In
fluid dynamicsIn physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
,
Airy wave theory (often referred to as
linear wave theory) gives a
linearisedA 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 general, nonlinear case....
description of the
propagationWave propagation is any of the ways in which waves travel.With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves....
of
gravity waveIn fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media which has the restoring force of gravity or buoyancy....
s on the surface of a homogeneous
fluidA fluid is a substance that continually deforms under an applied shear stress. All gases are fluids, but not all liquids are fluids. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
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 AirySir George Biddell Airy FRS was an English mathematician and astronomer, Astronomer Royal from 1835 to 1881. His many achievements include work on planetary orbits, measuring the mean density of the Earth, a method of solution of two-dimensional problems in solid mechanics and, in his role as...
in the 19
th century.
Airy wave theory is often applied in
ocean engineeringOcean Engineering is an ambiguously-defined discipline, but may refer to Oceanographic Engineering a term describing Marine Electronics Engineering applied to supporting the work of Oceanographers; or, may refer to Offshore Engineering, or Maritime Engineering, which is the branch of engineering...
and coastal engineering for the modelling of random
sea stateIn oceanography, a sea state is the general condition of the free surface on a large body of water—with respect to wind waves and swell—at a certain location and moment. A sea state is characterised by statistics, including the wave height, period, and power spectrum. The sea state varies with...
s — giving a description of the wave
kinematicsKinematics is the branch of classical mechanics that describes the motion of objects without consideration of the causes leading to the motion....
and
dynamicsDynamics may refer to:In Physics:*Dynamics , in physics, dynamics refers to time evolution of physical processes...
of high-enough accuracy for many purposes.
Further, several
second-orderPerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
nonlinear properties of surface gravity waves, and their propagation, can be estimated from its results. This linear theory is often used to get a quick and rough estimate of wave characteristics and their effects.
Description
Airy wave theory uses a
potential flowIn fluid dynamics, potential 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 approximation for several applications...
approach to describe the motion of gravity waves on a fluid surface. The use of — inviscid and irrotational — potential flow in water waves is remarkably successful, given its failure to describe many other fluid flows where it is often essential to take
viscosityViscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness." Thus, water is "thin," having a lower viscosity, while honey is "thick," having a higher viscosity...
,
vorticityVorticity is a concept used in fluid dynamics. In the simplest sense, vorticity is the tendency for elements of the fluid to "spin."More formally, vorticity can be related to the amount of "circulation" or "rotation" in a fluid.The average vorticity in a small region of fluid flow is equal to the...
,
turbulenceIn fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic, stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time. Flow that is not turbulent is called laminar flow...
and/or
flow separationAll solid objects travelling through a fluid acquire a boundary layer of fluid around them where viscous forces occur in the layer of fluid close to the solid surface. Boundary layers can be either laminar or turbulent...
into account. This is due to the fact that for the oscillatory part of the fluid motion, wave-induced vorticity is restricted to some thin oscillatory
Stokes boundary layerIn fluid dynamics, the Stokes boundary layer, or oscillatory boundary layer, refers to the boundary layer close to a solid wall in oscillatory flow of a viscous fluid...
s at the boundaries of the fluid domain.
Airy wave theory is often used in
ocean engineeringOcean Engineering is an ambiguously-defined discipline, but may refer to Oceanographic Engineering a term describing Marine Electronics Engineering applied to supporting the work of Oceanographers; or, may refer to Offshore Engineering, or Maritime Engineering, which is the branch of engineering...
and coastal engineering. Especially for random waves, sometimes called
wave turbulenceWave turbulence is a set of waves deviated far from thermal equilibrium. Such state is accompanied by dissipation. It is either decaying turbulence or requires external source of energy to sustain it...
, the evolution of the wave statistics — including the wave
spectrumA spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by...
— is predicted well over not too long distances (in terms of wavelengths) and in not too shallow water.
DiffractionDiffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings...
is one of the wave effects which can be described with Airy wave theory. Further, by using the WKBJ approximation,
wave shoalingIn fluid dynamics, wave shoaling is the effect in which surface waves on a water layer of decreasing depth change their wave height...
and
refractionRefraction is the change in direction of a wave due to a change in its velocity. This is most commonly observed when a wave passes from one medium to another...
can be predicted.
Earlier attempts to describe surface gravity waves using potential flow were made by, among others,
LaplacePierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...
, Poisson,
CauchyAugustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner. He also gave several important theorems in complex analysis and initiated the study of permutation...
and
KellandPhilip Kelland was a Scottish mathematician. He was known mainly for his great influence on the development of education in Scotland.-Early life:Kelland was born in 1808 in Dunster, Somerset, England...
. But
AirySir George Biddell Airy FRS was an English mathematician and astronomer, Astronomer Royal from 1835 to 1881. His many achievements include work on planetary orbits, measuring the mean density of the Earth, a method of solution of two-dimensional problems in solid mechanics and, in his role as...
was the first to publish the correct derivation and formulation in 1841. Soon after, in 1847, the linear theory of Airy was extended by
StokesSir George Gabriel Stokes, 1st Baronet FRS , was a mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...
for non-linear wave motion, correct up to third order in the wave steepness. Even before Airy's linear theory,
GerstnerFrantišek Josef Gerstner was a Bohemian physicist and engineer.Gerstner studied at the Jesuits gymnasium in Chomutov, after which he studied mathematics and astronomy at the Faculty of Philosophy in Prague between 1772 and 1777...
derived a nonlinear
trochoidthumb|290px|right|A [[cycloid]] generated by a rolling circleTrochoid is the word created by Gilles de Roberval for the curve described by a fixed point as a circle rolls along a straight line...
al wave theory in 1804, which however is not irrotational.
Airy wave theory is a linear theory for the propagation of waves on the surface of a potential flow and above a horizontal bottom. The free surface elevation
η(
x,
t) of one wave component is sinusoidal, as a function of horizontal position
x and time
t:
where
- a is the wave amplitude
Amplitude is the magnitude of change in the oscillating variable, with each oscillation, within an oscillating system. For instance, sound waves are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
in metreThe metre or meter is the basic unit of length in the International System of Units . Historically, the metre was defined by the French Academy of Sciences as the length between two marks on a platinum-iridium bar, which was designed to represent one ten-millionth of the distance from the Equator...
,
- cos is the cosine function,
- k is the angular wavenumber in radian
The radian is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees, or about 57°17′45″. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
per metre, related to the wavelengthIn physics, the wavelength of a sinusoidal wave is the spatial period of the wave – the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
λ as
- ω is the angular frequency
In physics , angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
in radian per secondThe second , sometimes abbreviated sec., is the name of a unit of time, and is the International System of Units base unit of time...
, related to the period T and frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency....
f by
The waves propagate along the water surface with the phase speed
cp:
The angular wavenumber
k and frequency
ω are not independent parameters (and thus also wavelength
λ and period
T are not independent), but are coupled. Surface gravity waves on a fluid are
dispersiveIn fluid dynamics, dispersion of water waves generally refers to frequency dispersion. Frequency dispersion means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, and forced by gravity and surface tension...
waves — exhibiting frequency dispersion — meaning that each wavenumber has its own frequency and phase speed.
Note that in engineering the
wave heightIn fluid dynamics, the wave height of a surface wave is the difference between the elevations of a crest and a neighbouring trough. Wave height is a term used by mariners, as well as in coastal, ocean engineering and naval engineering....
H — the difference in elevation between
crestA crest is the point on a wave with the maximum value or upward displacement within a cycle. A trough is the opposite of a crest, so the minimum or lowest point in a cycle....
and trough — is often used:
valid in the present case of linear periodic waves.
Underneath the surface, there is a fluid motion associated with the free surface motion. While the surface elevation shows a propagating wave, the fluid particles are in an
orbital motionIn physics, an orbit is the gravitationally curved path of one object around a point or another body, for example the gravitational orbit of a planet around a star....
. Within the framework of Airy wave theory, the orbits are in deep water closed
circleA circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the centre. The common distance of the points of a circle from its center is called its radius....
s, and in finite depth closed
ellipsoidAn ellipsoid is a type of quadric surface that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is...
s — with the ellipsoids becoming flatter near the bottom of the fluid layer. So while the wave propagates, the fluid particles just orbit (oscillate) around their
averageIn mathematics, an average, central tendency of a data set is a measure of the "middle" or "expected" value of the data set. There are many different descriptive statistics that can be chosen as a measurement of the central tendency of the data items. These include arithmetic mean, the median and...
position. With the propagating wave motion, the fluid particles transfer energy in the wave propagation direction, without having a mean velocity. The diameter of the orbits reduces with depth below the free surface. In deep water, the orbit's diameter is reduced to 4% of its free-surface value at a depth of half a wavelength.
In a similar fashion, there is also a pressure oscillation underneath the free surface, with wave-induced pressure oscillations reducing with depth below the free surface — in the same way as for the orbital motion of fluid parcels.
Flow problem formulation
The waves propagate in the horizontal direction, with
coordinateA coordinate is a number that determines the location of a point along some line or curve. A list of two, three, or more coordinates can be used to determine the location of a point on a surface, volume, or higher-dimensional domain....
x, and a fluid domain bound above by a free surface at
z =
η(
x,
t), with
z the vertical coordinate (positive in the upward direction) and
t being
timeTime is a component of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects...
. The level
z = 0 corresponds with the mean surface elevation. The
impermeablePermeability, permeable and semipermeable have several meanings:*Permeability , the degree of magnetization of a material in response to a magnetic field...
bed underneath the fluid layer is at
z = -
h. Further, the flow is assumed to be
incompressibleIn fluid mechanics or more generally continuum mechanics, an incompressible flow is solid or fluid flow in which the divergence of velocity is zero. This is more precisely termed isochoric flow. It is an idealization used to simplify analysis. In reality, all materials are compressible to some...
and irrotational — a good approximation of the flow in the fluid interior for waves on a liquid surface — and
potential theoryIn mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" arises from the fact that, in 19th-century physics, the fundamental forces of nature were believed to be derived from potentials which...
can be used to describe the flow. The velocity potential
Φ(
x,
z,
t) is related to the
flow velocityIn fluid dynamics the flow velocity, or velocity field, of a fluid is a vector field which is used to mathematically describe the motion of a fluid...
components
ux and
uz in the horizontal (
x) and vertical (
z) directions by:
Then, due to the continuity equation for an incompressible flow, the potential
Φ has to satisfy the Laplace equation:
Boundary conditions are needed at the bed and the free surface in order to close the system of equations. For their formulation within the framework of linear theory, it is necessary to specify what the base state (or
zeroth-order solutionPerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
) of the flow is. Here, we assume the base state is rest, implying the mean flow velocities are zero.
The bed being impermeable, leads to the
kinematicKinematics is the branch of classical mechanics that describes the motion of objects without consideration of the causes leading to the motion....
bed boundary-condition:
In case of deep water — by which is meant infinite water depth, from a mathematical point of view — the flow velocities have to go to zero in the
limit as the vertical coordinate goes to minus infinity:
z → -∞.
At the free surface, for
infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
waves, the vertical motion of the flow has to be equal to the vertical velocity of the free surface. This leads to the kinematic free-surface boundary-condition:
If the free surface elevation
η(
x,
t) was a known function, this would be enough to solve the flow problem. However, the surface elevation is an extra unknown, for which an additional boundary condition is needed. This is provided by
Bernoulli's equationIn fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...
for an unsteady potential flow. The pressure above the free surface is assumed to be constant. This constant pressure is taken equal to zero, without loss of generality, since the level of such a constant pressure does not alter the flow. After linearisation, this gives the dynamic free-surface boundary condition:
Because this is a linear theory, in both free-surface boundary conditions — the kinematic and the dynamic one, equations (3) and (4) — the value of
Φ and ∂
Φ/∂
z at the fixed mean level
z = 0 is used.
Solution for a progressive monochromatic wave
For a propagating wave of a single frequency — a monochromatic wave — the surface elevation is of the form:
The associated velocity potential, satisfying the Laplace equation (1) in the fluid interior, as well as the kinematic boundary conditions at the free surface (2), and bed (3), is:
with sinh and cosh the hyperbolic sine and hyperbolic cosine function, respectively.
But
η and
Φ also have to satisfy the dynamic boundary condition, which results in non-trivial (non-zero) values for the wave amplitude
a only if the linear
dispersion relationIn fluid dynamics, dispersion of water waves generally refers to frequency dispersion. Frequency dispersion means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, and forced by gravity and surface tension...
is satisfied:
with tanh the hyperbolic tangent. So angular frequency
ω and wavenumber
k — or equivalently period
T and wavelength
λ — cannot be chosen independently, but are related. This means that wave propagation at a fluid surface is an eigenproblem. When
ω and
k satisfy the dispersion relation, the wave amplitude
a can be chosen freely (but small enough for Airy wave theory to be a valid approximation).
Table of wave quantities
In the table below, several flow quantities and parameters according to Airy wave theory are given. The given quantities are for a bit more general situation as for the solution given above. Firstly, the waves may propagate in an arbitrary horizontal direction in the
x = (
x,
y) plane. The
wavenumberWavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters . Wavenumber is the spatial analog of frequency, that is, it is the measurement of the number of wavelengths per unit distance, or more commonly times that, or the number...
vector is
k, and is perpendicular to the cams of the
wave crestsA crest is the point on a wave with the maximum value or upward displacement within a cycle. A trough is the opposite of a crest, so the minimum or lowest point in a cycle....
. Secondly, allowance is made for a mean flow velocity
U, in the horizontal direction and uniform over (independent of) depth
z. This introduces a Doppler shift in the dispersion relations. At an Earth-fixed location, the
observed angular frequency (or
absolute angular frequency) is
ω. On the other hand, in a
frame of referenceA frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
moving with the mean velocity
U (so the mean velocity as observed from this reference frame is zero), the angular frequency is different. It is called the
intrinsic angular frequency (or
relative angular frequency), denoted as
σ. So in pure wave motion, with
U=
0, both frequencies
ω and
σ are equal. The wave number
k (and wave length
λ) are independent of the
frame of referenceA frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
, and have no Doppler shift (for monochromatic waves).
The table only gives the oscillatory parts of flow quantities — velocities, particle excursions and pressure — and not their mean value or drift.
The oscillatory particle excursions
ξx and
ξz are the time
integralIntegration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integralis defined informally...
s of the oscillatory flow velocities
ux and
uz respectively.
Water depth is classified into three regimes:
- deep water — for a water depth larger than half the wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave – the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
, h > ½ λ, the phase speed of the waves is hardly influenced by depth (this is the case for most wind waves on the sea and ocean surface),
- shallow water — for a water depth smaller than the wavelength divided by 20, h < λ, the phase speed of the waves is only dependent on water depth, and no longer a function of period
Period or periodic may refer to:-Language and literature:* Full stop, a punctuation mark indicating the end of a sentence or phrase* Periodic sentence, a sentence that is not grammatically complete until its end...
or wavelength; and
- intermediate depth — all other cases, λ < h < ½ λ, where both water depth and period (or wavelength) have a significant influence on the solution of Airy wave theory.
In the limiting cases of deep and shallow water, simplifying approximations to the solution can be made. While for intermediate depth, the full formulations have to be used.
| Properties of gravity waves on the surface of deep water, shallow water and at intermediate depth, according to Airy wave theory |
| quantity |
symbol |
units |
deep water( h > ½ λ ) |
shallow water( h < 0.05 λ ) |
intermediate depth( all λ and h ) |
| surface elevation |
|
mThe metre or meter is the basic unit of length in the International System of Units . Historically, the metre was defined by the French Academy of Sciences as the length between two marks on a platinum-iridium bar, which was designed to represent one ten-millionth of the distance from the Equator...
|
|
| wave phase |
|
rad The radian is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees, or about 57°17′45″. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
|
|
observed angular frequencyIn physics , angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
|
|
rad / s The second , sometimes abbreviated sec., is the name of a unit of time, and is the International System of Units base unit of time...
|
|
| intrinsic angular frequency |
|
rad / s |
|
| unit vector in the wave propagation direction |
|
– |
|
dispersion relationIn fluid dynamics, dispersion of water waves generally refers to frequency dispersion. Frequency dispersion means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, and forced by gravity and surface tension...
|
|
rad / s |
|
|
|
| phase speed |
|
m / s |
|
|
|
| group speed |
|
m / s |
|
|
|
| ratio |
|
– |
|
|
|
| horizontal velocity |
|
m / s |
|
|
|
| vertical velocity |
|
m / s |
|
|
|
| horizontal particle excursion |
|
m |
|
|
|
| vertical particle excursion |
|
m |
|
|
|
pressurePressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
oscillation |
|
N / m2 |
|
|
|
Surface tension effects
Due to
surface tensionSurface tension is a property of the surface of a liquid. It is what causes the surface portion of liquid to be attracted to another surface, such as that of another portion of liquid .Applying Newtonian physics to the forces that arise due to surface tension accurately predicts many liquid behaviors...
, the dispersion relation changes to:
with
γ the surface tension, with
SISi, si, or SI may refer to :- Places :* Mount Si, a mountain in state of Washington* Si County, county in Anhui, China* Si River, a river in China* Slovenia, a European nation Si, si, or SI may refer to (all SI unless otherwise stated):- Places :* Mount Si, a mountain in state of Washington* Si...
units in N/m
2. All above equations for linear waves remain the same, if the gravitational acceleration
g is replaced by
As a result of surface tension, the waves propagate faster. Surface tension only has influence for short waves, with wavelengths less than a few decimeters in case of a water–air interface. For very short wavelengths — two millimeter in case of the interface between air and water – gravity effects are negligible.
Interfacial waves
Surface gravity waves are a special case of interfacial waves, on the
interfaceAn interface is a surface forming a common boundary among two different phases, such as an insoluble solid and a liquid, two immiscible liquids or a liquid and an insoluble gas. The importance of the interface depends on which type of system is being treated: the bigger the quotient area/volume,...
between two fluids of different
densityThe density of a material is defined as its mass per unit volume. The symbol of density is ρ .- Formula :Mathematically:where: is the density, is the mass, is the volume....
. Consider two fluids separated by an interface, and without further boundaries. Then their dispersion relation becomes:
where
ρ and
ρ‘ are the densities of the two fluids, below (
ρ) and above (
ρ‘) the interface, respectively. For interfacial waves to exist, the lower layer has to be heavier than the upper one,
ρ > ρ‘. Otherwise, the interface is unstable and a Rayleigh–Taylor instability develops.
Second-order wave properties
Several
second-orderPerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
wave properties,
i.e. quadraticIn mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms...
in the wave amplitude
a, can be derived directly from Airy wave theory. They are of importance in many practical applications,
e.g forecasts of wave conditions. Using a WKBJ approximation, second-order wave properties also find their applications in describing waves in case of slowly-varying
bathymetryBathymetry is the study of underwater depth of the third dimension of lake or ocean floors. In other words, bathymetry is the underwater equivalent to hypsometry. The name comes from Greek βαθυς, deep, and μετρον, measure...
, and mean-flow variations of currents and surface elevation. As well as in the description of the wave and mean-flow interactions due to time and space-variations in amplitude, frequency, wavelength and direction of the wave field itself.
Table of second-order wave properties
In the table below, several second-order wave properties — as well as the dynamical equations they satisfy in case of slowly-varying conditions in space and time — are given. More details on these can be found below. The table gives results for wave propagation in one horizontal spatial dimension. Further on in this section, more detailed descriptions and results are given for the general case of propagation in two-dimensional horizontal space.
| Second-order quantities and their dynamics, using results of Airy wave theory |
| quantity |
symbol |
units |
formula |
| mean wave-energy density per unit horizontal area |
|
J The joule , named for James Prescott Joule, is the derived unit of energy in the International System of Units. It is the energy exerted by a force of one newton acting to move an object through a distance of one metre...
/ m2 |
|
radiation stress or excess horizontal momentumIn classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section "modern definitions of momentum" on this page...
fluxIn the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as the amount that flows through a unit area per unit time...
due to the wave motion |
|
N / m |
|
| wave action |
|
J·s / m2 |
|
| mean mass-flux due to the wave motion or the wave pseudo-momentum |
|
kg / (m·s) |
|
| mean horizontal mass-transport velocity |
|
m / s |
|
Stokes driftFor a pure wave motion in fluid dynamics, the Stokes drift velocity is the average velocity when following a specific fluid parcel as it travels with the fluid flow...
|
|
m / s |
|
| wave-energy propagation |
|
J / (m2·s) |
|
| wave action conservation |
|
J / m2 |
|
| wave-crest A crest is the point on a wave with the maximum value or upward displacement within a cycle. A trough is the opposite of a crest, so the minimum or lowest point in a cycle....
conservation |
|
rad / (m·s) |
with |
| mean mass conservation |
|
kg / (m2·s) |
|
| mean horizontal-momentum evolution |
|
N / m2 |
|
The last four equations describe the evolution of slowly-varying wave trains over
bathymetryBathymetry is the study of underwater depth of the third dimension of lake or ocean floors. In other words, bathymetry is the underwater equivalent to hypsometry. The name comes from Greek βαθυς, deep, and μετρον, measure...
in interaction with the mean flow, and can be derived from a variational principle: Whitham's average
LagrangianThe Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics. In classical mechanics, the...
method. In the mean horizontal-momentum equation,
d(
x) is the still water depth,
i.e. the bed underneath the fluid layer is located at
z = –
d. Note that the mean-flow velocity in the mass and momentum equations is the
mass transport velocity , including the splash-zone effects of the waves on horizontal mass transport, and not the mean
EulerianIn fluid dynamics and finite-deformation plasticity the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. Plotting the position of an individual parcel through time gives the...
velocity (e.g. as measured with a fixed flow meter).
Wave energy density
Wave energy is a quantity of primary interest, since it is a primary quantity that is transported with the wave trains. As can be seen above, many wave quantities like surface elevation and orbital velocity are oscillatory in nature with zero mean (within the framework of linear theory). In water waves, the most used energy measure is the mean wave energy density per unit horizontal area. It is the sum of the
kineticThe kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its current velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its...
and
potential energyPotential energy is energy stored within a physical system as a result of the position or configuration of the different parts of that system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do work in the process...
density, integrated over the depth of the fluid layer and averaged over the wave phase. Simplest to derive is the mean potential energy density per unit horizontal area
Epot of the surface gravity waves, which is the deviation of the potential energy due to the presence of the waves:
with an overbar denoting the mean value (which in the present case of periodic waves can be taken either as a time average or an average over one wavelength in space).
The mean kinetic energy density per unit horizontal area
Ekin of the wave motion is similarly found to be:
with
σ the intrinsic frequency, see the table of wave quantities. Using the dispersion relation, the result for surface gravity waves is:
As can be seen, the mean kinetic and potential energy densities are equal. This is a general property of energy densities of progressive linear waves in a conservative system.. Adding potential and kinetic contributions,
Epot and
Ekin, the mean energy density per unit horizontal area
E of the wave motion is:
In case of surface tension effects not being negligible, their contribution also adds to the potential and kinetic energy densities, giving
with
γ the
surface tensionSurface tension is a property of the surface of a liquid. It is what causes the surface portion of liquid to be attracted to another surface, such as that of another portion of liquid .Applying Newtonian physics to the forces that arise due to surface tension accurately predicts many liquid behaviors...
.
Wave action, wave energy flux and radiation stress
In general, there can be an energy transfer between the wave motion and the mean fluid motion. This means, that the wave energy density is not in all cases a conserved quantity (neglecting
dissipative effectsIn physics, dissipation embodies the concept of a dynamical system where important mechanical modes, such as waves or oscillations, lose energy over time, typically due to the action of friction or turbulence. The lost energy is converted into heat, raising the temperature of the system...
), but the total energy density — the sum of the energy density per unit area of the wave motion and the mean flow motion — is. However, there is for slowly-varying wave trains, propagating in slowly-varying
bathymetryBathymetry is the study of underwater depth of the third dimension of lake or ocean floors. In other words, bathymetry is the underwater equivalent to hypsometry. The name comes from Greek βαθυς, deep, and μετρον, measure...
and mean-flow fields, a similar and conserved wave quantity, the wave action :
with the action
fluxIn the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as the amount that flows through a unit area per unit time...
and the
group velocityThe group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....
vector. Action conservation forms the basis for many
wind wave modelIn fluid dynamics, wind wave modeling describes the effort to depict the sea state and predict the evolution of the energy of wind waves using numerical techniques...
s and
wave turbulenceWave turbulence is a set of waves deviated far from thermal equilibrium. Such state is accompanied by dissipation. It is either decaying turbulence or requires external source of energy to sustain it...
models. It is also the basis of coastal engineering models for the computation of
wave shoalingIn fluid dynamics, wave shoaling is the effect in which surface waves on a water layer of decreasing depth change their wave height...
. Expanding the above wave action conservation equation leads to the following evolution equation for the wave energy density:
with:
- is the mean wave energy density flux,
- is the radiation stress tensor
Tensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another...
and
- is the mean-velocity shear-rate
-Simple Shear:Shear rate for a fluid flowing between two fixed parallel plates is defined using the following equation:Where:* = The shear rate, measured in reciprocal seconds* = The velocity, measured in meters per second...
tensor.
In this equation in non-conservation form, the Frobenius inner product is the source term describing the energy exchange of the wave motion with the mean flow. Only in case the mean shear-rate is zero, the mean wave energy density is conserved. The two tensors and are in a
Cartesian coordinate systemA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length....
of the form:
with and the components of the wavenumber vector and similarly and the components in of the mean velocity vector .
Wave mass flux and wave momentum
The mean horizontal
momentumIn classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section "modern definitions of momentum" on this page...
per unit area induced by the wave motion — and also the wave-induced
mass fluxMass flux is the rate of mass flow across a unit area...
or mass
transportIn physics, chemistry, biology and engineering, a transport phenomenon is any of various mechanisms by which particles or quantities move from one place to another. The laws which govern transport connect a flux with a "motive force". Three common examples of transport phenomena are diffusion,...
— is:
which is an exact result for periodic progressive water waves, also valid for nonlinear waves. However, its validity strongly depends on the way how wave momentum and mass flux are defined.
StokesSir George Gabriel Stokes, 1st Baronet FRS , was a mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...
already identified two possible definitions of
phase velocityThe phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...
for periodic nonlinear waves:
- Stokes first definition of wave celerity
Celerity can refer to:*Speed, quickness.*Celerity BBS, a computer Bulletin Board System popular in the 1990s*Celerity, a power of supernatural quickness possessed by vampires in the roleplaying games Vampire: The Requiem and Vampire: The Masquerade...
(S1) — with the mean Eulerian flow velocityIn fluid dynamics and finite-deformation plasticity the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. Plotting the position of an individual parcel through time gives the...
equal to zero for all elevations z below the wave troughTrough may refer to:* Trough , a container for animal feed * Trough , a long depression less steep than a trench* Trough , an elongated region of low atmospheric pressure...
s, and
- Stokes second definition of wave celerity (S2) — with the mean mass transport equal to zero.
The above relation between wave momentum
M and wave energy density
E is valid within the framework of Stokes' first definition.
However, for waves perpendicular to a coast line or in closed laboratory wave channel, the second definition (S2) is more appropriate. These wave systems have zero mass flux and momentum when using the second definition. In contrast, according to Stokes' first definition (S1), there is a wave-induced mass flux in the wave propagation direction, which has to be balanced by a mean flow
U in the opposite direction — called the
undertowUndertow is a strong subsurface flow of water returning seaward from shore, often as result of wave action.Undertow may also refer to:* Undertow , a 2001 mystery novel by Warren Adler...
.
So in general, there are quite some subtleties involved. Therefore also the term pseudo-momentum of the waves is used instead of wave momentum.
Mass and momentum evolution equations
For slowly-varying
bathymetryBathymetry is the study of underwater depth of the third dimension of lake or ocean floors. In other words, bathymetry is the underwater equivalent to hypsometry. The name comes from Greek βαθυς, deep, and μετρον, measure...
, wave and mean-flow fields, the evolution of the mean flow can de described in terms of the mean mass-transport velocity defined as:
Note that for deep water, when the mean depth
h goes to infinity, the mean Eulerian velocity and mean transport velocity become equal.
The equation for mass conservation is:
In
fluid dynamicsIn physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
,
Airy wave theory (often referred to as
linear wave theory) gives a
linearisedA 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 general, nonlinear case....
description of the
propagationWave propagation is any of the ways in which waves travel.With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves....
of
gravity waveIn fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media which has the restoring force of gravity or buoyancy....
s on the surface of a homogeneous
fluidA fluid is a substance that continually deforms under an applied shear stress. All gases are fluids, but not all liquids are fluids. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
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 AirySir George Biddell Airy FRS was an English mathematician and astronomer, Astronomer Royal from 1835 to 1881. His many achievements include work on planetary orbits, measuring the mean density of the Earth, a method of solution of two-dimensional problems in solid mechanics and, in his role as...
in the 19
th century.
[Craik (2004).]
Airy wave theory is often applied in
ocean engineeringOcean Engineering is an ambiguously-defined discipline, but may refer to Oceanographic Engineering a term describing Marine Electronics Engineering applied to supporting the work of Oceanographers; or, may refer to Offshore Engineering, or Maritime Engineering, which is the branch of engineering...
and coastal engineering for the modelling of random
sea stateIn oceanography, a sea state is the general condition of the free surface on a large body of water—with respect to wind waves and swell—at a certain location and moment. A sea state is characterised by statistics, including the wave height, period, and power spectrum. The sea state varies with...
s — giving a description of the wave
kinematicsKinematics is the branch of classical mechanics that describes the motion of objects without consideration of the causes leading to the motion....
and
dynamicsDynamics may refer to:In Physics:*Dynamics , in physics, dynamics refers to time evolution of physical processes...
of high-enough accuracy for many purposes.
Further, several
second-orderPerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
nonlinear properties of surface gravity waves, and their propagation, can be estimated from its results. This linear theory is often used to get a quick and rough estimate of wave characteristics and their effects.
Description
Airy wave theory uses a
potential flowIn fluid dynamics, potential 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 approximation for several applications...
approach to describe the motion of gravity waves on a fluid surface. The use of — inviscid and irrotational — potential flow in water waves is remarkably successful, given its failure to describe many other fluid flows where it is often essential to take
viscosityViscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness." Thus, water is "thin," having a lower viscosity, while honey is "thick," having a higher viscosity...
,
vorticityVorticity is a concept used in fluid dynamics. In the simplest sense, vorticity is the tendency for elements of the fluid to "spin."More formally, vorticity can be related to the amount of "circulation" or "rotation" in a fluid.The average vorticity in a small region of fluid flow is equal to the...
,
turbulenceIn fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic, stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time. Flow that is not turbulent is called laminar flow...
and/or
flow separationAll solid objects travelling through a fluid acquire a boundary layer of fluid around them where viscous forces occur in the layer of fluid close to the solid surface. Boundary layers can be either laminar or turbulent...
into account. This is due to the fact that for the oscillatory part of the fluid motion, wave-induced vorticity is restricted to some thin oscillatory
Stokes boundary layerIn fluid dynamics, the Stokes boundary layer, or oscillatory boundary layer, refers to the boundary layer close to a solid wall in oscillatory flow of a viscous fluid...
s at the boundaries of the fluid domain.
Airy wave theory is often used in
ocean engineeringOcean Engineering is an ambiguously-defined discipline, but may refer to Oceanographic Engineering a term describing Marine Electronics Engineering applied to supporting the work of Oceanographers; or, may refer to Offshore Engineering, or Maritime Engineering, which is the branch of engineering...
and coastal engineering. Especially for random waves, sometimes called
wave turbulenceWave turbulence is a set of waves deviated far from thermal equilibrium. Such state is accompanied by dissipation. It is either decaying turbulence or requires external source of energy to sustain it...
, the evolution of the wave statistics — including the wave
spectrumA spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by...
— is predicted well over not too long distances (in terms of wavelengths) and in not too shallow water.
DiffractionDiffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings...
is one of the wave effects which can be described with Airy wave theory. Further, by using the WKBJ approximation,
wave shoalingIn fluid dynamics, wave shoaling is the effect in which surface waves on a water layer of decreasing depth change their wave height...
and
refractionRefraction is the change in direction of a wave due to a change in its velocity. This is most commonly observed when a wave passes from one medium to another...
can be predicted.
Earlier attempts to describe surface gravity waves using potential flow were made by, among others, LaplacePierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...
, Poisson, CauchyAugustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner. He also gave several important theorems in complex analysis and initiated the study of permutation...
and KellandPhilip Kelland was a Scottish mathematician. He was known mainly for his great influence on the development of education in Scotland.-Early life:Kelland was born in 1808 in Dunster, Somerset, England...
. But AirySir George Biddell Airy FRS was an English mathematician and astronomer, Astronomer Royal from 1835 to 1881. His many achievements include work on planetary orbits, measuring the mean density of the Earth, a method of solution of two-dimensional problems in solid mechanics and, in his role as...
was the first to publish the correct derivation and formulation in 1841.[ Soon after, in 1847, the linear theory of Airy was extended by Stokes]Sir George Gabriel Stokes, 1st Baronet FRS , was a mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...
for non-linear wave motion, correct up to third order in the wave steepness.[Stokes (1847).] Even before Airy's linear theory, GerstnerFrantišek Josef Gerstner was a Bohemian physicist and engineer.Gerstner studied at the Jesuits gymnasium in Chomutov, after which he studied mathematics and astronomy at the Faculty of Philosophy in Prague between 1772 and 1777...
derived a nonlinear trochoidthumb|290px|right|A [[cycloid]] generated by a rolling circleTrochoid is the word created by Gilles de Roberval for the curve described by a fixed point as a circle rolls along a straight line...
al wave theory in 1804, which however is not irrotational.
Airy wave theory is a linear theory for the propagation of waves on the surface of a potential flow and above a horizontal bottom. The free surface elevation η(x,t) of one wave component is sinusoidal, as a function of horizontal position x and time t:
where
- a is the wave amplitude
Amplitude is the magnitude of change in the oscillating variable, with each oscillation, within an oscillating system. For instance, sound waves are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
in metreThe metre or meter is the basic unit of length in the International System of Units . Historically, the metre was defined by the French Academy of Sciences as the length between two marks on a platinum-iridium bar, which was designed to represent one ten-millionth of the distance from the Equator...
,
- cos is the cosine function,
- k is the angular wavenumber in radian
The radian is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees, or about 57°17′45″. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
per metre, related to the wavelengthIn physics, the wavelength of a sinusoidal wave is the spatial period of the wave – the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
λ as
- ω is the angular frequency
In physics , angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
in radian per secondThe second , sometimes abbreviated sec., is the name of a unit of time, and is the International System of Units base unit of time...
, related to the period T and frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency....
f by
The waves propagate along the water surface with the phase speed cp:
The angular wavenumber k and frequency ω are not independent parameters (and thus also wavelength λ and period T are not independent), but are coupled. Surface gravity waves on a fluid are dispersiveIn fluid dynamics, dispersion of water waves generally refers to frequency dispersion. Frequency dispersion means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, and forced by gravity and surface tension...
waves — exhibiting frequency dispersion — meaning that each wavenumber has its own frequency and phase speed.
Note that in engineering the wave heightIn fluid dynamics, the wave height of a surface wave is the difference between the elevations of a crest and a neighbouring trough. Wave height is a term used by mariners, as well as in coastal, ocean engineering and naval engineering....
H — the difference in elevation between crestA crest is the point on a wave with the maximum value or upward displacement within a cycle. A trough is the opposite of a crest, so the minimum or lowest point in a cycle....
and trough — is often used:
valid in the present case of linear periodic waves.
Underneath the surface, there is a fluid motion associated with the free surface motion. While the surface elevation shows a propagating wave, the fluid particles are in an orbital motionIn physics, an orbit is the gravitationally curved path of one object around a point or another body, for example the gravitational orbit of a planet around a star....
. Within the framework of Airy wave theory, the orbits are in deep water closed circleA circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the centre. The common distance of the points of a circle from its center is called its radius....
s, and in finite depth closed ellipsoidAn ellipsoid is a type of quadric surface that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is...
s — with the ellipsoids becoming flatter near the bottom of the fluid layer. So while the wave propagates, the fluid particles just orbit (oscillate) around their averageIn mathematics, an average, central tendency of a data set is a measure of the "middle" or "expected" value of the data set. There are many different descriptive statistics that can be chosen as a measurement of the central tendency of the data items. These include arithmetic mean, the median and...
position. With the propagating wave motion, the fluid particles transfer energy in the wave propagation direction, without having a mean velocity. The diameter of the orbits reduces with depth below the free surface. In deep water, the orbit's diameter is reduced to 4% of its free-surface value at a depth of half a wavelength.
In a similar fashion, there is also a pressure oscillation underneath the free surface, with wave-induced pressure oscillations reducing with depth below the free surface — in the same way as for the orbital motion of fluid parcels.
Flow problem formulation
The waves propagate in the horizontal direction, with coordinateA coordinate is a number that determines the location of a point along some line or curve. A list of two, three, or more coordinates can be used to determine the location of a point on a surface, volume, or higher-dimensional domain....
x, and a fluid domain bound above by a free surface at z = η(x,t), with z the vertical coordinate (positive in the upward direction) and t being timeTime is a component of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects...
.[For the equations, solution and resulting approximations in deep and shallow water, see Dingemans (1997), Part 1, §2.1, pp. 38–45. Or: Phillips (1977), pp. 36–45.] The level z = 0 corresponds with the mean surface elevation. The impermeablePermeability, permeable and semipermeable have several meanings:*Permeability , the degree of magnetization of a material in response to a magnetic field...
bed underneath the fluid layer is at z = -h. Further, the flow is assumed to be incompressibleIn fluid mechanics or more generally continuum mechanics, an incompressible flow is solid or fluid flow in which the divergence of velocity is zero. This is more precisely termed isochoric flow. It is an idealization used to simplify analysis. In reality, all materials are compressible to some...
and irrotational — a good approximation of the flow in the fluid interior for waves on a liquid surface — and potential theoryIn mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" arises from the fact that, in 19th-century physics, the fundamental forces of nature were believed to be derived from potentials which...
can be used to describe the flow. The velocity potential Φ(x,z,t) is related to the flow velocityIn fluid dynamics the flow velocity, or velocity field, of a fluid is a vector field which is used to mathematically describe the motion of a fluid...
components ux and uz in the horizontal (x) and vertical (z) directions by:
Then, due to the continuity equation for an incompressible flow, the potential Φ has to satisfy the Laplace equation:
Boundary conditions are needed at the bed and the free surface in order to close the system of equations. For their formulation within the framework of linear theory, it is necessary to specify what the base state (or zeroth-order solutionPerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
) of the flow is. Here, we assume the base state is rest, implying the mean flow velocities are zero.
The bed being impermeable, leads to the kinematicKinematics is the branch of classical mechanics that describes the motion of objects without consideration of the causes leading to the motion....
bed boundary-condition:
In case of deep water — by which is meant infinite water depth, from a mathematical point of view — the flow velocities have to go to zero in the limit as the vertical coordinate goes to minus infinity: z → -∞.
At the free surface, for infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
waves, the vertical motion of the flow has to be equal to the vertical velocity of the free surface. This leads to the kinematic free-surface boundary-condition:
If the free surface elevation η(x,t) was a known function, this would be enough to solve the flow problem. However, the surface elevation is an extra unknown, for which an additional boundary condition is needed. This is provided by Bernoulli's equationIn fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...
for an unsteady potential flow. The pressure above the free surface is assumed to be constant. This constant pressure is taken equal to zero, without loss of generality, since the level of such a constant pressure does not alter the flow. After linearisation, this gives the dynamic free-surface boundary condition:
Because this is a linear theory, in both free-surface boundary conditions — the kinematic and the dynamic one, equations (3) and (4) — the value of Φ and ∂Φ/∂z at the fixed mean level z = 0 is used.
Solution for a progressive monochromatic wave
For a propagating wave of a single frequency — a monochromatic wave — the surface elevation is of the form:
The associated velocity potential, satisfying the Laplace equation (1) in the fluid interior, as well as the kinematic boundary conditions at the free surface (2), and bed (3), is:
with sinh and cosh the hyperbolic sine and hyperbolic cosine function, respectively.
But η and Φ also have to satisfy the dynamic boundary condition, which results in non-trivial (non-zero) values for the wave amplitude a only if the linear dispersion relationIn fluid dynamics, dispersion of water waves generally refers to frequency dispersion. Frequency dispersion means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, and forced by gravity and surface tension...
is satisfied:
with tanh the hyperbolic tangent. So angular frequency ω and wavenumber k — or equivalently period T and wavelength λ — cannot be chosen independently, but are related. This means that wave propagation at a fluid surface is an eigenproblem. When ω and k satisfy the dispersion relation, the wave amplitude a can be chosen freely (but small enough for Airy wave theory to be a valid approximation).
Table of wave quantities
In the table below, several flow quantities and parameters according to Airy wave theory are given.[ The given quantities are for a bit more general situation as for the solution given above. Firstly, the waves may propagate in an arbitrary horizontal direction in the x = (x,y) plane. The wavenumber]Wavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters . Wavenumber is the spatial analog of frequency, that is, it is the measurement of the number of wavelengths per unit distance, or more commonly times that, or the number...
vector is k, and is perpendicular to the cams of the wave crestsA crest is the point on a wave with the maximum value or upward displacement within a cycle. A trough is the opposite of a crest, so the minimum or lowest point in a cycle....
. Secondly, allowance is made for a mean flow velocity U, in the horizontal direction and uniform over (independent of) depth z. This introduces a Doppler shift in the dispersion relations. At an Earth-fixed location, the observed angular frequency (or absolute angular frequency) is ω. On the other hand, in a frame of referenceA frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
moving with the mean velocity U (so the mean velocity as observed from this reference frame is zero), the angular frequency is different. It is called the intrinsic angular frequency (or relative angular frequency), denoted as σ. So in pure wave motion, with U=0, both frequencies ω and σ are equal. The wave number k (and wave length λ) are independent of the frame of referenceA frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
, and have no Doppler shift (for monochromatic waves).
The table only gives the oscillatory parts of flow quantities — velocities, particle excursions and pressure — and not their mean value or drift.
The oscillatory particle excursions ξx and ξz are the time integralIntegration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integralis defined informally...
s of the oscillatory flow velocities ux and uz respectively.
Water depth is classified into three regimes:
- deep water — for a water depth larger than half the wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave – the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
, h > ½ λ, the phase speed of the waves is hardly influenced by depth (this is the case for most wind waves on the sea and ocean surface),
- shallow water — for a water depth smaller than the wavelength divided by 20, h < λ, the phase speed of the waves is only dependent on water depth, and no longer a function of period
Period or periodic may refer to:-Language and literature:* Full stop, a punctuation mark indicating the end of a sentence or phrase* Periodic sentence, a sentence that is not grammatically complete until its end...
or wavelength; and
- intermediate depth — all other cases, λ < h < ½ λ, where both water depth and period (or wavelength) have a significant influence on the solution of Airy wave theory.
In the limiting cases of deep and shallow water, simplifying approximations to the solution can be made. While for intermediate depth, the full formulations have to be used.
Properties of gravity waves on the surface of deep water, shallow water and at intermediate depth, according to Airy wave theory |
| quantity |
symbol |
units |
deep water( h > ½ λ ) |
shallow water( h < 0.05 λ ) |
intermediate depth( all λ and h ) |
| surface elevation |
|
mThe metre or meter is the basic unit of length in the International System of Units . Historically, the metre was defined by the French Academy of Sciences as the length between two marks on a platinum-iridium bar, which was designed to represent one ten-millionth of the distance from the Equator...
|
|
| wave phase |
|
rad The radian is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees, or about 57°17′45″. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
|
|
observed angular frequencyIn physics , angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
|
|
rad / s The second , sometimes abbreviated sec., is the name of a unit of time, and is the International System of Units base unit of time...
|
|
| intrinsic angular frequency |
|
rad / s |
|
| unit vector in the wave propagation direction |
|
– |
|
dispersion relationIn fluid dynamics, dispersion of water waves generally refers to frequency dispersion. Frequency dispersion means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, and forced by gravity and surface tension...
|
|
rad / s |
|
|
|
| phase speed |
|
m / s |
|
|
|
| group speed |
|
m / s |
|
|
|
| ratio |
|
– |
|
|
|
| horizontal velocity |
|
m / s |
|
|
|
| vertical velocity |
|
m / s |
|
|
|
| horizontal particle excursion |
|
m |
|
|
|
| vertical particle excursion |
|
m |
|
|
|
pressurePressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
oscillation |
|
N / m2 |
|
|
|
Surface tension effects
Due to surface tensionSurface tension is a property of the surface of a liquid. It is what causes the surface portion of liquid to be attracted to another surface, such as that of another portion of liquid .Applying Newtonian physics to the forces that arise due to surface tension accurately predicts many liquid behaviors...
, the dispersion relation changes to:[Phillips (1977), p. 37.]
with γ the surface tension, with SISi, si, or SI may refer to :- Places :* Mount Si, a mountain in state of Washington* Si County, county in Anhui, China* Si River, a river in China* Slovenia, a European nation Si, si, or SI may refer to (all SI unless otherwise stated):- Places :* Mount Si, a mountain in state of Washington* Si...
units in N/m2. All above equations for linear waves remain the same, if the gravitational acceleration g is replaced by
As a result of surface tension, the waves propagate faster. Surface tension only has influence for short waves, with wavelengths less than a few decimeters in case of a water–air interface. For very short wavelengths — two millimeter in case of the interface between air and water – gravity effects are negligible.
Interfacial waves
Surface gravity waves are a special case of interfacial waves, on the interfaceAn interface is a surface forming a common boundary among two different phases, such as an insoluble solid and a liquid, two immiscible liquids or a liquid and an insoluble gas. The importance of the interface depends on which type of system is being treated: the bigger the quotient area/volume,...
between two fluids of different densityThe density of a material is defined as its mass per unit volume. The symbol of density is ρ .- Formula :Mathematically:where: is the density, is the mass, is the volume....
. Consider two fluids separated by an interface, and without further boundaries. Then their dispersion relation becomes:
where ρ and ρ‘ are the densities of the two fluids, below (ρ) and above (ρ‘) the interface, respectively. For interfacial waves to exist, the lower layer has to be heavier than the upper one, ρ > ρ‘. Otherwise, the interface is unstable and a Rayleigh–Taylor instability develops.
Second-order wave properties
Several second-orderPerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
wave properties, i.e. quadraticIn mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms...
in the wave amplitude a, can be derived directly from Airy wave theory. They are of importance in many practical applications, e.g forecasts of wave conditions. Using a WKBJ approximation, second-order wave properties also find their applications in describing waves in case of slowly-varying bathymetryBathymetry is the study of underwater depth of the third dimension of lake or ocean floors. In other words, bathymetry is the underwater equivalent to hypsometry. The name comes from Greek βαθυς, deep, and μετρον, measure...
, and mean-flow variations of currents and surface elevation. As well as in the description of the wave and mean-flow interactions due to time and space-variations in amplitude, frequency, wavelength and direction of the wave field itself.
Table of second-order wave properties
In the table below, several second-order wave properties — as well as the dynamical equations they satisfy in case of slowly-varying conditions in space and time — are given. More details on these can be found below. The table gives results for wave propagation in one horizontal spatial dimension. Further on in this section, more detailed descriptions and results are given for the general case of propagation in two-dimensional horizontal space.
| Second-order quantities and their dynamics, using results of Airy wave theory |
| quantity |
symbol |
units |
formula |
| mean wave-energy density per unit horizontal area |
|
J The joule , named for James Prescott Joule, is the derived unit of energy in the International System of Units. It is the energy exerted by a force of one newton acting to move an object through a distance of one metre...
/ m2 |
|
radiation stress or excess horizontal momentumIn classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section "modern definitions of momentum" on this page...
fluxIn the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as the amount that flows through a unit area per unit time...
due to the wave motion |
|
N / m |
|
| wave action |
|
J·s / m2 |
|
| mean mass-flux due to the wave motion or the wave pseudo-momentum |
|
kg / (m·s) |
|
| mean horizontal mass-transport velocity |
|
m / s |
|
Stokes driftFor a pure wave motion in fluid dynamics, the Stokes drift velocity is the average velocity when following a specific fluid parcel as it travels with the fluid flow...
|
|
m / s |
|
| wave-energy propagation |
|
J / (m2·s) |
|
| wave action conservation |
|
J / m2 |
|
| wave-crest A crest is the point on a wave with the maximum value or upward displacement within a cycle. A trough is the opposite of a crest, so the minimum or lowest point in a cycle....
conservation |
|
rad / (m·s) |
with |
| mean mass conservation |
|
kg / (m2·s) |
|
| mean horizontal-momentum evolution |
|
N / m2 |
|
The last four equations describe the evolution of slowly-varying wave trains over bathymetryBathymetry is the study of underwater depth of the third dimension of lake or ocean floors. In other words, bathymetry is the underwater equivalent to hypsometry. The name comes from Greek βαθυς, deep, and μετρον, measure...
in interaction with the mean flow, and can be derived from a variational principle: Whitham's average LagrangianThe Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics. In classical mechanics, the...
method.[, p. 559.] In the mean horizontal-momentum equation, d(x) is the still water depth, i.e. the bed underneath the fluid layer is located at z = –d. Note that the mean-flow velocity in the mass and momentum equations is the mass transport velocity , including the splash-zone effects of the waves on horizontal mass transport, and not the mean EulerianIn fluid dynamics and finite-deformation plasticity the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. Plotting the position of an individual parcel through time gives the...
velocity (e.g. as measured with a fixed flow meter).
Wave energy density
Wave energy is a quantity of primary interest, since it is a primary quantity that is transported with the wave trains. As can be seen above, many wave quantities like surface elevation and orbital velocity are oscillatory in nature with zero mean (within the framework of linear theory). In water waves, the most used energy measure is the mean wave energy density per unit horizontal area. It is the sum of the kineticThe kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its current velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its...
and potential energyPotential energy is energy stored within a physical system as a result of the position or configuration of the different parts of that system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do work in the process...
density, integrated over the depth of the fluid layer and averaged over the wave phase. Simplest to derive is the mean potential energy density per unit horizontal area Epot of the surface gravity waves, which is the deviation of the potential energy due to the presence of the waves:[Phillips (1977), p. 39.]
with an overbar denoting the mean value (which in the present case of periodic waves can be taken either as a time average or an average over one wavelength in space).
The mean kinetic energy density per unit horizontal area Ekin of the wave motion is similarly found to be:
with σ the intrinsic frequency, see the table of wave quantities. Using the dispersion relation, the result for surface gravity waves is:
As can be seen, the mean kinetic and potential energy densities are equal. This is a general property of energy densities of progressive linear waves in a conservative system.[. Adding potential and kinetic contributions, Epot and Ekin, the mean energy density per unit horizontal area E of the wave motion is:]
In case of surface tension effects not being negligible, their contribution also adds to the potential and kinetic energy densities, giving[Phillips (1977), p. 38.]
with γ the surface tensionSurface tension is a property of the surface of a liquid. It is what causes the surface portion of liquid to be attracted to another surface, such as that of another portion of liquid .Applying Newtonian physics to the forces that arise due to surface tension accurately predicts many liquid behaviors...
.
Wave action, wave energy flux and radiation stress
In general, there can be an energy transfer between the wave motion and the mean fluid motion. This means, that the wave energy density is not in all cases a conserved quantity (neglecting dissipative effectsIn physics, dissipation embodies the concept of a dynamical system where important mechanical modes, such as waves or oscillations, lose energy over time, typically due to the action of friction or turbulence. The lost energy is converted into heat, raising the temperature of the system...
), but the total energy density — the sum of the energy density per unit area of the wave motion and the mean flow motion — is. However, there is for slowly-varying wave trains, propagating in slowly-varying bathymetryBathymetry is the study of underwater depth of the third dimension of lake or ocean floors. In other words, bathymetry is the underwater equivalent to hypsometry. The name comes from Greek βαθυς, deep, and μετρον, measure...
and mean-flow fields, a similar and conserved wave quantity, the wave action :[ Phillips (1977), p. 26.]
with the action fluxIn the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as the amount that flows through a unit area per unit time...
and the group velocityThe group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....
vector. Action conservation forms the basis for many wind wave modelIn fluid dynamics, wind wave modeling describes the effort to depict the sea state and predict the evolution of the energy of wind waves using numerical techniques...
s and wave turbulenceWave turbulence is a set of waves deviated far from thermal equilibrium. Such state is accompanied by dissipation. It is either decaying turbulence or requires external source of energy to sustain it...
models. It is also the basis of coastal engineering models for the computation of wave shoalingIn fluid dynamics, wave shoaling is the effect in which surface waves on a water layer of decreasing depth change their wave height...
. Expanding the above wave action conservation equation leads to the following evolution equation for the wave energy density:[Phillips (1977), p. 66.]
with:
- is the mean wave energy density flux,
- is the radiation stress tensor
Tensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another...
and
- is the mean-velocity shear-rate
-Simple Shear:Shear rate for a fluid flowing between two fixed parallel plates is defined using the following equation:Where:* = The shear rate, measured in reciprocal seconds* = The velocity, measured in meters per second...
tensor.
In this equation in non-conservation form, the Frobenius inner product is the source term describing the energy exchange of the wave motion with the mean flow. Only in case the mean shear-rate is zero, the mean wave energy density is conserved. The two tensors and are in a Cartesian coordinate systemA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length....
of the form:
with and the components of the wavenumber vector and similarly and the components in of the mean velocity vector .
Wave mass flux and wave momentum
The mean horizontal momentumIn classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section "modern definitions of momentum" on this page...
per unit area induced by the wave motion — and also the wave-induced mass fluxMass flux is the rate of mass flow across a unit area...
or mass transportIn physics, chemistry, biology and engineering, a transport phenomenon is any of various mechanisms by which particles or quantities move from one place to another. The laws which govern transport connect a flux with a "motive force". Three common examples of transport phenomena are diffusion,...
— is:[Phillips (1977), pp. 39–40 & 61.]
which is an exact result for periodic progressive water waves, also valid for nonlinear waves. However, its validity strongly depends on the way how wave momentum and mass flux are defined. StokesSir George Gabriel Stokes, 1st Baronet FRS , was a mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...
already identified two possible definitions of phase velocityThe phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...
for periodic nonlinear waves:[
]
- Stokes first definition of wave celerity
Celerity can refer to:*Speed, quickness.*Celerity BBS, a computer Bulletin Board System popular in the 1990s*Celerity, a power of supernatural quickness possessed by vampires in the roleplaying games Vampire: The Requiem and Vampire: The Masquerade...
(S1) — with the mean Eulerian flow velocityIn fluid dynamics and finite-deformation plasticity the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. Plotting the position of an individual parcel through time gives the...
equal to zero for all elevations z below the wave troughTrough may refer to:* Trough , a container for animal feed * Trough , a long depression less steep than a trench* Trough , an elongated region of low atmospheric pressure...
s, and
- Stokes second definition of wave celerity (S2) — with the mean mass transport equal to zero.
The above relation between wave momentum M and wave energy density E is valid within the framework of Stokes' first definition.
However, for waves perpendicular to a coast line or in closed laboratory wave channel, the second definition (S2) is more appropriate. These wave systems have zero mass flux and momentum when using the second definition. In contrast, according to Stokes' first definition (S1), there is a wave-induced mass flux in the wave propagation direction, which has to be balanced by a mean flow U in the opposite direction — called the undertowUndertow is a strong subsurface flow of water returning seaward from shore, often as result of wave action.Undertow may also refer to:* Undertow , a 2001 mystery novel by Warren Adler...
.
So in general, there are quite some subtleties involved. Therefore also the term pseudo-momentum of the waves is used instead of wave momentum.
Mass and momentum evolution equations
For slowly-varying bathymetryBathymetry is the study of underwater depth of the third dimension of lake or ocean floors. In other words, bathymetry is the underwater equivalent to hypsometry. The name comes from Greek βαθυς, deep, and μετρον, measure...
, wave and mean-flow fields, the evolution of the mean flow can de described in terms of the mean mass-transport velocity defined as:[Phillips (1977), pp. 61–63.]
Note that for deep water, when the mean depth h goes to infinity, the mean Eulerian velocity and mean transport velocity become equal.
The equation for mass conservation is:
In fluid dynamicsIn physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, Airy wave theory (often referred to as linear wave theory) gives a linearisedA 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 general, nonlinear case....
description of the propagationWave propagation is any of the ways in which waves travel.With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves....
of gravity waveIn fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media which has the restoring force of gravity or buoyancy....
s on the surface of a homogeneous fluidA fluid is a substance that continually deforms under an applied shear stress. All gases are fluids, but not all liquids are fluids. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....
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 AirySir George Biddell Airy FRS was an English mathematician and astronomer, Astronomer Royal from 1835 to 1881. His many achievements include work on planetary orbits, measuring the mean density of the Earth, a method of solution of two-dimensional problems in solid mechanics and, in his role as...
in the 19th century.[Craik (2004).]
Airy wave theory is often applied in ocean engineeringOcean Engineering is an ambiguously-defined discipline, but may refer to Oceanographic Engineering a term describing Marine Electronics Engineering applied to supporting the work of Oceanographers; or, may refer to Offshore Engineering, or Maritime Engineering, which is the branch of engineering...
and coastal engineering for the modelling of random sea stateIn oceanography, a sea state is the general condition of the free surface on a large body of water—with respect to wind waves and swell—at a certain location and moment. A sea state is characterised by statistics, including the wave height, period, and power spectrum. The sea state varies with...
s — giving a description of the wave kinematicsKinematics is the branch of classical mechanics that describes the motion of objects without consideration of the causes leading to the motion....
and dynamicsDynamics may refer to:In Physics:*Dynamics , in physics, dynamics refers to time evolution of physical processes...
of high-enough accuracy for many purposes.
Further, several second-orderPerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
nonlinear properties of surface gravity waves, and their propagation, can be estimated from its results. This linear theory is often used to get a quick and rough estimate of wave characteristics and their effects.
Description
Airy wave theory uses a potential flowIn fluid dynamics, potential 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 approximation for several applications...
approach to describe the motion of gravity waves on a fluid surface. The use of — inviscid and irrotational — potential flow in water waves is remarkably successful, given its failure to describe many other fluid flows where it is often essential to take viscosityViscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness." Thus, water is "thin," having a lower viscosity, while honey is "thick," having a higher viscosity...
, vorticityVorticity is a concept used in fluid dynamics. In the simplest sense, vorticity is the tendency for elements of the fluid to "spin."More formally, vorticity can be related to the amount of "circulation" or "rotation" in a fluid.The average vorticity in a small region of fluid flow is equal to the...
, turbulenceIn fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic, stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time. Flow that is not turbulent is called laminar flow...
and/or flow separationAll solid objects travelling through a fluid acquire a boundary layer of fluid around them where viscous forces occur in the layer of fluid close to the solid surface. Boundary layers can be either laminar or turbulent...
into account. This is due to the fact that for the oscillatory part of the fluid motion, wave-induced vorticity is restricted to some thin oscillatory Stokes boundary layerIn fluid dynamics, the Stokes boundary layer, or oscillatory boundary layer, refers to the boundary layer close to a solid wall in oscillatory flow of a viscous fluid...
s at the boundaries of the fluid domain.
Airy wave theory is often used in ocean engineeringOcean Engineering is an ambiguously-defined discipline, but may refer to Oceanographic Engineering a term describing Marine Electronics Engineering applied to supporting the work of Oceanographers; or, may refer to Offshore Engineering, or Maritime Engineering, which is the branch of engineering...
and coastal engineering. Especially for random waves, sometimes called wave turbulenceWave turbulence is a set of waves deviated far from thermal equilibrium. Such state is accompanied by dissipation. It is either decaying turbulence or requires external source of energy to sustain it...
, the evolution of the wave statistics — including the wave spectrumA spectrum is a condition that is not limited to a specific set of values but can vary infinitely within a continuum. The word saw its first scientific use within the field of optics to describe the rainbow of colors in visible light when separated using a prism; it has since been applied by...
— is predicted well over not too long distances (in terms of wavelengths) and in not too shallow water. DiffractionDiffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings...
is one of the wave effects which can be described with Airy wave theory. Further, by using the WKBJ approximation, wave shoalingIn fluid dynamics, wave shoaling is the effect in which surface waves on a water layer of decreasing depth change their wave height...
and refractionRefraction is the change in direction of a wave due to a change in its velocity. This is most commonly observed when a wave passes from one medium to another...
can be predicted.
Earlier attempts to describe surface gravity waves using potential flow were made by, among others, LaplacePierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...
, Poisson, CauchyAugustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner. He also gave several important theorems in complex analysis and initiated the study of permutation...
and KellandPhilip Kelland was a Scottish mathematician. He was known mainly for his great influence on the development of education in Scotland.-Early life:Kelland was born in 1808 in Dunster, Somerset, England...
. But AirySir George Biddell Airy FRS was an English mathematician and astronomer, Astronomer Royal from 1835 to 1881. His many achievements include work on planetary orbits, measuring the mean density of the Earth, a method of solution of two-dimensional problems in solid mechanics and, in his role as...
was the first to publish the correct derivation and formulation in 1841.[ Soon after, in 1847, the linear theory of Airy was extended by Stokes]Sir George Gabriel Stokes, 1st Baronet FRS , was a mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...
for non-linear wave motion, correct up to third order in the wave steepness.[Stokes (1847).] Even before Airy's linear theory, GerstnerFrantišek Josef Gerstner was a Bohemian physicist and engineer.Gerstner studied at the Jesuits gymnasium in Chomutov, after which he studied mathematics and astronomy at the Faculty of Philosophy in Prague between 1772 and 1777...
derived a nonlinear trochoidthumb|290px|right|A [[cycloid]] generated by a rolling circleTrochoid is the word created by Gilles de Roberval for the curve described by a fixed point as a circle rolls along a straight line...
al wave theory in 1804, which however is not irrotational.
Airy wave theory is a linear theory for the propagation of waves on the surface of a potential flow and above a horizontal bottom. The free surface elevation η(x,t) of one wave component is sinusoidal, as a function of horizontal position x and time t:
where
- a is the wave amplitude
Amplitude is the magnitude of change in the oscillating variable, with each oscillation, within an oscillating system. For instance, sound waves are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
in metreThe metre or meter is the basic unit of length in the International System of Units . Historically, the metre was defined by the French Academy of Sciences as the length between two marks on a platinum-iridium bar, which was designed to represent one ten-millionth of the distance from the Equator...
,
- cos is the cosine function,
- k is the angular wavenumber in radian
The radian is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees, or about 57°17′45″. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
per metre, related to the wavelengthIn physics, the wavelength of a sinusoidal wave is the spatial period of the wave – the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
λ as
- ω is the angular frequency
In physics , angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
in radian per secondThe second , sometimes abbreviated sec., is the name of a unit of time, and is the International System of Units base unit of time...
, related to the period T and frequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency....
f by
The waves propagate along the water surface with the phase speed cp:
The angular wavenumber k and frequency ω are not independent parameters (and thus also wavelength λ and period T are not independent), but are coupled. Surface gravity waves on a fluid are dispersiveIn fluid dynamics, dispersion of water waves generally refers to frequency dispersion. Frequency dispersion means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, and forced by gravity and surface tension...
waves — exhibiting frequency dispersion — meaning that each wavenumber has its own frequency and phase speed.
Note that in engineering the wave heightIn fluid dynamics, the wave height of a surface wave is the difference between the elevations of a crest and a neighbouring trough. Wave height is a term used by mariners, as well as in coastal, ocean engineering and naval engineering....
H — the difference in elevation between crestA crest is the point on a wave with the maximum value or upward displacement within a cycle. A trough is the opposite of a crest, so the minimum or lowest point in a cycle....
and trough — is often used:
valid in the present case of linear periodic waves.
Underneath the surface, there is a fluid motion associated with the free surface motion. While the surface elevation shows a propagating wave, the fluid particles are in an orbital motionIn physics, an orbit is the gravitationally curved path of one object around a point or another body, for example the gravitational orbit of a planet around a star....
. Within the framework of Airy wave theory, the orbits are in deep water closed circleA circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the centre. The common distance of the points of a circle from its center is called its radius....
s, and in finite depth closed ellipsoidAn ellipsoid is a type of quadric surface that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is...
s — with the ellipsoids becoming flatter near the bottom of the fluid layer. So while the wave propagates, the fluid particles just orbit (oscillate) around their averageIn mathematics, an average, central tendency of a data set is a measure of the "middle" or "expected" value of the data set. There are many different descriptive statistics that can be chosen as a measurement of the central tendency of the data items. These include arithmetic mean, the median and...
position. With the propagating wave motion, the fluid particles transfer energy in the wave propagation direction, without having a mean velocity. The diameter of the orbits reduces with depth below the free surface. In deep water, the orbit's diameter is reduced to 4% of its free-surface value at a depth of half a wavelength.
In a similar fashion, there is also a pressure oscillation underneath the free surface, with wave-induced pressure oscillations reducing with depth below the free surface — in the same way as for the orbital motion of fluid parcels.
Flow problem formulation
The waves propagate in the horizontal direction, with coordinateA coordinate is a number that determines the location of a point along some line or curve. A list of two, three, or more coordinates can be used to determine the location of a point on a surface, volume, or higher-dimensional domain....
x, and a fluid domain bound above by a free surface at z = η(x,t), with z the vertical coordinate (positive in the upward direction) and t being timeTime is a component of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects...
.[For the equations, solution and resulting approximations in deep and shallow water, see Dingemans (1997), Part 1, §2.1, pp. 38–45. Or: Phillips (1977), pp. 36–45.] The level z = 0 corresponds with the mean surface elevation. The impermeablePermeability, permeable and semipermeable have several meanings:*Permeability , the degree of magnetization of a material in response to a magnetic field...
bed underneath the fluid layer is at z = -h. Further, the flow is assumed to be incompressibleIn fluid mechanics or more generally continuum mechanics, an incompressible flow is solid or fluid flow in which the divergence of velocity is zero. This is more precisely termed isochoric flow. It is an idealization used to simplify analysis. In reality, all materials are compressible to some...
and irrotational — a good approximation of the flow in the fluid interior for waves on a liquid surface — and potential theoryIn mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" arises from the fact that, in 19th-century physics, the fundamental forces of nature were believed to be derived from potentials which...
can be used to describe the flow. The velocity potential Φ(x,z,t) is related to the flow velocityIn fluid dynamics the flow velocity, or velocity field, of a fluid is a vector field which is used to mathematically describe the motion of a fluid...
components ux and uz in the horizontal (x) and vertical (z) directions by:
Then, due to the continuity equation for an incompressible flow, the potential Φ has to satisfy the Laplace equation:
Boundary conditions are needed at the bed and the free surface in order to close the system of equations. For their formulation within the framework of linear theory, it is necessary to specify what the base state (or zeroth-order solutionPerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
) of the flow is. Here, we assume the base state is rest, implying the mean flow velocities are zero.
The bed being impermeable, leads to the kinematicKinematics is the branch of classical mechanics that describes the motion of objects without consideration of the causes leading to the motion....
bed boundary-condition:
In case of deep water — by which is meant infinite water depth, from a mathematical point of view — the flow velocities have to go to zero in the limit as the vertical coordinate goes to minus infinity: z → -∞.
At the free surface, for infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
waves, the vertical motion of the flow has to be equal to the vertical velocity of the free surface. This leads to the kinematic free-surface boundary-condition:
If the free surface elevation η(x,t) was a known function, this would be enough to solve the flow problem. However, the surface elevation is an extra unknown, for which an additional boundary condition is needed. This is provided by Bernoulli's equationIn fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...
for an unsteady potential flow. The pressure above the free surface is assumed to be constant. This constant pressure is taken equal to zero, without loss of generality, since the level of such a constant pressure does not alter the flow. After linearisation, this gives the dynamic free-surface boundary condition:
Because this is a linear theory, in both free-surface boundary conditions — the kinematic and the dynamic one, equations (3) and (4) — the value of Φ and ∂Φ/∂z at the fixed mean level z = 0 is used.
Solution for a progressive monochromatic wave
For a propagating wave of a single frequency — a monochromatic wave — the surface elevation is of the form:
The associated velocity potential, satisfying the Laplace equation (1) in the fluid interior, as well as the kinematic boundary conditions at the free surface (2), and bed (3), is:
with sinh and cosh the hyperbolic sine and hyperbolic cosine function, respectively.
But η and Φ also have to satisfy the dynamic boundary condition, which results in non-trivial (non-zero) values for the wave amplitude a only if the linear dispersion relationIn fluid dynamics, dispersion of water waves generally refers to frequency dispersion. Frequency dispersion means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, and forced by gravity and surface tension...
is satisfied:
with tanh the hyperbolic tangent. So angular frequency ω and wavenumber k — or equivalently period T and wavelength λ — cannot be chosen independently, but are related. This means that wave propagation at a fluid surface is an eigenproblem. When ω and k satisfy the dispersion relation, the wave amplitude a can be chosen freely (but small enough for Airy wave theory to be a valid approximation).
Table of wave quantities
In the table below, several flow quantities and parameters according to Airy wave theory are given.[ The given quantities are for a bit more general situation as for the solution given above. Firstly, the waves may propagate in an arbitrary horizontal direction in the x = (x,y) plane. The wavenumber]Wavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters . Wavenumber is the spatial analog of frequency, that is, it is the measurement of the number of wavelengths per unit distance, or more commonly times that, or the number...
vector is k, and is perpendicular to the cams of the wave crestsA crest is the point on a wave with the maximum value or upward displacement within a cycle. A trough is the opposite of a crest, so the minimum or lowest point in a cycle....
. Secondly, allowance is made for a mean flow velocity U, in the horizontal direction and uniform over (independent of) depth z. This introduces a Doppler shift in the dispersion relations. At an Earth-fixed location, the observed angular frequency (or absolute angular frequency) is ω. On the other hand, in a frame of referenceA frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
moving with the mean velocity U (so the mean velocity as observed from this reference frame is zero), the angular frequency is different. It is called the intrinsic angular frequency (or relative angular frequency), denoted as σ. So in pure wave motion, with U=0, both frequencies ω and σ are equal. The wave number k (and wave length λ) are independent of the frame of referenceA frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of motion of an observer.It may also refer to both an...
, and have no Doppler shift (for monochromatic waves).
The table only gives the oscillatory parts of flow quantities — velocities, particle excursions and pressure — and not their mean value or drift.
The oscillatory particle excursions ξx and ξz are the time integralIntegration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integralis defined informally...
s of the oscillatory flow velocities ux and uz respectively.
Water depth is classified into three regimes:
- deep water — for a water depth larger than half the wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave – the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
, h > ½ λ, the phase speed of the waves is hardly influenced by depth (this is the case for most wind waves on the sea and ocean surface),
- shallow water — for a water depth smaller than the wavelength divided by 20, h < λ, the phase speed of the waves is only dependent on water depth, and no longer a function of period
Period or periodic may refer to:-Language and literature:* Full stop, a punctuation mark indicating the end of a sentence or phrase* Periodic sentence, a sentence that is not grammatically complete until its end...
or wavelength; and
- intermediate depth — all other cases, λ < h < ½ λ, where both water depth and period (or wavelength) have a significant influence on the solution of Airy wave theory.
In the limiting cases of deep and shallow water, simplifying approximations to the solution can be made. While for intermediate depth, the full formulations have to be used.
Properties of gravity waves on the surface of deep water, shallow water and at intermediate depth, according to Airy wave theory |
| quantity |
symbol |
units |
deep water( h > ½ λ ) |
shallow water( h < 0.05 λ ) |
intermediate depth( all λ and h ) |
| surface elevation |
|
mThe metre or meter is the basic unit of length in the International System of Units . Historically, the metre was defined by the French Academy of Sciences as the length between two marks on a platinum-iridium bar, which was designed to represent one ten-millionth of the distance from the Equator...
|
|
| wave phase |
|
rad The radian is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees, or about 57°17′45″. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
|
|
observed angular frequencyIn physics , angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
|
|
rad / s The second , sometimes abbreviated sec., is the name of a unit of time, and is the International System of Units base unit of time...
|
|
| intrinsic angular frequency |
|
rad / s |
|
| unit vector in the wave propagation direction |
|
– |
|
dispersion relationIn fluid dynamics, dispersion of water waves generally refers to frequency dispersion. Frequency dispersion means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, and forced by gravity and surface tension...
|
|
rad / s |
|
|
|
| phase speed |
|
m / s |
|
|
|
| group speed |
|
m / s |
|
|
|
| ratio |
|
– |
|
|
|
| horizontal velocity |
|
m / s |
|
|
|
| vertical velocity |
|
m / s |
|
|
|
| horizontal particle excursion |
|
m |
|
|
|
| vertical particle excursion |
|
m |
|
|
|
pressurePressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
oscillation |
|
N / m2 |
|
|
|
Surface tension effects
Due to surface tensionSurface tension is a property of the surface of a liquid. It is what causes the surface portion of liquid to be attracted to another surface, such as that of another portion of liquid .Applying Newtonian physics to the forces that arise due to surface tension accurately predicts many liquid behaviors...
, the dispersion relation changes to:[Phillips (1977), p. 37.]
with γ the surface tension, with SISi, si, or SI may refer to :- Places :* Mount Si, a mountain in state of Washington* Si County, county in Anhui, China* Si River, a river in China* Slovenia, a European nation Si, si, or SI may refer to (all SI unless otherwise stated):- Places :* Mount Si, a mountain in state of Washington* Si...
units in N/m2. All above equations for linear waves remain the same, if the gravitational acceleration g is replaced by
As a result of surface tension, the waves propagate faster. Surface tension only has influence for short waves, with wavelengths less than a few decimeters in case of a water–air interface. For very short wavelengths — two millimeter in case of the interface between air and water – gravity effects are negligible.
Interfacial waves
Surface gravity waves are a special case of interfacial waves, on the interfaceAn interface is a surface forming a common boundary among two different phases, such as an insoluble solid and a liquid, two immiscible liquids or a liquid and an insoluble gas. The importance of the interface depends on which type of system is being treated: the bigger the quotient area/volume,...
between two fluids of different densityThe density of a material is defined as its mass per unit volume. The symbol of density is ρ .- Formula :Mathematically:where: is the density, is the mass, is the volume....
. Consider two fluids separated by an interface, and without further boundaries. Then their dispersion relation becomes:
where ρ and ρ‘ are the densities of the two fluids, below (ρ) and above (ρ‘) the interface, respectively. For interfacial waves to exist, the lower layer has to be heavier than the upper one, ρ > ρ‘. Otherwise, the interface is unstable and a Rayleigh–Taylor instability develops.
Second-order wave properties
Several second-orderPerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
wave properties, i.e. quadraticIn mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms...
in the wave amplitude a, can be derived directly from Airy wave theory. They are of importance in many practical applications, e.g forecasts of wave conditions. Using a WKBJ approximation, second-order wave properties also find their applications in describing waves in case of slowly-varying bathymetryBathymetry is the study of underwater depth of the third dimension of lake or ocean floors. In other words, bathymetry is the underwater equivalent to hypsometry. The name comes from Greek βαθυς, deep, and μετρον, measure...
, and mean-flow variations of currents and surface elevation. As well as in the description of the wave and mean-flow interactions due to time and space-variations in amplitude, frequency, wavelength and direction of the wave field itself.
Table of second-order wave properties
In the table below, several second-order wave properties — as well as the dynamical equations they satisfy in case of slowly-varying conditions in space and time — are given. More details on these can be found below. The table gives results for wave propagation in one horizontal spatial dimension. Further on in this section, more detailed descriptions and results are given for the general case of propagation in two-dimensional horizontal space.
| Second-order quantities and their dynamics, using results of Airy wave theory |
| quantity |
symbol |
units |
formula |
| mean wave-energy density per unit horizontal area |
|
J The joule , named for James Prescott Joule, is the derived unit of energy in the International System of Units. It is the energy exerted by a force of one newton acting to move an object through a distance of one metre...
/ m2 |
|
radiation stress or excess horizontal momentumIn classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section "modern definitions of momentum" on this page...
fluxIn the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as the amount that flows through a unit area per unit time...
due to the wave motion |
|
N / m |
|
| wave action |
|
J·s / m2 |
|
| mean mass-flux due to the wave motion or the wave pseudo-momentum |
|
kg / (m·s) |
|
| mean horizontal mass-transport velocity |
|
m / s |
|
Stokes driftFor a pure wave motion in fluid dynamics, the Stokes drift velocity is the average velocity when following a specific fluid parcel as it travels with the fluid flow...
|
|
m / s |
|
| wave-energy propagation |
|
J / (m2·s) |
|
| wave action conservation |
|
J / m2 |
|
| wave-crest A crest is the point on a wave with the maximum value or upward displacement within a cycle. A trough is the opposite of a crest, so the minimum or lowest point in a cycle....
conservation |
|
rad / (m·s) |
with |
| mean mass conservation |
|
kg / (m2·s) |
|
| mean horizontal-momentum evolution |
|
N / m2 |
|
The last four equations describe the evolution of slowly-varying wave trains over bathymetryBathymetry is the study of underwater depth of the third dimension of lake or ocean floors. In other words, bathymetry is the underwater equivalent to hypsometry. The name comes from Greek βαθυς, deep, and μετρον, measure...
in interaction with the mean flow, and can be derived from a variational principle: Whitham's average LagrangianThe Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics. In classical mechanics, the...
method.[, p. 559.] In the mean horizontal-momentum equation, d(x) is the still water depth, i.e. the bed underneath the fluid layer is located at z = –d. Note that the mean-flow velocity in the mass and momentum equations is the mass transport velocity , including the splash-zone effects of the waves on horizontal mass transport, and not the mean EulerianIn fluid dynamics and finite-deformation plasticity the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. Plotting the position of an individual parcel through time gives the...
velocity (e.g. as measured with a fixed flow meter).
Wave energy density
Wave energy is a quantity of primary interest, since it is a primary quantity that is transported with the wave trains. As can be seen above, many wave quantities like surface elevation and orbital velocity are oscillatory in nature with zero mean (within the framework of linear theory). In water waves, the most used energy measure is the mean wave energy density per unit horizontal area. It is the sum of the kineticThe kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its current velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its...
and potential energyPotential energy is energy stored within a physical system as a result of the position or configuration of the different parts of that system. It is called potential energy because it has the potential to be converted into other forms of energy, such as kinetic energy, and to do work in the process...
density, integrated over the depth of the fluid layer and averaged over the wave phase. Simplest to derive is the mean potential energy density per unit horizontal area Epot of the surface gravity waves, which is the deviation of the potential energy due to the presence of the waves:[Phillips (1977), p. 39.]
with an overbar denoting the mean value (which in the present case of periodic waves can be taken either as a time average or an average over one wavelength in space).
The mean kinetic energy density per unit horizontal area Ekin of the wave motion is similarly found to be:
with σ the intrinsic frequency, see the table of wave quantities. Using the dispersion relation, the result for surface gravity waves is:
As can be seen, the mean kinetic and potential energy densities are equal. This is a general property of energy densities of progressive linear waves in a conservative system.[. Adding potential and kinetic contributions, Epot and Ekin, the mean energy density per unit horizontal area E of the wave motion is:]
In case of surface tension effects not being negligible, their contribution also adds to the potential and kinetic energy densities, giving[Phillips (1977), p. 38.]
with γ the surface tensionSurface tension is a property of the surface of a liquid. It is what causes the surface portion of liquid to be attracted to another surface, such as that of another portion of liquid .Applying Newtonian physics to the forces that arise due to surface tension accurately predicts many liquid behaviors...
.
Wave action, wave energy flux and radiation stress
In general, there can be an energy transfer between the wave motion and the mean fluid motion. This means, that the wave energy density is not in all cases a conserved quantity (neglecting dissipative effectsIn physics, dissipation embodies the concept of a dynamical system where important mechanical modes, such as waves or oscillations, lose energy over time, typically due to the action of friction or turbulence. The lost energy is converted into heat, raising the temperature of the system...
), but the total energy density — the sum of the energy density per unit area of the wave motion and the mean flow motion — is. However, there is for slowly-varying wave trains, propagating in slowly-varying bathymetryBathymetry is the study of underwater depth of the third dimension of lake or ocean floors. In other words, bathymetry is the underwater equivalent to hypsometry. The name comes from Greek βαθυς, deep, and μετρον, measure...
and mean-flow fields, a similar and conserved wave quantity, the wave action :[ Phillips (1977), p. 26.]
with the action fluxIn the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as the amount that flows through a unit area per unit time...
and the group velocityThe group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....
vector. Action conservation forms the basis for many wind wave modelIn fluid dynamics, wind wave modeling describes the effort to depict the sea state and predict the evolution of the energy of wind waves using numerical techniques...
s and wave turbulenceWave turbulence is a set of waves deviated far from thermal equilibrium. Such state is accompanied by dissipation. It is either decaying turbulence or requires external source of energy to sustain it...
models. It is also the basis of coastal engineering models for the computation of wave shoalingIn fluid dynamics, wave shoaling is the effect in which surface waves on a water layer of decreasing depth change their wave height...
. Expanding the above wave action conservation equation leads to the following evolution equation for the wave energy density:[Phillips (1977), p. 66.]
with:
- is the mean wave energy density flux,
- is the radiation stress tensor
Tensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another...
and
- is the mean-velocity shear-rate
-Simple Shear:Shear rate for a fluid flowing between two fixed parallel plates is defined using the following equation:Where:* = The shear rate, measured in reciprocal seconds* = The velocity, measured in meters per second...
tensor.
In this equation in non-conservation form, the Frobenius inner product is the source term describing the energy exchange of the wave motion with the mean flow. Only in case the mean shear-rate is zero, the mean wave energy density is conserved. The two tensors and are in a Cartesian coordinate systemA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length....
of the form:
with and the components of the wavenumber vector and similarly and the components in of the mean velocity vector .
Wave mass flux and wave momentum
The mean horizontal momentumIn classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section "modern definitions of momentum" on this page...
per unit area induced by the wave motion — and also the wave-induced mass fluxMass flux is the rate of mass flow across a unit area...
or mass transportIn physics, chemistry, biology and engineering, a transport phenomenon is any of various mechanisms by which particles or quantities move from one place to another. The laws which govern transport connect a flux with a "motive force". Three common examples of transport phenomena are diffusion,...
— is:[Phillips (1977), pp. 39–40 & 61.]
which is an exact result for periodic progressive water waves, also valid for nonlinear waves. However, its validity strongly depends on the way how wave momentum and mass flux are defined. StokesSir George Gabriel Stokes, 1st Baronet FRS , was a mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...
already identified two possible definitions of phase velocityThe phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...
for periodic nonlinear waves:[
]
- Stokes first definition of wave celerity
Celerity can refer to:*Speed, quickness.*Celerity BBS, a computer Bulletin Board System popular in the 1990s*Celerity, a power of supernatural quickness possessed by vampires in the roleplaying games Vampire: The Requiem and Vampire: The Masquerade...
(S1) — with the mean Eulerian flow velocityIn fluid dynamics and finite-deformation plasticity the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. Plotting the position of an individual parcel through time gives the...
equal to zero for all elevations z below the wave troughTrough may refer to:* Trough , a container for animal feed * Trough , a long depression less steep than a trench* Trough , an elongated region of low atmospheric pressure...
s, and
- Stokes second definition of wave celerity (S2) — with the mean mass transport equal to zero.
The above relation between wave momentum M and wave energy density E is valid within the framework of Stokes' first definition.
However, for waves perpendicular to a coast line or in closed laboratory wave channel, the second definition (S2) is more appropriate. These wave systems have zero mass flux and momentum when using the second definition. In contrast, according to Stokes' first definition (S1), there is a wave-induced mass flux in the wave propagation direction, which has to be balanced by a mean flow U in the opposite direction — called the undertowUndertow is a strong subsurface flow of water returning seaward from shore, often as result of wave action.Undertow may also refer to:* Undertow , a 2001 mystery novel by Warren Adler...
.
So in general, there are quite some subtleties involved. Therefore also the term pseudo-momentum of the waves is used instead of wave momentum.
Mass and momentum evolution equations
For slowly-varying bathymetryBathymetry is the study of underwater depth of the third dimension of lake or ocean floors. In other words, bathymetry is the underwater equivalent to hypsometry. The name comes from Greek βαθυς, deep, and μετρον, measure...
, wave and mean-flow fields, the evolution of the mean flow can de described in terms of the mean mass-transport velocity defined as:[Phillips (1977), pp. 61–63.]
Note that for deep water, when the mean depth h goes to infinity, the mean Eulerian velocity and mean transport velocity become equal.
The equation for mass conservation is:
where h(x,t) is the mean water-depth, slowly varying in space and time.
Similarly, the mean horizontal momentum evolves as:
with d the still-water depth (the sea bed is at z=–d), is the wave radiation-stress tensorTensors are geometrical entities introduced into mathematics and physics to extend the notion of scalars, vectors, and matrices. Many physical quantities are naturally regarded, not as vectors themselves, but as correspondences between one set of vectors and another...
, is the identity matrixIn linear algebra, the identity matrix or unit matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
and is the dyadic productIn mathematics, in particular multilinear algebra, the dyadic productof two vectors, and , each having the same dimension, is the tensor product of the vectors and results in a tensor of order two and rank one...
:
Note that mean horizontal momentumIn classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section "modern definitions of momentum" on this page...
is only conserved if the sea bed is horizontal (i.e the still-water depth d is a constant), in agreement with Noether's theoremNoether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. This seminal theorem was proved by Emmy Noether in 1915 and published in 1918...
.
The system of equations is closed through the description of the waves. Wave energy propagation is described through the wave-action conservation equation (without dissipation and nonlinear wave interactions):[Phillips (1977), p. 66.]
The wave kinematics are described through the wave-crest conservation equation:
with the angular frequency ω a function of the (angular) wavenumberWavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters . Wavenumber is the spatial analog of frequency, that is, it is the measurement of the number of wavelengths per unit distance, or more commonly times that, or the number...
k, related through the dispersion relationIn fluid dynamics, dispersion of water waves generally refers to frequency dispersion. Frequency dispersion means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, and forced by gravity and surface tension...
. For this to be possible, the wave field must be coherentIn physics, coherence is a property of waves, that enables stationary interference. More generally, coherence describes all properties of the electronic correlation between physical quantities of a wave....
. By taking the curlIn vector calculus, the curl is a vector operator that describes the rotation of a vector field. At every point in the field, the curl is represented by a vector...
of the wave-crest conservation, it can be seen that an initially irrotational wavenumber field stays irrotational.
Stokes drift
When following a single particle in pure wave motion according to linear Airy wave theory the particles are in closed elliptical orbit. However, in nonlinear waves this is no longer the case and the particles exhibit a Stokes driftFor a pure wave motion in fluid dynamics, the Stokes drift velocity is the average velocity when following a specific fluid parcel as it travels with the fluid flow...
. The Stokes drift velocity , which is the Stokes drift after one wave cycle divided by the periodPeriod or periodic may refer to:-Language and literature:* Full stop, a punctuation mark indicating the end of a sentence or phrase* Periodic sentence, a sentence that is not grammatically complete until its end...
, can be estimated using the results of linear theory:[Phillips (1977), p. 44.]
so it varies as a function of elevaton. The given formula is for Stokes first definition of wave celerity. When is integratedIntegration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integralis defined informally...
over depth, the expression for the mean wave momentum is recovered.
See also
- 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...
— nonlinear theory for waves in shallow waterWhen 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,...
.
- Capillary wave
A capillary wave is a wave traveling along the phase boundary of a fluid, whose dynamics are dominated by the effects of surface tension.Capillary waves are common in nature and the home, and are often referred to as ripple...
— surface waves under the action of surface tensionSurface tension is a property of the surface of a liquid. It is what causes the surface portion of liquid to be attracted to another surface, such as that of another portion of liquid .Applying Newtonian physics to the forces that arise due to surface tension accurately predicts many liquid behaviors...
- Cnoidal wave
In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function cn, which is why they are coined cnoidal waves...
— nonlinear periodic waves in shallow water, solutions of the Korteweg–de Vries equationIn mathematics, the Korteweg–de Vries equation is a mathematical model of waves on shallow water surfaces. It is particularly famous 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...
- Mild-slope equation
In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries — like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave...
— refraction and diffraction of surface waves over varying depth
- Ocean surface wave
In fluid dynamics, wind waves or, more precisely, wind-generated waves are surface waves that occur on the free surface of oceans, seas, lakes, rivers, and canals or even on small puddles and ponds. They usually result from the wind blowing over a vast enough stretch of fluid surface. Some waves...
— real water waves as seen in the ocean and sea
- Wave power
Wave power is the transport of energy by ocean surface waves, and the capture of that energy to do useful work — for example for electricity generation, water desalination, or the pumping of water ....
— using ocean and sea waves for power generation.
Historical
. Also: "Trigonometry, On the Figure of the Earth, Tides and Waves", 396 pp.Reprinted in:
Further reading
Two parts, 967 pages. Originally published in 1879, the 6th extended edition appeared first in 1932. 504 pp.
External links
- Linear theory of ocean surface waves on WikiWaves.
- Water waves at MIT
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological research...
.