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Phase velocity
 
 
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The phase velocity (or phase speed) of a wave
Wave

A wave is a disturbance that propagates through space and time, usually with transference of energy. While a mechanical wave exists in a medium , waves of electromagnetic radiation can travel through vacuum, that is, without a medium....
 is the rate at which the phase
Phase (waves)

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....
 of the wave propagates in space. This is the speed
Speed

Speed is the rate of Motion , or equivalently the rate of change of distance.Speed is a Scalar quantity with dimensions length/time; the equivalent Vector quantity to speed is velocity....
 at which the phase of any one frequency
Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency....
 component of the wave travels. 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
Wavelength

In physics, wavelength is the distance between repeating units of a propagating wave of a given frequency. It is commonly designated by the Greek language letter lambda ....
 ? (lambda) and period
Period

Period or periodic may refer to:Language and literature* Full stop, a punctuation mark indicating the end of a sentence or phrase...
 T as

Or, equivalently, in terms of the wave's angular frequency
Angular frequency

In physics , angular frequency ? is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity....
 ω and wavenumber
Wavenumber

Wavenumber in most physics sciences is a wave property inverse related to wavelength, having SI units of reciprocal metre . Wavenumber is the space analog of frequency, that is, it is the measurement of the number of repeating units of a propagating wave per unit of space....
 k by

In a dispersive
Dispersion (optics)

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency.Media having such a property are termed dispersive media....
 medium, the phase velocity varies with frequency and is not necessarily the same as the group velocity
Group velocity

The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes propagate through space. For example, imagine what happens if a stone is thrown into the middle of a very still pond....
 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
Electromagnetic radiation

Electromagnetic radiation takes the form of wave propagation waves in a vacuum or in matter. EM radiation has an electric field and magnetic field component which oscillate in phase perpendicular to each other and to the direction of energy Wave propagation....
 may under certain circumstances (e.g.






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The phase velocity (or phase speed) of a wave
Wave

A wave is a disturbance that propagates through space and time, usually with transference of energy. While a mechanical wave exists in a medium , waves of electromagnetic radiation can travel through vacuum, that is, without a medium....
 is the rate at which the phase
Phase (waves)

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....
 of the wave propagates in space. This is the speed
Speed

Speed is the rate of Motion , or equivalently the rate of change of distance.Speed is a Scalar quantity with dimensions length/time; the equivalent Vector quantity to speed is velocity....
 at which the phase of any one frequency
Frequency

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency....
 component of the wave travels. 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
Wavelength

In physics, wavelength is the distance between repeating units of a propagating wave of a given frequency. It is commonly designated by the Greek language letter lambda ....
 ? (lambda) and period
Period

Period or periodic may refer to:Language and literature* Full stop, a punctuation mark indicating the end of a sentence or phrase...
 T as

Or, equivalently, in terms of the wave's angular frequency
Angular frequency

In physics , angular frequency ? is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity....
 ω and wavenumber
Wavenumber

Wavenumber in most physics sciences is a wave property inverse related to wavelength, having SI units of reciprocal metre . Wavenumber is the space analog of frequency, that is, it is the measurement of the number of repeating units of a propagating wave per unit of space....
 k by

In a dispersive
Dispersion (optics)

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency.Media having such a property are termed dispersive media....
 medium, the phase velocity varies with frequency and is not necessarily the same as the group velocity
Group velocity

The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes propagate through space. For example, imagine what happens if a stone is thrown into the middle of a very still pond....
 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
Electromagnetic radiation

Electromagnetic radiation takes the form of wave propagation waves in a vacuum or in matter. EM radiation has an electric field and magnetic field component which oscillate in phase perpendicular to each other and to the direction of energy Wave propagation....
 may under certain circumstances (e.g. in the case of anomalous dispersion) exceed the speed of light
Speed of light

The speed of light in an free space is an important physical constant usually written as c, with a value of 299,792,458 metres per second....
 in a vacuum, but this does not indicate any superluminal information or energy transfer. It was theoretically described by physicists such as Arnold Sommerfeld
Arnold Sommerfeld

Arnold Johannes Wilhelm Sommerfeld was a Germany theoretical physicist who pioneered developments in atomic physics and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics....
 and Leon Brillouin
Léon Brillouin

L?on Nicolas Brillouin was a France physicist. He was born in S?vres , France. His father, Marcel Brillouin, grand-father, ?leuth?re Mascart, and great-grand-father, Charles Briot, were physicists as well....
. See dispersion
Dispersion (optics)

In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency.Media having such a property are termed dispersive media....
 for a full discussion of wave velocities.

Matter wave phase

In quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
, particles also behave as waves with complex
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
 phases. By the de Broglie hypothesis
De Broglie hypothesis

In physics, the matter wave, aka de Broglie wave , is the wave-like nature of all matter . The de Broglie relations show that the wavelength is inversely proportional to the momentum of a particle and that the frequency is directly proportional to the particle's kinetic energy....
, we see that

.

Using relativistic
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
 relations for energy and momentum, we have

where E is the total energy of the particle (i.e. rest energy plus kinetic energy
Kinetic energy

The kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the mechanical work needed to accelerate a body of a given mass from rest to its current velocity....
 in kinematic sense), p the momentum
Momentum

In classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section Momentum#Modern definitions of momentum on this page....
, the Lorentz factor
Lorentz factor

The Lorentz factor or Lorentz term appears in several equations in special relativity, including time dilation, length contraction, and the relativistic mass formula....
, c the speed of light
Speed of light

The speed of light in an free space is an important physical constant usually written as c, with a value of 299,792,458 metres per second....
, and ß 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
Group velocity

The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes propagate through space. For example, imagine what happens if a stone is thrown into the middle of a very still pond....
 for more detail. Since the particle velocity for a massive particle according to special relativity
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#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
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
 range. The superluminal phase velocity does not violate special relativity, for it doesn't carry any information. See the article on signal velocity
Signal velocity

The signal velocity is the speed at which a wave carries information. It describes how quickly a message can be communicated between two separated parties....
 for detail.

See also

  • Group velocity
    Group velocity

    The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes propagate through space. For example, imagine what happens if a stone is thrown into the middle of a very still pond....
  • Wave propagation
    Wave propagation

    Wave propagation is any of the ways in which wave s travel.With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves....
  • Dispersion (optics)
    Dispersion (optics)

    In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency.Media having such a property are termed dispersive media....


External links

  • , a Java applet
  • , a Java applet by Paul Falstad
  • - Java applet showing the difference between group and phase velocity.