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The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in physics—such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics. Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.〔() (retrieved 13 Nov 2012).〕〔Gerard F Wheeler. (The Vibrating String Controversy, ) (retrieved 13 Nov 2012). Am. J. Phys., 1987, v55, n1, p33-37.〕〔For a special collection of the 9 groundbreaking papers by the three authors, see (First Appearance of the wave equation: D'Alembert, Leonhard Euler, Daniel Bernoulli. - the controversy about vibrating strings ) (retrieved 13 Nov 2012). Herman HJ Lynge and Son.〕〔For de Lagrange's contributions to the acoustic wave equation, can consult (Acoustics: An Introduction to Its Physical Principles and Applications ) Allan D. Pierce, Acoustical Soc of America, 1989; page 18.(retrieved 9 Dec 2012)〕 In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.〔Speiser, David. ''(Discovering the Principles of Mechanics 1600-1800 )'', p. 191 (Basel: Birkhäuser, 2008).〕 == Introduction == The wave equation is a hyperbolic partial differential equation. It typically concerns a time variable , one or more spatial variables , and a scalar function , whose values could model, for example, the mechanical displacement of a wave. The wave equation for is : where ∇2 is the (spatial) Laplacian and ''c'' is a fixed constant. Solutions of this equation describe propagation of disturbances out from the region at a fixed speed in one or in all spatial directions, as do physical waves from plane or localized sources; the constant ''c'' is identified with the propagation speed of the wave. This equation is linear. Therefore, the sum of any two solutions is again a solution: in physics this property is called the superposition principle. The wave equation alone does not specify a physical solution; a unique solution is usually obtained by setting a problem with further conditions, such as initial conditions, which prescribe the amplitude and phase of the wave. Another important class of problems occurs in enclosed spaces specified by boundary conditions, for which the solutions represent standing waves, or harmonics, analogous to the harmonics of musical instruments. The wave equation, and modifications of it, are also found in elasticity, quantum mechanics, plasma physics and general relativity. 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「wave equation」の詳細全文を読む スポンサード リンク
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