water waves
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of ...
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
s of different
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
s travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with
gravity
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
restoring force
In physics, the restoring force is a force that acts to bring a body to its equilibrium position. The restoring force is a function only of position of the mass or particle, and it is always directed back toward the equilibrium position of the s ...
s. As a result,
water
Water (chemical formula ) is an Inorganic compound, inorganic, transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living ...
with a
free surface
In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress,
such as the interface between two homogeneous fluids.
An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
is generally considered to be a
dispersive medium
In optics, and by analogy other branches of physics dealing with wave propagation, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency; sometimes the term chromatic dispersion is used for specificity to o ...
.
For a certain water depth, surface gravity waves – i.e. waves occurring at the air–water interface and gravity as the only force restoring it to flatness – propagate faster with increasing
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
. On the other hand, for a given (fixed) wavelength, gravity waves in deeper water have a larger phase speed than in shallower water. In contrast with the behavior of gravity waves,
capillary wave
A capillary wave is a wave traveling along the phase boundary of a fluid, whose dynamics and phase velocity are dominated by the effects of surface tension.
Capillary waves are common in nature, and are often referred to as ripples. The wav ...
s (i.e. only forced by surface tension) propagate faster for shorter wavelengths.
Besides frequency dispersion, water waves also exhibit amplitude dispersion. This is a
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
effect, by which waves of larger
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
have a different phase speed from small-amplitude waves.
capillary wave
A capillary wave is a wave traveling along the phase boundary of a fluid, whose dynamics and phase velocity are dominated by the effects of surface tension.
Capillary waves are common in nature, and are often referred to as ripples. The wav ...
elevation
The elevation of a geographic location is its height above or below a fixed reference point, most commonly a reference geoid, a mathematical model of the Earth's sea level as an equipotential gravitational surface (see Geodetic datum § Ver ...
''η( x, t )'' is given by:See Lamb (1994), §229, pp. 366–369.
:
where ''a'' is the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
(in metres) and ''θ = θ( x, t )'' is the phase function (in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s), depending on the horizontal position ( ''x'' , in metres) and time ( ''t'' , in seconds):See Whitham (1974), p.11.
: with and
where:
* ''λ'' is the
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
(in metres),
* ''T'' is the
period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Periodic sentence (or rhetorical period), a concept ...
(in seconds),
* ''k'' is the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
(in radians per metre) and
* ''ω'' is the
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
(in radians per second).
Characteristic phases of a water wave are:
* the upward zero-crossing at ''θ = 0'',
* the wave crest at ''θ = ''½'' π'',
* the downward zero-crossing at ''θ = π'' and
* the wave trough at ''θ = 1½ π''.
A certain phase repeats itself after an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
''m'' multiple of ''2π'': sin(''θ'') = sin(''θ+m•2π'').
Essential for water waves, and other wave phenomena in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, is that free propagating waves of non-zero amplitude only exist when the angular frequency ''ω'' and wavenumber ''k'' (or equivalently the wavelength ''λ'' and period ''T'' ) satisfy a
functional relationship
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...
: the frequency dispersion relationSee Phillips (1977), p. 37.
:
The dispersion relation has two solutions: ''ω = +Ω(k)'' and ''ω = −Ω(k)'', corresponding to waves travelling in the positive or negative ''x''–direction. The dispersion relation will in general depend on several other parameters in addition to the wavenumber ''k''. For gravity waves, according to linear theory, these are the acceleration by gravity ''g'' and the water depth ''h''. The dispersion relation for these waves is:
an
implicit equation
In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0.
An implicit func ...
with tanh denoting the hyperbolic tangent function.
An initial wave phase ''θ = θ0'' propagates as a
function of space and time Functions of space and time describe the evolution of a dynamic system or field, involving position (typically 'r') and time (t) (see Field equations
A classical field theory is a physical theory that predicts how one or more physical fields int ...
. Its subsequent position is given by:
:
This shows that the phase moves with the velocity:
:
which is called the phase velocity.
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
and with a constant
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
, propagates with the phase velocity, also called celerity or phase speed. While the phase velocity is a vector and has an associated direction, celerity or phase speed refer only to the magnitude of the phase velocity. According to linear theory for waves forced by gravity, the phase speed depends on the wavelength and the water depth. For a fixed water depth, long waves (with large wavelength) propagate faster than shorter waves.
In the left figure, it can be seen that shallow water waves, with wavelengths ''λ'' much larger than the water depth ''h'', travel with the phase velocity
:
with ''g'' the acceleration by gravity and ''cp'' the phase speed. Since this shallow-water phase speed is independent of the wavelength, shallow water waves do not have frequency dispersion.
Using another normalization for the same frequency dispersion relation, the figure on the right shows that for a fixed wavelength ''λ'' the phase speed ''cp'' increases with increasing water depth. Until, in deep water with water depth ''h'' larger than half the wavelength ''λ'' (so for ''h/λ > 0.5''), the phase velocity ''cp'' is independent of the water depth:
:
with ''T'' the wave
period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Periodic sentence (or rhetorical period), a concept ...
(the
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
of the
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
''f'', ''T=1/f'' ). So in deep water the phase speed increases with the wavelength, and with the period.
Since the phase speed satisfies ''cp = λ/T = λf'', wavelength and period (or frequency) are related. For instance in deep water:
:
The dispersion characteristics for intermediate depth are given below.
Group velocity
Interference
Interference is the act of interfering, invading, or poaching. Interference may also refer to:
Communications
* Interference (communication), anything which alters, modifies, or disrupts a message
* Adjacent-channel interference, caused by extr ...
of two sinusoidal waves with slightly different wavelengths, but the same
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
and propagation direction, results in a
beat pattern
Conducting is the art of directing a musical performance, such as an orchestral or choral concert. It has been defined as "the art of directing the simultaneous performance of several players or singers by the use of gesture." The primary duties ...
, called a wave group. As can be seen in the animation, the group moves with a group velocity ''cg'' different from the phase velocity ''cp'', due to frequency dispersion.
The group velocity is depicted by the red lines (marked ''B'') in the two figures above.
In shallow water, the group velocity is equal to the shallow-water phase velocity. This is because shallow water waves are not dispersive. In deep water, the group velocity is equal to half the phase velocity: ''cg = ½ cp''.See Phillips (1977), p. 25.
The group velocity also turns out to be the energy transport velocity. This is the velocity with which the mean wave energy is transported horizontally in a
narrow-band
Narrowband signals are signals that occupy a narrow range of frequencies or that have a small fractional bandwidth. In the audio spectrum, narrowband sounds are sounds that occupy a narrow range of frequencies. In telephony, narrowband is usu ...
wave field.See Lamb (1994), §237, pp. 382–384.
In the case of a group velocity different from the phase velocity, a consequence is that the number of waves counted in a wave group is different when counted from a snapshot in space at a certain moment, from when counted in time from the measured surface elevation at a fixed position. Consider a wave group of length ''Λg'' and group duration of ''τg''. The group velocity is:See Dingemans (1997), section 2.1.2, pp. 46–50.
:
The number of waves in a wave group, measured in space at a certain moment is: ''Λg / λ''. While measured at a fixed location in time, the number of waves in a group is: ''τg / T''. So the ratio of the number of waves measured in space to those measured in time is:
:
So in deep water, with ''cg = ½ cp'',See Lamb (1994), §236, pp. 380–382. a wave group has twice as many waves in time as it has in space.
The water surface elevation ''η(x,t)'', as a function of horizontal position ''x'' and time ''t'', for a bichromatic wave group of full modulation can be mathematically formulated as:
:
with:
* ''a'' the wave
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
of each frequency component in metres,
* ''k1'' and ''k2'' the
wave number
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
of each wave component, in radians per metre, and
* ''ω1'' and ''ω2'' the
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of each wave component, in radians per second.
Both ''ω1'' and ''k1'', as well as ''ω2'' and ''k2'', have to satisfy the dispersion relation:
: and
Using
trigonometric identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
, the surface elevation is written as:
:
The part between square brackets is the slowly varying amplitude of the group, with group wave number ''½ ( k1 − k2 )'' and group angular frequency ''½ ( ω1 − ω2 )''. As a result, the group velocity is, for the limit ''k1 → k2'' :
:
Wave groups can only be discerned in case of a narrow-banded signal, with the wave-number difference ''k1 − k2'' small compared to the mean wave number ''½ (k1 + k2)''.
Multi-component wave patterns
The effect of frequency dispersion is that the waves travel as a function of wavelength, so that spatial and temporal phase properties of the propagating wave are constantly changing. For example, under the action of gravity, water waves with a longer
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
travel faster than those with a shorter wavelength.
While two superimposed sinusoidal waves, called a bichromatic wave, have an
envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sh ...
which travels unchanged, three or more sinusoidal wave components result in a changing pattern of the waves and their envelope. A
sea state
In oceanography, 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 characterized by statistics, including the wave height, ...
– that is: real waves on the sea or ocean – can be described as a superposition of many sinusoidal waves with different wavelengths, amplitudes, initial phases and propagation directions. Each of these components travels with its own phase velocity, in accordance with the dispersion relation. The statistics of such a surface can be described by its
power spectrum
The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
.See Phillips (1977), p. 102.
Dispersion relation
In the table below, the dispersion relation ''ω2'' = 'Ω(k)''sup>2 between angular frequency ''ω = 2π / T'' and wave number ''k = 2π / λ'' is given, as well as the phase and group speeds.
Deep water corresponds with water depths larger than half the
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
, which is the common situation in the ocean. In deep water, longer period waves propagate faster and transport their energy faster. The deep-water group velocity is half the phase velocity. In shallow water, for wavelengths larger than twenty times the water depth, as found quite often near the coast, the group velocity is equal to the phase velocity.
History
The full linear dispersion relation was first found by
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
, although there were some errors in his solution for the linear wave problem.
The complete theory for linear water waves, including dispersion, was derived by
George Biddell Airy
Sir George Biddell Airy (; 27 July 18012 January 1892) was an English mathematician and astronomer, and the seventh Astronomer Royal from 1835 to 1881. His many achievements include work on planetary orbits, measuring the mean density of the E ...
and published in about 1840. A similar equation was also found by
Philip Kelland
Philip Kelland PRSE FRS (17 October 1808 – 8 May 1879) was an English mathematician. He was known mainly for his great influence on the development of education in Scotland.
Life
Kelland was born in 1808 the son of Philip Kelland (d.1847), ...
at around the same time (but making some mistakes in his derivation of the wave theory).
The shallow water (with small ''h / λ'') limit, ''ω2 = gh k2'', was derived by
Joseph Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia
In case of gravity–capillary waves, where surface tension affects the waves, the dispersion relation becomes:
:
with ''σ'' the surface tension (in N/m).
For a water–air interface (with and ) the waves can be approximated as pure capillary waves – dominated by surface-tension effects – for
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
s less than . For wavelengths above the waves are to good approximation pure surface gravity waves with very little surface-tension effects.
Interfacial waves
For two homogeneous layers of fluids, of mean thickness ''h'' below the interface and ''h′'' above – under the action of gravity and bounded above and below by horizontal rigid walls – the dispersion relationship ''ω2'' = Ω2(''k'') for gravity waves is provided by:
:
where again ''ρ'' and ''ρ′'' are the densities below and above the interface, while coth is the
hyperbolic cotangent
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
function. For the case ''ρ′'' is zero this reduces to the dispersion relation of surface gravity waves on water of finite depth ''h''.
When the depth of the two fluid layers becomes very large (''h''→∞, ''h′''→∞), the hyperbolic cotangents in the above formula approaches the value of one. Then:
:
Nonlinear effects
Shallow water
Amplitude dispersion effects appear for instance in the solitary wave: a single hump of water traveling with constant velocity in shallow water with a horizontal bed. Note that solitary waves are near-
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medi ...
s, but not exactly – after the interaction of two (colliding or overtaking) solitary waves, they have changed a bit in
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
and an oscillatory residual is left behind.See e.g.: The single soliton solution of the Korteweg–de Vries equation, of wave height ''H'' in water depth ''h'' far away from the wave crest, travels with the velocity:
:
So for this nonlinear gravity wave it is the total water depth under the wave crest that determines the speed, with higher waves traveling faster than lower waves. Note that solitary wave solutions only exist for positive values of ''H'', solitary gravity waves of depression do not exist.
Deep water
The linear dispersion relation – unaffected by wave amplitude – is for nonlinear waves also correct at the second order of the
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
expansion, with the orders in terms of the wave steepness (where ''a'' is wave
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
). To the third order, and for deep water, the dispersion relation isSee Lamb (1994), §250, pp. 417–420.
: so
This implies that large waves travel faster than small ones of the same frequency. This is only noticeable when the wave steepness is large.
Waves on a mean current: Doppler shift
Water waves on a mean flow (so a wave in a moving medium) experience a Doppler shift. Suppose the dispersion relation for a non-moving medium is:
:
with ''k'' the wavenumber. Then for a medium with mean
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
V, the dispersion relationship with Doppler shift becomes:See Phillips (1977), p. 24.
:
where k is the wavenumber vector, related to ''k'' as: ''k'' = , k, . The
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
k•V is equal to: k•V'' = kV ''cos ''α'', with ''V'' the length of the mean velocity vector V: ''V'' = , V, . And ''α'' the angle between the wave propagation direction and the mean flow direction. For waves and current in the same direction, k•V=''kV''.
Capillary wave
A capillary wave is a wave traveling along the phase boundary of a fluid, whose dynamics and phase velocity are dominated by the effects of surface tension.
Capillary waves are common in nature, and are often referred to as ripples. The wav ...
Dispersive water-wave models
*
Airy wave theory
In fluid dynamics, Airy wave theory (often referred to as linear wave theory) gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mea ...
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 ''cn''oidal waves. They are ...
Davey–Stewartson equation In fluid dynamics, the Davey–Stewartson equation (DSE) was introduced in a paper by to describe the evolution of a three-dimensional wave-packet on water of finite depth.
It is a system of partial differential equations for a complex ( wave-am ...
*
Kadomtsev–Petviashvili equation
In mathematics and physics, the Kadomtsev–Petviashvili equation (often abbreviated as KP equation) is a partial differential equation to describe nonlinear wave motion. Named after Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviash ...
Shallow water equations
The shallow-water equations (SWE) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface). ...
Trochoidal wave
In fluid dynamics, a trochoidal wave or Gerstner wave is an exact solution of the Euler equations for periodic surface gravity waves. It describes a progressive wave of permanent form on the surface of an incompressible fluid of infinite depth. ...