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Energy Density Waves

The vibrational energy per unit length along the string, or wave energy density [320] 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=}\,\frac{f^{{+}}(n)v^{+}(n)}{c}
\,\mathrel{\mathop=}\,\epsilon \left[v^{+}(n)\right]^2 \,\mathrel{\mathop=}\,\frac{\left[f^{{+}}(n)\right]^2}{K}, \\
W^{-}(n) &=& \frac{{\cal P}^{-}(n)}{c} \,\mathrel{\mathop=}\,-\frac{f^{{-}}(n)v^{-}(n)}{c}
\,\mathrel{\mathop=}\,\epsilon \left[v^{-}(n)\right]^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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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