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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
, a plane wave is a special case of
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (r ...
or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, the value of such a field can be written as :F(\vec x,t) = G(\vec x \cdot \vec n, t), where \vec n is a unit-length vector, and G(d,t) is a function that gives the field's value as dependent on only two real parameters: the time t, and the scalar-valued displacement d = \vec x \cdot \vec n of the point \vec x along the direction \vec n. The displacement is constant over each plane perpendicular to \vec n. The values of the field F may be scalars, vectors, or any other physical or mathematical quantity. They can be
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
, as in a complex exponential plane wave. When the values of F are vectors, the wave is said to be a
longitudinal wave Longitudinal waves are waves in which the vibration of the medium is parallel ("along") to the direction the wave travels and displacement of the medium is in the same (or opposite) direction of the wave propagation. Mechanical longitudinal wa ...
if the vectors are always collinear with the vector \vec n, 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 In mathematics and physics, a traveling plane wave is a special case of plane wave, namely a field (physics), field whose evolution in time can be described as simple translation of its values at a constant wave speed c, along a fixed direction of p ...
, whose evolution in time can be described as simple translation of the field at a constant wave speed c along the direction perpendicular to the wavefronts. Such a field can be written as :F(\vec x, t)=G\left(\vec x \cdot \vec n - c t\right)\, where G(u) is now a function of a single real parameter u = d - c t, that describes the "profile" of the wave, namely the value of the field at time t = 0, for each displacement d = \vec x \cdot \vec n. In that case, \vec n is called the direction of propagation. For each displacement d, the moving plane perpendicular to \vec n at distance d + c t from the origin is called a " wavefront". This plane travels along the direction of propagation \vec n with velocity c; 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 G(u)is a sinusoidal function. That is, :F(\vec x, t)=A \sin\left(2\pi f (\vec x \cdot \vec n - c t) + \varphi\right)\, The parameter A, which may be a scalar or a vector, is called the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of a ...
of the wave; the scalar coefficient f is its "spatial frequency"; and the scalar \varphi 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 :F(\vec x, t) = G(\vec x \cdot \vec n)\,S(t) where G is a function of one scalar parameter (the displacement d = \vec x \cdot \vec n) with scalar or vector values, and S is a scalar function of time. This representation is not unique, since the same field values are obtained if S and G are scaled by reciprocal factors. If , S(t), is bounded in the time interval of interest (which is usually the case in physical contexts), S and G can be scaled so that the maximum value of , S(t), is 1. Then , G(\vec x \cdot \vec n), will be the maximum field magnitude seen at the point \vec x.


Properties

A plane wave can be studied by ignoring the directions perpendicular to the direction vector \vec n; that is, by considering the function G(z,t) = F(z \vec n, t) as a wave in a one-dimensional medium. Any local operator,
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
or not, applied to a plane wave yields a plane wave. Any linear combination of plane waves with the same normal vector \vec n 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 \vec n; specifically, \nabla F(\vec x,t) = \vec n\partial_1 G(\vec x \cdot \vec n, t), where \partial_1 G is the partial derivative of G with respect to the first argument. The divergence of a vector-valued plane wave depends only on the projection of the vector G(d,t) in the direction \vec n. Specifically, : (\nabla \cdot F)(\vec x, t) \;=\; \vec n \cdot\partial_1G(\vec x \cdot \vec n, t) In particular, a transverse planar wave satisfies \nabla \cdot F = 0 for all \vec x and t.


See also

* Plane wave expansion * Rectilinear propagation * Wave equation * Weyl expansion


References


Sources

* *{{cite book , last=Jackson , first= John David , author-link= John David Jackson (physicist) , date= 1998 , title= Classical Electrodynamics , location=New York , publisher= Wiley , isbn= 9780471309321 , edition=3 Wave mechanics