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.

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:

:$U\, =\, \frac \left(\frac\right)^2\, =\, \frac,$

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.
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, 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.
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Notes

References

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

{{NonDimFluMech

Dimensionless numbers of fluid mechanics
Fluid dynamics
Water waves