Illustrated Physics
physics, drawn
Concept

Speed of sound — where it appears

The speed at which a small pressure disturbance travels through a gas, liquid or solid. It is the square root of the material's stiffness against compression divided by its density, which is why it is 343 m/s in air, about 1,480 m/s in water and faster still in steel.

Named by 8 essays across 5 fields — each of them below, with the objects they name alongside it.

The far end that has not been told yet. A rod 3 metres long, pushed at one end at time zero. On the left, position across and time upwards, for the two fastest disturbance speeds drawn here, with the light cone beside them: nothing may lean further to the right than that line, which crosses the rod in 10.0 nanoseconds. A rigid rod would be the vertical dashed line — the far end moving at the same instant as the near one — and it is not a limit that a hard material approaches. It is a signal at infinite speed. On the right, how long the far end actually waits, against how fast the disturbance travels, both logarithmic, with every material on it: 8.8 ms at 3.4e+2 m/s, 600.0 μs at 5.0e+3 m/s, 250.0 μs at 1.2e+4 m/s, 100.0 ns at 3.0e+7 m/s, 20.0 ns at 1.5e+8 m/s. The line has slope −1 and the light cone is a hard floor beneath it. Ordinary materials sit four to five decades above that floor, which is why rigidity is such a good approximation and why it is still not a limit: steel's delay is not small compared with light's, it is 6e+4 times larger. Everything usually derived from rigid bodies survives, because the delay is beneath notice in ordinary circumstances. What does not survive is the use of rigidity in an argument about simultaneity, which is where it does real damage: a rod pushed at one end is compressed for as long as the wave takes to cross it, and there is a frame in which its far end is still at rest while its near end is moving.

Nothing is allowed to be rigid

A rigid body would move its far end at the instant its near end was pushed, which is a signal at infinite speed. Relativity forbids it — not approximately, and not as a limit that a hard enough material approaches. What follows is a ceiling on how stiff matter may be, and that ceiling caps the mass of every neutron star.

relativity · Relativistic dynamics Water gets denser the deeper it is. The density of seawater at 0 °C and a salinity of 35 against depth, from the surface to the 10.99 km of the Challenger Deep, with the pressure integrated down the column from the equation of state rather than assumed. At the surface it is 1028.1 kg/m³. At the ocean's mean depth of 3.68 km it is 1045.3, 1.67 per cent more; at the bottom of the deepest trench it is 1076.5, 4.71 per cent more. An incompressible ocean would be the vertical line. The curve is almost straight; what curvature it has, too little to see here, is the water stiffening as it is squeezed: its secant bulk modulus rises by 3.24 pascals for every pascal of pressure.

The sea that stands lower because water gives

Water is the textbook incompressible fluid, and at the bottom of the ocean it is 4.7 per cent denser than at the top. That squeezing lowers the whole sea by about thirty metres, adds the weight of 266 metres of water to the pressure in the deepest trench, and speeds sound up so steadily with depth that it digs a channel in which a low note can cross an ocean.

fluids · Hydrostatics Sound cones that tip over. Time against position for sound in a fluid that speeds up from 0.55 to 1.3 times the speed of sound, drawn as the cone of places a sound made at each point can reach: its edges move at v − c and v + c. Upstream, where the flow is slow, the cones straddle the vertical and sound can go either way. At x = 0.20, where the flow reaches the speed of sound, the upstream edge stands vertical. Beyond it both edges point downstream: sound made there can no longer reach anywhere upstream of where it was made. That is a horizon, drawn in the same way as the light cones of a black hole, and it arises from nothing but the ordinary addition of a sound's speed to the speed of the water that carries it.

The river that sound cannot swim up

Sound travels at a fixed speed through the air or water carrying it, so a flowing medium adds its own speed to the sound's: faster downstream, slower upstream. That is the plainest velocity addition there is. Where the flow speeds up past the speed of sound, upstream-going sound is carried backwards, and nothing made beyond that point can be heard before it. The point is a horizon. Sound cones tip over there exactly as light cones do at a black hole, and the rate at which the flow speeds up plays the part of the surface gravity. A kitchen sink under a running tap makes the reverse kind of horizon, a white hole for ripples.

relativity · Velocity-addition The stiffness a two-solar-mass star demands. The heaviest neutron star the model matter can support, in solar masses, against the constant square of the sound speed assumed above 300 MeV/fm³, as a fraction of c². At the conformal value, one-third (dashed), it is 1.92 solar masses; the heaviest star the model can make rises to 2.97 at the causal limit. To hold up the 2.08 solar masses of J0740+6620 (shaded) the square of the sound speed must exceed 0.394 of c² somewhere above that density — above the conformal one-third, by this model's reckoning. The conclusion depends on the model's low-density part, which the next figure varies.

The sound speed a neutron star needs

Heat any gas without limit and the speed of sound in it climbs towards a ceiling: not the speed of light, but light's speed divided by the square root of three, the sound speed of a gas of massless particles. For decades it was a natural guess that no matter could beat that ceiling, since the densest matter ought to behave like a gas of free quarks. Neutron stars say otherwise. A star of twice the Sun's mass, packed into a ball twenty-five kilometres across, can only be held up if its core is stiffer than the ceiling allows — unless nuclear physics fails at densities where it is believed to hold. Somewhere inside the heaviest neutron stars, sound almost certainly travels faster than it can in any free gas.

relativity · Relativistic thermodynamics The speed of sound in dry sand rises with the load on it. The speeds of compressional and shear waves in a dry random packing of quartz grains, porosity 0.36, against the pressure on it, from the Hertz–Mindlin effective-medium theory, for coordination numbers from 6 to 9 (the bands). The pressure axis is logarithmic. Both speeds rise as the sixth root of the pressure. At 1 kPa — the load about six centimetres down in dry sand — compressional waves travel at 316–362 m/s; at 100 kPa, about six metres down, 681–780 m/s; at 10 MPa, 1468–1680. The dashed lines are the speed of sound in air, 343 m/s, and in water, 1,480 m/s. Near the surface dry sand carries sound no faster than air; measured shallow speeds are lower still, by up to a factor of two, because real contacts are fewer and weaker than the theory assumes.

The sound sand carries faster the deeper it goes

A solid carries sound at a speed set by what it is made of. A heap of quartz grains is made of quartz, and near its surface it carries sound more slowly than the air above it. The grains are not the solid; their contacts are, and a contact between two spheres grows stiffer the harder it is pressed. So the speed of sound in dry sand rises as the sixth root of the load on it — from about air's speed a few centimetres down to twice that at six metres — the paths of sound in a beach curve back up to the surface, and a water table is a mirror to one kind of wave and almost invisible to the other.

fluids · Granular matter A sound that survives only in a hot-electron plasma. The damping of an ion-acoustic wave at kλD = 0.3, as the fraction of its amplitude lost per radian of oscillation, −γ/ω, against the ratio of electron to ion temperature, for hydrogen and argon plasmas, on logarithmic axes: from the kinetic dispersion relation (solid) and from the weak-damping formula (dashed), which is valid only where the damping is small. With equal temperatures the wave loses 0.45 of its amplitude per radian in hydrogen — it is gone within a cycle. At Tₑ/Tᵢ = 10 the loss is 0.031; at 100 it is 0.0126, set by the electrons alone. Argon's floor is lower, 0.0020, because its ions are heavier and its electrons a smaller fraction of a sound wave's speed.

The sound a plasma carries on its electrons' heat

Sound in a gas needs two things: something heavy to be pushed, and a pressure to push it. In an ordinary gas the same molecules supply both. In a plasma they can be split — the ions carry the inertia and the electrons, a thousand times lighter and often a hundred times hotter, carry the pressure — and the result is a sound that travels at a speed set by one particle's temperature and the other's mass. It travels only while the two temperatures stay far apart. Warm the ions to match the electrons and the sound dies within a cycle, with nothing to rub against.

astrophysics · Plasma oscillation How a sphere of gas rings. The acoustic pressure against distance from the centre of a spherical cavity, for its three lowest radial modes, in which the gas breathes in and out in spherical shells: sin(zr/a)/(zr/a), with z = 4.4934, 7.7253, 10.9041 the roots of tan z = z that make the gas's motion vanish at the wall. Filled with argon at the triple point of water, 273.16 K, a sphere 62 mm in radius rings in these modes at 3551, 6104, 8616 Hz. Radial modes are the ones precision work uses: they have no motion along the wall, so the wall's roughness and the gas's viscosity there barely touch them, and their frequencies are set by the volume rather than the shape — a slightly imperfect sphere shifts them only at second order.

The note that fixed the kelvin

A sphere of argon, struck, rings at a pitch set by the speed of sound in it, and the speed of sound in argon contains only one quantity nobody could compute: the constant that converts a temperature into an energy. Argon's atoms can hold energy in exactly three ways, so its heat capacity is a count rather than a measurement, and its speed of sound is (5/3)kT/m with nothing left over. Ring the sphere at the temperature of melting ice, measure its size with radio waves in the same cavity, and Boltzmann's constant comes out to a part in a million. In 2019 that number stopped being measured and became the definition.

thermodynamics · Equipartition The shadow a thunderclap's sound cannot enter. Sound rays from a point 3 km up a lightning channel, on an afternoon with 25 °C at the ground and the air cooling 6.5 K per kilometre upwards, so that sound travels slower the higher it goes; the vertical scale is stretched 3.5 times. Every ray curves upwards. Rays aimed below the horizontal reach the ground, reflect and climb away; the last ray to touch the ground grazes it 23.1 km from the channel. Beyond that, shaded, no ray arrives at all: a listener's ears 2 m up are inside the shadow from 23.7 km. The flash is visible there and the thunder is not, though the air between is perfectly clear and absorbs the low notes of thunder hardly at all.

The thunder that passes over the listener's head

Lightning twenty-five kilometres away is plainly visible and usually silent, though the air between is clear and absorbs the low notes of thunder hardly at all. The thunder has not faded; it has gone over the listener's head. Daytime air cools upwards, sound slows as it climbs, and Huygens's wavelets, smaller at the top of every front than at the bottom, tilt each front towards the sky. Beyond the ray that just grazes the ground there is a shadow the sound cannot enter — and a hundred and fifty kilometres further out, the stratosphere can bend it back down.

waves · Huygens

Named alongside it

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

Equation of stateCausalityNeutron starRefractionAcoustic shadowAdiabatic indexAnalogue gravityAtmosphereBoltzmann constantBorn rigidityBulk modulusCharacteristics

All concepts