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