# Ursell Number

Ursell number, abbreviated as U, a dimensionless number, indicates the nonlinearity of long surface gravity waves on a fluid layer. It used in fluid dynamics to quantify the nonlinearity of wave interactions in a wave field. The Ursell number provides information about the importance of wave-wave interactions and is commonly used in the study of wave phenomena in various natural and engineered systems. This is used to determine the degree of nonlinearity in a wave system, particularly when the amplitude of the waves is not negligible compared to the wavelength. It's often used in the context of water waves, ocean waves, and other wave phenomena in fluid environments.

Key points about the Ursell number:

- A low Ursell number ($Ur≪1$) indicates weak nonlinear interactions between waves, meaning that wave-wave interactions have minimal effect on the wave field.
- A high Ursell number ($Ur≫1$) indicates strong nonlinear interactions between waves, implying that wave amplitudes are significant and wave-wave interactions play a substantial role.
- The Ursell number helps researchers understand the potential for wave steepening, wave breaking, and the formation of wave patterns due to nonlinear interactions.
- It's relevant in oceanography, coastal engineering, and the study of various fluid wave phenomena.

The Ursell number is just one of several dimensionless parameters used to characterize wave behavior. Its use provides insights into the complex behavior of waves in nonlinear systems and helps researchers and engineers predict the effects of wave interactions in various practical applications.

## Ursell Number formula |
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\(\large{ U = \frac {H} {h} \; \left( \frac {\lambda} {h} \right)^2 = \frac {H \; \lambda^2} {h^3} }\) | ||

Symbol |
English |
Metric |

\(\large{ U }\) = Ursell number | \(\large{dimensionless}\) | |

\(\large{ h }\) = mean water depth | \(\large{ft}\) | \(\large{m}\) |

\(\large{ H }\) = the wave height, the difference between the elevations of the wave crest and trough | \(\large{ft}\) | \(\large{m}\) |

\(\large{ \lambda }\) (Greek symbol lambda) = the wavelength, which has to be large compared to the depth, \(\large{\lambda \gg h}\) | \(\large{ft}\) | \(\large{m}\) |