Illustrated Physics
physics, drawn
Concept

Second sound — where it appears

Heat moving as a damped wave rather than by diffusion, possible where the heat flux relaxes over a finite time. It is the main way heat travels in superfluid helium and has been seen in very pure crystals at low temperature.

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

The speed of a wave that carries no pressure. Second-sound speed against temperature, computed from the two-fluid equations with the normal component treated as a phonon gas — which it is below about six-tenths of a kelvin. The upper line is ordinary sound at 238 m/s, which moves the two components together. The lower curve is the other mode, and at low temperature it sits at 137.4 m/s, which is 238/√3 to three figures: a result with no adjustable constant in it, and the reason to believe the two-fluid model rather than merely to use it. What oscillates in this wave is not the density — the two components move in opposite directions and their sum stays put — but the fraction that is normal, which is a temperature. So a temperature disturbance in helium II propagates, with a speed, a reflection and a resonance, where in every ordinary liquid it diffuses and has none of those. Above a kelvin the rotons take over from the phonons and the measured curve falls to about 20 m/s; the model here is the low-temperature one and it is drawn only where it holds.

The heat that arrives as a wave

Two fluids with two velocities give two wave equations, not one. In the first the components move together and the density oscillates, which is ordinary sound. In the second they move oppositely, the density stays put, and what oscillates is the temperature — so a heat pulse in liquid helium has a speed, a front and a reflection.

fluids · Superfluidity Heat that arrives everywhere at once, and heat that waits. The temperature spread from a point of heat released at time zero, in one dimension, at times 0.5, 2, 8 relaxation times, from the diffusion equation (dashed) and from Cattaneo's telegraph equation (solid) with the same diffusivity, distances in units of the signal speed times the relaxation time. At 0.5, the diffusion equation has put 62 per cent of the heat beyond the reach of any signal; the telegraph equation none, and 78 per cent is still in the two fronts; at 2, the diffusion equation has put 32 per cent of the heat beyond the reach of any signal; the telegraph equation none, and 37 per cent is still in the two fronts; at 8, the diffusion equation has put 5 per cent of the heat beyond the reach of any signal; the telegraph equation none, and 1.8 per cent is still in the two fronts. The fronts, drawn as the vertical marks, carry heat outward at the signal speed and decay; behind them the telegraph solution fills in and, after many relaxation times, becomes the diffusion equation's Gaussian.

The heat that arrives before it could

Release a pulse of heat at one point, and the diffusion equation — the law that has described heat flow since Fourier — says the temperature rises everywhere at once, however far away. A thermometer a kilometre off registers a change the instant the heat is released. That is an infinite signal speed, and relativity forbids it. The flaw is harmless for a kettle and fatal for a relativistic fluid. The repair is a relaxation time: heat flow takes a moment to respond to a temperature gradient. With it the equation carries heat at a finite speed, sends a sharp front ahead of the spreading warmth, and turns short-wavelength heat into a damped wave.

relativity · Relativistic thermodynamics

Named alongside it

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

DiffusionCausalityCounterflowEntropyHeat equationHeat transportLambda pointMean free pathNormal fluidPhononRelativistic hydrodynamicsRelaxation time

All concepts