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[[File:Sine wavelength.svg|thumb|right|Wavelength of a [[sine wave]], λ, can be measured between any two points with the same [[phase (waves)|phase]], such as between crests, or troughs, or corresponding [[zero crossing]]s as shown.]] 
In [[physics]], the '''wavelength''' of a [[sinusoidal wave]] is the spatial period of the wave—the distance over which the wave's shape repeats.<ref name=hecht>
{{cite book
 |first=Eugene
 |last=Hecht
 |year=1987
 |title=Optics
 |edition=2nd
 |publisher=Addison Wesley
 |isbn=0-201-11609-X
 |pages=15–16
 }}</ref>
It is usually determined by considering the distance between consecutive corresponding points of the same [[phase (waves)|phase]], such as crests, troughs, or [[zero crossing]]s, and is a characteristic of both traveling waves and [[standing wave]]s, as well as other spatial wave patterns.<ref name=Seaway>
{{cite book
 |title=Principles of physics
 |author=Raymond A. Serway, John W. Jewett
 |pages=404, 440
 |url=http://books.google.com/books?id=1DZz341Pp50C&pg=PA404
 |edition=4th
 |isbn=0-534-49143-X
 |publisher=Cengage Learning
 }}</ref><ref>
{{cite book
 | title=The surface physics of liquid crystals
 | author=A. A. Sonin
 | publisher=Taylor & Francis
 | year=1995
 | isbn=2-88124-995-7
 | page=17
}}</ref> Wavelength is commonly designated by the [[Greek letter]] ''[[lambda]]'' (λ). The concept can also be applied to periodic waves of non-sinusoidal shape.<ref name=hecht/><ref name=Flowers>
{{cite book
 |title=An introduction to numerical methods in C++
 |chapter=§21.2 Periodic functions
 |page=473
 |url=http://books.google.com/books?id=weYj75E_t6MC&pg=RA1-PA473
 |author=Brian Hilton Flowers
 |isbn=0-19-850693-7
 |year=2000
 |edition=2nd
 |publisher = Cambridge University Press
 }}</ref> 
The term ''wavelength'' is also sometimes applied to [[modulation|modulated]] waves, and to the sinusoidal [[envelope (mathematics)|envelopes]] of modulated waves or waves formed by [[Interference (wave propagation)|interference]] of several sinusoids.<ref>
{{cite book
 | title = Electromagnetic Theory for Microwaves and Optoelectronics
 | author = Keqian Zhang and Dejie Li
 | publisher = Springer, 
 | year = 2007
 | isbn = 978-3-540-74295-1
 | page = 533
 | url = http://books.google.com/books?id=3Da7MvRZTlAC&pg=PA533&dq=wavelength+modulated-wave+envelope
 }}</ref> The [[SI]] unit of wavelength is the [[meter]].

Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to [[frequency]] of the wave: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths.<ref>
{{cite book
 | title = In Quest of the Universe
 | author = Theo Koupelis and Karl F. Kuhn
 | publisher = Jones & Bartlett Publishers
 | year = 2007
 | isbn = 0-7637-4387-9
 | url = http://books.google.com/books?id=WwKjznJ9Kq0C&pg=PA102&dq=wavelength+lambda+light+sound+frequency+wave+speed }}</ref>

Examples of wave-like phenomena are [[sound wave]]s, [[light]], and [[water wave]]s. A [[sound]] wave is a variation in air [[sound pressure|pressure]], while in [[light]] and other [[electromagnetic radiation]] the strength of the [[electric field|electric]] and the [[magnetic field]] vary. Water waves are variations in the height of a body of water. In a crystal [[lattice vibration]], atomic positions vary.

Wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in sinusoidal waves over deep water a particle near the water's surface moves in a circle of the same diameter as the wave height, unrelated to wavelength.<ref name=Pinet>
{{cite book |title=Invitation to Oceanography
 |author = Paul R Pinet
 |url = http://books.google.com/books?id=6TCm8Xy-sLUC&pg=PA237
 |page = 237
 |publisher = Jones & Bartlett Publishers
 |isbn = 0-7637-5993-7
 |edition = 5th
 |year = 2008
 }}</ref>

==Sinusoidal waves==

In [[linear]] media, any wave pattern can be described in terms of the independent propagation of sinusoidal components. The wavelength ''λ'' of a sinusoidal waveform traveling at constant speed ''v'' is given by<ref name= Cassidy>
{{cite book
 |title=Understanding physics
 |author= David C. Cassidy, Gerald James Holton, Floyd James Rutherford
 |url=http://books.google.com/books?id=rpQo7f9F1xUC&pg=PA340
 |pages=339 ''ff''
 |isbn=0-387-98756-8
 |year=2002
 |publisher=Birkhäuser}}
</ref>

:<math>\lambda = \frac{v}{f}</math>,

where ''v'' is called the phase speed (magnitude of the [[phase velocity]]) of the wave and ''f'' is the wave's frequency. In a ''dispersive'' medium, the phase speed itself depends upon the frequency of the wave, making the [[dispersion relation|relationship between wavelength and frequency]] nonlinear.

In the case of [[electromagnetic radiation]]—such as light—in [[free space]], the phase speed is the [[speed of light]], about 3×10<sup>8</sup>&nbsp;m/s.  Thus the wavelength of a 100&nbsp;MHz electromagnetic (radio) wave is about: 3×10<sup>8</sup>&nbsp;m/s divided by 10<sup>8</sup>&nbsp;Hz = 3 metres. The wavelength of visible light ranges from deep [[red]], roughly 700 [[nanometre|nm]], to [[Violet (color)|violet]], roughly 400&nbsp;nm (for other examples, see [[electromagnetic spectrum]]).

For [[sound wave]]s in air, the [[speed of sound]] is 343&nbsp;m/s (at [[standard conditions for temperature and pressure|room temperature and atmospheric pressure]]). The wavelengths of sound frequencies audible to the human ear (20&nbsp;[[hertz|Hz]]–20&nbsp;kHz) are thus between approximately 17&nbsp;[[metre|m]] and 17&nbsp;[[millimetre|mm]], respectively. Note that the wavelengths in audible sound are much longer than those in visible light.

[[File:Waves in Box.svg|thumb|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.]]
[[File:Standing wave 2.gif|thumb|right|A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue)]]

===Standing waves===
A [[standing wave]] is an undulatory motion that stays in one place.  A sinusoidal standing wave includes stationary points of no motion, called [[node (physics)|nodes]], 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 of [[boundary 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.<ref>
{{cite book
 | title = The World of Physics
 | author = John Avison
 | publisher = Nelson Thornes
 | year = 1999
 | isbn = 978-0-17-438733-6
 | page = 460
 | url = http://books.google.com/books?id=DojwZzKAvN8C&pg=PA460&dq=%22standing+wave%22+wavelength
 }}</ref> Consequently, wavelength, period, and wave velocity are related just as for a traveling wave. For example, the [[Speed_of_light#Cavity_resonance|speed of light]] can 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 velocity ''v'' (in the x direction), frequency ''f'' and wavelength ''λ'' as:

:<math>  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 )</math>

where ''y'' is the value of the wave at any position ''x'' and time ''t'', and ''A'' is the [[amplitude]] of the wave.  They are also commonly expressed in terms of [[wavenumber]] ''k'' (2π times the reciprocal of wavelength) and [[angular frequency]] ''ω'' (2π times the frequency) as:

:<math>  y (x, \ t) = A \cos \left( kx - \omega t \right)  = A \cos \left(k(x - v t) \right) </math>

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

:<math> k = \frac{2 \pi}{\lambda} = \frac{2 \pi f}{v} = \frac{\omega}{v},</math>

or

:<math> \lambda = \frac{2 \pi}{k} = \frac{2 \pi v}{\omega} = \frac{v}{f}.</math>

In the second form given above, the phase {{nowrap|(''kx'' − ''ωt'')}} is often generalized to {{nowrap|('''k'''•'''r''' − ''ωt'')}}, by replacing the wavenumber ''k'' with a [[wave vector]] that specifies the direction and wavenumber of a [[plane wave]] in [[3-space]], parameterized by position vector '''r'''.  In that case, the wavenumber ''k'', the magnitude of '''k''', is still in the same relationship with wavelength as shown above, with ''v'' being 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; see [[plane wave]].  The typical convention of using the [[cosine]] phase instead of the [[sine]] phase when describing a wave is based on the fact that the cosine is the real part of the complex exponential in the wave
:<math>A e^{ i \left( kx - \omega t \right)}. </math>

===General media===
[[File:Wavelength & refractive index.JPG|thumb|Wavelength is decreased in a medium with slower propagation.]]
[[File:Refraction - Huygens-Fresnel principle.svg|right|thumb|Refraction: upon entering a medium where its speed is lower, the wave changes direction.]]
[[File:Light dispersion conceptual waves.gif|thumb|Separation of colors by a prism (click for animation)]]

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 in [[Vacuum#In_electromagnetism|vacuum]], 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 causes [[refraction]], or a change in direction of waves that encounter the interface between media at an angle.<ref name=mud>
To aid imagination, this bending of the wave often is compared to the analogy of a column of marching soldiers crossing from solid ground into mud. See, for example, {{cite book |title=Principles of Planetary Climate |url=http://books.google.com/books?id=bO_U8f5pVR8C&pg=PA327 |page=327 |year=2010 |author=Raymond T. Pierrehumbert |publisher=Cambridge University Press |isbn=0-521-86556-5 }}

</ref> For [[electromagnetic waves]], this change in the angle of propagation is governed by [[Snell'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 its ''[[refractive index]]'' according to
:<math>v = \frac{c}{n(\lambda_0)},</math>
where [http://physics.nist.gov/cgi-bin/cuu/Value?c ''c''] is the [[speed of light]] in vacuum and ''n''(λ<sub>0</sub>) is the refractive index of the medium at wavelength λ<sub>0</sub>, where the latter is measured in vacuum rather than in the medium. The corresponding wavelength in the medium is
:<math>\lambda = \frac{\lambda_0}{n(\lambda_0)}.</math>

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 vacuum wavelength is known as [[dispersion (optics)|dispersion]], and is also responsible for the familiar phenomenon in which light is separated into component colors by a [[dispersive prism|prism]]. Separation occurs when the refractive index inside the prism varies with wavelength, so different wavelengths propagate at different speeds inside the prism, causing them to [[refract]] at different angles. The mathematical relationship that describes how the speed of light within a medium varies with wavelength is known as a [[dispersion relation]].

====Nonuniform media====
[[File:Local wavelength.JPG|thumb|Various local wavelengths on a crest-to-crest basis in an ocean wave approaching shore<ref name=Pinet2/>]]
Wavelength can be a useful concept even if the wave is not [[periodic function|periodic]] in space. For example, in an ocean wave approaching shore, shown in the figure, the incoming wave undulates with a varying ''local'' wavelength 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.<ref name=Pinet2>
{{cite book |title=op. cit.
 |author = Paul R Pinet
 |url = http://books.google.com/books?id=6TCm8Xy-sLUC&pg=PA242
 |page = 242
 |isbn = 0-7637-5993-7
 }}</ref>

[[File:Cochlea wave animated.gif|right|thumb|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 (an ''inhomogeneous'' medium) 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 of [[differential equation]]s of such systems is often done approximately, using the ''[[WKB approximation|WKB method]]'' (also known as the ''Liouville–Green method'').  The method integrates phase through space using a local [[wavenumber]], which can be interpreted as indicating a "local wavelength" of the solution as a function of time and space.<ref>
{{cite book
 | title = Principles of Plasma Mechanics
 | author = Bishwanath Chakraborty
 | publisher = New Age International
 | isbn = 978-81-224-1446-2
 | page = 454
 | url = http://books.google.com/books?id=_MIdEiKqdawC&pg=PA454&dq=wkb+local-wavelength
 }}</ref><ref>
{{cite book
 | title = Time-frequency and time-scale methods: adaptive decompositions, uncertainty principles, and sampling
 | author = Jeffrey A. Hogan and Joseph D. Lakey
 | publisher = Birkhäuser
 | year = 2005
 | isbn = 978-0-8176-4276-1
 | page = 348
 | url = http://books.google.com/books?id=YOf0SRzxz3gC&pg=PA348&dq=wkb+local-wavelength
 }}</ref>
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 for [[conservation of energy]] in the wave.

====Crystals====

[[File:Wavelength indeterminacy.JPG|thumb |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 produces [[aliasing]] because the same vibration can be considered to have a variety of different wavelengths, as shown in the figure.<ref name=Putnis>See Figure 4.20 in {{cite book |author= A. Putnis |title=Introduction to mineral sciences |url=http://books.google.com/books?id=yMGzmOqYescC&pg=PA97 |page=97 |isbn=0-521-42947-1 |year=1992 |publisher=Cambridge University Press}} and Figure 2.3 in {{cite book |title=Introduction to lattice dynamics |author=Martin T. Dove |url=http://books.google.com/books?id=vM50l2Vf7HgC&pg=PA22 |page=22 |isbn=0-521-39293-4 |edition=4th |year=1993 |publisher=Cambridge University Press}}</ref> 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 the [[Brillouin zone]].<ref name=Razeghi>{{cite book |title=Fundamentals of solid state engineering 
|author=Manijeh Razeghi |pages=165 ''ff'' |url=http://books.google.com/books?id=6x07E9PSzr8C&pg=PA165 |isbn=0-387-28152-5 |year=2006 |publisher=Birkhäuser |edition=2nd}}</ref>

This indeterminacy in wavelength in solids is important in the analysis of wave phenomena such as [[energy bands]] and [[phonons|lattice vibrations]]. It is mathematically equivalent to the [[aliasing]] of a signal that is [[sampling (signal processing)|sampled]] at discrete intervals.

==More general waveforms==

[[File:Periodic waves in shallow water.png|right|thumb|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.<ref name=Rayleigh>
See {{cite book |title=Encyclopedia Britannica |author=[[Lord Rayleigh]] |chapter=Wave theory |url=http://books.google.com/books?id=r54UAAAAYAAJ&pg=PA422&dq=%22only+kind+of+wave+which+can+be+propagated+without+a+change+of+form%22&hl=en&sa=X&ei=npKWT83pL-vJiQLb19zeCQ&ved=0CDAQ6AEwAA#v=onepage&q=%22only%20kind%20of%20wave%20which%20can%20be%20propagated%20without%20a%20change%20of%20form%22&f=false |year=1890 |publisher=The Henry G Allen Company |edition=9th |page=422}}
</ref>  The wavelength (or alternatively [[wavenumber]] or [[wave 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 simplest [[Wave#Travelling_waves|traveling wave]] solutions, and more complex solutions can be built up by [[superposition principle|superposition]].

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 a [[cnoidal wave]],<ref name=Pilipchuk>
{{cite book
 |title=Nonlinear Dynamics: Between Linear and Impact Limits
 |author=Valery N. Pilipchuk
 |url=http://books.google.com/books?id=pqIlJNq-Ir8C&pg=PA127
 |page=127
 |chapter=Figure 4.4: Transition from quasi-harmonic to cnoidal wave
 |isbn=	3642127983
 |year=2010
 |publisher=Springer
 }}
</ref> a traveling wave so named because it is described by the [[Jacobi elliptic function]] of ''m''-th order, usually denoted as {{nowrap|''cn''(''x''; ''m'')}}.<ref name=Ludu>
{{cite book
 |title=Nonlinear Waves and Solitons on Contours and Closed Surfaces
 |author=Andrei Ludu
 |url=http://books.google.com/books?id=HIu9_8QKo6UC&pg=PA469
 |pages=469 ''ff''
 |chapter=§18.3 Special functions
 |isbn=	3642228941
 |year=2012
 |edition=2nd
 |publisher=Springer
 }}</ref>  Large-amplitude [[ocean wave]]s with certain shapes can propagate unchanged, because of properties of the nonlinear surface-wave medium.<ref>{{cite book
 | title = Nonlinear Ocean Waves and the Inverse Scattering Transform
 | author = Alfred Osborne
 | publisher = Academic Press
 | year = 2010
 | chapter=Chapter 1: Brief history and overview of nonlinear water waves
 | isbn = 0-12-528629-5
 | pages = 3 ''ff''
 | url = http://books.google.com/books?id=wdmsn9icd7YC&pg=PA3
 }}</ref> 

[[File:Nonsinusoidal wavelength.JPG|thumb|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 a ''periodic wave''.<ref name=McPherson>
{{cite book
 |title=Introduction to Macromolecular Crystallography
 |author=Alexander McPherson |url=http://books.google.com/books?id=o7sXm2GSr9IC&pg=PA77
 |page=77
 |chapter=Waves and their properties
 |isbn=0-470-18590-2
 |year=2009 
 |edition=2
 |publisher=Wiley
 }}</ref> Such waves are sometimes regarded as having a wavelength even though they are not sinusoidal.<ref name=may>
{{cite book |title=Fourier Analysis |url=http://books.google.com/books?id=gMPVFRHfgGYC&pg=PA1 |page=1 |isbn=1-118-16551-9 |year=2011 |publisher=John Wiley & Sons |author=Eric Stade}}
</ref> As shown in the figure, wavelength is measured between consecutive corresponding points on the waveform.

===Wave packets===
[[File:Wave packet (dispersion).gif|thumb|A propagating wave packet]]

{{Main|Wave packet}}

Localized [[wave packet]]s, "bursts" of wave action where each wave packet travels as a unit, find application in many fields of physics; the notion of a wavelength also may be applied to these wave packets.<ref>
{{cite book
 | title = Subquantum Kinetics: A Systems Approach to Physics and Cosmology
 | author = Paul A. LaViolette
 | publisher = Starlane Publications
 | year = 2003
 | isbn = 978-0-9642025-5-9
 | page = 80
 | url = http://books.google.com/books?id=8HQJAvA1EqkC&pg=PA80&dq=wave-packet-wavelength+de-Broglie-wavelength
 }}</ref>
The wave packet has an ''envelope'' that describes the overall amplitude of the wave; within the envelope, the distance between adjacent peaks or troughs is sometimes called a ''local wavelength''.<ref>
{{cite book
 | title = The Quantum Theory of Motion: An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics
 | author = Peter R. Holland
 | publisher = Cambridge University Press
 | year = 1995
 | isbn = 978-0-521-48543-2
 | page = 160
 | url = http://books.google.com/books?id=BsEfVBzToRMC&pg=PA160&dq=wave-packet+local-wavelength
 }}</ref><ref name=Cooper>
{{cite book
 |title=Introduction to partial differential equations with MATLAB
 |author=Jeffery Cooper
 |url=http://books.google.com/books?id=l0g2BcxOJVIC&pg=PA272
 |page=272 
 |isbn=0-8176-3967-5
 |publisher=Springer
 |year=1998
 |quote=The local wavelength λ of a dispersing wave is twice the distance between two successive zeros. ... the local wave length and the local wave number ''k'' are related by ''k'' = 2π / λ.
 }}</ref> An example is shown in the figure. In general, the ''envelope'' of the wave packet moves at a different speed than the constituent waves.<ref name= Fromhold>{{cite book |title=Quantum Mechanics for Applied Physics and Engineering |author=A. T. Fromhold |chapter=Wave packet solutions |pages=59 ''ff'' |quote=(p. 61) ... the individual waves move more slowly than the packet and therefore pass back through the packet as it advances 
 |url=http://books.google.com/books?id=3SOwc6npkIwC&pg=PA59
 |isbn=0-486-66741-3 |publisher=Courier Dover Publications
 |year=1991
 |edition=Reprint of Academic Press 1981
 }}</ref>

Using [[Fourier analysis]], wave packets can be analyzed into infinite sums (or integrals) of sinusoidal waves of different [[wavenumber]]s or wavelengths.<ref name=Manners>See, for example, Figs. 2.8–2.10 in
{{cite book
 | title = Quantum Physics: An Introduction
 | author = Joy Manners
 | publisher = CRC Press
 | year = 2000
 | isbn = 978-0-7503-0720-8
 | chapter=Heisenberg's uncertainty principle
 | pages = 53–56
 | url = http://books.google.com/books?id=LkDQV7PNJOMC&pg=PA54&dq=wave-packet+wavelengths
 }}</ref>

[[Louis de Broglie]] postulated that all particles with a specific value of [[momentum]] ''p'' have a wavelength ''&lambda; = h/p'', where ''h'' is [[Planck's constant]]. This hypothesis was at the basis of [[quantum mechanics]]. Nowadays, this wavelength is called the [[de Broglie wavelength]]. For example, the [[electron]]s in a [[cathode ray tube|CRT]] display have a De Broglie wavelength of about 10<sup>−13</sup> m. To prevent the [[wave function]] for such a particle being spread over all space, de Broglie proposed using wave packets to represent particles that are localized in space.<ref name=Marton>
{{cite book
 |title=Advances in Electronics and Electron Physics
 |page=271
 |url=http://books.google.com/books?id=g5q6tZRwUu4C&pg=PA271
 |isbn=0-12-014653-3 |year=1980 |publisher=Academic Press
 |volume=53 |editor=L. Marton and Claire Marton
 |author=Ming Chiang Li
 |chapter=Electron Interference
 }}</ref>  The spatial spread of the wave packet, and the spread of the [[wavenumber]]s of sinusoids that make up the packet, correspond to the uncertainties in the particle's position and momentum, the product of which is bounded by [[Heisenberg uncertainty principle]].<ref name=Manners/>

==Interference and diffraction==
===Double-slit interference===
{{main|Interference (wave propagation)}}

[[File:Interferometer path differences.JPG|thumb|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 the [[Interferometry|interferometer]]. A simple example is an experiment due to [[Thomas Young (scientist)|Young]] where light is passed through [[Double-slit experiment|two slits]].<ref name=Sluder>
{{cite book
 |title=Digital microscopy
 |author=Greenfield Sluder and David E. Wolf
 |url=http://books.google.com/books?id=H--zxc_N-jMC&pg=PA15
 |page=15
 |chapter=IV. Young's Experiment: Two-Slit Interference
 |isbn=0-12-374025-8
 |edition=3rd
 |year=2007
 |publisher=Academic Press
 }}</ref> 
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, ''s'' is large compared to the slit separation ''d'') then the paths are nearly parallel, and the path difference is simply ''d'' sin θ. Accordingly the condition for constructive interference is:<ref name=Halliday>
{{cite book
 |title=Fundamentals of Physics
 |url=http://books.google.com/books?id=RVCE4EUjDCgC&pg=PT965
 |page=965
 |chapter=§35-4 Young's interference experiment
 |author=Halliday, Resnick, Walker
 |isbn=81-265-1442-6
 |year=2008
 |edition=Extended 8th
 |publisher=Wiley-India
 }}</ref>

:<math> d \sin \theta = m \lambda \ , </math>

where ''m'' is an integer, and for destructive interference is:

:<math> d \sin \theta = (m + 1/2 )\lambda \ . </math>

Thus, if the wavelength of the light is known, the slit separation can be determined from the interference pattern or ''fringes'', and ''vice versa''.

For multiple slits, the pattern is <ref name=Harris>
{{cite book |author=Kordt Griepenkerl|title=Handbook of physics |url=http://books.google.com/books?id=c60mCxGRMR8C&pg=PA307 |pages=307 ''ff'' |editor=John W Harris, Walter Benenson, Horst Stöcker, Holger Lutz|chapter= §9.8.2 Diffraction by a grating |isbn=0-387-95269-1 |year=2002 |publisher=Springer}}
</ref>
:<math>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) \ , </math>
where ''q'' is the number of slits, and ''g'' is the grating constant. The first factor, ''I''<sub>1</sub>, 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 figure ''I''<sub>1</sub> has been set to unity, a very rough approximation.

It should be noted that the effect of interference is to ''redistribute'' the light, so the energy contained in the light is not altered, just where it shows up.<ref name= Murphy>
{{cite book
 |title=Fundamentals of light microscopy and electronic imaging
 |url=http://books.google.com/books?id=UFgdjxTULJMC&pg=PA64
 |page=64
 |author= Douglas B. Murphy
 |isbn=0-471-23429-X
 |year=2002
 |publisher=Wiley/IEEE
 }}</ref>

===Single-slit diffraction===

{{main|Diffraction|Diffraction formalism}}
[[File:Double-slit diffraction pattern.png|thumb|200px|Diffraction pattern of a double slit has a single-slit [[Envelope (waves)|envelope]].]]

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 called [[diffraction]].

Two types of diffraction are distinguished, depending upon the separation between the source and the screen: [[Fraunhofer diffraction]] or far-field diffraction at large separations and [[Fresnel diffraction]] or 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 (''Huygen's 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 a [[small-angle approximation]], the intensity spread ''S'' is related to position ''x'' via a squared [[sinc function]]:<ref>{{cite book
 | title = Optical scattering: measurement and analysis
 | edition = 2nd
 | author = John C. Stover
 | publisher = SPIE Press
 | year = 1995
 | isbn = 978-0-8194-1934-7
 | page = 64
 | url = http://books.google.com/books?id=ot0tjJL72uUC&pg=PA65&dq=single-slit+diffraction+sinc-function
 }}</ref>

:<math>S(u) = \mathrm{sinc}^2(u) = \left( \frac {\sin \pi u}{\pi u} \right) ^2 \ ; </math> &ensp;with&ensp; <math>u = \frac {x L}{\lambda R} \ , </math>

where ''L'' is the slit width, ''R'' is the distance of the pattern (on the screen) from the slit, and λ is the wavelength of light used. The function ''S'' has zeros where ''u'' is a non-zero integer, where are at ''x'' values at a separation proportion to wavelength.

===Diffraction-limited resolution===

{{main|Angular resolution|Diffraction-limited system}}

Diffraction is the fundamental limitation on the [[Angular resolution|resolving power]] of optical instruments, such as [[telescope]]s (including [[radiotelescope]]s) and [[microscopes]].<ref name=Saxby>
{{cite book
 |author=Graham Saxby
 |url=http://books.google.com/books?id=e5mC5TXlBw8C&pg=PA57
 |page=57 |title=The science of imaging
 |chapter=Diffraction limitation
 |isbn=0-7503-0734-X
 |year=2002
 |publisher=CRC Press
 }}</ref> 
For a circular aperture, the diffraction-limited image spot is known as an [[Airy disk]]; the distance ''x'' in the single-slit diffraction formula is replaced by radial distance ''r'' and the sine is replaced by 2''J''<sub>1</sub>, where ''J''<sub>1</sub> is a first order [[Bessel function]].<ref>
{{cite book
 | title = Introduction to Modern Optics
 | author = Grant R. Fowles
 | publisher = Courier Dover Publications
 | year = 1989
 | isbn = 978-0-486-65957-2
 | pages = 117–120
 | url = http://books.google.com/books?id=SL1n9TuJ5YMC&pg=PA119&dq=Airy-disk+Bessel+slit+diffraction+sin
 }}</ref>

The resolvable ''spatial'' size of objects viewed through a microscope is limited according to the [[Rayleigh 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 the [[numerical aperture]]:<ref>
{{cite book
 | title = Handbook of biological confocal microscopy
 | edition = 2nd 
 | author = James B. Pawley
 | publisher = Springer
 | year = 1995
 | isbn = 978-0-306-44826-3
 | page = 112
 | url = http://books.google.com/books?id=16Ft5k8RC-AC&pg=PA112
 }}</ref>

:<math>r_{Airy} = 1.22 \frac {\lambda}{2\mathrm{NA}} \ , </math>

where the numerical aperture is defined as <math>\mathrm{NA} = n \sin \theta\;</math> for θ being the half-angle of the cone of rays accepted by the [[microscope objective]].

The ''angular'' size of the central bright portion (radius to first null of the [[Airy disk]]) of the image diffracted by a circular aperture, a measure most commonly used for telescopes and cameras, is:<ref>
{{cite book
 | title = Reflecting Telescope Optics I: Basic Design Theory and Its Historical Development
 | author = Ray N. Wilson
 | publisher = Springer
 | year = 2004
 | isbn = 978-3-540-40106-3
 | page = 302
 | url = http://books.google.com/books?id=PuN7l2A2uzQC&pg=PA302&dq=telescope+diffraction-limited+resolution+sinc
 }}</ref>

:<math>\delta = 1.22 \frac {\lambda}{D} \ , </math>

where λ is the wavelength of the waves that are focused for imaging, ''D'' the [[entrance pupil]] diameter of the imaging system, in the same units, and the angular resolution δ is in radians.

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

==Subwavelength==

The term ''subwavelength'' is 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 term ''[[subwavelength-diameter optical fibre]]'' means an [[optical fibre]] whose 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 (see [[Rayleigh scattering]]).  Subwavelength [[aperture]]s are holes smaller than the wavelength of light propagating through them.  Such structures have applications in [[extraordinary optical transmission]], and [[zero-mode waveguide]]s, among other areas of [[photonics]].

''Subwavelength'' may also refer to a phenomenon involving subwavelength objects; for example, [[subwavelength imaging]].

==Angular wavelength==
A quantity related to the wavelength is the '''angular wavelength''' (also known as '''reduced wavelength'''), usually symbolized by ''ƛ'' (lambda-bar). It is equal to the "regular" wavelength "reduced" by a factor of 2π (''ƛ'' = ''λ''/2π). It is usually encountered in quantum mechanics, where it is used in combination with the [[reduced Planck constant]] (symbol ''ħ'', h-bar) and the [[angular frequency]] (symbol ''ω'') or [[angular wavenumber]] (symbol ''k'').

==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]]
* [[Spectrum]]
* [[Spectroscopy]]

==References==
{{reflist|2}}

==External links==
*[http://www.sengpielaudio.com/calculator-wavelength.htm Conversion: Wavelength to Frequency and vice versa – Sound waves and radio waves]
*[http://www.acoustics.salford.ac.uk/schools/index1.htm Teaching resource for 14–16 years on sound including wavelength]
*[http://www.magnetkern.de/spektrum.html The visible electromagnetic spectrum displayed in web colors with according wavelengths]

{{EMSpectrum}}

[[Category:Waves]]
[[Category:Concepts in physics]]
[[Category:Length]]