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).

## 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."

Mathematically, consider the inhomogeneous Helmholtz equation

$(\nabla ^{2}+k^{2})u=-f{\mbox{ in }}\mathbb {R} ^{n}$

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

$\lim _{|x|\to \infty }|x|^{\frac {n-1}{2}}\left({\frac {\partial }{\partial |x|}}-ik\right)u(x)=0$

uniformly in all directions

${\hat {x}}={\frac {x}{|x|}}$

(above, $i$  is the imaginary unit and $|\cdot |$  is the Euclidean norm). Here, it is assumed that the time-harmonic field is $e^{-i\omega t}u.$  If the time-harmonic field is instead $e^{i\omega t}u,$  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

$u_{\pm }(x)={\frac {e^{\pm ik|x-x_{0}|}}{4\pi |x-x_{0}|}}.$

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 $x_{0}.$