Acoustic Energy Density

The two forms of energy in a wave are kinetic and potential:

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

These are called the acoustic kinetic and potential energy densities, respectively.

In a plane wave, where $p=Rv$ and $I=pv$, we have

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

Thus, half of the acoustic intensity $I$ in a plane wave is kinetic, and the other half is potential:²

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

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.