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 [math]\displaystyle{ c }[/math], along a fixed direction of propagation [math]\displaystyle{ \vec n }[/math].
Such a field can be written as
- [math]\displaystyle{ F(\vec x, t)=G\left(\vec x \cdot \vec n - c t\right)\, }[/math]
where [math]\displaystyle{ G(u) }[/math] is a function of a single real parameter [math]\displaystyle{ u = d - c t }[/math]. The function [math]\displaystyle{ G }[/math] describes the profile of the wave, namely the value of the field at time [math]\displaystyle{ t = 0 }[/math], for each displacement [math]\displaystyle{ d = \vec x \cdot \vec n }[/math]. For each displacement [math]\displaystyle{ d }[/math], the moving plane perpendicular to [math]\displaystyle{ \vec n }[/math] at distance [math]\displaystyle{ d + c t }[/math] from the origin is called a wavefront. This plane too travels along the direction of propagation [math]\displaystyle{ \vec n }[/math] with velocity [math]\displaystyle{ c }[/math]; and the value of the field is then the same, and constant in time, at every one of its points.
The wave [math]\displaystyle{ F }[/math] may be a scalar or vector field; its values are the values of [math]\displaystyle{ G }[/math].
A sinusoidal plane wave is a special case, when [math]\displaystyle{ G(u) }[/math] is a sinusoidal function of [math]\displaystyle{ u }[/math].
Properties
A traveling plane wave can be studied by ignoring the dimensions of space perpendicular to the vector [math]\displaystyle{ \vec n }[/math]; that is, by considering the wave [math]\displaystyle{ F(z\vec n,t) = G(z - ct) }[/math] on a one-dimensional medium, with a single position coordinate [math]\displaystyle{ z }[/math].
For a scalar traveling plane wave in two or three dimensions, the gradient of the field is always collinear with the direction [math]\displaystyle{ \vec n }[/math]; specifically, [math]\displaystyle{ \nabla F(\vec x,t) = \vec n G'(\vec x \cdot \vec n - ct) }[/math], where [math]\displaystyle{ G' }[/math] is the derivative of [math]\displaystyle{ G }[/math]. Moreover, a traveling plane wave [math]\displaystyle{ F }[/math] of any shape satisfies the partial differential equation
- [math]\displaystyle{ \nabla F = -\frac{\vec n}{c}\frac{\partial F}{\partial t} }[/math]
Plane traveling waves are also special solutions of the wave equation in an homogeneous medium.
See also
- Spherical wave
- Spherical sinusoidal wave
- Standing wave
References
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