#jsDisabledContent { display:none; } My Account | Register | Help

# Ursell number

Article Id: WHEBN0019114012
Reproduction Date:

 Title: Ursell number Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Ursell number

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:

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 [5]

## 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.
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

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.