It’s a really detailed cartoon character expressing fear and amazement. Of course, you can grab this vector character and modify its look & feel to a more friendly and fresh. Just download the vector files and edit with any vector editing software like Adobe Illustrator. Available for instant and free download! File Format: Ai Author: moyicat. In physics, a wave vector (also spelled wavevector) is a vector which helps describe a wave.Like any vector, it has a 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 below). Vector Images and Royalty Free Clip Art and Illustrations, available from Dreamstime. Access our online collection of clip art images free, by registering with us today! Stock Photos Editorial Illustrations Videos Audio Free Photos Blog. Sign up for FREE or Sign in. Sign in Sign up for FREE. Are you looking for Water vectors or photos? We have 255499 free resources for you. Download on Freepik your photos, PSD, icons or vectors of Water.

In physics, a wave vector (also spelled wavevector) is a vector which helps describe a wave. Like any vector, it has a 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 below).

In the context of special relativity the wave vector can also be defined as a four-vector.

1Definitions
4In special relativity
- 4.1Lorentz transformation

Definitions[edit]

Wavelength of a sine wave, λ, can be measured between any two consecutive points with the same phase, such as between adjacent crests, or troughs, or adjacent zero crossings with the same direction of transit, as shown.

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.^[1] For this article, they will be called the 'physics definition' and the 'crystallography definition', respectively.

In both definitions below, the magnitude of the wave vector is represented by ${displaystyle k}$ ; the direction of the wave vector is discussed in the following section.

Physics definition[edit]

A perfect one-dimensional traveling wave follows the equation:

{displaystyle psi (x,t)=Acos(kx-omega t+varphi )}

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where:

x is position,
t is time,
${displaystyle psi }$ (a function of x and t) is the disturbance describing the wave (for example, for an ocean wave, ${displaystyle psi }$ would be the excess height of the water, or for a sound wave, ${displaystyle psi }$ would be the excess air pressure).
A is the amplitude of the wave (the peak magnitude of the oscillation),
${displaystyle varphi }$ is a 'phase offset' describing how two waves can be out of sync with each other,
${displaystyle omega }$ is the temporal angular frequency of the wave, describing how many oscillations it completes per unit of time, and related to the period ${displaystyle T}$ by the equation ${displaystyle omega =2pi /T}$ ,
${displaystyle k}$ is the spatial angular frequency (wavenumber) of the wave, describing how many oscillations it completes per unit of space, and related to the wavelength by the equation ${displaystyle k=2pi /lambda }$ .

${displaystyle k}$ is the magnitude of the wave vector. In this one-dimensional example, the direction of the wave vector is trivial: this wave travels in the +x direction with speed (more specifically, phase velocity) ${displaystyle omega /k}$ . In a multidimensional system, the scalar ${displaystyle kx}$ would be replaced by the vector dot product ${displaystyle {mathbf {k} }cdot {mathbf {r} }}$ , representing the wave vector and the position vector, respectively.

Crystallography definition[edit]

In crystallography, the same waves are described using slightly different equations.^[2] In one and three dimensions respectively:

{displaystyle psi (x,t)=Acos(2pi (kx-nu t)+varphi )}

{displaystyle psi left({mathbf {r} },tright)=Acos left(2pi ({mathbf {k} }cdot {mathbf {r} }-nu t)+varphi right)}

The differences between the above two definitions are:

The angular frequency ${displaystyle omega }$ is used in the physics definition, while the frequency ${displaystyle nu }$ is used in the crystallography definition. They are related by ${displaystyle 2pi nu =omega }$ . This substitution is not important for this article, but reflects common practice in crystallography.
The wavenumber ${displaystyle k}$ and wave vector k are defined differently: in the physics definition above, ${displaystyle k= {mathbf {k} } =2pi /lambda }$ , while in the crystallography definition below, ${displaystyle k= {mathbf {k} } =1/lambda }$ .

The direction of k is discussed in the next section.

Direction of the wave vector[edit]

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 normal direction to the surfaces of constant phase, also called wavefronts.

In a losslessisotropic 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 the 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 heterogeneous waves, these two species of surfaces differ in orientation. The wave vector is always perpendicular to surfaces of constant phase.

For example, when a wave travels through an anisotropic medium, such as 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.^[3]^[4]

In solid-state physics[edit]

In solid-state physics, the 'wavevector' (also called k-vector) of an electron or hole in a crystal is the wavevector of its quantum-mechanicalwavefunction. These electron waves are not ordinary sinusoidal waves, but they do have a kind of 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.^[5]

In special relativity[edit]

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 X) can be regarded as a one-parameter family of such hypersurfaces in spacetime. This variable X is a scalar function of position in spacetime. The derivative of this scalar is a vector that characterizes the wave, the four-wavevector.^[6]

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The four-wavevector is a wave four-vector that is defined, in Minkowski coordinates, as:

{displaystyle K^{mu }=left({frac {omega }{c}},{vec {k}}right)=left({frac {omega }{c}},{frac {omega }{v_{p}}}{hat {n}}right)=left({frac {2pi }{cT}},{frac {2pi {hat {n}}}{lambda }}right),}

where the angular frequency ${displaystyle {frac {omega }{c}}}$ is the temporal component, and the wavenumber vector ${displaystyle {vec {k}}}$ is the spatial component.

Alternately, the wavenumber ${displaystyle k}$ can be written as the angular frequency ${displaystyle omega }$ divided by the phase-velocity ${displaystyle v_{p}}$ , or in terms of inverse period ${displaystyle T}$ and inverse wavelength ${displaystyle lambda }$ .

When written out explicitly its contravariant and covariant forms are:

{displaystyle K^{mu }=left({frac {omega }{c}},k_{x},k_{y},k_{z}right),}

{displaystyle K_{mu }=left({frac {omega }{c}},-k_{x},-k_{y},-k_{z}right),}

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

{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 is null for massless (photonic) particles, where the rest mass ${displaystyle m_{o}=0}$

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

{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:

{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 the four-momentum as follows:

{displaystyle P^{mu }=left({frac {E}{c}},{vec {p}}right)=hbar K^{mu }=hbar left({frac {omega }{c}},{vec {k}}right)}

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

{displaystyle K^{mu }=left({frac {omega }{c}},{vec {k}}right)=left({frac {2pi }{c}}right)N^{mu }=left({frac {2pi }{c}}right)(nu ,c{vec {n}})}

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

{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 (c,{vec {u}})}

Lorentz transformation[edit]

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

{displaystyle Lambda ={begin{pmatrix}gamma &-beta gamma &0&0-beta gamma &gamma &0&00&0&1&00&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 transformation 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

{displaystyle k_{s}^{mu }=Lambda _{nu }^{mu }k_{mathrm {obs} }^{nu },}

and choosing just to look at the ${displaystyle mu =0}$ component results in

{displaystyle 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},}

${displaystyle {frac {omega _{s}}{c}},}$	${displaystyle =gamma {frac {omega _{mathrm {obs} }}{c}}-beta gamma k_{mathrm {obs} }^{1},}$
${displaystyle quad =gamma {frac {omega _{mathrm {obs} }}{c}}-beta gamma {frac {omega _{mathrm {obs} }}{c}}cos theta .,}$ where ${displaystyle cos theta ,}$ is the direction cosine of ${displaystyle k^{1}}$ wrt ${displaystyle k^{0},k^{1}=k^{0}cos theta .}$

{displaystyle {frac {omega _{mathrm {obs} }}{omega _{s}}}={frac {1}{gamma (1-beta cos theta )}},}

Source moving away (redshift)[edit]

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

{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)[edit]

To apply this to a situation where the source is moving straight towards the observer ( ${displaystyle theta =0}$ ), this becomes:

{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)[edit]

To apply this to a situation where the source is moving transversely with respect to the observer ( ${displaystyle theta =pi /2}$ ), this becomes:

{displaystyle {frac {omega _{mathrm {obs} }}{omega _{s}}}={frac {1}{gamma (1-0)}}={frac {1}{gamma }},}

References[edit]

^Physics definition example:Harris, Benenson, Stöcker (2002). Handbook of Physics. p. 288. ISBN978-0-387-95269-7.CS1 maint: multiple names: authors list (link). Crystallography definition example: Vaĭnshteĭn (1994). Modern Crystallography. p. 259. ISBN978-3-540-56558-1.
^Vaĭnshteĭn, Boris Konstantinovich (1994). Modern Crystallography. p. 259. ISBN978-3-540-56558-1.
^Fowles, Grant (1968). Introduction to modern optics. Holt, Rinehart, and Winston. p. 177.
^'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
^Donald H. Menzel (1960). '§10.5 Bloch waves'. Fundamental Formulas of Physics, Volume 2 (Reprint of Prentice-Hall 1955 2nd ed.). Courier-Dover. p. 624. ISBN978-0486605968.
^Wolfgang Rindler (1991). '§24 Wave motion'. Introduction to Special Relativity (2nd ed.). Oxford Science Publications. pp. 60–65. ISBN978-0-19-853952-0.

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