Editing Wavelength (section)

== 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 ''<math>v</math>'' is given by<ref name= Cassidy>
{{cite book
 |title=Understanding physics
 |author1=David C. Cassidy |author2=Gerald James Holton |author3=Floyd James Rutherford
 |url=https://books.google.com/books?id=rpQo7f9F1xUC&pg=PA340
 |pages=339 ''ff''
 |isbn=0-387-98756-8
 |year=2002
 |publisher=Birkhäuser
 }}</ref>
<math display="block">\lambda = \frac{v}{f}\,\,,</math>
where <math>v</math> is called the phase speed (magnitude of the [[phase velocity]]) of the wave and <math>f</math> 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 {{val|3|e=8|u=m/s}}. Thus the wavelength of a 100&nbsp;MHz electromagnetic (radio) wave is about: {{val|3|e=8|u=m/s}} divided by {{val|e=8|u=Hz}} = 3&nbsp;m. The wavelength of visible light ranges from deep [[red]], roughly 700&nbsp;[[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. Somewhat higher frequencies are used by [[bat]]s so they can resolve targets smaller than 17&nbsp;mm. 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]]), thus determining the allowed wavelengths. For example, for an electromagnetic wave, if the box has ideal conductive walls, the condition for nodes at the walls results because the conductive 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 = https://books.google.com/books?id=DojwZzKAvN8C&q=%22standing+wave%22+wavelength&pg=PA460
 }}</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 display="block">  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 display="block">  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 display="block"> k = \frac{2 \pi}{\lambda} = \frac{2 \pi f}{v} = \frac{\omega}{v},</math>
or
<math display="block"> \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 display="block">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 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.