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 .

The wavefronts of a traveling plane wave in three-dimensional space.

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.

The wave may be a scalar or vector field; its values are the values of .

A sinusoidal plane wave is a special case, when is a sinusoidal function of .


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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