Doppler effect calculator
Find the frequency an observer hears or sees when the source or observer moves, for sound and for light.
A clear answer. Here’s how.
For a wave in a still medium the observed frequency is f′ = f × (c + vo) ÷ (c − vs), where c is the wave speed, vo is the observer’s speed toward the source and vs is the source’s speed toward the observer. A source at or above the wave speed makes a shock wave, so the formula is rejected there. For light there is no medium and the relativistic result along the line of sight is f′ ÷ f = √((1 + β) ÷ (1 − β)) with β = v ÷ c, positive when source and observer approach; redshift z = f ÷ f′ − 1. Motion across the line of sight, wind and acceleration are not modelled.
How this is checked
4 reference checks passed These 4 cases were run when this page was built, on 20 October 2026, and all reproduce within the stated tolerance. The same cases are part of the automated tests that run on every change to Toolmoz. Engine version 1.
See the checks and where each expected answer comes from
Case 1
- Inputs
- Wave soundSource frequency (Hz) 440Source speed toward the observer (m/s, negative when moving away) 30Observer speed toward the source (m/s, negative when moving away) 0Wave speed in the medium (m/s) 343
- Expected, and what Toolmoz gave
- Observed frequency (Hz) expected 482.172524, Toolmoz 482.17252396166134Frequency ratio, observed ÷ source expected 1.0958, Toolmoz 1.095846645367412
- Where the expected answer comes from
- 440 × 343 ÷ (343 − 30) = 482.1725 Hz for a source approaching at 30 m/s (hand calculation)
- Allowed difference
- ±0.0001 absolute
Case 2
- Inputs
- Wave soundSource frequency (Hz) 440Source speed toward the observer (m/s, negative when moving away) -30Observer speed toward the source (m/s, negative when moving away) 0Wave speed in the medium (m/s) 343
- Expected, and what Toolmoz gave
- Observed frequency (Hz) expected 404.61126, Toolmoz 404.6112600536193
- Where the expected answer comes from
- 440 × 343 ÷ (343 + 30) = 404.6113 Hz for a source receding at 30 m/s (hand calculation)
- Allowed difference
- ±0.0001 absolute
Case 3
- Inputs
- Wave soundSource frequency (Hz) 440Source speed toward the observer (m/s, negative when moving away) 0Observer speed toward the source (m/s, negative when moving away) 20Wave speed in the medium (m/s) 343
- Expected, and what Toolmoz gave
- Observed frequency (Hz) expected 465.655977, Toolmoz 465.65597667638485
- Where the expected answer comes from
- 440 × (343 + 20) ÷ 343 = 465.6560 Hz for an observer approaching at 20 m/s (hand calculation)
- Allowed difference
- ±0.0001 absolute
Case 4
- Inputs
- Wave lightSource frequency (Hz) 100Relative speed as a fraction of the speed of light (positive when approaching) 0.5
- Expected, and what Toolmoz gave
- Frequency ratio, observed ÷ source expected 1.7320508, Toolmoz 1.7320508075688772Observed frequency (Hz) expected 173.20508, Toolmoz 173.20508075688772Shift direction expected Blueshift (approaching), Toolmoz Blueshift (approaching)
- Where the expected answer comes from
- √((1 + 0.5) ÷ (1 − 0.5)) = √3 = 1.7320508 for β = 0.5 approaching
- Allowed difference
- ±0.000001 absolute
What this shows: the engine reproduces independently worked values for these cases. It does not prove every possible input is right. To test an answer you were given, see how to check a number; to see what is and is not checked across Toolmoz, see the accuracy ledger.
See a worked example
Worked example
A fixed demonstration, separate from the values you enter above.
- Wave
- Sound or another wave in a medium
- Source frequency (Hz)
- 440
- Source speed toward the observer (m/s, negative when moving away)
- 30
- Observer speed toward the source (m/s, negative when moving away)
- 0
- Wave speed in the medium (m/s)
- 343
Example result
- Observed frequency (Hz)
- 482.1725
- Frequency ratio, observed ÷ source
- 1.095847
- Change in frequency (Hz)
- 42.1725
- Wavelength emitted at rest (m)
- 0.7795
- Wavelength in the medium as the source moves (m)
- 0.7114
- Source speed as a Mach number
- 0.087
Classical Doppler effect for a wave in a still medium: f′ = f × (c + vo) ÷ (c − vs), where c is the wave speed, vo is the observer’s speed toward the source and vs is the source’s speed toward the observer (negative when moving away). Speed of sound in air is about 343 m/s at 20 °C and 331 m/s at 0 °C. Wind, acceleration and motion off the line between source and observer are not modelled.