Illustrated Physics
physics, drawn
Concept

Characteristics — where it appears

The paths along which a disturbance travels in a wave or flow problem, found as the lines on which information is carried at the local signal speed. Drawn in time and position, they show where a signal can reach and where it cannot, including horizons.

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

A sine on its way to a vertical face. One period of a sine, followed by the equation whose only nonlinearity is that the local speed depends on the local height. The profiles are at σ = t/t_b of 0, 0.3, 0.6, 0.9, 0.995, each obtained by solving the implicit relation u = u₀(x − (c₀+βu)t) for u at every point by bisection — and checked against the partial differential equation itself, which it satisfies to 2.6e-7. The crest travels faster than the trough, so the descending front leans forward and the ascending one leans back; the wave stays exactly as tall as it started and exactly as long, and only its shape changes. The steepest gradient grows as one over (1 − σ) — measured here as 9.83 times its initial value at σ = 0.9, against ten — so it is infinite at σ = 1 and the curve has a vertical tangent. The picture cannot be drawn past that point, which is not a failure of the drawing: the solution genuinely becomes three-valued, and what actually happens is a jump whose width is set by the dissipation this equation does not contain.

The front that steepens until it cannot

In a linear medium every wave keeps its shape, because every part of it travels at the same speed. Let the speed depend on the height by even a little and the crest overtakes the trough, the front leans forward, and after a time that can be written down the wave demands two values at one place — which is where the description ends and a shock begins.

waves · Wave motion 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

Named alongside it

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

Analogue gravityDissipationEvent horizonHarmonic generationHawking temperatureHydraulic jumpLight coneNonlinear waveShock waveSimple waveSpeed of soundSuperposition

All concepts