The Doppler Effect: From Sirens to Redshift

Crests bunching up, crests spreading out

Any wave source — a speaker, a siren, a star — emits crests at a steady rate. If the source is stationary relative to you, those crests arrive at your ear or your telescope at exactly the rate they were emitted: a fixed frequency. The moment the source starts moving, that stops being true. A source moving toward you emits each new crest from a position slightly closer than the last one — the crests pile up in front of it, arrive more often, and you perceive a higher pitch or a shorter wavelength. A source moving away stretches the crests apart behind it, and you perceive a lower pitch or a longer wavelength. This is the Doppler effect, and the wave's actual propagation speed through its medium never changes — only the spacing between crests does.

A useful mental model is the ambulance. Its siren emits, say, 700 Hz constantly — the driver hears exactly 700 Hz the whole time, because there is no relative motion between the siren and the driver's ear. You, standing on the pavement as it races toward you at 30 m/s, hear something noticeably higher. The instant it passes and starts moving away, the pitch drops — almost instantly, because the source has crossed from "approaching" to "receding" in your frame at that exact moment.

The formula, worked through

For sound in still air, the frequency f′ you observe relates to the emitted frequency f by the speed of sound v, your own velocity v_o (positive toward the source), and the source's velocity v_s (positive toward you):

f' = f * (v + v_o) / (v - v_s)      v ~= 343 m/s in air at 20 C

Ambulance approaching at 30 m/s, observer still, f = 700 Hz:
  f' = 700 * (343 + 0) / (343 - 30) = 700 * 343/313 ~= 767 Hz

Ambulance receding at 30 m/s:
  f' = 700 * 343 / (343 + 30) ~= 644 Hz

The rule of thumb this encodes: every percent of the speed of sound the source closes on you shifts the perceived frequency up by roughly a percent. The effect is entirely ordinary physics, no relativity required, because sound propagates through a medium — air — that gives source and observer separately measurable speeds.

Where it quietly runs your life

Traffic radar guns send out a microwave pulse at a known frequency and measure the Doppler shift in the reflection off your car to compute your speed to within about 1 km/h. Doppler ultrasound sends a pulse into tissue and reads the shift in echoes from moving red blood cells to measure blood flow direction and velocity, which is how sonographers spot heart valve problems and deep-vein clots without surgery. Bats emit ultrasonic chirps and listen for shifted echoes off flying insects, resolving velocity differences of under a centimetre per second — enough to intercept prey in total darkness. Weather radar stations map the Doppler shift of reflections off raindrops across an entire storm to produce the wind-speed and rotation maps that flag a developing tornado before it touches down.

Light needs relativity

Sound's Doppler formula leans on a medium through which source and observer move at independently defined speeds. Light has no such medium — it travels at c in every reference frame, full stop — so the classical derivation cannot apply. The correct relativistic version depends only on the relative velocity β = v/c between source and observer:

f' = f * sqrt( (1 + beta) / (1 - beta) )      beta = v / c

approaching  -> shorter wavelength  -> blueshift
receding     -> longer wavelength   -> redshift

At everyday speeds this reduces almost exactly to the classical formula — the relativistic correction is swamped by measurement noise. For stars and galaxies moving at thousands of kilometres per second, it is not: the shift is large enough to read directly off a spectrum.

The redshift that revealed an expanding universe

In 1929, Edwin Hubble measured the spectral lines of distant galaxies and found that almost all of them were redshifted, and that the more distant a galaxy was, the greater its redshift. The conclusion was that the universe itself is expanding: distant galaxies are not so much racing away through space as being carried apart by the stretching of space itself, which is why cosmologists call it a cosmological redshift rather than a pure Doppler shift — the physical cause is different even though the observed signature looks similar. The Hubble constant, H₀ ≈ 70 km/s per megaparsec, quantifies the rate: a galaxy one megaparsec away recedes at roughly 70 km/s, one ten megaparsecs away at roughly 700 km/s, derived entirely from redshift measurements across thousands of galaxies. Even the oldest light in existence, the Cosmic Microwave Background emitted 380,000 years after the Big Bang as X-rays, has been redshifted by a factor of about 1,100 on its way to us — arriving today as microwave radiation at 2.7 K.

Frequently asked questions

Try it live

Everything above runs in your browser — open Doppler Effect and drag the source velocity while it is running to watch the wavefronts compress and stretch in real time. Nothing is installed, nothing is uploaded, the whole model lives in one tab.