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The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one 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 velocity is given in terms of the wavelength (lambda) and
time period The categorisation of the past into discrete, quantified named blocks of time is called periodization.Adam Rabinowitz. And kingIt’s about time: historical periodization and Linked Ancient World Data'. Study of the Ancient universe Papers, 2014 ...
as :v_\mathrm = \frac. Equivalently, in terms of the wave's
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
, which specifies angular change per unit of time, and wavenumber (or angular wave number) , which represent the angular change per unit of space, :v_\mathrm = \frac. To gain some basic intuition for this equation, we consider a propagating (cosine) wave . We want to see how fast a particular phase of the wave travels. For example, we can choose , the phase of the first crest. This implies , and so . Formally, we let the phase and see immediately that and . So, it immediately follows that : \frac = -\frac \frac = \frac. As a result we observe a inverse relation between the angular frequency and wavevector. If the wave has higher frequency oscillations, the
wave length 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 on the wave, such as two adjacent crests, tro ...
must be shortened for the phase velocity to remain constant. Additionally, the phase velocity of electromagnetic radiation may – under certain circumstances (for example
anomalous dispersion In optics, and by analogy other branches of physics dealing with wave propagation, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency; sometimes the term chromatic dispersion is used for specificity to o ...
) – 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 Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
and
Léon Brillouin Léon Nicolas Brillouin (; August 7, 1889 – October 4, 1969) was a French physicist. He made contributions to quantum mechanics, radio wave propagation in the atmosphere, solid state physics, and information theory. Early life Brillouin ...
.


Group velocity

The group velocity of a collection of waves is defined as : v_g = \frac . When multiple sinusoidal waves are propagating together, the resultant superposition of the waves can result in an "envelope" wave as well as a "carrier" wave that lies inside the envelope. This commonly appears in wireless communications, modulation, a change in amplitude and/or phase is employed to send data. To gain some intuition for this definition, we consider a superposition of (cosine) waves with their respective angular frequencies and wavevectors. :\begin f(x, t) &= \cos(k_1 x - \omega_1 t) + \cos(k_2 x - \omega_2 t)\\ &= 2\cos\left(\frac\right)\cos\left(\frac\right)\\ &= 2f_1(x,t)f_2(x,t). \end So, we have a product of two waves: an envelope wave formed by and a carrier wave formed by . We call the velocity of the envelope wave the group velocity. We see that the phase velocity of is : \frac. In the continuous differential case, this becomes the definition of the group velocity.


Refractive index

In the context of electromagnetics and optics, the frequency is some function of the wave number, so in general, the phase velocity and the group velocity depend on specific medium and frequency. The ratio between the speed of light ''c'' and the phase velocity ''v''''p'' is known as the refractive index, . In this way, we can obtain another form for group velocity for electromagnetics. Writing , a quick way to derive this form is to observe : k = \frac\omega n(\omega) \implies dk = \frac\left(n(\omega) + \omega \fracn(\omega)\right)d\omega. We can then rearrange the above to obtain : v_g = \frac = \frac. From this formula, we see that the group velocity is equal to the phase velocity only when the refractive index is a constant . When this occurs, the medium is called non-dispersive, as opposed to dispersive, where various properties of the medium depend on the frequency . The relation is known as the dispersion relation of the medium.


See also

* Cherenkov radiation *
Dispersion (optics) In optics, and by analogy other branches of physics dealing with wave propagation, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency; sometimes the term chromatic dispersion is used for specificity to o ...
* Group velocity *
Propagation delay Propagation delay is the time duration taken for a signal to reach its destination. It can relate to networking, electronics or physics. ''Hold time'' is the minimum interval required for the logic level to remain on the input after triggering e ...
*
Shear wave splitting Shear wave splitting, also called seismic birefringence, is the phenomenon that occurs when a Polarization (waves), polarized shear wave enters an anisotropic medium (Fig. 1). The incident shear wave splits into two polarized shear waves (Fig. 2). ...
* Wave propagation * Wave propagation speed * Planck constant * Speed of light * Matter wave#Phase velocity


References


Footnotes


Bibliography

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