Kelvin Wake

A duck, a boat and a supertanker leave the same V-shaped wake. In 1887 Lord Kelvin showed why: in deep water, waves of different lengths travel at different speeds, and only the waves whose crests keep pace with the ship stay behind it. A wave that travels at the angle \theta to the ship's path keeps pace when its speed equals U\cos\theta, where U is the ship's speed. In deep water that fixes its wavenumber to k_0/\cos^2\theta with k_0 = g/U^2.

This document adds up these steady waves: \eta(x, y) is the sum of 300 of them, one for each angle \theta(j) between -1.35 and 1.35 radians, with the weight w(j) fading the terms out at the ends of the range. Behind the ship the waves reinforce each other where their phase changes slowest with the angle, and they cancel everywhere else. The result shows the two families of the Kelvin pattern: the curved transverse waves across the ship's path, and the divergent waves that form the arms of the V.

The ship's speed U animates from 0.6 to 1.2. The waves grow four times longer, because the wavelength is 2\pi U^2/g, but the wake always stays inside the dashed lines y = \pm x/(2\sqrt{2}): a half-angle of \arcsin(1/3) \approx 19.47°, whatever the speed. (Fast boats in the real world show narrower wakes, because the hull size then matters; this model has a point source.)

After a Desmos graph by Fabrice Neyret.