Wavelength

For other uses, seeWavelength (disambiguation).

Inphysicsandmathematics,wavelengthorspatial periodof awaveorperiodic functionis the distance over which the wave's shape repeats.In other words, it is the distance between consecutive corresponding points of the samephaseon the wave, such as two adjacent crests, troughs, orzero crossings.Wavelength is a characteristic of both traveling waves andstanding waves,as well as other spatial wave patterns.Theinverseof the wavelength is called thespatial frequency.Wavelength is commonly designated by theGreek letterlambda(λ). The term "wavelength" is also sometimes applied tomodulatedwaves, and to the sinusoidalenvelopesof modulated waves or waves formed byinterferenceof several sinusoids.

The wavelength of asine wave,λ, can be measured between any two points with the samephase,such as between crests (on top), or troughs (on bottom), or correspondingzero crossingsas shown.

Assuming asinusoidal wavemoving at a fixed wave speed, wavelength is inversely proportional to thefrequencyof the wave: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths.

Wavelength depends on the medium (for example, vacuum, air, or water) that a wave travels through. Examples of waves aresound waves,light,water wavesand periodic electrical signals in aconductor.Asoundwave is a variation in airpressure,while inlightand otherelectromagnetic radiationthe strength of theelectricand themagnetic fieldvary. Water waves are variations in the height of a body of water. In a crystallattice vibration,atomic positions vary.

The range of wavelengths or frequencies for wave phenomena is called aspectrum.The name originated with thevisible light spectrumbut now can be applied to the entireelectromagnetic spectrumas well as to asound spectrumorvibration spectrum.

Sinusoidal waves

Inlinearmedia, any wave pattern can be described in terms of the independent propagation of sinusoidal components. The wavelengthλof a sinusoidal waveform traveling at constant speed$v$is given by

$\lambda ={\frac {v}{f}}\,\,,$

where$v$is called the phase speed (magnitude of thephase velocity) of the wave and$f$is the wave'sfrequency.In adispersive medium,the phase speed itself depends upon the frequency of the wave, making therelationship between wavelength and frequencynonlinear.

In the case ofelectromagnetic radiation—such as light—infree space,the phase speed is thespeed of light,about3×10⁸m/s.Thus the wavelength of a 100 MHz electromagnetic (radio) wave is about:3×10⁸m/sdivided by10⁸Hz= 3 m. The wavelength of visible light ranges from deepred,roughly 700nm,toviolet,roughly 400 nm (for other examples, seeelectromagnetic spectrum).

Forsound wavesin air, thespeed of soundis 343 m/s (atroom temperature and atmospheric pressure). The wavelengths of sound frequencies audible to the human ear (20Hz–20 kHz) are thus between approximately 17mand 17mm,respectively. Somewhat higher frequencies are used bybatsso they can resolve targets smaller than 17 mm. Wavelengths in audible sound are much longer than those in visible light.

Sinusoidal standing waves in a box that constrains the end points to be nodes will have an integer number of half wavelengths fitting in the box.

A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue)

Standing waves

Astanding waveis an undulatory motion that stays in one place. A sinusoidal standing wave includes stationary points of no motion, callednodes,and the wavelength is twice the distance between nodes.

The upper figure shows three standing waves in a box. The walls of the box are considered to require the wave to have nodes at the walls of the box (an example ofboundary conditions) determining which wavelengths are allowed. For example, for an electromagnetic wave, if the box has ideal metal walls, the condition for nodes at the walls results because the metal walls cannot support a tangential electric field, forcing the wave to have zero amplitude at the wall.

The stationary wave can be viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities.Consequently, wavelength, period, and wave velocity are related just as for a traveling wave. For example, thespeed of lightcan be determined from observation of standing waves in a metal box containing an ideal vacuum.

Mathematical representation

Traveling sinusoidal waves are often represented mathematically in terms of their velocityv(in the x direction), frequencyfand wavelengthλas:

$y(x,\ t)=A\cos \left(2\pi \left({\frac {x}{\lambda }}-ft\right)\right)=A\cos \left({\frac {2\pi }{\lambda }}(x-vt)\right)$

whereyis the value of the wave at any positionxand timet,andAis theamplitudeof the wave. They are also commonly expressed in terms ofwavenumberk(2π times the reciprocal of wavelength) andangular frequencyω(2π times the frequency) as:

$y(x,\ t)=A\cos \left(kx-\omega t\right)=A\cos \left(k(x-vt)\right)$

in which wavelength and wavenumber are related to velocity and frequency as:

$k={\frac {2\pi }{\lambda }}={\frac {2\pi f}{v}}={\frac {\omega }{v}},$

or

$\lambda ={\frac {2\pi }{k}}={\frac {2\pi v}{\omega }}={\frac {v}{f}}.$

In the second form given above, the phase(kx−ωt)is often generalized to(k⋅r−ωt),by replacing the wavenumberkwith awave vectorthat specifies the direction and wavenumber of aplane wavein3-space,parameterized by position vectorr.In that case, the wavenumberk,the magnitude ofk,is still in the same relationship with wavelength as shown above, withvbeing interpreted as scalar speed in the direction of the wave vector. The first form, using reciprocal wavelength in the phase, does not generalize as easily to a wave in an arbitrary direction.

Generalizations to sinusoids of other phases, and to complex exponentials, are also common; seeplane wave.The typical convention of using thecosinephase instead of thesinephase when describing a wave is based on the fact that the cosine is the real part of the complex exponential in the wave

$Ae^{i\left(kx-\omega t\right)}.$

General media

Wavelength is decreased in a medium with slower propagation.

Refraction: upon entering a medium where its speed is lower, the wave changes direction.

Separation of colors by a prism (click for animation if it is not already playing)

The speed of a wave depends upon the medium in which it propagates. In particular, the speed of light in a medium is less than invacuum,which means that the same frequency will correspond to a shorter wavelength in the medium than in vacuum, as shown in the figure at right.

This change in speed upon entering a medium causesrefraction,or a change in direction of waves that encounter the interface between media at an angle.Forelectromagnetic waves,this change in the angle of propagation is governed bySnell's law.

The wave velocity in one medium not only may differ from that in another, but the velocity typically varies with wavelength. As a result, the change in direction upon entering a different medium changes with the wavelength of the wave.

For electromagnetic waves the speed in a medium is governed by itsrefractive indexaccording to

$v={\frac {c}{n(\lambda _{0})}},$

wherecis thespeed of lightin vacuum andn(λ₀) is the refractive index of the medium at wavelength λ₀,where the latter is measured in vacuum rather than in the medium. The corresponding wavelength in the medium is

$\lambda ={\frac {\lambda _{0}}{n(\lambda _{0})}}.$

When wavelengths of electromagnetic radiation are quoted, the wavelength in vacuum usually is intended unless the wavelength is specifically identified as the wavelength in some other medium. In acoustics, where a medium is essential for the waves to exist, the wavelength value is given for a specified medium.

The variation in speed of light with wavelength is known asdispersion,and is also responsible for the familiar phenomenon in which light is separated into component colours by aprism.Separation occurs when the refractive index inside the prism varies with wavelength, so different wavelengths propagate at different speeds inside the prism, causing them torefractat different angles. The mathematical relationship that describes how the speed of light within a medium varies with wavelength is known as adispersion relation.

Nonuniform media

Various local wavelengths on a crest-to-crest basis in an ocean wave approaching shore

Wavelength can be a useful concept even if the wave is notperiodicin space. For example, in an ocean wave approaching shore, shown in the figure, the incoming wave undulates with a varyinglocalwavelength that depends in part on the depth of the sea floor compared to the wave height. The analysis of the wave can be based upon comparison of the local wavelength with the local water depth.

A sinusoidal wave travelling in a nonuniform medium, with loss

Waves that are sinusoidal in time but propagate through a medium whose properties vary with position (aninhomogeneousmedium) may propagate at a velocity that varies with position, and as a result may not be sinusoidal in space. The figure at right shows an example. As the wave slows down, the wavelength gets shorter and the amplitude increases; after a place of maximum response, the short wavelength is associated with a high loss and the wave dies out.

The analysis ofdifferential equationsof such systems is often done approximately, using theWKB method(also known as theLiouville–Green method). The method integrates phase through space using a localwavenumber,which can be interpreted as indicating a "local wavelength" of the solution as a function of time and space. This method treats the system locally as if it were uniform with the local properties; in particular, the local wave velocity associated with a frequency is the only thing needed to estimate the corresponding local wavenumber or wavelength. In addition, the method computes a slowly changing amplitude to satisfy other constraints of the equations or of the physical system, such as forconservation of energyin the wave.

Crystals

A wave on a line of atoms can be interpreted according to a variety of wavelengths.

Waves in crystalline solids are not continuous, because they are composed of vibrations of discrete particles arranged in a regular lattice. This producesaliasingbecause the same vibration can be considered to have a variety of different wavelengths, as shown in the figure.Descriptions using more than one of these wavelengths are redundant; it is conventional to choose the longest wavelength that fits the phenomenon. The range of wavelengths sufficient to provide a description of all possible waves in a crystalline medium corresponds to the wave vectors confined to theBrillouin zone.

This indeterminacy in wavelength in solids is important in the analysis of wave phenomena such asenergy bandsandlattice vibrations.It is mathematically equivalent to thealiasingof a signal that issampledat discrete intervals.

More general waveforms

Near-periodic waves over shallow water

The concept of wavelength is most often applied to sinusoidal, or nearly sinusoidal, waves, because in a linear system the sinusoid is the unique shape that propagates with no shape change – just a phase change and potentially an amplitude change.The wavelength (or alternativelywavenumberorwave vector) is a characterization of the wave in space, that is functionally related to its frequency, as constrained by the physics of the system. Sinusoids are the simplesttraveling wavesolutions, and more complex solutions can be built up bysuperposition.

In the special case of dispersion-free and uniform media, waves other than sinusoids propagate with unchanging shape and constant velocity. In certain circumstances, waves of unchanging shape also can occur in nonlinear media; for example, the figure shows ocean waves in shallow water that have sharper crests and flatter troughs than those of a sinusoid, typical of acnoidal wave,a traveling wave so named because it is described by theJacobi elliptic functionofm-th order, usually denoted ascn(x;m).Large-amplitudeocean waveswith certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium.

Wavelength of a periodic but non-sinusoidal waveform.

If a traveling wave has a fixed shape that repeats in space or in time, it is aperiodic wave.Such waves are sometimes regarded as having a wavelength even though they are not sinusoidal.As shown in the figure, wavelength is measured between consecutive corresponding points on the waveform.

Wave packets

A propagating wave packet

Main article:Wave packet

Localizedwave packets,"bursts" of wave action where each wave packet travels as a unit, find application in many fields of physics. A wave packet has anenvelopethat describes the overall amplitude of the wave; within the envelope, the distance between adjacent peaks or troughs is sometimes called alocal wavelength.An example is shown in the figure. In general, theenvelopeof the wave packet moves at a speed different from the constituent waves.

UsingFourier analysis,wave packets can be analyzed into infinite sums (or integrals) of sinusoidal waves of differentwavenumbersor wavelengths.

Louis de Brogliepostulated that all particles with a specific value ofmomentumphave a wavelengthλ = h/p,wherehisPlanck's constant.This hypothesis was at the basis ofquantum mechanics.Nowadays, this wavelength is called thede Broglie wavelength.For example, theelectronsin aCRTdisplay have a De Broglie wavelength of about 10⁻¹³m. To prevent thewave functionfor such a particle being spread over all space, de Broglie proposed using wave packets to represent particles that are localized in space.The spatial spread of the wave packet, and the spread of thewavenumbersof sinusoids that make up the packet, correspond to the uncertainties in the particle's position and momentum, the product of which is bounded byHeisenberg uncertainty principle.

Interference and diffraction

Double-slit interference

Main article:Interference (wave propagation)

Pattern of light intensity on a screen for light passing through two slits. The labels on the right refer to the difference of the path lengths from the two slits, which are idealized here as point sources.

When sinusoidal waveforms add, they may reinforce each other (constructive interference) or cancel each other (destructive interference) depending upon their relative phase. This phenomenon is used in theinterferometer.A simple example is an experiment due toYoungwhere light is passed throughtwo slits. As shown in the figure, light is passed through two slits and shines on a screen. The path of the light to a position on the screen is different for the two slits, and depends upon the angle θ the path makes with the screen. If we suppose the screen is far enough from the slits (that is,sis large compared to the slit separationd) then the paths are nearly parallel, and the path difference is simplydsin θ. Accordingly, the condition for constructive interference is:

$d\sin \theta =m\lambda \,$

wheremis an integer, and for destructive interference is:

$d\sin \theta =(m+1/2)\lambda \.$

Thus, if the wavelength of the light is known, the slit separation can be determined from the interference pattern orfringes,andvice versa.

For multiple slits, the pattern is

$I_{q}=I_{1}\sin ^{2}\left({\frac {q\pi g\sin \alpha }{\lambda }}\right)/\sin ^{2}\left({\frac {\pi g\sin \alpha }{\lambda }}\right)\,$

whereqis the number of slits, andgis the grating constant. The first factor,I₁,is the single-slit result, which modulates the more rapidly varying second factor that depends upon the number of slits and their spacing. In the figureI₁has been set to unity, a very rough approximation.

The effect of interference is toredistributethe light, so the energy contained in the light is not altered, just where it shows up.

Single-slit diffraction

Main articles:DiffractionandDiffraction formalism

Diffraction pattern of a double slit has a single-slitenvelope.

The notion of path difference and constructive or destructive interference used above for the double-slit experiment applies as well to the display of a single slit of light intercepted on a screen. The main result of this interference is to spread out the light from the narrow slit into a broader image on the screen. This distribution of wave energy is calleddiffraction.

Two types of diffraction are distinguished, depending upon the separation between the source and the screen:Fraunhofer diffractionor far-field diffraction at large separations andFresnel diffractionor near-field diffraction at close separations.

In the analysis of the single slit, the non-zero width of the slit is taken into account, and each point in the aperture is taken as the source of one contribution to the beam of light (Huygens' wavelets). On the screen, the light arriving from each position within the slit has a different path length, albeit possibly a very small difference. Consequently, interference occurs.

In the Fraunhofer diffraction pattern sufficiently far from a single slit, within asmall-angle approximation,the intensity spreadSis related to positionxvia a squaredsinc function:

$S(u)=\mathrm {sinc} ^{2}(u)=\left({\frac {\sin \pi u}{\pi u}}\right)^{2}\;$ with $u={\frac {xL}{\lambda R}}\,$

whereLis the slit width,Ris the distance of the pattern (on the screen) from the slit, and λ is the wavelength of light used. The functionShas zeros whereuis a non-zero integer, where are atxvalues at a separation proportion to wavelength.

Diffraction-limited resolution

Main articles:Angular resolutionandDiffraction-limited system

Diffraction is the fundamental limitation on theresolving powerof optical instruments, such astelescopes(includingradiotelescopes) andmicroscopes. For a circular aperture, the diffraction-limited image spot is known as anAiry disk;the distancexin the single-slit diffraction formula is replaced by radial distancerand the sine is replaced by 2J₁,whereJ₁is a first orderBessel function.

The resolvablespatialsize of objects viewed through a microscope is limited according to theRayleigh criterion,the radius to the first null of the Airy disk, to a size proportional to the wavelength of the light used, and depending on thenumerical aperture:

$r_{Airy}=1.22{\frac {\lambda }{2\mathrm {NA} }}\,$

where the numerical aperture is defined as$\mathrm {NA} =n\sin \theta \;$for θ being the half-angle of the cone of rays accepted by themicroscope objective.

Theangularsize of the central bright portion (radius to first null of theAiry disk) of the image diffracted by a circular aperture, a measure most commonly used for telescopes and cameras, is:

$\delta =1.22{\frac {\lambda }{D}}\,$

whereλis the wavelength of the waves that are focused for imaging,Dtheentrance pupildiameter of the imaging system, in the same units, and the angular resolutionδis in radians.

As with other diffraction patterns, the pattern scales in proportion to wavelength, so shorter wavelengths can lead to higher resolution.

Subwavelength

The termsubwavelengthis used to describe an object having one or more dimensions smaller than the length of the wave with which the object interacts. For example, the termsubwavelength-diameter optical fibremeans anoptical fibrewhose diameter is less than the wavelength of light propagating through it.

A subwavelength particle is a particle smaller than the wavelength of light with which it interacts (seeRayleigh scattering). Subwavelengthaperturesare holes smaller than the wavelength of light propagating through them. Such structures have applications inextraordinary optical transmission,andzero-mode waveguides,among other areas ofphotonics.

Subwavelengthmay also refer to a phenomenon involving subwavelength objects; for example,subwavelength imaging.

Angular wavelength

Relationship between wavelength, angular wavelength, and other wave properties.

A quantity related to the wavelength is theangular wavelength(also known asreduced wavelength), usually symbolized byƛ( "lambda-bar" orbarred lambda). It is equal to the ordinary wavelength reduced by a factor of 2π (ƛ=λ/2π), with SI units of meter per radian. It is the inverse ofangular wavenumber(k= 2π/λ). It is usually encountered in quantum mechanics, where it is used in combination with thereduced Planck constant(symbolħ,h-bar) and theangular frequency(symbolω= 2πf).

See also

• Emission spectrum
• Envelope (waves)
• Fraunhofer lines– dark lines in the solar spectrum, traditionally used as standard optical wavelength references
• Index of wave articles
• Length measurement
• Spectral line
• Spectroscopy
• Spectrum

External links

• Conversion: Wavelength to Frequency and vice versa – Sound waves and radio waves
• Teaching resource for 14–16 years on sound including wavelengthArchived2012-03-13 at theWayback Machine
• The visible electromagnetic spectrum displayed in web colors with according wavelengths