We have probably all seen charts of the decibel scale like the one below. But unless you know how the scale works, you might be led to think that a rock concert is twice as loud as a conversation. And instinctively, you might realise this cannot possibly be the case.
This post aims here to demystify the decibel scale. There is more than one layer of demystification to take place, too, so stop at whichever point suits you best (hopefully, though, not this point right here!).
Logarithmic scales
The first question is what decibels measure. We will base this post in the realm of acoustics, since it is probably in connection with sound that most people are familiar with this unit, but the principles can be applied to other applications such as signal levels in electronics, and amplification.
A tempting answer to our question might be ‘loudness’. Quite often, you will see use of the cunning term ‘sound level’ (well, I did mention ‘signal levels’ for electronics…). For those who think those two terms are synonymous, we will provide a discussion of the problem with the term ‘loudness’ toward the end of the post.
In fact, even for the term ‘sound level’, there are two different ways of calculating decibels, depending on the meaning you give to it. We will see this in the section ‘power quantities and field quantities’.
The next question is ‘how do the numbers on the scale work?’
Many scales are ‘linear’, so that equal shifts along the scale represent equal differences in the quantity concerned. So, when measuring length, the difference between 6 metres and 7 metres is the same number of metres (one) as the difference between 78 metres and 79 metres. In addition, 120 metres is twice as far as 60 m.
The decibel scale cannot be a ‘linear scale’ – we have already described that 120 dB must be much more than twice the sound level of 60 dB. In fact, it is a logarithmic scale. Logarithmic scales are common, but often people are not shown how they work, why they are used, or when they are in use – hence this post. A logarithmic scale doesn’t step up in equal intervals, but instead in equal ratios. The graph of decibels is labelled in increments of 20 dB. The difference from 40 dB to 60 dB is a step up of 100 times the power. So is the difference from 100 dB to 120 dB. But those two steps are not the same number of Watts of power.
In the same way, Jeff Bezos is approximately 1.5 times as wealthy as Mark Zuckerberg (time.com, 2018). The difference in their wealth is $40 billion. But if you are 1.5 times as wealthy as me, then you do not have $40 billion more than me.
Logarithmic scales are useful when trying to display values with wide variation. Indeed, the rock concert at 120 dB represents a rate of energy transfer to your eardrum ten billion times that provided by rustling leaves at 20 dB (you will see where ‘ten billion’ comes from soon).
Defining the decibel
We have described qualitatively how logarithmic scales work; to get any further with the decibel, we’ll need to define it properly…
When dealing with ‘power quantities’ the decibel (dB) expresses the ratio of two power values as 10 times the logarithm to base 10 of the ratio of the quantities.
We’ll see why the caveat ‘when dealing with power quantities’ is relevant soon. But there’s a more pressing issue. Some of you might not be all that used to logarithms (maybe skip the next section if you are).
Logarithms
The logarithm to base 10 of a quantity is the power you need to raise 10 to, to obtain that quantity.
For example, the logarithm of 1000 is 3, because you raise 10 to the power of 3 to get 1000. Likewise:
- the logarithm of 10,000 is 4, because you raise 10 to the power of 4 to get 10,000
- the logarithm of 100,000 is 5, because you raise 10 to the power of 5 to get 100,000
- the logarithm of 1,000,000 is 6, because you raise 10 to the power of 6 to get 1,000,000
Notice from the list above that as the ‘target number’ gets 10 times larger, the logarithm increases by one. This is the origin of the behaviour of logarithmic scales – ‘equal increases on the scale result in equal multiplications in the quantity’.
Aside: You can have a logarithm to any base, not just 10 – it’s just the power you raise that other base to. A very common base is ‘e’, which we have written about here on the main part of our website. But throughout this post, the base will always be 10, so we are just going to say ‘logarithm’ from now on, and we will mean ‘logarithm to the base 10’. (10 is a very sensible base, because we run our lives according to a decimal number system).
Returning to the decibel
Now we can make some headway with the definition ‘the decibel (dB) expresses the ratio of two power values as 10 times the logarithm to base 10 of the ratio of the quantities’.
Imagine two measured power values of 3500 and 350 (it doesn’t matter what the units are, as long as they are the same, because we are interested in the ratio. And the actual numbers are semi-random – all that counts is that one is ten times bigger than the other).
Then, applying the definition of decibels:
- the ratio of the two values is 10
- the logarithm of that ratio is 1
- Multiplying by 10 gives 10 dB
- One power value is 10 dB more than the other
That’s potentially misleading, because 10 times the power makes 10 dB difference, and you could easily jump to the wrong conclusion – you might think that 100 times the power makes 100 decibels difference. And it doesn’t – let’s see why, with new values 47,000 and 470:
- The ratio of the two values is 100
- The logarithm of that ratio is 2
- Multiplying by 10 gives 20 dB
- One power value is 20 dB more than the other
Every increase of 10 decibels represents a multiplication of the power by 10. An increase of 20 dB is two increases of ‘10 times’. And that makes ‘100 times’.
We can see now why the rock concert is 10 billion times the sound level of the rustling leaves. They are separated by 100 dB. 100 dB means 10 lots of 10 times the power, and 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10 billion…
The modified decibel chart below shows this more visually (except that the dB axis goes up in steps of 20, not 10, purely for visual clarity).
Power quantities and field quantities
It’s time to deal with our cunning use of the word ‘sound level’. Sound is a pressure wave, as we described in this previous post, which you may like to turn to if you are not ready for the following sections. ‘Sound level’ can refer to two different properties of that pressure wave.
One is the power, as described previously. The other is the amplitude (the maximum displacement from equilibrium). And, importantly, the power is proportional to the square of the amplitude. So to increase the power by a factor of 10, the amplitude need only increase by a factor of 3.16 (because 3.16 squared is ten).
It is possible to calculate dB from a ratio of amplitudes, rather than powers, but the fact that the ratio of the amplitudes will be the square root of the ratio of the powers makes a difference. To compensate, the number of the decibels is the logarithm of the ratio of the amplitudes, but multiplied by 20, instead of 10. Which of these calculations is most convenient often depends on relative ease of measurement, and in acoustics it is often easier to measure the amplitude of a wave (say, using a microphone and an oscilloscope) than to measure its power.
We will summarise that paragraph in two ways. The first is with the only equations in this post:
[latex]\mathrm{dB}=10\log(P_1/P_2)[/latex]
and
[latex]\mathrm{dB}=20\log(p_1/p_2)[/latex]
where (confusingly – sorry!) capital [latex]P[/latex] stands for power, and lower case [latex]p[/latex] stands for pressure.
The second way is with the diagram below, which gives a more visual representation of this behaviour.
In the diagram above, the sound at the top, compared to the reference sound in red, has a number of decibels calculated by [latex]10\log(10/1)[/latex], or [latex]20\log(3.16/1)[/latex]. Both versions give the answer ’+10 dB’.
Quantities that depend on amplitude, and require multiplication by 20 rather than 10 are known as field quantities (in fact, the standard ISO 80000 tells us we should call them root-power quantities, but I’m not convinced we’re all doing that yet). Quantities that depend on power are called power quantities (!). So how dB are calculated depends on whether you are dealing with a power quantity or a field quantity.
Doubling
The previous section explains a past confusion of my own. I spent ages hearing that ‘an increase of 3 dB is a doubling’ or ‘an increase of 6 dB is a doubling’ and got very confused. I wondered which was true. It took me ages to realise that they could both be true, depending on the nature of what is being doubled:
- +3dB is a doubling of power (approximately, but it is pretty close)
- +6 dB is a doubling of amplitude (which multiplies power by 4)
Once I realised that, my confusion evaporated fairly quickly. Again, the situations of +3 dB and +6 dB are represented visually below.
Decibels are always a ratio
All this discussion has concerned comparing two sound levels. The calculation of decibels requires a ratio.
So what do we mean when we describe an alarm clock as 80 dB. After all, we aren’t comparing alarm clocks with anything, are we? We’re just making a statement about alarm clocks.
Well, actually we are making a comparison. Use of decibels is always a comparison. When we use decibels to represent an absolute value, we have to decide upon a reference value and compare it to that. In the case of acoustics, the value 0 dB is taken to be the threshold of human hearing (and that is given a value, either in Watts or Pascals, depending on whether we are working with power or pressure). Then, for example an ‘absolute sound level’ of 20 dB is 100 times the power of that threshold.
Loudness
Notice that I haven’t said at any point that +20 dB means ‘100 times louder’. The reason for that is that pressures and powers are nice easy physical quantities. ‘Loudness’ is a perception, and as such it is partly physical, partly physiological and partly psychological (and consequently harder to deal with!). The sensitivity of human hearing depends upon frequency too, so the relationship between the power of a sound and how we perceive it varies with pitch. Also, to realise how difficult ‘loudness’ is as a concept, imagine me saying to you that one sound is ‘twice as loud’ as another. What does that even convey? ‘Louder’, sure, we could agree upon, but ‘twice as loud’? What does that mean? I’m not saying it’s meaningless, but it sure isn’t straightforward.
As a next step, if you are still interested, you could look up the unit ‘phons’ and ‘sones’. This University of New South Wales page is a good place to start.
Meanwhile, next time, we will look at another logarithmic scale – the earthquake magnitude scale, commonly known as the Richter scale.