Plane wave

In physics, a plane wave is a special case of a wave or field: a physical quantity whose value, at any moment, is constant through 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 dependent on only two real parameters: the time ${\displaystyle t}$, and the scalar-valued displacement ${\displaystyle d={\vec {x}}\cdot {\vec {n}}}$ of the point ${\displaystyle {\vec {x}}}$ along the direction ${\displaystyle {\vec {n}}}$. The displacement 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 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.