# 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

$F({\vec {x}},t)=G\left({\vec {x}}\cdot {\vec {n}}-ct\right)\,$ where $G(u)$ is a function of a single real parameter $u=d-ct$ . 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 $d+ct$ 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.