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Ship wakes: Kelvin or Mach angle?

M. Rabaud, F. Moisy

The V-shaped wakes behind objects moving on calm water is a fascinating wave phenomenon with important practical implications for the drag force on ships and for bank erosion along navigable waterways.

The wake pattern was first explained by Lord Kelvin, who by recognizing the dependence of the phase speed cφ of surface gravity waves on their wavelength (dispersion) predicted that the wake half-angle α = arcsin(1/3) = 19.47o should be independent of the object's velocity U.

However, Kelvin's analysis is called into question by numerous observations of wakes significantly narrower than he predicted, which have been rationalized by invoking finite-depth effects, nonlinear resonances or solitons, unsteady forcing, and visualisation biases.

Analysing a set of airborne images taken from the Google Earth database, we show that ship wakes undergo a transition from the classical Kelvin regime at low speeds to a previously unnoticed high-speed regime α = 1/U that resembles the Mach cone prediction α = arcsin(cφ/U) for supersonic airplanes.

Airborne images of ship wakes taken from the Google Earth database. (a), Cargo ship near Antwerpen, with α = 20o and Froude number Fr = 0.15. (b), Speed boat near Toronto, with α = 9o and Fr = 1.03. For each image, the wake angle α is defined from the slope of the line going through the brightest points resulting from sun glitter or whitecaps (yellow dotted line), which trace the maximum amplitude of the wake.

Log-log plot of the wake angle α as a function of the hull Froude number Fr = U / (gL)1/2. Red circles: angles measured from the 37 airborne images of the dataset. Blue line: model. Blue dotted line: asymptotic Mach law. Yellow squares: numerical simulations.

Capillary effects

For objects of small size, surface tension effects become important. We have performed experiments in the swimming-pool of Orsay and in a small-scale towing tank at laboratory FAST using surface-piercing vertical cylinders of various diameters D translated at constant velocity. We find that the angle of maximum wave amplitude still decreases as 1/U, with an additional dependence with respect to the Bond number Bo = Dc, where λc = 16 mm is the capillary length.

Two experiments showing the wake behind a vertical cylinder translated at constant velocity in deep water. Download movie: Low Res | High Res.


  • Mach-like capillary-gravity wakes
    F. Moisy and M. Rabaud, Phys. Rev. E 90, 023009 (2014). Editor's suggestion
    [Abstract | PDF]
  • Scaling of far-field wake angle of non-axisymmetric pressure disturbance
    F. Moisy and M. Rabaud, Phys. Rev. E 89, 063004 (2014).
    [Abstract | PDF]
  • Narrow ship wakes and wave drag for planing hulls
    M. Rabaud and F. Moisy, Ocean Eng. 90, 34 (2014).
    [Abstract | PDF]
  • Du neuf dans les sillages
    M. Rabaud, F. Moisy, Reflets de la Physique 39 10-13 (2014).
    [Abstract | PDF]
  • Ship wakes: Kelvin or Mach angle?
    M. Rabaud and F. Moisy, Phys. Rev. Lett. 110, 214503 (2013). Editor's suggestion
    [Abstract | PDF | Suppl. Material]

Research highlights

Last modification: January 06 2015, 02:00:59.