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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 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:
which is, apart from a constant 3 / (32 π^{2}), the ratio of the amplitudes of the second-order to the first-order term in the free surface elevation.^{[2]} The used parameters are:
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 non-linear theory for fairly long waves (λ > 7 h)^{[4]} — like the [5]
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