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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Elementary Physics, Mechanics, and Acoustics
       Wave Equation in Higher Dimensions

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Wave Equation in Higher Dimensions

The wave equation in 1D, 2D, or 3D may be written as

$\displaystyle \left(\nabla ^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} \right) z(\underline{x},t) \eqsp 0, \protect$ (B.47)

where, in 3D, $ z(\underline{x},t)$ denotes the amplitude of the wave at time $ t$ and position $ \underline{x}\in{\bf R}^3$, and

$\displaystyle \nabla ^2 \isdefs \nabla \cdot \nabla \isdefs \nabla ^T\nabla \eq... ...tial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} $

denotes the Laplacian operator in Euclidean coordinates. (In general coordinates, it is often denoted by $ \Delta$.) To investigate solutions of the wave equation, as pursued in §B.8.3 below, it is useful to first develop some simple expressions and notations for elementary waves in 2D and 3D.


Subsections
Previous: Air Absorption
Next: Plane Waves in Air

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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