Sommerfeld radiation condition

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." [A. Sommerfeld, "Partial Differential Equations in Physics", Academic Press, New York, New York, 1949.]

Mathematically, consider the inhomogeneous Helmholtz equation

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

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

: $lim\_.$

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

*cite book
last = Martin
first = P. A
title = Multiple scattering: interaction of time-harmonic waves with N obstacles
publisher = Cambridge; New York: Cambridge University Press
date = 2006
pages =
isbn = 0521865549

External links

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