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In mathematics and physics, a traveling plane wave is a special case of plane wave, namely a field whose evolution in time can be described as simple translation of its values at a constant wave speed c {\displaystyle c} , along a fixed direction of propagation n → {\displaystyle {\vec {n}}}.
Such a field can be written as
where G {\displaystyle G} is a function of a single real parameter u = d − c t {\displaystyle u=d-ct}. The function G {\displaystyle G} 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}}}. 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 too 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 wave F {\displaystyle F} may be a scalar or vector field; its values are the values of G {\displaystyle G}.