Wave vector

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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).

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

Definitions[edit]

See also: Traveling wave
Wavelength of a sine wave, λ, can be measured between any two consecutive points with the same phase, such as between adjacent crests, or troughs, or adjacent zero crossings with the same direction of transit, as shown.

Unfortunately, there are two common definitions of wave vector, which differ by a factor of 2π in their magnitudes. One definition is preferred in physics and related fields, while the other definition is preferred in crystallography and related fields.[1] For this article, they will be called the "physics definition" and the "crystallography definition", respectively.

Physics definition[edit]

A perfect one-dimensional traveling wave follows the equation:

\psi(x,t) = A \cos (k x - \omega t+\varphi)

where:

  • x is position,
  • t is time,
  • \psi (a function of x and t) is the disturbance describing the wave (for example, for an ocean wave, \psi would be the excess height of the water, or for a sound wave, \psi would be the excess air pressure).
  • A is the amplitude of the wave (the peak magnitude of the oscillation),
  • \varphi is a "phase offset" describing how two waves can be out of sync with each other,
  • \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\pi/\lambda.

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

Crystallography definition[edit]

In crystallography, the same waves are described using slightly different equations.[2] In one and three dimensions respectively:

\psi(x,t) = A \cos (2 \pi (k x - \nu t)+\varphi)
\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 \nu instead of angular frequency \omega is used. They are related by 2\pi \nu=\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=|{\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[edit]

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 a lossless 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. If the medium is lossy, the wave vector in general points in directions other than that of wave propagation. The condition for wave vector to point in the same direction in which the wave propagates is that the wave has to be homogeneous, which isn't necessarily satisfied when the medium is lossy. In a homogeneous wave, the surfaces of constant phase are also surfaces of constant amplitude. In case of inhomogeneous waves, these two species of surfaces differ in orientation. Wave vector is always perpendicular to surfaces of constant phase.

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.[3][4]

In solid-state physics[edit]

Main article: Bloch wave

In solid-state physics, the "wavevector" (also called k-vector) 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 wave for further details.[5]

In special relativity[edit]

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

The wave 4-vector is related to the four-momentum as follows:

p^\mu = (E/c, p_x, p_y, p_z) = (\hbar\omega/c, \hbar k_x, \hbar k_y, \hbar k_z) = \hbar k^\mu

Lorentz transformation[edit]

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 Ss and earth is in the observing frame, Sobs. 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 \mu = 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 k^1 wrt k^0,  k^1 = k^0 \cos \theta.

So

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

Source moving away[edit]

As an example, to apply this to a situation where the source is moving directly away from the observer (\theta=\pi), 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[edit]

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

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

See also[edit]

References[edit]

  1. ^ Physics definition example:Harris, Benenson, Stöcker (2002). Handbook of Physics. p. 288. ISBN 978-0-387-95269-7. . Crystallography definition example: Vaĭnshteĭn (1994). Modern Crystallography. p. 259. ISBN 978-3-540-56558-1. 
  2. ^ Vaĭnshteĭn, Boris Konstantinovich (1994). Modern Crystallography. p. 259. ISBN 978-3-540-56558-1. 
  3. ^ Fowles, Grant (1968). Introduction to modern optics. Holt, Rinehart, and Winston. p. 177. 
  4. ^ "This effect has been explained by Musgrave (1959) who has shown that the energy of an elastic wave in an anisotropic medium will not, in general, travel along the same path as the normal to the plane wavefront...", Sound waves in solids by Pollard, 1977. link
  5. ^ Donald H. Menzel (1960). "§10.5 Bloch waves". Fundamental Formulas of Physics, Volume 2 (Reprint of Prentice-Hall 1955 2nd ed.). Courier-Dover. p. 624. ISBN 0486605965. 

Further reading[edit]

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