Concept

Attenuation — where it appears

The loss of a wave's amplitude as it travels, by dissipation or by scattering. It is usually expressed as a rate per unit distance, and it often rises steeply with frequency, so that a medium filters out the shortest waves first.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

$A wave that travels and a wave that spreads. A harmonic pressure wave's amplitude and its instantaneous value along a tube, over two wavelengths of the inviscid wave, at four Womersley numbers. At α = 15 the wave marches on, a little weaker each wavelength. At α = 5 it is visibly damped. At α = 2 it is nearly gone within a wavelength. At α = 0.5 there is no wave to speak of: the disturbance falls away within a small fraction of the inviscid wavelength, as heat does into a wall.$

The pulse that has to travel

In a rigid tube the Womersley number decides the shape of an oscillating flow. Make the wall elastic and the pressure pulse has to travel, a second number appears — the tube's length in wavelengths — and the first number turns out to decide something more basic than the profile: whether the tube carries a wave at all, or only a disturbance that spreads like heat.

regimes · Womersley $Below the bubbles' resonance the mixture is slow; above its stop band it is faster than water. The phase speed of sound in water carrying millimetre air bubbles, against frequency, at void fractions of 10⁻⁴, 10⁻³ and 10⁻². At low frequency each is Wood's mixture speed with the gas isothermal — 312 m/s at 10⁻³, not the 366 an adiabatic gas would give. Approaching the bubbles' resonance near 3.2 kHz the speed falls further, then jumps through a stop band in which the wave hardly propagates, and above it the phase speed exceeds water's — at 10⁻³ 1640 m/s at 30 kHz — before returning to 1481 m/s from above.$

Above their resonance, bubbles make water faster

Wood's formula says a pinch of air makes water's sound slower than air's, and it is right at low frequency. Each bubble, though, is a spring with water for its mass, and the wave drives it. Below the bubbles' resonance the mixture is slow; through a band above it no sound propagates at all; and above that band the same bubbly water carries sound faster than pure water does. Even the slow end is not quite Wood's, because a slowly squeezed bubble keeps its heat.

compressible · Speed of sound $A spread of sizes smears the stop band and pushes the fast band up. The phase speed of sound against frequency in water carrying air bubbles at a void fraction of 10⁻³: all of one millimetre radius, and spread from 10 µm to 1 mm as n ∝ R^(−q) with q = 0, 3/2 and 10/3. Every cloud starts at Wood's speed, 312 m/s, because at low frequency every bubble is driven below its resonance. The single size jumps through its stop band near 3 to 13 kHz into a fast band. The distributions climb through a band as wide as their resonances, 3.2 kHz for the largest to 301 kHz for the smallest, and the more small bubbles they carry the later they emerge faster than water.$

The more small bubbles a cloud holds, the later it turns fast

Bubbles of one size give water three sound speeds: slow below their resonance, nothing in a stop band, faster than water above it. A cloud of many sizes has every resonance at once, so at any frequency its small bubbles soften it and its large ones stiffen it. The stop band becomes a wide band of attenuation, the fast band survives above it, and where the fast band starts is set by how much of the gas is in the smallest bubbles.

compressible · Speed of sound

Named alongside it

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

Model limit Bubble Dispersion Resonance Speed of sound Wave speed Acoustics Cavitation Compressibility Diffusion Dimensionless number Dispersion relation

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