- Ursell number
In

fluid dynamics , the**Ursell number**indicates thenonlinearity of long gravity surface-waves on afluid layer. Thisdimensionless parameter is named afterFritz Ursell , who discussed its significance in 1953. [*cite journal | first=F | last=Ursell | title=The long-wave paradox in the theory of gravity waves | journal=Proceedings of the Cambridge Philosophical Society | pages=685–694 | volume=49*]The Ursell number is derived from the Stokes' 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::$U,\; =,\; frac\{H\}\{h\}\; left(frac\{lambda\}\{h\}\; ight)^2,\; =,\; frac\{H,\; lambda^2\}\{h^3\},$

which is, apart from a constant 3 / (32 π

^{2}), the ratio of theamplitude s of the second-order to the first-order term in thefree surface elevation. [*Dingemans (1997), Part 1, §2.8.1, pp. 182–184.*] The used parameters are:

* "H" : thewave height , "i.e." the difference between the elevations of the wave crest and trough,

* "h" : the mean water depth, and

* "λ" : thewavelength , 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, [*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.*] linear wave theory is applicable. Otherwise (and most often) a non-linear theory for fairly long waves ("λ" > 7 "h") [*Dingemans (1997), Part 2, pp. 473 & 516.*] — like theKorteweg–de Vries equation or Boussinesq equations — has to be used.The parameter, with differentnormalisation , was already introduced byGeorge Gabriel Stokes in his historical paper on gravity surface waves of 1847.cite journal | first= G. G. | last=Stokes | year= 1847 | title= On the theory of oscillatory waves | journal= Transactions of the Cambridge Philosophical Society | volume= 8 | pages= 441–455

Reprinted in: cite book | first= G. G. | last=Stokes | year= 1880 | title= Mathematical and Physical Papers, Volume I | publisher= Cambridge University Press | pages= 197–229 | url= http://www.archive.org/details/mathphyspapers01stokrich ]**Notes****References*** In 2 parts, 967 pages.

* 722 pages.

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