Acoustic intensity and power

A basic characteristic of a source is the rate at which it transfers energy. If we multiply equation 3.2a by $c^{2}\rho'$,

$$c^{2}\rho'\frac{\partial \rho'}{\partial t} + \rho_{0}c^{2}\rho'\frac{\partial v}{\partial x} = 0$$ (13)

and note that $\rho'\partial \rho'/\partial t = \frac{1}{2}(\partial/\partial t){\rho'}^{2}$ and that $c^{2}\rho' = p'$,

$$\frac{c^{2}}{\rho_{0}}\frac{1}{2}\frac{\partial }{\partial t}{\rho'}^{2} + p'\frac{\partial v}{\partial x} = 0.$$

Multiplying the momentum equation 3.2b by $v$ gives

$$\rho_{0}v\frac{\partial v}{\partial t} + v\frac{\partial p'}{\partial x} = 0,$$

which can be rearranged:

$$\frac{1}{2}\rho_{0}\frac{\partial}{\partial t}v^{2} + v\frac{\partial p'}{\partial x} = 0.$$ (14)

Adding equations 3.13 and 3.14 gives a result for the energy transport in the sound field:

$$\frac{\partial}{\partial t}\left(\frac{1}{2}\rho_{0}v^{2} +\frac{... ...frac{c^{2}}{\rho_{0}}{\rho'}^{2}\right) + \frac{\partial}{\partial x}(p'v) = 0.$$ (15)

In equation 3.15, $\rho_{0}v^{2}/2$ is the kinetic energy per unit volume, $c^{2}/\rho_{0}{\rho'}^{2}/2$ is the potential energy per unit volume and $p'v$ is the acoustic intensity $I$ which is the rate of energy transport across unit area. Equation 3.15 is a statement of energy conservation for the system and says that the rate of change of energy in a region is equal to the net rate at which energy is carried into the region.

If apply the relationship between pressure and velocity, equation 3.12, the acoustic intensity is

$$I = \frac{p^{2}}{\rho c} + \frac{\partial}{\partial t} \left( \frac{f^{2}(t-R/c)}{2\rho R^{3}} \right).$$

If we average $I$ over time for a periodic wave, the second term has a mean value of zero and the resulting mean intensity is:

$$\bar{I} = \frac{\overline{p^{2}}}{\rho c}.$$ (16)