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In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant over any plane that is perpendicular to a fixed direction in space.[1]

For any position ${\displaystyle {\vec {x}}}$ in space and any time ${\displaystyle t}$, the value of such a field can be written as

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

where ${\displaystyle {\vec {n}}}$ is a unit-length vector, and ${\displaystyle G(d,t)}$ is a function that gives the field's value as from only two real parameters: the time ${\displaystyle t}For any position$${\displaystyle {\vec {x}}}$ in space and any time ${\displaystyle t}$, the value of such a field can be written as

where ${\displaystyle {\vec {n}}}$ is a unit-length vector, and ${\displaystyle G(d,t)}$ is a function that gives the field's value as from only two real parameters: the time ${\displaystyle t}$, and the displacement ${\displaystyle d={\vec {x}}\cdot {\vec {n}}}$ of the point ${\displaystyle {\vec {x}}}$ along the direction ${\displaystyle {\vec {n}}}$. The latter is constant over each plane perpendicular to ${\displaystyle {\vec {n}}}$.

The values of the field ${\displaystyle F}$ may be scalars, vectors, or any other physical or mathematical quantity. They can be complex numbers, as in a complex exponential plane wave.

When the values of ${\displaystyle F}$ are vectors, the wave is said to be a ${\displaystyle F}$ may be scalars, vectors, or any other physical or mathematical quantity. They can be complex numbers, as in a complex exponential plane wave.

When the values of ${\displaystyle F}$ are vectors, the wave is said to be a longitudinal wave if the vectors are always collinear with the vector ${\displaystyle {\vec {n}}}$, and a transverse wave if they are always orthogonal (perpendicular) to it.

Often the term "plane wave" refers specifically to a traveling plane wave, whose evolution in time can be described as simple translation of the field at a constant wave speed ${\displaystyle c}$ along the direction perpendicular to the wavefronts. 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 now a function of a single real parameter

### Sinusoidal plane wave

The term is also used, even more specifically, to mean a "monochromatic" or sinusoidal plane wave: a travelling plane wave whose profile ${\displaystyle G(u)}$is a sinusoidal function. That is,