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| In [[physics]], a '''wave vector''' (also spelled '''wavevector''') is a [[vector (geometric)|vector]] which helps describe a [[wave]]. Like any vector, it has a [[Euclidean vector|magnitude and direction]], both of which are important: Its magnitude is either the [[wavenumber]] or [[angular wavenumber]] of the wave (inversely proportional to the [[wavelength]]), and its direction is ordinarily the direction of [[wave propagation]] (but not always, see [[#Direction of the wave vector|below]]).
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| In the context of [[special relativity]] the wave vector can also be defined as a [[four-vector]].
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| ==Definitions==
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| {{see also|Traveling wave}}
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| [[File:Sine wavelength.svg|thumb|right|Wavelength of a [[sine wave]], ''λ'', can be measured between any two consecutive points with the same [[phase (waves)|phase]], such as between adjacent crests, or troughs, or adjacent [[zero crossing]]s with the same direction of transit, as shown.]]
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| Unfortunately, there are two common definitions of wave vector, which differ by a factor of 2π in their magnitudes. One definition is preferred in [[physics]] and related fields, while the other definition is preferred in [[crystallography]] and related fields.<ref>Physics definition example:{{cite book| url=http://books.google.com/books?id=c60mCxGRMR8C&pg=PA288 | title= Handbook of Physics| author= Harris, Benenson, Stöcker|page=288| isbn=978-0-387-95269-7| year=2002}}. Crystallography definition example: {{cite book| url=http://books.google.com/books?id=xjIGV_hPiysC&pg=PA259 | title=Modern Crystallography |author=Vaĭnshteĭn| page=259| isbn=978-3-540-56558-1| year=1994}}</ref> For this article, they will be called the "physics definition" and the "crystallography definition", respectively.
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| ===Physics definition===
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| A perfect one-dimensional [[traveling wave]] follows the equation:
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| :<math>\psi(x,t) = A \cos (k x - \omega t+\varphi)</math>
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| where:
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| *''x'' is position,
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| *''t'' is time,
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| *<math>\psi</math> (a function of ''x'' and ''t'') is the disturbance describing the wave (for example, for an [[ocean wave]], <math>\psi</math> would be the excess height of the water, or for a [[sound wave]], <math>\psi</math> would be the excess [[air pressure]]).
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| *''A'' is the [[amplitude]] of the wave (the peak magnitude of the oscillation),
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| *<math>\varphi</math> is a "phase offset" describing how two waves can be out of sync with each other,
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| *<math>\omega</math> is the [[angular frequency]] of the wave, related to how quickly it oscillates at a given point,
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| *<math>k</math> is the [[wavenumber]] (more specifically called angular wavenumber) of the wave, related to the [[wavelength]] by the equation <math>k=2\pi/\lambda</math>.
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| This wave travels in the +x direction with speed (more specifically, [[phase velocity]]) <math>\omega/k</math>.
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| ===Crystallography definition===
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| In [[crystallography]], the same waves are described using slightly different equations.<ref>{{cite book |url=http://books.google.com/books?id=xjIGV_hPiysC&pg=PA259|title= Modern Crystallography| page=259 |isbn=978-3-540-56558-1 |last=Vaĭnshteĭn|first=Boris Konstantinovich |year=1994}}</ref> In one and three dimensions respectively:
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| :<math>\psi(x,t) = A \cos (2 \pi (k x - \omega t)+\varphi)</math>
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| :<math>\psi \left({\mathbf r}, t \right) = A \cos \left(2\pi({\mathbf k} \cdot {\mathbf r} - \nu t) + \varphi \right)</math>
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| The differences are:
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| *The frequency <math>\nu</math> instead of angular frequency <math>\omega</math> is used. They are related by <math>2\pi \nu=\omega</math>. This substitution is not important for this article, but reflects common practice in crystallography.
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| *The wavenumber ''k'' and wave vector '''k''' are defined in a different way. Here, <math>k=|{\mathbf k}| = 1/\lambda</math>, while in the physics definition above, <math>k=|{\mathbf k}| = 2\pi/\lambda</math>.
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| The direction of '''k''' is discussed [[#Direction of the wave vector|below]].
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| ==Direction of the wave vector==
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| {{main|Group velocity}}
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| The direction in which the wave vector points must be distinguished from the "direction of [[wave propagation]]". The "direction of wave propagation" is the direction of a wave's energy flow, and the direction that a small [[wave packet]] will move, i.e. the direction of the [[group velocity]]. For light waves, this is also the direction of the [[Poynting vector]]. On the other hand, the wave vector points in the direction of [[phase velocity]]. In other words, the wave vector points in the [[surface normal|normal direction]] to the [[Wave front|surfaces of constant phase]], also called wave fronts.
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| In a lossless [[isotropy|isotropic medium]] such as air, any gas, any liquid, or some solids (such as [[glass]]), the direction of the wavevector is exactly the same as the direction of wave propagation. If the medium is lossy, the wave vector in general points in directions other than that of wave propagation. The condition for wave vector to point in the same direction in which the wave propagates is that the wave has to be homogeneous, which isn't necessarily satisfied when the medium is lossy. In a homogeneous wave, the surfaces of constant phase are also surfaces of constant amplitude. In case of inhomogeneous waves, these two species of surfaces differ in orientation. Wave vector is always perpendicular to surfaces of constant phase.
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| However, when a wave travels through an [[anisotropy|anisotropic medium]], such as [[crystal optics|light waves through an asymmetric crystal]] or sound waves through a [[sedimentary rock]], the wave vector may not point exactly in the direction of wave propagation.<ref name=fowles>{{cite book|last=Fowles|first=Grant|title=Introduction to modern optics|year=1968|publisher=Holt, Reinhart, and Winston|page=177}}</ref><ref name=pollard>"This effect has been explained by Musgrave (1959) who has shown that the energy of an elastic wave in an anisotropic medium will not, in general, travel along the same path as the normal to the plane wavefront...", ''Sound waves in solids'' by Pollard, 1977. [http://books.google.com/books?id=EOUNAQAAIAAJ link]</ref>
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| ==In solid-state physics==
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| {{main|Bloch wave}}
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| In [[solid-state physics]], the "wavevector" (also called '''k-vector''') of an [[electron]] or [[electron hole|hole]] in a [[crystal]] is the wavevector of its [[quantum mechanics|quantum-mechanical]] [[wavefunction]]. These electron waves are not ordinary [[sinusoidal]] waves, but they do have a kind of ''[[Envelope (waves)|envelope function]]'' which is sinusoidal, and the wavevector is defined via that envelope wave, usually using the "physics definition". See [[Bloch wave]] for further details.<ref>{{cite book |author=Donald H. Menzel |title=Fundamental Formulas of Physics, Volume 2 |url=http://books.google.com/books?id=-miofZvrH2sC&pg=PA624 |page=624 |chapter=§10.5 Bloch waves |isbn=0486605965 |year=1960 |edition=Reprint of Prentice-Hall 1955 2nd |publisher=Courier-Dover }}</ref>
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| ==In special relativity==
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| A beam of coherent, [[monochromatic]] light can be characterized by the (null) wave 4-vector
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| ::<math>k^\mu = \left(\frac{\omega}{c}, \vec{k} \right) \,</math>
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| which, when written out explicitly in its [[Covariance and contravariance of vectors|contravariant]] and [[covariance|covariant]] forms is
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| ::<math>k^\mu = \left(\frac{\omega}{c}, k_x, k_y, k_z \right)\, </math> and
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| ::<math>k_\mu = \left(\frac{\omega}{c}, -k_x, -k_y, -k_z \right) . \,</math>
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| The null character of the wave 4-vector gives a relation between the frequency and the magnitude of the spatial part of the wave 4-vector:
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| ::<math>k^\mu k_\mu = \left(\frac{\omega}{c}\right)^2 - k_x^2 - k_y^2 - k_z^2 \ = 0</math>
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| The wave 4-vector is related to the [[four-momentum]] as follows:
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| ::<math>p^\mu = (E/c, p_x, p_y, p_z) = (\hbar\omega/c, \hbar k_x, \hbar k_y, \hbar k_z) = \hbar k^\mu</math> | |
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| ===Lorentz transformation===
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| Taking the [[Lorentz transformation]] of the wave vector is one way to derive the [[relativistic Doppler effect]]. The Lorentz matrix is defined as
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| ::<math>\Lambda = \begin{pmatrix}
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| \gamma&-\beta \gamma&0&0 \\
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| -\beta \gamma&\gamma&0&0 \\
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| 0&0&1&0 \\
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| 0&0&0&1
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| \end{pmatrix}
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| </math>
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| In the situation where light is being emitted by a fast moving source and one would like to know the frequency of light detected in an earth (lab) frame, we would apply the Lorentz transformation as follows. Note that the source is in a frame ''S''<sup>s</sup> and earth is in the observing frame, ''S''<sup>obs</sup>.
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| Applying the lorentz transformation to the wave vector
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| ::<math>k^{\mu}_s = \Lambda^\mu_\nu k^\nu_{\mathrm{obs}} \,</math>
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| and choosing just to look at the <math>\mu = 0</math> component results in
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| ::<math>k^{0}_s = \Lambda^0_0 k^0_{\mathrm{obs}} + \Lambda^0_1 k^1_{\mathrm{obs}} + \Lambda^0_2 k^2_{\mathrm{obs}} + \Lambda^0_3 k^3_{\mathrm{obs}} \,</math>
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| ::{|
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| |<math>\frac{\omega_s}{c} \,</math>
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| |<math>= \gamma \frac{\omega_{\mathrm{obs}}}{c} - \beta \gamma k^1_{\mathrm{obs}} \,</math>
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| |-
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| |<math>\quad = \gamma \frac{\omega_{\mathrm{obs}}}{c} - \beta \gamma \frac{\omega_{\mathrm{obs}}}{c} \cos \theta. \,</math>
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| where <math> \cos \theta \,</math> is the direction cosine of <math>k^1</math> wrt <math>k^0, k^1 = k^0 \cos \theta. </math>
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| |}
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| So
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| ::{|cellpadding="2" style="border:2px solid #ccccff"
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| |<math>\frac{\omega_{\mathrm{obs}}}{\omega_s} = \frac{1}{\gamma (1 - \beta \cos \theta)} \,</math>
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| |}
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| ====Source moving away====
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| As an example, to apply this to a situation where the source is moving directly away from the observer (<math>\theta=\pi</math>), this becomes:
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| ::<math>\frac{\omega_{\mathrm{obs}}}{\omega_s} = \frac{1}{\gamma (1 + \beta)} = \frac{\sqrt{1-\beta^2}}{1+\beta} = \frac{\sqrt{(1+\beta)(1-\beta)}}{1+\beta} = \frac{\sqrt{1-\beta}}{\sqrt{1+\beta}} \,</math>
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| ====Source moving towards====
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| To apply this to a situation where the source is moving straight towards the observer (<math>\theta=0</math>), this becomes:
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| ::<math>\frac{\omega_{\mathrm{obs}}}{\omega_s} = \frac{\sqrt{1+\beta}}{\sqrt{1-\beta}} \,</math>
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| == See also ==
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| *[[Plane wave expansion]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| *{{cite book | author=Brau, Charles A. | title=Modern Problems in Classical Electrodynamics | publisher=Oxford University Press | year=2004 | isbn=0-19-514665-4}}
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| [[Category:Wave mechanics]]
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| [[Category:Vectors]]
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