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).

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

Definitions

, 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 $2\backslash pi$ 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: :$\backslash psi(x,t)\; =\; A\; \backslash cos\; (k\; x\; -\; \backslash omega\; t+\backslash varphi)$ where:

x is position,

t is time,

$\backslash psi$ (a function of x and t) is the disturbance describing the wave (for example, for an ocean wave, $\backslash psi$ would be the excess height of the water, or for a sound wave, $\backslash psi$ would be the excess air pressure).

A is the amplitude of the wave (the peak magnitude of the oscillation),

$\backslash varphi$ is a "phase offset" describing how two waves can be out of sync with each other,

$\backslash omega$ 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 $k=2\backslash pi/\backslash lambda$.

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

This formula is generalized to three dimensions by: :$\backslash psi\; \backslash left(\{\backslash mathbf\; r\},\; t\; \backslash right)\; =\; A\; \backslash cos\; \backslash left(\{\backslash mathbf\; k\}\; \backslash cdot\; \{\backslash mathbf\; r\}\; -\; \backslash omega\; t\; +\; \backslash varphi\; \backslash right)$ where:

r is the position vector in three-dimensional space,

$\backslash cdot$ 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: $|\{\backslash mathbf\; k\}|\; =\; 2\backslash pi/\backslash lambda$. 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: :$\backslash psi(x,t)\; =\; A\; \backslash cos\; (2\; \backslash pi\; (k\; x\; -\; \backslash nu\; t)+\backslash varphi)$ :$\backslash psi\; \backslash left(\{\backslash mathbf\; r\},\; t\; \backslash right)\; =\; A\; \backslash cos\; \backslash left(2\backslash pi(\{\backslash mathbf\; k\}\; \backslash cdot\; \{\backslash mathbf\; r\}\; -\; \backslash nu\; t)\; +\; \backslash varphi\; \backslash right)$ The differences are:

The frequency $\backslash nu$ instead of angular frequency $\backslash omega$ is used. They are related by $2\backslash pi\; \backslash nu=\backslash omega$. 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, $k=|\{\backslash mathbf\; k\}|\; =\; 1/\backslash lambda$, while in the physics definition above, $k=|\{\backslash mathbf\; k\}|\; =\; 2\backslash pi/\backslash lambda$.

The direction of k is discussed below.

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. For light waves, this is the direction of the Poynting vector.

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.

The general direction of the wave vector, which is true in both isotropic and anisotropic media, is that it points in the normal direction to the surfaces of constant phase, also called wave fronts.

In special relativity

A beam of coherent, monochromatic light can be characterized by the (null) wave 4-vector ::$k^\backslash mu\; =\; \backslash left(\backslash frac\{\backslash omega\}\{c\},\; \backslash vec\{k\}\; \backslash right)\; \backslash ,$ which, when written out explicitly in its contravariant and covariant forms is ::$k^\backslash mu\; =\; \backslash left(\backslash frac\{\backslash omega\}\{c\},\; k\_x,\; k\_y,\; k\_z\; \backslash right)\backslash ,$ and ::$k\_\backslash mu\; =\; \backslash left(\backslash frac\{\backslash omega\}\{c\},\; -k\_x,\; -k\_y,\; -k\_z\; \backslash right)\; .\; \backslash ,$

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:

::$k^\backslash mu\; k\_\backslash mu\; =\; \backslash frac\{\backslash omega\}\{c\}^2\; -\; k\_x^2\; -\; k\_y^2\; -\; k\_z^2\; \backslash \; =\; 0$

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 ::

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 ::$k^\{\backslash mu\}\_s\; =\; \backslash Lambda^\backslash mu\_\backslash nu\; k^\backslash nu\_\{\backslash mathrm\{obs\}\}\; \backslash ,$

and choosing just to look at the $\backslash mu\; =\; 0$ component results in ::$k^\{0\}\_s\; =\; \backslash Lambda^0\_0\; k^0\_\{\backslash mathrm\{obs\}\}\; +\; \backslash Lambda^0\_1\; k^1\_\{\backslash mathrm\{obs\}\}\; +\; \backslash Lambda^0\_2\; k^2\_\{\backslash mathrm\{obs\}\}\; +\; \backslash Lambda^0\_3\; k^3\_\{\backslash mathrm\{obs\}\}\; \backslash ,$ ::{| |$\backslash frac\{\backslash omega\_s\}\{c\}\; \backslash ,$ |$=\; \backslash gamma\; \backslash frac\{\backslash omega\_\{\backslash mathrm\{obs\}\}\}\{c\}\; -\; \backslash beta\; \backslash gamma\; k^1\_\{\backslash mathrm\{obs\}\}\; \backslash ,$ |- | |$\backslash quad\; =\; \backslash gamma\; \backslash frac\{\backslash omega\_\{\backslash mathrm\{obs\}\}\}\{c\}\; -\; \backslash beta\; \backslash gamma\; \backslash frac\{\backslash omega\_\{\backslash mathrm\{obs\}\}\}\{c\}\; \backslash cos\; \backslash theta.\; \backslash ,$

where $\backslash cos\; \backslash theta\; \backslash ,$ is the direction cosine of $k^1$ wrt $k^0,\; k^1\; =\; k^0\; \backslash cos\; \backslash theta.$ |} So ::{|cellpadding="2" style="border:2px solid #ccccff" |$\backslash frac\{\backslash omega\_\{\backslash mathrm\{obs\}\}\}\{\backslash omega\_s\}\; =\; \backslash frac\{1\}\{\backslash gamma\; (1\; -\; \backslash beta\; \backslash cos\; \backslash theta)\}\; \backslash ,$ |}

Source moving away

As an example, to apply this to a situation where the source is moving directly away from the observer ($\backslash theta=\backslash pi$), this becomes: ::$\backslash frac\{\backslash omega\_\{\backslash mathrm\{obs\}\}\}\{\backslash omega\_s\}\; =\; \backslash frac\{1\}\{\backslash gamma\; (1\; +\; \backslash beta)\}\; =\; \backslash frac\{\backslash sqrt\{1-\backslash beta^2\}\}\{1+\backslash beta\}\; =\; \backslash frac\{\backslash sqrt\{(1+\backslash beta)(1-\backslash beta)\}\}\{1+\backslash beta\}\; =\; \backslash frac\{\backslash sqrt\{1-\backslash beta\}\}\{\backslash sqrt\{1+\backslash beta\}\}\; \backslash ,$

Source moving towards

To apply this to a situation where the source is moving straight towards the observer ($\backslash theta=0$), this becomes: ::$\backslash frac\{\backslash omega\_\{\backslash mathrm\{obs\}\}\}\{\backslash omega\_s\}\; =\; \backslash frac\{\backslash sqrt\{1+\backslash beta\}\}\{\backslash sqrt\{1-\backslash beta\}\}\; \backslash ,$

References

Further Readings

Category:Wave mechanics Category:Vectors