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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Whitham equation ： ウィキペディア英語版 Whitham equation In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves: :$$ \frac + \alpha \eta \frac + \int_^ K(x-\xi)\, \frac\, \text\xi = 0. This integro-differential equation for the oscillatory variable ''η''(''x'',''t'') is named after Gerald Whitham who introduced it as a model to study breaking of non-linear dispersive water waves in 1967. For a certain choice of the kernel ''K''(''x'' − ''ξ'') it becomes the Fornberg–Whitham equation. ==Water waves== * For surface gravity waves, the phase speed ''c''(''k'') as a function of wavenumber ''k'' is taken as:〔 ::$$ c_\text(k) = \sqrt\, \tanh(kh)}, while $\backslash alpha\_\backslash text\; =\; \backslash frac\; \backslash sqrt\},$ :with ''g'' the gravitational acceleration and ''h'' the mean water depth. The associated kernel ''K''ww(''s'') is:〔 ::$$ K_\text(s) = \frac \int_^ c_\text(k)\, \text^\, \textk. * The Korteweg–de Vries equation emerges when retaining the first two terms of a series expansion of ''c''ww(''k'') for long waves with :〔 ::$$ c_\text(k) = \sqrt \left( 1 - \frac k^2 h^2 \right), $$ K_\text(s) = \sqrt \left( \delta(s) + \frac h^2\, \delta^(s) \right), $$ \alpha_\text = \frac \sqrt}, :with ''δ''(''s'') the Dirac delta function. * Bengt Fornberg and Gerald Whitham studied the kernel ''K''fw(''s'') – non-dimensionalised using ''g'' and ''h'': ::$K\_\backslash text(s)\; =\; \backslash frac12\; \backslash nu\; \backslash text^$ and $c\_\backslash text\; =\; \backslash frac,$ with $\backslash alpha\_\backslash text=\backslash frac32.$ :The resulting integro-differential equation can be reduced to the partial differential equation known as the Fornberg–Whitham equation:〔 ::$$ \left( \frac - \nu^2 \right) \left( \frac + \frac32\, \eta\, \frac \right) + \frac = 0. :This equation is shown to allow for peakon solutions – as a model for waves of limiting height – as well as the occurrence of wave breaking (shock waves, absent in e.g. solutions of the Korteweg–de Vries equation).〔〔 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「Whitham equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.