080 - Breaking Down the Decibel
I needed an expert to explain this one. TPA Contributor Madeleine Campbell breaks down the decibel.
Written by Madeleine Campbell
This fall I began pursuing my doctorate of audiology at City University New York. In one of my courses this semester, I learned about a fundamental audio topic that, as silly as it sounds, I hadn't previously given enough thought — the decibel!
Sound levels are measured in decibels, but what does this actually mean?
A decibel is a logarithmic unit of the ratio of two levels — the magnitude of a sound compared with a reference, which for our purposes would be the threshold of human hearing, the quietest possible sound we can perceive. Let’s unpack this!
As sound waves travel, the energy of air molecules propagates through the air. We can measure sound as power (in Watts) or force (in Pascals) over an area. Loudness is our perception of this!
(power = work/time and pressure = force/unit area)
Let’s start with power. Sound power is the rate at which that energy is transferred, measured with respect to the sound source. So when we say “sound power” or “acoustic power” we’re referring to the total power produced by the sound source in all directions. Again, this value is measured in Watts.
Sound intensity refers to the power per unit area, like Watts/m2 or Watts/cm2.
In absolute terms, acoustic power is quite small. For example, the acoustic power of a lawn mower is about .01 W. Compare this with an incandescent light bulb, which typically has 60 W of power. Our threshold of human hearing is 10-12 W — yes, one trillionth of a Watt.
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Again, when we talk about sound power level, we’re talking about the ratio of the power of a sound in comparison to our threshold of hearing. The absolute power of a given sound doesn’t mean anything on it’s own. Our threshold of hearing serves as a reference point upon which it can be based.
How, then, do we end up with decibels?
The human ear responds to a really wide range of intensity and pressure levels — a ratio of 1012 to 1! Instead of a linear value, it's much more convenient to express these relative magnitudes as a logarithmic value, which we call the bel, named after Alexander Graham Bell.
(N) bel = log10 (IX ÷ IR)
where N is number of bels, Ix is the intensity level of the sound in question and IR is our reference intensity, which, for our purposes, is the threshold of hearing, 1012Watts/m2
Simply put, this is the logarithm of the ratio.
On a linear scale, a change between two values is perceived by the difference between those values. For example, a change from 1 to 2 is perceived as the same amount of increase as a change from 2 to 3 or 3 to 4. On a logarithmic scale, a change between two values is perceived on the basis of their ratio. A change from 1 to 2 (1:2) is perceived as the same amount of increase as a change from 4:8 or 8:16.
Whereas a linear sequence of numbers might read: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, a logarithmic sequence would read -.01, -0.1, 1, 10, 100, 1000, 10,000, 100,000. Each increase is a power of 10.
Think about a volume knob. It’s easier to represent the range of magnitudes, from barely audible to very loud, as 1 to 10 rather than 1 to 10,000,000,000,000!
To avoid the excess use of decimals, we simplify the bel into the decibel, which is one-tenth of a bel. 1 bel = 10dB and 1 dB = 0.1 bel.
It’s important to account for this in our equation by multiplying the logarithm by 10 →
(N) dB = 10 log10 (IX ÷ IR)
This value represents the dB intensity level.
Let’s say we double the intensity of a sound — what is the change in dB? If we double the intensity, our ratio is 2:1, where 2 is the louder sound and 1 is the initial sound, the point of reference.
(N) dB = 10 log10 (2 ÷ 1)
(N) dB = 10 log10 (2)
(N) dB = 10 (.3) → 3dB
Therefore, doubling the intensity of a sound results in an increase of 3dB.
An increase of 10dB is actually increasing the intensity of a sound by a factor of 10.
As audio engineers, we more often look to the sound pressure level. Remember that acoustic power is a measure of the rate at which acoustic energy propagates through the air. This is to say that power takes time into account. On the other hand, sound pressure, the force per unit area, is an instantaneous measure. Pressure is measured in Pascals and is proportional to power2 — this is an important fact that impacts our equation!
(N) dB = 10 log10 (PX ÷ PR)2
Based on the rules of logarithms, we can move the 2 in front.
(N) dB SPL = 2 • 10 log10 (PX ÷ PR) → (N) dB = 20 log10 (PX ÷ PR)
In the same way that 10-12 reflects our threshold of hearing in Watts, a unit of power, 20μPa (micropascals) reflects our threshold of hearing of Pascals, a unit of pressure.
Let’s say we double the pressure of a sound — what is the change in dB?
(N) dB = 20 log10 (PX ÷ PR)
(N) dB = 20 log10 (2 ÷ 1)
(N) dB = 20 log10 (2)
(N) dB = 20 (.3) → 6dB
Doubling the pressure of a sound results in an increase of 6dB.
If these concepts are new to you and feel overwhelming, don’t worry! It’s a lot to take in, but when we know what “decibel” actually means, we can make more informed mixing decisions.
