physics Physics is the scientific study of matter, its Elementary particle, 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 whi ...

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

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, it is used to describe the velocity of a propagating front in the solution of hyperbolic partial differential equation.

Various velocities

Associated with propagation of a disturbance are several different velocities. For definiteness, consider an amplitude modulated electromagnetic carrier wave. The phase velocity is the speed of the underlying carrier wave. The group velocity is the 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 confusion by an apparent conflict with the

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

. 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. 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 exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...

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