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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 \eqsp
\frac{\partial^2}{\partial 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.



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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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