Math Department, Mission
College, Santa
Clara, California

Go to Math Dept Main
Page | Go to Mission
College Main Page

This paper was written as an assignment for Ian
Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material from this paper, please acknowledge
it.

To explore other such papers go to the Math G
Projects Page

Jake Foraker

Math G

Professor Walton

5 April 2010

The Power of Ten  -  Decibels

If
a tree falls in the woods and nobody is around to hear it fall, how many
decibels (dB) of sound pressure level (SPL) will it register? Given my recently
acquired and well-read knowledge of the decibel, an educated guess would put it
somewhere in the range of 110-130 dB.

“What
is this mumbo-jumbo ‘decibel’ jargon you speak of, Jake?” I
hear you ask… “And what the heck does it have to do with Math
G?” another curious reader ponders…

Well,
as it turns out, the concept of the decibel has plenty to do with math;
it’s one of those classic cross-subject interdisciplinary mathematical
ideas that has become the basis for the measurement of something humans
encounter at every moment of every day. That “something,” is
sound!

The
human ear has the astonishing capability of perceiving an incredibly large
range of sound pressure levels. According to Everest, if you were to sit in an
anechoic chamber for a long period of time, you would eventually be able to
hear the sound of air particles “pounding” against your eardrums,
revealing the threshold of human hearing (33). That is, we cannot detect
anything softer than air particles on the eardrum, or else it would simply be
“drowned out” by air particle noise (34). On the flipside of that,
we are capable of enduring a sonic beating, dancing in front of massive speakers
at a rock concert and surviving until the next day to tell our friends how loud
it was.

This
notion of perceivable extremes is precisely why the concept of the decibel and
its mathematical formula were created. We all know that sound is caused by
vibrations in a given atmosphere (gas, solid, or liquid), thus resulting in
variations in atmospheric pressure. These variations can be measured—more
on that in a moment. Moreover, the term sound
power is used as a description of “the amount of acoustic energy
produced per unit time by a sound source,” measured in watts (Prout/Bienvenue
139). Sounds heard in everyday life can be as tiny as 0.000,000,001 watt (a
soft whisper), or as deafening as 50,000,000 watts (a rocket engine blast).
Herein lies the brilliance of the decibel: it makes the measurement of such
drastic differences in value convenient by utilizing our good friend, the
logarithm.

Decibels,
similar to percentages, are a dimensionless unit. It is simply the measurement
of a ratio of between two units. In this case, it is the measurement of the
ratio between P, the sound power in watts, and Pref, the
reference power in watts.

dB = 10 log(P / Pref )

where

P = signal power

Pref  = reference power

Now, in
order to utilize this particular logarithmic function, we must first have an
agreed-upon reference power. Like I said before regarding the human threshold
of hearing, we can perceive a very wide range of sound pressures, and the
common agreed-upon reference uses the lowest perceivable measurement for
pressure, called micropascals (µPa).
One million micropascals are equal to one pascal
(Pa), and one pascal is the
equivalent to one newton (a unit of
force) per square meter (Prout/Bienvenue 140). Don’t get confused with
the specifics, just note that our “widely agreed-upon standard
rule” uses a miniscule vibration of air pressure which corresponds
roughly with the human threshold for hearing as the base reference for our
logarithmic scale of decibels.

“I
don’t get it. Is there a number for this abstract ‘threshold of
hearing’?”

Yes
sir. For sound in air the standard reference pressure is 20 µPa.  In real life, this is comparable to a
mosquito buzzing around your head at a close distance, as observed by F. Alton
Everest, author of The Master Handbook of Acoustics. To put it in
perspective, a jet engine 75 feet away is about 100,000,000 µPa. So, let
20 µPa = 0 dB. Now we have reference for which to base our logarithm.

As you
imagine hearing the jet engine and the mosquito, it is important to think about
what sounds fall in-between. How does an opera singer relate to a mosquito?
What about police car sirens? Where would they land on the pascal scale?

Notice
on the chart that the decibel levels stay in a manageable range of 0-140,
whereas the units of micropascals can confuse the reader in a sea of zeros.
That is the magic of the logarithm. Through multiplication of powers of ten,
the logarithm manages the values into friendly numbers. For example, the quotient
of sound intensities:

109 watts per square meter and

104 watts per square meter

is
100,000 units. It can be expressed as a difference of 50 decibels. Ten times
the log base ten of the ratio of powers.

Decibels and Logarithms

The decibel scale is a reflection
of the logarithmic response of the human ear to changes in sound intensity:

The logarithm to the base 10 used
in this expression is just the power of 10 of the quantity in brackets according
to the basic definition of the logarithm:

Examples:

Chart from R. Nave
of Georgia State
University

Making sense now? Not quite? How about
this…

All of the tens and logs come together full
circle. An increase in power by a factor of ten causes a +10 dB increase in
level.

An increase in power by a factor of 100 causes a
dB increase of 20…

An increase in power by a factor of 1000 causes
a dB increase of 30…

An increase in power by a factor
of 1,000,000 causes a dB increase of 60… The reason it is 60 and not 6 is
because in our original formula, dB = 10 log(P / Pref
), the ten in front of the log multiplies it by ten, hence creating the
“deci” bel. Without the ten, the formula would produce the solution
for bels (B), a less convenient unit considered too large for practical
applications (Nave).

Hypothetical Decibel
Scale

Type of Sound

Units of measurement

Decibel Scale Value

Rustle of leaves

10

10 dB

Whisper

100

20 dB

Soft conversation

1,000

30 dB

Average residence

10,000

40 dB

Average office

100,000

50 dB

Telephone conversation

10,000,000

70 dB

Heavy traffic

1,000,000,000

90 dB

Subway traffic

10,000,000,000

100 dB

Airplane engine

1,000,000,000,000

120 dB

Chart from
www.privateline.com

Speaking of bels, it is only appropriate to address
one of the founding fathers of the decibel, Alexander Graham Bell (1847-1922). Bell is most commonly known
for being the inventor of the telephone, but he also held a deep passion for
the study of acoustics. Both of Alexander’s parents were deaf, so his
work was greatly influenced by the mysteries of hearing and electronics. Originally
created by engineers at Bell Laboratories, the bel of the early days was known
as a “transmission unit” or TU. The early transmission
unit’s
original purpose was to show and measure the levels in which audio signals reduced over a one-mile stretch of telephone wire
(History). It was renamed to the “bel” (B) in 1923 in honor of the
great Alexander Graham Bell and his extensive contributions to the world of
telecommunications.

Along with the
ever-advancing technological world we live in, the decibel has become a widely
used unit of measurement for a multitude of scientific practices. Electronics
in particular utilizes the extensive abilities of the decibel in countless ways.
Just to name a few dB variations: dBu describes electric field intensity, dBµV
denotes voltage expressed in relation to one microvolt, dBm is a power level
above one milliwatt, and dBd has a complicated relationship with radio antenna
waves (“Field”).

The creation of the decibel has
certainly changed the way scientists, engineers, audiophiles and students alike
take measurements of extreme values. By using the mathematical power of the logarithm,
the decibel makes sense of very large ratios and turns them into numbers of
reasonable size. From the early bel (B) to the modern dBm used in wifi network
computing, it has been a fascinating ride tracing this widely misunderstood
dimensionless unit. It shall be an interesting voyage to watch the future
progression of the decibel, to see what realms of study it may demystify as our
world becomes increasingly advanced in all aspects of life.

.

Works Cited

1.      Everest,
F. Alton. The Master Handbook of Acoustics. Pennsylvania:
McGraw Hill, 1994.

2.      Prout,
James H. and Gordon Bienvenue. Acoustics For You. Florida:
Krieger, 1990.

3.      Nave,
Carl R. Hyperphysics. 2005. Georgia State University. 29 Mar. 2010

< http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html>.

4.      Farley,
Tom. “The Acoustic dB.” Privateline. 1 Apr. 2010

< http://www.privateline.com/decibel/decibel.html>.

5.      Wolfe,
Joe. dB: What is a Decibel? 1998. University of New
South Wales. 1 Apr. 2010

<
http://www.phys.unsw.edu.au/jw/dB.html#logs>.

6.      “Alexander
Graham Bell.” Solar Navigator. 1
Apr. 2010

<
http://www.solarnavigator.net/inventors/alexander_graham_bell.htm>.

7.      “Decibel
(dB).” Handbook for Acoustic Ecology. Simon Fraser University. 2 Apr. 2010.

< http://www.sfu.ca/sonic-studio/handbook/Decibel.html>.

8.      “Field
Intensity Units.” Softwright LLC. 1999. 2 Apr. 2010.

<http://www.softwright.com/faq/engineering/FIELD%20INTENSITY%20UNITS.html>.