Editing Dispersion relation (section)

==Dispersion==
{{main|Dispersion (optics)|Dispersion (water waves)|Acoustic dispersion}}
Dispersion occurs when sinusoidal waves of different wavelengths have different propagation velocities, so that a [[wave packet]] of mixed wavelengths tends to spread out in space. The speed of a plane wave, <math>v</math>, is a function of the wave's wavelength <math>\lambda</math>:

:<math>v = v(\lambda).</math>

The wave's speed, wavelength, and frequency, ''f'', are related by the identity

:<math>v(\lambda) = \lambda\ f(\lambda).</math>

The function <math> f(\lambda)</math> expresses the dispersion relation of the given medium. Dispersion relations are more commonly expressed in terms of the [[angular frequency]] <math>\omega=2\pi f</math> and [[wavenumber]] <math>k=2 \pi /\lambda</math>. Rewriting the relation above in these variables gives

:<math>\omega(k)= v(k) \cdot k.</math>

where we now view ''f'' as a function of ''k''. The use of ''ω''(''k'') to describe the dispersion relation has become standard because both the [[phase velocity]] ''ω''/''k'' and the [[group velocity]] ''dω''/''dk'' have convenient representations via this function.

The plane waves being considered can be described by

:<math>A(x, t) = A_0e^{2 \pi i \frac{x - v t}{\lambda}}= A_0e^{i (k x - \omega t)},</math>

where 
*''A'' is the amplitude of the wave,
*''A''<sub>0</sub> = ''A''(0, 0),
*''x'' is a position along the wave's direction of travel, and
*''t'' is the time at which the wave is described.