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Sommerfeld radiation condition

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[edit]

Arnold Sommerfeld defined the condition of radiation for a scalar field satisfying the Helmholtz equationas

"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

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

where ${\displaystyle n=2,3}$ is the dimension of the space, ${\displaystyle f}$ is a given function with compact support representing a bounded source of energy, and ${\displaystyle k>0}$ is a constant, called the wavenumber. A solution ${\displaystyle u}$ to this equation is called radiating if it satisfies the Sommerfeld radiation condition

${\displaystyle \lim _{|x|\to \infty }|x|^{\frac {n-1}{2}}\left({\frac {\partial }{\partial |x|}}-ik\right)u($x)=0}

uniformly in all directions

${\displaystyle {\hat {x}}={\frac {x}{|x|}}}$

(above, ${\displaystyle i}$ is the imaginary unit and ${\displaystyle |\cdot |}$ is the Euclidean norm). Here, it is assumed that the time-harmonic field is ${\displaystyle e^{-i\omega t}u.}$ If the time-harmonic field is instead ${\displaystyle e^{i\omega t}u,}$ one should replace ${\displaystyle -i}$ with ${\displaystyle +i}$ 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 ${\displaystyle x_{0}}$ in three dimensions, so the function ${\displaystyle f}$ in the Helmholtz equation is ${\displaystyle f($x)=\delta (x-x_{0}),} where ${\displaystyle \delta }$ is the Dirac delta function. This problem has an infinite number of solutions, for example, any function of the form

${\displaystyle u=cu_{+}+(1-c)u_{-}\,}$

where ${\displaystyle c}$ is a constant, and

${\displaystyle u_{\pm }($x)={\frac {e^{\pm ik|x-x_{0}|}}{4\pi |x-x_{0}|}}.}

Of all these solutions, only ${\displaystyle u_{+}}$ satisfies the Sommerfeld radiation condition and corresponds to a field radiating from ${\displaystyle x_{0}.}$ The other solutions are unphysical ^{[citation needed]}. For example, ${\displaystyle u_{-}}$ can be interpreted as energy coming from infinity and sinking at ${\displaystyle x_{0}.}$

See also[edit]

• Limiting absorption principle
• Limiting amplitude principle
• Nonradiation condition

References[edit]

• 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.

External links[edit]

• A.G. Sveshnikov (2001) [1994], "Radiation conditions", Encyclopedia of Mathematics, EMS Press