Sound waves spread out in three-dimensional space from the source of sound so the sound wave is three dimensional wave while the transverse wave in a string is one dimensional. We use power to define the strength of one dimensional waves, however intensity is used for three dimensional waves.
Intensity of a wave is the average rate of energy transfer per unit area perpendicular to the direction of the propagation of the wave. It's the same as the average power of the wave per unit area.
You can see in Figure 2 that a sound source emits sound waves in three dimensional space. The imaginary spherical surfaces of radius \(r_1\) and \(r_2\) enclose the source of sound.
The average power \(P_\text{av}\) through both spherical surfaces is the same. Now the intensity \(I_1\) through the spherical surface of radius \(r_1\) is \(I_1 = P_\text{av} /4\pi {r_1}^2\) and the intensity \(I_2\) through the spherical surface of radius \(r_2\) is \(I_2 = P_\text{av} /4\pi {r_2}^2\). Therefore,
\[\frac{{{I_1}}}{{{I_2}}} = \frac{{P_\text{av}/4\pi {r_1}^2}}{{P_\text{av}/4\pi {r_2}^2}} = \frac{{{r_2}^2}}{{{r_1}^2}} \tag{6} \label{6}\]
The above equation tells us that the intensity of a sound wave is inversely proportional to the square of distance from the source of sound. The above equation is called inverse square law for three dimensional waves.
