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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.

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:

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

which is, apart from a constant 3 / (32 π), the ratio of the amplitudes of the second-order to the first-order term in the free surface elevation. 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, λ ≫ h.

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

For long waves (λ ≫ h) with small Ursell number, U ≪ 32 π / 3 ≈ 100, linear wave theory is applicable. Otherwise (and most often) a non-linear theory for fairly long waves (λ > 7 h) – 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.

Notes

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. S2CID 121889662.
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.

References

• Dingemans, M. W. (1997). Water wave propagation over uneven bottoms. Advanced Series on Ocean Engineering. Vol. 13. World Scientific. p. 25769. ISBN 978-981-02-0427-3. In 2 parts, 967 pages.
• Svendsen, I. A. (2006). Introduction to nearshore hydrodynamics. Advanced Series on Ocean Engineering. Vol. 24. Singapore: World Scientific. ISBN 978-981-256-142-8. 722 pages.

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