Brass

The last key's horn unrolled, mouthpiece on the left, bell on the right, with the pressure waves running along it now. In the long narrow tubes (shaded) loud crests travel faster than troughs, so the outgoing wave's fronts grow steeper on the way to the bell: play loud and watch them turn into near-vertical shocks. The flare reflects the low harmonics back and lets the steep fronts out.

Three periods of the last note: the lips open once a period, the mouthpiece's pressure swings smoothly, and the same wave arrives at the bell steepened. Soft, the two have the same shape; loud, the bell's has a sudden jump each period, the shock that makes a trombone blare (Hirschberg et al., 1996).

The horn's input impedance |Z| at the mouthpiece (log), as the engine plays it, from half the note to its 16th harmonic, ticks at the note's harmonics. The lips buzz on one of its peaks: the slide or valves move them all, and the player picks the partial. The bell makes the peaks nearly harmonic and lets them fade above a cutoff, so the high notes are harder.

Soft, the harmonics fall away fast; loud, the shock adds treble all the way up, and the spectrum flattens. Turn the air's nonlinearity off and loud notes only get louder.

Watch a loud note's harmonics climb as it speaks: the treble arrives last, once the wave is strong enough to steepen.