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Rarefaction is the reduction of an item's density, the opposite of compression. Like compression, which can travel in waves (sound waves, for instance), rarefaction waves also exist in nature. A common rarefaction wave is the area of low relative pressure following a shock wave (see picture).
Rarefaction waves expand with time (much like sea waves spread out as they reach a beach); in most gases rarefaction waves keep the same overall profile ('shape') at all times throughout the wave's movement: it is a self-similar expansion. Each part of the wave travels at the local speed of sound, in the local medium. This expansion behaviour is in contrast to the behaviour of pressure increases, which get narrower with time, until they steepen into shock waves.
Physical examples
A natural example of rarefaction occurs in the layers of Earth's atmosphere. Because the atmosphere has mass, most atmospheric matter is nearer to the Earth due to the Earth's gravitation. Therefore, air at higher layers of the atmosphere is less dense, or rarefied, relative to air at lower layers. Thus rarefaction can refer either to a reduction in density over space at a single point of time, or a reduction of density over time for one particular area.
Rarefaction can be easily observed by compressing a spring and releasing it. Instead of seeing compressed loops seeming to move through the spring, spaced-out loops move through it: rarefaction waves.