Wave vector
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 fourvector.
Definitions
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 ; the direction of the wave vector is discussed in the following section.
Physics definition
A perfect onedimensional traveling wave follows the equation:
where:
 x is position,
 t is time,
 (a function of x and t) is the disturbance describing the wave (for example, for an ocean wave, would be the excess height of the water, or for a sound wave, would be the excess air pressure).
 A is the amplitude of the wave (the peak magnitude of the oscillation),
 is a "phase offset" describing how two waves can be out of sync with each other,
 is the temporal angular frequency of the wave, describing how many oscillations it completes per unit of time, and related to the period by the equation ,
 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 .
is the magnitude of the wave vector. In this onedimensional example, the direction of the wave vector is trivial: this wave travels in the +x direction with speed (more specifically, phase velocity) . In a multidimensional system, the scalar would be replaced by the vector dot product , representing the wave vector and the position vector, respectively.
Crystallography definition
In crystallography, the same waves are described using slightly different equations.^{[2]} In one and three dimensions respectively:
The differences between the above two definitions are:
 The angular frequency is used in the physics definition, while the frequency is used in the crystallography definition. They are related by . This substitution is not important for this article, but reflects common practice in crystallography.
 The wavenumber and wave vector k are defined differently: in the physics definition above, , while in the crystallography definition below, .
The direction of k is discussed in the next section.
Direction of the wave vector
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 wave fronts.
In a lossless isotropic 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 inhomogeneous waves, these two species of surfaces differ in orientation. 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 solidstate physics
In solidstate physics, the "wavevector" (also called kvector) of an electron or hole in a crystal is the wavevector of its quantummechanical wavefunction. 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
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 oneparameter 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 fourwavevector.^{[6]}
The fourwavevector is a wave fourvector that is defined, in Minkowski coordinates, as:
where the angular frequency is the temporal component, and the wavenumber vector is the spatial component.
Alternately, the wavenumber can be written as the angular frequency divided by the phasevelocity , or in terms of inverse period and inverse wavelength .
When written out explicitly its contravariant and covariant forms are:
In general, the Lorentz scalar magnitude of the wave fourvector is:
The fourwavevector is null for massless (photonic) particles, where the rest mass
An example of a null fourwavevector would be a beam of coherent, monochromatic light, which has phasevelocity

 {for lightlike/null}
which would have the following relation between the frequency and the magnitude of the spatial part of the fourwavevector:

 {for lightlike/null}
The fourwavevector is related to the fourmomentum as follows:
The fourwavevector is related to the fourfrequency as follows:
The fourwavevector is related to the fourvelocity as follows:
Lorentz transformation
Taking the Lorentz transformation of the fourwavevector is one way to derive the relativistic Doppler effect. The Lorentz matrix is defined as
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
and choosing just to look at the component results in


where is the direction cosine of wrt

So
As an example, to apply this to a situation where the source is moving directly away from the observer (), this becomes:
To apply this to a situation where the source is moving straight towards the observer (), this becomes:
To apply this to a situation where the source is moving transversely with respect to the observer (), this becomes:
See also
References
 ^ Physics definition example:Harris, Benenson, Stöcker (2002). Handbook of Physics. p. 288. ISBN 9780387952697.CS1 maint: Multiple names: authors list (link). Crystallography definition example: Vaĭnshteĭn (1994). Modern Crystallography. p. 259. ISBN 9783540565581.
 ^ Vaĭnshteĭn, Boris Konstantinovich (1994). Modern Crystallography. p. 259. ISBN 9783540565581.
 ^ 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 PrenticeHall 1955 2nd ed.). CourierDover. p. 624. ISBN 9780486605968.
 ^ Wolfgang Rindler (1991). "§24 Wave motion". Introduction to Special Relativity (2nd ed.). Oxford Science Publications. pp. 60–65. ISBN 9780198539520.
Further reading
 Brau, Charles A. (2004). Modern Problems in Classical Electrodynamics. Oxford University Press. ISBN 9780195146653.