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Concept
Wave vector
Summary
In physics, a wave vector (or wavevector) 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 (inversely proportional to the wavelength), 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 (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, 2pi 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 k for whichever is in use.
In the context of special relativity, wave vector can refer to a four-vector, in which the (angular) wave vector and (angular) frequency are combined.
Traveling wave
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 k and the angular wavenumber by k = . These are related by .
A sinusoidal traveling wave follows the equation
where:
r is position,
t is time,
ψ is a function of r and t describing the disturbance describing the wave (for example, for an ocean wave, ψ would be the excess height of the water, or for a sound wave, ψ would be the excess air pressure).
A is the amplitude of the wave (the peak magnitude of the oscillation),
φ is a phase offset,
ω is the (temporal) angular frequency of the wave, describing how many radians it traverses per unit of time, and related to the period t by the equation
k 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
Group velocity
The direction in which the wave vector points must be distinguished from the "direction of wave propagation".
Official source
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