Boussinesq approximation (water waves)
fluid dynamics, the Boussinesq approximation for water wavesis an approximationvalid for weakly non-linearand fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by John Scott Russellof the wave of translation(also known as solitary waveor soliton). 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 wavestakes into account the vertical structure of the horizontal and vertical flow velocity. This results in non-linear partial differential equations, called Boussinesq-type equations, which incorporate frequency dispersion (as opposite to the shallow water equations, which are not frequency-dispersive).
coastal engineering, Boussinesq-type equations are frequently used in computer modelsfor the simulationof water wavesin shallow seasand harbours.
shoalon a plane beach. This example combines several effects of waves and shallow water, including refraction, diffraction, shoaling and weak non-linearity.]
The essential idea in the Boussinesq approximation is the elimination of the vertical
coordinatefrom the flowequations, while retaining some of the influences of the vertical structure of the flowunder 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 Boussinesqin 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.
The steps in the Boussinesq approximation are:
Taylor expansionis made of the horizontal and vertical flow velocity(or velocity potential) around a certain elevation,
Taylor expansionis truncated to a finitenumber of terms,
*the conservation of mass (see
continuity equation) for an incompressible flowand the zero-curl condition for an irrotational floware used, to replace vertical partial derivativesof quantities in the Taylor expansionwith horizontal partial 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 resulting partial differential equationsare in terms of functions of the horizontal coordinates(and time).
As an example, consider
potential flowover a horizontal bed in the ("x,z") plane, with "x" the horizontal and "z" the vertical coordinate. The bed is located at "z = -h", where "h" is the meanwater depth. A Taylor expansionis made of the velocity potential"φ(x,z,t)" around the bed level "z = -h": [Dingemans (1997), p. 477.]
: where "φb(x,t)" is the velocity potential at the bed. Invoking
Laplace's equationfor "φ", as valid for incompressible flow, gives:
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
water waveson an incompressible fluidand irrotational flowin the ("x,z") plane, the boundary conditionsat the free surfaceelevation "z = η(x,t)" are: [Dingemans (1997), p. 475.]
where::"u" is the horizontal
flow velocitycomponent: "u = ∂φ / ∂x",:"w" is the vertical flow velocitycomponent: "w = ∂φ / ∂z",:"g" is the accelerationby gravity.
Now the Boussinesq approximation for the
velocity potential"φ", as given above, is applied in these boundary conditions. Further, in the resulting equations only the linearand quadratic terms with respect to "η" and "ub" are retained (with "ub = ∂φb / ∂x" the horizontal velocity at the bed "z = -h"). The cubic and higher order terms are assumed to be negligible. Then, the following partial differential equationsare obtained:;set A — Boussinesq (1872), equation (25)
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 single
partial differential equationfor the free surfaceelevation "η":;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 "h" and gravitational acceleration "g" for non-dimensionalization, this equation reads, after normalization: :
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, see
dispersion (water waves)]
Linear frequency dispersion
Water wavesof different wave lengths travel with different phase speeds, a phenomenon known as frequency dispersion. For the case of infinitesimalwave 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:Dingemans (1997), p. 521.]
wave number( "k = 2π / λ", with "λ" the wave length). The relative errorin 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, in engineeringapplications, set A is valid for wavelengths "λ" larger than 4 times the water depth "h".
The linear frequency dispersion characteristics of equation B are:
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 "k2 h2 > 3" equation B become physically meaningless, because there are no longer
real-valued solutionsof the phase speed. The original set of two partial differential equations(Boussinesq, 1872, equation 25, see set A above) does not have this shortcoming.
shallow water equationshave a relative error in the phase speed less than 4% for wave lengths "λ" in excess of 13 times the water depth "h".
There is an overwhelming amount of
mathematical modelswhich 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:
*improved frequency dispersion,
Taylor expansionaround different vertical elevations,
*dividing the fluid domain in layers, and applying the Boussinesq approximation in each layer separately,
internal waveson an interfacebetween fluid domains of different mass density,
*derivation from a
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:
Korteweg–de Vries equationfor wave propagationin one horizontal dimension,
Kadomtsev–Petviashvili equationfor wave propagationin two horizontal dimensions,
nonlinear Schrödinger equation(NLS equation) for the complex valued amplitudeof narrowbandwaves (slowly modulatedwaves).
*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
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