Plane wave

In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position in space and any time , the value of such a field can be written as where is a unit-length vector, and is a function that gives the field's value as dependent on only two real parameters: the time , and the scalar-valued displacement of the point along the direction . The displacement is constant over each plane perpendicular to . The values of the field may be scalars, vectors, or any other physical or mathematical quantity. They can be complex numbers, as in a complex exponential plane wave. When the values of are vectors, the wave is said to be a longitudinal wave if the vectors are always collinear with the vector , and a transverse wave if they are always orthogonal (perpendicular) to it. Often the term "plane wave" refers specifically to a traveling plane wave, whose evolution in time can be described as simple translation of the field at a constant wave speed along the direction perpendicular to the wavefronts. Such a field can be written as where is now a function of a single real parameter , that describes the "profile" of the wave, namely the value of the field at time , for each displacement . In that case, is called the direction of propagation. For each displacement , the moving plane perpendicular to at distance from the origin is called a "wavefront". This plane travels along the direction of propagation with velocity ; and the value of the field is then the same, and constant in time, at every one of its points. Sinusoidal plane wave The term is also used, even more specifically, to mean a "monochromatic" or sinusoidal plane wave: a travelling plane wave whose profile is a sinusoidal function. That is, The parameter , which may be a scalar or a vector, is called the amplitude of the wave; the scalar coefficient is its "spatial frequency"; and the scalar is its "phase".