Sommerfeld radiation condition

In mathematics, the Sommerfeld radiation condition is a concept from theory of differential equations and scattering theory used for choosing a particular solution to the Helmholtz equation. It was introduced by Arnold Sommerfeld in 1912[1] and is closely related to the limiting absorption principle (1905) and the limiting amplitude principle (1948).

FormulationEdit

Arnold Sommerfeld defined the condition of radiation for a scalar field satisfying the Helmholtz equation 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."[2]

Mathematically, consider the inhomogeneous Helmholtz equation

 

where   is the dimension of the space,   is a given function with compact support 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). 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. 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 alsoEdit

ReferencesEdit

  1. ^ A. Sommerfeld (1912). "Die Greensche Funktion der Schwingungslgleichung". Jahresbericht der Deutschen Mathematiker-Vereinigung. 21: 309–353.
  2. ^ A. Sommerfeld, Partial Differential Equations in Physics, Academic Press, New York, New York, 1949.

External linksEdit