6.2: Sound Intensity
- Page ID
- 134607
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Intensity and human hearing
Another way to describe the “size” of a sound is intensity. You can think of sound as the flow of energy from one place (the source) to another (the detector). Intensity describes how “concentrated” the flow of energy is. Mathematically, intensity is defined as the amount of power flow per unit area. Intensity is expressed in Watts per square meter (W/m2). The more intensity a sound has, the louder it seems (in general).
The intensity of sound waves is quite small and can vary over an incredible range. Sound from a jet engine at 30 yards has about the same intensity as the light coming from a typical light bulb and falling on a card 10 cm away- about 10 W/m2. The quietest sound humans can hear is about ten trillion times less intense than that (roughly 1 pW/m2). The table below shows approximate intensity values for some common sounds:
| Sound | Sound Intensity (in pW/m2) |
|---|---|
| Hearing threshold | 1 |
| Rice krispies | 1,000 |
| Conversation | 1,000,000 |
| Leaf blower | 1,000,000,000 |
| Loud indoor arena | 1,000,000,000,000 |
Intensity and pressure amplitude
A sound’s intensity and its pressure amplitude are closely related- the larger the amplitude, the greater the intensity. The equation that relates sound intensity (\(I\)) and pressure amplitude (\(p\) ) is:
\[I=\frac{p^2}{\rho c}\nonumber \]
In this equation, the Greek letter rho (\(\rho\)) represents the density of the medium in which sound is traveling and lower case cee (\(c\)) represents the speed of sound in the medium.
Notice the square in the equation. Doubling the amplitude does not double the intensity. Intensity is proportional to amplitude squared. Tripling the amplitude makes the intensity nine times larger- not three times, as you might expect.
For sounds in air, we can ignore the role that the medium plays in the relationship between amplitude and intensity. However, it is important to keep in mind when comparing sounds in one medium to sounds in another- two sounds with the same amplitude can have different intensities if the sounds are traveling in two different materials.
Intensity adds up
If pressure amplitude and intensity are so closely related, why bother with intensity? Simply put, the math for intensity is easier that the math for pressure. This is because intensity is built on the concept of energy.
First and foremost, intensities of sources add together- pressures don’t. The sound from two identical jackhammers (at any given distance) is twice as intense as the sound from one jackhammer. This is because two jackhammers put out twice as much sound energy as one. (Pressure amplitude does not follow this simple rule!)
The sound intensity inside a room with a loud vacuum cleaner is roughly 100 μW/m2. What is the sound intensity in a room with two loud vacuum cleaners?
Outdoors, sound spreads out. This means that intensity decreases as you get farther and farther from a source. The energy from the source is not lost- it simply is spread out over a larger area. As we will see later, the concept of energy conservation will allows us to show (mathematically) how quickly intensity decreases as you move away from a source.
200 μW/m2.