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/m^{2}. That's 0.0000000000000001 W per square
meter! The upper end of the dynamic range is 10^{-4},
or 0.0001 W/m^{2}. The total dynamic range of sound
intensity that we normally encounter is then 1,000,000,000,000
W/m^{2}; a trillion W/m^{2}! These are not
numbers we want to write out or try to keep track of.

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 Log_{10}
of a
number is simply the number of zeros after the "1" in that
huge number (1 trillion) above. For example,
Log_{10}(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 Log_{10} is just the
number of zeros plus one (to get to the whole number 1 and
not 0.1): Log_{10}(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 __deci__bel (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/m^{2}). 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/m^{2}. To calculate dB IL we use the formula below:

"IL" stands for "intensity level", "I" for the observed or measured intensity, and "I

For example, if you measure a sound intensity of 10^{-7}
W/m^{2}, how would you express that in dB IL?

10Log_{10}(10^{-7} / 10^{-12}) = ? dB IL

10Log_{10}(10^{5}) = ? dB IL

10(5) = 50 dB IL

10Log

10(5) = 50 dB IL

Now you try one:

10Log_{10}(10^{-15} / 10^{-12}) = ? dB IL

10Log_{10}(10^{-3}) = ? dB IL

10(-3) = -30 dB IL

10Log

10(-3) = -30 dB IL

Relative sound intensity is useful, but audiologists mostly
deal with sound pressure. Recall p = F/m^{2}, 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/I_{r}, however,
we substitute pressure:

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?

20Log_{10}(40 / 20) = ? dB SPL

20Log_{10}(2) = ? dB SPL

20(0.3) = 6 dB SPL

20Log

20(0.3) = 6 dB SPL

Now you try one:

20Log_{10}(5/20) = ? dB SPL

20Log_{10}(0.25) = ? dB SPL

20(-0.6) = -12 dB SPL

20Log

20(-0.6) = -12 dB SPL

Simple addition does **not** work for combining two
pressure levels in dB SPL. Remember that even though
dB SPL lets us work with nice-looking numbers, we are
actually dealing with the logarithm of a ratio.

Next Topic: Simple Harmonic Motion

REFERENCES

Speaks, C. E. (2017).