The Doppler effect

You have heard this without ever thinking about it. An ambulance races toward you with its siren screaming, and the moment it passes, the note suddenly drops. The siren never changed. What changed was that it was moving.

Sound travels as waves, a train of crests spreading out through the air like ripples on a pond. When the ambulance drives toward you, it is chasing after the waves it just made, so it catches up a little on each one. The crests arrive squashed together, closer in time, and your ear hears crowded crests as a higher pitch. Once it passes and starts driving away, it stretches the waves out behind it instead. Now the crests reach you spread apart, and you hear a lower note. That sudden drop as it goes by is the switch from squashed to stretched.

This is the Doppler effect, and it works on light too, not just sound. Light is waves as well, and the colour of light is really its pitch. Something rushing toward you has its light squashed a touch toward blue. Something rushing away has it stretched toward red.

That last part turned out to be one of the biggest clues in all of science. When astronomers looked at distant galaxies, the light from nearly all of them was stretched toward red. They are moving away from us. The whole universe is spreading out, and we know it because of a siren-drop written into starlight.

A wave has a speed, a wavelength (the distance between crests) and a frequency (how many crests pass a point each second). The three are tied together by \(v = f\lambda\). For sound in ordinary air the speed \(v\) is fixed by the air itself, about 343 metres per second, and does not care whether the thing making the sound is moving. That fixed speed is the whole reason the effect happens.

Picture a source giving off a crest, then travelling forward a little before it gives off the next one. Ahead of it, each new crest starts from slightly further along, so the gap between crests, the wavelength, is shorter. Behind it, the source has backed away between crests, so the gap is longer. Same waves, but bunched in front and spread behind. For a source moving at speed \(v_s\) through still air, the frequency you hear is

\[ f_{\text{obs}} = f_{\text{src}}\,\frac{v}{v \mp v_s}, \]

with the minus sign when it is coming toward you (higher pitch) and the plus sign when it is going away (lower pitch). A car horn at 500 hertz passing at highway speed shifts up and then down by roughly a tone, which is exactly the drop you hear.

Push the source faster and the crests ahead crowd closer and closer. When it reaches the speed of sound itself, it keeps pace with its own waves and they all pile onto a single wall of pressure at the front. Go faster still and that wall trails back into a cone. Fly through it and people on the ground hear the sharp crack of a sonic boom as the cone sweeps past.

Light does the same thing, with one twist: light has no air to travel through, and everyone measures it at the same speed \(c\). Motion toward a light source shifts its colour toward blue, motion away shifts it toward red. We use this everywhere. A police radar gun bounces a wave off your car and reads the shift to get your speed. Weather radar reads the shift off raindrops to see which way a storm is turning. An ultrasound machine reads it off your blood to watch it flow. And astronomers read the redshift of a star to tell how fast it is coming or going, which is how they weigh galaxies and find planets around other suns.

Source versus observer, and why the medium matters. For waves in a medium, like sound in air, it matters which thing is moving, because the wave speed is set by the medium. A moving source changes the emitted wavelength: \(f_{\text{obs}} = f_{\text{src}}\,v/(v - v_s)\) for approach. A moving observer instead sweeps through crests faster without changing their spacing: \(f_{\text{obs}} = f_{\text{src}}\,(v + v_o)/v\). The two formulas differ, and at low speeds both reduce to the same fractional shift \(\Delta f/f \approx v_{\text{rel}}/v\). Combine them for the general case where both move.

The Mach cone. When the source speed exceeds the wave speed, define the Mach number \(M = v_s/v\). The wavefronts can no longer outrun the source, and their common envelope is a cone whose half-angle \(\alpha\) satisfies \(\sin\alpha = 1/M\). The shock is that cone sweeping past; the boom is the pressure jump across it, not a one-off event at the moment of "breaking" the sound barrier but a continuous wake trailing any supersonic body.

Light is different, because there is no medium. No air carries light and no experiment finds a preferred frame, so the sound formulas cannot simply carry over. The relativistic Doppler shift for a source receding at speed \(v\), with \(\beta = v/c\), is \[ f_{\text{obs}} = f_{\text{src}}\sqrt{\frac{1-\beta}{1+\beta}}, \] and the redshift is \(1 + z = \sqrt{(1+\beta)/(1-\beta)}\). The extra square-root factor is time dilation: the moving source's own clock runs slow, so even a source crossing your line of sight, with no approach or recession at that instant, is redshifted. That transverse Doppler shift has no classical analogue and is a direct test of special relativity.

Cosmological redshift is not really a velocity. The redshift of a distant galaxy is usually written as a Doppler shift, and for nearby galaxies that is a fine approximation, giving Hubble's law \(v \approx H_0 d\). But the deeper picture is that space itself has expanded while the light was in flight, stretching its wavelength by the same factor the universe grew: \(1 + z = a(t_{\text{obs}})/a(t_{\text{emit}})\), where \(a\) is the scale factor. It is the expansion of space, not motion through it, and separating that cosmological stretch from a galaxy's own peculiar motion is a routine and important job in cosmology. See the Big Bang explainer.

Reading the shift is how we measure the sky. A spectral line sits at a known rest wavelength, so its observed shift gives a velocity to metres per second. Watch a star wobble, its light shifting blue then red on a cycle, and you have found an unseen planet tugging it: the radial-velocity method behind much of exoplanet science. The width of a line, blurred by the Doppler shifts of atoms moving every way at once, reads out a gas's temperature. And the same trick on the 21-centimetre line of hydrogen maps the rotation of whole galaxies, which is how the case for dark matter was first built.