The Doppler effect is one of the most observable phenomena in physics: the pitch of a passing siren, the red or blue tinge of starlight, the radar gun that catches speeding drivers, and the ultrasound that measures blood flow all rely on the same underlying principle — a wave’s perceived frequency shifts whenever the source and observer are in relative motion. This calculator applies the exact classical and relativistic Doppler formulas so you can compute any one unknown from the remaining knowns, see the full working, and understand where blueshift ends and redshift begins.
How it works
Classical Doppler (sound and mechanical waves)
For any wave travelling through a medium at speed v, the general formula relating the frequencies and velocities is:
f_observed = f_source × (v_wave + v_observer) / (v_wave + v_source)
where:
- v_wave is the wave’s speed through the medium (343 m/s for sound in air at 20 °C, 1480 m/s in water)
- v_observer is the observer’s speed relative to the medium — positive when moving toward the source, negative when moving away
- v_source is the source’s speed relative to the medium — positive when moving away from the observer, negative when moving toward
The calculator rearranges this expression algebraically to solve for whichever variable you leave unknown: observed frequency, source frequency, observer velocity, or source velocity.
Relativistic Doppler (light and EM waves)
Light has no medium; the relevant frame is just the relative velocity between source and observer. Special relativity gives:
f_observed = f_source × √((1 − β) / (1 + β))
where β = v_relative / c (positive when the source is receding). The factor √((1 − β) / (1 + β)) is called the relativistic Doppler factor. At low speeds (β ≪ 1) it reduces to the classical approximation (1 − β), confirming that both formulas agree for everyday velocities.
Redshift and blueshift
The dimensionless redshift parameter z is defined as:
z = (λ_observed − λ_source) / λ_source = (f_source − f_observed) / f_observed
A positive z (redshift) means the source is receding; a negative z (blueshift) means it is approaching. Cosmological redshifts measured by astronomers can exceed z = 10 for the earliest observable galaxies, corresponding to recession speeds many times the speed of light (apparent, due to space expansion, not object motion).
Worked example
An ambulance siren emits a tone at 800 Hz. The ambulance is travelling at 108 km/h (30 m/s) directly toward a stationary listener. Speed of sound is 343 m/s. What frequency does the listener hear?
- v_source = −30 m/s (moving toward the observer, so negative in our sign convention)
- v_observer = 0 m/s
f_obs = 800 × (343 + 0) / (343 + (−30))
f_obs = 800 × 343 / 313
f_obs ≈ 876.7 Hz
That is 76.7 Hz higher than the emitted tone — a noticeable rise. As the ambulance passes and recedes at the same speed:
f_obs = 800 × 343 / (343 + 30) ≈ 735.4 Hz
The difference between approach and recession tones is about 141 Hz — clearly audible as the classic “neeeow” drop.
| Scenario | f_source | v_source | f_observed | Shift |
|---|---|---|---|---|
| Ambulance approaching (30 m/s) | 800 Hz | −30 m/s | 876.7 Hz | +76.7 Hz |
| Ambulance receding (30 m/s) | 800 Hz | +30 m/s | 735.4 Hz | −64.6 Hz |
| Observer running toward source (5 m/s) | 440 Hz | 0 | 446.4 Hz | +6.4 Hz |
| Galaxy receding at 0.1c | 656 nm (Hα) | 0.1c | 718.3 nm | z = 0.095 |
Formula note
The classical formula assumes the wave travels in a stationary medium and that neither source nor observer exceeds the wave speed (Mach ≤ 1). It breaks down at supersonic speeds (the shock-wave / Mach-cone regime) and cannot model light, for which special relativity is mandatory. The relativistic formula is exact for all relative velocities below c and reduces to the classical formula when v ≪ c.
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