Boussinesq approximation (water waves)

In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph Boussinesq, who first derived them in response to the observation by John Scott Russell of the wave of translation (also known as solitary wave or soliton). The 1872 paper of Boussinesq introduces the equations now known as the Boussinesq equations.[1]

The Boussinesq approximation for water waves takes 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). In coastal engineering, Boussinesq-type equations are frequently used in computer models for the simulation of water waves in shallow seas and harbours.

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).

Simulation of periodic waves over an underwater shoal with a Boussinesq-type model. The waves propagate over an elliptic-shaped underwater shoal on a plane beach. This example combines several effects of waves and shallow water, including refraction, diffraction, shoaling and weak non-linearity.

Boussinesq approximation

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.

The steps in the Boussinesq approximation are:

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).

As an example, consider potential flow over 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 mean water depth. A Taylor expansion is made of the velocity potential φ(x,z,t) around the bed level z = −h:[2]

{\displaystyle {\begin{aligned}\varphi \,=\,&\varphi _{b}\,+\,(z+h)\,\left[{\frac {\partial \varphi }{\partial z}}\right]_{z=-h}\,+\,{\frac {1}{2}}\,(z+h)^{2}\,\left[{\frac {\partial ^{2}\varphi }{\partial z^{2}}}\right]_{z=-h}\,\\&+\,{\frac {1}{6}}\,(z+h)^{3}\,\left[{\frac {\partial ^{3}\varphi }{\partial z^{3}}}\right]_{z=-h}\,+\,{\frac {1}{24}}\,(z+h)^{4}\,\left[{\frac {\partial ^{4}\varphi }{\partial z^{4}}}\right]_{z=-h}\,+\,\cdots ,\end{aligned}}}

where φb(x,t) is the velocity potential at the bed. Invoking Laplace's equation for φ, as valid for incompressible flow, gives:

{\displaystyle {\begin{aligned}\varphi \,=\,&\left\{\,\varphi _{b}\,-\,{\frac {1}{2}}\,(z+h)^{2}\,{\frac {\partial ^{2}\varphi _{b}}{\partial x^{2}}}\,+\,{\frac {1}{24}}\,(z+h)^{4}\,{\frac {\partial ^{4}\varphi _{b}}{\partial x^{4}}}\,+\,\cdots \,\right\}\,\\&+\,\left\{\,(z+h)\,\left[{\frac {\partial \varphi }{\partial z}}\right]_{z=-h}\,-\,{\frac {1}{6}}\,(z+h)^{3}\,{\frac {\partial ^{2}}{\partial x^{2}}}\left[{\frac {\partial \varphi }{\partial z}}\right]_{z=-h}\,+\,\cdots \,\right\}\\=\,&\left\{\,\varphi _{b}\,-\,{\frac {1}{2}}\,(z+h)^{2}\,{\frac {\partial ^{2}\varphi _{b}}{\partial x^{2}}}\,+\,{\frac {1}{24}}\,(z+h)^{4}\,{\frac {\partial ^{4}\varphi _{b}}{\partial x^{4}}}\,+\,\cdots \,\right\},\end{aligned}}}

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 an incompressible fluid and irrotational flow in the (x,z) plane, the boundary conditions at the free surface elevation z = η(x,t) are:[3]

{\displaystyle {\begin{aligned}{\frac {\partial \eta }{\partial t}}\,&+\,u\,{\frac {\partial \eta }{\partial x}}\,-\,w\,=\,0\\{\frac {\partial \varphi }{\partial t}}\,&+\,{\frac {1}{2}}\,\left(u^{2}+w^{2}\right)\,+\,g\,\eta \,=\,0,\end{aligned}}}

where:

u is the horizontal flow velocity component: u = ∂φ / ∂x,
w is the vertical flow velocity component: w = ∂φ / ∂z,
g is the acceleration by 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 linear and 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 equations are obtained:

set A – Boussinesq (1872), equation (25)
{\displaystyle {\begin{aligned}{\frac {\partial \eta }{\partial t}}\,&+\,{\frac {\partial }{\partial x}}\,\left[\left(h+\eta \right)\,u_{b}\right]\,=\,{\frac {1}{6}}\,h^{3}\,{\frac {\partial ^{3}u_{b}}{\partial x^{3}}},\\{\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 x^{2}}}.\end{aligned}}}

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 equation for the free surface elevation η:

set B – Boussinesq (1872), equation (26)
${\displaystyle {\frac {\partial ^{2}\eta }{\partial t^{2}}}\,-\,gh\,{\frac {\partial ^{2}\eta }{\partial x^{2}}}\,-\,gh\,{\frac {\partial ^{2}}{\partial x^{2}}}\left({\frac {3}{2}}\,{\frac {\eta ^{2}}{h}}\,+\,{\frac {1}{3}}\,h^{2}\,{\frac {\partial ^{2}\eta }{\partial x^{2}}}\right)\,=\,0.}$

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:[4]

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

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.
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)

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 linear frequency dispersion characteristics for the above set A 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:

The relative 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, in engineering 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:[5]

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

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.[6]

For short waves with k2 h2 > 3 equation B become physically meaningless, because there are no longer real-valued solutions 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 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:

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

A simulation with a Boussinesq-type wave model of nearshore waves travelling towards a harbour entrance. The simulation is with the BOUSS-2D module of SMS.
Faster than real-time simulation with the Boussinesq module of Celeris, showing wave breaking and refraction near the beach. The model provides an interactive environment.

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

1. ^ 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").
2. ^ Dingemans (1997), p. 477.
3. ^ Dingemans (1997), p. 475.
4. ^ Johnson (1997), p. 219
5. ^ a b Dingemans (1997), p. 521.
6. ^ Dingemans (1997), p. 473 & 516.
7. ^ "Celeria.org - Celeris Boussinesq Wave Model". Celeria.org - Celeris Boussinesq Wave Model.
8. ^ "ISEC - Models". isec.nacse.org.
9. ^ "James T. Kirby, Funwave program". www1.udel.edu.

References

Airy wave theory

In fluid dynamics, Airy wave theory (often referred to as linear wave theory) gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.Airy wave theory is often applied in ocean engineering and coastal engineering for the modelling of random sea states – giving a description of the wave kinematics and dynamics of high-enough accuracy for many purposes. Further, several second-order nonlinear properties of surface gravity waves, and their propagation, can be estimated from its results. Airy wave theory is also a good approximation for tsunami waves in the ocean, before they steepen near the coast.

This linear theory is often used to get a quick and rough estimate of wave characteristics and their effects. This approximation is accurate for small ratios of the wave height to water depth (for waves in shallow water), and wave height to wavelength (for waves in deep water).

Boussinesq approximation

Boussinesq approximation may refer to several modelling concepts – as introduced by Joseph Valentin Boussinesq (1842–1929), a French mathematician and physicist known for advances in fluid dynamics:

Boussinesq approximation (buoyancy) for buoyancy-driven flows for small density differences in the fluid

Boussinesq approximation (water waves) for long waves propagating on the surface of a fluid layer under the action of gravity

Turbulence modeling and eddy viscosity: in modelling the turbulence Reynolds stresses, the Boussinesq approximation results in the use of an eddy viscosity concept

Dispersion (water waves)

In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium.

For a certain water depth, surface gravity waves – i.e. waves occurring at the air–water interface and gravity as the only force restoring it to flatness – propagate faster with increasing wavelength. On the other hand, for a given (fixed) wavelength, gravity waves in deeper water have a larger phase speed than in shallower water. In contrast with the behavior of gravity waves, capillary waves (i.e. only forced by surface tension) propagate faster for shorter wavelengths.

Besides frequency dispersion, water waves also exhibit amplitude dispersion. This is a nonlinear effect, by which waves of larger amplitude have a different phase speed from small-amplitude waves.

Fluid dynamics

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments on aircraft, determining the mass flow rate of petroleum through pipelines, predicting weather patterns, understanding nebulae in interstellar space and modelling fission weapon detonation.

Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as flow velocity, pressure, density, and temperature, as functions of space and time.

Before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, both of which can also be applied to gases.

Index of physics articles (B)

The index of physics articles is split into multiple pages due to its size.

To navigate by individual letter use the table of contents below.

Index of wave articles

This is a list of Wave topics.

Joseph Valentin Boussinesq

Joseph Valentin Boussinesq (pronounced [ʒɔzɛf valɑ̃tɛ̃ businɛsk]; 13 March 1842 – 19 February 1929) was a French mathematician and physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat.

Korteweg–de Vries equation

In mathematics, the Korteweg–de Vries (KdV) equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified. KdV can be solved by means of the inverse scattering transform. The mathematical theory behind the KdV equation is a topic of active research. The KdV equation was first introduced by Boussinesq (1877, footnote on page 360) and rediscovered by Diederik Korteweg and Gustav de Vries (1895).

Lille Laboratory of Mechanics

The Laboratoire de mécanique de Lille (LML) is a French research laboratory (UMR CNRS 8107) part of the Carnot institute ARTS. More than 200 people work in this laboratory which was created in 1985.

It supports academic activities in the following graduate schools :

Arts et Métiers ParisTech (ENSAM)

École centrale de Lille

University of Lille.It supports doctoral researches and hosts PhD doctoral candidates in relationship with the European Doctoral College Lille Nord de France.

Outline of oceanography

The following outline is provided as an overview of and introduction to Oceanography.

Waves and shallow water

When waves travel into areas of shallow water, they begin to be affected by the ocean bottom. The free orbital motion of the water is disrupted, and water particles in orbital motion no longer return to their original position. As the water becomes shallower, the swell becomes higher and steeper, ultimately assuming the familiar sharp-crested wave shape. After the wave breaks, it becomes a wave of translation and erosion of the ocean bottom intensifies.

Wind wave

In fluid dynamics, wind waves, or wind-generated waves, are water surface waves that occur on the free surface of the oceans and other bodies (like lakes, rivers, canals, puddles or ponds). They result from the wind blowing over an area of fluid surface. Waves in the oceans can travel thousands of miles before reaching land. Wind waves on Earth range in size from small ripples, to waves over 100 ft (30 m) high.When directly generated and affected by local waters, a wind wave system is called a wind sea. After the wind ceases to blow, wind waves are called swells. More generally, a swell consists of wind-generated waves that are not significantly affected by the local wind at that time. They have been generated elsewhere or some time ago. Wind waves in the ocean are called ocean surface waves.

Wind waves have a certain amount of randomness: subsequent waves differ in height, duration, and shape with limited predictability. They can be described as a stochastic process, in combination with the physics governing their generation, growth, propagation, and decay—as well as governing the interdependence between flow quantities such as: the water surface movements, flow velocities and water pressure. The key statistics of wind waves (both seas and swells) in evolving sea states can be predicted with wind wave models.

Although waves are usually considered in the water seas of Earth, the hydrocarbon seas of Titan may also have wind-driven waves.

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