The phase velocity of a

_{p} is known as the

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wave
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

is the rate at which the wave propagates in any medium. This is the velocity
The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

at which the phase of any one frequency
Frequency is the number of occurrences of a repeating event per unit of time
A unit of time is any particular time
Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparen ...

component of the wave travels. For such a component, any given phase of the wave (for example, the crest
Crest or CREST may refer to:
Buildings
*The Crest (Huntington, New York)
The Crest is a historic house on Eatons Neck in Suffolk County, New York. Although on the land mass of Eatons Neck, the house today is within the jurisdiction of the Incor ...

) will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase (waves), phase on the wave, such as two adja ...

(lambda) and time period
The categorization of the past into discrete, quantified named blocks of time is called periodization
Periodization is the process or study of categorizing the past into discrete, quantified named blocks of time.Adam Rabinowitz. It’s about ...

as
:$v\_\backslash mathrm\; =\; \backslash frac.$
Equivalently, in terms of the wave's angular frequency
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...

, which specifies angular change per unit of time, and wavenumber
In the physical science
Physical science is a branch of natural science that studies non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the "physical scienc ...

(or angular wave number) , which represents the proportionality between the angular frequency and the linear speed (speed of propagation) ,
:$v\_\backslash mathrm\; =\; \backslash frac.$
To understand where this equation comes from, considering a basic cosine wave
A sine wave or sinusoid is any of certain curve, mathematical curves that describe a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph of a function, graph. It occurs o ...

, . After time , the source has produced oscillations. After the same time, the initial wave front has propagated away from the source through space to the distance to fit the same number of oscillations, .
Thus the propagation velocity ''v'' is . The wave would have to propagate faster when higher frequency oscillations are distributed less densely in space unless the wave length
In , the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same on the wave, such as two adjacent crests, troughs, or s, ...

is compensatorily shortened. Formally, is the phase, where
:$\backslash frac\; =\; -\backslash frac.$
Since and , the wave velocity is .
Relation to group velocity, refractive index and transmission speed

Since a pure sine wave cannot convey any information, some change in amplitude or frequency, known asmodulation
In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the ''carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informatio ...

, is required. By combining two sines with slightly different frequencies and wavelengths,
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the amplitude becomes a sinusoid with ''phase'' speed . It is this modulation that represents the signal content. Since each amplitude ''envelope'' contains a group of internal waves, this speed is usually called the group velocity
in groups of gravity waves on the surface of deep water. The red square moves with the phase velocity, and the green circles propagate with the group velocity. In this deep-water case, ''the phase velocity is twice ...

, .
In a given medium, the frequency is some function of the wave number, so in general, the phase velocity and the group velocity depend on the frequency and on the medium. The ratio between the speed of light ''c'' and the phase velocity ''v''refractive index
In optics, the refractive index (also known as refraction index or index of refraction) of a optical medium, material is a dimensionless number that describes how fast EM radiation, light travels through the material. It is defined as
:n = \frac ...

, .
Taking the derivative of with respect to , would yield the group velocity
in groups of gravity waves on the surface of deep water. The red square moves with the phase velocity, and the green circles propagate with the group velocity. In this deep-water case, ''the phase velocity is twice ...

,
:$\backslash frac\; =\; \backslash frac\; -\; \backslash frac\backslash cdot\backslash frac~.$
except one cannot create a group with only a finite number of wave frequencies/wave vectors. (That is: the envelope in such a situation changes shape so rapidly that group velocity loses its meaning.) Noting that , indicates that the group speed is equal to the phase speed only when the refractive index is a constant , and in this case the phase speed and group speed are independent of frequency, .
Otherwise, both the phase velocity and the group velocity vary with frequency, and the medium is called dispersive; the relation is known as the dispersion relation
causes different colors to refract at different angles, splitting white light into a rainbow of colors.
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a ...

of the medium.
The phase velocity of electromagnetic radiation
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...

may – under certain circumstances (for example anomalous dispersion
A compact fluorescent lamp seen through an Amici prism ">Amici_prism.html" ;"title="compact fluorescent lamp seen through an Amici prism">compact fluorescent lamp seen through an Amici prism
In optics, dispersion is the phenomenon in which the ...

) – exceed the speed of light
The speed of light in vacuum
A vacuum is a space
Space is the boundless three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...

in a vacuum, but this does not indicate any superluminal
Faster-than-light (also superluminal, FTL or supercausal) communications
Communication (from Latin ''communicare'', meaning "to share") is the act of developing meaning among entities or groups through the use of sufficiently mutually underst ...

information or energy transfer. It was theoretically described by physicists such as Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German people, German theoretical physicist who pioneered developments in atomic physics, atomic and quantum physics, and also educated and mentored many students f ...

and Léon Brillouin.
See also

* Cherenkov radiation *Dispersion (optics)
In optics
Optics is the branch of that studies the behaviour and properties of , including its interactions with and the construction of that use or it. Optics usually describes the behaviour of , , and light. Because light is an , ot ...

*Group velocity
in groups of gravity waves on the surface of deep water. The red square moves with the phase velocity, and the green circles propagate with the group velocity. In this deep-water case, ''the phase velocity is twice ...

*Propagation delay
Propagation delay is the length of time taken for a signal to reach its destination. It can relate to networking, electronics
Electronics comprises the physics, engineering, technology and applications that deal with the emission, flow and con ...

* Shear wave splitting
*Wave propagation
Wave propagation is any of the ways in which wave
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, ...

*Wave propagation speedThe velocity factor (VF), also called wave propagation speed or velocity of propagation (VoP or of a transmission medium is the ratio of the speed at which a wavefront (of an electromagnetic signal, a radio signal, a light pulse in an optical fibre ...

*Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant
A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature an ...

*Speed of light
The speed of light in vacuum
A vacuum is a space
Space is the boundless three-dimensional
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...

*
References

Footnotes

Bibliography

*Crawford jr., Frank S. (1968). ''Waves (Berkeley Physics Course, Vol. 3)'', McGraw-Hill,Free online version

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