Ursell number

This dimensionless parameter is named after Fritz Ursell, who discussed its significance in 1953.

[1] 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: 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.

[2] The used parameters are: So the Ursell parameter U is the relative wave height H / h times the relative wavelength λ / h squared.

The parameter, with different normalisation, was already introduced by George Gabriel Stokes in his historical paper on surface gravity waves of 1847.