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.^{[1]}
The Ursell number is derived from the Stokes wave expansion, a perturbation series for nonlinear periodic waves, in the longwave limit of shallow water — when the wavelength is much larger than the water depth. Then the Ursell number U is defined as:
 U\, =\, \frac{H}{h} \left(\frac{\lambda}{h}\right)^2\, =\, \frac{H\, \lambda^2}{h^3},
which is, apart from a constant 3 / (32 π^{2}), the ratio of the amplitudes of the secondorder to the firstorder 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 π^{2} / 3 ≈ 100,^{[3]} linear wave theory is applicable. Otherwise (and most often) a nonlinear theory for fairly long waves (λ > 7 h)^{[4]} — like the [5]
Notes
 ^ Ursell, F (1953). "The longwave paradox in the theory of gravity waves". Proceedings of the Cambridge Philosophical Society 49 (4): 685–694.
 ^ Dingemans (1997), Part 1, §2.8.1, pp. 182–184.
 ^ This factor is due to the neglected constant in the amplitude ratio of the secondorder to firstorder 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. Advanced Series on Ocean Engineering 13. Singapore: World Scientific. In 2 parts, 967 pages.
 Svendsen, I. A. (2006). Introduction to nearshore hydrodynamics. Advanced Series on Ocean Engineering 24. Singapore: World Scientific. 722 pages.