Wave Vector

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:
:

\psi \left(t , z
ight) = A \cos \left(arphi + k z + \omega t
ight)

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:
:

\psi \left(t , {\mathbf r} 
ight) = A \cos \left(arphi + {\mathbf k} \cdot {\mathbf r} + \omega t
ight)

.

IN SPECIAL RELATIVITY

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

k^\mu = \left(rac{\omega}{c}, ec{k} 
ight) \,

which, when written out explicitly in its Covariant and Contravariant forms is
::

k^\mu = \left(rac{\omega}{c}, k^1, k^2, k^3 
ight)\,

and
::

k_\mu = \left(rac{\omega}{c}, -k_1, -k_2, -k_3 
ight) . \,

The magnitude of this wave vector is then
::

k^2 = k^\mu k_\mu = k^0 k_0 - k^1 k_1 - k^2 k_2 - k^3 k_3 \,

:::::

=rac{\omega^2}{c^2} - ec{k}^2 = 0. \,

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

k = rac{\omega}{c}. \,

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
::

\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 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
::

k^{\mu}_s = \Lambda^\mu_
u k^
u_{\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}} \,

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Source moving towards

To apply this to a situation where the source is moving straight towards the observer (

heta=0

), this becomes:
::

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

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Information About

Wave Vector