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.
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).
Now the Boussinesq approximation for the
velocity potential, as given above, is applied in these
boundary conditions. Further, in the resulting equations only the
linear and
quadratic terms with respect to and are retained (with the horizontal velocity at the bed ). The
cubic and higher order terms are assumed to be negligible. Then, the following
partial differential equations are obtained:
set A – Boussinesq (1872), equation (25)
This set of equations has been derived for a flat horizontal bed, i.e. the mean depth is a constant independent of position . When the right-hand sides of the above equations are set to zero, they reduce to the
shallow water equations.
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 :
set B – Boussinesq (1872), equation (26)
From the terms between brackets, the importance of nonlinearity of the equation can be expressed in terms of the
Ursell number.
In
dimensionless quantities, using the water depth and gravitational acceleration for non-dimensionalization, this equation reads, after
normalization:[4]
with:
: the dimensionless surface elevation,
: the dimensionless time, and
: the dimensionless horizontal position.
Linear frequency dispersion
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
relative error in the phase speed for set A, as compared with
linear theory for water waves, is less than 4% for a relative wave number . So, in
engineering applications, set A is valid for wavelengths larger than 4 times the water depth .
The relative error in the phase speed for equation B is less than 4% for , equivalent to wave lengths longer than 7 times the water depth , called fairly long waves.[6]
For short waves with equation B become physically meaningless, because there are no longer
real-valuedsolutions of the
phase speed.
The original set of two
partial differential equations (Boussinesq, 1872, equation 25, see set A above) does not have this shortcoming.
The
shallow water equations have a relative error in the phase speed less than 4% for wave lengths in excess of 13 times the water depth .
Boussinesq-type equations and extensions
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. So it is more appropriate 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 simultaneously in opposing directions, it is often advantageous to only consider waves traveling in one direction. 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. These are approximate solutions of the Boussinesq equation.
Numerical models
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.[9] Most numerical models employ
finite-difference,
finite-volume or
finite element techniques for the
discretization of the model equations. 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).
Notes
^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").
Johnson, R.S. (1997). A modern introduction to the mathematical theory of water waves. Cambridge Texts in Applied Mathematics. Vol. 19. Cambridge University Press.
ISBN0-521-59832-X.
Kirby, J.T. (2003). "Boussinesq models and applications to nearshore wave propagation, surfzone processes and wave-induced currents". In Lakhan, V.C. (ed.). Advances in Coastal Modeling. Elsevier Oceanography Series. Vol. 67. Elsevier. pp. 1–41.
ISBN0-444-51149-0.
Peregrine, D.H. (1972). "Equations for water waves and the approximations behind them". In Meyer, R.E. (ed.). Waves on Beaches and Resulting Sediment Transport. Academic Press. pp. 95–122.
ISBN0-12-493250-9.