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' 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:

\[U\, =\, \frac{H}{h} \left(\frac{\lambda}{h}\right)^2\, =\, \frac{H\, \lambda^2}{h^3},\]

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]

Notes

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  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.
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References

  • Script error In 2 parts, 967 pages.
  • Script error 722 pages.
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