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Boussinesq approximation (water waves)
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 of the solitary wave made by John Scott Russell. The 1872 paper of Boussinesq, for the first time introducing the equations now known as the Boussinesq equations, starts with:
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.
Additional recommended knowledge
The essential idea in the Boussinesq approximation is the eliminatation 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 was first done by Joseph Boussinesq in 1871 to construct an approximate solution for the solitary wave (or wave of translation), and subsequently in 1872 to derive the equations now known as the Boussinesq equations.
The steps in the Boussinesq approximation are:
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 z=0, around which level we make a Taylor expansion of the velocity potential φ(x,z,t):
where φb(x,t) is the velocity potential at the bed. Invoking Laplace's equation for φ, as valid for incompressible flow, we get:
since the vertical velocity is zero at the horizontal bed z=0. This series may subsequently be truncated to a finite number of terms.
Original Boussinesq equations
Now the Boussinesq approximation for the velocity potential φ, as given above, is applied in these boundary conditions. Further, the resulting equations are only allowed to have linear and quadratic terms of η and ub (the x-derivative of φb). Then, the following partial differential equations are obtained:
This set of equations has been derived for a flat horizontal bed, i.e. the mean depth h is a constant independent of position x. When the right-hand sides of the above equations are set to zero, they reduce to the shallow water equations.
The linear frequency dispersion characteristics of this set A of equations is:
with c the phase speed and k the wave number.
Under some additional approximations, the above set A can be reduced to a single partial differential equation for the free surface elevation η:
The linear frequency dispersion characteristics of this equation B are:
For kh > 1/3 the approximation becomes physically meaningless because there are no longer real-valued solutions. The original set of two partial differential equations (Boussinesq, 1872, equation 25, see set A above) does not have this shortcoming.
There is an overwhelming amount of mathematical models which are referred to as Boussinesq equations. This may easily lead to confusion, since often they are loosely referenced to as the Boussinesq equations, while in fact a variant thereof is considered. So it is more appropiate to call them Boussinesq-type equations. Strictly speaking, the Boussinesq equations is the above mentioned set B, since it is used in the analysis in the remainder of his 1872 paper.
Some directions, into which the Boussinesq equations have been extended, are:
Further approximations for one-way wave propagation
While the Boussinesq equations allow for waves traveling in any direction, it is often advantageous to only consider waves traveling in one direction. Under small additional assumptions, the Boussinesq equations reduce to
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Boussinesq_approximation_(water_waves)". A list of authors is available in Wikipedia.|