# Bels and Decibels

Written by Travis M. Moore
Last edited 20-Sep-2019

When we talk about the range of pressures over which human beings can hear, we must consider the low end (the softest sound a person can hear) and the high end (the most intense sound a person can hear without causing damage). All of the pressures between the low and high end make up the dynamic range of human hearing.

The dynamic range of sound intensities humans routinely experience is enormous. The softest audible sound is roughly 10-16 W/m2. That's 0.0000000000000001 W per square meter! The upper end of the dynamic range is 10-4, or 0.0001 W/m2. The total dynamic range of sound intensity that we normally encounter is then 1,000,000,000,000 W/m2; a trillion W/m2! These are not numbers we want to write out or try to keep track of.

### Bels and Decibels

How do we shrink large numbers? One way is to take the logarithm. Specifically for acoustics, the base 10 logarithm works well. But what's a logarithm? Despite its cryptic name, the Log10 of a number is simply the number of zeros after the "1" in that huge number (1 trillion) above. For example, Log10(1,000,000,000,000) = 12. Much better! A more formal definition of a logarithm is the power to which a base number (e.g., 10) must be raised to result in a specified number. So to get 1,000,000,000,000 the number 1 must be raised to the power of 12. In other words, the decimal point of 1.0 must be moved 12 times.

When the number is smaller than one, the Log10 is just the number of zeros plus one (to get to the whole number 1 and not 0.1): Log10(0.0001) = 4 (move the decimal 4 times to reach 1.0). We now have a dynamic range of 4 to 16! This scale is called the Bel scale, after Alexander Graham Bell, who was involved with the invention of the first telephone. Unfortunately, this scale is too small to be of much use. We would be using a lot of decimals to refer to small changes in sound pressure (e.g., 6.3478 Bels).

A simple solution to expanding the Bel scale is to multiply it by 10; hence decibel (dB). We also want to specify we measured the sound intensity, so we append "IL" for "intensity level" on the end: dB IL. The range of numbers we use to quantify sound intensity is now around 120 dB IL.

In acoustics it is often useful to refer to the intensity of one sound compared to another. This gives us some reference of intensity for a starting point. Intensity expressed in reference to a standard intensity is known as relative sound intensity. (Contrast this with absolute sound intensity in W/m2). Because audiologists are often interested in measuring hearing thresholds, it makes sense to refer to the intensity of a very soft sound. The standard reference is 1X10-12 W/m2. To calculate dB IL we use the formula below:

For example, if you measure a sound intensity of 10-7 W/m2, how would you express that in dB IL?

Now you try one:

10Log10(10-15 / 10-12) = ? dB IL
10Log10(10-3) = ? dB IL
10(-3) = -30 dB IL

### Decibels and Sound Pressure

Relative sound intensity is useful, but audiologists mostly deal with sound pressure. Recall p = F/m2, and the units of pressure are Pa. In fact, because sound pressures are so small, we use micro Pascals (μPa). It so happens that intensity is proportional to pressure squared. So we can keep the outline of the formula for dB IL. Instead of measured intensity over a reference intensity I/Ir, however, we substitute pressure:

dB sound pressure = 10Log10(p2/pr2)
We can even clean this up a bit and multiply the exponents out of the equation. We are then left with: dB pressure level = 20Log10( p/pr). To specify pressure level, we add "SPL" for "sound pressure level": dB SPL.

It is important to note that sound pressure level is also a relative quantity (i.e., compared to a reference). The standard reference pressure used when discussing sounds pertinent to humans is 20 μPa. Let's walk through an example.

Suppose you measure a sound with a relative pressure of 40 μPa. What is this in dB SPL?

Now you try one:

20Log10(5/20) = ? dB SPL
20Log10(0.25) = ? dB SPL
20(-0.6) = -12 dB SPL