- Boussinesq approximation (water waves)
In

fluid dynamics , the**Boussinesq approximation**forwater waves is anapproximation valid for weaklynon-linear and fairly 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 the**Boussinesq equations**. [*This paper (Boussinesq, 1872) starts with: "Touts les ingénieurs connaissants les belles expériences de J. Scott Russell and 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").*]The Boussinesq approximation for

water waves takes into account the vertical structure of the horizontal and verticalflow velocity . This results innon-linear partial differential equations , called Boussinesq-type equations, which incorporate frequency dispersion (as opposite to theshallow water equations , which are not frequency-dispersive).In

coastal engineering , Boussinesq-type equations are frequently used incomputer models for thesimulation ofwater waves in shallowseas andharbours .

[shoal on a plane beach. This example combines several effects ofwaves and shallow water , includingrefraction ,diffraction , shoaling and weaknon-linearity .]**Boussinesq approximation**The essential idea in the Boussinesq approximation is the elimination of the vertical

coordinate from theflow equations, while retaining some of the influences of the vertical structure of theflow 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 by

Joseph 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 of functions of the horizontalcoordinates (andtime ).As an example, consider

potential flow over a horizontal bed in the ("x,z") plane, with "x" the horizontal and "z" the verticalcoordinate . The bed is located at "z = -h", where "h" is themean water depth. ATaylor expansion is made of thevelocity potential "φ(x,z,t)" around the bed level "z = -h": [*Dingemans (1997), p. 477.*]:$varphi,\; =,\; varphi\_b,\; +,\; z,\; left\; [\; frac\{partial\; varphi\}\{partial\; z\; \}\; ight]\; \_\{z=-h\},\; +,\; frac\{1\}\{2\},\; z^2,\; left\; [\; frac\{partial^2\; varphi\}\{partial\; z^2\}\; ight]\; \_\{z=-h\},\; +,\; frac\{1\}\{6\},\; z^3,\; left\; [\; frac\{partial^3\; varphi\}\{partial\; z^3\}\; ight]\; \_\{z=-h\},\; +,\; frac\{1\}\{24\},\; z^4,\; left\; [\; frac\{partial^4\; varphi\}\{partial\; z^4\}\; ight]\; \_\{z=-h\},\; +,\; cdots,$ where "φ

_{b}(x,t)" is the velocity potential at the bed. InvokingLaplace's equation for "φ", as valid forincompressible flow , gives::$egin\{align\}\; varphi,\; =,\; left\{,\; varphi\_b,\; -,\; frac\{1\}\{2\},\; z^2,\; frac\{partial^2\; varphi\_b\}\{partial\; x^2\},\; +,\; frac\{1\}\{24\},\; z^4,\; frac\{partial^4\; varphi\_b\}\{partial\; x^4\},\; +,\; cdots,\; ight\},\; +,\; left\{,\; z,\; left\; [\; frac\{partial\; varphi\}\{partial\; z\}\; ight]\; \_\{z=-h\},\; -,\; frac\{1\}\{6\},\; z^3,\; frac\{partial^2\}\{partial\; x^2\},\; left\; [\; frac\{partial\; varphi\}\{partial\; z\}\; ight]\; \_\{z=-h\},\; +,\; cdots,\; ight\}\; \backslash \backslash \; =,\; left\{,\; varphi\_b,\; -,\; frac\{1\}\{2\},\; z^2,\; frac\{partial^2\; varphi\_b\}\{partial\; x^2\},\; +,\; frac\{1\}\{24\},\; z^4,\; frac\{partial^4\; varphi\_b\}\{partial\; x^4\},\; +,\; cdots,\; ight\},\; end\{align\}$

since the vertical velocity "∂φ / ∂z" is zero at the — impermeable — horizontal bed "z = -h". This series may subsequently be truncated to a finite number of terms.

**Original Boussinesq equations****Derivation**For

water waves on anincompressible fluid andirrotational flow in the ("x,z") plane, theboundary conditions at thefree surface elevation "z = η(x,t)" are: [*Dingemans (1997), p. 475.*]:$egin\{align\}\; frac\{partial\; eta\}\{partial\; t\},\; +,\; u,\; frac\{partial\; eta\}\{partial\; x\},\; -,\; w,\; =,\; 0\; \backslash \backslash \; frac\{partial\; varphi\}\{partial\; t\},\; +,\; frac\{1\}\{2\},\; left(\; u^2\; +\; w^2\; ight),\; +,\; g,\; eta,\; =,\; 0,\; end\{align\}$

where::"u" is the horizontal

flow velocity component: "u = ∂φ / ∂x",:"w" is the verticalflow velocity component: "w = ∂φ / ∂z",:"g" is theacceleration bygravity .Now the Boussinesq approximation for the

velocity potential "φ", as given above, is applied in theseboundary conditions . Further, in the resulting equations only thelinear and quadratic terms with respect to "η" and "u_{b}" are retained (with "u_{b}= ∂φ_{b}/ ∂x" the horizontal velocity at the bed "z = -h"). The cubic and higher order terms are assumed to benegligible . Then, the followingpartial differential equations are obtained:;set A — Boussinesq (1872), equation (25):$egin\{align\}\; frac\{partial\; eta\}\{partial\; t\},\; +,\; frac\{partial\}\{partial\; x\},\; left\; [\; left(\; h\; +\; eta\; ight),\; u\_b\; ight]\; ,\; =,\; frac\{1\}\{6\},\; h^3,\; frac\{partial^3\; u\_b\}\{partial\; x^3\},\; \backslash \backslash \; frac\{partial\; u\_b\}\{partial\; t\},\; +,\; u\_b,\; frac\{partial\; u\_b\}\{partial\; x\},\; +,\; g,\; frac\{partial\; eta\}\{partial\; x\},\; =,\; frac\{1\}\{2\},\; h^2,\; frac\{partial^3\; u\_b\}\{partial\; t,\; partial^2\; x\}.end\{align\}$

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 .Under some additional approximations, but at the same order of accuracy, the above set

**A**can be reduced to a singlepartial differential equation for thefree surface elevation "η":;set B — Boussinesq (1872), equation (26):$frac\{partial^2\; eta\}\{partial\; t^2\},\; -,\; g\; h,\; frac\{partial^2\; eta\}\{partial\; x^2\},\; -,\; g\; h,\; frac\{partial^2\}\{partial\; x^2\}\; left(\; frac\{3\}\{2\},\; frac\{eta^2\}\{h\},\; +,\; frac\{1\}\{3\},\; h^2,\; frac\{partial^2\; eta\}\{partial\; x^2\}\; ight),\; =,\; 0.$

From the terms between brackets, the importance of nonlinearity of the equation can be expressed in terms of the

Ursell number .Indimensionless quantities , using the water depth "h" and gravitational acceleration "g" for non-dimensionalization, this equation reads, afternormalization : :$frac\{partial^2\; psi\}\{partial\; au^2\},\; -,\; frac\{partial^2\; psi\}\{partial\; xi^2\},\; -,\; frac\{partial^2\}\{partial\; xi^2\}\; left(,\; frac\{1\}\{2\},\; psi^2,\; +,\; frac\{partial^2\; psi\}\{partial\; xi^2\},\; ight),\; =,\; 0,$with:

[

^{2}/(gh)"as a function of relative wave number "kh".**A**= Boussinesq (1872), equation (25),**B**= Boussinesq (1872), equation (26),**C**= full linear wave theory, seedispersion (water waves) ]**Linear frequency dispersion**Water waves of differentwave length s travel with differentphase speed s, a phenomenon known as frequency dispersion. For the case ofinfinitesimal waveamplitude , 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 validapproximation .The linear frequency dispersion characteristics for the above set

**A**of equations are:Dingemans (1997), p. 521.]:$c^2,\; =;\; g\; h,\; frac\{\; 1,\; +,\; frac\{1\}\{6\},\; k^2\; h^2\; \}\{\; 1,\; +,\; frac\{1\}\{2\},\; k^2\; h^2\; \},$

with:

*"c" thephase speed ,

*"k" thewave number ( "k = 2π / λ", with "λ" thewave length ). Therelative error in the phase speed "c" for set**A**, as compared with linear theory for water waves, is less than 4% for a relative wave number "kh < ½ π". So, inengineering applications, set**A**is valid for wavelengths "λ" larger than 4 times the water depth "h".The linear frequency dispersion characteristics of equation

**B**are::$c^2,\; =,\; g\; h,\; left(\; 1,\; -,\; frac\{1\}\{3\},\; k^2\; h^2\; ight).$

The relative error in the phase speed for equation

**B**is less than 4% for "kh < 2π / 7", equivalent to wave lengths "λ" longer than 7 times the water depth "h", called**fairly long waves**. [*Dingemans (1997), p. 473 & 516.*] For short waves with "k^{2}h^{2}> 3" equation**B**become physically meaningless, because there are no longerreal-valued solutions of thephase speed . The original set of twopartial 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 "λ" in excess of 13 times the water depth "h".**Extensions**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 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:

*varyingbathymetry ,

*improved frequency 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 .**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:

*theKorteweg–de Vries equation forwave propagation in one horizontaldimension ,

*theKadomtsev–Petviashvili equation forwave propagation in two horizontaldimension s,

*thenonlinear Schrödinger equation (NLS equation) for the complex valuedamplitude ofnarrowband waves (slowlymodulated waves).**References***cite journal | author= J. Boussinesq | authorlink=Joseph Boussinesq | year= 1871 | title= Théorie de l'intumescence liquide, applelée onde solitaire ou de translation, se propageant dans un canal rectangulaire |journal= Comptes Rendus de l'Academie des Sciences | volume= 72 | pages= 755–759 | url= http://gallica.bnf.fr/ark:/12148/bpt6k3029x/f759

*cite journal | author= J. Boussinesq | authorlink=Joseph Boussinesq | year= 1872 | title= 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= Journal de Mathématique Pures et Appliquées, Deuxième Série | volume= 17 | pages= 55–108 | url= http://visualiseur.bnf.fr/ConsulterElementNum?O=NUMM-16416&Deb=63&Fin=116&E=PDF

*cite book | author= M.W. Dingemans | year= 1997 | title=Wave propagation over uneven bottoms | series= Advanced Series on Ocean Engineering**13**| publisher= World Scientific, Singapore | url= http://www.worldscibooks.com/engineering/1241.html | id= ISBN 981-02-0427-2 "See Part 2, Chapter 5".

*cite journal | author= D.H. Peregrine | authorlink=Howell Peregrine | year= 1967 | title= Long waves on a beach | journal= Journal of Fluid Mechanics | volume= 27 | issue= 4 | pages= 815–827 |url= http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=381506# | doi= 10.1017/S0022112067002605

*cite conference | author= D.H. Peregrine | authorlink=Howell Peregrine | year=1972 | title= Equations for water waves and the approximations behind them | booktitle= Waves on Beaches and Resulting Sediment Transport | editor= Ed. R.E. Meyer | publisher= Academic Press | pages= 95–122 | id= ISBN 0124932509**Notes**

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