Illustrated Physics
physics, drawn
Concept

Bessel functions — where it appears

Functions that solve the wave equation on a circle or in a medium whose stiffness grows linearly with distance. They oscillate and slowly decay, and they give the notes of a drum and a hanging chain, and the share of a frequency-modulated tone left at its central line.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A better glass is harder to hit. The half-width of the resonance in cents, and its ring-up time, against Q at 600 Hz. One falls as the other rises, and their product is fixed. A glass of Q = 100 has a window of ±8.6 cents, inside what a steady voice can hold, and it gains only a hundredfold. A glass of Q = 1000 gains ten times as much, and asks for ±0.9 cents held for over a second. Where a voice's steadiness crosses the window is about the Q at which a voice stops being able to use a further increase, and that is near Q = 173.

The glass a voice can break

A singer can shatter a wine glass, and it is not done by loudness. A good glass answers a tone a thousand times more strongly than a steady push, and it charges for that three times over: the note must be within about a cent, it must be held for over a second, and the glass must be flawed enough to fail. A voice's vibrato turns out to matter far less than its aim.

waves · Resonance The pulse behaves like a body falling at half of g. The height of a pulse going down a uniform rope 10 m long, hanging free, from the integrated wave equation with tension set by the weight below each point, against the path x = (√L − √g t/2)² that the local speed √(gx) gives. It is a parabola: constant acceleration of g/2, the same on a rope of any mass, any material. So a pulse takes 2.02 s to go down 10 m, against a stone's 1.43 s. A falling stone has speed √(2gx) after dropping x, and the pulse has √(gx) with x the height left: the parabola is the same with the acceleration halved, and the pulse is slowing rather than speeding up.

The pulse that climbs a rope at half of g

Flick the top of a hanging rope, and the pulse that travels down it slows at a steady rate of half of g, whatever the rope is made of and however heavy it is. Send it back up from the free end and it comes back altered: not the same way up and not upside down, but something else, with a lobe of each sign. A hanging rope is the simplest medium in which a wave's speed varies, and it has several surprises to give.

waves · Wave motion

Named alongside it

The objects these essays reach for when they reach for this one.

FractureFrequency modulationImpedanceQuality factorResonanceTensionWave equationWave speed

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