Editing Wavelength (section)

=== 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 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 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=https://books.google.com/books?id=bO_U8f5pVR8C&pg=PA327 |page=327 |year=2010 |author=Raymond T. Pierrehumbert |publisher=Cambridge University Press |isbn=978-0-521-86556-2 }}
</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 display="block">v = \frac{c}{n(\lambda_0)},</math>
where ''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 display="block">\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 wavelength is known as [[dispersion (optics)|dispersion]], and is also responsible for the familiar phenomenon in which light is separated into component colours 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.svg|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 = https://books.google.com/books?id=6TCm8Xy-sLUC&pg=PA242
 |page = 242
 |isbn = 978-0-7637-5993-3
 |year = 2009
 |publisher = Jones & Bartlett Learning
 }}</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 = https://books.google.com/books?id=_MIdEiKqdawC&q=wkb+local-wavelength&pg=PA454
 | year = 2007
 }}</ref><ref>
{{cite book
 | title = Time-frequency and time-scale methods: adaptive decompositions, uncertainty principles, and sampling
 | author1=Jeffrey A. Hogan
 | author2=Joseph D. Lakey
 | name-list-style=amp | publisher = Birkhäuser
 | year = 2005
 | isbn = 978-0-8176-4276-1
 | page = 348
 | url = https://books.google.com/books?id=YOf0SRzxz3gC&q=wkb+local-wavelength&pg=PA348
 }}</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=https://archive.org/details/introductiontomi00putn |url-access= registration |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=https://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=https://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.