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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 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 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.
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 Sellmeier equation is an empirical relationship between refractive index and wavelength for a particular transparent medium. The equation is used to determine the dispersion of light in the medium. It was first proposed in 1872 by Wolfgang Sellmeier and was a development of the work of Augustin Cauchy on Cauchy's equation for modelling ...
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.
This equation was studied in Benjamin, Bona, and Mahony as an improvement of the Korteweg–de Vries equation (KdV equation) for modeling long surface gravity waves of small amplitude – propagating uni-directionally in 1+1 dimensions. They show the stability and uniqueness of solutions to the BBM equation.
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 .