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A partial differential equation (PDE) extends ODEs by adding one or more independent variables (usually spatial variables). For example, the wave equation for the ideal vibrating string adds one spatial dimension $ x$ (along the axis of the string) and may be written as follows:

$\displaystyle K\, y''(x,t) = \epsilon \, {\ddot y}(t)$   (Restoring Force = Inertial Force)$\displaystyle , \protect$ (2.1)

where $ y(x,t)$ denotes the transverse displacement of the string at position $ x$ along the string and time $ t$ , and $ y'(x,t)\isdeftext \partial y(x,t)/\partial x$ denotes the partial derivative of $ y$ with respect to $ x$ .2.7 The physical parameters in this case are string tension $ K$ and string mass-density $ \epsilon $ . This PDE is the starting point for both digital waveguide models (Chapter 6) and finite difference schemesC.2.1).

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University