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 by John Scott Russell of the wave of translation (also known as solitary wave or soliton).

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).

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.

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 an example, consider potential flow over a horizontal bed in the

, as valid for incompressible flow, gives: since the vertical velocity

This series may subsequently be truncated to a finite number of terms.

For water waves on an incompressible fluid and irrotational flow in the

plane, the boundary conditions at the free surface elevation

Further, in the resulting equations only the linear and quadratic terms with respect to

The cubic and higher order terms are assumed to be negligible.

Under some additional approximations, but at the same order of accuracy, the above set A can be reduced to a single partial differential equation for the free surface elevation

for non-dimensionalization, this equation reads, after normalization:[4] with: Water waves of different wave lengths travel with different phase speeds, a phenomenon known as frequency dispersion.

For the case of infinitesimal wave amplitude, the terminology is linear frequency dispersion.

The frequency dispersion characteristics of a Boussinesq-type of equation can be used to determine the range of wave lengths, for which it is a valid approximation.

The linear frequency dispersion characteristics for the above set A of equations are:[5] with: The relative error in the phase speed

equation B become physically meaningless, because there are no longer real-valued solutions of the phase speed.

The shallow water equations have a relative error in the phase speed less than 4% for wave lengths

There are an overwhelming number 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.

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.

Under small additional assumptions, the Boussinesq equations reduce to: Besides solitary wave solutions, the Korteweg–de Vries equation also has periodic and exact solutions, called cnoidal waves.

For the simulation of wave motion near coasts and harbours, numerical models – both commercial and academic – employing Boussinesq-type equations exist.

Some commercial examples are the Boussinesq-type wave modules in MIKE 21 and SMS.

Some of the free Boussinesq models are Celeris,[7] COULWAVE,[8] and FUNWAVE.

Scientific reviews and intercomparisons of several Boussinesq-type equations, their numerical approximation and performance are e.g. Kirby (2003), Dingemans (1997, Part 2, Chapter 5) and Hamm, Madsen & Peregrine (1993).