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Ursell number

File:Sine wave amplitude.svg
Wave characteristics.

In fluid dynamics, the Ursell number indicates the nonlinearity of long surface gravity waves on a fluid layer. This dimensionless parameter is named after Fritz Ursell, who discussed its significance in 1953.[1]

The Ursell number is derived from the Stokes wave expansion, a perturbation series for nonlinear periodic waves, in the long-wave limit of shallow water — when the wavelength is much larger than the water depth. Then the Ursell number U is defined as:

<math>U\, =\, \frac{H}{h} \left(\frac{\lambda}{h}\right)^2\, =\, \frac{H\, \lambda^2}{h^3},</math>

which is, apart from a constant 3 / (32 π2), the ratio of the amplitudes of the second-order to the first-order term in the free surface elevation.[2] The used parameters are:

  • H : the wave height, i.e. the difference between the elevations of the wave crest and trough,
  • h : the mean water depth, and
  • λ : the wavelength, which has to be large compared to the depth, λh.

So the Ursell parameter U is the relative wave height H / h times the relative wavelength λ / h squared.

For long waves (λh) with small Ursell number, U ≪ 32 π2 / 3 ≈ 100,[3] linear wave theory is applicable. Otherwise (and most often) a non-linear theory for fairly long waves (λ > 7 h)[4] — like the Korteweg–de Vries equation or Boussinesq equations — has to be used. The parameter, with different normalisation, was already introduced by George Gabriel Stokes in his historical paper on surface gravity waves of 1847.[5]


  1. ^ Ursell, F (1953). "The long-wave paradox in the theory of gravity waves". Proceedings of the Cambridge Philosophical Society 49 (4): 685–694. Bibcode:1953PCPS...49..685U. doi:10.1017/S0305004100028887. 
  2. ^ Dingemans (1997), Part 1, §2.8.1, pp. 182–184.
  3. ^ This factor is due to the neglected constant in the amplitude ratio of the second-order to first-order terms in the Stokes' wave expansion. See Dingemans (1997), p. 179 & 182.
  4. ^ Dingemans (1997), Part 2, pp. 473 & 516.
  5. ^ Stokes, G. G. (1847). "On the theory of oscillatory waves". Transactions of the Cambridge Philosophical Society 8: 441–455. 
    Reprinted in: Stokes, G. G. (1880). Mathematical and Physical Papers, Volume I. Cambridge University Press. pp. 197–229. 


  • Dingemans, M. W. (1997). Water wave propagation over uneven bottoms. Advanced Series on Ocean Engineering 13. Singapore: World Scientific. ISBN 981-02-0427-2.  In 2 parts, 967 pages.
  • Svendsen, I. A. (2006). Introduction to nearshore hydrodynamics. Advanced Series on Ocean Engineering 24. Singapore: World Scientific. ISBN 981-256-142-0.  722 pages.