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Watching ducks on a pond one may wonder how these beautiful wake patterns arise. William Thomson (the famous physicist Lord Kelvin) found already in 1887 the remarkable fact that the angle at which the wake fans out is always the same, regardless whether the wake is caused by a duck, a boat, or a ship, as long as the water is deep as compared with the lengths of the generated waves [1].

Wakes of ducks on a pond |
Wakes of two boats on Avon Gorge. Photo: Arpingstone, source |

While the mathematical treatment of water waves is difficult, see the Wikipedia article, some results are easy to obtain, and in particular the shape of the wake can be explained using only elementary maths [2].

Whereas sound waves propagate with the same velocity irrespective of their length, with the well known effect of shock waves when flying objects exceed sound velocity, in deep water the velocity of the waves depends on the wavelength. The longer the waves, the faster they propagate. This can be found from a simple dimensional analysis.

What are the quantities on which the velocity of the waves may depend? Clearly the wavelength λ, the depth of the water H, the earths gravitation g, the density of the water ρ, the surface tension σ, or what else? The surface tension affects only very short waves, here we may ignore it. If the water is deep, so to say infinitely deep, the depth cannot occur in an expression for the wave velocity.

If we have an equation like

v = something,

then this “something” must have the dimension of a velocity, i.e. length divided by time. The square root of wavelength times gravitational acceleration has this dimension, thus we conclude that the velocity is proportional to this root:

v ∝ √gλ .

(The exact expression for the phase velocity is
v_{φ} = √gλ ⁄ (2π) .)

The phase velocity is, roughly speaking, the velocity of the individual wave crests. A wave train of finite length, however, moves with the group velocity which in this case is only half the phase velocity, v_{g} = v_{φ}/2. (This follows from the above equation and the general expression for the group velocity, see e.g. this Wikipedia article.) If we look at a group of waves, following a particular crest, we see that it moves forward faster than the whole bunch, gets weaker as it approaches its front, and vanishes. But from behind, a new crest comes up, gets stronger, and weaker again as it approaches the leading edge …

The phase velocity of gravity waves depends on the wavelength and the group velocity is half the phase velocity, that is all what is necessary to understand the gross features of the wakes.

Strongly powered fast boats apparently arouse narrower wakes. This has been discussed recently by Rabaud and Moisy [3], see also [4], [5] and is due to the fact that waves longer than the boat's hull are excited the less the longer they are.

[1] William Thomson (1887): "On ship waves", Institution of Mechanical Engineers, Proceedings, 38:409–434 pp. 641–649.

[2] Frank S. Crawford: Elementary derivation of the wake pattern of a boat. Am. J. Phys. 52, 782 (1984); (abstract).

[3] Marc Rabaud and Frédéric Moisy: Ship wakes: Kelvin or Mach angle? arXiv.org > physics > arXiv:1304.2653 or PRL 110, 214503 (2013)

[4] Lord Kelvin Wipes Out on Speed Boat Wakes? By Adrian Cho, May 9, 2013 link

[5] Reawakening the Kelvin wake. By Hamish Johnston link