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Sommerfeld radiation condition

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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[1] 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

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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."[2]

Mathematically, consider the inhomogeneous Helmholtz equation

where is the dimension of the space, is a given function with compact support representing a bounded source of energy, and is a constant, called the wavenumber. A solution to this equation is called radiating if it satisfies the Sommerfeld radiation condition

uniformly in all directions

(above, is the imaginary unit and is the Euclidean norm). Here, it is assumed that the time-harmonic field is If the time-harmonic field is instead one should replace with 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 in three dimensions, so the function in the Helmholtz equation is where is the Dirac delta function. This problem has an infinite number of solutions, for example, any function of the form

where is a constant, and

Of all these solutions, only satisfies the Sommerfeld radiation condition and corresponds to a field radiating from The other solutions are unphysical [citation needed]. For example, can be interpreted as energy coming from infinity and sinking at [3]

See also

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Notes

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References

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  • Sommerfeld, A. (1912). "Die Greensche Funktion der Schwingungslgleichung". Jahresbericht der Deutschen Mathematiker-Vereinigung. 21: 309–352. ISSN 0012-0456.
  • Sommerfeld, Arnold (1967). Partial Differential Equations in Physics. ISBN 0-12-654656-8.
  • Martin, P. A. (2006). Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles. Cambridge University Press. doi:10.1017/cbo9780511735110. ISBN 978-0-521-86554-8.
  • Schot, Steven H (1992). "Eighty years of Sommerfeld's radiation condition". Historia Mathematica. 19 (4): 385–401. doi:10.1016/0315-0860(92)90004-U.
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