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 â€" 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 Ï€²), 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 Ï€² / 3 ≈ 100, linear wave theory is applicable. Otherwise (and most often) a non-linear theory for fairly long waves (λ > 7 h) â€" like the Kortewegâ€"de Vries equation or Boussinesq equations â€" 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

References

• Dingemans, M. W. (1997). Water wave propagation over uneven bottoms. Advanced Series on Ocean Engineering 13. Singapore: World Scientific. ISBN 981-02-0427-2.  In 2 parts, 967 pages.
• Svendsen, I. A. (2006). Introduction to nearshore hydrodynamics. Advanced Series on Ocean Engineering 24. Singapore: World Scientific. ISBN 981-256-142-0.  722 pages.