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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 r ...
, 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 (res ...
or
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
: 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 Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
parameters: the time t, and the scalar-valued
displacement Displacement may refer to: Physical sciences Mathematics and Physics * Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
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 form a ...
, 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 waves ...
if the vectors are always collinear with the vector \vec n, and a
transverse wave In physics, a transverse wave is a wave whose oscillations are perpendicular to the direction of the wave's advance. This is in contrast to a longitudinal wave which travels in the direction of its oscillations. Water waves are an example of t ...
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 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 In physics, the wavefront of a time-varying '' wave field'' is the set ( locus) of all points having the same '' phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal fr ...
". 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 In physics, a sinusoidal (or monochromatic) plane wave is a special case of plane wave: a field whose value varies as a sinusoidal function of time and of the distance from some fixed plane. For any position \vec x in space and any time t, the val ...
: a travelling plane wave whose profile G(u)is a
sinusoidal A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in m ...
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 amplit ...
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 Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
, 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 Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administrat ...
,
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 In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
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 In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves: e^ = \sum_^\infty (2 \ell + 1) i^\ell j_\ell(k r) P_\ell(\hat \cdot \hat), where * is the imaginary unit, * is a wave vector of length , * ...
*
Rectilinear propagation Rectilinear propagation describes the tendency of electromagnetic waves (light) to travel in a straight line. Light does not deviate when travelling through a homogeneous medium, which has the same refractive index throughout; otherwise, light su ...
*
Wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
*
Weyl expansion In physics, the Weyl expansion, also known as the Weyl identity or angular spectrum expansion, expresses an outgoing spherical wave as a linear combination of plane waves. In a Cartesian coordinate system, it can be denoted as :\frac=\frac \int_^ ...


References


Sources

* *{{cite book , last=Jackson , first= John David , author-link= John David Jackson (physicist) , date= 1998 , title=
Classical Electrodynamics Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fi ...
, location=New York , publisher=
Wiley Wiley may refer to: Locations * Wiley, Colorado, a U.S. town * Wiley, Pleasants County, West Virginia, U.S. * Wiley-Kaserne, a district of the city of Neu-Ulm, Germany People * Wiley (musician), British grime MC, rapper, and producer * Wiley Mil ...
, isbn= 9780471309321 , edition=3 Wave mechanics