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:
\( {\displaystyle U={\frac {H}{h}}\left({\frac {\lambda }{h}}\right)^{2}\,=\,{\frac {H\,\lambda ^{2}}{h^{3}}},} \)
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]
Notes
Ursell, F (1953). "The long-wave paradox in the theory of gravity waves". Proceedings of the Cambridge Philosophical Society. 49 (4): 685–694. Bibcode:1953PCPS...49..685U. doi:10.1017/S0305004100028887. S2CID 121889662.
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.
References
Dingemans, M. W. (1997). "Water wave propagation over uneven bottoms". NASA Sti/Recon Technical Report N. Advanced Series on Ocean Engineering. 13: 25769. Bibcode:1985STIN...8525769K. ISBN 978-981-02-0427-3. In 2 parts, 967 pages.
Svendsen, I. A. (2006). Introduction to nearshore hydrodynamics. Advanced Series on Ocean Engineering. Vol. 24. Singapore: World Scientific. ISBN 978-981-256-142-8. 722 pages.
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Dimensionless numbers in fluid mechanics
Archimedes Atwood Bagnold Bejan Biot Bond Brinkman Capillary Cauchy Chandrasekhar Damköhler Darcy Dean Deborah Dukhin Eckert Ekman Eötvös Euler Froude Galilei Graetz Grashof Görtler Hagen Iribarren Kapitza Keulegan–Carpenter Knudsen Laplace Lewis Mach Marangoni Morton Nusselt Ohnesorge Péclet Prandtl
magnetic turbulent Rayleigh Reynolds
magnetic Richardson Roshko Rossby Rouse Schmidt Scruton Sherwood Shields Stanton Stokes Strouhal Stuart Suratman Taylor Ursell Weber Weissenberg Womersley
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