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Phase velocity
Phase velocity in periodic gravity waves on the surface of deep water. The red dot moves with the phase
velocity, and is located at a fixed wave phase: the crest for the case shown.
The phase velocity (or phase speed) of a wave is the rate at which the phase of the wave propagates in space.
This is the speed 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 speed is given in terms of the wavelength λ (lambda) and period T as
vP 
λ
T
Or, equivalently, in terms of the wave's angular frequency ω and wavenumber k by
vP 
ω
k
In a dispersive medium, the phase velocity varies with frequency and is not necessarily the same as the group
velocity of the wave, which is the rate that changes in amplitude (known as the envelope of the wave) will
propagate.
The phase velocity of electromagnetic radiation may under certain circumstances (e.g. in the case of anomalous
dispersion) 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 and Leon Brillouin.
See dispersion for a full discussion of wave velocities.
Matter wave phase
In quantum mechanics, particles also behave as waves with complex phases. By the de Broglie hypothesis, we
see that
vP 
E  E

p  p
Using relativistic relations for energy and momentum, we have
E γ mc2 c2 c
vP  


p
γ mv
v
β
where E is the total energy of the particle (i.e. rest energy plus kinetic energy in kinematic sense), p the
momentum, γ the Lorentz factor, c the speed of light, and β the velocity as a fraction of c. The variable v can
either be taken to be the velocity of the particle or the group velocity of the corresponding matter wave. See the
article on group velocity for more detail. Since the particle velocity v < c for a massive particle according to
special relativity, phase velocity of matter waves always exceed c, i.e.
vP  c
and as we can see, it approaches c when the particle velocity is in the relativistic range. The superluminal phase
velocity does not violate special relativity, for it doesn't carry any information. See the article on signal velocity
for detail.
Group velocity
Frequency dispersion in bichromatic groups of gravity waves on the surface of deep water. The red dot moves
with the phase velocity, and the green dots propagate with the group velocity. In this deep-water case, the
phase velocity is twice the group velocity. The red dot overtakes two green dots, when moving from the left to
the right of the figure.
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.
For gravity surface-waves, the water particle velocities are much smaller than the phase velocity, in most cases.
Albert Einstein first explained the wave-particle duality of light in 1905. Louis de Broglie hypothesized that
any particle should also exhibit such a duality. The velocity of a particle, he concluded then (but may be
questioned today, see above), should always equal the group velocity of the corresponding wave. De Broglie
deduced that if the duality equations already known for light were the same for any particle, then his hypothesis
would hold. This means that
vg 
d d ( E ) dE


dk d ( p ) dp
where
E is the total energy of the particle,
p is its momentum,
is Planck's constant.
For a free non-relativistic particle it follows that
dE d  p 2  p

 v
vg 

dp dp  2m  m
where
m is the mass of the particle and
and v its velocity.
Also in special relativity we find that
dE d
vg 

dp dp
 (p c
2
2
m c
2
4
  ( p c
pc 2
2
2
 m2c 4
 m

p
( p (mc)) 2  1


p
mv

v
m
m
where
m is the rest mass of the particle,
c is the speed of light in a vacuum,
γ is the Lorentz factor.
and v is the velocity of the particle regardless of wave behavior.
As a result we can, both in relativistic and non-relativistic quantum physics, identify the group velocity of a
particle's wave function with the particle velocity.
Quantum mechanics has very accurately demonstrated this hypothesis, and the relation has been shown
explicitly for particles as large as molecules.
Front velocity
In physics, Front velocity is the speed at which the first rise of a pulse above zero moves forward.
In mathematics, it is also used to describe the velocity of a possibly propagating front in the solution of
hyperbolic partial differential equation.
Signal velocity
The signal velocity is the speed at which a wave carries information. It describes how quickly a message can be
communicated (using any particular method) between two separated parties. Every signal velocity is always
slower than (or equal to) the speed of a light pulse in a vacuum (by Special Relativity).
In the vast majority of cases, signal velocity is equal to group velocity (the speed of a short "pulse" or of a
wave-packet's middle or "envelope"). However, in a few special cases (e.g., media designed to amplify the
front-most parts of a pulse and then attenuate the back section of the pulse), group velocity can exceed even the
vacuum speed of light.
Signal velocity is always[citation needed] the same as front velocity.
For light (electromagnetic waves) in vacuum, the signal velocity is the same as the group velocity and the phase
velocity (the speed of any individual crest of the wave).
In electronic circuits, signal velocity is one member of a group of five closely related parameters. In these
circuits, signals are usually treated as operating in TEM mode. That is, the fields are perpendicular to the
direction of transmission and perpendicular to each other. Given this presumption, the quantities: signal
velocity, the product of dielectric constant and magnetic permeability, characteristic impedance, inductance of a
structure, and capacitance of that structure, are all related such that if you know any two, you can calculate the
rest. In a uniform medium if the permeability is constant, then variation of the signal velocity will be dependent
only on variation of the dielectric constant.
In a transmission line, signal velocity is the reciprocal of the square root of the capacitance-inductance product,
where inductance and capacitance are typically expressed as per-unit length. In circuit boards made of FR4
material, the signal velocity is typically about six inches per nanosecond. In these boards, permeability is
usually constant and dielectric constant often varies from location to location, causing variations in signal
velocity. As data rates increase, these variations become a major concern for computer manufacturers.