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Doppler Effect Calculator

Physics

Or start from a worked scenario:

The frequency in the source's own rest frame.
Speed of the wave in the medium at rest.
Relative to the medium, not to the observer.
Also relative to the medium.
Positive blows from source toward observer; negative is a headwind.
0° is head-on; 90° is closest approach, where the classical shift is exactly zero.
Mach 0 to Mach 3 — watch the wavefronts pile up and the frequency hand off to cone geometry.
0 to 10.
Observed frequency f′
547.9233 Hz
Shift Δf
+47.9233 Hz
Mach number
0.0875
Up-shift (approaching)regime: subsonicv = 343.0000 m/s, entered directly
M = 0.0875

The same closing speed, two different answers

Every classical speed is measured relative to the undisturbed medium, not relative to the other party. That is why a moving source and a moving observer at the same closing speed give different answers.

500.0000 HzEmitted f547.9233 HzSource moves543.7318 HzObserver moves
ScenarioClosing speedObserved f′
Source moves toward a still observer30.0000 m/s547.9233 Hz
Observer moves toward a still source30.0000 m/s543.7318 Hz
Difference—4.1915 Hz (0.7709 %)
Closed form of the gap, f β²/(1 − β)β = 0.08754.1915 Hz

Wavelength in the medium: 0.6860 m at rest, 0.6260 m as observed (Δλ = -0.0600 m).

Worked solution

General classical relation

f′ = f (v + w + v_o) / (v + w − v_s)

Substitute

f′ = 500 × (343 + 0) / (343 − 30)

Result

f′ = 500 × 343 / 313 = 547.9233 Hz

Reference: the 343 m/s shorthand is dry air at about 20 °C, where the temperature formula gives 343.2146 m/s. Angles are converted with π/180 = 0.017453293 rad per degree, and inverse sines come back as 57.295780 degrees per radian.

Pitch intervals use 12 log₂(f′/f), so a doubling of frequency is 12 semitones — one octave — and pass-by drops are reported in semitones as well as hertz.

About This Tool

Doppler Effect Calculator – Sirens, Radar Guns and Receding Galaxies

Everyone has heard an ambulance drop in pitch as it goes past, which makes the Doppler effectone of the most widely encountered pieces of physics and one of the most widely misunderstood. The usual intuition — that only the relative speed between source and listener matters — is simply false for sound. This Doppler shift calculator works the classical relation for a wave in a medium, inverts a measured shift back into a speed, hands off to shock-cone geometry at Mach 1, and switches to the relativistic formula the moment the wave is light.

The formula, and why the numerator and denominator differ

The general classical relation is f′ = f (v + w + v_o) / (v + w − v_s), with every speed measured relative to the undisturbed medium and wthe medium’s own velocity from source to observer. A moving source appears in the denominator; a moving observer appears in the numerator. Those are genuinely different functions, and they agree only to first order.

Take a 500 Hz siren and air at 343 m/s. A source approaching at 30 m/s gives 500 × 343 / 313 = 547.9233 Hz. An observer approaching a stationary source at the very same 30 m/s gives 500 × 373 / 343 = 543.7318 Hz. Identical closing speed, a 4.1915 Hz difference, and it is audible. Writing β = 30/343, the moving-observer case is exactly f(1 + β) while the moving-source case is f/(1 − β) = f(1 + β + β² + …). The gap is f β²/(1 − β) = 4.1915 Hz — exactly what the table shows.

Sound has a rest frame; light does not
Air picks out a preferred frame, so “the source is moving” and “the observer is moving” are physically different situations rather than two descriptions of one. For light there is no medium and no preferred frame, and the two cases become indistinguishable in principle.

Wind, and the myth that it changes the pitch

Wind does not shift a frequency by itself. It changes the effective wave speedalong the source–observer line, so you substitute v → v + w everywhere. With nothing moving, the formula collapses to f′ = f (v + w)/(v + w) = f for any wind whatsoever. With the same 30 m/s approaching source, a 10 m/s tailwind gives 546.4396 Hz and a 10 m/s headwind gives 549.5050 Hz: wind only ever modulates a shift real motion has already produced.

Turning a shift back into a speed

Invert the relation and a measurement falls out. A 500 Hz source heard at 550 Hz implies v_s = v(1 − f/f′) = 31.1818 m/s, or 112.25 km/h. A radar gun is different: it illuminates the target and receives the reflection, so the shift is applied twice and the beat note is Δf = 2 v f cos θ / c. A car at 30 m/s under a 24.125 GHz K-band beam returns 4828.34 Hz— an audible tone, which is why early guns had a loudspeaker.

Two classic errors this tool blocks
Applying a one-way formula to a reflected signal halves the answer, and an off-axis beam always under-reads by cos θ. The cosine effect can never over-state a speed, which is why it always favours the driver.

What happens at Mach 1

When v_s = vthe denominator is exactly zero. That is not a bug to trap: the source is keeping pace with its own wavefronts, so every crest emitted over the whole approach arrives at once, as a single pressure discontinuity. Above Mach 1 there is no observed frequency ahead of the source at all — it arrives before its sound does — and the useful output becomes geometry: μ = arcsin(1/M). At Mach 1.1662 the cone half-angle is 59.0370°, and at Mach 2 it is exactly 30°. The cone narrowsas speed rises, which is the opposite of most people’s intuition.

Light, redshift and the geometric mean

For an electromagnetic wave the correct relation is f′ = f √((1 − β)/(1 + β)) when receding. At β = 0.1 the hydrogen-alpha line stretches from 656.281 nm to 725.5459 nm, a redshift of z = 0.10554. The two naive classical treatments would give factors of 0.909091 and 0.900000, differing by 1 % — enough to reveal who was “really” moving, which no experiment has ever managed. The true factor, 0.904534, is exactly the geometric mean of the two, the missing piece being the 1/γ of time dilation applied symmetrically. At closest approach that leaves a pure transverse shift of 1/γ = 0.994987 with no line-of-sight motion at all.

From redshift to recession velocity

The inverse is β = ((1 + z)² − 1)/((1 + z)² + 1). The quasar 3C 273 at z = 0.158 gives 43 665 km/s, while the shortcut v = zc over-reads by 8.48 %. At z = 3 the shortcut returns 3c, which is impossible; the correct answer is β = 15/17. Note that at cosmological distances the redshift comes from the expansion of space rather than motion through it, so the velocity is a textbook conversion rather than a peculiar velocity — a distinction worth keeping in mind whenever a redshift calculator quotes kilometres per second.

Frequently Asked Questions

Is the Doppler Effect Calculator free?

Yes, Doppler Effect Calculator is totally free :)

Can I use the Doppler Effect Calculator offline?

Yes, you can install the webapp as PWA.

Is it safe to use Doppler Effect Calculator?

Yes, any data related to Doppler Effect Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

How does this Doppler effect calculator work?

Choose whether the wave is mechanical (sound in a medium) or electromagnetic (light in vacuum), then enter the emitted frequency and the speeds involved. The classical branch evaluates f′ = f (v + w + v_o) / (v + w − v_s) with every speed measured relative to the undisturbed medium, and the relativistic branch evaluates f′ = f √((1 ± β)/(1 ∓ β)) with no medium at all. Everything is computed in SI units and converted to your display units only at the last step, so no answer is ever built from a rounded intermediate figure.

Why do a moving source and a moving observer give different answers?

Because sound travels in a medium, and the medium picks out a rest frame. A source at 30 m/s toward a still observer gives 500 × 343/313 = 547.9233 Hz, while an observer at 30 m/s toward a still source gives 500 × 373/343 = 543.7318 Hz — the same 30 m/s closing speed and a 4.1915 Hz difference. Algebraically the moving-observer case is exactly f(1 + β) while the moving-source case is f/(1 − β) = f(1 + β + β² + …), so they agree only in the first-order term. The gap is exactly f β²/(1 − β).

What happens when the source reaches the speed of sound?

The denominator (v − v_s) goes to zero, which is a physical event rather than a calculation error. The source is keeping pace with its own wavefronts, so every crest emitted over the whole approach arrives at the listener simultaneously as a single pressure discontinuity — the sonic boom. The calculator detects this before dividing and hands off to the Mach-cone model instead of printing infinity. Above Mach 1 there is no single observed frequency ahead of the source at all, so it reports the cone half-angle μ = arcsin(1/M) instead.

Why can I not use the classical formula for light?

The classical formula needs a medium to measure speeds against, and light has none. If you applied the two classical formulas to a source receding at 0.1 c you would get factors of 0.909091 and 0.900000, differing by about 1 % — which would let you determine who was really moving. No experiment has ever been able to do that. The correct relativistic factor, 0.904534, is exactly the geometric mean of the two, the difference being the extra 1/γ of time dilation applied symmetrically to both.

Why is a radar gun's shift double the ordinary Doppler shift?

A radar or ultrasound beam is shifted twice: once when the moving target receives it as a moving observer, and again when the target re-radiates it as a moving source. The beat note is therefore Δf = 2 v f cos θ / c, and a car at 30 m/s illuminated at 24.125 GHz returns a 4828.34 Hz beat — an audible tone, which is why early radar guns had a loudspeaker. Applying a one-way formula to a reflected signal halves the answer, and an off-axis beam always under-reads by cos θ.

Is the recession velocity from a redshift a real speed?

For nearby objects, yes. For distant galaxies the redshift is produced by the expansion of space rather than by motion through it, so the figure is the standard special-relativistic textbook conversion rather than a peculiar velocity. What the calculator does show clearly is that the shortcut v = zc fails quickly: at z = 0.158 it over-reads by 8.48 %, and at z = 3 it returns 3c, which is impossible. The correct inversion is β = ((1 + z)² − 1)/((1 + z)² + 1), giving 15/17 at z = 3.