STUDY #29 · 2026 · IN OBSERVATION

Faraday Waves

A model-driven visual study of parametric surface waves — the pattern a shaken liquid chooses for itself.

MOVING IMAGE — THE WAVELENGTH ARC ω 2.83 → 5.6 → 2.83 · A 1.7 → 0.72 A_c · box held at 87.9 ℓ_c · ν 0.03

WHAT IS THIS

A layer of liquid is shaken vertically. Nothing is stirred, nothing is poured in: the whole experiment is one number going up and down. Below a threshold acceleration the surface stays flat. Above it, a standing pattern grows — and it repeats only every second drive cycle. Michael Faraday reported this in 1831; the half-frequency response is the signature.

Each Fourier mode of the surface turns out to obey a damped Mathieu equation: the shaking modulates gravity, gravity sets the stiffness, and a mode whose natural frequency is half the drive frequency is pumped once per cycle in phase with itself. That resonance condition, together with the gravity–capillary dispersion relation, is what selects the wavelength — no length scale is written into the simulation. This study runs that model as a second-order-in-time field on the GPU, with the nonlocal dispersion operator reduced to a polynomial in the Laplacian so that a shader can evaluate it.

the square lattice — the study's face; a 27 Hz shake answered at 13.5 Hz in 17 mm cells ω 2.83 (27 Hz) · A 1.6 A_c · ν 0.03 · c₂ 1.2 / c₃ 2.4 · mercury

Motif parametric instability / subharmonic response / wavelength selection / dark-mirror relief

Method A small simulator was generated and modified with AI assistance, then ported to a real-time GPU (GLSL) renderer. The visual output was selected through parameter exploration.

Observation The threshold and the selected wavelength are not chosen anywhere in the code — they fall out of the damped Mathieu equation each surface mode obeys, and both engines reproduce the exact Floquet exponents of that equation (Python to 5e-6, the shader to 6e-4). What stayed with me is how sharp the threshold is: at 0.90 of it the field decays to nothing, at 1.15 it grows without bound, and everything worth looking at lives between those two numbers.

Reference T. Brooke Benjamin & F. Ursell, "The stability of the plane free surface of a liquid in vertical periodic motion," Proceedings of the Royal Society A, vol.225, 505-515 (1954); Michael Faraday, Philosophical Transactions of the Royal Society of London, vol.121, 299-340 (1831).

Tools Python / NumPy / SciPy / three.js / React / GLSL / ffmpeg / AI coding assistant

Year 2026

This is not a scientific simulation result, but a visual interpretation of the phenomenon.

SAME FAMILY, NOW DRIVEN

Chladni's plate is told its pattern by its edges. Faraday's liquid picks its own.

Study #03 — Chladni Study #29 — Faraday

Where the pattern comes from the plate's eigenmodes — the boundary decides the figure the liquid decides — dispersion plus resonance pick the wavelength, whatever the dish

The response at the driving frequency — a mode rung up at resonance at half the driving frequency — the surface repeats every second cycle

What kind of event a resonance — always present, merely louder near a mode an instability — nothing below threshold, exponential growth above it

The image sand gathers where nothing moves — the nodes, held still the moving slopes themselves — a dark mirror that never rests

PARAMETERS EXPLORED

param meaning effect on the image

ω drive angular frequency — the response is always ω/2 the one lever over the wavelength: faster is finer (in water, a 15 → 61 Hz shake takes the cells from 12.4 to 3.1 capillary lengths)

A / A_c drive amplitude over the exact Floquet threshold for that setting below 1 the surface stays a perfect mirror; the moment it crosses 1 the pattern grows; far past 2 the order breaks and the surface boils

ν viscosity — damping γ(k) = 2νk² sets the height of the threshold and the sharpness of the pattern; thicker liquids answer fatter, blunter, less selectively

c₂ / c₃ saturation terms — the model's nonlinearity (quadratic = up-down asymmetry, cubic = arrest) stripes or lattice: with c₂ = 0 every setting gives stripes — the square lattice only stands once c₂ is switched on

A₂, χ a second drive tone at 2ω with phase χ the two-frequency route toward hexagons and quasi-patterns (Edwards-Fauve); in this implementation's range it stayed subharmonic stripes

palette the liquid — reflectance F0 and body colour mercury / ink / oil / gallium / milk; reflection only, no transmitted light — kept deliberately apart from

Each image below records its exact parameter set.

THE MATHEMATICS the model behind the images

One equation, once per surface mode. The shaking modulates gravity; every mode becomes a pumped, damped oscillator; the only waves that survive are the ones whose own frequency is exactly half the driving frequency.

\ddot h_k + 2\gamma(k)\,\dot h_k + \left[\,\omega_0^2(k) - A\,k\tanh(kH)\cos(\omega t)\,\right] h_k = N_k

Each Fourier mode of the surface obeys a damped Mathieu equation (Benjamin & Ursell 1954): the vertical shaking modulates effective gravity, and gravity sets the stiffness. A mode is pumped in phase with itself exactly when its natural frequency is half the drive frequency.

\omega_0^2(k) = \left(k + k^3\right)\tanh(kH), \qquad \gamma(k) = 2\nu k^2, \qquad \omega_0(k_c) = \tfrac{\omega}{2}

The gravity-capillary dispersion relation in capillary units (for water, 1 = 2.73 mm) and the resonance condition that selects the wavelength — no length scale is written into the code. The home setting ω = 2.83 is a 27 Hz shake answered at 13.5 Hz in 17 mm cells.

A_c(k) = \frac{4\,\omega_0(k)}{k\tanh(kH)}\,\sqrt{\gamma(k)^2 + \big(\tfrac{\omega}{2} - \omega_0(k)\big)^{2}}

The weak-damping neutral curve; its minimum gives the critical acceleration and the critical wavenumber. The exact Floquet computation lands on it to 5.07e-4 relative error at ν = 0.005, the gap shrinking monotonically as damping drops — confirming the formula as the weak-damping limit.

A visual interpretation of the model, not a claim of scientific precision. The linear theory is exact; the saturation term — quadratic plus cubic in the slope u = k_c h — is a model, written in slope units so the saturated steepness stays comparable across drive frequencies. On the GPU the nonlocal dispersion operator is reduced to a polynomial in the Laplacian, exact at k_c, so the shader keeps the resonance in the same place.

SELECTED STILLS — 4

the square lattice — a 27 Hz shake answered at 13.5 Hz in 17 mm cells ω 2.83 (27 Hz) · A 1.6 A_c · ν 0.03 · c₂ 1.2 / c₃ 2.4 · mercury

the fine net — the same liquid answering a faster shake with smaller cells ω 6.0 (58 Hz) · A 1.7 A_c · ν 0.05 · c₂ 0.9 / c₃ 1.8 · gallium

the rolls — the setting that can only make stripes, made into the picture ω 2.83 · A 1.6 A_c · ν 0.03 · c₂ 0 / c₃ 1.2 · ink

the swell — the gravity side of the dispersion curve; a slow shake answered in long waves ω 1.9 · A 1.7 A_c · ν 0.02 · c₂ 0 / c₃ 1.0 · mercury

SIGNATURE — THE HALF BEAT

Nothing in the setup contains a wavelength. The liquid picks it — and answers on every other shake.

The drive is one number going up and down, yet the surface refuses to follow its rhythm: the pattern needs two drive periods to come back to itself. In Floquet terms the dominant multiplier is a negative real number — measured μ = −1.055 here — so one period maps the surface onto minus itself: a period doubling. The saturated field's spectrum peaks exactly at ω/2, with eleven times more power at the half beat than at the drive in the Python engine, fifteen in the shader.

That answer was checked before any image was kept. The exact Floquet neutral curve closes onto the weak-damping analytic threshold monotonically as viscosity drops; the field's monodromy matrix reproduces the Mathieu equation's to 5.3e-6; and the wavelength that grows out of broadband noise lands on the dispersion prediction to about one percent across a 4.4× range of wavenumber. The threshold itself is razor thin: at 0.90 of it the field decays to nothing, at 1.15 it grows without bound — everything worth looking at lives between those two numbers.

COLOUR = A DARK MIRROR

The surface is a mirror, so the picture is not the pattern — it is the slope of the pattern, reflecting a dark room. That is also how the experiment is photographed: a Faraday cell is lit from the side and shot against black, because transmitted light washes the relief away. Every bright filament here is a place where the surface tilts just enough to bounce the lamp into the camera.

The palettes are grounded in real liquids through one number, the normal-incidence reflectance F0: mercury (~0.72) and gallium (~0.62) are metals and come back hard and neutral; ink, oil and milk are dielectrics (~0.03–0.05) and show almost nothing except at grazing angles, where Fresnel takes over and the edges light up. The lengths are real too: in these units 1 = the capillary length of water, 2.73 mm, and the study's home setting is a 27 Hz shake answering at 13.5 Hz with 17 mm cells.

ink — a dielectric that returns almost nothing until Fresnel takes over at the grazing edges ω 2.83 · A 1.6 A_c · c₂ 0 · ink (F0 ≈ 0.03–0.05)

These are artistic approximations of published optical constants, not measurements.

REFERENCES

Michael Faraday. "On a Peculiar Class of Acoustical Figures; and on Certain Forms Assumed by Groups of Particles upon Vibrating Elastic Surfaces." Philosophical Transactions of the Royal Society of London, vol.121, 299-340 (1831).
T. Brooke Benjamin, F. Ursell. "The stability of the plane free surface of a liquid in vertical periodic motion." Proceedings of the Royal Society A, vol.225, no.1163, 505-515 (1954).
Wenbin Zhang, Jorge Viñals. "Pattern formation in weakly damped parametric surface waves." Journal of Fluid Mechanics, vol.336, 301-330 (1997).
K. Kumar, Laurette S. Tuckerman. "Parametric instability of the interface between two fluids." Journal of Fluid Mechanics, vol.279, 49-68 (1994).
W. Stuart Edwards, Stéphane Fauve. "Patterns and quasi-patterns in the Faraday experiment." Journal of Fluid Mechanics, vol.278, 123-148 (1994).

INTERACTIVE STUDY

Shake a dish of liquid straight up and down. Below a threshold the surface stays a mirror. Above it, a pattern appears — and it answers on every other shake: the beat readout ticks once per drive period, the surface completes its cycle every two. Nothing here chooses the wavelength — it follows from the liquid's own dispersion and the shake frequency you set, so a faster shake buys a finer wave, and shake strength is dialled in units of the surface's own threshold. It is a deliberately simplified instrument — purely cubic saturation, so it shows rolls, not the finished works' square lattices — capped at a 160² patch with a few curated knobs and no export, separate from the full engine used to author the finished works.

SIMPLIFIED INSTRUMENTFARADAY · DAMPED MATHIEU FIELD · ω/2 LIVE

This interactive study is not intended as a scientifically validated reproduction. It is a visual interpretation generated from an implemented model and curated parameter exploration — and it is a deliberately simplified instrument, separate from the full engine used to author the finished works.