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The Lossy 1D Wave Equation

The ideal vibrating string.

Sources of loss in a vibrating string:

  1. Yielding terminations
  2. Drag due to air viscosity
  3. Internal bending friction

Simplest case: Add a term proportional to velocity:

$\displaystyle Ky''= \epsilon {\ddot y}\underbrace{ + \mu{\dot y}}_{\hbox{new}}

More generally,

$\displaystyle Ky''= \epsilon {\ddot y}
+ \sum_{\stackrel{m=0}{m\;\hbox{\large odd}}}^{M-1}\mu_m
\frac{\partial^m y(t,x)}{\partial t^m}

where $ \mu_m$ may be determined indirectly by measuring linear damping versus frequency

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Download WaveEquations.pdf
Download WaveEquations_2up.pdf
Download WaveEquations_4up.pdf

``Distributed Modeling in Discrete Time'', by Julius O. Smith III and Nelson Lee,
REALSIMPLE Project — work supported by the Wallenberg Global Learning Network .
Released 2008-06-05 under the Creative Commons License (Attribution 2.5), by Julius O. Smith III and Nelson Lee
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University