Traveling plane wave
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 , along a fixed direction of propagation .
Such a field can be written as
where is a function of a single real parameter . The function describes the profile of the wave, namely the value of the field at time , for each displacement . For each displacement , the moving plane perpendicular to at distance from the origin is called a wavefront. This plane too 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.
A traveling plane wave can be studied by ignoring the dimensions of space perpendicular to the vector ; that is, by considering the wave on a one-dimensional medium, with a single position coordinate .
For a scalar traveling plane wave in two or three dimensions, the gradient of the field is always collinear with the direction ; specifically, , where is the derivative of . Moreover, a traveling plane wave of any shape satisfies the partial differential equation
Plane traveling waves are also special solutions of the wave equation in an homogeneous medium.
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