Editing Phase velocity (section)

{{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>