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/m^{2}, the total sound intensity in the room is twice that amount, or 200μW/m^{2}.

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]

So, to increase the sound * level *by 30 dB, the sound

**must increase by a factor of [latex]10^3[/latex]! Since intensities add, that means one thousand 80 dB vacuums are needed to produce a sound level of 110 dB.**

*intensity*Do unit conversions (if needed) then plug in numbers: No unit conversions are needed.

Reflect on the answer:

- This might seem like a lot of vacuum cleaners, but keep in mind that large changes in sound intensity are needed to make small changes in sound level.
- Notice that each application of the rule
the intensity by ten. Three applications of the rule add 30 dB to the sound level, but multiply the intensity by ten three times in succession. This leads a 1000-fold increase in intensity (10*10*10).**multiplies**

### Other rules of thumb for dB

In the science of sound, there are lots of rules of thumb for decibels- too many to cover here. Some are based on intensity; others are based on pressure, amplifier power, output voltage- you name it. While all of the rules *look* different, all decibel levels have a lot in common:

- All are based on logarithms.
- All are relative measures, comparing a measurement to a reference level.
- All are linked to SIL. In most cases, the dB levels are adjusted so that they match SIL.

**The equations**

Here are the equations, for those who want to see the math. The equation for sound intensity level (SIL) is:

[latex]SIL = (10 dB) \log {\dfrac{I}{I_0}}[/latex]

where is the intensity of the sound and is the intensity of the reference sound to which your sound is being compared. If the reference level is the threshold of hearing, [latex]I_0=1 \dfrac{pW}{m^2}[/latex].

The equation for SPL is:

[latex]SPL = (20 dB) \log {\dfrac{p}{p_0}}[/latex]

where is the sound pressure of the sound and is the sound pressure for the reference sound. If the threshold of hearing is the reference, [latex]p_0=20 \mu Pa[/latex].

**The SIL of a sound always equals its SPL (unless you “mix and match” reference levels)**, even though the equations *look* different. The equations look different because pressure and intensity are not the same thing. Mathematically, the “extra” factor of two in the SPL equation exactly compensates for the square in [latex]I=\frac {p^2}{\rho c}[/latex].

### Stop to think Answer

The 80 dB sound is far more intense than the two 40 dB sources working together. Two identical 40 dB sources produce a sound level of only 43 dB. (Doubling the intensity only leads to a 3 dB increase in the sound level).