Sommerfeld radiation condition

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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

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

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 $e - i ω t u .$ If the time-harmonic field is instead $e i ω t u,$ 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 $f (x) = δ(x - x 0),$ where $δ$ is the Dirac delta function. This problem has an infinite number of solutions. All solutions have 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 $x 0 .$

[edit] References

^ A. Sommerfeld, Partial Differential Equations in Physics, Academic Press, New York, New York, 1949.

Martin, P. A (2006). Multiple scattering: interaction of time-harmonic waves with N obstacles. Cambridge; New York: Cambridge University Press. ISBN 0521865549.