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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 Eikonal equation ： ウィキペディア英語版 Eikonal equation The eikonal equation (from German ''Eikonal'', which is from Greek εἰκών, image〔The Oxford English Dictionary. 2nd ed. 1989. OED Online. Oxford University Press. 4 April 2000 http://dictionary.oed.com/cgi/entry/00292404〕〔Evans, L. C., ''Partial Differential Equations'', AMS Graduate Texts in Mathematics, Vol. 19, pg. 93.〕) is a non-linear partial differential equation encountered in problems of wave propagation, when the wave equation is approximated using the WKB theory. It is derivable from Maxwell's equations of electromagnetics, and provides a link between physical (wave) optics and geometric (ray) optics. The eikonal equation is of the form : $|\; \backslash nabla\; u(x)|=F(x),\; \backslash \; x\backslash in\; \backslash Omega$ subject to $u|\_=0$, where $\backslash Omega$ is an open set in $\backslash mathbb^n$ with well-behaved boundary, $F(x)$ is a function with positive values, $\backslash nabla$ denotes the gradient and |·| is the Euclidean norm. Here, the right-hand side $F(x)$ is typically supplied as known input. Physically, the solution $u(x)$ is the shortest time needed to travel from the boundary $\backslash partial\; \backslash Omega$ to $x$ inside $\backslash Omega,$ with $F(x)$ being the time cost (not speed) at $x$. In the special case when $F=1$, the solution gives the signed distance from $\backslash partial\; \backslash Omega$. One fast computational algorithm to approximate the solution to the eikonal equation is the fast marching method. ==Physical interpretation== The physical meaning of the eikonal equation is related to the formula : $E\; =\; -\backslash nabla\; V$ where $E$ is the electric field strength and $V$ is the electric potential. There is a similar equation for velocity potential in fluid flow and temperature in heat transfer. The physical meaning of this equation in the electromagnetic example is that any charge in the region is pushed to move at right angles to the lines of constant potential, and along lines of force determined by the field of the E vector and the sign of the charge. Ray optics and electromagnetism are related by the fact that the eikonal equation gives a second electromagnetic formula of the same form as the potential equation above where the line of constant potential has been replaced by a line of constant phase and the force lines have been replaced by normal vectors coming out of the constant phase line at right angles. The magnitude of these normal vectors is given by the square root of the relative permittivity. The line of constant phase can be considered the edge of one of the advancing light waves. The normal vectors are the rays the light is traveling down in ray optics. 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「Eikonal equation」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.