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== Special waves == === Sine waves === {{excerpt|Sine wave}} === Plane waves === {{Main|Plane wave}} A [[plane wave]] is a kind of wave whose value varies only in one spatial direction. That is, its value is constant on a plane that is perpendicular to that direction. Plane waves can be specified by a vector of unit length <math>\hat n</math> indicating the direction that the wave varies in, and a wave profile describing how the wave varies as a function of the displacement along that direction (<math>\hat n \cdot \vec{x}</math>) and time (<math>t</math>). Since the wave profile only depends on the position <math>\vec{x}</math> in the combination <math>\hat n \cdot \vec{x}</math>, any displacement in directions perpendicular to <math>\hat n</math> cannot affect the value of the field. Plane waves are often used to model [[electromagnetic waves]] far from a source. For electromagnetic plane waves, the electric and magnetic fields themselves are transverse to the direction of propagation, and also perpendicular to each other. === Standing waves === {{Main|Standing wave|Acoustic resonance|Helmholtz resonance|Organ pipe}} [[File:Standing wave.gif|class=skin-invert-image|thumb|right|300px|Standing wave. The red dots represent the wave [[Node (physics)|nodes]].]] A standing wave, also known as a ''stationary wave'', is a wave whose [[Envelope (waves)|envelope]] remains in a constant position. This phenomenon arises as a result of [[Interference (wave propagation)|interference]] between two waves traveling in opposite directions. The ''sum'' of two counter-propagating waves (of equal amplitude and frequency) creates a ''standing wave''. Standing waves commonly arise when a boundary blocks further propagation of the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave. For example, when a [[violin]] string is displaced, transverse waves propagate out to where the string is held in place at the [[Bridge (instrument)|bridge]] and the [[Nut (string instrument)|nut]], where the waves are reflected back. At the bridge and nut, the two opposed waves are in [[antiphase]] and cancel each other, producing a [[node (physics)|node]]. Halfway between two nodes there is an [[antinode]], where the two counter-propagating waves ''enhance'' each other maximally. There is no net [[Energy transfer|propagation of energy]] over time. <gallery> Image:Standing waves on a string.gif|One-dimensional standing waves; the [[fundamental frequency|fundamental]] mode and the first 5 [[overtone]]s Image:Drum vibration mode01.gif|A two-dimensional [[Vibrations of a circular drum|standing wave on a disk]]; this is the fundamental mode. Image:Drum vibration mode21.gif|A [[Vibrations of a circular drum|standing wave on a disk]] with two nodal lines crossing at the center; this is an overtone. </gallery> === Solitary waves === {{Main|Soliton}} [[File:Soliton hydro.jpg|thumb|[[Solitary wave (water waves)|Solitary wave]] in a laboratory [[wave channel]]]] A '''soliton''' or '''solitary wave''' is a self-reinforcing [[wave packet]] that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of [[nonlinearity|nonlinear]] and [[dispersion relation|dispersive effects]] in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive [[partial differential equation]]s describing physical systems.
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