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).
Periodic waves in the Boussinesq approximation, shown in a vertical cross section in the wave propagation direction. Notice the flat troughs and sharp crests, due to the wave nonlinearity. This case (drawn on scale) shows a wave with the wavelength equal to 39.1 m, the wave height is 1.8 m (i.e. the difference between crest and trough elevation), and the mean water depth is 5 m, while the gravitational acceleration is 9.81 m/s2.
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).
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.
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 h and gravitational acceleration g for non-dimensionalization, this equation reads, after normalization:
: the dimensionless surface elevation,
: the dimensionless time, and
: the dimensionless horizontal position.
Linear phase speed squared c2/(gh) as a function of relative wave number kh. A = Boussinesq (1872), equation (25), B = Boussinesq (1872), equation (26), C = full linear wave theory, see dispersion (water waves)
The shallow water equations have a relative error in the phase speed less than 4% for wave lengths λ in excess of 13 times the water depth h.
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:
^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").
^"James T. Kirby, Funwave program". www1.udel.edu.
Boussinesq, J. (1871). "Théorie de l'intumescence liquide, applelée onde solitaire ou de translation, se propageant dans un canal rectangulaire". Comptes Rendus de l'Académie des Sciences. 72: 755–759.
Boussinesq, J. (1872). "Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond". Journal de Mathématiques Pures et Appliquées. Deuxième Série. 17: 55–108.
Dingemans, M.W. (1997). Wave propagation over uneven bottoms. Advanced Series on Ocean Engineering 13. World Scientific, Singapore. ISBN 978-981-02-0427-3. Archived from the original on 2012-02-08. Retrieved 2008-01-21. See Part 2, Chapter 5.
Hamm, L.; Madsen, P.A.; Peregrine, D.H. (1993). "Wave transformation in the nearshore zone: A review". Coastal Engineering. 21 (1–3): 5–39. doi:10.1016/0378-3839(93)90044-9.
Johnson, R.S. (1997). A modern introduction to the mathematical theory of water waves. Cambridge Texts in Applied Mathematics. 19. Cambridge University Press. ISBN 0-521-59832-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. 67. Elsevier. pp. 1–41. ISBN 0-444-51149-0.
Peregrine, D.H. (1967). "Long waves on a beach". Journal of Fluid Mechanics. 27 (4): 815–827. Bibcode:1967JFM....27..815P. doi:10.1017/S0022112067002605.
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. ISBN 0-12-493250-9.