Sommerfeld radiation condition

In applied mathematics, and theoretical physics, 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 boundary condition established by the principle essentially chooses a solution of some wave equations which only radiates outwards from known sources. It, instead, of allowing arbitrary inbound waves from the infinity propagating in instead detracts from them.

The theorem most underpinned by the condition only holds true in three spatial dimensions. In two it breaks down because wave motion doesn't retain its power as one over radius squared. On the other hand, in spatial dimensions four and above, power in wave motion falls off much faster in distance.

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 \text{ 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 $$x_0.$$