physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...

, front velocity is the speed at which the first rise of a pulse above zero moves forward. In

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, it is used to describe the velocity of a propagating front in the solution of

hyperbolic partial differential equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be ...

Various velocities

Associated with propagation of a disturbance are several different velocities. For definiteness, consider an

amplitude modulated Amplitude modulation (AM) is a modulation technique used in electronic communication, most commonly for transmitting messages with a radio wave. In amplitude modulation, the amplitude (signal strength) of the wave is varied in proportion to t ...

electromagnetic carrier wave. The

phase velocity The phase velocity of a wave is the rate at which the wave 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, ...

is the speed of the underlying carrier wave. The

group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...

is the speed of the modulation or

envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...

. 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 confusion by an apparent conflict with the

theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...

. That observation led to consideration of what constitutes a ''signal''. By definition, a signal involves new information or an element of 'surprise' that cannot be predicted from the wave motion at an earlier time. One possible form for a signal (at the point of emission) is:

f(t) = u(t) \sin \omega t \, ,

where ''u''(''t'') is the

Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argum ...

. 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 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 fo ...

''c'' in any medium. In fact, the earliest appearance of the front of an electromagnetic disturbance (the

precursor Precursor or Precursors may refer to: * Precursor (religion), a forerunner, predecessor ** The Precursor, John the Baptist Science and technology * Precursor (bird), a hypothesized genus of fossil birds that was composed of fossilized parts of u ...

) travels at the ''front velocity'', which is ''c'', ''no matter what the medium''. However, the process always starts from zero amplitude and builds up.

References

Wave mechanics {{classicalmechanics-stub