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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Sommerfeld radiation condition ： ウィキペディア英語版 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 :$$ (\nabla^2 + k^2) u = -f \mbox \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 : $\backslash lim\_\; |x|^\}\; \backslash left(\; \backslash frac\; -\; ik\; \backslash right)\; u(x)\; =\; 0$ uniformly in all directions :$\backslash hat\; =\; \backslash frac$ (above, $i$ is the imaginary unit and $|\backslash cdot|$ is the Euclidean norm). Here, it is assumed that the time-harmonic field is $e^u.$ If the time-harmonic field is instead $e^u,$ 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 : $u\_(x)\; =\; \backslash frac.$ 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== 〔 * * "Eighty years of Sommerfeld’s radiation condition", Steven H. Schot, ''Historia Mathematica'' 19, #4 (November 1992), pp. 385-401, . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sommerfeld radiation condition」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.