Intensity of waves
Intensity
The intensity of a wave is the power (time rate of energy transfer) the wave transfers through a unit area perpendicular to the direction of propagation of the wave:
\[I := \frac{P}{A}.\] If the power changes over the surface, we must define the intensity for a very small surface for which the power can be approximated as constant and thus the simple ratio becomes a derivative: \[I := \frac{\mathrm{d}P}{\mathrm{d}A}.\]
Example: intensity of a speaker
Let us examine how the intensity associated with a 10-W speaker decays as a function of the distance from the speaker. For simplicity, let us consider the speaker point-like and assume that spherical waves originate from this point. This means that at a distance \(r\) from the source, the power of \(P = 10\,\mathrm{W}\) is evenly distributed over the surface of a sphere of radius \(r,\) and this power is transported in directions that are perpendicular to this surface everywhere. The corresponding surface area is
\[A = 4\pi r^2,\]
so the intensity can be obtained as
\[I = \frac{P}{A} = \frac{10\,\mathrm{W}}{4 \pi r^2},\]
that is, it decays in inverse proportion to the square of the distance from the source.