Dimensionless number indicating the nonlinearity of long surface gravity waves on a fluid layer.
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:
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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:
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") – 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.
Notes.
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