Sommerfeld radiation condition

In applied 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).

Formulation

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."A. Sommerfeld, ''Partial Differential Equations in Physics'', Academic Press, New York, New York, 1949.

Mathematically, consider the inhomogeneous Helmholtz equation

:$(\nabla^2 + k^2) u = -f \text \mathbb R^n$

where $n=2, 3$ is the dimension of the space, $f$ is a given function with compact support representing a bounded source of energy, and $k>0$ is a constant, called the ''wavenumber''. A solution $u$ to this equation is called ''radiating'' if it satisfies the Sommerfeld radiation condition

: $\lim_ , x, ^ \left( \frac - ik \right) u(x) = 0$

uniformly in all directions

:$\hat = \frac$

(above, $i$ is the imaginary unit and $, \cdot,$ is the Euclidean norm). Here, it is assumed that the time-harmonic field is $e^u.$ If the time-harmonic field is instead $e^u,$ one should replace $-i$ with $+i$ 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 $x_0$ in three dimensions, so the function $f$ in the Helmholtz equation is $f(x)=\delta(x-x_0),$ where $\delta$ is the Dirac delta function. This problem has an infinite number of solutions, for example, any function of the form

:$u = cu_+ + (1-c) u_- \,$

where $c$ is a constant, and

: $u_(x) = \frac.$

Of all these solutions, only $u_+$ satisfies the Sommerfeld radiation condition and corresponds to a field radiating from $x_0.$ The other solutions are unphysical . For example, $u_$ can be interpreted as energy coming from infinity and sinking at $x_0.$

See also

* Limiting absorption principle
* Limiting amplitude principle
* 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