Phase velocity in periodic gravity waves on the surface of deep water. In Mathematics, a periodic function is a function that repeats its values after some definite period has been added to its Independent variable Ocean surface waves are Surface waves that occur on the Free surface of the Ocean. The red dot moves with the phase velocity, and is located at a fixed wave phase: the crest for the case shown. A crest is the point on a Wave with the greatest positive value or upward displacement in a Cycle.

The phase velocity (or phase speed) of a wave is the rate at which the phase of the wave propagates in space. A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 This is the speed at which the phase of any one frequency component of the wave travels. Speed is the rate of motion, or equivalently the rate of change in position often expressed as Distance d traveled per unit of Frequency is a measure of the number of occurrences of a repeating event per unit Time. For such a component, any given phase of the wave (for example, the crest) will appear to travel at the phase velocity. The phase speed is given in terms of the wavelength λ (lambda) and period T as

Or, equivalently, in terms of the wave's angular frequency ω and wavenumber k by

In a dispersive medium, the phase velocity varies with frequency and 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. In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. Do not confuse with Angular velocity In Physics (specifically Mechanics and Electrical engineering) angular frequency Wavenumber in most physical sciences is a Wave property inversely related to Wavelength, having SI units of reciprocal meters In Optics, dispersion is the phenomenon in which the Phase velocity of a wave depends on its frequency 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

The phase velocity of electromagnetic radiation may under certain circumstances (e. Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. 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. In Optics, dispersion is the phenomenon in which the Phase velocity of a wave depends on its frequency It was theoretically described by physicists such as Arnold Sommerfeld and Leon Brillouin. Arnold Johannes Wilhelm Sommerfeld (5 December 1868 &ndash 26 April 1951 was a German theoretical Physicist who pioneered developments in atomic Léon Nicolas Brillouin ( August 7, 1889 &ndash December 1969 was a French physicist See dispersion for a full discussion of wave velocities. In Optics, dispersion is the phenomenon in which the Phase velocity of a wave depends on its frequency

Matter wave phase

In quantum mechanics, particles also behave as waves with complex phases. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted By the de Broglie hypothesis, we see that

. 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

Using relativistic relations for energy and momentum, we have

where E is the total energy of the particle (i. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In Classical physics, the total energy of an object is the sum of its Potential energy and its Kinetic energy. 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 kinetic energy of an object is the extra Energy which it possesses due to its motion Kinematics ( Greek κινειν, kinein, to move is a branch of Classical mechanics which describes the motion of objects without In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product The Lorentz factor or Lorentz term appears in several equations in Special relativity, including Time dilation, Length contraction, and the 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. 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 Since the particle velocity v < c for a massive particle according to special relativity, phase velocity of matter waves always exceed c, i. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial e.

,

and as we can see, it approaches c when the particle velocity is in the relativistic range. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial The superluminal phase velocity does not violate special relativity, for it doesn't carry any information. See the article on signal velocity for detail. The signal velocity is the speed at which a Wave carries information

See also

• Group velocity
• Wave propagation

References

External links

• Subluminal, a Java applet
• Group and Phase Velocity - Java applet showing the difference between group and phase velocity.

Velocities of Waves
Phase velocity | Group velocity | Front velocity | Velocity of energy transfer | Signal velocity | Information velocity