\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\)

Decibels

When a wave hits a receiver (like a microphone or your eardrum), the energy in the wave is transferred to the receiver. Because the total energy received depends in part on the area of the receiver and on time, we would rather talk about the intensity of the wave (cf Intensity) which we measure in watts per square meter.

For sound, intensity is a measure of how loud a sound is, and the intensity of the sounds we encounter in everyday life vary greatly. The threshold of hearing (the quietest sound we can hear) is $10^{-12}\u{W/m^2}$. In a quiet room, the intensity might be $I=10^{-8}\u{W/m^2}$. Inside a moving car, the intensity is 10,000 times greater, or $10^{-4}\u{W/m^2}$, while a loud rock concert is 10,000 times greater than that, or $1\u{W/m^2}$. Instead of using scientific notation to talk about intensities, it would be simpler to just give the exponent on the $10$.

Fortunately, we have a math function which will do just that: the log function (see Logarithms). If we look at the log (base 10) of the intensity, it is $\log_{10}I=-12$ at the threshold of hearing, $-8$ for the quiet room, $-4$ for the moving car, and $0$ for the rock concert. We can then add 12 to all these numbers (so that the threshold of hearing is $0$) to get the sound intensity level: $$\beta = (\log_{10}I+12)\u{B}$$ which is measured in the unit bels (B), named after Alexander Graham Bell. It is much more common (for reasons I don't understand) to measure sound intensity levels in decibels, so the more common equation is

Some more examples of decibel levels are shown to the right.

Multiple Objects

Suppose we have a single object that creates a sound with intensity $I$ and decibel level $\beta_1=10(\log_{10}I+12)$. Intensity, like energy, is additive, so if we had $N$ copies of that object, it will create a sound with intensity of $NI$. Decibels work differently, however: $$\beta_N = 10\u{dB}(\log_{10}NI+12)$$ $$=10\u{dB}(\log_{10}N+\log_{10}I+12)$$ $$=10\u{dB}(\log_{10}I+12) + 10\u{dB}\log_{10}N $$

For instance, if there are $N=2$ objects that each produce a sound of $\beta_1=40\u{dB}$, then the decibel level of both is $$40\u{dB}+10\log_{10}2 = 40+10(0.301) = 40+3 = 43\u{dB}$$ If there are $N=5$ objects, then we add $10\log_{10}5 = 7dB$ to the initial level, and so forth.

Distance

The sound intensity level you hear depends on how far away you are from the source. If you hear a sound with decibel level $\beta$ at a distance $r$, then at a different distance $r'$, the sound intensity level is

For instance, if you move twice as far away from the source, then the sound intensity level decreases by $20\log_{10}{2r\over r}=20\log_{10}2$ by 6 decibels.