Wave vector

Wave vector

A wave vector is a vector representation of a wave. The wave vector has magnitude indicating wavenumber (reciprocal of wavelength), and the direction of the vector indicates the direction of wave propagation.

The wave vector is most useful for generalizing the equation of a single wave into a description of a family of waves. As long as the family of waves all travel in the same direction and with the same wavelength, a single wave vector is valid for the entire family. The most common case of a family of waves that meets these requirements is the plane wave, in which the family of waves is also coherent, i.e. all the waves have the same phase.

For example, a common representation of a single wave at a single point in space is::psi left(t ight) = A cos left(varphi + omega t ight)where "A" is the amplitude, "ω" is the angular frequency, and "φ" is the starting phase of the wave (the independent variable "t" is time).

In order to generalize the equation to all points in the one-dimensional space of the direction of propagation, we add in an additional phase offset term: :psi left(t , z ight) = A cos left(varphi + k z + omega t ight)where "k" is the angular wavenumber ("2π/λ") and the new independent variable "z" is the distance along the wave.

Now, as long as we are dealing with a simple family of waves, with identical direction, wavelength, and phase (i.e. a plane wave), we can easily extend the formula by substituting the wave vector k for the wavenumber "k", and the location in space vector r for the variable "z"::psi left(t , {mathbf r} ight) = A cos left(varphi + {mathbf k} cdot {mathbf r} + omega t ight)

Wavenumber is also closely related to wavelength by the following formula::k = frac{2pi}{lambda}

In special relativity

A wave packet of nearly monochromatic light can be characterized by the wave vector::k^mu = left(frac{omega}{c}, vec{k} ight) ,which, when written out explicitly in its contravariant and covariant forms is ::k^mu = left(frac{omega}{c}, k^1, k^2, k^3 ight), and ::k_mu = left(frac{omega}{c}, -k_1, -k_2, -k_3 ight) . ,

The magnitude of this wave vector is then::k^2 = k^mu k_mu = k^0 k_0 - k^1 k_1 - k^2 k_2 - k^3 k_3 ,:::::=frac{omega^2}{c^2} - vec{k}^2 = 0. ,

That last step where it equals zero, is a result of the fact that, for light in vacuum,::k = frac{omega}{c}. ,

Lorentz transform

Taking the Lorentz transform of the wave vector is one way to derive the Relativistic Doppler effect. The Lorentz matrix is defined as::Lambda = egin{pmatrix}gamma&-eta gamma&0&0 \-eta gamma&gamma&0&0 \0&0&1&0 \0&0&0&1end{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 transform as follows. Note that the source is in a frame "S"s and earth is in the observing frame, "S"obs.Applying the lorentz transformation to the wave vector::k^{mu}_s = Lambda^mu_ u k^ u_{mathrm{obs ,

and choosing just to look at the mu = 0 component results in::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 ,::So::

ource moving away

As an example, to apply this to a situation where the source is moving directly away from the observer ( heta=pi), this becomes:::frac{omega_{mathrm{obs}{omega_s} = frac{1}{gamma (1 + eta)} = frac{sqrt{1-eta^2{1+eta} = frac{sqrt{(1+eta)(1-eta){1+eta} = frac{sqrt{1-eta{sqrt{1+eta ,

ource moving towards

To apply this to a situation where the source is moving straight towards the observer ( heta=0), this becomes:::frac{omega_{mathrm{obs}{omega_s} = frac{sqrt{1+eta{sqrt{1-eta ,

References

*cite book | author=Brau, Charles A. | title=Modern Problems in Classical Electrodynamics | publisher=Oxford University Press | year=2004 | id=ISBN 0-19-514665-4


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • wave vector — bangos vektorius statusas T sritis Standartizacija ir metrologija apibrėžtis Apibrėžtį žr. priede. priedas( ai) Grafinis formatas atitikmenys: angl. wave vector vok. Wellenvektor, m; Wellenzahlvektor, m rus. волновой вектор, m pranc. vecteur… …   Penkiakalbis aiškinamasis metrologijos terminų žodynas

  • wave vector — bangos vektorius statusas T sritis fizika atitikmenys: angl. wave vector vok. Wellenvektor, m rus. волновой вектор, m pranc. vecteur d’onde, m …   Fizikos terminų žodynas

  • wave-vector space — bangos vektorių erdvė statusas T sritis fizika atitikmenys: angl. k space; wave vector space vok. k Raum, m; Wellenvektorraum, m rus. k пространство, n; пространство волновых векторов, n pranc. espace k, m; espace vecteur d’onde, m …   Fizikos terminų žodynas

  • wave vector — noun A vector whose direction is that of the direction of propagation of a wave, and whose magnitude is inversely proportional to its wavelength …   Wiktionary

  • reduced wave vector — redukuotasis bangos vektorius statusas T sritis fizika atitikmenys: angl. reduced wave vector vok. reduzierter Wellenvektor, m rus. приведённый волновой вектор, m pranc. vecteur d’onde réduit, m …   Fizikos terminų žodynas

  • Wave function — Not to be confused with the related concept of the Wave equation Some trajectories of a harmonic oscillator (a ball attached to a spring) in classical mechanics (A B) and quantum mechanics (C H). In quantum mechanics (C H), the ball has a wave… …   Wikipedia

  • Wave–particle duality — Quantum mechanics Uncertainty principle …   Wikipedia

  • Vector synthesis — is a type of audio synthesis introduced by Sequential Circuits in the Prophet VS synthesizer during 1986. The concept was subsequently used by Yamaha in the SY22/TG33 and similar instruments and by Korg in the Wavestation. Vector synthesis… …   Wikipedia

  • Vector Prime (Transformers) — Vector Prime is a fictional character from the toyline, animated series and comics.Transformers: CybertronTransformers character name = Vector Prime caption = Knight Warrior of Cybertron and Defender of Spacetime affiliation =Autobot subgroup =… …   Wikipedia

  • Vector — may refer to: In mathematics * Euclidean vector, a geometric entity endowed with both length and direction, an element of a Euclidean vector space * Coordinate vector, in linear algebra, an explicit representation of an element of any abstract… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”