In the context of special relativity the wave vector can also be defined as a four-vector.
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.
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. For this article, they will be called the "physics definition" and the "crystallography definition", respectively.
In both definitions below, the magnitude of the wave vector is represented by ; the direction of the wave vector is discussed in the following section.
(a function of x and t) is the disturbance describing the wave (for example, for an ocean wave, would be the excess height of the water, or for a sound wave, would be the excess air pressure).
A is the amplitude of the wave (the peak magnitude of the oscillation),
is a phase offset describing how two waves can be out of sync with each other,
is the temporal angular frequency of the wave, describing how many oscillations it completes per unit of time, and related to the period by the equation ,
is the spatial angular frequency (wavenumber) of the wave, describing how many oscillations it completes per unit of space, and related to the wavelength by the equation .
is the magnitude of the wave vector. In this one-dimensional example, the direction of the wave vector is trivial: this wave travels in the +x direction with speed (more specifically, phase velocity) . In a multidimensional system, the scalar would be replaced by the vector dot product , representing the wave vector and the position vector, respectively.
In crystallography, the same waves are described using slightly different equations. In one and three dimensions respectively:
The differences between the above two definitions are:
The angular frequency is used in the physics definition, while the frequency is used in the crystallography definition. They are related by . This substitution is not important for this article, but reflects common practice in crystallography.
The wavenumber and wave vector k are defined differently: in the physics definition above, , while in the crystallography definition below, .
In a losslessisotropic medium such as air, any gas, any liquid, amorphous solids (such as glass), and cubic crystals the direction of the wavevector is exactly 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 condition for the 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 anisotropic. In a homogeneous wave, the surfaces of constant phase are also surfaces of constant amplitude. In case of heterogeneous waves, these two species of surfaces differ in orientation. The wave vector is always perpendicular to surfaces of constant phase.
A moving wave surface in special relativity may be regarded as a hypersurface (a 3D subspace) in spacetime, formed by all the events passed by the wave surface. A wavetrain (denoted by some variable X) can be regarded as a one-parameter family of such hypersurfaces in spacetime. This variable X is a scalar function of position in spacetime. The derivative of this scalar is a vector that characterizes the wave, the four-wavevector.
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
and choosing just to look at the component results in
where is the direction cosine of with respect to
Source moving away (redshift)
As an example, to apply this to a situation where the source is moving directly away from the observer (), this becomes:
Source moving towards (blueshift)
To apply this to a situation where the source is moving straight towards the observer (), this becomes:
^Physics definition example:Harris, Benenson, Stöcker (2002). Handbook of Physics. p. 288. ISBN 978-0-387-95269-7.CS1 maint: multiple names: authors list (link). Crystallography definition example: Vaĭnshteĭn (1994). Modern Crystallography. p. 259. ISBN 978-3-540-56558-1.
^Vaĭnshteĭn, Boris Konstantinovich (1994). Modern Crystallography. p. 259. ISBN 978-3-540-56558-1.
^Fowles, Grant (1968). Introduction to modern optics. Holt, Rinehart, and Winston. p. 177.
^"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
^Donald H. Menzel (1960). "§10.5 Bloch wave". Fundamental Formulas of Physics, Volume 2 (Reprint of Prentice-Hall 1955 2nd ed.). Courier-Dover. p. 624. ISBN 978-0486605968.
^Wolfgang Rindler (1991). "§24 Wave motion". Introduction to Special Relativity (2nd ed.). Oxford Science Publications. pp. 60–65. ISBN 978-0-19-853952-0.
Brau, Charles A. (2004). Modern Problems in Classical Electrodynamics. Oxford University Press. ISBN 978-0-19-514665-3.