Doppler Effect Calculator

Introduction

A siren drops in pitch the instant the ambulance passes you, even though the siren itself never changes. That drop is the Doppler effect: the wavefronts pile up ahead of a moving source and stretch out behind it, so the frequency your ear receives depends on how the source and you are moving relative to each other, not just on the frequency that was emitted. The same geometry applies to a police radar gun, a weather radar tracking a storm cell, a bat echolocating a moth, and the redshift that tells astronomers a galaxy is receding.

This calculator handles both halves of the effect. In sound mode it takes a source frequency and the two velocities and returns the observed frequency, the shift, the observed wavelength and whether the pitch rises or falls. In light mode it converts a rest wavelength and an observed wavelength into a redshift and a radial velocity, which is the measurement astronomers actually take. The wave-side groundwork — speed, frequency and wavelength — is in the Wave Speed Calculator and the Wavelength Calculator; if you are working on the loudness side of a siren instead of its pitch, the Decibel Calculator is the right tool.

How to Use

Pick the wave type first. Sound mode takes velocities in metres per second; light mode takes wavelengths in nanometres. Both accept positive numbers only, and the speeds are signed by direction: an observer speed is positive when the observer moves toward the source, and a source speed is positive when the source moves toward the observer.

Example 1 — a stationary siren. A source emits 440 Hz in air at 343 m/s, and nobody is moving. Enter 440, 343, 0 and 0. The observed frequency comes back at 440.00 Hz, the shift at 0.00 Hz, and the pitch reads Unchanged. This is the sanity check worth doing before any other case: if the rest case drifts, something is wrong with the inputs.

Example 2 — the ambulance coming at you. Keep the source at 440 Hz and 343 m/s, and set the source speed to 34.3 m/s toward the observer, which is about 76 mph. The calculator returns 488.89 Hz, a shift of +48.89 Hz, and Approaching - higher pitch. The observed wavelength shortens to 0.702 m. That is the pitch you hear before the pass, and it is a rise of more than a semitone.

Example 3 — you moving toward the source. Set the source speed back to 0 and the observer speed to 34.3 m/s. The result is 484.03 Hz. The two cases are not symmetric: moving the source compresses the medium ahead of it, while moving the observer only changes how often the observer meets the same wavefronts, so the source-moving case gives the larger shift at the same speed.

Example 4 — a redshift. Switch to Light (redshift), enter a rest wavelength of 656.3 nm (the hydrogen-alpha line) and an observed wavelength of 660 nm. The redshift is 0.005638 and the radial velocity is about 1,684 km/s, with the direction reported as Redshift - receding. For a blueshift, enter a shorter observed wavelength and the velocity is reported the same way while the direction flips.

Working Example 2 by hand. Multiply the emitted frequency by the wave speed and divide by the speed-adjusted wave speed. With 440 Hz and 343 m/s: 440 times 343 gives 150,920, and the denominator is 343 minus 34.3, which is 308.7. Dividing 150,920 by 308.7 gives 488.89 Hz. The shift is the difference from 440, or 48.89 Hz, and the observed wavelength is the corrected speed divided by the emitted frequency: 308.7 divided by 440 gives 0.702 m. Every number the calculator reports can be reproduced this way in about thirty seconds with a pencil, which is the fastest way to check whether a suspicious result came from the physics or from the input.

The Formula

For a source and an observer moving along the line between them, the observed frequency is the emitted frequency scaled by how the two speeds compare with the wave speed:

Here v is the speed of the wave in the medium, v_o is the observer's speed and v_s the source's speed. The sign convention is the one that does the work: v_o enters with a plus when the observer approaches the source, and v_s enters as a subtraction when the source approaches the observer. Both terms have a physical reading. In the numerator, an approaching observer sweeps through more wavefronts per second. In the denominator, an approaching source has already crowded the wavefronts closer together before they leave, so the effective wavelength in front of it is shorter:

That is why the same speed produces a bigger shift when the source moves. Divide the two shifts at 34.3 m/s into the numbers from the examples: the source case moves 48.89 Hz, the observer case moves 44.03 Hz. As the source speed approaches the wave speed the denominator approaches zero and the observed frequency runs away — the calculator refuses that input rather than returning infinity.

For light there is no medium to measure against, so the same absolute speeds are useless and the observable is the wavelength ratio. The redshift is defined as a fractional change:

and the recession speed follows from the relativistic Doppler relation, which is what the calculator inverts:

At small redshifts this is indistinguishable from the classical v = c z, but the difference grows quickly: at z = 0.0056 the two agree to four significant figures, while at z = 1 the classical formula would report c and the relativistic one reports about 0.6c.

Reference Table: Speed of Sound by Medium

The formula is only as good as the wave speed you feed it, and that speed depends on the medium and its temperature. Dry air at 20 °C carries sound at 343 m/s; every degree of warming adds roughly 0.6 m/s, which is why a cold morning makes a marching band sound flat across a field.

The spread matters for practical numbers. A siren at 34.3 m/s produces a 48.8 Hz shift in air at 20 °C, but the same siren heard through water would barely shift at all, because the wave speed is four times larger and dominates both terms. Whenever a Doppler result looks implausible, check the wave speed before checking the arithmetic.

Practical Tips

• Fix the sign convention before typing. The most common error is entering an approaching source as a negative number. Approaching means positive for both speeds in this calculator; a negative value means receding, and it will lower the observed frequency.
• Use the medium's speed, not a memorised constant. 343 m/s is air at 20 °C. At 0 °C it is 331 m/s, and in helium it is 972 m/s; a 4% error in v propagates almost one-to-one into the shift.
• Remember the asymmetry. At equal speeds the moving source shifts more than the moving observer, so do not swap the two entries and expect the same answer.
• For light, work in wavelengths. Rest and observed wavelengths are what spectrographs report; converting to frequency first adds a step and invites rounding errors. Keep four or more significant figures in the wavelengths.
• Do not push the source past the wave speed. A supersonic source produces a shock cone rather than a shifted tone; the calculator flags that case instead of returning a meaningless number.
• Cross-check with the rest case. If entering zero for both speeds does not return the emitted frequency, either a sign is wrong or the wave speed is missing.

Limitations

The sound formula assumes a uniform, still medium and motion along the line of sight. Wind, temperature gradients and crosswinds add their own frequency shifts and are not modelled. Transverse motion — an object passing at right angles — produces a second-order shift that this linear formula ignores; that is why the pitch changes fastest at the moment of the pass and then settles.

The formula also assumes steady motion. A siren that accelerates, or a whistle on a swinging pendulum, has a velocity that changes continuously, so the observed frequency sweeps rather than stepping from one value to another. During a passing manoeuvre, the component of velocity along the line of sight falls smoothly through zero at the crossing point, which is why the pitch glides rather than jumping.

The light mode assumes the observed shift comes entirely from radial motion, with no gravitational redshift and no cosmological expansion. For nearby stars and galaxies the difference is small; for distant galaxies, the measured redshift includes expansion and should be interpreted through cosmology rather than as a plain velocity. The returned velocity is the relativistic Doppler value, not a proper distance.

Finally, the calculator works with a single moving source and a single moving observer. Doppler radar, ultrasound blood-flow measurement and bat echolocation all involve reflections, where the shift applies twice — once on the way out and once on the way back — so the observed shift doubles.

Frequently Asked Questions

❓ What is the Doppler effect in one sentence?

✅ It is the change in observed frequency caused by relative motion between a wave source and an observer: wavefronts bunch up when the two approach and stretch out when they separate.

❓ Why does the pitch drop after an ambulance passes?

✅ Before the pass the source is moving toward you, which compresses the wavefronts and raises the frequency. The moment it passes, the same velocity now stretches them, so the frequency falls by roughly the same amount it had risen.

❓ How do I enter a receding source?

✅ Use a negative source speed. In this calculator a positive source speed means the source approaches the observer; a negative value means it moves away, and the observed frequency drops below the emitted one.

❓ What speed of sound should I use?

✅ 343 m/s for dry air at 20 °C, which is the default. Use 331 m/s at 0 °C, about 1482 m/s for water at room temperature, or the value for whatever medium the wave actually travels in.

❓ Why is the shift bigger when the source moves instead of the observer?

✅ A moving source changes the wavelength in the medium before the wave leaves it, so the compression acts on the wave itself. A moving observer only changes how often the observer encounters the same wavefronts.

❓ Can the observed frequency be infinite?

✅ Only if the source reaches the wave speed, where the denominator of the formula goes to zero. Real sources that reach that speed create a shock wave, not an infinite tone, so the calculator rejects speeds at or above the wave speed.

❓ What is redshift and how is it different from the sound case?

✅ Redshift is the light equivalent: the observed wavelength is longer than the rest wavelength, which happens when the source recedes. Because light needs no medium, the useful formula is the relativistic one based on the wavelength ratio.

❓ How do I calculate radial velocity from a redshift?

✅ Use v = c times ((1 + z) squared minus 1) divided by ((1 + z) squared plus 1). For small redshifts this is essentially c times z, which is why the shortcut works for nearby objects.

❓ Does this work for radar and ultrasound?

✅ The math applies, but those systems measure reflections, so the shift happens twice and the observed shift doubles. Divide the measured shift by two before using the formula to recover the target speed.

❓ Why is weather radar useful if it uses Doppler?

✅ Rain drops reflect the beam, and the frequency shift of the return gives their velocity toward or away from the radar. That is how a radar map distinguishes winds moving toward the station from winds moving away.

❓ What about the Doppler effect for light from a rotating star?

✅ Rotation makes one limb approach and the other recede, so the spectral lines broaden rather than shift as a whole. The width of that broadening gives the rotation speed, and each side individually follows this calculator.