STUDY #34 · 2026 · IN OBSERVATION

Kelvin Wake — Ship Waves

A model-driven visual study of ship waves and the angle that will not move.

MOVING IMAGE — THE DEPTH ARC U 3 held for all 20 s · deep → Fr 0.985 → 1.68 · σ 5.5 · ν 0.06 · satellite · every change of angle is the floor's doing

WHAT IS THIS

The V-shaped wake behind a ship looks like it should depend on the ship. It does not: in deep water the envelope always spreads at arcsin(1/3) = 19.47° from the track, whatever the speed — Kelvin proved it in 1887. A racing ferry and a rowing duck drag the same wedge; speed only picks the wavelength woven inside (λ = 2πU²/g on the track).

The model is the linear water-wave equation: every Fourier mode of the surface is an independent oscillator with the dispersion law ω² = gk·tanh(kh), forced by a moving, volume-free pressure bump — the hull displaces water, it does not remove it. This study runs it in real time on the GPU, with the nonlocal deep-water operator rebuilt as a saturated cubic of blurred Laplacians so that a shader can hold the wedge.

the satellite wedge — the study's face; transverse arcs and the festooned cusp line in sun-glitter blues U 3 · deep water · σ 5.5 · ν 0.06 · satellite

Motif Kelvin wake / dispersive water waves / the 19°28′ envelope / Froude transition

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 In deep water the measured wake half-angle stayed within half a degree of arcsin(1/3) across a 1.55× speed range, while the track wavelength followed 2πU²/g to better than 1% — the angle belongs to the water, the wavelength to the ship. Draining the sea opened the same wedge along the stationary-phase curve and folded it into the Mach cone past Froude 1; and the wake's 2-D spectrum sat on the curve ω(|k|) = kₓU — the wake photographed in spectral space is the dispersion relation itself.

Reference Lord Kelvin, "On ship waves," Proceedings of the Institution of Mechanical Engineers, vol.38, 409-434 (1887); T. H. Havelock, Proceedings of the Royal Society of London A, vol.81, 398-430 (1908).

Tools Python / NumPy / 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 ENGINE, THE DRIVE NOW RUNS

Faraday's liquid is shaken in place. Kelvin's is merely passed through — and answers with a wedge.

Study #29 — Faraday Study #34 — Kelvin Wake

The drive a shaken floor — gravity itself modulated, everywhere at once a running pressure point — one hull, forcing the surface directly

The resonance parametric — a mode pumped in phase with itself at half the drive frequency station-keeping — crests whose speed matches the hull's shadow, c(k) = U·cosθ

What is selected one wavelength out of noise — an instability with a threshold a whole curve of wavelengths — every direction its own k; their envelope is the wedge

The pattern stands in the laboratory, breathing at the half beat stands in the ship's frame — steady, dragged across the sea

PARAMETERS EXPLORED

param meaning effect on the image

U hull speed writes the wavelength inside the wedge (λ = 2πU²/g on the track) — and in deep water refuses to touch the angle

h water depth the only dial that turns the wedge: Fr = U/√(gh) → 1 opens it toward 90°, and past 1 it folds into the Mach cone asin(1/Fr)

σ hull size — the width of the pressure bump reweights the excited band: a fine hull raises the divergent-wave lace, a fat hull leaves only the transverse swell; the geometry never moves

ν viscosity — damping γ(k) = 2νk² the wake's lifetime; old waves and fine waves age first, which is what keeps the wings finite

palette the observation channel satellite sun-glitter / dusk / abyss / SAR backscatter / chart ink — the ways wakes are actually seen

Each image below records its exact parameter set.

THE MATHEMATICS the model behind the images

One oscillator per wavelength, one moving bump to excite them, and one ratio that decides the whole picture.

\ddot{\hat\eta}_{\mathbf k} + 2\gamma(k)\,\dot{\hat\eta}_{\mathbf k} + \omega^2(k)\,\hat\eta_{\mathbf k} = \hat f(\mathbf k, t), \qquad \omega^2(k) = g\,k\tanh(kh)

c_g = \tfrac{1}{2}\,c_p \;\Longrightarrow\; \sin\alpha = \tfrac{1}{3}, \qquad \lambda_t = \frac{2\pi U^2}{g}, \qquad \sin\theta_{\text{Mach}} = \frac{1}{\mathrm{Fr}}

P(x) = a_1 x + a_2 x^2 + a_3 x^3, \qquad x = \big(-\nabla^2_{(9)}\big)\circ F_{13}^{\,2}

A visual interpretation of the model, not a claim of scientific precision. Block one is the engine: every surface mode an independent damped oscillator (the same second-order skeleton as Study #29, the drive now a running, volume-free pressure point). Block two is Kelvin's argument: in deep water energy travels at exactly half the crest speed, so the envelope angle is fixed and the hull's speed cancels; finite depth breaks the ratio, and past Froude 1 the wedge is a Mach cone. Block three is the GPU water: deep-water dispersion is nonlocal and no polynomial in the Laplacian can follow √k, so the operator variable is saturated — a 9-point Laplacian seen through a squared Fejér smoother — and the cubic is pinned by three exact geometry locks (transverse wavelength, cusp wavenumber, cusp group speed), solved identically in Python and JavaScript to 6.6e-12. The honest cost: the shader's caustic sits about 1.2° inside the exact 19.47°; the speed-invariance survives exactly.

SELECTED STILLS — 4

the satellite wedge — transverse arcs, the cusp festoon and a white hull U 3 · deep · σ 5.5 · ν 0.06 · satellite

the dusk glide — the same angle under a low bronze sun U 2.6 · deep · σ 3.8 · sun alt 0.16 · dusk

the radar V — bright backscatter on a black sea, the wake as orbit actually records it U 3 · deep · σ 5 · relief 2.6 · SAR

the Mach torrent — a shallow racing floor; the transverse system dead, one shock front left U 5 · h 8 · Fr 1.77 · σ 13.9 · satellite

PROCESS — PARAMETER SWEEPS

The claim, run as a ladder — four hull speeds spanning 1.6×, the source Froude-scaled the way model basins scale their hulls. The wavelength on the track quadruples, 25 to 65 px, tracking 2πU² exactly; the measured half-angle stays on 19.47° to within half a degree. The angle belongs to the water, the wavelength to the ship.

the U ladder — four speeds, one angle deep water · U 2.0 / 2.4 / 2.8 / 3.2 · measured 18.54 / 19.69 / 19.64 / 19.84° vs arcsin(1/3) = 19.47° · λ_t 25 → 65 px

SIGNATURE — THE STUBBORN ANGLE

Speed cannot turn the wedge. Depth can.

A wave can keep station with the hull only if its crests match the hull's shadow: c(k) = U·cosθ. In deep water, energy travels at exactly half that speed — and the envelope of every half-speed messenger, taken over all directions, is a wedge of sin α = 1/3. The hull's speed cancels out of the geometry. Measured here: half-angles 18.98–19.78° across a 1.55× speed range against Kelvin's 19.4712°, while the track wavelength obeyed 2πU²/g to better than one percent.

Only depth breaks the one-half ratio. Over a shallow floor the same wedge opens along Havelock's stationary-phase curve — measured 24.3° against a predicted 24.3° at Fr 0.8 — passes wide open at Fr = 1, and past it folds into the water's own sonic boom, the Mach cone asin(1/Fr). And photographed in spectral space, the whole wake — transverse arcs, divergent lace, cusp — sits on one curve, ω(|k|) = kₓU, to median 1.7%: the wake is the dispersion relation, drawn on water.

the depth arc surveyed — the same shallow floor crossed faster and faster; the wedge opens along the stationary-phase curve, passes wide open near Fr = 1, and folds into the Mach cone h 12 · Fr 0.55 → 1.9 · theory vs measured annotated per panel

COLOUR = THE CHANNELS WAKES ARE SEEN THROUGH

The palettes are the channels through which wakes are actually seen: the sun-glitter blues of satellite photography — the glint density is the wave-slope statistics — the low bronze sun of dusk, the black-green backscatter of SAR radar, where a ship's wake writes a bright V on a dark sea from orbit, and the Prussian-blue wash of a nautical chart.

The scalloped bright edge of the wedge is the cusp caustic, where the transverse and divergent wave systems meet and reinforce — the reason the wings of a real wake are their own brightest line. The hull itself is kept to a single white highlight; gold on this page is reserved for Kelvin's constant.

the SAR channel — tilted-surface backscatter, the wake as radar sees it U 3 · deep · relief 2.6 · SAR

All palettes are artistic approximations of these observation channels, not measurements.

REFERENCES

Lord Kelvin (Sir William Thomson). "On ship waves." Proceedings of the Institution of Mechanical Engineers, vol.38, no.1, 409-434 (1887). https://doi.org/10.1243/PIME_PROC_1887_038_028_02
T. H. Havelock. "The propagation of groups of waves in dispersive media, with application to waves on water produced by a travelling disturbance." Proceedings of the Royal Society of London A, vol.81, no.549, 398-430 (1908).
M. Rabaud, F. Moisy. "Ship wakes: Kelvin or Mach angle?." Physical Review Letters, vol.110, 214503 (2013).

INTERACTIVE STUDY

The whole theorem fits under one finger. Hold the sea and the hull opens its throttle ×1.8 — in deep water the measured needle shrugs at arcsin(1/3) = 19°28′, and only the depth slider can swing it, through wide open at Fr = 1 to the Mach cone. The wedge angles here are exact — 96 station-keeping waves of the true dispersion — while the water itself is a portrait. It is a deliberately simplified instrument, capped in resolution with a few curated knobs and no export, separate from the full engine used to author the finished works.

SIMPLIFIED INSTRUMENTKELVIN WAKE · 96 STATIONARY WAVES · THE PROTRACTOR

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.