Plane wave

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

, the value of such a field can be written as

is a function that gives the field's value as dependent on only two real parameters: the time

The displacement is constant over each plane perpendicular to

may be scalars, vectors, or any other physical or mathematical quantity.

, 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

is now a function of a single real parameter

, that describes the "profile" of the wave, namely the value of the field at time

This plane travels along the direction of propagation

; and the value of the field is then the same, and constant in time, at every one of its points.

[2] The term is also used, even more specifically, to mean a "monochromatic" or sinusoidal plane wave: a travelling plane wave whose profile

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.

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

is a function of one scalar parameter (the displacement

is a scalar function of time.

This representation is not unique, since the same field values are obtained if

is bounded in the time interval of interest (which is usually the case in physical contexts),

can be scaled so that the maximum value of

will be the maximum field magnitude seen at the point

A plane wave can be studied by ignoring the directions perpendicular to the direction vector

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

For a scalar plane wave in two or three dimensions, the gradient of the field is always collinear with the direction

The divergence of a vector-valued plane wave depends only on the projection of the vector

In particular, a transverse planar wave satisfies