What this calculator does
Find out what pitch you will hear when the thing making a sound, the listener, or both are moving. When an ambulance races past, its siren seems to drop in pitch the moment it passes. That is the Doppler effect. When the source and the listener move closer together, more wave crests reach you each second, so the pitch rises. When they move apart, it falls. The source itself never changes its tone.
It works for sound or any other wave traveling through a medium (the air, water or other material the wave moves through). Textbooks disagree on which direction counts as positive, so you choose the sign convention. The working then shows exactly how your speeds are plugged in.
Where it's used
Why this math matters outside the classroom:
- Passing sirens. Listeners hear an ambulance siren drop in pitch as it passes because its waves bunch up ahead of it and spread out behind.
- Speed radar. Police radar measures a car's speed from the frequency shift of microwaves bounced off the moving vehicle.
- Weather radar. Doppler weather radar reads wind speed and rotation inside storms from shifts in the returning signal.
- Astronomy. Astronomers spot distant galaxies moving away by the redshift in their light, and find planets by a star's tiny wobble.
- Pitch shift. A 500 Hz source approaching at 20 m/s in air at 343 m/s sounds like 500 × 343 ÷ 323, about 531 Hz.
How to use it
- Choose a sign convention. The first, 'positive = moving toward the other', is the easiest: approaching speeds are positive and receding speeds negative, for both the source and the observer.
- Enter the emitted frequency and the speed of the wave in the medium, for example 343 m/s for sound in air.
- Enter the source and observer velocities relative to the air (or other medium), using 0 for anything standing still.
- Read the observed frequency, the shift in hertz, as a percentage and in musical semitones.
What the inputs mean
- Sign convention for the speeds
- 'Positive = moving toward the other': a source driving toward the listener and a listener walking toward the source both get positive speeds. 'Positive = the direction from source to observer': one shared axis points from the source to the observer; a source moving along it approaches, but an observer moving along it retreats, so the calculator flips the observer's sign for you.
- Emitted frequency (f)
- The frequency the source actually produces, as heard by someone moving with it.
- Wave speed in the medium (v)
- About 343 m/s for sound in air at 20 °C, about 1,480 m/s in water.
- Source velocity (vₛ)
- Must be slower than the wave speed in either direction. At or above it, the waves bunch into a shock wave and the formula breaks down.
- Observer velocity (vₒ)
- An observer can approach faster than the wave speed, but cannot move away that fast and still hear the wave.
Formula and method
f′ = f × (v + vₒ) ÷ (v − vₛ) vₒ > 0: observer moving toward the source vₛ > 0: source moving toward the observer Semitones = 12 × log₂(f′ ÷ f)
Here f is the frequency the source makes and f′ ("f prime") is the frequency you hear, both in hertz (waves per second). v is the speed of the wave in the medium, vₛ is the source's speed and vₒ is the observer's speed. log₂ is the base-2 logarithm, which counts how many times a number doubles.
A moving source squeezes the waves ahead of it, shortening the wavelength (the distance between crests) to (v − vₛ) ÷ f. A moving observer meets those crests at a relative speed of v + vₒ. Dividing that speed by the wavelength gives the frequency heard.
The two motions are not interchangeable. A source approaching at 30 m/s raises the pitch more than an observer approaching at 30 m/s. That is because the air carries the sound, so speeds are measured against the air.
Worked example
A car horn sounds at 440 Hz while the car drives toward you at 30 m/s (about 67 mph). You stand still and the air is at 20 °C, so v = 343 m/s.
- f′ = 440 × (343 + 0) ÷ (343 − 30) = 440 × 343 ÷ 313 = 482.17 Hz.
- Shift: 482.17 − 440 = 42.17 Hz, or about 1.58 semitones higher.
Result: You hear about 482.17 Hz while the car approaches. After it passes (vₛ = −30 m/s) the pitch drops to about 404.61 Hz.
Understanding the result
A positive shift means the source and observer are closing on each other; a negative shift means they are separating. The sudden drop as a vehicle passes is the switch from one to the other.
One semitone is a frequency ratio of about 1.059, so a shift of a couple of semitones is easily heard. Everyday traffic at highway speeds produces shifts of roughly one to three semitones.
Assumptions and limitations
- Motion is along the straight line joining the source and observer. For motion at an angle, use only the velocity component along that line.
- The medium (air) is still. With wind, measure both velocities relative to the moving air.
- Speeds are well below the speed of light.
- Light and radio waves in a vacuum need the relativistic Doppler formula, which depends only on the relative speed of source and observer.
- At or above the wave speed the source creates a shock wave and the calculator returns an error instead of a frequency.
- Echo problems, such as radar or ultrasound, involve two shifts (out and back) and need the formula applied twice.
Frequently asked questions
Why does the pitch change suddenly as a vehicle passes?
While it approaches, the velocity component toward you is positive and the pitch is raised. As it passes, that component swings through zero to negative, so the pitch drops below the true tone. The closer it passes, the quicker the drop.
Which sign convention should I use?
Either gives the same answer if applied consistently. Choose 'positive = toward the other' if you think about speeds as approaching and moving away. Choose the axis option if your textbook draws a single x-axis from the source to the observer.
Is it the same if the listener moves instead of the source?
Close, but not equal. With v = 343 m/s and a speed of 30 m/s, a moving source gives 482.17 Hz from 440 Hz, while a moving listener gives 478.48 Hz. The difference grows as speeds approach the speed of sound.
What happens at the speed of sound?
The waves in front of the source pile up on top of each other into a single pressure front, the sonic boom. The simple formula would divide by zero, so the calculator reports a shock wave instead.
How this calculator is checked
Method: Classical Doppler effect for waves in a medium: f′ = f(v + vₒ)/(v − vₛ), approach velocities positive. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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