4.2 Energy, Power, and Intensity

4.2.1 Energy Density

An infinitesimal element of the medium of mass has kinetic energy in the acoustic limit

$\begin{displaymath} dK = \frac{1}{2} dm \dot{\delta}_x^2 \end{displaymath}$

from which we obtain the kinetic energy density at the position of that element by dividing by its volume ,

$\begin{displaymath} w_k = \frac{1}{2} \rho_o \dot{\delta}_x^2 \end{displaymath}$ (4.24)
The medium is subject to a restoring pressure which stores potential energy per unit volume

$\begin{displaymath} w_u = -\int_0^{V} p \frac{dV}{V} = \frac{1}{B} \int_0^p pdp \end{displaymath}$

which according to Eq. 4.12 can be written

$\begin{displaymath} w_u = \frac{p^2}{2B} = \frac{1}{2}B\delta_x'^2 \end{displaymath}$ (4.25)
The total instantaneous energy density is then

$\begin{displaymath} w(x,t) = \frac{1}{2} \rho_o \dot{\delta}_x^2(x,t) + \frac{1}{2} B \delta_x'^2(x,t) \end{displaymath}$ (4.26)
Plane Waves
- The plane wave solutions (Eq. 4.9) to the acoustic wave equation with Eq. 4.26 yield
  
  $\begin{displaymath} w(x,t) = \left(\frac{1}{2} \rho_o \omega^2 + \frac{1}{2} B k^2 \right) \delta_{xo}^2 \cos^2 (kx - \omega t + \phi) \end{displaymath}$ (4.27)
- The phase velocity expression of Eq. 4.14 reveals that the two terms in Eq. 4.27 are equal, giving
  
  $\begin{displaymath} w(x,t) = w_o \cos^2 (kx - \omega t + \phi) \end{displaymath}$ (4.28)
  
  with
  
  $\begin{displaymath} w_o = \rho_o \omega^2 \delta_{xo}^2 = \frac{p_o^2}{B} \end{displaymath}$ (4.29)
- Often, in the context of energy, power, and intensity, we are more concerned with average quantities than instantaneous ones. The $\sin^2$ and $\cos^2$ functions both have an average value of $\frac{1}{2}$ over an integer number of periods. Hence,
  
  $\begin{displaymath} \bar{w} = \frac{1}{2} \rho_o \omega^2 \delta_{xo}^2 = \frac{p_o^2}{2B} \end{displaymath}$ (4.30)

4.2.2 Intensity

The intensity is defined as the energy per unit area transmitted by the wave past a given position per unit time. We can quantify it by considering the work done by the wave per unit time on an area perpendicular to its velocity,

$\begin{displaymath} dW = F d \delta_x = F \dot{\delta_x} dt \end{displaymath}$

which we can divide by both and the cross sectional area to which the force is applied to find the instantaneous intensity

$\begin{displaymath} I(x,t) = p(x,t) \dot{\delta}_x(x,t) \end{displaymath}$ (4.31)
Typically, when we refer to the intensity, we mean the average intensity $\bar{I}$ .
Plane Waves
- Eq.'s 4.9, 4.16, and 4.31 give an instantaneous intensity
  
  $\begin{displaymath} I(x,t) = I_o \sin^2 (kx - \omega t + \phi) \end{displaymath}$ (4.32)
  
  with
  
  $\begin{displaymath} I_o = \sqrt{\rho_o B} \omega^2 \delta_{xo}^2 = \frac{p_o^2}{\sqrt{\rho_o B}} \end{displaymath}$ (4.33)
- The average intensity is
  
  $\begin{displaymath} \bar{I} = \frac{I_o}{2} = \frac{1}{2} \sqrt{\rho_o B} \omega^2 \delta_{xo}^2 = \frac{p_o^2}{2 \sqrt{\rho_o B}} \end{displaymath}$ (4.34)
- Comparisons of Eq.'s 4.14, 4.30, and 4.34 reveal
  
  $\begin{displaymath} I(x,t) = v_p w(x,t) \end{displaymath}$ (4.35)
  
  and
  
  $\begin{displaymath} \bar{I} = v_p \bar{w} \end{displaymath}$ (4.36)

4.2.3 Decibels

The range of human hearing spans 12 orders of magnitude in intensity, from the threshold of human hearing defined as $I_t = 1\mathrm{e}{-12}$ W/m and the threshold of pain at $I \approx 1$ W/m.
A quantity spanning orders of magnitude is conveniently measured on a logarithmic scale. The decibel scale is a logarithmic sound level scale related to intensity by

$\begin{displaymath} I\mathrm{[dB]} = 10 \log \left( \frac{I}{I_t} \right) \end{displaymath}$ (4.37)

and which collapses the 12 order of magnitude range of human hearing in W/m into the range 0-120 dB.
The average human cannot distinguish between sound levels that differ by less than one decibel (1 dB).
Decibels are also used to describe amplification and attenuation ratios. In these cases, two intensities and (neither of which is likely to be ) are compared via

$\begin{displaymath} \Delta I = 10 \log\left( \frac{I'}{I} \right) \end{displaymath}$ (4.38)

For example...
- ...if the intensity of a sound is attenuated by a factor of 100 as it passes through a door, the change in sound level is -20 dB, and
- if the intensity of a sound is amplified by a factor of 1000 by a microphone/amplifier/speaker system, the change in sound level is 30 dB.

Next: 4.3 Reflection, Refraction, and Up: 4. Acoustic Waves Previous: 4.1 The Wave Equation