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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, a wave vector (or wavevector) is a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
used in describing a
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the
wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
of the wave (inversely proportional to the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
), and its direction is perpendicular to the wavefront. In isotropic media, this is also the direction of
wave propagation Wave propagation is any of the ways in which waves travel. Single wave propagation can be calculated by 2nd order wave equation ( standing wavefield) or 1st order one-way wave equation. With respect to the direction of the oscillation relative to ...
. A closely related vector is the angular wave vector (or angular 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 of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
to refer to the angular wave vector simply as the ''wave vector'', in contrast to, for example,
crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
. It is also common to use the symbol ''k'' for whichever is in use. In the context of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
, ''wave vector'' can refer to a
four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
, in which the (angular) wave vector and (angular) frequency are combined.


Definition

For this article, the terms ''wave vector'' and ''angular wave vector'' are used with distinct meanings to avoid confusion. Here, the wave vector is denoted by and the wavenumber by , and the angular wave vector is denoted by k and the angular wavenumber by . These are related by . A sinusoidal
traveling wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
follows the equation :\psi(\mathbf,t) = A \cos (\mathbf \cdot \mathbf - \omega t + \varphi) , where: * r is position, * ''t'' is time, * \psi is a function of r and ''t'' describing the disturbance describing the wave (for example, for an
ocean wave In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction o ...
, \psi would be the excess height of the water, or for a
sound wave In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
, \psi would be the excess
air pressure Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The Standard atmosphere (unit), standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equival ...
). * ''A'' is the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
of the wave (the peak magnitude of the oscillation), * \varphi is a
phase offset In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it v ...
, * \omega is the (temporal)
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of the wave, describing how many oscillations it completes per unit of time, and related to the
period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Periodic sentence (or rhetorical period), a concept ...
T by the equation \omega=2\pi/T, * \mathbf is the angular wave vector of the wave, describing how many oscillations it completes per unit of distance, and related to the
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
by the equation , \mathbf, =2\pi/\lambda. The equivalent equation using the wave vector and frequency is : \psi \left( \mathbf, t \right) = A \cos \left(2\pi(\overset \cdot - \nu t) + \varphi \right) , where: * \nu is the frequency * is the wave vector


Direction of the wave vector

The direction in which the wave vector points must be distinguished from the "direction of
wave propagation Wave propagation is any of the ways in which waves travel. Single wave propagation can be calculated by 2nd order wave equation ( standing wavefield) or 1st order one-way wave equation. With respect to the direction of the oscillation relative to ...
". The "direction of wave propagation" is the direction of a wave's energy flow, and the direction that a small
wave packet In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of diffe ...
will move, i.e. the direction of the
group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...
. For light waves in vacuum, this is also the direction of the
Poynting vector In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or '' power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt ...
. On the other hand, the wave vector points in the direction of
phase velocity The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
. In other words, the wave vector points in the normal direction to the surfaces of constant phase, also called
wavefronts In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
. In a
lossless Lossless compression is a class of data compression that allows the original data to be perfectly reconstructed from the compressed data with no loss of information. Lossless compression is possible because most real-world data exhibits statistic ...
isotropic medium such as air, any gas, any liquid,
amorphous solids In condensed matter physics and materials science, an amorphous solid (or non-crystalline solid, glassy solid) is a solid that lacks the long-range order that is characteristic of a crystal. Etymology The term comes from the Greek ''a'' ("wit ...
(such as
glass Glass is a non-crystalline, often transparent, amorphous solid that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most often formed by rapid cooling (quenching) of ...
), and
cubic crystal In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of ...
s 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 an anisotropic medium, such as light waves through an asymmetric crystal or sound waves through a
sedimentary rock Sedimentary rocks are types of rock that are formed by the accumulation or deposition of mineral or organic particles at Earth's surface, followed by cementation. Sedimentation is the collective name for processes that cause these particles ...
, the wave vector may not point exactly in the direction of wave propagation."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
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In solid-state physics

In
solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...
, the "wavevector" (also called k-vector) of an
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
or
hole A hole is an opening in or through a particular medium, usually a solid body. Holes occur through natural and artificial processes, and may be useful for various purposes, or may represent a problem needing to be addressed in many fields of en ...
in a
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
is the wavevector of its quantum-mechanical
wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
. These electron waves are not ordinary
sinusoidal A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in m ...
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's theorem In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who d ...
for further details.


In special relativity

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. The four-wavevector is a wave
four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
that is defined, in Minkowski coordinates, as: :K^\mu = \left(\frac, \vec\right) = \left(\frac, \frac\hat\right) = \left(\frac, \frac\right) \, where the angular frequency \frac is the temporal component, and the wavenumber vector \vec is the spatial component. Alternately, the wavenumber k can be written as the angular frequency \omega divided by the phase-velocity v_p, or in terms of inverse period T and inverse wavelength \lambda. When written out explicitly its contravariant and covariant forms are: :\begin K^\mu &= \left(\frac, k_x, k_y, k_z \right)\, \\ K_\mu &= \left(\frac, -k_x, -k_y, -k_z \right) \end In general, the Lorentz scalar magnitude of the wave four-vector is: :K^\mu K_\mu = \left(\frac\right)^2 - k_x^2 - k_y^2 - k_z^2 = \left(\frac\right)^2 = \left(\frac\right)^2 The four-wavevector is
null Null may refer to: Science, technology, and mathematics Computing * Null (SQL) (or NULL), a special marker and keyword in SQL indicating that something has no value * Null character, the zero-valued ASCII character, also designated by , often use ...
for massless (photonic) particles, where the rest mass m_o = 0 An example of a null four-wavevector would be a beam of coherent,
monochromatic A monochrome or monochromatic image, object or color scheme, palette is composed of one color (or lightness, values of one color). Images using only Tint, shade and tone, shades of grey are called grayscale (typically digital) or Black and wh ...
light, which has phase-velocity v_p = c :K^\mu = \left(\frac, \vec\right) = \left(\frac, \frac\hat\right) = \frac\left(1, \hat\right) \, which would have the following relation between the frequency and the magnitude of the spatial part of the four-wavevector: :K^\mu K_\mu = \left(\frac\right)^2 - k_x^2 - k_y^2 - k_z^2 = 0 The four-wavevector is related to the
four-momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
as follows: :P^\mu = \left(\frac, \vec\right) = \hbar K^\mu = \hbar\left(\frac, \vec\right) The four-wavevector is related to the
four-frequency The four-frequency of a massless particle, such as a photon, is a four-vector defined by :N^a = \left( \nu, \nu \hat \right) where \nu is the photon's frequency and \hat is a unit vector in the direction of the photon's motion. The four-frequency ...
as follows: :K^\mu = \left(\frac, \vec\right) = \left(\frac\right)N^\mu = \left(\frac\right)\left(\nu, \nu \vec\right) The four-wavevector is related to the
four-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, spacet ...
as follows: :K^\mu = \left(\frac, \vec\right) = \left(\frac\right)U^\mu = \left(\frac\right) \gamma \left(c, \vec\right)


Lorentz transformation

Taking the
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
of the four-wavevector is one way to derive the
relativistic Doppler effect The relativistic Doppler effect is the change in frequency (and wavelength) of light, caused by the relative motion of the source and the observer (as in the classical Doppler effect), when taking into account effects described by the special rel ...
. The Lorentz matrix is defined as :\Lambda = \begin \gamma & -\beta \gamma & 0 & 0 \\ -\beta \gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end 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 :k^_s = \Lambda^\mu_\nu k^\nu_ and choosing just to look at the \mu = 0 component results in :\begin k^_s &= \Lambda^0_0 k^0_ + \Lambda^0_1 k^1_ + \Lambda^0_2 k^2_ + \Lambda^0_3 k^3_ \\ pt \frac &= \gamma \frac - \beta \gamma k^1_ \\ &= \gamma \frac - \beta \gamma \frac \cos \theta. \end where \cos \theta is the direction cosine of k^1 with respect to k^0, k^1 = k^0 \cos \theta. So :


Source moving away (redshift)

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


Source moving towards (blueshift)

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


Source moving tangentially (transverse Doppler effect)

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


See also

*
Plane wave expansion In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves: e^ = \sum_^\infty (2 \ell + 1) i^\ell j_\ell(k r) P_\ell(\hat \cdot \hat), where * is the imaginary unit, * is a wave vector of length , * ...
*
Plane of incidence In describing reflection and refraction in optics, the plane of incidence (also called the incidence plane or the meridional plane) is the plane which contains the surface normal and the propagation vector of the incoming radiation. (In wave optic ...


References


Further reading

* {{Authority control Wave mechanics Vectors (mathematics and physics)