Traveling plane wave

The wavefronts of a traveling plane wave in three-dimensional space.

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 $\overrightarrow{n}$.

Such a field can be written as

$F(\overrightarrow{x},t)=G(\overrightarrow{x}\cdot \overrightarrow{n}-ct)$

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 $d=\overrightarrow{x}\cdot \overrightarrow{n}$. For each displacement $d$, the moving plane perpendicular to $\overrightarrow{n}$ at distance $d+ct$ from the origin is called a wavefront. This plane too travels along the direction of propagation $\overrightarrow{n}$ 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 $\overrightarrow{n}$; that is, by considering the wave $F(z\overrightarrow{n},t)=G(z-ct)$ 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 $\overrightarrow{n}$; specifically, $\mathrm{\nabla }F(\overrightarrow{x},t)=\overrightarrow{n}{G}^{\prime }(\overrightarrow{x}\cdot \overrightarrow{n}-ct)$, where $G^{\prime }$ is the derivative of $G$. Moreover, a traveling plane wave $F$ of any shape satisfies the partial differential equation

$\mathrm{\nabla }F=-\frac{\overrightarrow{n}}{c}\frac{\partial F}{\partial t}$

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

See also

• Spherical wave
• Standing wave