厄塞爾數

厄塞爾數（Ursell number）是流體動力學中的無量綱，表示流體層中長的表面重力波的非線性程度，得名自1953年發現此重要性的弗里茨·厄塞爾（英語：Fritz Ursell）^[1]。

厄塞爾數是推導自史托克波（英語：Stokes wave），一個針對非線性週期波的攝動序列，在淺水（英語：waves and shallow water）的長波極限－其波長遠大於水深時，Ursell數U可以定義如下：

U\,=\,{\frac {H}{h}}\left({\frac {\lambda }{h}}\right)^{2}\,=\,{\frac {H\,\lambda ^{2}}{h^{3}}},

若不考慮常數3 / (32 π²)的話，上述公式就是自由表面提昇振幅中，二次項和一次項的比例，^[2] 有用到的參數有

H：波高（英語：Wave height），也就是波峰和波谷之間的高度差。
h：平均水深
λ：波長，需遠大於深度，也就是λ ≫ h.

因此厄塞爾數U是相對波高H / h乘以相對波長的平方。

針對厄塞爾數小（U ≪ 32 π² / 3 ≈ 100）的長波（λ ≫ h）^[3]，可以用線性的波理論求解。否則（多半是通常）若針對比較長的波（λ > 7 h）^[4]，需使用像KdV方程或博欣內斯克方程等非線性的理論。此參數（經過不同的正規化）已由喬治·斯托克斯寫在他1847年的表面重力波論文中^[5]。

腳註

^ Ursell, F. The long-wave paradox in the theory of gravity waves. Proceedings of the Cambridge Philosophical Society. 1953, 49 (4): 685–694. Bibcode:1953PCPS...49..685U. doi:10.1017/S0305004100028887.
^ Dingemans (1997), Part 1, §2.8.1, pp. 182–184.
^ 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.
^ Dingemans (1997), Part 2, pp. 473 & 516.
^ Stokes, G. G. On the theory of oscillatory waves. Transactions of the Cambridge Philosophical Society. 1847, 8: 441–455.
Reprinted in: Stokes, G. G. Mathematical and Physical Papers, Volume I. Cambridge University Press. 1880: 197–229.

參考資料

Dingemans, M. W. Water wave propagation over uneven bottoms. Advanced Series on Ocean Engineering 13. Singapore: World Scientific. 1997. ISBN 981-02-0427-2. In 2 parts, 967 pages.
Svendsen, I. A. Introduction to nearshore hydrodynamics. Advanced Series on Ocean Engineering 24. Singapore: World Scientific. 2006. ISBN 981-256-142-0. 722 pages.

[1] Ursell, F. The long-wave paradox in the theory of gravity waves. Proceedings of the Cambridge Philosophical Society. 1953, 49 (4): 685–694. Bibcode:1953PCPS...49..685U. doi:10.1017/S0305004100028887.

[2] Dingemans (1997), Part 1, §2.8.1, pp. 182–184.

[3] 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.

[4] Dingemans (1997), Part 2, pp. 473 & 516.

[Stokes1847-5] Stokes, G. G. On the theory of oscillatory waves. Transactions of the Cambridge Philosophical Society. 1847, 8: 441–455.
Reprinted in: Stokes, G. G. Mathematical and Physical Papers, Volume I. Cambridge University Press. 1880: 197–229.

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