Traveling plane wave

In mathematics and physics, a traveling plane wave[1] 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

describes the profile of the wave, namely the value of the field at time

, the moving plane perpendicular to

from the origin is called a wavefront.

This plane too travels along the direction of propagation

; and the value of the field is then the same, and constant in time, at every one of its points.

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

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

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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