Traveling plane wave

In mathematics and physics, a traveling plane wave^[1] 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 ${\displaystyle c}$, along a fixed direction of propagation ${\displaystyle {\vec {n}}}$.

Such a field can be written as

${\displaystyle F({\vec {x}},t)=G\left({\vec {x}}\cdot {\vec {n}}-ct\right)\,}$

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

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

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

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

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

${\displaystyle \nabla F=-{\frac {\vec {n}}{c}}{\frac {\partial F}{\partial t}}}$

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

• Spherical wave
• Standing wave