Sommerfeld radiation condition

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 k0 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.

Notes

References