Wave vector


In physics, a wave vector is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave, and its direction is perpendicular to the wavefront. In isotropic media, this is also the direction of wave propagation.
A closely related vector is the angular wave vector, with a typical unit being radian per metre. The wave vector and angular wave vector are related by a fixed constant of proportionality, 2 radians per cycle.
It is common in several fields of physics to refer to the angular wave vector simply as the wave vector, in contrast to, for example, crystallography. It is also common to use the symbol for whichever is in use.
In the context of special relativity, a wave four-vector can be defined, combining the wave vector and frequency.

Definition

The terms wave vector and angular wave vector have distinct meanings. Here, the wave vector is denoted by and the wavenumber by. The angular wave vector is denoted by and the angular wavenumber by. These are related by.
A sinusoidal traveling wave follows the equation
where:
  • is position,
  • is time,
  • is a function of and describing the disturbance describing the wave.
  • is the amplitude of the wave,
  • is a phase offset,
  • is the angular frequency of the wave, describing how many radians it traverses per unit of time, and related to the period by the equation
  • is the angular wave vector of the wave, describing how many radians it traverses per unit of distance, and related to the wavelength by the equation
The equivalent equation using the wave vector and frequency is
where:
  • 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". 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 in vacuum, 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, and cubic crystals, 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, the wave vector may not point exactly in the direction of wave propagation.

In solid-state physics

In solid-state physics, the "wavevector" 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 in spacetime, formed by all the events passed by the wave surface. A wavetrain can be regarded as a one-parameter family of such hypersurfaces in spacetime. This variable 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 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 phase-velocity, 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 four-vector is:
The four-wavevector is null for massless particles, where the rest mass
An example of a null four-wavevector would be a beam of coherent, monochromatic light, which has phase-velocity
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 frame, we would apply the Lorentz transformation as follows. Note that the source is in a frame and earth is in the observing frame,.
Applying the Lorentz transformation to the wave vector
and choosing just to look at the component results in
where is the direction cosine of with respect to
So