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Rewriting the relation above in these variables gives = (). 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
In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating total energy (which is also called relativistic energy) to invariant mass (which is also called rest mass) and momentum. It is the extension of mass–energy equivalence for bodies or systems with non-zero momentum.
Each propagates at generally different speeds determined by the important function called the dispersion relation. The use of the explicit form ω ( k ) is standard, since the phase velocity ω / k and the group velocity d ω /d k usually have convenient representations by this function.
The relationship between frequency (proportional to energy) and wavenumber or velocity (proportional to momentum) is called a dispersion relation. Light waves in a vacuum have linear dispersion relation between frequency: ω = c k {\displaystyle \omega =ck} .
A nonlinear dispersion relation (NDR) is a dispersion relation that assigns the correct phase velocity to a nonlinear wave structure. As an example of how diverse and intricate the underlying description can be, we deal with plane electrostatic wave structures ϕ ( x − v 0 t ) {\displaystyle \phi (x-v_{0}t)} which propagate with v 0 ...
The function ω(k), which gives ω as a function of k, is known as the dispersion relation. If ω is directly proportional to k, then the group velocity is exactly equal to the phase velocity. A wave of any shape will travel undistorted at this velocity.
Dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency. [1] Sometimes the term chromatic dispersion is used to refer to optics specifically, as opposed to wave propagation in general. A medium having this common property may be termed a dispersive medium.
To gain some basic intuition for this equation, we consider a propagating (cosine) wave A cos(kx − ωt). We want to see how fast a particular phase of the wave travels. For example, we can choose kx - ωt = 0, the phase of the first crest. This implies kx = ωt, and so v = x / t = ω / k.