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In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by John Scott Russell of the wave of translation (also known as solitary wave or soliton). The 1872 paper of Boussinesq introduces the equations now known as the Boussinesq equations.〔This paper (Boussinesq, 1872) starts with: ''"Tous les ingénieurs connaissent les belles expériences de J. Scott Russell et M. Basin sur la production et la propagation des ondes solitaires"'' (''"All engineers know the beautiful experiments of J. Scott Russell and M. Basin on the generation and propagation of solitary waves"'').〕 The Boussinesq approximation for water waves takes into account the vertical structure of the horizontal and vertical flow velocity. This results in non-linear partial differential equations, called Boussinesq-type equations, which incorporate frequency dispersion (as opposite to the shallow water equations, which are not frequency-dispersive). In coastal engineering, Boussinesq-type equations are frequently used in computer models for the simulation of water waves in shallow seas and harbours. While the Boussinesq approximation is applicable to fairly long waves – that is, when the wavelength is large compared to the water depth – the Stokes expansion is more appropriate for short waves (when the wavelength is of the same order as the water depth, or shorter). ==Boussinesq approximation== The essential idea in the Boussinesq approximation is the elimination of the vertical coordinate from the flow equations, while retaining some of the influences of the vertical structure of the flow under water waves. This is useful because the waves propagate in the horizontal plane and have a different (not wave-like) behaviour in the vertical direction. Often, as in Boussinesq's case, the interest is primarily in the wave propagation. This elimination of the vertical coordinate was first done by Joseph Boussinesq in 1871, to construct an approximate solution for the solitary wave (or wave of translation). Subsequently, in 1872, Boussinesq derived the equations known nowadays as the Boussinesq equations. The steps in the Boussinesq approximation are: *a Taylor expansion is made of the horizontal and vertical flow velocity (or velocity potential) around a certain elevation, *this Taylor expansion is truncated to a finite number of terms, *the conservation of mass (see continuity equation) for an incompressible flow and the zero-curl condition for an irrotational flow are used, to replace vertical partial derivatives of quantities in the Taylor expansion with horizontal partial derivatives. Thereafter, the Boussinesq approximation is applied to the remaining flow equations, in order to eliminate the dependence on the vertical coordinate. As a result, the resulting partial differential equations are in terms of functions of the horizontal coordinates (and time). As an example, consider potential flow over a horizontal bed in the (''x,z'') plane, with ''x'' the horizontal and ''z'' the vertical coordinate. The bed is located at , where ''h'' is the mean water depth. A Taylor expansion is made of the velocity potential ''φ(x,z,t)'' around the bed level :〔Dingemans (1997), p. 477.〕 : where ''φb(x,t)'' is the velocity potential at the bed. Invoking Laplace's equation for ''φ'', as valid for incompressible flow, gives: : since the vertical velocity is zero at the – impermeable – horizontal bed . This series may subsequently be truncated to a finite number of terms. 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「Boussinesq approximation (water waves)」の詳細全文を読む スポンサード リンク
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