In physics, a wave is a disturbance that transfers energy through matter or space, with little or no associated mass transport. Waves consist, instead, of oscillations or vibrations of a physical medium or a field, around relatively fixed locations. There are two main types of waves: mechanical and electromagnetic. Mechanical waves propagate through a physical matter, whose substance is being deformed. Restoring forces then reverse the deformation. For example, sound waves propagate via air molecules colliding with their neighbors. When the molecules collide, they also bounce away from each other (a restoring force). This keeps the molecules from continuing to travel in the direction of the wave. Electromagnetic waves
Electromagnetic waves do not require a medium. Instead, they consist of periodic oscillations of electrical and magnetic fields originally generated by charged particles, and can therefore travel through a vacuum. These types vary in wavelength, and include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays
X-rays and gamma rays. Waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematical form of this equation varies depending on the type of wave. Further, the behavior of particles in quantum mechanics are described by waves. In addition, gravitational waves also travel through space, which are a result of a vibration or movement in gravitational fields. A wave can be transverse, where a disturbance creates oscillations that are perpendicular to the propagation of energy transfer, or longitudinal: the oscillations are parallel to the direction of energy propagation. While mechanical waves can be both transverse and longitudinal, all electromagnetic waves are transverse in free space.
1 General features 2 Mathematical description of one-dimensional waves
3 Sinusoidal waves 4 Plane waves 5 Standing waves 6 Physical properties
6.1 Transmission and media 6.2 Absorption 6.3 Reflection 6.4 Interference 6.5 Refraction 6.6 Diffraction 6.7 Polarization 6.8 Dispersion
7 Mechanical waves
7.1 Waves on strings
7.2 Acoustic waves
8 Electromagnetic waves 9 Quantum mechanical waves
9.1 Schrödinger equation 9.2 Dirac equation 9.3 de Broglie waves
10 Gravity waves 11 Gravitational waves 12 See also
12.1 Waves in general
12.1.1 Parameters 12.1.2 Waveforms
12.2 Electromagnetic waves 12.3 In fluids 12.4 In quantum mechanics 12.5 In relativity 12.6 Other specific types of waves 12.7 Related topics
13 References 14 Sources 15 External links
Surface waves in water showing water ripples
A single, all-encompassing definition for the term wave is not
straightforward. A vibration can be defined as a back-and-forth motion
around a reference value. However, a vibration is not necessarily a
wave. An attempt to define the necessary and sufficient
characteristics that qualify a phenomenon as a wave results in a
The term wave is often intuitively understood as referring to a
transport of spatial disturbances that are generally not accompanied
by a motion of medium occupying this space as a whole In a wave, the
energy of a vibration is moving away from the source in the form of a
disturbance within the surrounding medium (Hall 1982, p. 8).
However, this motion is problematic for a standing wave (for example,
a wave on a string), where energy is moving in both directions
equally, or for electromagnetic (e.g., light) waves in a vacuum, where
the concept of medium does not apply and interaction with a target is
the key to wave detection and practical applications. There are water
waves on the ocean surface; gamma waves and light waves emitted by the
Sun; microwaves used in microwave ovens and in radar equipment; radio
waves broadcast by radio stations; and sound waves generated by radio
receivers, telephone handsets and living creatures (as voices), to
mention only a few wave phenomena.
It may appear that the description of waves is closely related to
their physical origin for each specific instance of a wave process.
For example, acoustics is distinguished from optics in that sound
waves are related to a mechanical rather than an electromagnetic wave
transfer caused by vibration. Concepts such as mass, momentum,
inertia, or elasticity, become therefore crucial in describing
acoustic (as distinct from optic) wave processes. This difference in
origin introduces certain wave characteristics particular to the
properties of the medium involved. For example, in the case of air:
vortices, radiation pressure, shock waves etc.; in the case of solids:
Rayleigh waves, dispersion; and so on....
Other properties, however, although usually described in terms of
origin, may be generalized to all waves. For such reasons, wave theory
represents a particular branch of physics that is concerned with the
properties of wave processes independently of their physical
origin. For example, based on the mechanical origin of acoustic
waves, a moving disturbance in space–time can exist if and only if
the medium involved is neither infinitely stiff nor infinitely
pliable. If all the parts making up a medium were rigidly bound, then
they would all vibrate as one, with no delay in the transmission of
the vibration and therefore no wave motion. On the other hand, if all
the parts were independent, then there would not be any transmission
of the vibration and again, no wave motion. Although the above
statements are meaningless in the case of waves that do not require a
medium, they reveal a characteristic that is relevant to all waves
regardless of origin: within a wave, the phase of a vibration (that
is, its position within the vibration cycle) is different for adjacent
points in space because the vibration reaches these points at
different times repeatedly .
Mathematical description of one-dimensional waves
Animation of two waves, the green wave moves to the right while blue wave moves to the left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves. Note that f(x,t) + g(x,t) = u(x,t)
direction in space. E.g., let the positive
direction be to the right, and the negative
direction be to the left. with constant amplitude
with constant velocity
independent of wavelength (no dispersion) independent of amplitude (linear media, not nonlinear).
with constant waveform, or shape
This wave can then be described by the two-dimensional functions
u ( x , t ) = F ( x − v t )
displaystyle u(x,t)=F(x-v t)
traveling to the right)
u ( x , t ) = G ( x + v t )
displaystyle u(x,t)=G(x+v t)
traveling to the left)
or, more generally, by d'Alembert's formula:
u ( x , t ) = F ( x − v t ) + G ( x + v t ) .
representing two component waveforms
traveling through the medium in opposite directions. A generalized representation of this wave can be obtained as the partial differential equation
displaystyle frac 1 v^ 2 frac partial ^ 2 u partial t^ 2 = frac partial ^ 2 u partial x^ 2 .,
General solutions are based upon Duhamel's principle.
Sine, square, triangle and sawtooth waveforms.
The form or shape of F in d'Alembert's formula involves the argument x
− vt. Constant values of this argument correspond to constant values
of F, and these constant values occur if x increases at the same rate
that vt increases. That is, the wave shaped like the function F will
move in the positive x-direction at velocity v (and G will propagate
at the same speed in the negative x-direction).
In the case of a periodic function F with period λ, that is, F(x + λ
− vt) = F(x − vt), the periodicity of F in space means that a
snapshot of the wave at a given time t finds the wave varying
periodically in space with period λ (the wavelength of the wave). In
a similar fashion, this periodicity of F implies a periodicity in time
as well: F(x − v(t + T)) = F(x − vt) provided vT = λ, so an
observation of the wave at a fixed location x finds the wave
undulating periodically in time with period T = λ/v.
Illustration of the envelope (the slowly varying red curve) of an amplitude-modulated wave. The fast varying blue curve is the carrier wave, which is being modulated.
u ( x , t ) = A ( x , t ) sin ( k x − ω t + ϕ ) ,
displaystyle u(x,t)=A(x,t)sin(kx-omega t+phi ) ,
A ( x , t )
displaystyle A(x, t)
is the amplitude envelope of the wave,
is the wavenumber and
is the phase. If the group velocity
displaystyle v_ g
(see below) is wavelength-independent, this equation can be simplified as:
u ( x , t ) = A ( x −
t ) sin ( k x − ω t + ϕ ) ,
displaystyle u(x,t)=A(x-v_ g t)sin(kx-omega t+phi ) ,
showing that the envelope moves with the group velocity and retains
its shape. Otherwise, in cases where the group velocity varies with
wavelength, the pulse shape changes in a manner often described using
an envelope equation.
The red dot moves with the phase velocity, while the green dots propagate with the group velocity
There are two velocities that are associated with waves, the phase
velocity and the group velocity.
displaystyle v_ mathrm p = frac lambda T .
A wave with the group and phase velocities going in different directions
This section duplicates the scope of other sections, specifically, Sinusoidal wave and Frequency. (July 2015)
Sinusoidal waves correspond to simple harmonic motion.
Mathematically, the most basic wave is the (spatially) one-dimensional sine wave (or harmonic wave or sinusoid) with an amplitude
described by the equation:
u ( x , t ) = A sin ( k x − ω t + ϕ ) ,
displaystyle u(x,t)=Asin(kx-omega t+phi ) ,
is the maximum amplitude of the wave, maximum distance from the highest point of the disturbance in the medium (the crest) to the equilibrium point during one wave cycle. In the illustration to the right, this is the maximum vertical distance between the baseline and the wave.
is the space coordinate
is the time coordinate
is the wavenumber
is the angular frequency
is the phase constant.
The units of the amplitude depend on the type of wave. Transverse mechanical waves (e.g., a wave on a string) have an amplitude expressed as a distance (e.g., meters), longitudinal mechanical waves (e.g., sound waves) use units of pressure (e.g., pascals), and electromagnetic waves (a form of transverse vacuum wave) express the amplitude in terms of its electric field (e.g., volts/meter). The wavelength
is the distance between two sequential crests or troughs (or other equivalent points), generally is measured in meters. A wavenumber
, the spatial frequency of the wave in radians per unit distance (typically per meter), can be associated with the wavelength by the relation
displaystyle k= frac 2pi lambda .,
is the time for one complete cycle of an oscillation of a wave. The frequency
is the number of periods per unit time (per second) and is typically measured in hertz denoted as Hz. These are related by:
displaystyle f= frac 1 T .,
In other words, the frequency and period of a wave are reciprocals. The angular frequency
represents the frequency in radians per second. It is related to the frequency or period by
ω = 2 π f =
displaystyle omega =2pi f= frac 2pi T .,
of a sinusoidal waveform traveling at constant speed
is given by:
displaystyle lambda = frac v f ,
is called the phase speed (magnitude of the phase velocity) of the wave and
is the wave's frequency.
This section should include a summary of Plane wave. See Wikipedia:Summary style for information on how to incorporate it into this article's main text. (July 2015)
Standing waves Main articles: Standing wave, Acoustic resonance, Helmholtz resonator, and Organ pipe
A standing wave, also known as a stationary wave, is a wave that remains in a constant position. This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions. The sum of two counter-propagating waves (of equal amplitude and frequency) creates a standing wave. Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave. For example, when a violin string is displaced, transverse waves propagate out to where the string is held in place at the bridge and the nut, where the waves are reflected back. At the bridge and nut, the two opposed waves are in antiphase and cancel each other, producing a node. Halfway between two nodes there is an antinode, where the two counter-propagating waves enhance each other maximally. There is no net propagation of energy over time.
One-dimensional standing waves; the fundamental mode and the first 5 overtones.
A two-dimensional standing wave on a disk; this is the fundamental mode.
A standing wave on a disk with two nodal lines crossing at the center; this is an overtone.
Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a prism
Waves exhibit common behaviors under a number of standard situations, e. g. Transmission and media Main articles: Rectilinear propagation, Transmittance, and Transmission medium Waves normally move in a straight line (i.e. rectilinearly) through a transmission medium. Such media can be classified into one or more of the following categories:
A bounded medium if it is finite in extent, otherwise an unbounded medium A linear medium if the amplitudes of different waves at any particular point in the medium can be added A uniform medium or homogeneous medium if its physical properties are unchanged at different locations in space An anisotropic medium if one or more of its physical properties differ in one or more directions An isotropic medium if its physical properties are the same in all directions
Sinusoidal traveling plane wave entering a region of lower wave velocity at an angle, illustrating the decrease in wavelength and change of direction (refraction) that results.
The phenomenon of polarization arises when wave motion can occur simultaneously in two orthogonal directions. Transverse waves can be polarized, for instance. When polarization is used as a descriptor without qualification, it usually refers to the special, simple case of linear polarization. A transverse wave is linearly polarized if it oscillates in only one direction or plane. In the case of linear polarization, it is often useful to add the relative orientation of that plane, perpendicular to the direction of travel, in which the oscillation occurs, such as "horizontal" for instance, if the plane of polarization is parallel to the ground. Electromagnetic waves propagating in free space, for instance, are transverse; they can be polarized by the use of a polarizing filter. Longitudinal waves, such as sound waves, do not exhibit polarization. For these waves there is only one direction of oscillation, that is, along the direction of travel. Dispersion
Schematic of light being dispersed by a prism. Click to see animation.
Main articles: Dispersion relation, Dispersion (optics), and
Dispersion (water waves)
A wave undergoes dispersion when either the phase velocity or the
group velocity depends on the wave frequency. Dispersion is most
easily seen by letting white light pass through a prism, the result of
which is to produce the spectrum of colours of the rainbow. Isaac
Newton performed experiments with light and prisms, presenting his
findings in the
displaystyle v= sqrt frac T mu ,,
where the linear density μ is the mass per unit length of the string. Acoustic waves Acoustic or sound waves travel at speed given by
displaystyle v= sqrt frac B rho _ 0 ,,
or the square root of the adiabatic bulk modulus divided by the
ambient fluid density (see speed of sound).
Ripples on the surface of a pond are actually a combination of transverse and longitudinal waves; therefore, the points on the surface follow orbital paths. Sound—a mechanical wave that propagates through gases, liquids, solids and plasmas; Inertial waves, which occur in rotating fluids and are restored by the Coriolis effect; Ocean surface waves, which are perturbations that propagate through water.
Main article: Seismic waves
Main article: Shock wave
A shock wave is a type of propagating disturbance. When a wave moves
faster than the local speed of sound in a fluid, it is a shock wave.
Like an ordinary wave, a shock wave carries energy and can propagate
through a medium; however, it is characterized by an abrupt, nearly
discontinuous change in pressure, temperature and density of the
Waves of traffic, that is, propagation of different densities of motor vehicles, and so forth, which can be modeled as kinematic waves
Metachronal wave refers to the appearance of a traveling wave produced by coordinated sequential actions.
A propagating wave packet; in general, the envelope of the wave packet moves at a different speed than the constituent waves.
displaystyle lambda = frac h p ,
where h is Planck's constant, and p is the magnitude of the momentum of the particle. This hypothesis was at the basis of quantum mechanics. Nowadays, this wavelength is called the de Broglie wavelength. For example, the electrons in a CRT display have a de Broglie wavelength of about 10−13 m. A wave representing such a particle traveling in the k-direction is expressed by the wave function as follows:
, t = 0 ) = A
k ⋅ r
displaystyle psi (mathbf r , t=0)=A e^ imathbf kcdot r ,
where the wavelength is determined by the wave vector k as:
displaystyle lambda = frac 2pi k ,
and the momentum by:
displaystyle mathbf p =hbar mathbf k .
However, a wave like this with definite wavelength is not localized in space, and so cannot represent a particle localized in space. To localize a particle, de Broglie proposed a superposition of different wavelengths ranging around a central value in a wave packet, a waveform often used in quantum mechanics to describe the wave function of a particle. In a wave packet, the wavelength of the particle is not precise, and the local wavelength deviates on either side of the main wavelength value. In representing the wave function of a localized particle, the wave packet is often taken to have a Gaussian shape and is called a Gaussian wave packet. Gaussian wave packets also are used to analyze water waves. For example, a Gaussian wavefunction ψ might take the form:
ψ ( x , t = 0 ) = A exp
displaystyle psi (x, t=0)=A exp left(- frac x^ 2 2sigma ^ 2 +ik_ 0 xright) ,
at some initial time t = 0, where the central wavelength is related to
the central wave vector k0 as λ0 = 2π / k0. It is well known from
the theory of Fourier analysis, or from the Heisenberg uncertainty
principle (in the case of quantum mechanics) that a narrow range of
wavelengths is necessary to produce a localized wave packet, and the
more localized the envelope, the larger the spread in required
f ( x ) =
displaystyle f(x)=e^ -x^ 2 /(2sigma ^ 2 ) ,
( k ) = σ
displaystyle tilde f (k)=sigma e^ -sigma ^ 2 k^ 2 /2 .
The Gaussian in space therefore is made up of waves:
f ( x ) =
( k )
i k x
d k ;
displaystyle f(x)= frac 1 sqrt 2pi int _ -infty ^ infty tilde f (k)e^ ikx dk ;
that is, a number of waves of wavelengths λ such that kλ = 2 π.
The parameter σ decides the spatial spread of the Gaussian along the
x-axis, while the
Animation showing the effect of a cross-polarized gravitational wave on a ring of test particles
Index of wave articles
Waves in general
Creeping wave, a wave diffracted around a sphere Evanescent wave Longitudinal wave Periodic travelling wave Sine wave Square wave Standing wave Transverse wave
Airy wave theory, in fluid dynamics
Capillary wave, in fluid dynamics
Cnoidal wave, in fluid dynamics
Edge wave, a surface gravity wave fixed by refraction against a rigid
Faraday wave, a type of wave in liquids
Gravity wave, in fluid dynamics
In quantum mechanics
Gravitational wave, in relativity theory Relativistic wave equations, wave equations that consider special relativity pp-wave spacetime, a set of exact solutions to Einstein's field equation
Other specific types of waves
Alfvén wave, in particle science Atmospheric wave, a periodic disturbance in the fields of atmospheric variables Fir wave, a forest configuration Lamb waves, in solid materials Rayleigh waves, surface acoustic waves that travel on solids Spin wave, in magnetism Spin-density wave, in solid materials Trojan wave packet, in particle science Waves in plasmas, in particle science
Index of wave articles
List of waves named after people
Shallow water equations
Shive wave machine
^ Lev A. Ostrovsky & Alexander I. Potapov (2001). Modulated waves:
theory and application. Johns Hopkins University Press.
^ Michael A. Slawinski (2003). "
Fleisch, D.; Kinnaman, L. (2015). A student's guide to waves.
Cambridge, UK: Cambridge University Press.
Campbell, Murray; Greated, Clive (2001). The musician's guide to
acoustics (Repr. ed.). Oxford: Oxford University Press.
French, A.P. (1971). Vibrations and Waves (M.I.T. Introductory physics
series). Nelson Thornes. ISBN 0-393-09936-9.
Hall, D. E. (1980). Musical Acoustics: An Introduction. Belmont,
California: Wadsworth Publishing Company.
ISBN 0-534-00758-9. .
Hunt, Frederick Vinton (1978). Origins in acoustics. Woodbury, NY:
Published for the Acoustical Society of America through the American
Institute of Physics. ISBN 978-0300022209.
Ostrovsky, L. A.; Potapov, A. S. (1999). Modulated Waves, Theory and
Applications. Baltimore: The Johns Hopkins University Press.
ISBN 0-8018-5870-4. .
Griffiths, G.; Schiesser, W. E. (2010). Traveling
Find more aboutWaveat's sister projects
Definitions from Wiktionary Media from Wikimedia Commons Quotations from Wikiquote
Interactive Visual Representation of Waves
v t e
Velocities of waves
Phase Group Front Signal
v t e
Patterns in nature
Crack Dune Foam Meander Phyllotaxis Soap bubble Symmetry
in crystals Quasicrystals in flowers in biology
Pattern formation Biology
Natural selection Camouflage Mimicry Sexual selection
Chaos theory Fractal Logarithmic spiral
Crystal Fluid dynamics Plateau's laws Self-organization
Plato Pythagoras Empedocles Leonardo Fibonacci
Adolf Zeising Ernst Haeckel Joseph Plateau Wilson Bentley D'Arcy Wentworth Thompson
On Growth and Form
The Chemical Basis of Morphogenesis
Aristid Lindenmayer Benoît Mandelbrot
How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
GND: 4065310-9 N