Plane wave expansion
Encyclopedia
In physics
the plane wave expansion expresses a plane wave as a sum of spherical waves,
where
. The wave vector
has length
and the vector 
has length 
. The angle between the vectors 
and
is 
. The functions 
are
Sperical Bessel functions and
are
Legendre polynomials.
With the spherical harmonic
addition theorem the equation can be
rewritten as
where
and 
are the spherical coordinates of the vectors
and
, respectively, and the functions 
are spherical harmonics
.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
the plane wave expansion expresses a plane wave as a sum of spherical waves,
where
. The wave vectorWave vector
In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave , and its direction is ordinarily the direction of wave propagation In...
has length
and the vector has length . The angle between the vectors and is . The functions areSperical Bessel functions and
areLegendre polynomials.
With the spherical harmonic
Spherical Harmonic
Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro. It tells the story of Dyhianna Selei , the Ruby Pharaoh of the Skolian Imperialate, as she strives to reform her government and reunite her family in the aftermath of a devastating interstellar...
addition theorem the equation can be
rewritten as
where
and are the spherical coordinates of the vectors
and
, respectively, and the functions are spherical harmonics
Spherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...
.
Applications
The plane wave expansion is applied in- AcousticsAcousticsAcoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...
- OpticsOpticsOptics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
- Quantum scattering theory