Phase Velocity
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The phase velocity of a
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
is the rate at which the wave propagates in any medium. This is the
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
at which the phase of any one
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
component of the wave travels. For such a component, any given phase of the wave (for example, the
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) 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 on the wave, such as two adjacent crests, tro ...
(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 In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
(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, tr ...
must be shortened for the phase velocity to remain constant. Additionally, the phase velocity of
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic field, electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, inf ...
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 The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in a vacuum, but this does not indicate any
superluminal Faster-than-light (also FTL, superluminal or supercausal) travel and communication are the conjectural propagation of matter or information faster than the speed of light (). The special theory of relativity implies that only particles with zero ...
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 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 ...
, 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 optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
, . 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 In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the d ...
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 is any of the ways in which waves travel. Single wave propagation can be calculated by 2nd order wave equation ( standing wavefield) or 1st order one-way wave equation. With respect to the direction of the oscillation relative to ...
*
Wave propagation speed The 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 fibr ...
*
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
*
Speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
* 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