In
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) an ...
, the Boussinesq approximation for
water waves
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of ...
is an
approximation
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
valid for weakly
non-linear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
and
fairly long waves. The approximation is named after
Joseph Boussinesq, who first derived them in response to the observation by
John Scott Russell
John Scott Russell FRSE FRS FRSA (9 May 1808, Parkhead, Glasgow – 8 June 1882, Ventnor, Isle of Wight) was a Scottish civil engineer, naval architect and shipbuilder who built '' Great Eastern'' in collaboration with Isambard Kingdom Bru ...
of the
wave of translation (also known as
solitary wave or
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
). The 1872 paper of Boussinesq introduces the equations now known as the Boussinesq equations.
The Boussinesq approximation for
water waves
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of ...
takes into account the vertical structure of the horizontal and vertical
flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the ...
. This results in
non-linear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ho ...
, called Boussinesq-type equations, which incorporate
frequency dispersion (as opposite to the
shallow water equations
The shallow-water equations (SWE) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface). ...
, which are not frequency-dispersive). In
coastal engineering
Coastal engineering is a branch of civil engineering concerned with the specific demands posed by constructing at or near the coast, as well as the development of the coast itself.
The hydrodynamic impact of especially waves, tides, storm sur ...
, Boussinesq-type equations are frequently used in
computer models for the
simulation
A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the ...
of
water waves
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of ...
in
shallow seas and
harbours
A harbor (American English), harbour (British English; see spelling differences), or haven is a sheltered body of water where ships, boats, and barges can be docked. The term ''harbor'' is often used interchangeably with ''port'', which is a ...
.
While the Boussinesq approximation is applicable to fairly long waves – that is, when the
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
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).
Boussinesq approximation
The essential idea in the Boussinesq approximation is the elimination of the vertical
coordinate
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
from the flow equations, while retaining some of the influences of the vertical structure of the flow under
water waves
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of ...
. 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:
*a
Taylor expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
is made of the horizontal and vertical
flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the ...
(or
velocity potential) around a certain
elevation
The elevation of a geographic location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface (see Geodetic datum § V ...
,
*this
Taylor expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
is truncated to a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which ...
number of terms,
*the conservation of mass (see
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
) for an
incompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An e ...
and the zero-
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was ...
condition for an
irrotational flow
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not c ...
are used, to replace vertical
partial derivatives
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
of quantities in the
Taylor expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
with horizontal
partial derivatives
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
.
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
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ho ...
are in terms of
functions of the horizontal
coordinates
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is si ...
(and
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...
).
As an example, consider
potential flow
In fluid dynamics, potential flow (or ideal flow) describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid app ...
over a horizontal bed in the
plane, with
the horizontal and
the vertical
coordinate
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
. The bed is located at
, where
is the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arith ...
water depth. A
Taylor expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
is made of the
velocity potential around the bed level
:
:
where
is the velocity potential at the bed. Invoking
Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \na ...
for
, as valid for
incompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An e ...
, gives:
:
since the vertical velocity
is zero at the – impermeable – horizontal bed
. This series may subsequently be truncated to a finite number of terms.
Original Boussinesq equations
Derivation
For
water waves
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of ...
on an
incompressible fluid and
irrotational flow
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not c ...
in the
plane, the
boundary conditions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
at the
free surface elevation
are:
:
where:
*
is the horizontal
flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the ...
component:
,
*
is the vertical
flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the ...
component:
,
*
is the
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
by
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the str ...
.
Now the Boussinesq approximation for the
velocity potential , as given above, is applied in these
boundary conditions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
. Further, in the resulting equations only the
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear re ...
and
quadratic terms with respect to
and
are retained (with
the horizontal velocity at the bed
). The
cubic and higher order terms are assumed to be negligible. Then, the following
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ho ...
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
is a constant independent of position
. When the right-hand sides of the above equations are set to zero, they reduce to the
shallow water equations
The shallow-water equations (SWE) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface). ...
.
Under some additional approximations, but at the same order of accuracy, the above set A can be reduced to a single
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
for the
free surface elevation
:
;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
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
, using the water depth
and gravitational acceleration
for non-dimensionalization, this equation reads, after
normalization
Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to:
* Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...
:
:
with:
Linear frequency dispersion
Water waves
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of ...
of different
wave length
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
s travel with different
phase speed
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
s, a phenomenon known as
frequency dispersion. For the case of
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refe ...
wave
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
, 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
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
.
The linear
frequency dispersion characteristics for the above set A of equations are:
[Dingemans (1997), p. 521.]
:
with:
*
the
phase speed
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
,
*
the
wave number
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
(
, with
the
wave length
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
).
The
relative error
The approximation error in a data value is the discrepancy between an exact value and some ''approximation'' to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute er ...
in the phase speed
for set A, as compared with
linear theory for water waves, is less than 4% for a relative wave number
. So, in
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
applications, set A is valid for wavelengths
larger than 4 times the water depth
.
The linear
frequency dispersion characteristics of equation B are:
[
:
The relative error in the phase speed for equation B is less than 4% for , equivalent to wave lengths longer than 7 times the water depth , called fairly long waves.
For short waves with equation B become physically meaningless, because there are no longer real-valued solutions of the ]phase speed
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
.
The original set of two partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ho ...
(Boussinesq, 1872, equation 25, see set A above) does not have this shortcoming.
The shallow water equations
The shallow-water equations (SWE) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface). ...
have a relative error in the phase speed less than 4% for wave lengths in excess of 13 times the water depth .
Boussinesq-type equations and extensions
There are an overwhelming number of mathematical models
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, b ...
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:
*varying bathymetry
Bathymetry (; ) is the study of underwater depth of ocean floors (''seabed topography''), lake floors, or river floors. In other words, bathymetry is the underwater equivalent to hypsometry or topography. The first recorded evidence of water ...
,
*improved frequency dispersion,
*improved non-linear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
behavior,
*making a Taylor expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
around different vertical elevations,
*dividing the fluid domain in layers, and applying the Boussinesq approximation in each layer separately,
*inclusion of wave breaking
In fluid dynamics, a breaking wave or breaker is a wave whose amplitude reaches a critical level at which large amounts of wave energy transform into turbulent kinetic energy. At this point, simple physical models that describe wave dynamic ...
,
*inclusion of surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) t ...
,
*extension to internal waves on an interface
Interface or interfacing may refer to:
Academic journals
* ''Interface'' (journal), by the Electrochemical Society
* '' Interface, Journal of Applied Linguistics'', now merged with ''ITL International Journal of Applied Linguistics''
* '' Int ...
between fluid domains of different mass density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
,
*derivation from a variational principle
In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations, which concerns finding functions that optimize the values of quantities that depend on those funct ...
.
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:
*the Korteweg–de Vries equation for wave propagation
Wave propagation is any of the ways in which waves travel. Single wave propagation can be calculated by 2nd order wave equation (standing wavefield) or 1st order one-way wave equation.
With respect to the direction of the oscillation relative to ...
in one horizontal dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
,
*the Kadomtsev–Petviashvili equation
In mathematics and physics, the Kadomtsev–Petviashvili equation (often abbreviated as KP equation) is a partial differential equation to describe nonlinear wave motion. Named after Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvil ...
for (near uni-directional) wave propagation
Wave propagation is any of the ways in which waves travel. Single wave propagation can be calculated by 2nd order wave equation (standing wavefield) or 1st order one-way wave equation.
With respect to the direction of the oscillation relative to ...
in two horizontal dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
s,
*the nonlinear Schrödinger equation
In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlin ...
(NLS equation) for the complex-valued amplitude of narrowband
Narrowband signals are signals that occupy a narrow range of frequencies or that have a small fractional bandwidth. In the audio spectrum, narrowband sounds are sounds that occupy a narrow range of frequencies. In telephony, narrowband is usu ...
waves (slowly modulated
In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the ''carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informa ...
waves).
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
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
Short Message/Messaging Service, commonly abbreviated as SMS, is a text messaging service component of most telephone, Internet and mobile device systems. It uses standardized communication protocols that let mobile devices exchange short text ...
. Some of the free Boussinesq models are Celeris, COULWAVE, and FUNWAVE. Most numerical models employ finite-difference, finite-volume or finite element
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
techniques for the discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerica ...
of the model equations. Scientific reviews and intercomparisons of several Boussinesq-type equations, their numerical approximation and performance are e.g. , and .
Notes
References
*
*
* ''See Part 2, Chapter 5''.
*
*
*
*
*
{{physical oceanography
Fluid dynamics
Water waves
Equations of fluid dynamics