Doppler Effect Calculator

What the Doppler effect calculator does

When a wave source and an observer move relative to each other, the observed frequency shifts. This calculator works that shift out from the emitted frequency, the wave speed, and the speeds of the source and the receiver, and it can solve for any one of those quantities when you know the rest.

Below is what the Doppler effect is, the equation behind it, how the directions of motion enter, and a worked example.

How to use it

1. Choose what to calculate: the observed frequency, the emitted frequency, the wave speed, or either velocity.
2. Enter the known values, giving the source and receiver velocities with the right sign for their direction of motion.
3. Press Calculate for the result, or Reset to clear it.

What the Doppler effect is

The Doppler effect is the change in the frequency of a wave that an observer measures when the source of the wave, the observer, or both are moving. If they are approaching each other, the observed frequency rises above what the source emits; if they are moving apart, it falls below. The wave itself is unchanged at its source, but the motion squeezes or stretches the cycles as they reach the observer.

The everyday example is the siren of a passing ambulance. As it races toward you, its pitch sounds higher than normal, then as it passes and speeds away, the pitch drops noticeably. The siren is sounding exactly the same note the whole time, but the ambulance's motion compresses the sound waves ahead of it and stretches them behind, so you hear a higher pitch coming and a lower one going. This calculator quantifies that shift for any wave and any motion.

The equation it uses

The observed frequency depends on the emitted frequency and the speeds involved through the Doppler relationship:

f = f₀ × (v + v_r) ÷ (v + v_s)

Here f is the observed frequency, f₀ is the emitted frequency, v is the speed of the wave in the medium, v_r is the velocity of the receiver, and v_s is the velocity of the source. The motion of the receiver enters in the top of the fraction and the motion of the source in the bottom, which reflects that they affect the observed frequency in different ways. The calculator rearranges this same equation to solve for whichever quantity you leave unknown.

Approaching and receding

The directions of motion are what decide whether the frequency rises or falls, and they enter the formula through the signs of the two velocities. The velocities are measured along the line joining the source and the observer, with a consistent direction taken as positive, so motion one way counts as positive and the other way as negative. The reliable guide is the physics itself: any motion that brings the source and observer together raises the observed frequency, and any motion that carries them apart lowers it.

This gives you a built-in check. If you enter velocities for an approaching source and the result comes out lower than the emitted frequency, the signs need flipping, because approach must raise the pitch. The size of the shift grows with the speeds: the faster the relative motion, the larger the change in frequency. At everyday speeds the shift is small, a slight change in a siren's pitch, but for fast-moving sources it becomes dramatic, and this calculator gives the exact figure once the directions are set correctly.

Where the Doppler effect shows up

The Doppler effect is far more than a curiosity of passing sirens; it is a workhorse of science and technology. Police speed guns and weather radar bounce waves off moving objects and read the frequency shift of the echo to measure how fast cars are travelling or how rain is moving in a storm. Medical ultrasound uses the same principle to measure the flow of blood through vessels and the beating of a heart.

The effect reaches to the edge of the universe. Because light is a wave too, the Doppler effect shifts the light of moving stars and galaxies: objects rushing away from us have their light stretched toward the red end of the spectrum, the famous redshift, while those approaching are blueshifted. Measuring these shifts is how astronomers discovered that the universe is expanding and how they detect planets orbiting distant stars. The same simple relationship behind a changing siren underlies all of it, and this calculator captures its core.

Units, scope, and precision

The calculator takes frequencies in hertz and up, and velocities in metres per second and other units, with a default wave speed for sound in air and the option of the speed of light for electromagnetic waves. It uses the classical Doppler relationship, which is exact for sound and an excellent approximation for light at ordinary speeds; very near the speed of light, the fuller relativistic Doppler formula is needed, since time dilation then contributes too. For everyday motion the classical result is accurate. Results carry several significant figures.

A worked example

Suppose an ambulance sounds a 700 hertz siren and approaches you at 30 metres per second, while you stand still, with sound travelling at 343 metres per second.

As it approaches, the compression of the sound waves raises the observed frequency to about 767 hertz. The instant it passes and begins to recede, the waves stretch out and the frequency drops to about 644 hertz. That sudden fall of more than 120 hertz as the ambulance goes by is exactly the drop in pitch you hear, and in the calculator you would capture it by entering the source velocity with opposite signs for the approach and the recession.

Questions people ask

What is the Doppler effect?

The change in a wave's observed frequency when the source, the observer, or both are moving. Approaching motion raises the frequency; receding motion lowers it, as with a passing siren.

How do you calculate the Doppler shift?

Use f = f₀(v + v_r)/(v + v_s), with the emitted frequency, the wave speed, and the receiver and source velocities. The velocities carry signs for their direction of motion.

Why does the pitch rise then fall as something passes?

Because while it approaches, the waves are compressed and the frequency is higher, and once it recedes, the waves stretch and the frequency drops. The shift reverses the moment it passes you.

Does the Doppler effect work for light?

Yes. Moving light sources are redshifted when receding and blueshifted when approaching. This is how astronomers measure the motion of stars and galaxies, though near light speed a relativistic version of the formula is required.

References

A quick note on where the physics comes from. The Doppler effect for sound and its formula are standard physics, set out in OpenStax's University Physics and in Georgia State University's HyperPhysics. The SI units follow the US National Institute of Standards and Technology. The HyperPhysics link is worth a quick click to confirm it lands where you expect.

1. OpenStax, University Physics Volume 1, Section 17.8, The Doppler Effect. https://openstax.org/books/university-physics-volume-1/pages/17-8-the-doppler-effect
2. HyperPhysics, Doppler Effect. http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/doppler.html
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI). https://www.nist.gov/pml/special-publication-811