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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
The linear dispersion relation – unaffected by wave amplitude – is for nonlinear waves also correct at the second order of the perturbation theory expansion, with the orders in terms of the wave steepness k a (where a is wave amplitude). To the third order, and for deep water, the dispersion relation is [19]
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.
Dispersion. Light waves of all frequencies travel at the same speed of light while matter wave velocity varies strongly with frequency. The relationship between frequency (proportional to energy) and wavenumber or velocity (proportional to momentum) is called a dispersion relation.
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.
We derive the ion acoustic wave dispersion relation for a linearized fluid description of a plasma with electrons and ion species. We write each quantity as X = X 0 + δ ⋅ X 1 {\displaystyle X=X_{0}+\delta \cdot X_{1}} where subscript 0 denotes the "zero-order" constant equilibrium value, and 1 denotes the first-order perturbation.
The dispersion relation can be written as an expression for the frequency (squared), but it is also common to write it as an expression for the index of refraction: = (). The full equation is typically given as follows: [4]
In mathematics, a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive. In this context, dispersion means that waves of different wavelength propagate at different phase velocities.