## Sound and Vibration Basics

# dB - the decibel

**The decibel** (abbreviated dB) confuses many people, perhaps because they assume it is an absolute unit or level of sound.

The dB is not a unit in the sense that a metre or a kilogram are well-defined units of distance and weight. A decibel is the relationship or ratio between two sound levels, for example the measured sound pressure level and the minimum sound pressure level a person with good hearing can detect.

Why use decibels, why not stick to the real units that are directly measurable? Hopefully the reason will be clear when we understand how sensitive the human ear is.

The smallest sound we can hear has a sound power level of about 0.000000000001 watt/sq. metre and the threshold of pain is around 1 watt/sq. metre, which is a range of a million million to 1. So how do you describe these levels and the stages between in meaningful numbers?

Enter Alexander Graham Bell, the Scottish telephone engineer, who suggested simply converting these enormous numbers into logarithms so the threshold of hearing would be 0 and the threshold of pain would be 12 and call them Bels. This was adopted for a while, but it was soon found that compressing such a wide range down to 12 'units' was going too far the other way, it was therefore agreed to multiply the answer by 10 and call them decibels i.e. 1 bel = 10 decibels. This meant the 'normal' range would be from 0 to 120 dB, a much more sensible arrangement.

How does this work out in the real world? If we put 10 identical noise sources in a room, then there is 10 times as much sound energy so the measured sound level, sometimes called the decibel level or the dB level increases by 10 dB, which is quite logical. Similarly if we had doubled the sound power i.e. two machines, then the measured increase would only be 3 dB.

In the first case we had 10 times as much power and the log of 10 is 1 bel or 10 dB. Similarly twice as much power or a factor of 2 gives a log of 0.3 bels or 3 dB.

It follows therefore that if one machine = 90 dB, then 2 = 93 dB and 10 machines = 100 dB. If you then doubled the number of sources from 10 to 20 the measured level would only increase by a further 3 dB to 103 dB.

You would need to cram 100 machines into the room to increase the level to 110 dB or a 1000 machines to measure 120 dB, which is the threshold of pain for most people.

Conversely if you switched off one machine in a room containing 100 machines you would never notice, or measure the difference because you would have to switch off 50 before the level came down by 3 dB.

If 1 = 90 dB : 2 = 93 dB : 10 = 100 dB : 20 = 103 dB : 50 = 107 dB : 100 = 110 dB level or 110 decibels.

### Adding decibels, rounded to 0.1 dBs

If dB difference = | 0 dB | 1 dB | 2 dB | 3 dB | 4 dB | 5 dB | 6 dB | 7 dB | 8 dB | 9 dB | 10 dB | 15 dB | 20 dB |

then add ** | 3 | 2.5 | 2.1 | 1.8 | 1.5 | 1.2 | 1.0 | 0.8 | 0.6 | 0.5 | 0.4 | 0.1 | 0 |

Examples

90 dB + 80 dB = 90.4 dB

60 dB + 64 dB = 65.5 dB

25 dB + 32 dB = 32.8 dB

You may wish to check these figures for yourself using a calculator, the one included with Windows is ideal, select the Scientific option to include the log. options.

The following is for readers who are not conversant with logarithmic calculations.

Say you want to add 80 dB and 80 dB to check the 3 dB Rule and see the precise value.

Enter 8 into the calculator, the Bel value not the decibel value *

Click 10^{x} to calculate and see the anti-log value 100000000 or 10^{8}

Press +

Enter 8

Click 10^{x} to calculate and see the second anti-log value 100000000

Click = to see the total 200000000

Click log to calculate the total number of Bels = 8.3010299566.......

Finally multiply by 10 to get back to decibels = 83.010 ≈ 83 dB

* In practice we multiple Bels by 10 to give the more useful range 0 to 120 dB, see the earlier notes on this page.