Jump to content

Traveling plane wave

From Wikipedia, the free encyclopedia
The wavefronts of a traveling plane wave in three-dimensional space.

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 . 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 .

Properties

[edit]

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.

See also

[edit]

References

[edit]
  1. ^ Tohyama, Mikio (2011), Tohyama, Mikio (ed.), "Waves and Speed of Sound in the Air", Sound and Signals, Berlin, Heidelberg: Springer, pp. 89–102, doi:10.1007/978-3-642-20122-6_6#citeas, ISBN 978-3-642-20122-6, retrieved 2024-08-05