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Physical Audio Signal Processing
    Digital Waveguide Theory
       Alternative Wave Variables
          Energy Density Waves

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Energy Density Waves

The vibrational energy per unit length along the string, or wave energy density [317] is given by the sum of potential and kinetic energy densities:

$\displaystyle W(t,x) \isdefs \underbrace{\frac{1}{2} Ky'^2(t,x)}_{\mbox{potential}} + \underbrace{\frac{1}{2} \epsilon {\dot y}^2(t,x)}_{\mbox{kinetic}}$ (C.50)

Sampling across time and space, and substituting traveling wave components, one can show in a few lines of algebra that the sampled wave energy density is given by

$\displaystyle W(t_n,x_m) \isdefs W^{+}(n-m) + W^{-}(n+m),$ (C.51)

where

\begin{eqnarray*} W^{+}(n) &=& \frac{{\cal P}^{+}(n)}{c} \,\mathrel{\mathop=}\,\... ...ht]^2 \,\mathrel{\mathop=}\,\frac{\left[f^{{-}}(n)\right]^2}{K}. \end{eqnarray*}

Thus, traveling power waves (energy per unit time) can be converted to energy density waves (energy per unit length) by simply dividing by $ c$, the speed of propagation. Quite naturally, the total wave energy in the string is given by the integral along the string of the energy density:

$\displaystyle {\cal E}(t) \,\mathrel{\mathop=}\,\int_{x=-\infty}^\infty W(t,x)dx \approx \sum_{m = -\infty}^\infty W(t,x_m)X$ (C.52)

In practice, of course, the string length is finite, and the limits of integration are from the $ x$ coordinate of the left endpoint to that of the right endpoint, e.g., 0 to $ L$.

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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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