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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. 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 π2), the ratio of the amplitudes of the second-order to the first-order term in the free surface elevation.〔Dingemans (1997), Part 1, §2.8.1, pp. 182–184.〕 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 π2 / 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 the Korteweg–de Vries equation or Boussinesq equations — has to be used. The parameter, with different normalisation, was already introduced by George Gabriel Stokes in his historical paper on surface gravity waves of 1847.〔 Reprinted in: 〕 == Notes == 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ursell number」の詳細全文を読む スポンサード リンク
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