Open Ocean Simulator

Superposition

To first order the ocean surface obeys a linear wave equation, so overlapping wave systems simply add. At every point the elevation is the sum of every component of every train plus the wind sea:

Where crests coincide the surface reinforces — constructive interference. Where a crest meets a trough they cancel. Open the 2D view to watch the sum being taken along the centre line.

Wave trains are focused groups

A real swell is not one sine wave but a band of frequencies. Each train here bundles 7 components spread over ±bandwidth around its peak frequency, with phases arranged so they all line up at one point — a focused wave group, the same trick wave tanks use to make rogue waves to order.

In deep water each frequency travels at its own speed (dispersion, ω² = g·k, so c = g/2πf), so after focusing the components drift apart and the group flattens out — then, because the components are evenly spaced in frequency, they drift back into step every 1/Δf seconds. Wider bandwidth means a sharper, shorter-lived focus. Turn dispersion off and every component of a train shares one speed: the group becomes a rigid profile that never disperses.

The wind sea

Underneath the trains is an ambient wind sea: a JONSWAP spectrum for the chosen wind speed (60 km fetch), realised as 48–144 random-phase components (by render quality) with a Gaussian directional spread about the wind heading. Only waves longer than ~16 m are kept, so the mesh can resolve them: about a fifth of the spectrum survives at 3 m/s, half at 4 m/s and three quarters at 8 m/s, so light winds leave the sea nearly calm. It is part of the sea state, so its energy counts in H_s — a stronger wind raises the bar a rogue has to clear while also adding random crests that sometimes help.

Crossing seas and spread

Trains with different headings cross: their crest lines only fully reinforce where they intersect, and that intersection sweeps across the sea. Crossing wind sea and swell is a documented rogue mechanism and the leading explanation for the Draupner wave (25.6 m in a 12 m sea, 1 January 1995). Spread fans a train's components across a range of headings, making it short-crested: focus then happens at a point rather than along a whole line, so events are rarer but more realistic.

Significant wave height and the rogue criterion

A sea state is characterised by the significant wave height, four times the standard deviation of the surface:

A wave is rogue when its crest-to-trough height, measured between successive zero down-crossings, exceeds twice the significant height:

The detector samples the central 600 × 320 m of the sea on an 8 m grid 30 times a second, including the sea time between samples, and scans every row and column for zero-crossing waves. The beacon marks the crest of the tallest one when it crosses the line. Crests climbing toward the H_s elevation warm up, and glow above it — rogue territory.

What is visual only

Crest steepness adds Gerstner-style horizontal displacement that sharpens crests and flattens troughs; foam appears where the surface folds, and whitecaps where the short wind waves drawn below the mesh's ~16 m steepen past a slope of about 0.3. Splash ripples from clicking the water are decorative packets. None of these enter H_s or the detector; the buoy, however, rides the rendered surface, ripples included.

Why rogues are rare

Two pure sine waves can never exceed 2·H_s. It takes many components conspiring: in still water even a fully focused group on its own peaks around 1.7–1.8 H_s and never crosses the line. The wind sea's random crests riding on a focus can tip a lone group over it now and then, but the big, repeatable events need two groups to refocus at the same place at the same time — which, with periods of 100 s and 136 s at the default settings, happens in irregular bursts a minute or two apart.

Recipe to hunt one: keep the amplitudes comparable, keep bandwidths near 20%, spreads small, and let it run. Watch the H_max/H_s meter approach the red tick.