Illustrated Physics
physics, drawn
Concept

Pattern formation — where it appears

The spontaneous appearance of regular spatial structure — stripes, rings, spots or cells — in a system that started uniform and is driven uniformly. It arises from an instability that favours one length scale, set for example by a nucleation threshold in Liesegang rings or by a dispersion relation in Faraday waves.

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

Bands of precipitate laid down in a tube of gel. A tube of gel holding one salt, B, with a second salt, A, diffusing in from its left end, computed at the end of the run: the shading shows where solid precipitate has formed, darker for more. The product of their reaction precipitates only where its concentration first passes a nucleation threshold, 0.5 in the units used, and then wherever solid is already present. The solid is not spread smoothly. It forms in 23 sharp bands, close together near the entrance and further and further apart down the tube — the last two are 26.0 units apart, against 0.8 between the first two. Each band holds more precipitate than the last, in proportion to its distance. Nothing in the gel was patterned to begin with.

The rings a reaction lays down in a gel

Let one salt soak into a tube of jelly that holds another, and where they meet a solid should form, spreading smoothly behind the advancing front. It does not. It forms in sharp, separate bands, close together near the entrance and further and further apart, each one a fixed multiple of the distance of the last. Raphael Liesegang saw the rings in 1896 and they have been found since in rocks, in agate, in gallstones and in microchips. Nothing in the jelly is patterned to begin with. The pattern comes from a solid that will not start until the solution is overloaded, and that then empties the solution around it.

thermodynamics · Diffusion Where shaking makes the surface unstable. For water 1 cm deep shaken vertically at 50 Hz, the regions of wavenumber and shaking acceleration in which a surface wave grows rather than decays, computed mode by mode from the Floquet multipliers of its damped Mathieu equation (inside the curves, unstable). The largest tongue reaches down to 0.80 m/s² — 0.081 g — at k = 633 m⁻¹, the wave whose own frequency is exactly half the drive's: a wavelength of 9.9 mm. A second, narrower tongue sits at k = 1066 m⁻¹, where the wave's frequency equals the drive's, and needs much harder shaking. Without damping both tongues would reach down to zero; viscosity lifts their tips, most for the shortest waves.

The ripples that answer at half the shaking

Shake a dish of water up and down fifty times a second and nothing happens — until the shaking passes a threshold, when the whole surface erupts into a pattern of standing ripples, squares or stripes, each crest rising and falling twenty-five times a second, at exactly half the frequency of the shaking. Michael Faraday saw it in 1831. The size of the ripples is not set by the dish. It is chosen by the water itself, as the one wavelength whose natural frequency is half the drive's, and the threshold is set by how quickly the water's own viscosity would kill that wave.

waves · Standing waves

Named alongside it

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

Capillary waveChemical precipitationDiffusionDispersionFaraday wavesMathieu equationNucleationParametric resonanceReaction diffusionScaling lawSelf organisationStanding wave

All concepts