Wave 1

WaveThis article is about waves in the scientific sense. For waves on the surface of the ocean or lakes, see Wind wave.For other uses of wave or waves, see Wave (disambiguation).

Surface waves in water

In physics, a wave is disturbance oroscillation that travels through matter/space,accompanied by a transfer of energy. Wavemotion transfers energy from one point toanother, often with no permanentdisplacement of the particles of themedium—that is, with little or no associatedmass transport. They consist, instead, ofoscillations or vibrations around almostfixed locations. Waves are described by awave equation which sets out how thedisturbance proceeds over time. Themathematical form of this equation variesdepending on the type of wave.

There are two main types of waves. Mechanical waves propagate through a medium, and the substance of thismedium is deformed. The deformation reverses itself owing to restoring forces resulting from its deformation. Forexample, sound waves propagate via air molecules colliding with their neighbors. When air molecules collide, theyalso bounce away from each other (a restoring force). This keeps the molecules from continuing to travel in thedirection of the wave.

The second main type of wave, electromagnetic waves, do not require a medium. Instead, they consist of periodicoscillations of electrical and magnetic fields generated by charged particles, and can therefore travel through avacuum. These types of waves vary in wavelength, and include radio waves, microwaves, infrared radiation, visiblelight, ultraviolet radiation, X-rays, and gamma rays.Further, the behavior of particles in quantum mechanics are described by waves. In addition, gravitational waves alsotravel through space, which are a result of a vibration or movement in gravitational fields.A wave can be transverse or longitudinal depending on the direction of its oscillation. Transverse waves occur whena disturbance creates oscillations perpendicular (at right angles) to the propagation (the direction of energy transfer).Longitudinal waves occur when the oscillations are parallel to the direction of propagation. While mechanical wavescan be both transverse and longitudinal, all electromagnetic waves are transverse.

General featuresA single, all-encompassing definition for the term wave is not straightforward. A vibration can be defined as aback-and-forth motion around a reference value. However, a vibration is not necessarily a wave. An attempt todefine the necessary and sufficient characteristics that qualify a phenomenon to be called a wave results in a fuzzyborder line.The term wave is often intuitively understood as referring to a transport of spatial disturbances that are generally not accompanied by a motion of the 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 1980, p. 8). However, this notion 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

Wave 2

not apply and interaction with a target is the key to wave detection and practical applications. There are water waveson the ocean surface; gamma waves and light waves emitted by the Sun; microwaves used in microwave ovens andin 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 awave process. For example, acoustics is distinguished from optics in that sound waves are related to a mechanicalrather than an electromagnetic wave transfer caused by vibration. Concepts such as mass, momentum, inertia, orelasticity, become therefore crucial in describing acoustic (as distinct from optic) wave processes. This difference inorigin introduces certain wave characteristics particular to the properties of the medium involved. For example, inthe case of air: vortices, radiation pressure, shock waves etc.; in the case of solids: Rayleigh waves, dispersion; andso on.Other properties, however, although usually described in terms of origin, may be generalized to all waves. For suchreasons, wave theory represents a particular branch of physics that is concerned with the properties of waveprocesses independently of their physical origin. For example, based on the mechanical origin of acoustic waves, amoving disturbance in space–time can exist if and only if the medium involved is neither infinitely stiff nor infinitelypliable. If all the parts making up a medium were rigidly bound, then they would all vibrate as one, with no delay inthe 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 statementsare meaningless in the case of waves that do not require a medium, they reveal a characteristic that is relevant to allwaves regardless of origin: within a wave, the phase of a vibration (that is, its position within the vibration cycle) isdifferent for adjacent points in space because the vibration reaches these points at different times.Similarly, wave processes revealed from the study of waves other than sound waves can be significant to theunderstanding of sound phenomena. A relevant example is Thomas Young's principle of interference (Young, 1802,in Hunt 1992, p. 132). This principle was first introduced in Young's study of light and, within some specificcontexts (for example, scattering of sound by sound), is still a researched area in the study of sound.

Mathematical description of one-dimensional waves

Wave equationMain articles: Wave equation and D'Alembert's formulaConsider a traveling transverse wave (which may be a pulse) on a string (the medium). Consider the string to have asingle spatial dimension. Consider this wave as traveling

Wavelength λ, can be measured between any twocorresponding points on a waveform

• in the direction in space. E.g., let the positive direction beto the right, and the negative direction be to the left.

• with constant amplitude • with constant velocity , where is

• independent of wavelength (no dispersion)• independent of amplitude (linear media, not nonlinear).

• with constant waveform, or shapeThis wave can then be described by the two-dimensional functions

(waveform traveling to theright)

(waveform traveling to the left)or, more generally, by d'Alembert's formula:

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representing two component waveforms and traveling through the medium in opposite directions. Ageneralized representation of this wave can be obtained[1] as the partial differential equation

General solutions are based upon Duhamel's principle.

Wave forms

Sine, square, triangle and sawtooth waveforms.

The form or shape of F in d'Alembert's formulainvolves the argument x − vt. Constant values ofthis argument correspond to constant values ofF, and these constant values occur if x increasesat the same rate that vt increases. That is, thewave shaped like the function F will move in thepositive x-direction at velocity v (and G willpropagate at the same speed in the negativex-direction).

In the case of a periodic function F with periodλ, that is, F(x + λ − vt) = F(x − vt), theperiodicity of F in space means that a snapshotof the wave at a given time t finds the wavevarying periodically in space with period λ (thewavelength of the wave). In a similar fashion,this periodicity of F implies a periodicity in timeas 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.

Amplitude and modulation

Illustration of the envelope (the slowly varyingred curve) of an amplitude-modulated wave. Thefast varying blue curve is the carrier wave, which

is being modulated.

Main article: Amplitude modulationSee also: Frequency modulation and Phase modulationThe amplitude of a wave may be constant (in which case the wave is ac.w. or continuous wave), or may be modulated so as to vary with timeand/or position. The outline of the variation in amplitude is called theenvelope of the wave. Mathematically, the modulated wave can bewritten in the form:

where is the amplitude envelope of the wave, is thewavenumber and is the phase. If the group velocity (see below)is wavelength-independent, this equation can be simplified as:

showing that the envelope moves with the group velocity and retains its shape. Otherwise, in cases where the groupvelocity varies with wavelength, the pulse shape changes in a manner often described using an envelope equation.

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Phase velocity and group velocity

Frequency dispersion in groups of gravity waveson the surface of deep water. The red dot moves

with the phase velocity, and the green dotspropagate with the group velocity.

Main articles: Phase velocity and Group velocityThere are two velocities that are associated with waves, the phasevelocity and the group velocity. To understand them, one mustconsider several types of waveform. For simplification, examination isrestricted to one dimension.

This shows a wave with the Group velocity andPhase velocity going in different directions.

The most basic wave (a form of plane wave) may be expressed in theform:

which can be related to the usual sine and cosine forms using Euler'sformula. Rewriting the argument, ,

makes clear that this expression describes a vibration of wavelengthtraveling in the x-direction with a constant phase velocity

.

The other type of wave to be considered is one with localized structure described by an envelope, which may beexpressed mathematically as, for example:

where now A(k1) (the integral is the inverse fourier transform of A(k1)) is a function exhibiting a sharp peak in aregion of wave vectors Δk surrounding the point k1 = k. In exponential form:

with Ao the magnitude of A. For example, a common choice for Ao is a Gaussian wave packet:[2]

where σ determines the spread of k1-values about k, and N is the amplitude of the wave.The exponential function inside the integral for ψ oscillates rapidly with its argument, say φ(k1), and where it variesrapidly, the exponentials cancel each other out, interfere destructively, contributing little to ψ. However, anexception occurs at the location where the argument φ of the exponential varies slowly. (This observation is the basisfor the method of stationary phase for evaluation of such integrals.) The condition for φ to vary slowly is that its rateof change with k1 be small; this rate of variation is:

where the evaluation is made at k1 = k because A(k1) is centered there. This result shows that the position x where thephase changes slowly, the position where ψ is appreciable, moves with time at a speed called the group velocity:

The group velocity therefore depends upon the dispersion relation connecting ω and k. For example, in quantummechanics the energy of a particle represented as a wave packet is E = ħω = (ħk)2/(2m). Consequently, for that wavesituation, the group velocity is

Wave 5

showing that the velocity of a localized particle in quantum mechanics is its group velocity. Because the groupvelocity varies with k, the shape of the wave packet broadens with time, and the particle becomes less localized. Inother words, the velocity of the constituent waves of the wave packet travel at a rate that varies with theirwavelength, so some move faster than others, and they cannot maintain the same interference pattern as the wavepropagates.

Sinusoidal waves

Sinusoidal waves correspond to simple harmonicmotion.

Mathematically, the most basic wave is the (spatially) one-dimensionalsine wave (or harmonic wave or sinusoid) with an amplitude described by the equation:

where• is the maximum amplitude of the wave, maximum distance from

the highest point of the disturbance in the medium (the crest) to theequilibrium point during one wave cycle. In the illustration to theright, 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) havean amplitude expressed as a distance (e.g., meters), longitudinal mechanical waves (e.g., sound waves) use units ofpressure (e.g., pascals), and electromagnetic waves (a form of transverse vacuum wave) express the amplitude interms of its electric field (e.g., volts/meter).The wavelength is the distance between two sequential crests or troughs (or other equivalent points), generally ismeasured in meters. A wavenumber , the spatial frequency of the wave in radians per unit distance (typically permeter), can be associated with the wavelength by the relation

The period is the time for one complete cycle of an oscillation of a wave. The frequency is the number ofperiods per unit time (per second) and is typically measured in hertz. These are related by:

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

The wavelength of a sinusoidal waveform traveling at constant speed is given by:

where is called the phase speed (magnitude of the phase velocity) of the wave and is the wave's frequency.Wavelength can be a useful concept even if the wave is not periodic in space. For example, in an ocean wave approaching shore, the incoming wave undulates with a varying local wavelength that depends in part on the depth

Wave 6

of the sea floor compared to the wave height. The analysis of the wave can be based upon comparison of the localwavelength with the local water depth.Although arbitrary wave shapes will propagate unchanged in lossless linear time-invariant systems, in the presenceof dispersion the sine wave is the unique shape that will propagate unchanged but for phase and amplitude, making iteasy to analyze. Due to the Kramers–Kronig relations, a linear medium with dispersion also exhibits loss, so the sinewave propagating in a dispersive medium is attenuated in certain frequency ranges that depend upon the medium.[3]

The sine function is periodic, so the sine wave or sinusoid has a wavelength in space and a period in time.The sinusoid is defined for all times and distances, whereas in physical situations we usually deal with waves thatexist for a limited span in space and duration in time. Fortunately, an arbitrary wave shape can be decomposed intoan infinite set of sinusoidal waves by the use of Fourier analysis. As a result, the simple case of a single sinusoidalwave can be applied to more general cases. In particular, many media are linear, or nearly so, so the calculation ofarbitrary wave behavior can be found by adding up responses to individual sinusoidal waves using the superpositionprinciple to find the solution for a general waveform. When a medium is nonlinear, the response to complex wavescannot be determined from a sine-wave decomposition.

Plane wavesMain article: Plane wave

Standing wavesMain articles: Standing wave, Acoustic resonance, Helmholtz resonator and Organ pipe

Standing wave in stationary medium. The red dots represent the wave nodes

A standing wave, also known as a stationarywave, is a wave that remains in a constantposition. This phenomenon can occurbecause the medium is moving in theopposite direction to the wave, or it canarise in a stationary medium as a result ofinterference between two waves traveling inopposite directions.

The sum of two counter-propagating waves (of equal amplitude and frequency) creates a standing wave. Standingwaves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, andtherefore introducing a counter-propagating wave. For example when a violin string is displaced, transverse wavespropagate out to where the string is held in place at the bridge and the nut, where the waves are reflected back. At thebridge and nut, the two opposed waves are in antiphase and cancel each other, producing a node. Halfway betweentwo nodes there is an antinode, where the two counter-propagating waves enhance each other maximally. There is nonet propagation of energy over time.

One-dimensionalstanding waves; the

fundamental mode andthe first 5 overtones.

A two-dimensional standing wave on a disk;this is the fundamental mode.

A standing wave on a disk with two nodallines crossing at the center; this is an

overtone.

Wave 7

Physical properties

Light beam exhibiting reflection,refraction, transmission and

dispersion when encountering aprism

Waves exhibit common behaviors under a number of standard situations, e. g.

Transmission and media

Main articles: Rectilinear propagation, Transmittance and Transmission mediumWaves normally move in a straight line (i.e. rectilinearly) through a transmissionmedium. Such media can be classified into one or more of the followingcategories:

• 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

AbsorptionMain articles: Absorption (acoustics) and Absorption (electromagnetic radiation)Absorption of waves mean, if a kind of wave strikes a matter, it will be absorbed by the matter. When a wave withthat same natural frequency impinges upon an atom, then the electrons of that atom will be set into vibrationalmotion. If a wave of a given frequency strikes a material with electrons having the same vibrational frequencies, thenthose electrons will absorb the energy of the wave and transform it into vibrational motion.

ReflectionMain article: Reflection (physics)When a wave strikes a reflective surface, it changes direction, such that the angle made by the incident wave and linenormal to the surface equals the angle made by the reflected wave and the same normal line.

InterferenceMain article: Interference (wave propagation)Waves that encounter each other combine through superposition to create a new wave called an interference pattern.Important interference patterns occur for waves that are in phase.

Wave 8

RefractionMain article: Refraction

Sinusoidal traveling plane wave entering a region oflower wave velocity at an angle, illustrating thedecrease in wavelength and change of direction

(refraction) that results.

Refraction is the phenomenon of a wave changing its speed.Mathematically, this means that the size of the phase velocitychanges. Typically, refraction occurs when a wave passes fromone medium into another. The amount by which a wave isrefracted by a material is given by the refractive index of thematerial. The directions of incidence and refraction are related tothe refractive indices of the two materials by Snell's law.

Diffraction

Main article: DiffractionA wave exhibits diffraction when it encounters an obstacle thatbends the wave or when it spreads after emerging from anopening. Diffraction effects are more pronounced when the size ofthe obstacle or opening is comparable to the wavelength of the wave.

PolarizationMain article: Polarization (waves)

A wave is polarized if it oscillates in one direction or plane. A wavecan be polarized by the use of a polarizing filter. The polarization of atransverse wave describes the direction of oscillation in the planeperpendicular to the direction of travel.Longitudinal waves such as sound waves do not exhibit polarization.For these waves the direction of oscillation is along the direction of

travel.

DispersionMain articles: 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 thespectrum of colours of the rainbow. Isaac Newton performed experiments with light and prisms, presenting hisfindings in the Opticks (1704) that white light consists of several colours and that these colours cannot bedecomposed any further.

Wave 9

Mechanical wavesMain article: Mechanical wave

Waves on stringsMain article: Vibrating stringThe speed of a transverse wave traveling along a vibrating string ( v ) is directly proportional to the square root of thetension of the string ( T ) over the linear mass density ( μ ):

where the linear density μ is the mass per unit length of the string.

Acoustic wavesAcoustic or sound waves travel at speed given by

or the square root of the adiabatic bulk modulus divided by the ambient fluid density (see speed of sound).

Water waves

Main article: Water waves• Ripples on the surface of a pond are actually a combination of

transverse and longitudinal waves; therefore, the points on thesurface 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.

Seismic wavesMain article: Seismic waves

Shock waves

Main article: Shock waveSee also: Sonic boom and Cherenkovradiation

Other

• Waves of traffic, that is, propagation ofdifferent densities of motor vehicles, andso forth, which can be modeled askinematic waves[4]

• Metachronal wave refers to theappearance of a traveling wave producedby coordinated sequential actions.

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• It is worth noting that the mass-energy equivalence equation can be solved for this form: .

Electromagnetic waves

Main articles: Electromagnetic radiation and Electromagneticspectrum

(radio, micro, infrared, visible, uv)An electromagnetic wave consists of two waves that areoscillations of the electric and magnetic fields. An electromagneticwave travels in a direction that is at right angles to the oscillationdirection of both fields. In the 19th century, James Clerk Maxwellshowed that, in vacuum, the electric and magnetic fields satisfy thewave equation both with speed equal to that of the speed of light.From this emerged the idea that light is an electromagnetic wave. Electromagnetic waves can have differentfrequencies (and thus wavelengths), giving rise to various types of radiation such as radio waves, microwaves,infrared, visible light, ultraviolet and X-rays.

Quantum mechanical wavesMain article: Schrödinger equationSee also: Wave functionThe Schrödinger equation describes the wave-like behavior of particles in quantum mechanics. Solutions of thisequation are wave functions which can be used to describe the probability density of a particle.

A propagating wave packet; in general, theenvelope of the wave packet moves at a different

speed than the constituent waves.

de Broglie waves

Main articles: Wave packet and Matter waveLouis de Broglie postulated that all particles with momentum have awavelength

where h is Planck's constant, and p is the magnitude of the momentumof the particle. This hypothesis was at the basis of quantum mechanics.Nowadays, this wavelength is called the de Broglie wavelength. Forexample, the electrons in a CRT display have a de Broglie wavelengthof about 10−13 m.

A wave representing such a particle traveling in the k-direction is expressed by the wave function as follows:

where the wavelength is determined by the wave vector k as:

and the momentum by:

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

Wave 11

around a central value in a wave packet, a waveform often used in quantum mechanics to describe the wave functionof a particle. In a wave packet, the wavelength of the particle is not precise, and the local wavelength deviates oneither 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 andis called a Gaussian wave packet.[5] Gaussian wave packets also are used to analyze water waves.For example, a Gaussian wavefunction ψ might take the form:

at some initial time t = 0, where the central wavelength is related to the central wave vector k0 as λ0 = 2π / k0. It iswell known from the theory of Fourier analysis, or from the Heisenberg uncertainty principle (in the case of quantummechanics) that a narrow range of wavelengths is necessary to produce a localized wave packet, and the morelocalized the envelope, the larger the spread in required wavelengths. The Fourier transform of a Gaussian is itself aGaussian. Given the Gaussian:

the Fourier transform is:

The Gaussian in space therefore is made up of waves:

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 Fourier transform shows aspread in wave vector k determined by 1/σ. That is, the smaller the extent in space, the larger the extent in k, andhence in λ = 2π/k.

Animation showing the effect of a cross-polarizedgravitational wave on a ring of test particles

Gravitational waves

Main article: Gravitational waveResearchers believe that gravitational waves also travel through space,although gravitational waves have never been directly detected. Not tobe confused with gravity waves, gravitational waves are disturbancesin the curvature of spacetime, predicted by Einstein's theory of generalrelativity.

WKB method

Main article: WKB methodSee also: Slowly varying envelope approximationIn a nonuniform medium, in which the wavenumber k can depend onthe location as well as the frequency, the phase term kx is typically replaced by the integral of k(x)dx, according tothe WKB method. Such nonuniform traveling waves are common in many physical problems, including themechanics of the cochlea and waves on hanging ropes.

Wave 12

References[1][1] For an example derivation, see the steps leading up to eq. (17) in[2][2] See, for example, Eq. 2(a) in[3][3] See Eq. 5.10 and discussion in ; Eq. 6.36 and associated discussion in ; and Eq. 3.5 in[4][4] And:[5][5] See for example and ,.

Sources• Campbell, Murray; Greated, Clive (2001). The musician's guide to acoustics (Repr. ed.). Oxford: Oxford

University Press. ISBN 978-0198165057.• French, A.P. (1971). Vibrations and Waves (M.I.T. Introductory physics series). Nelson Thornes.

ISBN 0-393-09936-9. OCLC  163810889 (http:/ / www. worldcat. org/ oclc/ 163810889).• 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 Wave Analysis of Partial Differential Equations: Numerical and

Analytical Methods with Matlab and Maple. Academic Press. ISBN 9780123846532.

External links• Interactive Visual Representation of Waves (http:/ / resonanceswavesandfields. blogspot. com/ 2007/ 08/

true-waves. html)• Science Aid: Wave properties—Concise guide aimed at teens (http:/ / www. scienceaid. co. uk/ physics/ waves/

properties. html)• Simulation of diffraction of water wave passing through a gap (http:/ / www. phy. hk/ wiki/ englishhtm/

Diffraction. htm)• Simulation of interference of water waves (http:/ / www. phy. hk/ wiki/ englishhtm/ Interference. htm)• Simulation of longitudinal traveling wave (http:/ / www. phy. hk/ wiki/ englishhtm/ Lwave. htm)• Simulation of stationary wave on a string (http:/ / www. phy. hk/ wiki/ englishhtm/ StatWave. htm)• Simulation of transverse traveling wave (http:/ / www. phy. hk/ wiki/ englishhtm/ TwaveA. htm)• Sounds Amazing—AS and A-Level learning resource for sound and waves (http:/ / www. acoustics. salford. ac.

uk/ feschools/ )• chapter from an online textbook (http:/ / www. lightandmatter. com/ html_books/ lm/ ch19/ ch19. html)• Simulation of waves on a string (http:/ / www. physics-lab. net/ applets/ mechanical-waves)• of longitudinal and transverse mechanical wave (http:/ / www. cbu. edu/ ~jvarrian/ applets/ waves1/ lontra_g.

htm-simulation)• MIT OpenCourseWare 8.03: Vibrations and Waves (http:/ / ocw. mit. edu/ courses/ physics/

8-03-physics-iii-vibrations-and-waves-fall-2004/ ) Free, independent study course with video lectures,assignments, lecture notes and exams.

Velocities of waves

Phase velocity • Group velocity • Front velocity • Signal velocity

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Image Sources, Licenses and ContributorsFile:2006-01-14 Surface waves.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:2006-01-14_Surface_waves.jpg  License: GNU Free Documentation License  Contributors: RogerMcLassusFile:Nonsinusoidal wavelength.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Nonsinusoidal_wavelength.JPG  License: Creative Commons Attribution-Sharealike 3.0 Contributors: Brews ohareFile:Waveforms.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Waveforms.svg  License: GNU Free Documentation License  Contributors: OmegatronFile:Wave packet.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wave_packet.svg  License: Public Domain  Contributors: Oleg AlexandrovImage:Wave group.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Wave_group.gif  License: Creative Commons Attribution-Share Alike  Contributors: KraaiennestImage:Wave opposite-group-phase-velocity.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Wave_opposite-group-phase-velocity.gif  License: Creative Commons Attribution 3.0 Contributors: Geek3File:Simple harmonic motion animation.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Simple_harmonic_motion_animation.gif  License: Public Domain  Contributors:User:Evil_saltineFile:Standing wave.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Standing_wave.gif  License: Public Domain  Contributors: BrokenSegue, Cdang, Joolz, Kersti Nebelsiek,LucasVB, Mike.lifeguard, Pieter Kuiper, PtjImage:Standing waves on a string.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Standing_waves_on_a_string.gif  License: Creative Commons Attribution-Sharealike 3.0 Contributors: User:AdjwilleyImage:Drum vibration mode01.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Drum_vibration_mode01.gif  License: Public Domain  Contributors: Oleg AlexandrovImage:Drum vibration mode21.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Drum_vibration_mode21.gif  License: Public Domain  Contributors: Oleg AlexandrovFile:Light dispersion of a mercury-vapor lamp with a flint glass prism IPNr°0125.jpg  Source:http://en.wikipedia.org/w/index.php?title=File:Light_dispersion_of_a_mercury-vapor_lamp_with_a_flint_glass_prism_IPNr°0125.jpg  License: unknown  Contributors: D-Kuru, Daniele PugliesiFile:Wave refraction.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Wave_refraction.gif  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Dicklyon (Richard F.Lyon)File:Circular.Polarization.Circularly.Polarized.Light Circular.Polarizer Creating.Left.Handed.Helix.View.svg  Source:http://en.wikipedia.org/w/index.php?title=File:Circular.Polarization.Circularly.Polarized.Light_Circular.Polarizer_Creating.Left.Handed.Helix.View.svg  License: Public Domain  Contributors:Dave3457File:Shallow water wave.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Shallow_water_wave.gif  License: Creative Commons Attribution-Share Alike  Contributors: KraaiennestFile:Transonico-en.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Transonico-en.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: Cmprince,Cobatfor, Ignacio Icke, Pieter Kuiper, Rocket000File:Onde electromagnétique.png  Source: http://en.wikipedia.org/w/index.php?title=File:Onde_electromagnétique.png  License: Creative Commons Attribution-ShareAlike 3.0 Unported Contributors: ploufandsplashFile:Wave packet (dispersion).gif  Source: http://en.wikipedia.org/w/index.php?title=File:Wave_packet_(dispersion).gif  License: Public Domain  Contributors: Cdang, Chenspec, Fffred, KerstiNebelsiekFile:GravitationalWave CrossPolarization.gif  Source: http://en.wikipedia.org/w/index.php?title=File:GravitationalWave_CrossPolarization.gif  License: Public Domain  Contributors: Originaluploader was MOBle at en.wikipedia

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