Kelvin ship-wake · 19.47°

Why does every boat — and every duck — drag the same V-shaped wake behind it, and why doesn't its angle change with speed?

▶ Run the simulationSee the measured result

Units: degrees — half-angle α of the Kelvin deep-water wake cusp to the sailing track; sin α = 1/3, full opening 2α = 38.94° (Kelvin 1887; Lighthill §3.10)

How the lab tests it

Deep-water gravity waves are DISPERSIVE: ω=√(gk), so energy travels at half the phase speed (c_g=½c_p). A wave component whose crests make angle θ with the sailing line phase-locks to a disturbance at speed V only if c_p=V·cosθ; its energy packet then sits at bearing β(θ)=atan[½cosθ sinθ/(1−½cos²θ)] off the track. Build the packet cloud from the dispersion relation ONLY, and read the wake edge as the MAXIMUM bearing over θ (a caustic, where the packet density diverges). Even the ½ that fixes the result is recovered by central-differencing ω(k), not coded — and the shown speed is cycled to re-measure the angle live.

What it checks

the Kelvin half-angle α = arcsin(1/3) = 19.47° (full wake 38.94°), recovered to ~1e-8 as the envelope maximum with no 19.47°, 1/3 or arcsin coded — and, the whole point, INDEPENDENT of ship speed: the packet-cloud coordinates scale linearly with V (a fast ship's wake is physically larger) yet the caustic edge sits at the same 19.47° across a 50× speed range (spread <1e-5°), because a slow duck and a fast ship both radiate the same self-similar gravity-wave spectrum. Dispersion is what fixes it: run the SAME construction with non-dispersive waves (c_g=c_p) and the wake opens all the way to ~90° — no finite caustic — and sweeping the group/phase ratio from ½→1 walks the angle from 19.47° up toward 90°. Contrast ?world=doppler, whose Mach cone sin μ=1/M NARROWS as the source speeds up; the Kelvin angle is the dispersive sibling that stays put. Kelvin (W. Thomson), 1887.

Kelvin ship-wake angle, wake wavelength & deep-water calculator

Every boat, duck and dropped pebble drags the same V behind it, and the V is the same width whether the boat crawls or races. That is the fact this page computes. A steadily moving disturbance on deep water radiates gravity waves, and those waves are dispersive: long ones travel faster than short ones, and a packet of them carries its energy at HALF the speed its crests move. Only the components whose crests keep pace with the boat survive, and their energy packets — each dawdling at half the speed of the crest that made it — crowd against a limiting bearing. That bearing is the wake edge. Write the group-to-phase ratio as f and this page's whole subject is one line: sin α = f/(2−f). At f = ½ it returns exactly a third, and a third is 19.47°. The boat's speed is nowhere in that sentence, which is why the angle is universal: a duck and a destroyer wear the same V, and only the WAVELENGTH tells them apart — it scales as V², so a destroyer's wake is enormously bigger and not one degree wider. This page prints the constant every way it can be had without a library: as the ratio f/(2−f), which at f = ½ is bit-identical to ⅓; as tan α = f/(2√(1−f)), one correctly-rounded square root, which at f = ½ lands one ulp from √2/4 and the page says which of the two is nearer the truth; and as the caustic's own generating angle θ* = 45° − α/2, a fact this lab had not written down, cross-checked against an independent arccosine that agrees to the last bit rather than by construction. Only the DEGREES need an approximated function, and that is stated where it belongs rather than buried: a hand-built arctangent was measured against the engine's arcsine before being rejected — it is ten times worse at the one value this page exists to print, and worse on twenty-one thousand of forty thousand inputs — so the library call is kept, quarantined by name with a pinned count, and its cost disclosed. Two directions are about the water rather than the angle. The wake's wavelength directly astern is 2πV²/g and the cusp's own wavelength is exactly two thirds of it; both are refused rather than rounded once the ship is fast enough that V² leaves the representable range. And the deep-water assumption is itself checked: gravity waves only behave this way when the depth beats half a wavelength, and a boat in water shallower than that stops making a Kelvin wake at all and starts making a Mach cone — which is the same rival this lab already falsified, f = 1, arriving as real physics rather than as a strawman. At f = 1 the bearing collapses to the exact identity β = 90° − θ, which has no interior maximum at all, and that is why the lab's own rival run returned 89.50°: not a measurement of anything, but the smallest angle its θ grid contained, read back. The last direction scores this lab's recovered numbers against the closed form and revises none of them.