Illustrated Physics
physics, drawn
Concept

Acoustic impedance — where it appears

The ratio of sound pressure to the velocity of the medium it moves, for a plane wave equal to density times sound speed. It decides how much sound crosses a boundary: air and water differ 3,600-fold, so 99.9 per cent of sound reflects between them.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

What a real coating leaves behind, and over what range. Reflectance against wavelength for a glass surface of index 1.52 in air, uncoated and with quarter-wave layers of 2 different indices, each a quarter of a wave thick at 550 nm. The ideal index is the geometric mean, 1.2329, and the layer made of it takes the reflectance to zero at the design wavelength exactly. No durable solid has that index: magnesium fluoride at 1.38 is the usual compromise and it leaves 1.26% at the design wavelength against 4.26% bare — a reduction of 3.4× rather than a removal. Both curves rise away from the design wavelength, because the thickness is a quarter of a wave only there, and the useful band is wide but not unlimited: the better coating stays under a quarter of the bare reflectance from 415 to 780 nm. The purple cast of a coated lens is that residual — the ends of the visible reflecting while the middle does not.

The layer that makes a reflection vanish

A wave meeting a step in impedance reflects, and nothing can be done about the step. Put a third medium between the two, a quarter of a wavelength thick and of exactly the intermediate impedance, and the reflection stops existing — not reduced, cancelled.

waves · Impedance A lever that makes air and water meet. The fraction of sound power crossing from air into water through an ideal lossless transformer that multiplies pressure by a factor n, against n on a logarithmic axis. With no transformer, n = 1, it is 0.11 per cent. It rises as the transformer makes the water look less stiff and air look stiffer, reaching everything at n = 60, and falls again beyond, where the water is made to look softer than air. The middle ear's 22 gives 43 per cent in this picture — a gain of 26 dB over no transformer at all. The curve is flat near its top: a lever within a factor of two of the ideal still passes about two-thirds of the power, which is why a mechanism need not be precise to be effective.

The lever that lets air speak to water

Sound in air striking a surface of water is almost entirely reflected: 99.9 per cent bounces off, which is why a swimmer underwater hears the world above as a muffled murmur. The inner ear is filled with fluid, and the eardrum faces air, so every land animal has the problem of getting sound across that boundary. The answer is three tiny bones and a difference in area, which together multiply the pressure about twenty-fold and turn a thousandth of the power into nearly half. The middle ear is an impedance transformer, the mechanical twin of the one in every radio, and it works best over just the band where speech carries its meaning.

waves · Impedance A bottle as a mass on a spring. A 750 ml bottle with a neck 8 cm long and 19 mm across, and its equivalent: the plug of air in the neck, 32 milligrams, moving in and out as a piston on the air in the body, which acts as a spring of stiffness 15 N/m because squeezing it raises its pressure. The plug's mass and the spring's stiffness give a frequency of 109 Hz, the note heard when air is blown across the mouth. The effective neck length includes about one and a half radii of air beyond its two ends that moves with the plug. Nothing in the body's air moves much — it is compressed and released almost uniformly, because the wavelength, three metres, is far larger than the bottle. That is what makes the resonator lumped: one mass, one spring, one note.

The note a bottle sings whatever its shape

Blow across the mouth of an empty wine bottle and it sounds a low note, about 109 hertz for a 750-millilitre bottle. Its wavelength is three metres, ten times the bottle's height, so it is not a standing wave in the bottle the way a flute's note is a standing wave in the flute. It is a mass bouncing on a spring: the plug of air in the neck, thirty milligrams of it, riding on the springiness of the air in the body. The note depends on the body's volume and the neck's size and on nothing about the body's shape, which is why a sphere, a cube and a tall cylinder of the same volume all sing the same note — and why a car driven with one window open throbs at eighteen hertz.

waves · Resonance The mass law: how much sound a limp wall stops. The transmission loss of a wall that only has mass — no stiffness — for sound arriving square-on, against frequency on a logarithmic axis, for surface masses of 10 kg/m², 20 kg/m², 40 kg/m². It is 10 log(1 + (ωm/2ρc)²) decibels. At 500 Hz, 10 kg/m² stops 31.6 dB, 20 kg/m² stops 37.6 dB, 40 kg/m² stops 43.6 dB. Doubling the mass or the frequency adds six decibels, and nothing else about the wall matters: it is the wall's inertia, set against air's characteristic impedance ρc = 413 rayl, that turns the sound back.

The wall that stops sound by its weight

A wall between two rooms keeps out sound not by absorbing it but by being hard to shake. The sound's pressure pushes the wall back and forth, the moving wall becomes a loudspeaker on the far side, and how much gets through depends almost entirely on the wall's mass per square metre against the frequency: six decibels more for every doubling of either. Read as impedance, a wall is a mass placed in series with the air, and the same reading explains the frequency at which a stiff pane of glass nearly vanishes and why two thin walls with a gap beat one thick one.

waves · Impedance A light panel that blocks one band as a heavy one would. Sound transmission loss at normal incidence against frequency, on a logarithmic frequency axis. Dashed: a limp panel of 2 kg/m², following the mass law, 13.4 dB at 300 Hz. Dotted: the same panel made 3 kg/m² by adding 1 kg/m² of plain mass, 16.8 dB. Solid: the 2 kg/m² panel with the same added mass hung on springs tuned to 300 Hz (damping 0.02). It blocks 35 dB at 300 Hz — the loss the mass law would give a wall 26 kg/m² heavy — and just above, at 367 Hz, falls to 5.2 dB, far below even the bare panel's 15.1: there the panel lets sound through more easily than if the resonators were not there.

The wall made heavy only at one note

A wall stops sound by being hard to shake, and how hard a plain wall is to shake is its weight: that is the mass law, and it makes quiet walls heavy. Hang small masses on springs inside a light panel, tuned to one frequency, and near that frequency the panel becomes many times harder to shake than it weighs, because the hidden masses swing against it. A two-kilogram panel then blocks a tone as a wall thirteen times heavier would. The price comes just above the tuning, where the panel's dynamic mass passes through zero and it lets sound through more easily than if the masses were not there.

waves · Impedance

Named alongside it

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

ResonanceAdiabatic compressionAntireflectionBand gapBandwidthBending waveBragg mirrorCoincidenceDampingDecibelDestructive interferenceDynamic mass

All concepts