Acoustic Energy Density

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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 and position $\underline{x}$ $w_v(t,\underline{x})$ and $w_p(t,\underline{x})$ , respectively, we can write them in terms velocity and wave impedance $R=\rho c$ as follows:

$\begin{eqnarray*} w_v &=& \frac{1}{2} \rho v^2 = \frac{1}{2c} R v^2 \quad\left(\... ...ad\left(\frac{\mbox{\small Energy}}{\mbox{\small Volume}}\right) \end{eqnarray*}$

More specifically, and 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 and position $\underline{x}$ . Thus, half of the acoustic intensity in a plane wave is kinetic, and the other half is potential:^F.12

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Physical Audio Signal Processing
    Elementary Physics, Mechanics, and Acoustics
       Properties of Gases
          Acoustic Energy Density