# Ursell number

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:

which is, apart from a constant 3 / (32 π^{2}), 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,*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^{2} / 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

- 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. - Dingemans (1997), Part 1, §2.8.1, pp. 182–184.
- 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.
- Dingemans (1997), Part 2, pp. 473 & 516.
- 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".
*Nasa Sti/recon Technical Report N*. Advanced Series on Ocean Engineering.**13**: 25769. Bibcode:1985STIN...8525769K. ISBN 978-981-02-0427-3. In 2 parts, 967 pages. - Svendsen, I. A. (2006).
*Introduction to nearshore hydrodynamics*. Advanced Series on Ocean Engineering.**24**. Singapore: World Scientific. ISBN 978-981-256-142-8. 722 pages.