Sonic Booms, Musical Scales & Acoustics

Three last stories about sound: what happens when something outruns it, why some notes sound good together, and how to build a hall where every seat hears clearly.

Faster than sound: the sonic boom#

The Doppler formula $\nu' = v\nu/(v - u_s)$ blows up when the source speed reaches the speed of sound. Beyond that, the source outruns its own sound. Each wavefront spreads from the point where it was emitted, but the source is always ahead of all of them. The fronts pile up along a cone with the source at its tip: a shock wave.

In a time t the source moves $u_s t$ while a wavefront it emitted at the start grows to radius $vt$. The cone touches all these spheres, so its half-angle θ satisfies

The cone carries a sudden, large pressure jump. When it sweeps over you, you hear a sharp bang: the sonic boom. A supersonic aircraft drags its cone along the whole time it flies faster than sound, so everyone along its path hears a boom, not just people below the point where it “broke the sound barrier”.

Musical scale#

Certain pairs of notes sound pleasant together when their frequencies are in simple ratios, such as 2 : 1, 3 : 2 or 5 : 4. A musical scale is a set of notes chosen so that many such pairs appear. The diatonic scale has eight notes; the last has twice the frequency of the first, and the gap between them is an octave.

Starting from a different first note just multiplies every frequency by the same factor, so the scale sounds the same, only higher or lower.

Acoustics of buildings#

An auditorium should let every listener hear clearly. Several problems need care:

Even distribution

Large flat or concave walls and ceilings can focus sound into some spots and leave dead spots elsewhere. Suitably curved, often convex, surfaces spread the reflected sound evenly across the audience.

Echoes

A strong reflection arriving more than about 0.1 s after the direct sound is heard as a separate echo. Distant flat walls are covered with absorbing materials or broken up so that their reflections don’t return as distinct echoes.

Reverberation

After a source stops, sound keeps bouncing around the hall and dies away gradually. This lingering is reverberation.

Long reverberation blurs speech, as each syllable overlaps the last. Curtains, carpets, soft seats and a full audience absorb sound and shorten it. But too short a reverberation time makes a hall sound “dead” and the music thin. A good hall is designed for the right value for its use: shorter for lectures, longer for concerts.

Amplification

Loudspeakers must be placed so that the sound reaching a listener from the speaker doesn’t arrive noticeably before or after the sound from the stage; electronic delays help. If a microphone picks up its own loudspeaker’s output, the sound circulates and grows into a loud howl or ringing: feedback.