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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Digital Waveguide Theory
       Alternative Wave Variables
          Total Energy in a Rigidly Terminated String

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Total Energy in a Rigidly Terminated String

The total energy $ {\cal E}$ in a length $ L$, rigidly terminated, freely vibrating string can be computed as

$\displaystyle {\cal E}(t)$ $\displaystyle \isdef$ $\displaystyle \int_{0}^L W(t,x)dx = \int_{t_0}^{t_0+2L/c}{\cal P}(\tau,x) d\tau$ (C.54)
  $\displaystyle \approx$ $\displaystyle \sum_{m=0}^{N/2-1} W(t_n,x_m)X = \sum_{n=1}^{N}{\cal P}(t_0 + t_n,x_m) T$ (C.55)

for any $ x\in[0,L]$. Since the energy never decays, $ t$ and $ t_0$ are also arbitrary. Thus, because free vibrations of a doubly terminated string must be periodic in time, the total energy equals the integral of power over any period at any point along the string.
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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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