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 over any plane that is perpendicular to a fixed direction in space. In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant over any plane that is perpendicular to a fixed direction in space. For any position x → {displaystyle {vec {x}}} in space and any time t {displaystyle t} , the value of such a field can be written as where n → {displaystyle {vec {n}}} is a unit-length vector, and G ( d , t ) {displaystyle G(d,t)} is a function that gives the field's value as from only two real parameters: the time t {displaystyle t} , and the displacement d = x → ⋅ n → {displaystyle d={vec {x}}cdot {vec {n}}} of the point x → {displaystyle {vec {x}}} along the direction n → {displaystyle {vec {n}}} . The latter is constant over each plane perpendicular to n → {displaystyle {vec {n}}} . The values of the field F {displaystyle F} 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 F {displaystyle F} are vectors, the wave is said to be a longitudinal wave if the vectors are always collinear with the vector n → {displaystyle {vec {n}}} , 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 c {displaystyle c} along the direction perpendicular to the wavefronts. Such a field can be written as where G ( u ) {displaystyle G(u)} is now a function of a single real parameter u = d − c t {displaystyle u=d-ct} , that describes the 'profile' of the wave, namely the value of the field at time t = 0 {displaystyle t=0} , for each displacement d = x → ⋅ n → > {displaystyle d={vec {x}}cdot {vec {n}}>} . In that case, n → {displaystyle {vec {n}}} is called the direction of propagation. For each displacement d {displaystyle d} , the moving plane perpendicular to n → {displaystyle {vec {n}}} at distance d + c t {displaystyle d+ct} from the origin is called a 'wavefront'. This plane travels along the direction of propagation n → {displaystyle {vec {n}}} with velocity c {displaystyle c} ; and the value of the field is then the same, and constant in time, at every one of its points. The term is also used, even more specifically, to mean a 'monochromatic' or sinusoidal plane wave: a travelling plane wave whose profile G ( u ) {displaystyle G(u)} is a sinusoidal function. That is, The parameter A {displaystyle A} , which may be a scalar or a vector, is called the amplitude of the wave; the scalar coefficient f {displaystyle f} is its 'spatial frequency'; and the scalar φ {displaystyle varphi } is its 'phase'.