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."[1]

Mathematically, consider the inhomogeneous Helmholtz equation

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

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

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

uniformly in all directions

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

(above, ${\displaystyle i}$ is the imaginary unit and ${\displaystyle |\cdot |}$ is the Euclidean norm). Here, it is assumed that the time-harmonic field is ${\displaystyle e^{-i\omega t}u.}$ If the time-harmonic field is instead ${\displaystyle e^{i\omega t}u,}$ one should replace ${\displaystyle -i}$ with ${\displaystyle +i}$ 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 ${\displaystyle x_{0}}$ in three dimensions, so the function ${\displaystyle f}$ in the Helmholtz equation is ${\displaystyle f(x)=\delta (x-x_{0}),}$ where ${\displaystyle \delta }$ is the Dirac delta function. This problem has an infinite number of solutions, for example, any function of the form

${\displaystyle u=cu_{+}+(1-c)u_{-}\,}$

where ${\displaystyle c}$ is a constant, and

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

Of all these solutions, only ${\displaystyle u_{+}}$ satisfies the Sommerfeld radiation condition and corresponds to a field radiating from ${\displaystyle x_{0}.}$ The other solutions are unphysical. For example, ${\displaystyle u_{-}}$ can be interpreted as energy coming from infinity and sinking at ${\displaystyle x_{0}.}$