# Plane wave

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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.

Physics is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

In physics, a field is a physical quantity, represented by a number or tensor, that has a value for each point in space-time. For example, on a weather map, the surface temperature is described by assigning a real number to each point on a map; the temperature can be considered at a fixed point in time or over some time interval, to study the dynamics of temperature change. A surface wind map, assigning a vector to each point on a map that describes the wind velocity at that point, would be an example of a 1-dimensional tensor field, i.e. a vector field. Field theories, mathematical descriptions of how field values change in space and time, are ubiquitous in physics. For instance, the electric field is another rank-1 tensor field, and the full description of electrodynamics can be formulated in terms of two interacting vector fields at each point in space-time, or as a single-rank 2-tensor field theory.

## Contents

For any position ${\displaystyle {\vec {x}}}$ in space and any time ${\displaystyle t}$, the value of such a field can be written as

${\displaystyle F({\vec {x}},t)=G({\vec {x}}\cdot {\vec {n}},t),}$

where ${\displaystyle {\vec {n}}}$ is a unit-length vector, and ${\displaystyle G(d,t)}$ is a function that gives the field's value as from only two real parameters: the time ${\displaystyle t}$, and the displacement ${\displaystyle d={\vec {x}}\cdot {\vec {n}}}$ of the point ${\displaystyle {\vec {x}}}$ along the direction ${\displaystyle {\vec {n}}}$. The latter is constant over each plane perpendicular to ${\displaystyle {\vec {n}}}$.

In mathematics, a unit vector in a normed vector space is a vector of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat": . The term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as d. Two 2D direction vectors, d1 and d2 are illustrated. 2D spatial directions represented this way are numerically equivalent to points on the unit circle.

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as 2. Included within the irrationals are the transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.

The values of the field ${\displaystyle F}$ 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 ${\displaystyle F}$ are vectors, the wave is said to be a longitudinal wave if the vectors are always collinear with the vector ${\displaystyle {\vec {n}}}$, and a transverse wave if they are always orthogonal (perpendicular) to it.

Longitudinal waves are waves in which the displacement of the medium is in the same direction as, or the opposite direction to, the direction of propagation of the wave. Mechanical longitudinal waves are also called compressional or compression waves, because they produce compression and rarefaction when traveling through a medium, and pressure waves, because they produce increases and decreases in pressure.

In physics, a transverse wave is a moving wave whose oscillations are perpendicular to the direction of the wave.

## 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${\displaystyle c}$ along the direction perpendicular to the wavefronts. Such a field can be written as

In mathematics and physics, a traveling plane wave is a special case of plane wave, namely a field whose evolution in time can be described as simple translation of its values at a constant wave speed , along a fixed direction of propagation .

${\displaystyle F({\vec {x}},t)=G\left({\vec {x}}\cdot {\vec {n}}-ct\right)\,}$

where ${\displaystyle G(u)}$ is now a function of a single real parameter ${\displaystyle u=d-ct}$, that describes the "profile" of the wave, namely the value of the field at time ${\displaystyle t=0}$, for each displacement ${\displaystyle d={\vec {x}}\cdot {\vec {n}}>}$. In that case, ${\displaystyle {\vec {n}}}$ is called the direction of propagation. For each displacement ${\displaystyle d}$, the moving plane perpendicular to ${\displaystyle {\vec {n}}}$ at distance ${\displaystyle d+ct}$ from the origin is called a "wavefront". This plane travels along the direction of propagation ${\displaystyle {\vec {n}}}$ with velocity ${\displaystyle c}$; and the value of the field is then the same, and constant in time, at every one of its points.

In physics, a wavefront of a time-varying field is the set (locus) of all points where the wave has the same phase of the sinusoid. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal frequency.

### 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 ${\displaystyle G(u)}$is a sinusoidal function. That is,

${\displaystyle F({\vec {x}},t)=A\sin \left(2\pi f({\vec {x}}\cdot {\vec {n}}-ct)+\varphi \right)\,}$

The parameter ${\displaystyle A}$, which may be a scalar or a vector, is called the amplitude of the wave; the scalar coefficient ${\displaystyle f}$ is its "spatial frequency"; and the scalar ${\displaystyle \varphi }$ 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

${\displaystyle F({\vec {x}},t)=G({\vec {x}}\cdot {\vec {n}})\,S(t)}$

where ${\displaystyle G}$ is a function of one scalar parameter (the displacement ${\displaystyle d={\vec {x}}\cdot {\vec {n}}}$) with scalar or vector values, and ${\displaystyle S}$ is a scalar function of time.

This representation is not unique, since the same field values are obtained if ${\displaystyle S}$ and ${\displaystyle G}$ are scaled by reciprocal factors. If ${\displaystyle |S(t)|}$ is bounded in the time interval of interest (which is usually the case in physical contexts), ${\displaystyle S}$ and ${\displaystyle G}$ can be scaled so that the maximum value of ${\displaystyle |S(t)|}$ is 1. Then ${\displaystyle |G({\vec {x}}\cdot {\vec {n}})|}$ will be the maximum field magnitude seen at the point ${\displaystyle {\vec {x}}}$.

## Properties

A plane wave can be studied by ignoring the directions perpendicular to the direction vector ${\displaystyle {\vec {n}}}$; that is, by considering the function ${\displaystyle G(z,t)=F(z{\vec {n}},t)}$ 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 ${\displaystyle {\vec {n}}}$ 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 ${\displaystyle {\vec {n}}}$; specifically, ${\displaystyle \nabla F({\vec {x}},t)={\vec {n}}\partial _{1}G({\vec {x}}\cdot {\vec {n}},t)}$, where ${\displaystyle \partial _{1}G}$ is the partial derivative of ${\displaystyle G}$ with respect to the first argument.

The divergence of a vector-valued plane wave depends only on the projection of the vector ${\displaystyle G(d,t)}$ in the direction ${\displaystyle {\vec {n}}}$. Specifically,

${\displaystyle (\nabla \cdot F)({\vec {x}},t)\;=\;{\vec {n}}\cdot \partial _{1}G({\vec {x}}\cdot {\vec {n}},t)}$

In particular, a transverse planar wave satisfies ${\displaystyle \nabla \cdot F=0}$ for all ${\displaystyle {\vec {x}}}$ and ${\displaystyle t}$.

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