Concept

Acoustics — where it appears

The study of small-amplitude pressure waves travelling through a fluid at the speed of sound, and of how they are made, carried and absorbed. In a flow it separates the compressible part of the motion that propagates from the vortical part that is carried along with the fluid.

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

An oscillation with no mean, and the steady flow it drives. The steady second-order velocity through a Stokes layer, in units of U U′/ω. The first-order flow averages to zero at every height; the average of its own nonlinear term does not, and the pale curve is that forcing. Integrating it twice across the layer, with no slip at the wall and no stress at the top, gives a steady velocity that rises through the layer and settles at -0.749998 — Rayleigh's −3/4, which was not put in anywhere. Beyond about five layer thicknesses nothing more happens, which is why the number is a boundary condition for the flow outside.

An oscillation with somewhere to go

Shake a fluid back and forth over a body and it develops a steady circulation that never reverses. The driving flow has no mean at all; the mean of its own nonlinear term does, and integrating that twice across the oscillatory layer gives a slip velocity of exactly three-quarters of U dU/dx over the frequency.

viscous · Streaming

Dry air makes a slow bang. The 10-to-90 per cent rise time of a steady shock against relative humidity, for jumps of 25, 50 and 90 pascals, all below the strength at which a discontinuity returns. Each falls roughly as the nitrogen relaxation time does, and doubles when the jump is halved. The rule marks the thermoviscous rise time of the 50-pascal shock, a few microseconds — two to three orders of magnitude below any of the curves.

Oxygen makes a boom's crack, nitrogen its rise time

The shock at the front of a sonic boom is not the viscous shock of a textbook, a few microseconds thick. It is spread over a large fraction of a millisecond by the vibrational relaxation of the air's molecules, and the two gases do different jobs. Oxygen, fast, removes the discontinuity for any boom weaker than about ninety pascals and decides how much of the front is left at the frequencies the ear weighs most; nitrogen, slow, sets the rise time that is measured. Water vapour speeds both, so a dry day makes a softer bang.

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

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

Named alongside it

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

Dispersion Model limit Attenuation Averaging Boundary layer Bubble Cavitation Compressibility Discontinuity Mixing Nonlinear steepening Nonlinearity

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