Ursell number explained

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.^[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:

U = (H)/(\left( h) (λ)/(h) \right) ^{2} = (H λ ^{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.^[2] 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,^[3] linear wave theory is applicable. Otherwise (and most often) a non-linear theory for fairly long waves (λ > 7 h)^[4] – 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.^[5]

References

• Water wave propagation over uneven bottoms . World Scientific . M. W. . Dingemans . 1997 . Advanced Series on Ocean Engineering . 13 . 25769 . 978-981-02-0427-3 . In 2 parts, 967 pages.
• Book: Svendsen, I. A. . Introduction to nearshore hydrodynamics . 2006 . Advanced Series on Ocean Engineering . 24 . World Scientific . Singapore . 978-981-256-142-8 . 722 pages.

Notes and References

1. F . Ursell . The long-wave paradox in the theory of gravity waves . Proceedings of the Cambridge Philosophical Society . 685–694 . 4 . 49 . 10.1017/S0305004100028887 . 1953. 1953PCPS...49..685U . 121889662 .
2. Dingemans (1997), Part 1, §2.8.1, pp. 182–184.
3. 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.
4. Dingemans (1997), Part 2, pp. 473 & 516.
5. G. G. . Stokes . 1847 . On the theory of oscillatory waves . Transactions of the Cambridge Philosophical Society . 8 . 441–455 .
   Reprinted in: Book: Stokes, G. G. . 1880 . Mathematical and Physical Papers, Volume I . Cambridge University Press . 197–229 .