PHYS 140
Sound Intensity
The “loudness” of a sound is determined by the amplitude of the sound wave. Physicists
characterize loudness by measuring sound intensity (I) and sound intensity level (L).
Sound intensity is measured in Watts/m2. A Watt (abbreviated W) is the unit of power.
What is power?
As you know, waves carry energy. Thus, a speaker, musical instrument, or other source
of sound emits energy. Power is defined to be the amount of energy emitted per unit
time.
It is measured in Watts, where 1 Watt = 1 Joule / second. (A Joule is a small unit of
energy. A household lightbulb might emit 60 Joules per second, or 60 Watts.)
Fun fact: Humans release energy in the form of heat. The power emitted by a typical
person is about the same as your average lightbulb. (The reason humans don’t glow like
lightbulbs is that the heat is released in the form of infrared light, which is invisible.)
Sound Intensity
The lower threshold of human hearing is about I = 10-12 W/ m2. This is a very tiny
number: 1 divided by 1,000,000,000,000 (1 followed by 12 zeroes.) A typical rock
concert has a sound intensity of about I = 1 W/m2.

How much higher is the sound intensity of a rock concert (I2) compared to the
softest sound a human can hear (I1)?
Whispering has a sound intensity of about 10-8 W/m2.

How much higher is the sound intensity of a rock concert compared to a whisper?
Sound Intensity Level (decibel level)
Rock concerts, while loud, don’t really sound more than a million times louder than a
whisper. “Sound intensity” is an awkward concept because it doesn’t really reflect the
perceived loudness of sound. This is why the decibel (dB) scale was invented.
“Sound intensity level” is abbreviated “L” and measured in decibels. L = 0 dB
corresponds to the threshold of human hearing, and L = 120 dB corresponds to a sound
intensity of 1 W/m2. Most sounds you encounter on an everyday basis fall between 0 and
100 dB.
Humans usually perceive an increase of 10 dB to be a doubling of “loudness.”
The decibel scale is logarithmic.
What is a Logarithm?
Taking the logarithm of a number is like extracting the exponent.
1) Examples:
Log(1012) = 12
Log(100) = Log(102) = 2
Log(10) = Log(101) = 1
Log(1) = 0
2) Now do these ones:
Log(108) =
Log(1000) =
3) Use google calculator or a physical calculator to do these:
Log(5) =
Log(78) =
4) More examples:
Log(105/107) = Log(105-7) = Log(10-2) = -2
Log(10-5/10-7) = Log(10-5+7) = Log(102) = 2
5) Do these without a calculator:
Log(10-2/10-4) =
Log(102/10-4) =
Log(10-8/108) =
Log(100/1012) =
Log(10/10-12) =
Random Fact:
Earthquake strength is also measured on a logarithmic scale (the Richter scale). Just as a 70 dB
noise is 10 times more intense than a 60 dB noise, an earthquake of magnitude Richter 7.0 is 10
times as powerful as one of magnitude Richter 6.0.
People often find logarithmic scales like the Richter scale and the Decibel scale counter-intuitive.
“I have no proof of this, but I think the decibel was invented in a bar, late one night, by a
committee of drunken electrical engineers who wanted to take revenge on the world for their total
lack of dancing partners. Apart from the calculation problems, the use of intensity for measuring
sounds is indirect and overly complicated.” Powell, How Music Works
How to calculate sound intensity level
To calculate sound intensity level in dB, you take something’s sound intensity (I2) and
compare it to the lowest audible sound intensity (I1 = 10-12 W/ m2) using a logarithm:
L (in decibels) = 10 Log(I2/ I1).
1) What is the sound intensity level, in decibels, of a sound that corresponds to I2 =
10-5 W/m2? (Don’t use a calculator; you can do these on your own!)
1) What is the sound intensity level, in dB, of a sound that corresponds to.1 W/m2?
(It may help to write .1 in scientific notation.)
2) What is the sound intensity level, in dB, of a sound that corresponds to 100
W/m2?
3) What is the sound intensity if a sound has L = 30 dB?
4) What is the sound intensity if a sound has L = 60 dB?
6) How much more intense is an 80 dB sound compared to a 70 dB sound? (On a log
scale such as the decibel scale, an increase of 10 dB means that the sound is 10 times as
intense.)
7) How much more intense is 80 dB compared to 50 dB?
8) Let’s find out what happens to the decibel level (L) when you double the intensity of a
sound (I).
Sound intensity level (L) in decibels = 10 Log(I2/ I1). = 10 Log (2) = ______________.
(Here we assumed that I2 = 2 I1, so that I2/ I1 = 2.)
Thus, doubling the intensity of a sound increases the decibel level by _____________.
9) Sounds softer than 0dB are hard for humans to hear, but that doesn’t mean they don’t
exist. They have negative values for the decibel level.
What sound intensity level (L) would correspond to a sound intensity of 10-13 W/m2?
Frequency Response
We’ve been discussing perceived sound intensity / loudness as if it’s independent of
frequency. In practice, however, the perceived loudness of a sound depends on
frequency, because human hearing is more sensitive to some frequencies than others.
Human hearing is very good at detecting frequencies centered around 4000 Hz (this is a
very high frequency—about that of the highest note on a piano). On the other hand,
humans are bad a detecting low frequencies (the lowest note on a piano is 28 Hz).
To be perceived as having the same loudness, a note with f = 28 Hz must have a much
higher intensity than a note with f = 4000 Hz.
Example: According to your textbook, a 50 Hz sound played at 43 dB sounds about as
loud as a 4000 Hz sound played at 2 dB.
Activity 5: Computer experiment
For this activity, you’ll need a computer and head-phones. Go to this site:
http://www.phys.unsw.edu.au/jw/hearing.html .
Follow the instructions below the sound grid; the site explains how to create your “curve
of equal loudness.”
Then answer the following questions:
1) What frequency did you find to be the loudest? (ie, what note or notes was at the
bottom of your curve?)
2) If humans perceived all sounds of a given decibel level as equally loud, what would
everyone’s “curve of equal loudness” look like?
By the way—You might wonder why a low note on a piano has roughly the same loudness as a high note.
After all, shouldn’t the high note subjectively seem much louder, given the facts above? I looked this up
and it turns out that the higher notes on a piano are produced using much thinner strings than the lower
notes on a piano, and this decreases their relative loudness. In fact, the thinness of the upper-frequency
strings decreases the loudness so much that piano manufacturers have to give each note more than one
string. For these reason, even though there are 88 notes on a standard piano, it can have over 200 strings.