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.For any position
x
→
{\displaystyle {\vec {x}}}
in space and any time
t
{\displaystyle t}
, the value of such a field can be written as
F
(
x
→
,
t
)
=
G
(
x
→
⋅
n
→
,
t
)
,
{\displaystyle F({\vec {x}},t)=G({\vec {x}}\cdot {\vec {n}},t),}
where
n
→
{\displaystyle {\vec {n}}}
is a unit-length vector, and
G
(
d
,
t
)
{\displaystyle G(d,t)}
is a function that gives the field's value as from only two real parameters: the time
t
{\displaystyle t}
, and the displacement
d
=
x
→
⋅
n
→
{\displaystyle d={\vec {x}}\cdot {\vec {n}}}
of the point
x
→
{\displaystyle {\vec {x}}}
along the direction
n
→
{\displaystyle {\vec {n}}}
. The latter is constant over each plane perpendicular to
n
→
{\displaystyle {\vec {n}}}
.
The values of the field
F
{\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
F
{\displaystyle F}
are vectors, the wave is said to be a longitudinal wave if the vectors are always collinear with the vector
n
→
{\displaystyle {\vec {n}}}
, and a transverse wave if they are always orthogonal (perpendicular) to it.