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A standing wave, also known as a stationary wave, is a wave  that remains in a constant position. This phenomenon can occur because  the medium is moving in the opposite direction to the wave, or it can  arise in a stationary medium as a result of 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 and the 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. Halfway between two nodes there is an antinode, where the two counter-propagating waves enhance each other maximally. There is no net propagation of energy over time.
A plane wave (also spelled planewave) is a constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant amplitude normal to the phase velocity vector.
For example, a localized source such as an antenna produces a field that is approximately a plane wave in its far-field region.  Equivalently, for propagation in a homogeneous medium over lengthscales  much longer than the wavelength, the "rays" in the limit where ray optics is valid correspond locally to approximate plane waves.