Loudness perception
43 Decibels (the math)
Avoiding the calculator
Even though decibels are based on logarithms, you can do many calculations with decibels without a calculator. The key is the two simple rules of thumb shown below. These rules allow you to move back and forth between intensity and sound intensity level without a calculator:
| Change in intensity | Change in intensity level |
| Multiply/divide by two | Add/subtract 3 dB |
| Multiply/divide by ten | Add/subtract 10 dB |
According to the first rule, if the sound intensity doubles, the SIL goes up by 3 dB. According to the second rule, if the intensity is multiplied by ten, the SIL goes up by 10 dB. Notice that multiplication/division for intensity “turns into” addition/subtraction for SIL (and vice versa).
Combining sources
When you have multiple sources of sound in a room, the intensities add together. For example, if there are two loud vacuum cleaners in a room, and each produces a sound intensity of 100μW/m2, the total sound intensity in the room is twice that amount, or 200μW/m2.
However, when sources combine, the sound levels do not add together. Two 80 dB vacuums create a sound level of just 83 dB! Going from one vacuum to two does double the sound intensity, but doubling the intensity only increases the sound level by 3 dB. (Remember the rules of thumb!) If you look at a decibel chart for common sounds (like the one in the last section), you realize that two 80 dB vacuums could not possibly create a sound level of 160 dB- the combined sound would have an SIL greater than a jet at takeoff (130 dB)!
Stop to think
Which is more intense: the sound from a single 80 dB source or the sound from two 40 dB sources?
The following example shows you how to apply the rules of thumb to a problem.
Example: Lots of vacuums!
QUESTION:
How many 80 dB vacuum cleaners must be operating at once to create a sound level of 110 dB?
SOLUTION:
Identify important physics concept: Sound intensities (not sound levels) add. We’ll also need the second rule of thumb from the chart below to connect sound intensity with sound level:
| Change in intensity | Change in intensity level |
| Multiply/divide by two | Add/subtract 3 dB |
| Multiply/divide by ten | Add/subtract 10 dB |
List known and unknown quantities (with letter names and units):
[latex]SIL=80 \: dB[/latex] (for one vacuum)
[latex]SIL=110 \: dB[/latex] (for ??? vacuums)
I will call the sound intensity produced by a single 80 dB vacuum cleaner “1 unit” of intensity.
[latex]I=1 \: unit[/latex] (for one vacuum)
[latex]I=?\: units[/latex] (for # of vacs needed for 110 dB)
Do the algebra: Getting from 80 dB to 110 dB is a matter of applying the second rule in the chart three times. Each time the SIL goes up by ten dB, the intensity is ten times what it was before:
[latex]I=10 \: units[/latex] when [latex]SIL=90 \: dB[/latex]
[latex]I=100 \: units[/latex] when [latex]SIL=100 \: dB[/latex]
[latex]I=1000 \: units[/latex] when [latex]SIL=110 \: dB[/latex]
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