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Energy Density Waves

Energy density = potential + kinetic energy densities:

$\displaystyle W(t,x) \mathrel{\stackrel{\mathrm{\Delta}}{=}}
\underbrace{\frac{1}{2} Ky'^2(t,x)}_{\hbox{potential}}
+
\underbrace{\frac{1}{2} \epsilon {\dot y}^2(t,x)}_{\hbox{kinetic}}
$

Sampled wave energy density can be expressed as

$\displaystyle W(t_n,x_m) \mathrel{\stackrel{\mathrm{\Delta}}{=}}W^{+}(n-m) + W^{-}(n+m)
$

where

\begin{eqnarray*}
W^{+}(n) &=& \frac{{\cal P}^{+}(n)}{c} = \frac{f^{{+}}(n)v^{+}(n)}{c}
= \epsilon \left[v^{+}(n)\right]^2 = \frac{\left[f^{{+}}(n)\right]^2}{K} \\
W^{-}(n) &=& \frac{{\cal P}^{-}(n)}{c} = -\frac{f^{{-}}(n)v^{-}(n)}{c}
= \epsilon \left[v^{-}(n)\right]^2 = \frac{\left[f^{{-}}(n)\right]^2}{K} \nonumber
\end{eqnarray*}

Total wave energy in string of length $ L$ :

$\displaystyle {\cal E}(t) = \int_{x=0}^L W(t,x)dx
\approx \sum_{m = 0}^{L/X-1} W(t,x_m)X
$


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``Choice of Wave Variables in Digital Waveguide Models'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2019-02-05 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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