Kelvin ship-wake angle

Partly settled. Some of it is understood, some is still argued about.

Kelvin derived in 1887 that the envelope of gravity waves behind any moving disturbance in deep water should form a wedge of exactly 19.47 degrees, independent of speed — an elegant, purely geometric result of dispersion. For over a century this was treated as essentially exact, but satellite and aerial images of large fast ships and boats seemed to show narrower wakes. In 2013 Rabaud and Moisy showed with real wake images and simulations that above a Froude number of about 0.5 (relative to hull length) the wake angle instead decreases with speed, resembling a Mach cone from the finite size of the hull, with a crossover between the two regimes. Subsequent papers have debated exactly how universal this transition is and how strongly it depends on hull shape and wave drag, so the picture is broadly accepted but still being refined rather than fully closed.

A textbook 'universal constant' angle turns out to secretly depend on how fast you push it, revealing a hidden Mach-cone regime behind ordinary boat wakes.

Explanations that circulated

• Kelvin's 19.47-degree angle is a universal law obeyed by every wake at every speed — false; from analysis of airborne wake images, the observed angle instead narrows roughly as 1/speed for fast ships, more like a Mach cone than a fixed Kelvin angle.

References

• Ship Wakes: Kelvin or Mach Angle?, Marc Rabaud, Frédéric Moisy, Physical Review Letters, 2013