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Encyclopedia > Phase velocity $v_mathrm{p} = frac{omega}{k}$

Note that the phase velocity is not necessarily the same as the group velocity of the wave, which is the rate that changes in amplitude (known as the envelope of the wave) will propagate. The group velocity of a wave is the velocity with which the variations in the shape of the waves amplitude (known as the modulation or envelope of the wave) propagate through space. ...

The phase velocity of electromagnetic radiation may under certain circumstances (e.g. in the case of anomalous dispersion) exceed the speed of light in a vacuum, but this does not indicate any superluminal information or energy transfer. It was theoretically described by physicists such as Arnold Sommerfeld and Leon Brillouin. See dispersion for a full discussion of wave velocities. Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ... Dispersion of a light beam in a prism. ... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... Faster-than-light (also superluminal or FTL) communications and travel are staples of the science fiction genre. ... Arnold Johannes Wilhelm Sommerfeld (December 5, 1868 in KÃ¶nigsberg, East Prussia â€“ April 26, 1951 in Munich, Germany) was a German physicist who introduced the fine-structure constant in 1919. ... Léon N. Brillouin ( August 7, 1889- 1969) was a French physicist. ... Dispersion of a light beam in a prism. ...

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In quantum mechanics, particles also behave as waves with complex phases. By the de Broglie hypothesis, we see that For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In physics, the de Broglie hypothesis is the statement that all matter (any object) has a wave-like nature (wave-particle duality). ... $v_mathrm{p} = frac{omega}{k} = frac{E/hbar}{p/hbar} = frac{E}{p}$.

Using relativistic relations for energy and momentum, we have Albert Einsteins theory of relativity is a set of two theories in physics: special relativity and general relativity. ... $v_mathrm{p} = frac{E}{p} = frac{gamma m c^2}{gamma m v} = frac{c^2}{v} = frac{c}{beta}$

where E is the total energy of the particle (i.e. rest energy plus kinetic energy in kinematic sense), p the momentum, γ the Lorentz factor, c the speed of light, and β the velocity as a fraction of c. The variable v can either be taken to be the velocity of the particle or the group velocity of the corresponding matter wave. See the article on group velocity for more detail. Since the particle velocity v < c for a massive particle according to special relativity, phase velocity of matter waves always exceed c, i.e. In classical physics, the total energy of an object is the sum of its potential energy and its kinetic energy. ... The term mass in special relativity is used in a couple of different ways, occasionally leading to a great deal of confusion. ... The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ... In physics, kinematics is the branch of mechanics concerned with the motions of objects without being concerned with the forces that cause the motion. ... This article is about momentum in physics. ... It has been suggested that Lorentz term be merged into this article or section. ... The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation, including visible light, in a vacuum. ... The group velocity of a wave is the velocity with which the variations in the shape of the waves amplitude (known as the modulation or envelope of the wave) propagate through space. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... $v_mathrm{p} > c ,$,

and as we can see, it approaches c when the particle velocity is in the relativistic range. The superluminal phase velocity does not violate special relativity, for it doesn't carry any information. See the article on signal velocity for detail. Albert Einsteins theory of relativity is a set of two theories in physics: special relativity and general relativity. ... Faster-than-light (also superluminal or FTL) communications and travel are staples of the science fiction genre. ... The signal velocity of a wave is the speed at which a pulse travels through a medium. ... Results from FactBites:

 Phase Velocity - Microwave Encyclopedia - Microwaves101.com (150 words) Phase velocity is the speed at which microwave energy travels along a transmission line. Velocity for a traveling wave is equal to the speed of light divided by the square root of the effective relative permittivity. Phase velocity can also be devived from the inductance and capacitance per unit length of a transmission line.
 Phase velocity - Wikipedia, the free encyclopedia (305 words) The phase velocity of a wave is the rate at which the phase of the wave propagates in space. Note that the phase velocity is not necessarily the same as the group velocity of the wave, which is the rate that changes in amplitude (known as the envelope of the wave) will propagate. The phase velocity of electromagnetic radiation may under certain circumstances exceed the speed of light in a vacuum, but this does not indicate any superluminal information or energy transfer.
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