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<title>Wave vector</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Wave vector</span></span>

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</p><p>In <a href="Physics" title="Physics">physics</a>, a <b>wave vector</b> (or <b>wavevector</b>) is a <a href="Vector_(geometric)" class="mw-redirect" title="Vector (geometric)">vector</a> used in describing a <a href="Wave" title="Wave">wave</a>, with a typical unit being cycle per metre. It has a <a href="Euclidean_vector" title="Euclidean vector">magnitude and direction</a>. Its magnitude is the <a href="Wavenumber" title="Wavenumber">wavenumber</a> of the wave (inversely proportional to the <a href="Wavelength" title="Wavelength">wavelength</a>), and its direction is perpendicular to the <a href="Wavefront" title="Wavefront">wavefront</a>. In isotropic media, this is also the direction of <a href="Wave_propagation" class="mw-redirect" title="Wave propagation">wave propagation</a>.

</p><p>A closely related vector is the <b>angular wave vector</b> (or <b>angular wavevector</b>), with a typical unit being radian per metre. The wave vector and angular wave vector are related by a fixed constant of proportionality, 2<span class="texhtml mvar" style="font-style:italic;">π</span> radians per cycle.

</p><p>It is common in several fields of <a href="Physics" title="Physics">physics</a> to refer to the angular wave vector simply as the <i>wave vector</i>, in contrast to, for example, <a href="Crystallography" title="Crystallography">crystallography</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> It is also common to use the symbol <span class="texhtml mvar" style="font-style:italic;"><b>k</b></span> for whichever is in use.

</p><p>In the context of <a href="Special_relativity" title="Special relativity">special relativity</a>, a <i><a href="Wave_four-vector" class="mw-redirect" title="Wave four-vector">wave four-vector</a></i> can be defined, combining the (angular) wave vector and (angular) frequency.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Traveling_wave" class="mw-redirect" title="Traveling wave">Traveling wave</a></div>

<p>The terms <i>wave vector</i> and <i>angular wave vector</i> have distinct meanings. Here, the wave vector is denoted by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\boldsymbol {\nu }}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\boldsymbol {\nu }}}}</annotation>

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</math></span><img src="./579ef165890cc30b5edcac5374334d32bd28b4c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.413ex; height:2.176ex;" alt="{\displaystyle {\tilde {\boldsymbol {\nu }}}}" loading="lazy"></span> and the wavenumber by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\nu }}=\left|{\tilde {\boldsymbol {\nu }}}\right|}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\nu }}=\left|{\tilde {\boldsymbol {\nu }}}\right|}</annotation>

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</math></span><img src="./c09f22c7100c599d3440efc7bd49a6735a2123aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.122ex; height:2.843ex;" alt="{\displaystyle {\tilde {\nu }}=\left|{\tilde {\boldsymbol {\nu }}}\right|}" loading="lazy"></span>. The angular wave vector is denoted by <span class="texhtml"><b>k</b></span> and the angular wavenumber by <span class="texhtml"><i>k</i> = |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><b>k</b></span>|</span>. These are related by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} =2\pi {\tilde {\boldsymbol {\nu }}}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {k} =2\pi {\tilde {\boldsymbol {\nu }}}}</annotation>

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</math></span><img src="./cbe45c56de656fa61860c8dbb6db5bfaf598c0aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.417ex; height:2.176ex;" alt="{\displaystyle \mathbf {k} =2\pi {\tilde {\boldsymbol {\nu }}}}" loading="lazy"></span>.

</p><p>A sinusoidal <a href="Traveling_wave" class="mw-redirect" title="Traveling wave">traveling wave</a> follows the equation

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r} ,t)=A\cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi ),}</annotation>

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</math></span><img src="./c0ec4fbbc225706edd391bdd7d1f3d6b9ebb0f35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.772ex; height:2.843ex;" alt="{\displaystyle \psi (\mathbf {r} ,t)=A\cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi ),}" loading="lazy"></span></dd></dl>

<p>where:

</p>

<ul><li><span class="texhtml"><b>r</b></span> is position,</li>

<li><span class="texhtml mvar" style="font-style:italic;">ψ</span> is a function of <span class="texhtml"><b>r</b></span> and <span class="texhtml mvar" style="font-style:italic;">t</span> describing the disturbance describing the wave (for example, for an <a href="Ocean_wave" class="mw-redirect" title="Ocean wave">ocean wave</a>, <span class="texhtml mvar" style="font-style:italic;">ψ</span> would be the excess height of the water, or for a <a href="Sound_wave" class="mw-redirect" title="Sound wave">sound wave</a>, <span class="texhtml mvar" style="font-style:italic;">ψ</span> would be the excess <a href="Air_pressure" class="mw-redirect" title="Air pressure">air pressure</a>).</li>

<li><span class="texhtml mvar" style="font-style:italic;">A</span> is the <a href="Amplitude" title="Amplitude">amplitude</a> of the wave (the peak magnitude of the oscillation),</li>

<li><span class="texhtml mvar" style="font-style:italic;">φ</span> is a <a href="Phase_offset" class="mw-redirect" title="Phase offset">phase offset</a>,</li>

<li><span class="texhtml mvar" style="font-style:italic;">ω</span> is the (temporal) <a href="Angular_frequency" title="Angular frequency">angular frequency</a> of the wave, describing how many radians it traverses per unit of time, and related to the <a href="Period_(physics)" class="mw-redirect" title="Period (physics)">period</a> <span class="texhtml mvar" style="font-style:italic;">T</span> by the equation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\tfrac {2\pi }{T}},}">

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<annotation encoding="application/x-tex">{\displaystyle \omega ={\tfrac {2\pi }{T}},}</annotation>

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</math></span><img src="./24142484ec8af2185cd5d2cd587150f88c39b8e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.791ex; height:3.509ex;" alt="{\displaystyle \omega ={\tfrac {2\pi }{T}},}" loading="lazy"></span></li>

<li><span class="texhtml"><b>k</b></span> is the angular wave vector of the wave, describing how many radians it traverses per unit of distance, and related to the <a href="Wavelength" title="Wavelength">wavelength</a> by the equation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {k} |={\tfrac {2\pi }{\lambda }}.}">

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<annotation encoding="application/x-tex">{\displaystyle |\mathbf {k} |={\tfrac {2\pi }{\lambda }}.}</annotation>

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</math></span><img src="./fa44518f15565e9a8424eab5fa8ff29750265726.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.05ex; height:3.676ex;" alt="{\displaystyle |\mathbf {k} |={\tfrac {2\pi }{\lambda }}.}" loading="lazy"></span></li></ul>

<p>The equivalent equation using the wave vector and frequency is<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle \psi \left(\mathbf {r} ,t\right)=A\cos \left(2\pi \left({\tilde {\boldsymbol {\nu }}}\cdot {\mathbf {r} }-ft\right)+\varphi \right),}</annotation>

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</math></span><img src="./118447e4276a8380341c335d9b37635eeb2cc9df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.686ex; height:2.843ex;" alt="{\displaystyle \psi \left(\mathbf {r} ,t\right)=A\cos \left(2\pi \left({\tilde {\boldsymbol {\nu }}}\cdot {\mathbf {r} }-ft\right)+\varphi \right),}" loading="lazy"></span></dd></dl>

<p>where:

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is the frequency</li>

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\boldsymbol {\nu }}}}</annotation>

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</math></span><img src="./579ef165890cc30b5edcac5374334d32bd28b4c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.413ex; height:2.176ex;" alt="{\displaystyle {\tilde {\boldsymbol {\nu }}}}" loading="lazy"></span> is the wave vector</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Direction_of_the_wave_vector">Direction of the wave vector</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Group_velocity" title="Group velocity">Group velocity</a></div>

<p>The direction in which the wave vector points must be distinguished from the "direction of <a href="Wave_propagation" class="mw-redirect" title="Wave propagation">wave propagation</a>". The "direction of wave propagation" is the direction of a wave's energy flow, and the direction that a small <a href="Wave_packet" title="Wave packet">wave packet</a> will move, i.e. the direction of the <a href="Group_velocity" title="Group velocity">group velocity</a>. For light waves in vacuum, this is also the direction of the <a href="Poynting_vector" title="Poynting vector">Poynting vector</a>. On the other hand, the wave vector points in the direction of <a href="Phase_velocity" title="Phase velocity">phase velocity</a>. In other words, the wave vector points in the <a href="Surface_normal" class="mw-redirect" title="Surface normal">normal direction</a> to the <a href="Wave_front" class="mw-redirect" title="Wave front">surfaces of constant phase</a>, also called <a href="Wavefronts" class="mw-redirect" title="Wavefronts">wavefronts</a>.

</p><p>In a <a href="Attenuation" title="Attenuation">lossless</a> <a href="Isotropy" title="Isotropy">isotropic medium</a> such as air, any gas, any liquid, <a href="Amorphous_solids" class="mw-redirect" title="Amorphous solids">amorphous solids</a> (such as <a href="Glass" title="Glass">glass</a>), and <a href="Cubic_crystal" class="mw-redirect" title="Cubic crystal">cubic crystals</a>, the direction of the wavevector is the same as the direction of wave propagation. If the medium is anisotropic, the wave vector in general points in directions other than that of the wave propagation. The wave vector is always perpendicular to surfaces of constant phase.

</p><p>For example, when a wave travels through an <a href="Anisotropy" title="Anisotropy">anisotropic medium</a>, such as <a href="Crystal_optics" title="Crystal optics">light waves through an asymmetric crystal</a> or sound waves through a <a href="Sedimentary_rock" title="Sedimentary rock">sedimentary rock</a>, the wave vector may not point exactly in the direction of wave propagation.<sup id="cite_ref-fowles_4-0" class="reference"><a href="#cite_note-fowles-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-pollard_5-0" class="reference"><a href="#cite_note-pollard-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="In_solid-state_physics">In solid-state physics</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Bloch's_theorem" title="Bloch's theorem">Bloch's theorem</a></div>

<p>In <a href="Solid-state_physics" title="Solid-state physics">solid-state physics</a>, the "wavevector" (also called <b>k-vector</b>) of an <a href="Electron" title="Electron">electron</a> or <a href="Electron_hole" title="Electron hole">hole</a> in a <a href="Crystal" title="Crystal">crystal</a> is the wavevector of its <a href="Quantum_mechanics" title="Quantum mechanics">quantum-mechanical</a> <a href="Wavefunction" class="mw-redirect" title="Wavefunction">wavefunction</a>. These electron waves are not ordinary <a href="Sinusoidal" class="mw-redirect" title="Sinusoidal">sinusoidal</a> waves, but they do have a kind of <i><a href="Envelope_(waves)" title="Envelope (waves)">envelope function</a></i> which is sinusoidal, and the wavevector is defined via that envelope wave, usually using the "physics definition". See <a href="Bloch's_theorem" title="Bloch's theorem">Bloch's theorem</a> for further details.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="In_special_relativity">In special relativity</h2></div>

<p>A moving wave surface in special relativity may be regarded as a hypersurface (a 3D subspace) in spacetime, formed by all the events passed by the wave surface. A wavetrain (denoted by some variable <span class="texhtml mvar" style="font-style:italic;">X</span>) can be regarded as a one-parameter family of such hypersurfaces in spacetime. This variable <span class="texhtml mvar" style="font-style:italic;">X</span> is a scalar function of position in spacetime. The derivative of this scalar is a vector that characterizes the wave, the four-wavevector.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>The four-wavevector is a wave <a href="Four-vector" title="Four-vector">four-vector</a> that is defined, in <a href="Minkowski_space" title="Minkowski space">Minkowski coordinates</a>, as:

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<annotation encoding="application/x-tex">{\displaystyle K^{\mu }=\left({\frac {\omega }{c}},{\vec {k}}\right)=\left({\frac {\omega }{c}},{\frac {\omega }{v_{p}}}{\hat {n}}\right)=\left({\frac {2\pi }{cT}},{\frac {2\pi {\hat {n}}}{\lambda }}\right)\,}</annotation>

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</math></span><img src="./8b60cd8f2e2057c2e19dbcbaf28cf4d01a67cfb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.117ex; height:6.176ex;" alt="{\displaystyle K^{\mu }=\left({\frac {\omega }{c}},{\vec {k}}\right)=\left({\frac {\omega }{c}},{\frac {\omega }{v_{p}}}{\hat {n}}\right)=\left({\frac {2\pi }{cT}},{\frac {2\pi {\hat {n}}}{\lambda }}\right)\,}" loading="lazy"></span></dd></dl>

<p>where the angular frequency <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\omega }{c}}}">

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</math></span><img src="./39e4fe2a638d6cf973aedb7a2a1fd42ebef234a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.858ex; height:3.009ex;" alt="{\displaystyle {\tfrac {\omega }{c}}}" loading="lazy"></span> is the temporal component, and the wavenumber vector <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {k}}}">

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</math></span><img src="./5ccd4b98d198d6538010ae815ee1199baabd3493.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.843ex;" alt="{\displaystyle {\vec {k}}}" loading="lazy"></span> is the spatial component.

</p><p>Alternately, the wavenumber <span class="texhtml mvar" style="font-style:italic;">k</span> can be written as the angular frequency <span class="texhtml mvar" style="font-style:italic;">ω</span> divided by the <a href="Phase_velocity" title="Phase velocity">phase-velocity</a> <span class="texhtml mvar" style="font-style:italic;">v<sub>p</sub></span>, or in terms of inverse period <span class="texhtml mvar" style="font-style:italic;">T</span> and inverse wavelength <span class="texhtml mvar" style="font-style:italic;">λ</span>.

</p><p>When written out explicitly its <a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">contravariant</a> and <a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">covariant</a> forms are:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}K^{\mu }&=\left({\frac {\omega }{c}},k_{x},k_{y},k_{z}\right)\,\\[4pt]K_{\mu }&=\left({\frac {\omega }{c}},-k_{x},-k_{y},-k_{z}\right)\end{aligned}}}</annotation>

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</math></span><img src="./388b3644b7ce389d19a5e0a46b8e49d870972897.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:27.608ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}K^{\mu }&=\left({\frac {\omega }{c}},k_{x},k_{y},k_{z}\right)\,\\[4pt]K_{\mu }&=\left({\frac {\omega }{c}},-k_{x},-k_{y},-k_{z}\right)\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>In general, the <a href="Lorentz_scalar" title="Lorentz scalar">Lorentz scalar</a> magnitude of the wave four-vector is:

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<annotation encoding="application/x-tex">{\displaystyle K^{\mu }K_{\mu }=\left({\frac {\omega }{c}}\right)^{2}-k_{x}^{2}-k_{y}^{2}-k_{z}^{2}=\left({\frac {\omega _{o}}{c}}\right)^{2}=\left({\frac {m_{o}c}{\hbar }}\right)^{2}}</annotation>

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</math></span><img src="./235683e415b64fcf97e4503b0d6f6e79e9adadb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:53.241ex; height:5.343ex;" alt="{\displaystyle K^{\mu }K_{\mu }=\left({\frac {\omega }{c}}\right)^{2}-k_{x}^{2}-k_{y}^{2}-k_{z}^{2}=\left({\frac {\omega _{o}}{c}}\right)^{2}=\left({\frac {m_{o}c}{\hbar }}\right)^{2}}" loading="lazy"></span></dd></dl>

<p>The four-wavevector is <a href="Causal_structure#Tangent_vectors" title="Causal structure">null</a> for <a href="Massless_particle" title="Massless particle">massless</a> (photonic) particles, where the rest mass <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{o}=0}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle m_{o}=0}</annotation>

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</math></span><img src="./4a52271347b64283cd6572332dbdc05ecf0bdb0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.331ex; height:2.509ex;" alt="{\displaystyle m_{o}=0}" loading="lazy"></span>

</p><p>An example of a null four-wavevector would be a beam of coherent, <a href="Monochromatic" class="mw-redirect" title="Monochromatic">monochromatic</a> light, which has phase-velocity <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{p}=c}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle v_{p}=c}</annotation>

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</math></span><img src="./cfdfaae1f8c99e7d78a0b3449365e7d3a7c81f73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.292ex; height:2.343ex;" alt="{\displaystyle v_{p}=c}" loading="lazy"></span>

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<annotation encoding="application/x-tex">{\displaystyle K^{\mu }=\left({\frac {\omega }{c}},{\vec {k}}\right)=\left({\frac {\omega }{c}},{\frac {\omega }{c}}{\hat {n}}\right)={\frac {\omega }{c}}\left(1,{\hat {n}}\right)\,}</annotation>

</semantics>

</math></span><img src="./a3f2cddcdb236202baf82635b17c86189e39362a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.14ex; height:4.843ex;" alt="{\displaystyle K^{\mu }=\left({\frac {\omega }{c}},{\vec {k}}\right)=\left({\frac {\omega }{c}},{\frac {\omega }{c}}{\hat {n}}\right)={\frac {\omega }{c}}\left(1,{\hat {n}}\right)\,}" loading="lazy"></span> {for light-like/null}</dd></dl>

<p>which would have the following relation between the frequency and the magnitude of the spatial part of the four-wavevector:

</p>

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<annotation encoding="application/x-tex">{\displaystyle K^{\mu }K_{\mu }=\left({\frac {\omega }{c}}\right)^{2}-k_{x}^{2}-k_{y}^{2}-k_{z}^{2}=0}</annotation>

</semantics>

</math></span><img src="./13aa817d6beb14226572fdc2458465504aaa0ff5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.421ex; height:5.176ex;" alt="{\displaystyle K^{\mu }K_{\mu }=\left({\frac {\omega }{c}}\right)^{2}-k_{x}^{2}-k_{y}^{2}-k_{z}^{2}=0}" loading="lazy"></span> {for light-like/null}</dd></dl>

<p>The four-wavevector is related to the <a href="Four-momentum" title="Four-momentum">four-momentum</a> as follows:

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<annotation encoding="application/x-tex">{\displaystyle P^{\mu }=\left({\frac {E}{c}},{\vec {p}}\right)=\hbar K^{\mu }=\hbar \left({\frac {\omega }{c}},{\vec {k}}\right)}</annotation>

</semantics>

</math></span><img src="./20802ef48b661328a0ebc5f421ef4b085643b450.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.352ex; height:6.176ex;" alt="{\displaystyle P^{\mu }=\left({\frac {E}{c}},{\vec {p}}\right)=\hbar K^{\mu }=\hbar \left({\frac {\omega }{c}},{\vec {k}}\right)}" loading="lazy"></span></dd></dl>

<p>The four-wavevector is related to the <a href="Four-frequency" title="Four-frequency">four-frequency</a> as follows:

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<annotation encoding="application/x-tex">{\displaystyle K^{\mu }=\left({\frac {\omega }{c}},{\vec {k}}\right)=\left({\frac {2\pi }{c}}\right)N^{\mu }=\left({\frac {2\pi }{c}}\right)\left(\nu ,\nu {\vec {n}}\right)}</annotation>

</semantics>

</math></span><img src="./164484d9dec923e75c984a88e937cfe7b3074ebd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.242ex; height:6.176ex;" alt="{\displaystyle K^{\mu }=\left({\frac {\omega }{c}},{\vec {k}}\right)=\left({\frac {2\pi }{c}}\right)N^{\mu }=\left({\frac {2\pi }{c}}\right)\left(\nu ,\nu {\vec {n}}\right)}" loading="lazy"></span></dd></dl>

<p>The four-wavevector is related to the <a href="Four-velocity" title="Four-velocity">four-velocity</a> as follows:

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<annotation encoding="application/x-tex">{\displaystyle K^{\mu }=\left({\frac {\omega }{c}},{\vec {k}}\right)=\left({\frac {\omega _{o}}{c^{2}}}\right)U^{\mu }=\left({\frac {\omega _{o}}{c^{2}}}\right)\gamma \left(c,{\vec {u}}\right)}</annotation>

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</math></span><img src="./28c8604d06aabaf795a95811bd32f5bb8bc9e652.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.049ex; height:6.176ex;" alt="{\displaystyle K^{\mu }=\left({\frac {\omega }{c}},{\vec {k}}\right)=\left({\frac {\omega _{o}}{c^{2}}}\right)U^{\mu }=\left({\frac {\omega _{o}}{c^{2}}}\right)\gamma \left(c,{\vec {u}}\right)}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Lorentz_transformation">Lorentz transformation</h3></div>

<p>Taking the <a href="Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a> of the four-wavevector is one way to derive the <a href="Relativistic_Doppler_effect" title="Relativistic Doppler effect">relativistic Doppler effect</a>. The Lorentz matrix is defined as

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<annotation encoding="application/x-tex">{\displaystyle \Lambda ={\begin{pmatrix}\gamma &-\beta \gamma &\ 0\ &\ 0\ \\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}</annotation>

</semantics>

</math></span><img src="./ee243bdff2697f8eb1978b6a89bc697c72db4912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:29.95ex; height:12.509ex;" alt="{\displaystyle \Lambda ={\begin{pmatrix}\gamma &-\beta \gamma &\ 0\ &\ 0\ \\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}" loading="lazy"></span></dd></dl>

<p>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 <span class="texhtml"><i>S</i><sup>s</sup></span> and earth is in the observing frame, <span class="texhtml"><i>S</i><sup>obs</sup></span>.

Applying the Lorentz transformation to the wave vector

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<annotation encoding="application/x-tex">{\displaystyle k_{s}^{\mu }=\Lambda _{\nu }^{\mu }k_{\mathrm {obs} }^{\nu }}</annotation>

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</math></span><img src="./fa65b3a2a52d536f8456f45273649fa7783552f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.197ex; height:3.176ex;" alt="{\displaystyle k_{s}^{\mu }=\Lambda _{\nu }^{\mu }k_{\mathrm {obs} }^{\nu }}" loading="lazy"></span></dd></dl>

<p>and choosing just to look at the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mu =0}</annotation>

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</math></span><img src="./3753282c0ad2ea1e7d63f39425efd13c37da3169.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="{\displaystyle \mu =0}" loading="lazy"></span> component results in

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}k_{s}^{0}&=\Lambda _{0}^{0}k_{\mathrm {obs} }^{0}+\Lambda _{1}^{0}k_{\mathrm {obs} }^{1}+\Lambda _{2}^{0}k_{\mathrm {obs} }^{2}+\Lambda _{3}^{0}k_{\mathrm {obs} }^{3}\\[3pt]{\frac {\omega _{s}}{c}}&=\gamma {\frac {\omega _{\mathrm {obs} }}{c}}-\beta \gamma k_{\mathrm {obs} }^{1}\\&=\gamma {\frac {\omega _{\mathrm {obs} }}{c}}-\beta \gamma {\frac {\omega _{\mathrm {obs} }}{c}}\cos \theta .\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./467f60737b832c1dc0ed86fcdb2b60180d38dde9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:41.635ex; height:13.843ex;" alt="{\displaystyle {\begin{aligned}k_{s}^{0}&=\Lambda _{0}^{0}k_{\mathrm {obs} }^{0}+\Lambda _{1}^{0}k_{\mathrm {obs} }^{1}+\Lambda _{2}^{0}k_{\mathrm {obs} }^{2}+\Lambda _{3}^{0}k_{\mathrm {obs} }^{3}\\[3pt]{\frac {\omega _{s}}{c}}&=\gamma {\frac {\omega _{\mathrm {obs} }}{c}}-\beta \gamma k_{\mathrm {obs} }^{1}\\&=\gamma {\frac {\omega _{\mathrm {obs} }}{c}}-\beta \gamma {\frac {\omega _{\mathrm {obs} }}{c}}\cos \theta .\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \cos \theta }</annotation>

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</math></span><img src="./611e5c70de1d1cf4ebc3b70d2b5467f45d17a483.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.589ex; height:2.176ex;" alt="{\displaystyle \cos \theta }" loading="lazy"></span> is the direction cosine of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{1}}">

<msup>

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</math></span><img src="./8ebebaf6063bd8619056d7d23f407dd6a59468c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.265ex; height:2.676ex;" alt="{\displaystyle k^{1}}" loading="lazy"></span> with respect to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{0},k^{1}=k^{0}\cos \theta .}">

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<annotation encoding="application/x-tex">{\displaystyle k^{0},k^{1}=k^{0}\cos \theta .}</annotation>

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</math></span><img src="./e527aedc0042a8e334127a940f1612f2d5d704a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.551ex; height:3.009ex;" alt="{\displaystyle k^{0},k^{1}=k^{0}\cos \theta .}" loading="lazy"></span>

</p><p>So

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {1}{\gamma (1-\beta \cos \theta )}}}</annotation>

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</math></span><img src="./c86dbb0dfff5265031ad5e8ba529fe9bc0e4bd75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.215ex; height:6.009ex;" alt="{\displaystyle {\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {1}{\gamma (1-\beta \cos \theta )}}}" loading="lazy"></span>

</td></tr></tbody></table></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Source_moving_away_(redshift)">Source moving away (redshift)</h4></div>

<p>As an example, to apply this to a situation where the source is moving directly away from the observer (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\pi }">

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<annotation encoding="application/x-tex">{\displaystyle \theta =\pi }</annotation>

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</math></span><img src="./ab4db588619489e27efb50a1d0d5aa016c49ce15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.521ex; height:2.176ex;" alt="{\displaystyle \theta =\pi }" loading="lazy"></span>), this becomes:

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<annotation encoding="application/x-tex">{\displaystyle {\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 }}}}</annotation>

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</math></span><img src="./d4469e2529c0ae74b9c58e232f91aaccfd05411a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:62.031ex; height:7.509ex;" alt="{\displaystyle {\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 }}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Source_moving_towards_(blueshift)">Source moving towards (blueshift)</h4></div>

<p>To apply this to a situation where the source is moving straight towards the observer (<span class="texhtml"><i>θ</i> = 0</span>), this becomes:

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<annotation encoding="application/x-tex">{\displaystyle {\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 }}}}</annotation>

</semantics>

</math></span><img src="./55088f178a23ab7796f0bb740bb495b70266503e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:62.031ex; height:7.509ex;" alt="{\displaystyle {\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 }}}}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Source_moving_tangentially_(transverse_Doppler_effect)">Source moving tangentially (transverse Doppler effect)</h4></div>

<p>To apply this to a situation where the source is moving transversely with respect to the observer (<span class="texhtml"><i>θ</i> = <i>π</i>/2</span>), this becomes:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {1}{\gamma (1-0)}}={\frac {1}{\gamma }}}</annotation>

</semantics>

</math></span><img src="./cf7331fb3c48de7311667b065cc2214f28612a01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.266ex; height:6.009ex;" alt="{\displaystyle {\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {1}{\gamma (1-0)}}={\frac {1}{\gamma }}}" loading="lazy"></span></dd></dl>

<ul><li><a href="Plane-wave_expansion" title="Plane-wave expansion">Plane-wave expansion</a></li>

<li><a href="Plane_of_incidence" title="Plane of incidence">Plane of incidence</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</div>

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Physics example: <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><cite id="CITEREFHarris,_Benenson,_Stöcker2002" class="citation book cs1">Harris, Benenson, Stöcker (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=c60mCxGRMR8C&pg=PA288"><i>Handbook of Physics</i></a>. p. 288. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-95269-7</bdi>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: multiple names: authors list (link)</span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Crystallography example: <cite id="CITEREFVaĭnshteĭn1994" class="citation book cs1">Vaĭnshteĭn (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xjIGV_hPiysC&pg=PA259"><i>Modern Crystallography</i></a>. p. 259. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-56558-1</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFVaĭnshteĭn1994" class="citation book cs1">Vaĭnshteĭn, Boris Konstantinovich (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xjIGV_hPiysC&pg=PA259"><i>Modern Crystallography</i></a>. p. 259. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-56558-1</bdi>.</cite></span>

</li>

<li id="cite_note-fowles-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-fowles_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFFowles1968" class="citation book cs1">Fowles, Grant (1968). <i>Introduction to modern optics</i>. Holt, Rinehart, and Winston. p. 177.</cite></span>

</li>

<li id="cite_note-pollard-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-pollard_5-0">^</a></b></span> <span class="reference-text">"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 ...", <i>Sound waves in solids</i> by Pollard, 1977. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=EOUNAQAAIAAJ">link</a></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFDonald_H._Menzel1960" class="citation book cs1">Donald H. Menzel (1960). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-miofZvrH2sC&pg=PA624">"§10.5 Bloch wave"</a>. <i>Fundamental Formulas of Physics, Volume 2</i> (Reprint of Prentice-Hall 1955 2nd ed.). Courier-Dover. p. 624. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0486605968</bdi>.</cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{cite book}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFWolfgang_Rindler1991" class="citation book cs1">Wolfgang Rindler (1991). "§24 Wave motion". <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontosp0000rind/page/60"><i>Introduction to Special Relativity</i></a> (2nd ed.). Oxford Science Publications. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontosp0000rind/page/60">60–65</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-853952-0</bdi>.</cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFBrau,_Charles_A.2004" class="citation book cs1">Brau, Charles A. (2004). <i>Modern Problems in Classical Electrodynamics</i>. Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-514665-3</bdi>.</cite></li></ul>

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