Influid dynamics, theUrsell number indicates thenonlinearity of longsurface gravity waves on afluid layer. Thisdimensionless parameter is named afterFritz Ursell, who discussed its significance in 1953.^[1]

The Ursell number is derived from theStokes wave expansion, aperturbation series for nonlinearperiodic waves, in the long-wavelimit ofshallow water – when thewavelength is much larger than the water depth. Then the Ursell numberU 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 theamplitudes of the second-order to the first-order term in thefree surface elevation.^[2]The used parameters are:

• H : thewave height,i.e. the difference between the elevations of the wavecrest andtrough,
• h : the mean water depth, and
• λ : the wavelength, which has to be large compared to the depth,λ ≫h.

So the Ursell parameterU is the relative wave heightH /h times the relative wavelengthλ /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 theKorteweg–de Vries equation orBoussinesq equations – has to be used.The parameter, with different normalisation, was already introduced byGeorge Gabriel Stokes in his historical paper on surface gravity waves of 1847.^[5]

• Dingemans, M. W. (1997).Water wave propagation over uneven bottoms. Advanced Series on Ocean Engineering. Vol. 13. World Scientific. p. 25769.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.

• Dimensionless numbers of fluid mechanics
• Fluid dynamics
• Water waves

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