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
F ( x → , t ) = G ( x → ⋅ n → − c t ) {\displaystyle F({\vec {x}},t)=G\left({\vec {x}}\cdot {\vec {n}}-ct\right)\,}
where G ( u ) {\displaystyle G(u)} 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} . A sinusoidal plane wave is a special case, when G ( u ) {\displaystyle G(u)} is a sinusoidal function of u {\displaystyle u} .
Properties A traveling plane wave can be studied by ignoring the dimensions of space perpendicular to the vector n → {\displaystyle {\vec {n}}} ; that is, by considering the wave F ( z n → , t ) = G ( z − c t ) {\displaystyle F(z{\vec {n}},t)=G(z-ct)} on a one-dimensional medium, with a single position coordinate z {\displaystyle z} . For a scalar traveling plane wave in two or three dimensions, the gradient of the field is always collinear with the direction n → {\displaystyle {\vec {n}}} ; specifically, ∇ F ( x → , t ) = n → G ′ ( x → ⋅ n → − c t ) {\displaystyle \nabla F({\vec {x}},t)={\vec {n}}G'({\vec {x}}\cdot {\vec {n}}-ct)} , where G ′ {\displaystyle G'} is the derivative of G {\displaystyle G} . Moreover, a traveling plane wave F {\displaystyle F} of any shape satisfies the partial differential equation
∇ F = − n → c ∂ F ∂ t {\displaystyle \nabla F=-{\frac {\vec {n}}{c}}{\frac {\partial F}{\partial t}}}
Plane traveling waves are also special solutions of the wave equation in an homogeneous medium.
See also Spherical wave Standing wave
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