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In particular, the dispersion parameter D defined above is obtained from only one derivative of the group velocity. Higher derivatives are known as higher-order dispersion . [ 6 ] [ 7 ] These terms are simply a Taylor series expansion of the dispersion relation β ( ω ) of the medium or waveguide around some particular frequency.
In optics, group-velocity dispersion (GVD) is a characteristic of a dispersive medium, used most often to determine how the medium affects the duration of an optical pulse traveling through it. Formally, GVD is defined as the derivative of the inverse of group velocity of light in a material with respect to angular frequency , [ 1 ] [ 2 ]
Dispersion of waves on water was studied by Pierre-Simon Laplace in 1776. [ 7 ] The universality of the Kramers–Kronig relations (1926–27) became apparent with subsequent papers on the dispersion relation's connection to causality in the scattering theory of all types of waves and particles.
For common optical glasses, the refractive index calculated with the three-term Sellmeier equation deviates from the actual refractive index by less than 5×10 −6 over the wavelengths' range [5] of 365 nm to 2.3 μm, which is of the order of the homogeneity of a glass sample. [6]
In condensed matter physics, a Kohn anomaly (also called the Kohn effect [1]) is an anomaly in the dispersion relation of a phonon branch in a metal. For a specific wavevector, the frequency (and thus the energy) of the associated phonon is considerably lowered, and there is a discontinuity in its derivative. In extreme cases (that can happen ...
The wake has equation: [17] (,) = + / / (/) (/) / The wavefront itself also involves increasingly higher derivatives of the Dirac delta function. This means that a general Huygens' principle – the wave displacement at a point ( t , x ) {\displaystyle (t,x)} in spacetime depends only on the state at points on characteristic rays passing ( t ...
A third-order derivative term representing dispersion of wavenumbers are often encountered in many applications. The disperseively modified Kuramoto–Sivashinsky equation, which is often called as the Kawahara equation, [14] is given by [15] + + + + =
The convection–diffusion equation can be derived in a straightforward way [4] from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume: + =, where j is the total flux and R is a net ...