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

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

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.