1. Exploring Loudness.
You hear a loud noise. Which wave property or properties (frequency, wavelength, wave speed, amplitude, wave number, phase, etc.) do you think is related to how loud it is?
Section 7.11 Wave Intensity
Exercises Warm-up Activity
Consider a transverse wave on a horizontal string. Any point along the string moves up and down as a wave passes through it. The average speed of a point on the string is related to the amplitude of the wave and also to how quickly the motion changes from up to down – thus the frequency.
\begin{equation*}
K = \frac{1}{2}mv^2
\end{equation*}
\begin{equation*}
E \propto f^2 A^2
\end{equation*}
The kinetic energy of the mass at that point depends on the square of the speed. We can infer that the energy of the wave goes as the square of the amplitude, and the square of the frequency.
Definition 7.11.1. Intensity.
The intensity of a wave emanating from a point source:
\begin{equation*}
I = \frac{P}{a}
\end{equation*}
where \(P\) is power and \(a\) is the surface area of a sphere.
We define the intensity \(I\) of a wave to be the power delivered by the wave, divided by the area that the wave impinges upon. The units of intensity are Watts per square meter.
For a point source wave, emanating as a spherical wave front, the area is the surface area of a sphere. Intensity takes the form:
\begin{equation*}
I = \frac{P}{4\pi r^2}
\end{equation*}
We can visualize this by considering the energy passing through a small area of a sphere of radius \(r\text{.}\) The same amount of energy must pass through a larger area of a sphere of radius \(2r\text{.}\) Since the surface area of the larger sphere increases by a factor of \(r^2\text{,}\) the energy passing through the same area must be \(\frac{1}{4}\) as much.
Exercises Activities
1. Explanation - Distance and Intensity.
Your friend is a distance \(d\) from a point source of sound and measures the intensity of the sound wave to be \(I\text{.}\) How far away from the source of the wave do you need to be to measure the intensity to be \(\frac{I}{2}\text{?}\)
References References
[1]
Figure 7.11.2 created by Kathryn Hadley.
