Illustrated Physics
physics, drawn
Concept

Faraday waves — where it appears

Standing waves that appear on the surface of a liquid shaken up and down, oscillating at half the shaking frequency. They arise by parametric resonance above a threshold acceleration, and their wavelength follows the liquid's capillary–gravity dispersion relation.

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

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 The film that thins too slowly. The thickness of the air film under a silicone-oil drop of radius 0.38 mm pressed onto its pool, against time, on logarithmic axes, by Reynolds's law: dh/dt ∝ −h³, so the film thins quickly at first and then ever more slowly, its thickness falling only as the inverse square root of time once it is thin. The shaded band is one bounce, 4.1 ms, in which the film falls from 10 µm to 1240 nm; the dashed line is the 100 nm at which van der Waals forces pull the surfaces together, reached after 0.65 s. The air does not have to be held up; it has only to be slow to leave, and the cube of the thickness makes it slower and slower the thinner it gets. A vibrating bath that lifts the drop off every few milliseconds refills the film each time.

The drop that will not merge

A drop of oil let fall onto a pool of the same oil ought to vanish into it at once: merging releases surface energy, and nothing holds the two apart. Instead it bounces. A film of air is trapped between drop and pool, and air, though it offers no strength at all, is slow to leave a gap a few micrometres thick: the film takes the better part of a second to drain to the hundred nanometres at which the surfaces can touch, while a bounce lasts four milliseconds. Shake the pool and the drop can bounce for ever, in rhythms that double as the shaking grows — and, near the pool's own instability, it starts to walk across the surface on the waves of its own making.

fluids · Surface tension

Named alongside it

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

Air filmCapillary timeCapillary-waveCoalescenceDispersionLubrication theoryMathieu equationParametric resonancePattern formationPeriod doublingPilot waveStanding wave

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