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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 mild-slope equation describes the combined effects of
diffraction
Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
and
refraction
In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomeno ...
for
water wave
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 t ...
s propagating over
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 de ...
and due to lateral boundaries—like
breakwater
Breakwater may refer to:
* Breakwater (structure), a structure for protecting a beach or harbour
Places
* Breakwater, Victoria, a suburb of Geelong, Victoria, Australia
* Breakwater Island
Breakwater Island () is a small island in the Palme ...
s and
coastline
The coast, also known as the coastline or seashore, is defined as the area where land meets the ocean, or as a line that forms the boundary between the land and the coastline. The Earth has around of coastline. Coasts are important zones in n ...
s. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used 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 surges ...
to compute the wave-field changes near
harbour
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 ...
s and
coast
The coast, also known as the coastline or seashore, is defined as the area where land meets the ocean, or as a line that forms the boundary between the land and the coastline. The Earth has around of coastline. Coasts are important zones in n ...
s.
The mild-slope equation models the propagation and transformation of water waves, as they travel through waters of varying depth and interact with lateral boundaries such as
cliff
In geography and geology, a cliff is an area of rock which has a general angle defined by the vertical, or nearly vertical. Cliffs are formed by the processes of weathering and erosion, with the effect of gravity. Cliffs are common on co ...
s,
beach
A beach is a landform alongside a body of water which consists of loose particles. The particles composing a beach are typically made from rock, such as sand, gravel, shingle, pebbles, etc., or biological sources, such as mollusc shel ...
es,
seawall
A seawall (or sea wall) is a form of coastal defense constructed where the sea, and associated coastal processes, impact directly upon the landforms of the coast. The purpose of a seawall is to protect areas of human habitation, conservation ...
s and breakwaters. As a result, it describes the variations in 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 amplit ...
, or equivalently
wave height
In fluid dynamics, the wave height of a surface wave is the difference between the elevations of a crest and a neighboring trough. ''Wave height'' is a term used by mariners, as well as in coastal, ocean and naval engineering.
At sea, the term ' ...
. From the wave amplitude, the amplitude of the
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 f ...
oscillations underneath the water surface can also be computed. These quantities—wave amplitude and flow-velocity amplitude—may subsequently be used to determine the wave effects on coastal and offshore structures, ships and other floating objects,
sediment transport
Sediment transport is the movement of solid particles (sediment), typically due to a combination of gravity acting on the sediment, and/or the movement of the fluid in which the sediment is entrained. Sediment transport occurs in natural system ...
and resulting
bathymetric
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 de ...
changes of the sea bed and coastline, mean flow fields and
mass transfer
Mass transfer is the net movement of mass from one location (usually meaning stream, phase, fraction or component) to another. Mass transfer occurs in many processes, such as absorption, evaporation, drying, precipitation, membrane filtration, ...
of dissolved and floating materials. Most often, the mild-slope equation is solved by computer using methods from
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
.
A first form of the mild-slope equation was developed by
Eckart Eckart is a German surname, and may refer to:
* Anselm Eckart (1721–1809), German Jesuit missionary
* Carl Eckart
* Dennis E. Eckart (born 1950), American lawyer, former member of the U.S. House of Representatives
* Dietrich Eckart (1868–1923), ...
in 1952, and an improved version—the mild-slope equation in its classical formulation—has been derived independently by Juri Berkhoff in 1972. Thereafter, many modified and extended forms have been proposed, to include the effects of, for instance:
wave–current interaction
In fluid dynamics, wave–current interaction is the interaction between surface gravity waves and a mean flow. The interaction implies an exchange of energy, so after the start of the interaction both the waves and the mean flow are affected.
Fo ...
, wave
nonlinearity
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 other ...
, steeper sea-bed slopes,
bed friction and
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 dynamics ...
. Also
parabolic approximations to the mild-slope equation are often used, in order to reduce the computational cost.
In case of a constant depth, the mild-slope equation reduces to the
Helmholtz equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation
\nabla^2 f = -k^2 f,
where is the Laplace operator (or "Laplacian"), is the eigenv ...
for wave diffraction.
Formulation for monochromatic wave motion
For
monochromatic
A monochrome or monochromatic image, object or color scheme, palette is composed of one color (or lightness, values of one color). Images using only Tint, shade and tone, shades of grey are called grayscale (typically digital) or Black and wh ...
waves according to
linear theory
In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.
Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.
As a mathematical abstraction or ...
—with the
free surface
In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress,
such as the interface between two homogeneous fluids.
An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
elevation given as
and the waves propagating on a fluid layer of
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 ''arithme ...
water depth
—the mild-slope equation is:
where:
*
is the
complex-valued amplitude
In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to ...
of the free-surface elevation
*
is the horizontal position;
*
is the
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of the monochromatic wave motion;
*
is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
;
*
means taking the
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
of the quantity between braces;
*
is the horizontal
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
operator;
*
is the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
operator;
*
is the
wavenumber
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 ...
;
*
is 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, ...
of the waves and
*
is the
group speed
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space.
For example, if a stone is thrown into the middl ...
of the waves.
The phase and group speed depend on the
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the d ...
, and are derived from
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 mea ...
as:
where
*
is
Earth's gravity
The gravity of Earth, denoted by , is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation).
It is a vector quantity ...
and
*
is the
hyperbolic tangent
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the un ...
.
For a given angular frequency
, the wavenumber
has to be solved from the dispersion equation, which relates these two quantities to the water depth
.
Transformation to an inhomogeneous Helmholtz equation
Through the transformation
the mild slope equation can be cast in the form of an
inhomogeneous Helmholtz equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation
\nabla^2 f = -k^2 f,
where is the Laplace operator (or "Laplacian"), is the eigenvalu ...
:
[
where is the ]Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
.
Propagating waves
In spatially coherent
Coherence, coherency, or coherent may refer to the following:
Physics
* Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference
* Coherence (units of measurement), a deri ...
fields of propagating waves, it is useful to split the complex amplitude
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
in its amplitude and phase, both real valued:
where
* is the amplitude or absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of and
* is the wave phase, which is the argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
of
This transforms the mild-slope equation in the following set of equations (apart from locations for which is singular):[
where
* is the ]average
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...
wave-energy density per unit horizontal area (the sum of the kinetic
Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to:
* Kinetic theory of gases, Kinetic theory, describing a gas as particles in random motion
* Kinetic energy, the energy of an object that it possesses due to i ...
and potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potentia ...
densities),
* is the effective wavenumber vector, with components
* is the effective group velocity
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space.
For example, if a stone is thrown into the middl ...
vector,
* is the fluid 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 ...
, and
* is the acceleration by the Earth's gravity
The gravity of Earth, denoted by , is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation).
It is a vector quantity ...
.
The last equation shows that wave energy is conserved in the mild-slope equation, and that the wave energy is transported in the -direction normal to the wave crest
Crest or CREST may refer to:
Buildings
*The Crest (Huntington, New York), a historic house in Suffolk County, New York
*"The Crest", an alternate name for 63 Wall Street, in Manhattan, New York
*Crest Castle (Château Du Crest), Jussy, Switzerla ...
s (in this case of pure wave motion without mean currents).[ The effective group speed is different from the group speed
The first equation states that the effective wavenumber is ]irrotational
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 ...
, a direct consequence of the fact it is the derivative of the wave phase , a scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
. The second equation is the eikonal equation
An eikonal equation (from Greek εἰκών, image) is a non-linear first-order partial differential equation that is encountered in problems of wave propagation.
The classical eikonal equation in geometric optics is a differential equation of ...
. It shows the effects of diffraction on the effective wavenumber: only for more-or-less progressive waves, with the splitting into amplitude and phase leads to consistent-varying and meaningful fields of and . Otherwise, ''κ''2 can even become negative. When the diffraction effects are totally neglected, the effective wavenumber ''κ'' is equal to , and the geometric optics
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
approximation for wave refraction
In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomeno ...
can be used.[
When is used in the mild-slope equation, the result is, apart from a factor :
Now both the real part and the imaginary part of this equation have to be equal to zero:
The effective wavenumber vector is ''defined'' as the gradient of the wave phase:
and its ]vector length
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is ze ...
is
Note that is an irrotational
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 ...
field, since the curl of the gradient is zero:
Now the real and imaginary parts of the transformed mild-slope equation become, first multiplying the imaginary part by :
The first equation directly leads to the eikonal equation above for , while the second gives:
which—by noting that in which the angular frequency is a constant for time-harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
motion—leads to the wave-energy conservation equation.
Derivation of the mild-slope equation
The mild-slope equation can be derived by the use of several methods. Here, we will use a variational approach.[ The fluid is assumed to be ]inviscid
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
and incompressible
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 flow is assumed to be irrotational
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 ...
. These assumptions are valid ones for surface gravity waves, since the effects of vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wit ...
and viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inte ...
are only significant in the Stokes boundary layer
In fluid dynamics, Stokes problem also known as Stokes second problem or sometimes referred to as Stokes boundary layer or Oscillating boundary layer is a problem of determining the flow created by an oscillating solid surface, named after Sir Geor ...
s (for the oscillatory part of the flow). Because the flow is irrotational, the wave motion can be described using 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 appr ...
theory.
Luke's variational principle
Luke's Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
formulation gives a variational formulation for 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 other ...
surface gravity waves.
For the case of a horizontally unbounded domain with a constant 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 ...
, a free fluid surface at and a fixed sea bed at Luke's variational principle uses the Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
where is the horizontal Lagrangian density
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
, given by:
where is the velocity potential, with the 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 f ...
components being and in the , and directions, respectively.
Luke's Lagrangian formulation can also be recast into a Hamiltonian formulation in terms of the surface elevation and velocity potential at the free surface.
Taking the variations of with respect to the potential and surface elevation leads to the Laplace 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 = \nab ...
for in the fluid interior, as well as all the boundary conditions both on the free surface as at the bed at
Linear wave theory
In case of linear wave theory, the vertical integral in the Lagrangian density is split into a part from the bed to the mean surface at and a second part from to the free surface . Using a Taylor series
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 serie ...
expansion for the second integral around the mean free-surface elevation and only retaining quadratic terms in and the Lagrangian density for linear wave motion becomes
The term in the vertical integral is dropped since it has become dynamically uninteresting: it gives a zero contribution to the Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
s, with the upper integration limit now fixed. The same is true for the neglected bottom term proportional to in the potential energy.
The waves propagate in the horizontal plane, while the structure of the potential is not wave-like in the vertical -direction. This suggests the use of the following assumption on the form of the potential
with normalisation at the mean free-surface elevation
Here is the velocity potential at the mean free-surface level Next, the mild-slope assumption is made, in that the vertical shape function changes slowly in the -plane, and horizontal derivatives of can be neglected in the flow velocity. So:
As a result:
with
The Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
s for this Lagrangian density are, with representing either or
Now is first taken equal to and then to
As a result, the evolution equations for the wave motion become:[
with the horizontal gradient operator: where superscript denotes the ]transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
.
The next step is to choose the shape function and to determine and
Vertical shape function from Airy wave theory
Since the objective is the description of waves over mildly sloping beds, the shape function is chosen according to 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 mea ...
. This is the linear theory of waves propagating in constant depth The form of the shape function is:[
with now in general not a constant, but chosen to vary with and according to the local depth and the linear dispersion relation:][
Here a constant angular frequency, chosen in accordance with the characteristics of the wave field under study. Consequently, the integrals and become:][
The following time-dependent equations give the evolution of the free-surface elevation and free-surface potential ][
From the two evolution equations, one of the variables or can be eliminated, to obtain the time-dependent form of the mild-slope equation:][
and the corresponding equation for the free-surface potential is identical, with replaced by The time-dependent mild-slope equation can be used to model waves in a narrow band of frequencies around
]
Monochromatic waves
Consider monochromatic waves with complex amplitude and angular frequency :
with and chosen equal to each other, Using this in the time-dependent form of the mild-slope equation, recovers the classical mild-slope equation for time-harmonic wave motion:[
]
Applicability and validity of the mild-slope equation
The standard mild slope equation, without extra terms for bed slope and bed curvature, provides accurate results for the wave field over bed slopes ranging from 0 to about 1/3. However, some subtle aspects, like the amplitude of reflected waves, can be completely wrong, even for slopes going to zero. This mathematical curiosity has little practical importance in general since this reflection becomes vanishingly small for small bottom slopes.
Notes
References
*, 2 Parts, 967 pages.
*
*, 740 pages.
*
*
{{physical oceanography
Coastal geography
Equations of fluid dynamics
Water waves