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\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.

References[edit]

  1. ^ 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 0-521-86554-9. 
  • "Eighty years of Sommerfeld’s radiation condition", Steven H. Schot, Historia Mathematica 19, #4 (November 1992), pp. 385-401, doi:10.1016/0315-0860(92)90004-U.

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