DOPPLER AND SHOCK WAVES

In 1842, an Austrian physicist named Christian Doppler published a short paper arguing that the colour of starlight should depend on how fast the star was moving toward or away from Earth. The reasoning took about a page. He got the details of the stellar colour wrong — stars emit too broad a spectrum for the effect to be visible — but the underlying mechanism was one of the most reusable ideas in physics.

Three years later, a Dutch meteorologist named Christophe Buys Ballot decided to check. He rented an open railway flatcar, stood a group of trumpet players on it, and had them hold a single sustained note as the train thundered past a platform of trained musicians with perfect pitch. The listeners on the platform wrote down what they heard.

As the train approached, the note was sharper. As it receded, the note was flatter. The same trumpets, played the same way, sounded like two different notes depending on which side of the platform the train was on.

Doppler was right. The effect now bears his name. And the same geometry — wavefronts squeezed together in front of a moving source, stretched apart behind it — explains everything from a police siren to a supersonic jet to the expansion of the universe.

Think about a source that emits one spherical wavefront every T seconds. It sits at the origin and pulses: ping, ping, ping. The wavefronts spread outward as concentric circles, each one T · c metres further out than the last, where c is the wave speed in the medium.

Now set the source moving. Between one ping and the next, it has travelled a distance v · T. So the second wavefront is centred not at the origin, but a little further along. The third even further. The wavefronts bunch up ahead of the source — compressed — and spread out behind — stretched.

That is the whole of the Doppler effect in one sentence.

Slide the speed above c. The geometry breaks: wavefronts pile up along a cone. We'll come back to that in § 05.

If the source moves toward a stationary observer at speed $v_s$, each new wavefront starts closer than the last. The spacing between consecutive wavefronts — the observed wavelength — drops from $cT$ to $(c - v_s)T$.

Frequency is the reciprocal of period, and the observed period is the spacing divided by c:

The minus sign is the whole story. As $v_s$ creeps toward $c$, the denominator shrinks toward zero and the observed frequency diverges. When $v_s$ exceeds $c$, the formula becomes negative — physics telling you that the model no longer applies. Something else happens. We'll get there.

For the ambulance passing at 30 m/s (≈108 km/h) emitting a 1000 Hz siren: the observed pitch approaching is $1000 \cdot 343 / (343 - 30) \approx 1096$ Hz. Receding, it's $1000 \cdot 343 / (343 + 30) \approx 920$ Hz. A perfect fifth apart — close enough to the musical difference you actually hear.

Now flip the setup. The source sits still; the observer moves toward it at speed $v_o$. You might guess the formula is the same with $v_o$ instead of $v_s$. It isn't.

A moving observer doesn't change the spacing of the wavefronts in the air. It sweeps through them faster. The relative speed between observer and wavefront is now $c + v_o$, and the crossing rate — the observed frequency — becomes:

Combine both cases when both are moving:

These two denominators — $(c - v_s)$ and $c$ alone for the observer — are not symmetric. Plug in $v_s = 30$ and $v_o = 0$: you get about 1096 Hz. Plug in $v_s = 0$ and $v_o = 30$: you get about 1087 Hz. Same relative speed, different answer.

This will matter in the next section when we turn to light, which has no medium.

Sound lives in air. Take the air away and sound dies. Light doesn't need a medium — experiments from the 1880s onward tried to find one ("the luminiferous ether") and failed. Einstein's 1905 paper removed the ether entirely.

With no medium, there's no preferred frame. The asymmetry from § 04 has to vanish. The relativistic Doppler formula for light — which we state here without derivation — is symmetric in the relative velocity $v$ alone:

Positive $\beta$ means the source and observer are separating. The wavelength stretches and we call it a redshift. Negative $\beta$: they're approaching, and the light blueshifts. The shift is usually quoted as a dimensionless number $z$:

For $\beta \ll 1$ this reduces to $z \approx v/c$, which is the version astronomers mostly use.

In 1912, an American astronomer named Vesto Slipher pointed a spectroscope at the Andromeda nebula and measured its redshift — the first such measurement of a galaxy outside the Milky Way. He went on to measure dozens more, and almost all of them were redshifted — meaning almost all of them were moving away from us. A decade later, Hubble-era astronomers would stitch Slipher's velocities together with distance measurements and discover that the universe is expanding. Slipher did the hard part first.

Now return to the scene in § 02 with the source speed above $c$. Every wavefront still expands at $c$, but the source itself is outrunning them. Each new wavefront is emitted outside all the older ones. The wavefronts pile up on a cone whose tip is the source and whose surface is the envelope of every sphere it has emitted.

This is the Mach cone, named for Ernst Mach, who worked out the geometry in 1887 from photographs of a bullet in flight. The half-angle $\theta$ of the cone depends only on how much faster than sound the source is going. Call that ratio the Mach number, $M = v / c$:

At Mach 1 the cone is flat — the wavefronts all pile onto a vertical wall right in front of the source. This is the "sound barrier," and it's where the pressure gradient becomes extreme. Past Mach 1, the cone tightens as $M$ grows. At Mach 2, the half-angle is exactly 30°. At Mach 10 it's about 5.7°.

A sonic boom is what you hear when that cone sweeps past you. It's not "the sound of the jet breaking the sound barrier" — that's folklore. The jet is always producing the cone as long as it's supersonic. The boom is the moment the cone's edge arrives at your ear.

Once you see the moving-source picture, you notice the Doppler effect everywhere.

One piece of geometry, written down on one page in 1842, supplies the technique by which half of modern observational physics measures speed. There is no trick here. Every one of these is exactly the picture from § 02.

We've assumed all along that the medium transmits every frequency at the same speed. For sound in air and light in vacuum, that's fine. But glass bends blue light more sharply than red. Deep water sends long waves faster than short ones. When wave speed depends on wavelength — when a medium is dispersive — a pulse falls apart as it travels.

That is the next story.