# Ursell number

﻿
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.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Ursell — is a surname and may refer to:* Fritz Ursell (born 1923), British mathematician, * Harold Ursell (1907 ndash;1969), English mathematician.ee also* Ursell function * Ursell number …   Wikipedia

• Fritz Ursell — Infobox Scientist name = Fritz J. Ursell image width = caption = birth date = 1923 birth place = Düsseldorf, Germany death date = death place = residence = citizenship = nationality = ethnicity = field = Applied mathematics work institution =… …   Wikipedia

• Nusselt number — In heat transfer at a boundary (surface) within a fluid, the Nusselt number is the ratio of convective to conductive heat transfer across (normal to) the boundary. Named after Wilhelm Nusselt, it is a dimensionless number. The conductive… …   Wikipedia

• Rayleigh number — In fluid mechanics, the Rayleigh number for a fluid is a dimensionless number associated with buoyancy driven flow (also known as free convection or natural convection). When the Rayleigh number is below the critical value for that fluid, heat… …   Wikipedia

• Magnetic Reynolds number — The Magnetic Reynolds number is a dimensionless group that occurs in magnetohydrodynamics. It gives an estimate of the effects of magnetic advection to magnetic diffusion, and is typically defined by: where U is a typical velocity scale of the… …   Wikipedia

• Morton number — For Morton number in number theory, see Morton number (number theory). In fluid dynamics, the Morton number (Mo) is a dimensionless number used together with the Eötvös number to characterize the shape of bubbles or drops moving in a surrounding… …   Wikipedia

• Keulegan–Carpenter number — The Keulegan–Carpenter number is important for the computation of the wave forces on offshore platforms. In fluid dynamics, the Keulegan–Carpenter number, also called the period number, is a dimensionless quantity describing the relative… …   Wikipedia

• Ohnesorge number — The Ohnesorge number, Oh, is a dimensionless number that relates the viscous forces to inertial and surface tension forces. It is defined as: Where μ is the liquid viscosity ρ is the liquid density σ is the surface tension L is the characteristic …   Wikipedia

• Dean number — The Dean number is a dimensionless group in fluid mechanics, which occurs in the study of flow in curved pipes and channels. It is named after the British scientist W. R. Dean, who studied such flows in the 1920s (Dean, 1927, 1928). Definition… …   Wikipedia

• Deborah number — The Deborah number is a dimensionless number, often used in rheology to characterize the fluidity of materials under specific flow conditions. It was originally proposed by Markus Reiner, a professor at Technion in Israel, inspired by a verse in… …   Wikipedia

### Share the article and excerpts

Do a right-click on the link above