Plane wave

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 formula_1 in space and any time formula_2, the value of such a field can be written as
where formula_4 is a unit-length vector, and formula_5 is a function that gives the field's value as from only two real parameters: the time formula_2, and the displacement formula_7 of the point formula_1 along the direction formula_4. The latter is constant over each plane perpendicular to formula_4.
The values of the field formula_11 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 formula_11 are vectors, the wave is said to be a longitudinal wave if the vectors are always collinear with the vector formula_4, and a transverse wave if they are always orthogonal (perpendicular) to it.
Special types.
Traveling plane wave.
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 formula_14 along the direction perpendicular to the wavefronts. Such a field can be written as 
where formula_16 is now a function of a single real parameter formula_17, that describes the "profile" of the wave, namely the value of the field at time formula_18, for each displacement formula_7. In that case, formula_4 is called the direction of propagation. For each displacement formula_21, the moving plane perpendicular to formula_4 at distance formula_23 from the origin is called a "wavefront". This plane travels along the direction of propagation formula_4 with velocity formula_14; and the value of the field is then the same, and constant in time, at every one of its points. 
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 formula_16is a sinusoidal function. That is,
The parameter formula_28, which may be a scalar or a vector, is called the amplitude of the wave; the scalar coefficient formula_29 is its "spatial frequency"; and the scalar formula_30 is its "phase".
A true plane wave cannot physically exist, because it would have to fill all space. Nevertheless, the plane wave model is important and widely used in physics. The waves emitted by any source with finite extent into a large homogeneous region of space can be well approximated by plane waves when viewed over any part of that region that is sufficiently small compared to its distance from the source. That is the case, for example, of the light waves from a distant star that arrive at a telescope.
Plane standing wave.
A standing wave is a field whose value can be expressed as the product of two functions, one depending only on position, the other only on time. A plane standing wave, in particular, can be expressed as
where formula_32 is a function of one scalar parameter (the displacement formula_7) with scalar or vector values, and formula_34 is a scalar function of time. 
This representation is not unique, since the same field values are obtained if formula_34 and formula_32 are scaled by reciprocal factors. If formula_37 is bounded in the time interval of interest (which is usually the case in physical contexts), formula_34 and formula_32 can be scaled so that the maximum value of formula_37 is 1. Then formula_41 will be the maximum field magnitude seen at the point formula_1.
Properties.
A plane wave can be studied by ignoring the directions perpendicular to the direction vector formula_4; that is, by considering the function formula_44 as a wave in a one-dimensional medium. 
Any local operator, linear or not, applied to a plane wave yields a plane wave. Any linear combination of plane waves with the same normal vector formula_4 is also a plane wave.
For a scalar plane wave in two or three dimensions, the gradient of the field is always collinear with the direction formula_4; specifically, formula_47, where formula_48 is the partial derivative of formula_32 with respect to the first argument. 
The divergence of a vector-valued plane wave depends only on the projection of the vector formula_5 in the direction formula_4. Specifically,
In particular, a transverse planar wave satisfies formula_53 for all formula_1 and formula_2.