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 ${\vec {n}}$.

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

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.

See also

• Spherical wave
• Standing wave