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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' 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:
- 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]
Notes
- ^
Ursell, F (1953). "The long-wave
paradox in the theory of gravity waves". Proceedings of the
Cambridge Philosophical Society 49: 685–694.
doi:10.1017/S0305004100028887.
- ^
Dingemans (1997), Part 1, §2.8.1, pp. 182–184.
- ^
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.
- ^
Dingemans (1997), Part 2, pp. 473 & 516.
- ^ Stokes, G. G. (1847). "On the theory of
oscillatory waves". Transactions of the Cambridge Philosophical
Society 8: 441–455.
Reprinted in: Stokes, G. G. (1880). Mathematical and Physical
Papers, Volume I. Cambridge University Press.
pp. 197–229. http://www.archive.org/details/mathphyspapers01stokrich.
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 25 6142
0.
722 pages.