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Acoustic Energy Density

The two forms of energy in a wave are kinetic and potential. Denoting them at a particular time $ t$ and position $ \underline{x}$ by $ w_v(t,\underline{x})$ and $ w_p(t,\underline{x})$ , respectively, we can write them in terms of velocity $ v$ and wave impedance $ R=\rho c$ as follows:

\begin{eqnarray*}
w_v &=& \frac{1}{2} \rho v^2 \eqsp \frac{1}{2c} R v^2 \quad\left(\frac{\mbox{\small Energy}}{\mbox{\small Volume}}\right)\\ [10pt]
w_p &=& \frac{1}{2} \frac{p^2}{\rho c^2} \eqsp \frac{1}{2c} \frac{p^2}{R} \quad\left(\frac{\mbox{\small Energy}}{\mbox{\small Volume}}\right)
\end{eqnarray*}

More specifically, $ w_v$ and $ w_p$ may be called the acoustic kinetic energy density and the acoustic potential energy density, respectively.

At each point in a plane wave, we have $ p(t,\underline{x})=R\,v(t,\underline{x})$ (pressure equals wave-impedance times velocity), and so

\begin{eqnarray*}
w_v &=& \frac{1}{2c} R v^2 = \frac{1}{2}\cdot \frac{I}{c}\\
w_p &=& \frac{1}{2c} \frac{p^2}{R} = \frac{1}{2} \cdot \frac{I}{c},
\end{eqnarray*}

where $ I(t,\underline{x})\isdef p(t,\underline{x})\,v(t,\underline{x})$ denotes the acoustic intensity (pressure times velocity) at time $ t$ and position $ \underline{x}$ . Thus, half of the acoustic intensity $ I$ in a plane wave is kinetic, and the other half is potential:B.30

$\displaystyle \frac{I}{c} = w = w_v+w_p = 2w_v = 2w_p
$

Note that acoustic intensity $ I$ has units of energy per unit area per unit time while the acoustic energy density $ w=I/c$ has units of energy per unit volume.


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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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