# Sommerfeld radiation condition

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 $x_{0}.$ 