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 four-vector.
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. 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 formula_1; the direction of the wave vector is discussed in the following section.
Physics definition.
A perfect one-dimensional traveling wave follows the equation:
where:
formula_1 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) formula_13. In a multidimensional system, the scalar formula_14 would be replaced by the vector dot product formula_15, representing the wave vector and the position vector, respectively.
Crystallography definition.
In crystallography, the same waves are described using slightly different equations. In one and three dimensions respectively:
The differences between the above two definitions are:
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 wavefronts.
In a lossless isotropic medium such as air, any gas, any liquid, amorphous solids (such as glass), and cubic crystals the direction of the wavevector is exactly 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 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 anisotropic. 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.
In solid-state physics.
In solid-state physics, the "wavevector" (also called k-vector) of an electron or hole in a crystal is the wavevector of its quantum-mechanical 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's theorem 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 that is defined, in Minkowski coordinates, as:
where the angular frequency formula_24 is the temporal component, and the wavenumber vector formula_25 is the spatial component.
Alternately, the wavenumber formula_1 can be written as the angular frequency formula_7 divided by the phase-velocity formula_28, or in terms of inverse period formula_8 and inverse wavelength formula_30.
When written out explicitly its contravariant and covariant forms are: 
In general, the Lorentz scalar magnitude of the wave four-vector is:
The four-wavevector is null for massless (photonic) particles, where the rest mass formula_33
An example of a null four-wavevector would be a beam of coherent, monochromatic light, which has phase-velocity formula_34
which would have the following relation between the frequency and the magnitude of the spatial part of the four-wavevector:
The four-wavevector is related to the four-momentum as follows:
The four-wavevector is related to the four-frequency as follows:
The four-wavevector is related to the four-velocity as follows:
Lorentz transformation.
Taking the Lorentz transformation of the four-wavevector 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 formula_42 component results in
where formula_44 is the direction cosine of formula_45 wrt formula_46
So
Source moving away (redshift).
As an example, to apply this to a situation where the source is moving directly away from the observer (formula_47), this becomes:
Source moving towards (blueshift).
To apply this to a situation where the source is moving straight towards the observer (formula_49), this becomes:
Source moving tangentially (transverse Doppler effect).
To apply this to a situation where the source is moving transversely with respect to the observer (formula_51), this becomes: