Front velocity
Encyclopedia
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
.
electromagnetic carrier wave. The phase velocity
is the wave speed of the carrier. The group velocity
is the wave speed of the modulation or envelope
. Initially it was thought that the group velocity coincided with the speed at which information traveled. However, it turns out that this speed can exceed the speed of light in some circumstances, causing a conflict with the theory of relativity
. That observation led to consideration of what constitutes a signal.
A signal involves new information or an element of 'surprise' that cannot be predicted from the wave motion at an earlier time. Thus, a possible form for a signal is:
where u(t) is the Heaviside step function
. Using such a form for a signal, it can be shown, subject to the (expected) condition that the refractive index of any medium tends to one as the frequency tends to infinity, that the wave discontinuity, called the front, propagates at a speed less than or equal to the speed of light
c in any medium. In fact, the earliest appearance of the front travels at the front velocity c, no matter what the medium. However, the process always starts from zero amplitude and builds up.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, front velocity is the speed at which the first rise of a pulse above zero moves forward.
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, it is also used to describe the velocity of a possibly propagating front in the solution of hyperbolic partial differential equation
Hyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation that, roughly speaking, has a well-posed initial value problem for the first n−1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along...
.
Various velocities
Associated with propagation of a disturbance are several different velocities. For definiteness, consider an amplitude modulatedAmplitude modulation
Amplitude modulation is a technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. AM works by varying the strength of the transmitted signal in relation to the information being sent...
electromagnetic carrier wave. The phase velocity
Phase velocity
The phase velocity 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 will appear to travel at the phase velocity...
is the wave speed of the carrier. The group velocity
Group velocity
The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....
is the wave speed of the modulation or envelope
Envelope detector
An envelope detector is an electronic circuit that takes a high-frequency signal as input and provides an output which is the "envelope" of the original signal. The capacitor in the circuit stores up charge on the rising edge, and releases it slowly through the resistor when the signal falls...
. Initially it was thought that the group velocity coincided with the speed at which information traveled. However, it turns out that this speed can exceed the speed of light in some circumstances, causing a conflict with the theory of relativity
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
. That observation led to consideration of what constitutes a signal.
A signal involves new information or an element of 'surprise' that cannot be predicted from the wave motion at an earlier time. Thus, a possible form for a signal is:
where u(t) is the Heaviside step function
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by H , is a discontinuous function whose value is zero for negative argument and one for positive argument....
. Using such a form for a signal, it can be shown, subject to the (expected) condition that the refractive index of any medium tends to one as the frequency tends to infinity, that the wave discontinuity, called the front, propagates at a speed less than or equal to the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
c in any medium. In fact, the earliest appearance of the front travels at the front velocity c, no matter what the medium. However, the process always starts from zero amplitude and builds up.