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The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics.
The group velocity is the rate at which the wave envelope, i.e. the changes in amplitude, propagates. The wave envelope is the profile of the wave amplitudes; all transverse displacements are bound by the envelope profile.
Different fields of application have different definitions for the term. All the meanings are very similar in concept: In chemistry, the transmission coefficient refers to a chemical reaction overcoming a potential barrier; in optics and telecommunications it is the amplitude of a wave transmitted through a medium or conductor to that of the incident wave; in quantum mechanics it is used to ...
The units of the amplitude depend on the type of wave, but are always in the same units as the oscillating variable. A more general representation of the wave equation is more complex, but the role of amplitude remains analogous to this simple case. For waves on a string, or in a medium such as water, the amplitude is a displacement.
For a monochromatic propagating electromagnetic wave, such as a plane wave or a Gaussian beam, if E is the complex amplitude of the electric field, then the time-averaged energy density of the wave, travelling in a non-magnetic material, is given by: = | |, and the local intensity is obtained by multiplying this expression by the wave velocity
In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. [1] The envelope thus generalizes the concept of a constant amplitude into an instantaneous amplitude. The figure illustrates a modulated sine wave varying between an upper envelope and a lower envelope. The envelope function may be a ...
The wave vector and angular wave vector are related by a fixed constant of proportionality, 2 π radians per cycle. It is common in several fields of physics to refer to the angular wave vector simply as the wave vector, in contrast to, for example, crystallography. [1] [2] It is also common to use the symbol k for whichever is in use.
It turns out that the original relativistic wave equations and their solutions are still needed to build the Hilbert space. Moreover, the free fields operators, i.e. when interactions are assumed not to exist, turn out to (formally) satisfy the same equation as do the fields (wave functions) in many cases.