   August 9, 2016

# Why Decibels?

Let’s face it, the decibel, universally used by acousticians, can be a confusing concept.  Even some of my very distinguished colleagues in other technical fields cannot get comfortable with it.   So as a public service in this blog I will attempt to explain how it works and why it is a natural way to describe sound.

My own introduction to the dB was problematic.  Upon leaving graduate school I was fortunate enough to have been offered a job at Bolt Beranek and Newman (BBN), Acentech’s prestigious forebear.   I had studied physics but I had not spent one hour in an acoustics course, but I was able to get to work in BBN’s underwater acoustics projects as I discovered that acoustics is the application of physical science to the everyday world.  But before I could do anything I had to get used to the decibel, which I knew existed but was as confused by it as many people initially are.  This took some time.

Before we begin we need to note that the dB is mostly used as a single number to describe the  amplitude or “strength” of a sound.  As such, it cannot describe anything about its other important properties such as frequency, temporal nature, and so on.  For our purposes let’s posit that the dB represents all of a sound’s energy summed over all its frequencies.  A more detailed discussion of the frequency spectrum of sound could be the topic of another blog.

So where does the dB come from?  The answer is in two parts, one based on the physics of sound in the natural world and the other on the biological response to this sound.  The physics is simple:  Sound in air is oscillating air pressure and our perception of sound is proportional to its intensity which is proportional to the square of the pressure. The key observation is that the range of sound intensity in the environment is astonishingly large.  For example, at the threshold for human hearing the pressure-squared is on the order of 4 x 10-10 Pa2, where Pa is the abbreviation of Pascals, the measure of pressure in the metric system. Sound in a quiet room is on the order of 10-6 Pa2 and louder sounds such a diesel truck are on the order of 1 Pa2.  The loudest rock concerts? As high as 100 Pa2. This range of sound pressures is a factor of 1011 = 100,000,000,000.   This is a rough estimate of the number of stars in the galaxy!

How can we think and talk intuitively about such huge differences using our normal way of thinking, which is most comfortable with linear quantities?  How could you plot them on a graph?  We could perhaps just refer to the exponents in the quantities, referring to these levels as “exponent -6” or “+2”.  What is amazing is that this is almost exactly how Mother Nature has solved the problem:  Our auditory system has evolved to perceive sound in proportion to the logarithm of the sound power. This is equivalent to converting sound levels to exponents, as for example log(102) = +2.  An equivalent description is that the auditory system perceives equal ratios of sound equally rather than equal differences. This is because equal ratios have the same logarithm.  For example, a change from 1 to 2 would appear to be the same increase as a change of 2 to 4.  If our auditory system worked linearly, we would say the first was a change of 2-1 = 1 and the second as a change of 4-2 = 2, which is much larger.

The decibel system is based on this principle, with the man-made addition of a scaling factor of 10 and a reference value for the pressure pref, so that the dB measure of a sound pressure level (SPL) of pressure p is defined by the familiar equation: Where pref is chosen to be 20 x 10-6 Pa, which is a measure of the threshold of human hearing, so that the sound we can just barely hear is at a SPL of 0 dB.  (This is a little like Daniel Fahrenheit’s 1724 assignment of zero degrees to the coldest temperature he could create in his lab. The “bel” part is named for Alexander Graham Bell.)  On this scale our rock concert comes in at about 130 dB, so we have converted the range of sound pressure from 11 orders of magnitude to a range of a values on the order of 100, which is something we can deal with intuitively.  The factor of ten in the definition determines the usual scaling laws, such as a difference of two in intensity is the equivalent of 3 dB, etc. One price that has to be paid for the logarithmic form is the sometimes confusing and anti-intuitive rules about combining decibels, so that for example two sources at, say, 100 dB when added give a sound pressure level of 103 dB and not 200 dB.

Another interesting example of evolution’s choice of logarithms is found in astronomy.  Ancient astronomers grouped stars according to their apparent brightness, assigning the brightest they could see to stellar magnitude 1 and the dimmest to stellar magnitude 6.  Today we understand that these are logarithmic-based perceptions of the actual light flux.  The modern definition of stellar magnitudes is where F is the optical flux from a star or other astronomical object with Fres as a reference value set to approximate the early assignments.  Modern astronomy uses this to extend the system to magnitudes less than zero for inherently bright objects and greater than 6 to describe the light from stars too dim to be seen by the naked eye.  Here at least “magnitudes” doesn’t sound as obscure as “decibels.”

To summarize, the decibel is a natural way to describe sound that works in the same way as our logarithmic perception of sound that has evolved to deal with the extraordinary range of sounds in the environment.  If it sometimes can seem mystifying, this is because our rational minds think linearly while our biological audio (and optical) systems work logarithmically.  Keep this in mind next time you find yourself confused by the humble decibel.

### One response to “Why Decibels?”

1. Jeff Burnett says:

Thank you!
Very clear thinking.

Jeff Burnett (newly retired WSU Pullman Architecture professor – acoustics course)