Doppler Effect Calculator
Find the frequency a listener hears when a sound source or the listener is moving, work backwards to the source's frequency or speed, or switch to light to turn a speed into a redshift z — or a redshift into a speed. Sign conventions are stated, the wavefronts are drawn, and the working is written out. Free, no sign-up.
343 m/s in air at 20 °C; 331 m/s at 0 °C.
Convention: the observer's speed counts as positive toward the source, and the source's speed as positive toward the observer. Air is taken to be at rest.
Results
767.1 Hz
Frequency heard
+9.585%
Change in frequency
643.7 Hz
If both directions were reversed
0.08746
Source speed as a Mach number
The working
- f′ = f × (v + 0) ÷ (v − 30) = 700 × (343 + 0) ÷ (343 − 30) = 767.1 Hz
How to Use the Doppler Effect Calculator
- 1. Choose sound or light. Sound needs a medium and uses the classical formula; light travels through empty space and needs the relativistic one.
- 2. For sound, pick what to find. Choose the frequency heard, the frequency emitted, or the source's speed, then enter the speed of sound (343 m/s in air at 20 °C), the frequencies, and each party's speed and direction.
- 3. For light, enter what you know. Give a speed, a redshift z, or the observed and emitted wavelengths. Add a rest wavelength to see where a spectral line lands.
- 4. Read the results and the working. You get the shifted value, the percentage change, the Mach number or β, a drawing of the wavefronts and each formula with your numbers in it.
The Doppler Formula for Sound
When a sound source and a listener move relative to the air, the frequency heard is f′ = f × (v + vₒ) ÷ (v − vₛ). Here f is the frequency the source emits, v is the speed of sound in the medium, vₒ is the observer's speed toward the source and vₛ is the source's speed toward the observer. Either speed is negative if the motion is away. An approaching source squeezes the wavefronts together ahead of it (a higher pitch), and a receding source stretches them out (a lower pitch).
| Situation | Formula | Result |
|---|---|---|
| Source approaching, observer still | f′ = f × v ÷ (v − vₛ) | Higher pitch |
| Source receding, observer still | f′ = f × v ÷ (v + vₛ) | Lower pitch |
| Observer approaching, source still | f′ = f × (v + vₒ) ÷ v | Higher pitch |
| Observer receding, source still | f′ = f × (v − vₒ) ÷ v | Lower pitch |
The two cases are not symmetric. A moving source changes the wavelength reaching a stationary observer; a moving observer meets the same wavelength at a different rate. They give slightly different shifts for the same speed, and the formula keeps them separate.
Worked Example: An Ambulance Siren
A siren emits 700 Hz and drives at 30 m/s (108 km/h) past a standing listener, with the speed of sound at 343 m/s. These are the starting values:
- Approaching: f′ = 700 × 343 ÷ (343 − 30) = 767.1 Hz, 9.6% higher.
- Receding: f′ = 700 × 343 ÷ (343 + 30) = 643.7 Hz, 8.0% lower. The calculator shows this as "if both directions were reversed".
- Walking toward a 440 Hz speaker at 1.5 m/s: f′ = 440 × (343 + 1.5) ÷ 343 = 441.9 Hz, a shift of only 0.44%, too small to hear.
The same formula run backwards finds the source's speed. If a 700 Hz siren is heard at 767 Hz by someone standing still, the source speed is v − f × v ÷ f′ = 343 − 700 × 343 ÷ 767, which is about 30 m/s toward the listener. When the source reaches the speed of sound the denominator becomes zero; beyond that, the waves pile into a shock front, the sonic boom, and the formula stops applying.
Doppler Shift and Redshift for Light
Light needs no medium, so only the relative velocity matters, and the shift follows from special relativity. For a source receding along the line of sight at speed v, with β = v ÷ c, the observed wavelength is longer by the factor 1 + z = √((1 + β) ÷ (1 − β)); an approaching source gives a negative z, a blueshift. Inverting, β = ((1 + z)² − 1) ÷ ((1 + z)² + 1). At low speed z ≈ v/c, but the classical estimate fails near the speed of light.
| Redshift z | Speed as a fraction of c | Speed (km/s) |
|---|---|---|
| 0.01 | 0.00995 | 2,983 |
| 0.1 | 0.0950 | 28,487 |
| 0.5 | 0.385 | 115,305 |
| 1 | 0.600 | 179,875 |
| 2 | 0.800 | 239,834 |
| 5 | 0.946 | 283,587 |
The starting example is a source receding at 30,000 km/s (β = 0.1001): relativity gives z = 0.1056, so a spectral line emitted at 656.3 nm is seen at 725.6 nm, while the classical estimate z ≈ v/c = 0.1001 is 5.3% too low.
Redshifts of distant galaxies
For nearby sources the Doppler formula describes a real motion. For galaxies at cosmological distances the observed redshift is mostly the stretching of space itself, so astronomers quote z directly and treat the speed from this formula as a convenient equivalent, not a literal velocity. The calculator also does not model the transverse Doppler effect, which applies when the motion is sideways to the line of sight.
Frequently Asked Questions
- What is the Doppler effect formula?
- For sound, f′ = f × (v + vₒ) ÷ (v − vₛ), with v the speed of sound, vₒ the observer's speed toward the source and vₛ the source's speed toward the observer. Use negative values for motion away. For light, 1 + z = √((1 + β) ÷ (1 − β)) for a receding source.
- Why does an ambulance siren drop in pitch as it passes?
- While it approaches, each wavefront is emitted closer to you than the last, so they arrive more often and you hear a higher frequency. After it passes, each wavefront starts further away, so they arrive less often and the pitch is lower. The siren itself never changes.
- What is redshift and what does z mean?
- Redshift z is the fractional increase in wavelength: z = (λ_observed − λ_emitted) ÷ λ_emitted. A spectral line at 500 nm seen at 550 nm has z = 0.1. A negative z means the light is shifted toward the blue.
- Can a source move faster than the speed of sound in the Doppler formula?
- No. At the speed of sound the formula gives an infinite frequency, and faster than that the waves overlap into a cone-shaped shock wave. The Mach number, shown in the results, is the source speed divided by the speed of sound.
- Does the speed of the observer or the source matter more?
- In sound, they differ slightly. For the same speed, a moving source shifts the frequency a little more than a moving observer, because the speed enters the denominator. For light the two cases are identical, as only the relative velocity matters.
- How are Doppler shifts used in practice?
- Police and weather radar measure the shift of reflected microwaves to find speeds and wind; medical ultrasound uses it to measure blood flow; and astronomers use spectral line shifts to measure the speeds of stars, planets that tug on their star, and galaxies.