Illustrated Physics
physics, drawn
Concept

Shock wave — where it appears

A thin front across which pressure, density and speed jump almost discontinuously, formed when the faster parts of a strong wave overtake the slower ones. It travels faster than sound in the gas ahead, dissipates energy as it goes, and is heard as a crack or a boom.

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 The same energy in less and less whip. The speed of the moving part of a whip 2 m long and weighing 250 g, against how far the loop the arm threw into it still has to go to the tip, on a logarithmic scale, if the 15 J the arm gave it stays in the part still moving. Three whips: uniform, tapering linearly, and tapering faster, each ending in a 0.2 g cracker. At the handle every part moves at 11 m/s. Half a metre from the tip the uniform whip is at 22 m/s and the faster taper at 72; the uniform whip passes the speed of sound (dashed) only within its last millimetre, the tapers 1.6 cm and 1.5 cm from it. All three end at the same ceiling, √(2E/m) = 387 m/s, set by the cracker alone: the taper decides how gradually the speed builds, not how fast the end goes.

The crack at the end of a whip

An arm moves a whip's handle at ten metres a second, and a fifth of a second later its tip passes the speed of sound. Nothing in between is a lever or a spring. The arm throws a loop into the whip, the part behind the loop comes to rest, and the energy stays with the part still moving, which grows lighter as the loop runs out towards the tip. Energy shared by less and less mass means speed, and what caps it is the tip's own mass, the tuft of string at the end. The crack is a sonic boom, made by something the size of a shoelace's end.

mechanics · Momentum

Named alongside it

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

CharacteristicsConservation of energyDissipationFolded chainHarmonic generationKinetic energyThe Mach coneMomentumNonlinear waveSimple waveSonic boomSuperposition

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