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While periodic travelling waves have been known as solutions of the wave equation since the 18th century, their study in nonlinear systems began in the 1970s. A key early research paper was that of Nancy Kopell and Lou Howard [1] which proved several fundamental results on periodic travelling waves in reaction–diffusion equations.
For an incident wave traveling from one medium (where the wave speed is c 1) to another medium (where the wave speed is c 2), one part of the wave will transmit into the second medium, while another part reflects back into the other direction and stays in the first medium. The amplitude of the transmitted wave and the reflected wave can be ...
Periodic waves oscillate repeatedly about an equilibrium (resting) value at some frequency. When the entire waveform moves in one direction, it is said to be a travelling wave; by contrast, a pair of superimposed periodic waves traveling in opposite directions makes a standing wave. In a standing wave, the amplitude of vibration has nulls at ...
The wave envelope is the profile of the wave amplitudes; all transverse displacements are bound by the envelope profile. Intuitively the wave envelope is the "global profile" of the wave, which "contains" changing "local profiles inside the global profile".
These solutions move with constant speed without changing their shape and are of the form u(x, t) = û(ξ) with ξ = x − ct, where c is the speed of the travelling wave. Note that while travelling waves are generically stable structures, all non-monotonous stationary solutions (e.g. localized domains composed of a front-antifront pair) are ...
A looped animation of a wave packet propagating without dispersion: the envelope is maintained even as the phase changes. In physics, a wave packet (also known as a wave train or wave group) is a short burst of localized wave action that travels as a unit, outlined by an envelope.
Solitary wave in a laboratory wave channel. In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is strongly stable, in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such localized wave packets.
The term is also used, even more specifically, to mean a "monochromatic" or sinusoidal plane wave: a travelling plane wave whose profile () is a sinusoidal function. That is, (,) = (() +) The parameter , which may be a scalar or a vector, is called the amplitude of the wave; the scalar coefficient is its "spatial frequency"; and the scalar is its "phase shift".