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 and is closely related to the limiting absorption principle (1905) and the limiting amplitude principle (1948).
- "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."
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
- A. Sommerfeld (1912). "Die Greensche Funktion der Schwingungslgleichung". Jahresbericht der Deutschen Mathematiker-Vereinigung. 21: 309–353.
- A. Sommerfeld, Partial Differential Equations in Physics, Academic Press, New York, New York, 1949.
- Martin, P. A (2006). Multiple scattering: interaction of time-harmonic waves with N obstacles. Cambridge; New York: Cambridge University Press. ISBN 0-521-86554-9.
- "Eighty years of Sommerfeld’s radiation condition", Steven H. Schot, Historia Mathematica 19, #4 (November 1992), pp. 385-401, doi:10.1016/0315-0860(92)90004-U.