Information about Wave Vector

A wave vector is a vector that specifies the wavenumber and direction of propagation for a wave. The magnitude of the wave vector indicates the wavenumber. The orientation of the wave vector indicates the direction of wave propagation.

For example consider a plane wave. A common representation of the oscillation at time (t) and a single point in space (z) along the direction of propagation is:

where A is the amplitude, φ is the starting phase of the wave, k is the angular wavenumber, and ω is the angular frequency. We can easily extend the formula by substituting the dot product of the wave vector k and the position vector r for the scalar product of the wavenumber k and the variable z as follows:

.

In special relativity

A wave packet of nearly monochromatic light can be characterized by the wave vector

:

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

: and

:

The magnitude of this wave vector is then

:

::::

That last step where it equals zero, is a result of the fact that, for light,

:

Lorentz transform

Taking the Lorentz transform 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 transform 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

:

and choosing just to look at the component results in

:

: So

:

Source moving away

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

:

Source moving towards

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

:

References

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

spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector can be thought of as an arrow in Euclidean space, drawn from an initial point A pointing to a terminal point B.
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Wavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters (m⁻¹). Wavenumber is the spatial analogue of frequency.
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Wavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters (m⁻¹). Wavenumber is the spatial analogue of frequency.
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Wavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters (m⁻¹). Wavenumber is the spatial analogue of frequency.
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angular frequency ω (also referred to by the terms angular speed, radial frequency, and radian frequency) is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity.
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dot product, also known as the scalar product, is an operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. It is the standard inner product of the Euclidean space.
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Wavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters (m⁻¹). Wavenumber is the spatial analogue of frequency.
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Monochrome comes from the two Greek words mono (μoνο, meaning "only" or "alone"), and chroma (χρωμα, meaning "colour"). A monochromatic object has a single colour.
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Covariance and contravariance may refer to:

• Covariance and contravariance of vectors, in mathematics and theoretical physics
• Covariance and contravariance of functors, in category theory

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Covariance and contravariance may refer to:

• Covariance and contravariance of vectors, in mathematics and theoretical physics
• Covariance and contravariance of functors, in category theory

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In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other.
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relativistic Doppler effect is the change in frequency (and wavelength) of light, caused by the relative motion of the source and the observer (as in the classical Doppler effect), when taking into account effects of the special theory of relativity.
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