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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Digital Waveguide Models
       Ideal Vibrating String
          Sampled Traveling-Wave Solution

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Sampled Traveling-Wave Solution

As discussed in more detail in Appendix C, the continuous traveling-wave solution to the wave equation given in Eq.$ \,$(6.2) can be sampled to yield

$\displaystyle y(nT,mX)$ $\displaystyle =$ $\displaystyle y_r(nT-mX/c) + y_l(nT+mX/c)$   $\displaystyle \mbox{(set $X=cT$)}$  
  $\displaystyle =$ $\displaystyle y_r(nT-mT) + y_l(nT+mT)$  
  $\displaystyle \isdef$ $\displaystyle y^{+}(n-m) + y^{-}(n+m) \protect,$ (7.3)

where $ T$ denotes the time sampling interval in seconds, $ X=cT$ denotes the spatial sampling interval in meters, and $ y^{+}$ and $ y^{-}$ are defined for notational convenience.

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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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