In
applied mathematics, the Sommerfeld radiation condition is a concept from
theory of differential equations and
scattering theory
In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunli ...
used for choosing a particular solution to the
Helmholtz equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation
\nabla^2 f = -k^2 f,
where is the Laplace operator (or "Laplacian"), is the eigenv ...
. It was introduced by
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
in 1912
and is closely related to the
limiting absorption principle In mathematics, the limiting absorption principle (LAP) is a concept from operator theory and scattering theory that consists of choosing the "correct" resolvent of a linear operator at the essential spectrum based on the behavior of the resolven ...
(1905) and the
limiting amplitude principle
In mathematics, the limiting amplitude principle is a concept from operator theory and scattering theory used for choosing a particular solution to the Helmholtz equation. The choice is made by considering a particular time-dependent problem of the ...
(1948).
Formulation
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
defined the condition of radiation for a scalar field satisfying the
Helmholtz equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation
\nabla^2 f = -k^2 f,
where is the Laplace operator (or "Laplacian"), is the eigenv ...
as
: "the sources must be sources, not sinks of energy. The energy which is radiated from the sources must scatter to infinity; no energy may be radiated from infinity into ... the field."
[A. Sommerfeld, ''Partial Differential Equations in Physics'', Academic Press, New York, New York, 1949.]
Mathematically, consider the inhomogeneous
Helmholtz equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation
\nabla^2 f = -k^2 f,
where is the Laplace operator (or "Laplacian"), is the eigenv ...
:
where
is the dimension of the space,
is a given function with
compact support
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
representing a bounded source of energy, and
is a constant, called the ''wavenumber''. A solution
to this equation is called ''radiating'' if it satisfies the Sommerfeld radiation condition
:
uniformly in all directions
:
(above,
is the
imaginary unit and
is the
Euclidean norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
). Here, it is assumed that the time-harmonic field is
If the time-harmonic field is instead
one should replace
with
in the Sommerfeld radiation condition.
The Sommerfeld radiation condition is used to solve uniquely the Helmholtz equation. For example, consider the problem of radiation due to a point source
in three dimensions, so the function
in the Helmholtz equation is
where
is the
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
. This problem has an infinite number of solutions, for example, any function of the form
:
where
is a constant, and
:
Of all these solutions, only
satisfies the Sommerfeld radiation condition and corresponds to a field radiating from
The other solutions are unphysical . For example,
can be interpreted as energy coming from infinity and sinking at
See also
*
Limiting absorption principle In mathematics, the limiting absorption principle (LAP) is a concept from operator theory and scattering theory that consists of choosing the "correct" resolvent of a linear operator at the essential spectrum based on the behavior of the resolven ...
*
Limiting amplitude principle
In mathematics, the limiting amplitude principle is a concept from operator theory and scattering theory used for choosing a particular solution to the Helmholtz equation. The choice is made by considering a particular time-dependent problem of the ...
*
Nonradiation condition
References
*
* "Eighty years of Sommerfeld’s radiation condition", Steven H. Schot, ''Historia Mathematica'' 19, #4 (November 1992), pp. 385–401, .
External links
* {{springer, id=r/r077060, title=Radiation conditions, author=A.G. Sveshnikov
Radiation
Boundary conditions