Kelvin Ship-Wave Pattern

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The displacement produced by a ship moving at constant speed produces a Kelvin wave pattern behind it.

Contributed by: Kay Herbert (September 2011)
Open content licensed under CC BY-NC-SA



The Kelvin ship-wave pattern can be calculated from an integral of the form , where and are the surface coordinates of the wave pattern. The integrand is oscillatory with stationary points.

Although Mathematica is able to perform the integration, here it is performed numerically using a midpoint formula to avoid the singularities at the limits. The stationary points are automatically handled with an asymptotic formula derived from analytical solutions at the stationary points.

The result is very robust and calculations can be done in seconds rather than minutes on a core i3 laptop.

For more information, see §36.13 Kelvin’s Ship-Wave Pattern" in [1].


[1] National Instutute of Standards and Technology, "Digital Library of Functions." (Jul 1, 2010)

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