Information about Phase Velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave will propagate. You could pick one particular phase of the wave (for example the crest) and it would appear to travel at the phase velocity. The phase velocity is given in terms of the wave's angular frequency ω and wave vector k by
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 (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.
Using relativistic relations for energy and momentum, we have
where E is the total energy of the particle (i.e. rest energy plus kinetic energy in kinematic sense), p the momentum,
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 
for a massive particle according to special relativity, phase velocity of matter waves always exceed c, i.e.
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.
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 (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.
Matter wave phase
In quantum mechanics, particles also behave as waves with complex phases. By the de Broglie hypothesis, we see that
.
Using relativistic relations for energy and momentum, we have
where E is the total energy of the particle (i.e. rest energy plus kinetic energy in kinematic sense), p the momentum,
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 for a massive particle according to special relativity, phase velocity of matter waves always exceed c, i.e.
,
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.
External links
- Subluminal, a Java applet
- Group and Phase Velocity - Java applet showing the difference between group and phase velocity.
See also
References
- Tipler, Paul A. and Ralph A. Llewellyn (2003). Modern Physics. 4th ed. New York; W. H. Freeman and Company. ISBN 0-7167-4345-0. 222-3 pp.
- Leon Brillouin "Wave Propagation And Group Velocity" Academic Press Inc., New York and London (1960) ISBN 0-1213-4968-3.
- Main, Iain G. (1988).Vibrations and Waves in Physics. 2nd ed. New York; Cambridge University Press. ISBN 0-5212-7846-5. 214-6 pp.
Velocities of Waves ){kind=small}
|
|---|
| Phase velocity | Group velocity | Front velocity | Velocity of energy transfer | Signal velocity | Information velocity |
wave is a mode of energy transfer from one place to another, often with little or no permanent displacement of the particles of the medium (i.e. little or no associated mass transport); instead there are oscillations around almost fixed positions.
..... Click the link for more information.
..... Click the link for more information.
phase can be readily understood in terms of simple harmonic motion. The same concept applies to wave motion, viewed either at a point in space over an interval of time or across an interval of space at a moment in time.
..... Click the link for more information.
..... Click the link for more information.
angular frequency ω (also referred to by the terms angular speed, radial frequency, and radian frequency) is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity.
..... Click the link for more information.
..... Click the link for more information.
A wave vector is a vector that specifies the wavenumber and direction of propagation for a wave. The magnitude of the wave vector indicates the wavenumber. The orientation of the wave vector indicates the direction of wave propagation.
For example consider a plane wave.
..... Click the link for more information.
For example consider a plane wave.
..... Click the link for more information.
The group velocity of a wave is the velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope of the wave) propagate through space.
..... Click the link for more information.
..... Click the link for more information.
Electromagnetic (EM) radiation is a self-propagating wave in space with electric and magnetic components. These components oscillate at right angles to each other and to the direction of propagation, and are in phase with each other.
..... Click the link for more information.
..... Click the link for more information.
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".[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum.
..... Click the link for more information.
..... Click the link for more information.
Faster-than-light (also superluminal or FTL) communications and travel refer to the propagation of information or matter faster than the speed of light. "True" FTL, in which matter exceeds the speed of light in its own local region, is considered to be impossible by
..... Click the link for more information.
..... Click the link for more information.
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld
Born November 5 1868
Königsberg, Province of Prussia
..... Click the link for more information.
Arnold Johannes Wilhelm Sommerfeld
Born November 5 1868
Königsberg, Province of Prussia
..... Click the link for more information.
Léon Nicolas Brillouin (August 7,1889 – 1969) was a French physicist. He was born in Sèvres (near Paris), France. His father, Marcel Brillouin, was a physicist as well.
..... Click the link for more information.
..... Click the link for more information.
dispersion is the phenomenon that the phase velocity of a wave depends on its frequency.[1] In a prism, dispersion causes the spatial separation of a white light into spectral components of different wavelengths.
..... Click the link for more information.
..... Click the link for more information.
quantum mechanics is the study of the relationship between energy quanta (radiation) and matter, in particular that between valence shell electrons and photons. Quantum mechanics is a fundamental branch of physics with wide applications in both experimental and theoretical physics.
..... Click the link for more information.
..... Click the link for more information.
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 = −1.
..... Click the link for more information.
where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
..... Click the link for more information.
In physics, the de Broglie hypothesis (pronounced /brœj/, as French breuil, close to "broy") is the statement that all matter (any object) has a wave-like nature (wave-particle duality).
..... Click the link for more information.
..... Click the link for more information.
Relativity may refer to:
..... Click the link for more information.
Physics
- Special relativity, a theory of physics formulated by Albert Einstein
- General relativity, Einstein's theory of gravitation
..... Click the link for more information.
In classical physics, the total energy of an object is the sum of its potential energy and its kinetic energy. Note that since all other forms of energy can be derived from these two types, the total energy is effectively the theoretical maximum amount of energy that could be taken
..... Click the link for more information.
..... Click the link for more information.
invariant mass or intrinsic mass or proper mass or just mass is a measurement or calculation of the mass of an object that is the same for all frames of reference.
..... Click the link for more information.
..... Click the link for more information.
kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its current velocity.
..... Click the link for more information.
..... Click the link for more information.
Kinematics (Greek κινειν,kinein, to move) is a branch of mechanics which describes the motion of objects without the consideration of the masses or forces that bring about the motion.
..... Click the link for more information.
..... Click the link for more information.
momentum (pl. momenta; SI unit kg m/s, or, equivalently, N•s) is the product of the mass and velocity of an object. For more accurate measures of momentum, see the section "modern definitions of momentum" on this page.
..... Click the link for more information.
..... Click the link for more information.
The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula. Because of its ubiquity, physicists generally represent it with the shorthand symbol γ.
..... Click the link for more information.
..... Click the link for more information.
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".[1] It is the speed of all electromagnetic radiation, including visible light, in a vacuum.
..... Click the link for more information.
..... Click the link for more information.
The group velocity of a wave is the velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope of the wave) propagate through space.
..... Click the link for more information.
..... Click the link for more information.
special theory of relativity was proposed in 1905 by Albert Einstein in his article "On the Electrodynamics of Moving Bodies". Some three centuries earlier, Galileo's principle of relativity had stated that all uniform motion was relative, and that there was no absolute and
..... Click the link for more information.
..... Click the link for more information.
Relativity may refer to:
..... Click the link for more information.
Physics
- Special relativity, a theory of physics formulated by Albert Einstein
- General relativity, Einstein's theory of gravitation
..... Click the link for more information.
Faster-than-light (also superluminal or FTL) communications and travel refer to the propagation of information or matter faster than the speed of light. "True" FTL, in which matter exceeds the speed of light in its own local region, is considered to be impossible by
..... Click the link for more information.
..... Click the link for more information.
The signal velocity of a wave is the speed at which a pulse travels through a medium. The signal velocity is usually defined from the position of half-maximum intensity of the pulse.
..... Click the link for more information.
..... Click the link for more information.
The group velocity of a wave is the velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope of the wave) propagate through space.
..... Click the link for more information.
..... Click the link for more information.
Wave propagation is any of the ways in which waves travel through a medium (waveguide).
With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves.
..... Click the link for more information.
With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves.
..... Click the link for more information.
Léon Nicolas Brillouin (August 7,1889 – 1969) was a French physicist. He was born in Sèvres (near Paris), France. His father, Marcel Brillouin, was a physicist as well.
..... Click the link for more information.
..... Click the link for more information.
This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.
Herod_Archelaus
