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, 2π 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, a wave four-vector can be defined, combining the (angular) wave vector and (angular) frequency.
Definition
The terms wave vector and angular wave vector have distinct meanings. Here, the wave vector is denoted by ν ~ {\displaystyle {\tilde {\boldsymbol {\nu }}}} and the wavenumber by ν ~ = | ν ~ | {\displaystyle {\tilde {\nu }}=\left|{\tilde {\boldsymbol {\nu }}}\right|} . The angular wave vector is denoted by k and the angular wavenumber by k = |k|. These are related by k = 2 π ν ~ {\displaystyle \mathbf {k} =2\pi {\tilde {\boldsymbol {\nu }}}} . A sinusoidal traveling wave follows the equation
ψ ( r , t ) = A cos ( k ⋅ r − ω t + φ ) , {\displaystyle \psi (\mathbf {r} ,t)=A\cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi ),}
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 ω = 2 π T , {\displaystyle \omega ={\tfrac {2\pi }{T}},}
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 | k | = 2 π λ . {\displaystyle |\mathbf {k} |={\tfrac {2\pi }{\lambda }}.}
The equivalent equation using the wave vector and frequency is
ψ ( r , t ) = A cos ( 2 π ( ν ~ ⋅ r − f t ) + φ ) , {\displaystyle \psi \left(\mathbf {r} ,t\right)=A\cos \left(2\pi \left({\tilde {\boldsymbol {\nu }}}\cdot {\mathbf {r} }-ft\right)+\varphi \right),}
where:
f {\displaystyle f} is the frequency
ν ~ {\displaystyle {\tilde {\boldsymbol {\nu }}}} 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 (such as glass), 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
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