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Wave vector

From Wikipedia, the free encyclopedia
Vector describing a wave; often its propagation direction

Inphysics, awave vector (orwavevector) is avector used in describing awave, with a typical unit being cycle per metre. It has amagnitude and direction. Its magnitude is thewavenumber of the wave (inversely proportional to thewavelength), and its direction is perpendicular to thewavefront. In isotropic media, this is also the direction ofwave propagation.

A closely related vector is theangular wave vector (orangular wavevector), with a typical unit being radian per metre. The wave vector and angular wave vector are related by a fixed constant of proportionality, 2π radians per cycle.

It is common in several fields ofphysics to refer to the angular wave vector simply as thewave vector, in contrast to, for example,crystallography.[1][2] It is also common to use the symbolk for whichever is in use.

In the context ofspecial relativity, awave four-vector can be defined, combining the (angular) wave vector and (angular) frequency.

Definition

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See also:Traveling wave
Wavelength of asine wave,λ, can be measured between any two consecutive points with the samephase, such as between adjacent crests, or troughs, or adjacentzero crossings with the same direction of transit, as shown.

The termswave vector andangular wave vector have distinct meanings. Here, the wave vector is denoted byν~{\displaystyle {\tilde {\boldsymbol {\nu }}}} and the wavenumber byν~=|ν~|{\displaystyle {\tilde {\nu }}=\left|{\tilde {\boldsymbol {\nu }}}\right|}. The angular wave vector is denoted byk and the angular wavenumber byk = |k|. These are related byk=2πν~{\displaystyle \mathbf {k} =2\pi {\tilde {\boldsymbol {\nu }}}}.

A sinusoidaltraveling wave follows the equation

ψ(r,t)=Acos⁡(k⋅r−ωt+φ),{\displaystyle \psi (\mathbf {r} ,t)=A\cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi ),}

where:

The equivalent equation using the wave vector and frequency is[3]

ψ(r,t)=Acos⁡(2π(ν~⋅r−ft)+φ),{\displaystyle \psi \left(\mathbf {r} ,t\right)=A\cos \left(2\pi \left({\tilde {\boldsymbol {\nu }}}\cdot {\mathbf {r} }-ft\right)+\varphi \right),}

where:

Direction of the wave vector

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Main article:Group velocity

The direction in which the wave vector points must be distinguished from the "direction ofwave propagation". The "direction of wave propagation" is the direction of a wave's energy flow, and the direction that a smallwave packet will move, i.e. the direction of thegroup velocity. For light waves in vacuum, this is also the direction of thePoynting vector. On the other hand, the wave vector points in the direction ofphase velocity. In other words, the wave vector points in thenormal direction to thesurfaces of constant phase, also calledwavefronts.

In alosslessisotropic medium such as air, any gas, any liquid,amorphous solids (such asglass), andcubic crystals, 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.

For example, when a wave travels through ananisotropic medium, such aslight waves through an asymmetric crystal or sound waves through asedimentary rock, the wave vector may not point exactly in the direction of wave propagation.[4][5]

In solid-state physics

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Main article:Bloch's theorem

Insolid-state physics, the "wavevector" (also calledk-vector) of anelectron orhole in acrystal is the wavevector of itsquantum-mechanicalwavefunction. These electron waves are not ordinarysinusoidal waves, but they do have a kind ofenvelope function which is sinusoidal, and the wavevector is defined via that envelope wave, usually using the "physics definition". SeeBloch's theorem for further details.[6]

In special relativity

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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 variableX) can be regarded as a one-parameter family of such hypersurfaces in spacetime. This variableX is a scalar function of position in spacetime. The derivative of this scalar is a vector that characterizes the wave, the four-wavevector.[7]

The four-wavevector is a wavefour-vector that is defined, inMinkowski coordinates, as:

Kμ=(ωc,k→)=(ωc,ωvpn^)=(2πcT,2πn^λ){\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)\,}

where the angular frequencyωc{\displaystyle {\tfrac {\omega }{c}}} is the temporal component, and the wavenumber vectork→{\displaystyle {\vec {k}}} is the spatial component.

Alternately, the wavenumberk can be written as the angular frequencyω divided by thephase-velocityvp, or in terms of inverse periodT and inverse wavelengthλ.

When written out explicitly itscontravariant andcovariant forms are:

Kμ=(ωc,kx,ky,kz)Kμ=(ωc,−kx,−ky,−kz){\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}}}

In general, the Lorentz scalar magnitude of the wave four-vector is:

KμKμ=(ωc)2−kx2−ky2−kz2=(ωoc)2=(mocℏ)2{\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}}

The four-wavevector isnull formassless (photonic) particles, where the rest massmo=0{\displaystyle m_{o}=0}

An example of a null four-wavevector would be a beam of coherent,monochromatic light, which has phase-velocityvp=c{\displaystyle v_{p}=c}

Kμ=(ωc,k→)=(ωc,ωcn^)=ωc(1,n^){\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)\,} {for light-like/null}

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

KμKμ=(ωc)2−kx2−ky2−kz2=0{\displaystyle K^{\mu }K_{\mu }=\left({\frac {\omega }{c}}\right)^{2}-k_{x}^{2}-k_{y}^{2}-k_{z}^{2}=0} {for light-like/null}

The four-wavevector is related to thefour-momentum as follows:

Pμ=(Ec,p→)=ℏKμ=ℏ(ωc,k→){\displaystyle P^{\mu }=\left({\frac {E}{c}},{\vec {p}}\right)=\hbar K^{\mu }=\hbar \left({\frac {\omega }{c}},{\vec {k}}\right)}

The four-wavevector is related to thefour-frequency as follows:

Kμ=(ωc,k→)=(2πc)Nμ=(2πc)(ν,νn→){\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)}

The four-wavevector is related to thefour-velocity as follows:

Kμ=(ωc,k→)=(ωoc2)Uμ=(ωoc2)γ(c,u→){\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)}

Lorentz transformation

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Taking theLorentz transformation of the four-wavevector is one way to derive therelativistic Doppler effect. The Lorentz matrix is defined as

Λ=(γ−βγ 0  0 −βγγ0000100001){\displaystyle \Lambda ={\begin{pmatrix}\gamma &-\beta \gamma &\ 0\ &\ 0\ \\-\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}

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 frameSs and earth is in the observing frame,Sobs.Applying the Lorentz transformation to the wave vector

ksμ=Λνμkobsν{\displaystyle k_{s}^{\mu }=\Lambda _{\nu }^{\mu }k_{\mathrm {obs} }^{\nu }}

and choosing just to look at theμ=0{\displaystyle \mu =0} component results in

ks0=Λ00kobs0+Λ10kobs1+Λ20kobs2+Λ30kobs3ωsc=γωobsc−βγkobs1=γωobsc−βγωobsccos⁡θ.{\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}}}

wherecos⁡θ{\displaystyle \cos \theta } is the direction cosine ofk1{\displaystyle k^{1}} with respect tok0,k1=k0cos⁡θ.{\displaystyle k^{0},k^{1}=k^{0}\cos \theta .}

So

ωobsωs=1γ(1−βcos⁡θ){\displaystyle {\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {1}{\gamma (1-\beta \cos \theta )}}}

Source moving away (redshift)

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As an example, to apply this to a situation where the source is moving directly away from the observer (θ=π{\displaystyle \theta =\pi }), this becomes:

ωobsωs=1γ(1+β)=1−β21+β=(1+β)(1−β)1+β=1−β1+β{\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 }}}}

Source moving towards (blueshift)

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To apply this to a situation where the source is moving straight towards the observer (θ = 0), this becomes:

ωobsωs=1γ(1−β)=1−β21−β=(1+β)(1−β)1−β=1+β1−β{\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 }}}}

Source moving tangentially (transverse Doppler effect)

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To apply this to a situation where the source is moving transversely with respect to the observer (θ =π/2), this becomes:

ωobsωs=1γ(1−0)=1γ{\displaystyle {\frac {\omega _{\mathrm {obs} }}{\omega _{s}}}={\frac {1}{\gamma (1-0)}}={\frac {1}{\gamma }}}

See also

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References

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  1. ^Physics example:Harris, Benenson, Stöcker (2002).Handbook of Physics. p. 288.ISBN 978-0-387-95269-7.{{cite book}}: CS1 maint: multiple names: authors list (link)
  2. ^Crystallography example:Vaĭnshteĭn (1994).Modern Crystallography. p. 259.ISBN 978-3-540-56558-1.
  3. ^Vaĭnshteĭn, Boris Konstantinovich (1994).Modern Crystallography. p. 259.ISBN 978-3-540-56558-1.
  4. ^Fowles, Grant (1968).Introduction to modern optics. Holt, Rinehart, and Winston. p. 177.
  5. ^"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.link
  6. ^Donald H. Menzel (1960)."§10.5 Bloch wave".Fundamental Formulas of Physics, Volume 2 (Reprint of Prentice-Hall 1955 2nd ed.). Courier-Dover. p. 624.ISBN 978-0486605968.
  7. ^Wolfgang Rindler (1991). "§24 Wave motion".Introduction to Special Relativity (2nd ed.). Oxford Science Publications. pp. 60–65.ISBN 978-0-19-853952-0.

Further reading

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  • Brau, Charles A. (2004).Modern Problems in Classical Electrodynamics. Oxford University Press.ISBN 978-0-19-514665-3.
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