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 $$c$$, along a fixed direction of propagation $$\vec n$$.

Such a field can be written as
 * $$F(\vec x, t)=G\left(\vec x \cdot \vec n - c t\right)\,$$

where $$G(u)$$ is a function of a single real parameter $$u = d - c t$$. The function $$G$$ describes the profile of the wave, namely the value of the field at time $$t = 0$$, for each displacement $$d = \vec x \cdot \vec n$$. For each displacement $$d$$, the moving plane perpendicular to $$\vec n$$ at distance $$d + c t$$ from the origin is called a wavefront. This plane too travels along the direction of propagation $$\vec n$$ with velocity $$c$$; and the value of the field is then the same, and constant in time, at every one of its points.

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

A sinusoidal plane wave is a special case, when $$G(u)$$ is a sinusoidal function of $$u$$.

Properties
A traveling plane wave can be studied by ignoring the dimensions of space perpendicular to the vector $$\vec n$$; that is, by considering the wave $$F(z\vec n,t) = G(z - ct)$$ on a one-dimensional medium, with a single position coordinate $$z$$.

For a scalar traveling plane wave in two or three dimensions, the gradient of the field is always collinear with the direction $$\vec n$$; specifically, $$\nabla F(\vec x,t) = \vec n G'(\vec x \cdot \vec n - ct)$$, where $$G'$$ is the derivative of $$G$$. Moreover, a traveling plane wave $$F$$ of any shape satisfies the partial differential equation
 * $$\nabla F = -\frac{\vec n}{c}\frac{\partial F}{\partial t}$$

Plane traveling waves are also special solutions of the wave equation in an homogeneous medium.