Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Series

Speed of sound — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A source at Mach 0.55, and the sound it has already made. Each circle is one pressure pulse, expanding at the speed of sound from the point the source was at when it left. Below the speed of sound every circle still contains the source, so the pressure disturbance reaches every point of the fluid before the source does. Above it the circles have an envelope, and outside that envelope nothing has been heard at all.

    What a signal travels at

    Air finds out that an aeroplane is coming, and it finds out at a definite speed. That speed does not depend on how hard the air is squeezed or on how much of it there is — only on how hot it is — and every result in compressible flow is downstream of that one fact.

    part 1 · compressible
  2. A source at Mach 2.00, and the sound it has already made. Each circle is one pressure pulse, expanding at the speed of sound from the point the source was at when it left. Below the speed of sound every circle still contains the source, so the pressure disturbance reaches every point of the fluid before the source does. Above it the circles have an envelope, and outside that envelope nothing has been heard at all.

    When the warning cannot arrive

    Supersonic flow is not fast flow. It is flow in which the fluid ahead has been told nothing, because the body is outrunning its own pressure signals — and that single change turns an equation of one type into an equation of another.

    part 2 · compressible
  3. The sound speed of a mixture, against how much of it is gas. Wood's formula for air in water. Both ends are the pure phases at 343 and 1,481 metres a second; in between the mixture takes the water's inertia and the air's springiness and the speed collapses to twenty-four metres a second — a fourteenth of the slower constituent.

    Slower than either of them

    Sound travels at 343 metres a second in air and 1,481 in water. In a mixture of the two it travels at twenty-four, because the mixture takes the water's inertia and the air's springiness — and one per cent of air by volume is enough to take water down to a twelfth of its own speed.

    part 3 · compressible
  4. 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.

    part 4 · compressible
  5. 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.

    part 5 · compressible

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