Information about Plane Wave
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The wavefronts of a plane wave traveling in space.
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The real part of a plane wave travelling up.
By extension, the term is also used to describe waves that are approximately plane waves in a localized region of space. For example, a localized source such as an antenna produces a field that is approximately a plane wave in its far-field region. Equivalently, the "rays" in the limit where ray optics is valid (i.e. for propagation in a homogeneous medium over lengthscales much longer than the wavelength) correspond locally to approximate plane waves.
Mathematically, a plane wave is a solution to the wave equation of the following form:
where i is the imaginary unit, k is the wave vector, ω is the angular frequency, and a is the (complex) amplitude. (The above form of the plane wave uses the physics time convention; in the engineering time convention,
is used instead of 
in the exponent.) The physical solution is usually found by taking the real part of this expression.
This is the solution for a scalar wave equation in a homogeneous medium. For vector wave equations, such as the ones describing electromagnetic radiation or waves in an elastic solid, the solution for a homogeneous medium is similar:
multiplied by a constant vector a. (For example, in electromagnetism a is typically the vector for the electric field, magnetic field, or vector potential.) A transverse wave is one in which the amplitude vector is orthogonal to k (e.g. for electromagnetic waves in an isotropic medium), whereas a longitudinal wave is one in which the amplitude vector is parallel to k (e.g. for acoustic waves in a gas or fluid).
In this equation, the function ω(k) is the dispersion relation of the medium, with the ratio ω/|k| giving the magnitude of the phase velocity and dω/dk giving the group velocity. For electromagnetism in an isotropic medium with index of refraction n, the phase velocity is c/n (which equals the group velocity only if the index is not frequency-dependent).
The form of the planewave solution is actually a general consequence of translational symmetry. More generally, for periodic structures (i.e. with discrete translational symmetry), the solutions take the form of Bloch waves, most famously in crystalline atomic materials but also in photonic crystals and other periodic wave equations. As another generalization, for structures that are only uniform along one direction x (such as a waveguide along the x direction), the solutions (waveguide modes) are of the form
multiplied by some amplitude function 
. (This is a special case of a separable partial differential equation.)
(The term is used in the same way for telecommunication, e.g. in Federal Standard 1037C and MIL-STD-188.)
References
- J. D. Jackson, Classical Electrodynamics (Wiley: New York, 1998).
Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
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Wave propagation is any of the ways in which waves travel through a medium (waveguide).
With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves.
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With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves.
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Electromagnetic (EM) radiation is a self-propagating wave in space with electric and magnetic components. These components oscillate at right angles to each other and to the direction of propagation, and are in phase with each other.
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- :For the notion of wave front in functional analysis, see wave front set.
In optics and physics, a wavefront is the locus (a line, or, in a wave propagating in 3 dimensions a surface) of points having the same phase.
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phase can be readily understood in terms of simple harmonic motion. The same concept applies to wave motion, viewed either at a point in space over an interval of time or across an interval of space at a moment in time.
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amplitude is a nonnegative scalar measure of a wave's magnitude of oscillation, that is, the magnitude of the maximum disturbance in the medium during one wave cycle.
Sometimes this distance is called the peak amplitude
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Sometimes this distance is called the peak amplitude
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The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave will propagate.
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antenna is a transducer designed to transmit or receive radio waves which are a class of electromagnetic waves. In other words, antennas convert radio frequency electrical currents into electromagnetic waves and vice versa.
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. The near field is that part of the radiated field nearest to the antenna. Beyond the near field is the infinite far field. The concept of near and far fields is different in modeling the propagation and coupling principles in mathematical terms and for the convenience and
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Optics (ὀπτική appearance or look in Ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter.
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The wave equation is an important second-order linear partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics.
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In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . Its precise definition is dependent upon the particular method of extension.
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A wave vector is a vector that specifies the wavenumber and direction of propagation for a wave. The magnitude of the wave vector indicates the wavenumber. The orientation of the wave vector indicates the direction of wave propagation.
For example consider a plane wave.
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For example consider a plane wave.
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angular frequency ω (also referred to by the terms angular speed, radial frequency, and radian frequency) is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity.
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In mathematics, a complex number is a number of the form
where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector.
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In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are
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Electromagnetic (EM) radiation is a self-propagating wave in space with electric and magnetic components. These components oscillate at right angles to each other and to the direction of propagation, and are in phase with each other.
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Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles.
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electric field. This electric field exerts a force on other electrically charged objects. The concept of electric field was introduced by Michael Faraday.
The electric field is a vector field with SI units of newtons per coulomb (N C−1
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The electric field is a vector field with SI units of newtons per coulomb (N C−1
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magnetic field is a field that permeates space and which exerts a magnetic force on moving electric charges and magnetic dipoles. Magnetic fields surround electric currents, magnetic dipoles, and changing electric fields.
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In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential, which is a scalar field whose negative gradient is a given vector field.
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transverse wave is a wave that causes vibration in the medium in a perpendicular direction to its own motion. For example: if a wave moves along the x-axis, its disturbances are in the yz-plane.
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In mathematics, orthogonal, as a simple adjective, not part of a longer phrase, is a generalization of perpendicular. It means at right angles, from the Greek ὀρθός orthos
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Isotropy (the opposite of anisotropy) is the property of being independent of direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented.
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Longitudinal waves are waves that have vibrations along or parallel to their direction of travel. They include waves in which the motion of the medium is in the same direction as the motion of the wave.
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In physics, the dispersion relation is the relation between the energy of a system and its corresponding momentum. For example, for massive particles in free space, the dispersion relation can easily be calculated from the definition of kinetic energy:
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The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the velocity at which the phase of any one frequency component of the wave will propagate.
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The group velocity of a wave is the velocity with which the variations in the shape of the wave's amplitude (known as the modulation or envelope of the wave) propagate through space.
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a: Ta(p) = p + a.
In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation.
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In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation.
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