Every crest is born at the back and dies at the front.

A wave travelling across deep water moves at one speed. The crests inside it move at another — twice as fast. Watch one long enough and you see it rise at the wave's trailing edge, march forward through it, and flatten out off the front.

What you're seeing

A single pure wave would be an endless train of identical ripples, going on forever in both directions. A real wave is a burst — a patch of ripples with calm water either side. To build one you have to add together many pure waves of slightly different wavelengths, which cancel everywhere except in one small region. That patch is the packet, drawn here inside its dim outline.

Now the question that matters: how fast does this thing travel? It turns out there are two answers, and they are different numbers. The packet — the patch of disturbance, the thing carrying the energy — moves at one speed. Any single crest inside it moves at another. The green triangle tracks the packet. The amber dot rides one particular crest. Watch them separate.

On real deep water the crests go exactly twice as fast as the packet. So a crest appears out of flat water at the packet's trailing edge, grows as it travels forward through the packet, peaks in the middle, shrinks, and dies as it runs out of the front. Then another does the same. The shape moves; the ripples inside it are passing through.

The dial is the reason. It sets how much a wave's frequency depends on its wavelength — the dispersion: how much the water sorts waves by size, letting some outrun others. Set it to exactly 1.00 and there is no sorting at all. Every component travels at the same speed, so the packet and its crests move together and the dot holds its place beside the triangle forever. That is light in a vacuum, or sound in air.

Push the dial above 1.00 and the drift reverses. The packet now outruns its own crests, and each one slides backwards through it, born at the front and dying at the back. This is what fine capillary ripples do. Press ride along with the packet to sit in the packet's own frame, where the only motion left is the crests streaming through.

The rule, exactly. The packet is a sum of cosines with wavenumbers spread around k₀, under the dispersion relation ω(k) = A k^p The packet travels at the group velocity and the crests at the phase velocity: v_g = dω/dk = p A k^p−1, v_p = ω/k = A k^p−1 so their ratio is exactly p. Deep-water gravity waves have ω = √(gk), giving p = ½ and the familiar factor of two. Verified in node (improve/verify/93-group-velocity.js, 23 checks). Both speeds are measured from the simulated field, never assumed: the packet by its intensity-weighted centroid, a crest by parabolic refinement of a local maximum. Across five regimes both match theory to better than 0.06%, and the measured ratio reproduces p itself. Four negative controls. At p = 1 the measured drift of a crest relative to the packet is 2×10⁻⁵ over eight time units — essentially zero. The drift is required to flip sign across p = 1, not merely to be non-zero. The naive prediction that a packet travels at its crests' speed is shown to be wrong by 50% at p = ½. And the measurement error falls monotonically from 0.085% to 0.005% as the packet is made more monochromatic, confirming finite bandwidth as the error source rather than a systematic bias.

← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 93.