when the source is moving
An ambulance siren sounds higher as it approaches and lower as it drives away — even though the siren plays the same tone the whole time. See the waves that cause this, and find out what happens when the source catches up with its own sound.
f′ = f · c − u·cos ψc − v·cos φ
f — source frequency, c = 343 m/s — speed of sound in air (20 °C), v — source speed, φ — angle between the source's direction of motion and the direction to the listener at the moment the wave was emitted, u — listener speed (ψ — its angle).
Stars and galaxies move too. Their light is stretched when they recede and squeezed when they approach — and the giveaway is the dark lines in the spectrum, the “barcode” of the elements.
| Line | λ₀ (lab) | λ′ (object) | Δλ | Range |
|---|
λ′ = λ · √1 + β1 − β β = v/c
z = λ′/λ − 1. For small speeds z ≈ v/c, and the formula turns into the classical one: Δλ/λ ≈ v/c. Speed of light c = 299,792 km/s.
The Doppler effect is not just a textbook curiosity — every day it measures car speeds, blood flow and wind in clouds, and helps with navigation. The same trick works everywhere: compare the wave sent out with the wave received.
A Doppler speed camera sends out microwaves at about 24 GHz. The wave reflected from an approaching car comes back higher by Δf ≈ 2·v·f/c. The shift is doubled because the car first receives the wave as a moving receiver and then sends it back as a moving source.
The probe sends out ultrasound that bounces off flowing blood cells. The color shows the direction of flow. Where a vessel runs perpendicular to the beam there is almost no color — because cos 90° = 0.
Weather radars (in Poland, the POLRAD network) measure not only where it is raining but also how fast the drops are moving toward or away from the radar. A pair of neighboring patches — one green, one red — reveals a rotating storm cloud (a mesocyclone), which can spawn a tornado.
Bats emit ultrasound, dolphins series of clicks. From the echo delay they work out the distance (d = c·Δt/2), and from the Doppler shift whether the prey is approaching and how fast. Try the “Bat” scenario in Sound mode.
A GPS satellite races along at about 3.9 km/s. When it rises, its signal arrives raised by as much as about 5 kHz; when it sets, lowered; and at the zenith the shift drops to zero. The receiver has to account for this just to “lock on” to the signal.
The characteristic sound of a car flying past is pure Doppler. At 300 km/h the pitch before the pass is about 1.64 times higher than after it — a difference of about 8.6 semitones, almost a major sixth.
The source emits wave crests at equal time intervals. When it moves, each new crest starts from a slightly different place. In front of the source the crests bunch up — the wave is shorter and the sound higher. Behind the source they spread out — the sound is lower. The source itself plays the same all the time; only what reaches the ear changes.
When the source travels at the speed of sound, the crests pile up into a single wall — the sound barrier. Above Mach 1 the source outruns its own waves, and their envelope forms a Mach cone with half-angle θ, where sin θ = c/v. When the cone sweeps across the ground, you hear a sonic boom.
Light is a wave too, so the motion of its source changes its wavelength. When a star recedes, the waves arrive stretched and the spectral lines shift toward the red; when it approaches — toward the violet. At ordinary stellar speeds the shift is a fraction of a nanometer, so you can't see the color change with the naked eye — but the sharp absorption lines of hydrogen, sodium or calcium reveal the motion to within meters per second.
For light only the relative speed matters, and at high speeds you need the formula from relativity: λ′ = λ·√((1+β)/(1−β)). Note: the shift of distant galaxies and quasars is actually caused by the expansion of the Universe (cosmological redshift) — here, for simplicity, we calculate it with the Doppler formula.
Every time a wave bounces off something that moves, its frequency changes. Just compare the wave sent out with the reflected one to learn the speed — without touching anything. On reflection the shift is doubled (Δf ≈ 2·v·f/c), because the reflecting object is first a moving receiver and then a moving source.
The formula is the same everywhere; only the waves change: microwaves in radars, ultrasound in medicine and in bats, radio waves in satellite navigation, laser light in wind measurements.
Three tasks to complete on this page — the page will check for itself whether you succeeded.
0/3In Sound mode, speed up the source with the slider or the → key to above Mach 1 and watch the waves merge into a Mach cone.
Set the speed so that Ania (in front) hears a sound exactly one octave higher than the source emits: f′ = 2·f (±1%). Hint: c/(c − v) = 2.
In Light mode, move the Hβ line (486.1 nm) exactly to where Hα (656.3 nm) sits in the lab, ±1 nm. This is exactly why astronomers always look at many lines at once!