Wave vector

"k-vector" redirects here. For the concept in solid-state physics, see Crystal momentum.

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).

The wave vector can also be defined as a four-vector in the context of special relativity.

Definitions

See also: Traveling wave

Wavelength of a sine wave, λ, can be measured between any two points with the same phase, such as between crests, or troughs, or corresponding zero crossings as shown.

Unfortunately, there are two common definitions of wave vector which differ by a factor of 2π in their magnitudes. In general, 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.

Physics definition

A perfect one-dimensional traveling wave follows the equation:

ψ(x,t) = Acos(kx − ωt + φ)

where:

• x is position,
• t is time,
• ψ (a function of x and t) is 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" describing how two waves can be out of sync with each other,
• ω is the angular frequency of the wave, related to how quickly it oscillates at a given point,
• k is the wavenumber (more specifically called angular wavenumber) of the wave, related to the wavelength by the equation k = 2π / λ.

This wave travels in the +x direction with speed (more specifically, phase velocity) ω / k.

This formula is generalized to three dimensions by:

$\psi \left({\mathbf r}, t \right) = A \cos \left({\mathbf k} \cdot {\mathbf r} - \omega t + \varphi \right)$

where:

• r is the position vector in three-dimensional space,
• $\cdot$ is the vector dot product.
• k is the wave vector.

This formula describes a plane wave. The magnitude of the wave vector is the angular wavenumber as in the one-dimensional case above: $|{\mathbf k}| = 2\pi/\lambda$. The direction of the wave vector is ordinarily the direction that the plane wave is traveling, but it can differ slightly in an anisotropic medium. (See below).

Crystallography definition

In crystallography, the same waves are described using slightly different equations. In one and three dimensions respectively:

ψ(x,t) = Acos(2π(kx − νt) + φ)

$\psi \left({\mathbf r}, t \right) = A \cos \left(2\pi({\mathbf k} \cdot {\mathbf r} - \nu t) + \varphi \right)$

The differences are:

• The frequency ν instead of angular frequency ω is used. They are related by 2πν = ω. This substitution is not important for this article, but reflects common practice in crystallography.
• The wavenumber k and wave vector k are defined in a different way. Here, $k=|{\mathbf k}| = 1/\lambda$, while in the physics definition above, $k=|{\mathbf k}| = 2\pi/\lambda$.

The direction of k is discussed below.

Direction of the wave vector

Main article: Group velocity

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 an 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.

However, 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 special relativity

A beam of coherent, monochromatic light can be characterized by the (null) wave 4-vector

$k^\mu = \left(\frac{\omega}{c}, \vec{k} \right) \,$

which, when written out explicitly in its contravariant and covariant forms is

$k^\mu = \left(\frac{\omega}{c}, k_x, k_y, k_z \right)\,$ and

$k_\mu = \left(\frac{\omega}{c}, -k_x, -k_y, -k_z \right) . \,$

The null character of the wave 4-vector gives a relation between the frequency and the magnitude of the spatial part of the wave 4-vector:

$k^\mu k_\mu = \left(\frac{\omega}{c}\right)^2 - k_x^2 - k_y^2 - k_z^2 \ = 0$

Lorentz transformation

Taking the Lorentz transformation of the wave vector is one way to derive the relativistic Doppler effect. The Lorentz matrix is defined as

$\Lambda = \begin{pmatrix} \gamma&-\beta \gamma&0&0 \\ -\beta \gamma&\gamma&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{pmatrix}$

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

$k^{\mu}_s = \Lambda^\mu_\nu k^\nu_{\mathrm{obs}} \,$

and choosing just to look at the μ = 0 component results in

$k^{0}_s = \Lambda^0_0 k^0_{\mathrm{obs}} + \Lambda^0_1 k^1_{\mathrm{obs}} + \Lambda^0_2 k^2_{\mathrm{obs}} + \Lambda^0_3 k^3_{\mathrm{obs}} \,$

$\frac{\omega_s}{c} \,$	$= \gamma \frac{\omega_{\mathrm{obs}}}{c} - \beta \gamma k^1_{\mathrm{obs}} \,$
	$\quad = \gamma \frac{\omega_{\mathrm{obs}}}{c} - \beta \gamma \frac{\omega_{\mathrm{obs}}}{c} \cos \theta. \,$ where $\cos \theta \,$ is the direction cosine of k1 wrt k0,k1 = k0cos θ.

So

$\frac{\omega_{\mathrm{obs}}}{\omega_s} = \frac{1}{\gamma (1 - \beta \cos \theta)} \,$

Source moving away

As an example, to apply this to a situation where the source is moving directly away from the observer (θ = π), this becomes:

$\frac{\omega_{\mathrm{obs}}}{\omega_s} = \frac{1}{\gamma (1 + \beta)} = \frac{\sqrt{1-\beta^2}}{1+\beta} = \frac{\sqrt{(1+\beta)(1-\beta)}}{1+\beta} = \frac{\sqrt{1-\beta}}{\sqrt{1+\beta}} \,$

Source moving towards

To apply this to a situation where the source is moving straight towards the observer (θ = 0), this becomes:

$\frac{\omega_{\mathrm{obs}}}{\omega_s} = \frac{\sqrt{1+\beta}}{\sqrt{1-\beta}} \,$

See also

• Plane wave expansion

Further Readings

• Brau, Charles A. (2004). Modern Problems in Classical Electrodynamics. Oxford University Press. ISBN 0-19-514665-4.