BOOM

What you're looking at

The red dot is the source. Every fraction of a second it sends out a wave, and each wave spreads outward at the same speed no matter how fast the source is moving. That last part is the whole trick: the waves don't inherit the source's speed.

Top view

A flat slice through the source. Drag the green listener anywhere to move it. When a wavefront reaches the listener it flashes, and the panel below counts how often that happens.

3D shock cone

The same waves, drawn as spheres in real space, with a plane flying above the ground. Drag to spin the camera. The orange patch on the ground is the boom carpet — everywhere the shock has already swept past.

One thing to know

Only the most recent handful of wavefronts are drawn. The source has been emitting the whole time and the listener still receives every wave — the older circles are hidden so the picture stays readable, which is also why the V extends past the last circle you can see. It is riding on wavefronts that are still out there.

Who moves?

Switch Moving to Listener and the source parks in the middle while the listener drives straight through it. The pitch still rises and falls. Same effect, opposite cause — which is the cleanest way to show that a passing siren never actually changes its note.

The listener runs head-on, so the pitch is flat on the way in and flat on the way out with one step at the crossing. Drag it off the centre line and the flat sections start to drift, because the angle to the source is now swinging the whole way past.

Things to try

The listener starts on the source's path, so the source comes straight at it and then goes straight away. That makes the note flat on the way in, flat on the way out, and changes in a single step as it passes. Drag the listener off the line and both halves start to drift, because now the angle to the source is swinging the whole way past — the general case, and a good second step once the clean one has landed.

Set the speed to 0. Every wave is a perfect circle around the source and the listener hears exactly the rate you set — no Doppler shift at all. Now drag the listener toward the source: the pitch climbs, even though the source never moved. Drag it away and the pitch drops. It takes motion to shift a pitch, but it doesn't matter who is doing the moving.

Creep up to 1.00. The wavefronts crowd against each other right at the nose of the source.

Go past 1.00 and watch the V form. Notice it gets narrower the faster you go.

Park the listener ahead of a supersonic source: nothing arrives at all until the cone sweeps over it. That silence is why you never hear a fast jet coming.

Then keep listening. The source doesn't go quiet after the boom — it comes back twice at once. One sound is the jet carrying on away from you. The other is the approach you missed, arriving late and in reverse. Both start as a shriek at the bang and slide downward.

Doppler Effect & Shock Waves

A source sends out a wave at a steady rate. Speed the source up and the waves bunch together in front and spread out behind — that's the Doppler effect. Push it past the wave speed and the wavefronts stack into a V: a bow wave on water, a shock cone in air. One rule, three phenomena.

Simple and Full

The simulation opens in Simple, which is the version to teach from. On screen it shows nothing but expanding rings and the source that made them — no cone lines, no path, no origin dots. The V is there all the same, as the shape the rings themselves make, and a class that finds it that way owns it in a way that a class shown a drawn line does not. Tick Bow wave / shock cone once they have seen it, to name what they found.

Every ring stays exactly as bright as it started, however large it grows. Real wave amplitude does drop as the energy spreads around a longer and longer circumference — but that belongs with energy, not here, and a ring that dims as it expands reads to a student as a wave petering out. The source makes waves only while it is on screen, and the next run begins once the last of them has expanded past the edge, so a run is one complete pass with nothing left over.

It also plays one steady tone at a constant volume, so the pitch change is the only thing moving. Simple works that pitch out from the geometry rather than timing wavefronts as they land, which matters more than it sounds: a rate timed from intervals cannot change faster than one interval, so a head-on pass comes out as a second-long slide instead of the clean step it should be. Send the listener straight through a parked source and the note is dead flat coming in, dead flat going out, and changes in one move at the pass. The same holds with the source doing the moving, which is why the listener starts on its path rather than off to one side. Full goes back to timing real arrivals; the two agree to well under a percent everywhere else.

Full adds three things that are true but demanding: a longer wavefront trail, volume that falls off with distance, and the second sound stream a supersonic source delivers after its boom. Worth showing to a class that has already got the main idea, and worth leaving off until then.

The one rule behind all of it

Waves travel at a speed set by the medium, not by whatever made them. Sound moves through air at about 340 m/s whether it came from a parked car or a fighter jet. So once a wavefront leaves the source, the source can't catch up to its own old waves — it can only run into the space where the next ones will go.

Doppler effect (source slower than the waves)

Moving forward between one wave and the next, the source shortens the gap ahead of it and stretches the gap behind. Someone in front receives waves more often — higher pitch. Someone behind receives them less often — lower pitch. The source itself never changed its note. Only the spacing changed.

If you hear nothing at all, check the listener panel — ahead of a supersonic source, silence is the correct answer until the shock arrives.

In Full detail, volume also falls off with distance the way real sound does, so a source on its way out of the field fades rather than stopping.

The listener readout is measured the honest way: the simulation just counts wavefront arrivals and reports the rate. No formula is applied to fake the result — which is also why the listener goes quiet ahead of a supersonic source and comes back after the boom without any of that being programmed in as a special case.

It doesn't matter who is moving

Set Moving to Listener and the source sits still while the listener drives past. The waves leave in perfect circles, evenly spaced, exactly as they would if nobody moved at all — and the listener still hears a high note coming in and a low note going out. Nothing about the sound changed. What changed is how quickly the listener runs into wavefronts.

This is the version worth showing before anyone writes a formula, because it kills the most common wrong answer outright. A siren does not drop its pitch as it passes you. It plays one steady note the entire time.

The two cases are not quite identical once you look closely. A moving source changes the wave spacing; a moving listener changes the rate of meeting waves that are already evenly spaced. They give slightly different formulas, which is why the full Doppler expression has the source term in the denominator and the listener term in the numerator. At everyday speeds the difference is far too small to hear.

Bow waves and shock cones (source faster than the waves)

Now the source outruns its own waves. Each circle is left behind, and their common tangent line is a sharp V. On a pond that V is the wake behind a duck or a boat. In air it's a cone, and the pressure jump across it is the crack you hear as a sonic boom.

The half-angle of the V follows from a triangle: in the time the source travels a distance v·t, its wave has only spread c·t, so sin θ = c / v. Faster source, narrower cone.

What you hear after the boom

The boom is not the end of the sound — it's the beginning of it. Before the shock arrives you hear nothing, because no wave has reached you yet. The moment it arrives, a whole flight's worth of backed-up sound lands at once: that's the bang.

After that, two sounds arrive together. One is the source continuing away from you, dropping in pitch the usual way. The other is the part of the flight that happened before the boom, finally catching up — and because the source outran those waves, they reach you in the reverse of the order they were made. You hear the approach played backwards, on top of the departure. Turn the sound on with the supersonic preset and you can hear both voices peel apart from the bang.

Mathematically, t = t₀ + d(t₀)/c has one solution below the wave speed and two above it. Every sound you hear corresponds to a solution of that equation, so above Mach 1 you hear everything twice.

Why a boom is a sweep, not a bang in one spot

The cone is dragged along with the plane, so where it cuts the ground it traces a curve that travels with it. Everyone under that path hears one boom as the edge passes over them — which is why a single supersonic pass leaves a long strip of boom on the ground rather than a single loud point.

Classroom sequence

  1. Speed 0. Concentric circles. Establish that the source's rate and the listener's rate match.
  2. Speed 0.6. Ask which side is bunched before revealing the listener readout.
  3. Speed 1.0. The pile-up. "What would happen if it went faster?"
  4. Speed 1.6. The V appears. Trace it back to the individual circles.
  5. 3D view. Same physics, now with a ground to boom.