A wave is a disturbance that propagates through space and time, usually with transference of energy. A mechanical wave is a wave that propagates or travels through a medium due to the restoring forces it produces upon deformation. For example, when a sound wave is traveling through the air, air molecules slam into their neighbors, which pushes their neighbors into their neighbors (and so on); but when air molecules collide with their neighbors, they also bounce away from them back in the direction they came from. These collisions provide a restoring force that keeps the molecules from actually traveling with the wave.
In many areas of science, the idea of a wave is used metaphorically. If an ocean wave is seen as a prototype wave, it is the basis for the metaphor—the surface of water undulating up and down. However, upon investigating a sound wave, its air does not undulate up and down (as the ocean surface did). Instead, an abstraction is made; if we could look at the air molecules, they would be bunching together (in compressions) and then spreading apart (in rarefactions). Thus, the medium itself is not undulating up and down, but its density is (and its pressure is). When we speak of waves in physics, therefore, we are often speaking metaphorically, in an abstraction, of a periodic fluctuation of a specific characteristic. The characteristics that oscillate could be density, pressure, electrical or magnetic polarities or other (sometimes exotic) characteristics.
As well, it is believed that gravitational waves travel through space; gravitational waves have never been directly detected but are believed to exist. (See gravitational radiation.)
Agreeing on a single, all-encompassing definition for the term wave is non-trivial. A vibration can be defined as a back-and-forth motion around a reference value. However, a vibration is not necessarily a wave. Defining the necessary and sufficient characteristics that qualify a phenomenon to be called a wave is, at least, flexible.
It may be seen that the description of waves is accompanied by a heavy reliance on physical origin when describing any specific instance of a wave process. For example, acoustics is distinguished from optics in that sound waves are related to a mechanical rather than an electromagnetic wave-like transfer / transformation of vibratory energy. Concepts such as mass, momentum, inertia, or elasticity, become therefore crucial in describing acoustic (as distinct from optic) wave processes. This difference in origin introduces certain wave characteristics particular to the properties of the medium involved (for example, in the case of air: vortices, radiation pressure, shock waves, etc., in the case of solids: Rayleigh waves, dispersion, etc., and so on).
Similarly, wave processes revealed from the study of waves other than sound waves can be significant to the understanding of sound phenomena. A relevant example is Thomas Young's principle of interference (Young, 1802, in Hunt 1992, p. 132). This principle was first introduced in Young's study of light and, within some specific contexts (for example, scattering of sound by sound), is still a researched area in the study of sound.
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