Editing Phase velocity

{{Short description|Rate at which the phase of the wave propagates in space}}
[[Image:Wave group.gif|frame|[[Dispersion (water waves)|Frequency dispersion]] in groups of [[gravity wave]]s on the surface of deep water. The {{colorbull|#dd0000|square|size=150}} red square moves with the phase velocity, and the {{colorbull|#77ac30|circle|size=150}} green circles propagate with the [[group velocity]]. In this deep-water case, ''the phase velocity is twice the group velocity''. The red square overtakes two green circles when moving from the left to the right of the figure.{{paragraph}}
New waves seem to emerge at the back of a wave group, grow in amplitude until they are at the center of the group, and vanish at the wave group front.{{paragraph}}
For surface gravity waves, the water particle velocities are much smaller than the phase velocity, in most cases.]]
[[File:Wave packet propagation (phase faster than group, nondispersive).gif|thumb|Propagation of a [[wave packet]] demonstrating a phase velocity greater than the group velocity.]]

[[Image:Wave opposite-group-phase-velocity.gif|thumb|right|This shows a wave with the group velocity and phase velocity going in different directions.<ref name=nemirovsky2012negative>{{cite journal|last=Nemirovsky|first=Jonathan|author2=Rechtsman, Mikael C|author3=Segev, Mordechai|title=Negative radiation pressure and negative effective refractive index via dielectric birefringence|journal=Optics Express|date=9 April 2012|volume=20|issue=8|pages=8907–8914|doi=10.1364/OE.20.008907|bibcode=2012OExpr..20.8907N|pmid=22513601|doi-access=free}}</ref> The group velocity is positive (i.e., the [[Envelope (waves)|envelope]] of the wave moves rightward), while the phase velocity is negative (i.e., the peaks and troughs move leftward).]]

The '''phase velocity''' of a [[wave]] is the rate at which the wave [[Wave propagation|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 (physics)|crest]]) will appear to travel at the phase velocity. The phase velocity is given in terms of the [[wavelength]] {{mvar|λ}} (lambda) and [[Wave period|time period]] {{mvar|T}} as
:<math>v_\mathrm{p} = \frac{\lambda}{T}.</math>

Equivalently, in terms of the wave's [[angular frequency]] {{mvar|ω}}, which specifies angular change per unit of time, and [[wavenumber]] (or angular wave number) {{mvar|k}}, which represent the angular change per unit of space,

:<math>v_\mathrm{p} = \frac{\omega}{k}.</math>

To gain some basic intuition for this equation, we consider a propagating (cosine) wave {{math|''A'' cos(''kx'' − ''ωt'')}}. We want to see how fast a particular phase of the wave travels. For example, we can choose {{math|''kx'' - ''ωt'' {{=}} 0}}, the phase of the first crest. This implies {{math| ''kx'' {{=}} ω''t''}}, and so {{math| ''v'' {{=}} ''x'' / ''t'' {{=}} ''ω'' / ''k''}}.

Formally, we let the phase {{math|φ {{=}} ''kx'' - ''ωt''}} and see immediately that {{math| ω {{=}} -dφ / d''t''}} and {{math| ''k'' {{=}} dφ / d''x''}}. So, it immediately follows that

:<math> \frac{\partial x}{\partial t} = -\frac{ \partial \phi }{\partial t} \frac{\partial x}{\partial \phi} = \frac{\omega}{k}.</math>

As a result, we observe an inverse relation between the angular frequency and [[wavevector]]. If the wave has higher frequency oscillations, the [[wave vector|wavelength]] must be shortened for the phase velocity to remain constant.<ref name="mathpages1">{{cite web|url=http://www.mathpages.com/home/kmath210/kmath210.htm |title=Phase, Group, and Signal Velocity |publisher=Mathpages.com |access-date=2011-07-24}}</ref> Additionally, the phase velocity of [[electromagnetic radiation]] may – under certain circumstances (for example [[anomalous dispersion]]) – exceed the [[speed of light]] in vacuum, but this does not indicate any [[superluminal]] information or energy transfer.{{citation needed|date=April 2020}} It was theoretically described by physicists such as [[Arnold Sommerfeld]] and [[Léon Brillouin]].

The previous definition of phase velocity has been demonstrated for an isolated wave. However, such a definition can be extended to a beat of waves, or to a signal composed of multiple waves. For this it is necessary to mathematically write the beat or signal as a low frequency envelope multiplying a carrier. Thus the phase velocity of the carrier determines the phase velocity of the wave set.<ref name="electroagenda">{{cite web|url=https://electroagenda.com/en/phase-velocity-waves-and-signals/ |title=Phase Velocity: Waves and Signals |publisher=electroagenda.com }}</ref>

== Group velocity ==

[[File:Wavepacket1.gif|thumb|right|A superposition of 1D plane waves (blue) each traveling at a different phase velocity (traced by blue dots) results in a Gaussian wave packet (red) that propagates at the group velocity (traced by the red line).]]

The [[group velocity]] of a collection of waves is defined as

:<math> v_g = \frac{\partial \omega} {\partial k}.</math>

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 communication when [[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 {{mvar|f(x, t)}} with their respective angular frequencies and wavevectors.

:<math>\begin{align}
f(x, t) &= \cos(k_1 x - \omega_1 t) + \cos(k_2 x - \omega_2 t)\\
&= 2\cos\left(\frac{(k_2-k_1)x-(\omega_2-\omega_1)t}{2}\right)\cos\left(\frac{(k_2+k_1)x-(\omega_2+\omega_1)t}{2}\right)\\
&= 2f_1(x,t)f_2(x,t).
\end{align}</math>

So, we have a product of two waves: an envelope wave formed by {{math| ''f''<sub>1</sub> }} and a carrier wave formed by {{math| ''f''<sub>2</sub> }}. We call the velocity of the envelope wave the group velocity. We see that the '''phase velocity''' of {{math| ''f''<sub>1</sub> }} is  
:<math> \frac{\omega_2 - \omega_1}{k_2-k_1}.</math>
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 {{math|''ω''(''k'')}} 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''<sub>''p''</sub> is known as the [[refractive index]], {{math|''n'' {{=}} ''c'' / ''v''<sub>''p''</sub> {{=}} ''ck'' / ''ω''}}.

In this way, we can obtain another form for group velocity for electromagnetics. Writing {{math| ''n'' {{=}} ''n''(ω)}}, a quick way to derive this form is to observe
:<math> k = \frac{1}{c}\omega n(\omega) \implies dk = \frac{1}{c}\left(n(\omega) + \omega \frac{\partial}{\partial \omega}n(\omega)\right)d\omega.</math>
We can then rearrange the above to obtain 
:<math> v_g = \frac{\partial w}{\partial k} = \frac{c}{n+\omega\frac{\partial n}{\partial \omega}}.</math>

From this formula, we see that the group velocity is equal to the phase velocity only when the refractive index is independent of frequency <math display=inline>\partial n / \partial\omega = 0</math>. When this occurs, the medium is called non-dispersive, as opposed to [[dispersion (optics)|dispersive]], where various properties of the medium depend on the frequency {{mvar|ω}}. The relation <math>\omega(k)</math> is known as the [[dispersion relation]] of the medium.

== See also ==
{{Div col|colwidth=25em}}
*[[Cherenkov radiation]]
*[[Dispersion (optics)]]
*[[Group velocity]]
*[[Propagation delay]]
*[[Shear wave splitting]]
*[[Wave propagation]]
*[[Wave propagation speed]]
*[[Planck constant]]
*[[Speed of light]]
*[[Matter wave#Phase velocity]]
{{div col end}}

==References==

===Footnotes===
{{reflist}}

===Bibliography===
{{refbegin}}
*Crawford jr.,  Frank S. (1968).  ''Waves (Berkeley Physics Course, Vol. 3)'', McGraw-Hill,  {{ISBN|978-0070048607}} [https://archive.org/details/Waves_371 Free online version]
*{{citation | author-link=Léon Brillouin | last=Brillouin |first=Léon |title=Wave Propagation And Group Velocity |publisher=Academic Press Inc. |location=New York and London |year=1960 |isbn=978-0-12-134968-4 }}
*{{citation |last=Main |first=Iain G. |year=1988 |title=Vibrations and Waves in Physics |edition=2nd |location=New York |publisher=Cambridge University Press |isbn=978-0-521-27846-1 |pages=214–216 }}
*{{citation |last1=Tipler |first1=Paul A. |first2=Ralph A. |last2=Llewellyn |year=2003 |title=Modern Physics |edition=4th |location=New York |publisher=W. H. Freeman and Company |isbn=978-0-7167-4345-3 |pages=222–223 }}
{{refend}}

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[[Category:Wave mechanics]]