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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 velocity potential ： ウィキペディア英語版 velocity potential A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788. It is used in continuum mechanics, when a continuum occupies a simply-connected region and is irrotational. In such a case, : $\backslash nabla\; \backslash times\; \backslash mathbf\; =0,$ where $\backslash mathbf$ denotes the flow velocity. As a result, $\backslash mathbf$ can be represented as the gradient of a scalar function $\backslash Phi\backslash ;$: :$\backslash mathbf\; =\; \backslash nabla\; \backslash Phi\backslash \; =\; \backslash frac\; \backslash mathbf\; +$ \frac \mathbf + \frac \mathbf . $\backslash Phi\backslash ;$ is known as a velocity potential for $\backslash mathbf$. A velocity potential is not unique. If $a\backslash ;$ is a constant, or a function solely of the temporal variable, then $\backslash Phi+a(t)\backslash ;$ is also a velocity potential for $\backslash mathbf\backslash ;$. Conversely, if $\backslash Psi\backslash ;$ is a velocity potential for $\backslash mathbf\backslash ;$ then $\backslash Psi=\backslash Phi+b\backslash ;$ for some constant, or a function solely of the temporal variable $b(t)\backslash ;$. In other words, velocity potentials are unique up to a constant, or a function solely of the temporal variable. If a velocity potential satisfies Laplace equation, the flow is incompressible ; one can check this statement by, for instance, developing $\backslash nabla\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash mathbf)$ and using, thanks to the Clairaut-Schwarz's theorem, the commutation between the gradient and the laplacian operators. Unlike a stream function, a velocity potential can exist in three-dimensional flow. ==Usage in acoustics== In theoretical acoustics, it is often desirable to work with the acoustic wave equation of the velocity potential $\backslash Phi\backslash ;$ instead of pressure $p\backslash ;$ and/or particle velocity $\backslash mathbf\backslash ;$. :$\backslash nabla\; ^2\; \backslash Phi\; -\; =\; 0$ Solving the wave equation for either $p\backslash ;$ field or $\backslash mathbf\backslash ;$ field doesn't necessarily provide a simple answer for the other field. On the other hand, when $\backslash Phi\backslash ;$ is solved for, not only is $\backslash mathbf\backslash ;$ found as given above, but $p\backslash ;$ is also easily found – from the (linearised) Bernoulli equation for irrotational and unsteady flow – as : $p\; =\; -\backslash rho\; \backslash Phi$. 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「velocity potential」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.