# Ursell number

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Ursell number

In fluid dynamics, the Ursell number indicates the nonlinearity of long gravity surface-waves on a fluid layer. This dimensionless parameter is named after Fritz Ursell, who discussed its significance in 1953. [cite journal | first=F | last=Ursell | title=The long-wave paradox in the theory of gravity waves | journal=Proceedings of the Cambridge Philosophical Society | pages=685–694 | volume=49]

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:

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

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. [Dingemans (1997), Part 1, §2.8.1, pp. 182–184.] 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 π2 / 3 ≈ 100, [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.] linear wave theory is applicable. Otherwise (and most often) a non-linear theory for fairly long waves ("λ" > 7 "h") [Dingemans (1997), Part 2, pp. 473 & 516.] — 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 gravity surface waves of 1847.cite journal | first= G. G. | last=Stokes | year= 1847 | title= On the theory of oscillatory waves | journal= Transactions of the Cambridge Philosophical Society | volume= 8 | pages= 441–455
Reprinted in: cite book | first= G. G. | last=Stokes | year= 1880 | title= Mathematical and Physical Papers, Volume I | publisher= Cambridge University Press | pages= 197–229 | url= http://www.archive.org/details/mathphyspapers01stokrich ]

Notes

References

* In 2 parts, 967 pages.
* 722 pages.

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