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- In physics, a sinusoidal (or monochromatic) plane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane. For any position in space and any time , the value of such a field can be written as where is a unit-length vector, the direction of propagation of the wave, and "" denotes the dot product of two vectors. The parameter , which may be a scalar or a vector, is called the amplitude of the wave; the coefficient , a positive scalar, its spatial frequency; and the adimensional scalar , an angle in radians, is its initial phase or phase shift. The scalar quantity gives the (signed) displacement of the point from the plane that is perpendicular to and goes through the origin of the coordinate system. This quantity is constant over each plane perpendicular to . At time , the field varies with the displacement as a sinusoidal function The spatial frequency is the number of full cycles per unit of length along the direction . For any other value of , the field values are displaced by the distance in the direction . That is, the whole field seems to travel in that direction with velocity . For each displacement , the moving plane perpendicular to at distance from the origin is called a wavefront. This plane lies at distance from the origin when , and travels in the direction also with speed ; and the value of the field is then the same, and constant in time, at every one of its points. A sinusoidal plane wave could be a suitable model for a sound wave within a volume of air that is small compared to the distance of the source (provided that there are no echos from nearly objects). In that case, would be a scalar field, the deviation of air pressure at point and time , away from its normal level. At any fixed point , the field will also vary sinusoidally with time; it will be a scalar multiple of the amplitude , between and When the amplitude is a vector orthogonal to , the wave is said to be transverse. Such waves may exhibit polarization, if can be oriented along two non-collinear directions. When is a vector collinear with , the wave is said to be longitudinal. These two possibilities are exemplified by the S (shear) waves and P (pressure) waves studied in seismology. The formula above gives a purely "kinematic" description of the wave, without reference to whatever physical process may be causing its motion. In a mechanical or electromagnetic wave that is propagating through an isotropic medium, the vector of the apparent propagation of the wave is also the direction in which energy or momentum is actually flowing. However, the two directions may be different in an anisotropic medium. (en)
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