c + vo) / (c - vs)
Formula page →Observed frequency with any combination of source and observer motion.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| f' | Observed frequency | hertz | The frequency the listener hears, in hertz. |
| f | Source frequency | hertz | The frequency the source emits, in hertz. |
| c | Speed of sound | m/s | About 343 metres per second in air at room temperature. |
| vo | Observer speed | m/s | Positive when the observer moves toward the source. |
| vs | Source speed | m/s | Positive when the source moves toward the observer. |
f' = f x c / (c - vs)
Formula page →With a stationary listener the numerator is simply the speed of sound.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| f' | Observed frequency | hertz | The frequency heard, in hertz. |
| vs | Source speed | m/s | Positive when approaching. |
| c | Speed of sound | m/s | About 343 metres per second in air. |
f' = f x (c + vo) / c
Formula page →With a stationary source the denominator is simply the speed of sound.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| f' | Observed frequency | hertz | The frequency heard, in hertz. |
| vo | Observer speed | m/s | Positive when moving toward the source. |
| c | Speed of sound | m/s | About 343 metres per second in air. |
How To Calculate The Doppler Effect
- 1
Write down the emitted frequency and the speed of sound
The siren emits 440 hertz and sound travels at 343 metres per second. Both numbers are in SI units, so no conversion is needed.
- 2
Assign signs to the two speeds
Use positive for motion toward the other party and negative for motion away. With both at rest the observer speed and source speed are zero.
- 3
Build the numerator and denominator
The numerator is the speed of sound plus the observer speed, which is 343 plus 0. The denominator is the speed of sound minus the source speed, which is 343 minus 0.
- 4
Divide and compare
440 times 343 divided by 343 is 440 hertz, so no shift appears when nothing moves. Any motion makes the fraction differ from one and the pitch shifts accordingly.
Examples
Example 1: A 440 hertz siren approaching at 30 metres per second
- Source frequency
- 440
- Source speed
- 30
- Source direction
- Toward the observer
- Observer speed
- 0
- Observer direction
- Toward the source
- Speed of sound
- 343
| Step | Calculation | Result |
|---|---|---|
| Source speed toward the observer | 30 | 30 |
| Denominator | 343 - 30 | 313 |
| Numerator | 343 + 0 | 343 |
| Observed frequency | 440 x 343 / 313 | 482.1725 |
| Shift | 482.1725 - 440 | 42.1725 |
Result: A 440 hertz siren approaching at 30 metres per second is heard at about 482.1725 hertz, a shift of 42.1725 hertz upward. Once it passes and moves away at the same speed the same formula gives a lower pitch.
Example 2: A listener moving toward a stationary 440 hertz siren at 20 metres per second
- Source frequency
- 440
- Source speed
- 0
- Source direction
- Toward the observer
- Observer speed
- 20
- Observer direction
- Toward the source
- Speed of sound
- 343
| Step | Calculation | Result |
|---|---|---|
| Denominator with a still source | 343 - 0 | 343 |
| Numerator | 343 + 20 | 363 |
| Observed frequency | 440 x 363 / 343 | 465.656 |
| Shift | 465.656 - 440 | 25.656 |
Result: Walking toward a stationary 440 hertz siren at 20 metres per second raises the heard pitch to about 465.656 hertz, a shift of 25.656 hertz. The motion changes how often crests are met, not the wavelength in the air.
Calculator
Observed frequency
440
- Shift
- 0
- Shift as a percentage
- 0.00%
- Wavelength in front of the source
- 0.7795
- Wavelength behind the source
- 0.7795
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the The Doppler Effect calculator page.
Common Mistakes
Putting the source speed in the numerator
Source motion belongs in the denominator because it changes the wavelength. Adding 30 to 343 instead of subtracting it predicts a lower pitch for an approaching siren.
Ignoring the direction signs
Approaching means the source speed is subtracted and the observer speed added. Treating both as positive regardless of direction gives the same answer for approach and recession.
Using the wrong speed of sound
The value 343 metres per second assumes air near room temperature. Using a value for water or steel changes the shift by orders of magnitude.
Forgetting to convert kilometres per hour
Divide by 3.6 to get metres per second. A car at 100 kilometres per hour is 27.78 metres per second, and skipping the conversion makes the shift look negligible.
Applying the formula at or above the speed of sound
When the source reaches the speed of sound the denominator approaches zero and the formula breaks down. A shock wave forms instead, and the simple ratio no longer describes the sound.
Expecting the same shift for moving source and moving observer
The two are not symmetric in a medium. A source at 30 metres per second and an observer at 30 metres per second give slightly different shifts.
FAQ
What is the Doppler effect formula for sound?
Observed frequency equals emitted frequency times the speed of sound plus the observer speed, divided by the speed of sound minus the source speed, with speeds signed toward each other.
Why does a siren drop in pitch as it passes?
While approaching, the source compresses the waves and raises the pitch. Once it passes, the same motion stretches the waves behind it, so the pitch falls below the emitted value.
Does the Doppler effect apply to light as well as sound?
Yes, but light needs the relativistic version because its speed is the same for all observers. Receding galaxies show a redshift, which is how cosmic expansion is measured.
What speed of sound should I use?
About 343 metres per second in air at 20 degrees Celsius. The value rises roughly 0.6 metres per second for each extra degree, and it is far higher in water or steel.
What happens when the source moves faster than sound?
The source outruns its own waves and they pile into a shock front, producing a sonic boom. The simple ratio is no longer valid because the denominator would be zero or negative.
How is the Doppler effect used in medicine?
Ultrasound measures the shift of sound reflected from moving blood cells. The size and sign of the shift reveal both the speed and the direction of blood flow.
References
- [1]Wikipedia, Doppler effect — https://en.wikipedia.org/wiki/Doppler_effect
- [2]Wikipedia, Speed of sound — https://en.wikipedia.org/wiki/Speed_of_sound
- [3]Wikipedia, Sonic boom — https://en.wikipedia.org/wiki/Sonic_boom
- [4]Wikipedia, Doppler ultrasonography — https://en.wikipedia.org/wiki/Doppler_ultrasonography