Ursell number

{{short description|Dimensionless number indicating the nonlinearity of long surface gravity waves on a fluid layer.}}

File:Sine wave amplitude.svg

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.{{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 | issue=4 | volume=49 | doi=10.1017/S0305004100028887 | year=1953|bibcode = 1953PCPS...49..685U | s2cid=121889662 }}

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:

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

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.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 surface gravity 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= [https://archive.org/details/mathphyspapers01stokrich/page/n214 197]–229 | url= https://archive.org/details/mathphyspapers01stokrich }}