Influid dynamics, theBoussinesq approximation forwater waves is anapproximation valid for weaklynon-linear andfairly long waves. The approximation is named afterJoseph Boussinesq, who first derived them in response to the observation byJohn Scott Russell of thewave of translation (also known assolitary wave orsoliton). The 1872 paper of Boussinesq introduces the equations now known as theBoussinesq equations.^[1]

The Boussinesq approximation forwater waves takes into account the vertical structure of the horizontal and verticalflow velocity. This results innon-linearpartial differential equations, calledBoussinesq-type equations, which incorporatefrequency dispersion (as opposite to theshallow water equations, which are not frequency-dispersive). Incoastal engineering, Boussinesq-type equations are frequently used incomputer models for thesimulation ofwater waves inshallowseas andharbours.

While the Boussinesq approximation is applicable to fairly long waves – that is, when thewavelength is large compared to the water depth – theStokes 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 verticalcoordinate from the flow equations, while retaining some of the influences of the vertical structure of the flow underwater 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 byJoseph Boussinesq in 1871, to construct an approximate solution for the solitary wave (orwave of translation). Subsequently, in 1872, Boussinesq derived the equations known nowadays as the Boussinesq equations.

The steps in the Boussinesq approximation are:

• aTaylor expansion is made of the horizontal and verticalflow velocity (orvelocity potential) around a certainelevation,
• thisTaylor expansion is truncated to afinite number of terms,
• the conservation of mass (seecontinuity equation) for anincompressible flow and the zero-curl condition for anirrotational flow are used, to replace verticalpartial derivatives of quantities in theTaylor expansion with horizontalpartial 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 resultingpartial differential equations are in terms offunctions of the horizontalcoordinates (andtime).

As an example, considerpotential flow over a horizontal bed in the${\displaystyle (x,z)}$ plane, with${\displaystyle x}$ the horizontal and${\displaystyle z}$ the verticalcoordinate. The bed is located at${\displaystyle z=-h}$, where${\displaystyle h}$ is themean water depth. ATaylor expansion is made of thevelocity potential${\displaystyle \phi (x,z,t)}$ around the bed level${\displaystyle z=-h}$:^[2]

where${\displaystyle {\phi }_b(x,t)}$ is the velocity potential at the bed. InvokingLaplace's equation for${\displaystyle \phi }$, as valid forincompressible flow, gives:

since the vertical velocity${\displaystyle \mathrm{\partial }\phi /\mathrm{\partial }z}$ is zero at the – impermeable – horizontal bed${\displaystyle z=-h}$. This series may subsequently be truncated to a finite number of terms.

Forwater waves on anincompressible fluid andirrotational flow in the${\displaystyle (x,z)}$ plane, theboundary conditions at thefree surface elevation${\displaystyle z=\eta (x,t)}$ are:^[3]

where:

• ${\displaystyle u}$ is the horizontalflow velocity component:${\displaystyle u=\mathrm{\partial }\phi /\mathrm{\partial }x}$,
• ${\displaystyle w}$ is the verticalflow velocity component:${\displaystyle w=\mathrm{\partial }\phi /\mathrm{\partial }z}$,
• ${\displaystyle g}$ is theacceleration bygravity.

Now the Boussinesq approximation for thevelocity potential${\displaystyle \phi }$, as given above, is applied in theseboundary conditions. Further, in the resulting equations only thelinear andquadratic terms with respect to${\displaystyle \eta }$ and${\displaystyle u_b}$ are retained (with${\displaystyle u_b=\mathrm{\partial }{\phi }_b/\mathrm{\partial }x}$ the horizontal velocity at the bed${\displaystyle z=-h}$). Thecubic and higher order terms are assumed to be negligible. Then, the followingpartial 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${\displaystyle h}$ is a constant independent of position${\displaystyle x}$. When the right-hand sides of the above equations are set to zero, they reduce to theshallow water equations.

Under some additional approximations, but at the same order of accuracy, the above setA can be reduced to a singlepartial differential equation for thefree surface elevation${\displaystyle \eta }$:

set B – Boussinesq (1872), equation (26)

${\displaystyle \frac{{\mathrm{\partial }}^2\eta }{\mathrm{\partial }{t}^2}-gh\frac{{\mathrm{\partial }}^2\eta }{\mathrm{\partial }{x}^2}-gh\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}\left(\frac{3}{2}\frac{{\eta }^2}{h}+\frac{1}{3}{h}^2\frac{{\mathrm{\partial }}^2\eta }{\mathrm{\partial }{x}^2}\right)=0.}$

From the terms between brackets, the importance of nonlinearity of the equation can be expressed in terms of theUrsell number.Indimensionless quantities, using the water depth${\displaystyle h}$ and gravitational acceleration${\displaystyle g}$ for non-dimensionalization, this equation reads, afternormalization:^[4]

${\displaystyle \frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{\tau }^2}-\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{\xi }^2}-\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{\xi }^2}\left(3{\psi }^2+\frac{{\mathrm{\partial }}^2\psi }{\mathrm{\partial }{\xi }^2}\right)=0,}$

with:

${\displaystyle \psi =\frac{1}{2}\frac{\eta }{h}}$	: the dimensionless surface elevation,
${\displaystyle \tau =\sqrt{3}t\sqrt{\frac{g}{h}}}$	: the dimensionless time, and
${\displaystyle \xi =\sqrt{3}\frac{x}{h}}$	: the dimensionless horizontal position.

Water waves of differentwave lengths travel with differentphase speeds, a phenomenon known asfrequency dispersion. For the case ofinfinitesimal waveamplitude, the terminology islinear 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 validapproximation.

The linearfrequency dispersion characteristics for the above setA of equations are:^[5]

${\displaystyle c^2=gh\frac{1+\frac{1}{6}{k}^2{h}^2}{1+\frac{1}{2}{k}^2{h}^2},}$

with:

• ${\displaystyle c}$ thephase speed,
• ${\displaystyle k}$ thewave number (${\displaystyle k=2\pi /\lambda }$, with${\displaystyle \lambda }$ thewave length).

Therelative error in the phase speed${\displaystyle c}$ for setA, as compared withlinear theory for water waves, is less than 4% for a relative wave number. So, inengineering applications, setA is valid for wavelengths${\displaystyle \lambda }$ larger than 4 times the water depth${\displaystyle h}$.

The linearfrequency dispersion characteristics of equationB are:^[5]

${\displaystyle c^2=gh\left(1-\frac{1}{3}{k}^2{h}^2\right).}$

The relative error in the phase speed for equationB is less than 4% for, equivalent to wave lengths${\displaystyle \lambda }$ longer than 7 times the water depth${\displaystyle h}$, calledfairly long waves.^[6]

For short waves with${\displaystyle k^2{h}^2>3}$ equationB become physically meaningless, because there are no longerreal-valuedsolutions of thephase speed. The original set of twopartial differential equations (Boussinesq, 1872, equation 25, see setA above) does not have this shortcoming.

Theshallow water equations have a relative error in the phase speed less than 4% for wave lengths${\displaystyle \lambda }$ in excess of 13 times the water depth${\displaystyle h}$.

There are an overwhelming number ofmathematical models which are referred to as Boussinesq equations. This may easily lead to confusion, since often they are loosely referenced to asthe Boussinesq equations, while in fact a variant thereof is considered. So it is more appropriate to call themBoussinesq-type equations. Strictly speaking,the Boussinesq equations is the above-mentioned setB, 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:

• varyingbathymetry,
• improvedfrequency dispersion,
• improvednon-linear behavior,
• making aTaylor expansion around different verticalelevations,
• dividing the fluid domain in layers, and applying the Boussinesq approximation in each layer separately,
• inclusion ofwave breaking,
• inclusion ofsurface tension,
• extension tointernal waves on aninterface between fluid domains of differentmass density,
• derivation from avariational principle.

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:

• theKorteweg–de Vries equation forwave propagation in one horizontaldimension,
• theKadomtsev–Petviashvili equation for (near uni-directional)wave propagation in two horizontaldimensions,
• thenonlinear Schrödinger equation (NLS equation) for thecomplex-valued amplitude ofnarrowband waves (slowlymodulated waves).

Besides solitary wave solutions, the Korteweg–de Vries equation also has periodic and exact solutions, calledcnoidal waves. These are approximate solutions of the Boussinesq equation.

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 inMIKE 21 andSMS. Some of the free Boussinesq models are Celeris,^[7] COULWAVE,^[8] and FUNWAVE.^[9] Most numerical models employfinite-difference,finite-volume orfinite element techniques for thediscretization 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) andHamm, Madsen & Peregrine (1993).

• 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 Engineering13. World Scientific, Singapore.ISBN 978-981-02-0427-3. Archived fromthe original on 2012-02-08. Retrieved2008-01-21.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.Bibcode:1993CoasE..21....5H.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. Vol. 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. Vol. 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.S2CID 119385147.
• 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.

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