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."{{sfn | Sommerfeld | 1967 | p=189}}

Mathematically, consider the inhomogeneous Helmholtz equation

:$$

(\nabla^2 + k^2) u = -f \text{ 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

: $\backslash lim\_$

uniformly in all directions

:$\backslash hat\{x\}\; =\; \backslash frac\{x\}$

(above, $i$ is the imaginary unit and $|\backslash cdot|$ is the Euclidean norm). Here, it is assumed that the time-harmonic field is $e^\{-i\backslash omega\; t\}u.$ If the time-harmonic field is instead $e^\{i\backslash omega\; t\}u,$ 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 $f(x)=\backslash delta(x-x\_0),$ where $\backslash delta$ is the Dirac delta function. This problem has an infinite number of solutions, for example, any function of the form

:$u\; =\; cu\_+\; +\; (1-c)\; u\_-\; \backslash ,$

where $c$ is a constant, and

: $u\_\{\backslash pm\}(x)\; =\; \backslash frac\{e^\{\backslash pm\; ik|x-x\_0|\}\}\{4\backslash pi\; |x-x\_0|\}.$

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 {{citation needed|date=April 2022}}. For example, $u\_\{-\}$ can be interpreted as energy coming from infinity and sinking at $x\_0.${{sfn | Schot | 1992 | p=394}}

• Limiting absorption principle
• Limiting amplitude principle
• Nonradiation condition

{{reflist}}

• {{cite journal | last=Sommerfeld | first=A. | title=Die Greensche Funktion der Schwingungsgleichung. | journal=Jahresbericht der Deutschen Mathematiker-Vereinigung | volume=21 | issn=0012-0456 | pages=309–352 | url=https://gdz.sub.uni-goettingen.de/id/PPN37721857X_0021?tify=%7B%22view%22:%22info%22,%22pages%22:%5B317%5D%7D | year=1912}}
• {{cite book | last=Sommerfeld | first=Arnold | title=Partial Differential Equations in Physics | date=1967 | isbn=0-12-654656-8}}
• {{cite book | last=Martin | first=P. A. | title=Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles | publisher=Cambridge University Press | date=2006 | isbn=978-0-521-86554-8 | doi=10.1017/cbo9780511735110}}
• {{cite journal | last=Schot | first=Steven H | title=Eighty years of Sommerfeld's radiation condition | journal=Historia Mathematica | volume=19 | issue=4 | date=1992 | doi=10.1016/0315-0860(92)90004-U | pages=385–401}}

• {{springer|id=r/r077060|title=Radiation conditions|author=A.G. Sveshnikov}}

Category:Radiation

Category:Boundary conditions