Patterns Shaken Out of Oil

Wrinkled Water, 1831

In 1831 Michael Faraday published a long paper on vibrating surfaces in the Philosophical Transactions of the Royal Society. He put water, milk, white of egg, ink and mercury on a vibrating plate and saw the liquid wrinkle into small, regular cells.

He counted carefully: each little heap formed again only once for every two vibrations of the plate. The liquid moved at half the plate’s frequency.

Model: an aluminium bath 180 mm across inside, with 4 mm of silicone oil, on an electrodynamic shaker

Half the Frequency

The dish goes up, down, up and down before a crest on the surface is back where it started. Two shakes of the dish, one rise and fall of the oil. This is a subharmonic response.

The real motion is too fast to follow: the dish shakes 80 times a second and the surface rises and falls 40 times. Here it is slowed many times over, and the dish’s travel is drawn larger than life.

Slowed down; the dish’s travel exaggerated

Like a Swing

Someone on a swing crouches at the top and stands at the bottom, twice in every full swing. The swing goes higher with no one pushing it. This is parametric resonance: what changes is a property of the system itself, not a push from outside.

A shaken dish works the same way. Seen from the dish, gravity grows and shrinks once with every shake. A ripple that rises and falls on its own, with a period of exactly two shakes, is strengthened again and again.

In 1954 Brooke Benjamin and Fritz Ursell wrote the equation for this: a ripple of each wavelength obeys a Mathieu equation. Its solutions grow inside a series of tongue-shaped regions. The widest is where the ripple’s frequency is half the shaking frequency.

A Threshold

Without friction, even the gentlest shaking would slowly grow the matching ripple. Oil is viscous and a ripple loses energy all the time. It grows only if the shaking puts in more than it loses.

So the pattern has a threshold acceleration. For 20 cSt silicone oil shaken at 80 Hz, the measured threshold is 4.2 times gravity. Below it the surface is flat; above it the pattern fills the dish within seconds.

Stripes, Squares, Hexagons

Just past the threshold, the surface is a few sets of standing waves laid on top of each other. How many sets and in which directions sets the pattern: one set makes stripes, two at right angles make squares, three at 60 degrees make hexagons.

Which one appears depends on the oil’s viscosity and the frequency. In 1996 Arshad Kudrolli and Jerry Gollub mapped it in a dish 32 cm across: below about 25 Hz mostly hexagons; at higher frequencies, squares in thin oil and stripes in thick oil.

Faster Shaking, Shorter Waves

The spacing of the pattern is set by the waves on the oil. A wave’s frequency and its length are tied together: long waves are pulled flat by gravity, short ones by surface tension.

The faster the shaking, the shorter the wave that is picked. On silicone oil, shaking at 20 Hz gives waves about 18 mm long; at 80 Hz, about 4.8 mm. Above about 40 Hz, surface tension matters more than gravity.

Two Frequencies

Shake with two frequencies at once and the pattern can be richer. In 1993 William Edwards and Stéphan Fauve shook a layer of 100 cSt oil at two frequencies and, at certain ratios, saw a pattern with twelvefold symmetry that never repeats.

It is like a two-dimensional quasicrystal: turned through 30 degrees it looks the same, but no shift lines it up with itself. Drawn here: six sets of standing waves, 30 degrees apart.

A Drop That Does Not Land

Below the threshold the surface stays flat. Dip a toothpick into the oil and pull it out, and it leaves a drop less than a millimetre across. The drop falls onto the shaking surface, and instead of merging, it bounces.

Each time it comes down, a thin film of air is trapped between drop and bath. Before the air can drain away, the drop is thrown up again. It can bounce like this for a long time, and every landing sends a small ring of waves across the surface.

Drop 0.8 mm across; magnified and slowed down

A Drop That Walks

The closer the shaking is to the threshold, the longer the drop’s waves last. Close enough, the drop lands on the slope of the wave it made last time and is pushed to one side. It starts to walk across the bath, about a centimetre a second.

In 2005 Yves Couder and his colleagues reported these walking drops. What drives a drop is the wave it leaves behind, and that wave keeps a record of where it has been. The nearer the threshold, the longer the memory.

20 cSt silicone oil at 80 Hz, just below threshold; the drop’s waves exaggerated

An Analogy

A particle guided by its own wave recalls Louis de Broglie’s pilot-wave picture of quantum mechanics. Walking drops show some quantum-like behaviour: in a rotating bath their circular orbits come only in certain sizes, and a drop sometimes crosses a shallow barrier.

But this is not quantum mechanics. It happens on a flat surface only; the wavelength is set by the shaking, not by how fast the drop moves; and later, more careful experiments did not reproduce an early report of double-slit interference. It is a classical analogy you can see, a help for thinking, not an explanation.