waves wikia

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Wave

From Wikipedia, the free encyclopediaThis article is about waves in the scientific sense. For waves on the surface of the ocean or lakes, seeWindwave.For other uses, seeWave (disambiguation).

Surface wavesinwater

Inphysics,a waveis disturbance or oscillation (of a physical quantity), that travels through matter or space,accompanied by a transfer of energy. Wave motiontransfersenergyfrom one point to another, often withno permanent displacement of the particles of the mediumthat is, with little or no associated masstransport. They consist, instead, ofoscillationsor vibrations around almost fixed locations. Waves aredescribed by awave equationwhich sets out how the disturbance proceeds over time. The mathematicalform of this equation varies depending on the type of wave.

There are two main types of waves.Mechanical wavespropagate through a medium, and the substance ofthis medium is deformed. The deformation reverses itself owing torestoring forcesresulting from itsdeformation. For example, sound waves propagate via air molecules colliding with their neighbors. When

air molecules collide, they also bounce away from each other (a restoring force). This keeps the moleculesfrom continuing to travel in the direction of the wave.

The second main type of wave,electromagnetic waves,do not require a medium. Instead, they consist ofperiodic oscillations of electrical and magnetic fields generated by charged particles, and can therefore travelthrough avacuum.These types of waves vary in wavelength, and includeradio waves,microwaves,infraredradiation,visible light,ultraviolet radiation,X-rays,andgamma rays.

Further, the behavior of particles inquantum mechanicsare described by waves. In addition,gravitationalwavesalso travel through space, which are a result of a vibration or movement in gravitational fields.

A wave can betransverseorlongitudinaldepending on the direction of its oscillation. Transverse wavesoccur when a disturbance creates oscillations perpendicular (at right angles) to the propagation (the directionof energy transfer). Longitudinal waves occur when the oscillations areparallelto the direction of

propagation. While mechanical waves can be both transverse and longitudinal, all electromagnetic waves aretransverse in free space.

Contents

1 General features

2 Mathematical description of one-dimensional waveso

2.1 Wave equationo 2.2 Wave formso 2.3 Amplitude and modulationo 2.4 Phase velocity and group velocity
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wiki/Restoring_forcehttp://en.wikipedia.org/wiki/Mechanical_wavehttp://en.wikipedia.org/wiki/Wave_equationhttp://en.wikipedia.org/wiki/Oscillationhttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Waterhttp://en.wikipedia.org/wiki/Surface_wavehttp://en.wikipedia.org/wiki/Wave_%28disambiguation%29http://en.wikipedia.org/wiki/Wind_wavehttp://en.wikipedia.org/wiki/Wind_wave


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3 Sinusoidal waves 4 Plane waves

5 Standing waves 6 Physical properties

o 6.1 Transmission and mediao 6.2 Absorptiono 6.3 Reflectiono 6.4 Interferenceo

6.5 Refractiono 6.6 Diffractiono 6.7 Polarizationo 6.8 Dispersion

7 Mechanical waveso 7.1 Waves on stringso 7.2 Acoustic waveso 7.3 Water waveso 7.4 Seismic waveso 7.5 Shock waveso

7.6 Other 8 Electromagnetic waves 9 Quantum mechanical waves

o 9.1 de Broglie waves 10 Gravity waves

11 Gravitational waves 12 WKB method

13 See also 14 References

15 Sources 16 External links

General features

A single, all-encompassing definition for the term waveis not straightforward. Avibrationcan be defined asa back-and-forthmotion around a reference value. However, a vibration is not necessarily a wave. Anattempt to define the necessary and sufficient characteristics that qualify aphenomenonto be called a waveresults in a fuzzy border line.

The term waveis often intuitively understood as referring to a transport of spatial disturbances that aregenerally not accompanied by a motion of the medium occupying this space as a whole. In a wave, the

energyof avibrationis moving away from the source in the form of a disturbance within the surroundingmedium (Hall 1980,p. 8). However, this notion is problematic for astanding wave(for example, a wave ona string), whereenergyis moving in both directions equally, or for electromagnetic (e.g., light) waves in avacuum,where the concept of medium does not apply and interaction with a target is the key to wavedetection and practical applications. There arewater waveson the ocean surface;gamma wavesandlightwavesemitted by the Sun;microwavesused in microwave ovens and inradarequipment;radio waves

broadcast by radio stations; andsound wavesgenerated by radio receivers, telephone handsets and livingcreatures (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 specificinstance of a wave process. For example,acousticsis distinguished fromopticsin that sound waves are

related to a mechanical rather than an electromagnetic wave transfer caused byvibration.Concepts such asmass,momentum,inertia,orelasticity,become therefore crucial in describing acoustic (as distinct fromoptic) 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 wavesetc.; in the case of solids:Rayleigh waves,dispersion;and so on.
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Inertiahttp://en.wikipedia.org/wiki/Inertiahttp://en.wikipedia.org/wiki/Inertiahttp://en.wikipedia.org/wiki/Elasticity_%28physics%29http://en.wikipedia.org/wiki/Elasticity_%28physics%29http://en.wikipedia.org/wiki/Elasticity_%28physics%29http://en.wikipedia.org/wiki/Vortexhttp://en.wikipedia.org/wiki/Vortexhttp://en.wikipedia.org/wiki/Vortexhttp://en.wikipedia.org/wiki/Radiation_pressurehttp://en.wikipedia.org/wiki/Radiation_pressurehttp://en.wikipedia.org/wiki/Radiation_pressurehttp://en.wikipedia.org/wiki/Shock_waveshttp://en.wikipedia.org/wiki/Shock_waveshttp://en.wikipedia.org/wiki/Shock_waveshttp://en.wikipedia.org/wiki/Rayleigh_waveshttp://en.wikipedia.org/wiki/Rayleigh_waveshttp://en.wikipedia.org/wiki/Rayleigh_waveshttp://en.wikipedia.org/wiki/Dispersion_%28chemistry%29http://en.wikipedia.org/wiki/Dispersion_%28chemistry%29http://en.wikipedia.org/wiki/Dispersion_%28chemistry%29http://en.wikipedia.org/wiki/Dispersion_%28chemistry%29http://en.wikipedia.org/wiki/Rayleigh_waveshttp://en.wikipedia.org/wiki/Shock_waveshttp://en.wikipedia.org/wiki/Radiation_pressurehttp://en.wikipedia.org/wiki/Vortexhttp://en.wikipedia.org/wiki/Elasticity_%28physics%29http://en.wikipedia.org/wiki/Inertiahttp://en.wikipedia.org/wiki/Momentumhttp://en.wikipedia.org/wiki/Masshttp://en.wikipedia.org/wiki/Vibrationhttp://en.wikipedia.org/wiki/Opticshttp://en.wikipedia.org/wiki/Acousticshttp://en.wikipedia.org/wiki/Sound_waveshttp://en.wikipedia.org/wiki/Radio_waveshttp://en.wikipedia.org/wiki/Radarhttp://en.wikipedia.org/wiki/Microwaveshttp://en.wikipedia.org/wiki/Light_waveshttp://en.wikipedia.org/wiki/Light_waveshttp://en.wikipedia.org/wiki/Gamma_rayshttp://en.wikipedia.org/wiki/Water_waveshttp://en.wikipedia.org/wiki/Vacuumhttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Standing_wavehttp://en.wikipedia.org/wiki/Wave#CITEREFHall1980http://en.wikipedia.org/wiki/Vibrationhttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Phenomenonhttp://en.wikipedia.org/wiki/Vibrationhttp://en.wikipedia.org/wiki/Wave#External_linkshttp://en.wikipedia.org/wiki/Wave#Sourceshttp://en.wikipedia.org/wiki/Wave#Referenceshttp://en.wikipedia.org/wiki/Wave#See_alsohttp://en.wikipedia.org/wiki/Wave#WKB_methodhttp://en.wikipedia.org/wiki/Wave#Gravitational_waveshttp://en.wikipedia.org/wiki/Wave#Gravity_waveshttp://en.wikipedia.org/wiki/Wave#de_Broglie_waveshttp://en.wikipedia.org/wiki/Wave#Quantum_mechanical_waveshttp://en.wikipedia.org/wiki/Wave#Electromagnetic_waveshttp://en.wikipedia.org/wiki/Wave#Otherhttp://en.wikipedia.org/wiki/Wave#Shock_waveshttp://en.wikipedia.org/wiki/Wave#Seismic_waveshttp://en.wikipedia.org/wiki/Wave#Water_waveshttp://en.wikipedia.org/wiki/Wave#Acoustic_waveshttp://en.wikipedia.org/wiki/Wave#Waves_on_stringshttp://en.wikipedia.org/wiki/Wave#Mechanical_waveshttp://en.wikipedia.org/wiki/Wave#Dispersionhttp://en.wikipedia.org/wiki/Wave#Polarizationhttp://en.wikipedia.org/wiki/Wave#Diffractionhttp://en.wikipedia.org/wiki/Wave#Refractionhttp://en.wikipedia.org/wiki/Wave#Interferencehttp://en.wikipedia.org/wiki/Wave#Reflectionhttp://en.wikipedia.org/wiki/Wave#Absorptionhttp://en.wikipedia.org/wiki/Wave#Transmission_and_mediahttp://en.wikipedia.org/wiki/Wave#Physical_propertieshttp://en.wikipedia.org/wiki/Wave#Standing_waveshttp://en.wikipedia.org/wiki/Wave#Plane_waveshttp://en.wikipedia.org/wiki/Wave#Sinusoidal_waves


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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 ofphysicsthat is concerned with the propertiesof wave processes independently of their physical origin.[1]For example, based on the mechanical origin ofacoustic waves, a moving disturbance in spacetime can exist if and only if the medium involved is neitherinfinitely stiff nor infinitely pliable. If all the parts making up a medium were rigidly bound, then they wouldall vibrate as one, with no delay in the transmission of the vibration and therefore no wave motion. On theother hand, if all the parts were independent, then there would not be any transmission of the vibration andagain, no wave motion. Although the above statements are meaningless in the case of waves that do notrequire a medium, they reveal a characteristic that is relevant to all waves regardless of origin: within awave, thephaseof a vibration (that is, its position within the vibration cycle) is different for adjacent pointsin 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 isThomas Young's principle of interference(Young, 1802, inHunt 1992,p. 132). This principle was first introduced in Young's study oflightand,within some specific contexts (for example,scatteringof sound by sound), is still a researched area in thestudy of sound.

Mathematical description of one-dimensional waves

Wave equation

Main articles:Wave equationandD'Alembert's formula

Consider a travelingtransverse wave(which may be apulse)on a string (the medium). Consider the stringto have a single spatial dimension. Consider this wave as traveling

Wavelength, can be measured between any two corresponding points on a waveform

in the direction in space. E.g., let the positive direction be to the right, and the negativedirection be to the left.

with constantamplitude with constant velocity , where is

o

independent ofwavelength(nodispersion)o independent of amplitude (linearmedia, notnonlinear).[2] with constantwaveform,or shape

This wave can then be described by the two-dimensional functions

(waveform traveling to the right)

(waveform traveling to the left)

or, more generally, byd'Alembert's formula:[3]

representing two component waveforms and traveling through the medium in opposite directions. Ageneralized representation of this wave can be obtained[4]as thepartial differential equation
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General solutions are based uponDuhamel's principle.[5]

Wave forms

Sine,square,triangleandsawtoothwaveforms.

The form or shape ofFind'Alembert's formulainvolves the argumentx vt. Constant values of thisargument correspond to constant values ofF, and these constant values occur ifxincreases at the same ratethat vtincreases. That is, the wave shaped like the functionFwill move in the positivex-direction atvelocity v(and Gwill propagate at the same speed in the negativex-direction).[6]

In the case of a periodic functionFwith period, that is,F(x + vt) =F(x vt), the periodicity ofFinspace means that a snapshot of the wave at a given time tfinds the wave varying periodically in space with

period(thewavelengthof the wave). In a similar fashion, this periodicity ofFimplies a periodicity in timeas well:F(x v(t + T)) =F(x vt) provided vT=, so an observation of the wave at a fixed locationxfindsthe wave undulating periodically in time with period T = /v.[7]

Amplitude and modulation

Illustration of the envelope(the slowly varying red curve) of an amplitude-modulated wave. The fast varyingblue curve is the carrierwave, which is being modulated.Main article:Amplitude modulationSee also:Frequency modulationandPhase modulation

The amplitude of a wave may be constant (in which case the wave is a c.w.orcontinuous wave), or may be

modulatedso as to vary with time and/or position. The outline of the variation in amplitude is called theenvelopeof the wave. Mathematically, themodulated wavecan be written in the form:[8][9][10]
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where is the amplitude envelope of the wave, is thewavenumberand is thephase.If thegroupvelocity (see below) is wavelength-independent, this equation can be simplified as:[11]

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

envelope equation.[11][12]

Phase velocity and group velocity

Main articles:Phase velocityandGroup velocitySee also:Envelope (waves) Phase and group velocity

Frequency dispersionin groups ofgravity waveson the surface of deep water. The red dot moves with thephase velocity,and the green dots propagate with thegroup velocity.

There are two velocities that are associated with waves, thephase velocityand thegroup velocity.To

understand them, one must consider several types of waveform. For simplification, examination is restrictedto one dimension.

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

The most basic wave (a form ofplane wave)may be expressed in the form:

which can be related to the usual sine and cosine forms usingEuler's formula.Rewriting the argument,

, makes clear that this expression describes a vibration of wavelength

traveling in thex-direction with a constantphase velocity .[13]

The other type of wave to be considered is one with localized structure described by anenvelope,whichmay be expressed mathematically as, for example:

where nowA(k1)(the integral is the inverse fourier transform of A(k1)) is a function exhibiting a sharp peakin a region of wave vectors ksurrounding the point k1= k. In exponential form:

withAothe magnitude ofA. For example, a common choice forAois aGaussian wave packet:[14]
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p://en.wikipedia.org/wiki/Wave#cite_note-Chen-11http://en.wikipedia.org/wiki/Wave#cite_note-Chen-11http://en.wikipedia.org/wiki/Wave#cite_note-Chen-11http://en.wikipedia.org/wiki/Group_velocityhttp://en.wikipedia.org/wiki/Group_velocityhttp://en.wikipedia.org/wiki/Phase_%28waves%29http://en.wikipedia.org/wiki/Wavenumber


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where determines the spread of k1-values about k, andNis the amplitude of the wave.

The exponential function inside the integral for oscillates rapidly with its argument, say (k1), and where itvaries rapidly, the exponentials cancel each other out,interferedestructively, contributing little to .[13]However, an exception occurs at the location where the argument of the exponential varies slowly. (This

observation is the basis for the method ofstationary phasefor evaluation of such integrals.[15]

)The conditionfor to vary slowly is that its rate of change with k1be small; this rate of variation is:[13]

where the evaluation is made at k1= kbecauseA(k1)is centered there. This result shows that the positionxwhere the phase changes slowly, the position where is appreciable, moves with time at a speed called the

group velocity:

The group velocity therefore depends upon thedispersion relationconnecting and k. For example, inquantum mechanics the energy of a particle represented as a wave packet isE= = (k)2/(2m).Consequently, for that wave situation, the group velocity is

showing that the velocity of a localized particle in quantum mechanics is its group velocity.[13]

Because thegroup velocity varies with k, the shape of the wave packet broadens with time, and the particle becomes lesslocalized.[16]In other words, the velocity of the constituent waves of the wave packet travel at a rate thatvaries with their wavelength, so some move faster than others, and they cannot maintain the sameinterference patternas the wave propagates.

Sinusoidal waves

Sinusoidal waves correspond tosimple harmonic motion.

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

where

is the maximumamplitudeof the wave, maximum distance from the highest point of thedisturbance in the medium (the crest) to the equilibrium point during one wave cycle. In theillustration to the right, this is the maximum vertical distance between the baseline and the wave.
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is the spacecoordinate is the time coordinate

is thewavenumber is theangular frequency

is thephase constant.

The units of the amplitude depend on the type of wave. Transverse mechanical waves (e.g., a wave on astring) have an amplitude expressed as adistance(e.g., meters), longitudinal mechanical waves (e.g., soundwaves) use units of pressure (e.g., pascals), and electromagnetic waves (a form of transverse vacuum wave)express the amplitude in terms of itselectric field(e.g., volts/meter).

Thewavelength is the distance between two sequential crests or troughs (or other equivalent points),generally is measured in meters. Awavenumber , the spatial frequency of the wave inradiansper unitdistance (typically per meter), can be associated with the wavelength by the relation

Theperiod is the time for one complete cycle of an oscillation of a wave. Thefrequency is the numberof periods per unit time (per second) and is typically measured inhertz.These are related by:

In other words, the frequency and period of a wave are reciprocals.

Theangular frequency represents the frequency in radians per second. It is related to the frequency orperiod by

The wavelength of a sinusoidal waveform traveling at constant speed is given by:[17]

where is called the phase speed (magnitude of thephase velocity)of the wave and is the wave's

frequency.Wavelength can be a useful concept even if the wave is notperiodicin space. For example, in an oceanwave approaching shore, the incoming wave undulates with a varying localwavelength that depends in parton the depth of the sea floor compared to the wave height. The analysis of the wave can be based uponcomparison of the local wavelength with the local water depth.[18]

Although arbitrary wave shapes will propagate unchanged in losslesslinear time-invariant systems,in thepresence of dispersion thesine waveis the unique shape that will propagate unchanged but for phase andamplitude, making it easy to analyze.[19]Due to theKramersKronig relations,a linear medium withdispersion also exhibits loss, so the sine wave propagating in a dispersive medium is attenuated in certain

frequency ranges that depend upon the medium.[20]

Thesine functionis periodic, so thesine waveorsinusoid has awavelengthin space and a period in time.[21][22]

The sinusoid is defined for all times and distances, whereas in physical situations we usually deal withwaves that exist for a limited span in space and duration in time. Fortunately, an arbitrary wave shape can be
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decomposed into an infinite set of sinusoidal waves by the use ofFourier analysis.As a result, the simplecase of a single sinusoidal wave can be applied to more general cases.[23][24]In particular, many media arelinear,or nearly so, so the calculation of arbitrary wave behavior can be found by adding up responses toindividual sinusoidal waves using thesuperposition principleto find the solution for a general waveform.[25]When a medium isnonlinear,the response to complex waves cannot be determined from a sine-wavedecomposition.

Plane wavesMain article:Plane wave

Standing waves

Main articles:Standing wave,Acoustic resonance,Helmholtz resonatorandOrgan pipe

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

A standing wave, also known as astationary wave, is a wave that remains in a constant position. Thisphenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arisein a stationary medium as a result ofinterferencebetween two waves traveling in opposite directions.

Thesumof two counter-propagating waves (of equal amplitude and frequency) creates astanding wave.Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causingwave reflection, and therefore introducing a counter-propagating wave. For example when aviolinstring isdisplaced, transverse waves propagate out to where the string is held in place at thebridgeand thenut,where the waves are reflected back. At the bridge and nut, the two opposed waves are inantiphaseandcancel each other, producing anode.Halfway between two nodes there is anantinode,where the twocounter-propagating waves enhanceeach other maximally. There is no netpropagation of energyover time.

One-dimensional standing waves; thefundamentalmode and the first 5overtones.

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

Astanding wave on a diskwith two nodal lines crossing at the center; this is an overtone.
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Physical properties

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,TransmittanceandTransmission medium

Waves normally move in a straight line (i.e. rectilinearly) through atransmission medium.Such media canbe classified into one or more of the following categories:

A bounded mediumif it is finite in extent, otherwise an unbounded medium

A linear mediumif the amplitudes of different waves at any particular point in the medium can beadded

A uniform mediumor homogeneous mediumif its physical properties are unchanged at different

locations in space

An anisotropic mediumif one or more of its physical properties differ in one or more directions An isotropic mediumif its physical properties are thesamein all directions

Absorption

Main articles:Absorption (acoustics)andAbsorption (electromagnetic radiation)

Absorption of waves mean, if a kind of wave strikes a matter, it will be absorbed by the matter. When awave with that same natural frequency impinges upon an atom, then the electrons of that atom will be setinto vibrational motion. If a wave of a given frequency strikes a material with electrons having the same

vibrational frequencies, then those electrons will absorb the energy of the wave and transform it intovibrational motion.

Reflection

Main article:Reflection (physics)

When a wave strikes a reflective surface, it changes direction, such that the angle made by theincident waveand linenormalto the surface equals the angle made by the reflected wave and the same normal line.

Interference

Main article:Interference (wave propagation)
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Waves that encounter each other combine throughsuperpositionto create a new wave called aninterferencepattern.Important interference patterns occur for waves that are inphase.

Refraction

Main article:Refraction

Sinusoidal traveling plane wave entering a region of lower 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 thephase velocitychanges. Typically, refraction occurs when a wave passes from onemediuminto another. Theamount by which a wave is refracted by a material is given by therefractive indexof the material. Thedirections of incidence and refraction are related to the refractive indices of the two materials bySnell's law.

Diffraction

Main article:Diffraction

A wave exhibits diffraction when it encounters an obstacle that bends the wave or when it spreads afteremerging from an opening. Diffraction effects are more pronounced when the size of the obstacle or openingis comparable to the wavelength of the wave.

Polarization

Main article:Polarization (waves)

A wave is polarized if it oscillates in one direction or plane. A wave can be polarized by the use of apolarizing filter. The polarization of a transverse 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 ofoscillation is along the direction of travel.

Dispersion

Schematic of light being dispersed by a prism. Click to see animation.
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Main articles:Dispersion (optics)andDispersion (water waves)

A wave undergoes dispersion when either thephase velocityor thegroup velocitydepends on the wavefrequency. Dispersion is most easily seen by letting white light pass through aprism,the result of which isto produce the spectrum of colours of the rainbow.Isaac Newtonperformed experiments with light and

prisms, presenting his findings in theOpticks(1704) that white light consists of several colours and thatthese colours cannot be decomposed any further.[26]

Mechanical waves

Main article:Mechanical wave

Waves on strings

Main article:Vibrating string

The speed of a transverse wave traveling along avibrating string( v) is directly proportional to the squareroot of thetensionof the string ( T) over thelinear mass density():

where the linear densityis the mass per unit length of the string.

Acoustic waves

Acoustic orsoundwaves travel at speed given by

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

Water waves

Main article:Water waves

Rippleson the surface of a pond are actually a combination of transverse and longitudinal waves;therefore, the points on the surface follow orbital paths.

Sounda mechanical wave that propagates through gases, liquids, solids and plasmas; Inertial waves,which occur in rotating fluids and are restored by theCoriolis effect; Ocean surface waves,which are perturbations that propagate through water.

Seismic waves

Main article:Seismic waves

Shock waves
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Main article:Shock waveSee also:Sonic boomandCherenkov radiation

Other

Waves oftraffic,that is, propagation of different densities of motor vehicles, and so forth, which canbe modeled as kinematic waves[27]

Metachronal waverefers to the appearance of a traveling wave produced by coordinated sequentialactions.

It is worth noting that themass-energy equivalenceequation can be solved for this form:.

Electromagnetic waves

Main articles:Electromagnetic radiationandElectromagnetic spectrum

(radio, micro, infrared, visible, uv)

An electromagnetic wave consists of two waves that are oscillations of theelectricandmagneticfields. An

electromagnetic wave travels in a direction that is at right angles to the oscillation direction of both fields. Inthe 19th century,James Clerk Maxwellshowed that, invacuum,the electric and magnetic fields satisfy thewave equationboth with speed equal to that of thespeed of light.From this emerged the idea thatlightis anelectromagnetic wave. Electromagnetic waves can have different frequencies (and thus wavelengths), givingrise to various types of radiation such asradio waves,microwaves,infrared,visible light,ultravioletandX-rays.

Quantum mechanical waves

Main article:Schrdinger equation

See also:Wave function

TheSchrdinger equationdescribes the wave-like behavior of particles inquantum mechanics.Solutions ofthis equation arewave functionswhich can be used to describe the probability density of a particle.
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A propagating wave packet; in general, the envelopeof the wave packet moves at a different speed than the

constituent waves.[28]

de Broglie waves

Main articles:Wave packetandMatter wave

Louis de Brogliepostulated that all particles withmomentumhave a wavelength

where hisPlanck's constant,andpis the magnitude of themomentumof the particle. This hypothesis was atthe basis ofquantum mechanics.Nowadays, this wavelength is called thede Broglie wavelength.Forexample, theelectronsin aCRTdisplay have a de Broglie wavelength of about 1013m.

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 thewave vectorkas:

and the momentum by:

However, a wave like this with definite wavelength is not localized in space, and so cannot represent aparticle localized in space. To localize a particle, de Broglie proposed a superposition of differentwavelengths ranging around a central value in awave packet,[29]a waveform often used inquantum

mechanicsto describe thewave functionof a particle. In a wave packet, the wavelength of the particle is notprecise, and the local wavelength deviates on either side of the main wavelength value.

In representing the wave function of a localized particle, thewave packetis often taken to have aGaussianshapeand is called a Gaussian wave packet.[30]Gaussian wave packets also are used to analyze waterwaves.[31]

For example, a Gaussian wavefunction might take the form:[32]

at some initial time t= 0, where the central wavelength is related to the central wave vector k0as 0= 2 / k0.It is well known from the theory ofFourier analysis,[33]or from theHeisenberg uncertainty principle(in the
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case of quantum mechanics) that a narrow range of wavelengths is necessary to produce a localized wavepacket, and the more localized the envelope, the larger the spread in required wavelengths. TheFouriertransformof a Gaussian is itself a Gaussian.[34]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 such that k = 2 .

The parameter decides the spatial spread of the Gaussian along the x-axis, while the Fourier transformshows a spread inwave vectorkdetermined by 1/. That is, the smaller the extent in space, the larger theextent in k, and hence in = 2/k.

Animation showing the effect of a cross-polarized gravitational wave on a ring oftest particles

Gravity waves

Gravity wavesare waves generated in a fluid medium or at the interface between two media when the force

of gravity or buoyancy tries to restore equilibrium. A ripple on a pond is one example.

Gravitational waves

Main article:Gravitational wave

Researchers believe thatgravitational wavesalso travel through space, although gravitational waves havenever been directly detected. Gravitational waves are disturbances in the curvature ofspacetime,predicted

by Einstein's theory ofgeneral relativity.

WKB method

Main article:WKB methodSee also:Slowly varying envelope approximation
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In a nonuniform medium, in which the wavenumber kcan depend on the location as well as the frequency

waves wikia

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