Modeling Sound
Background
Sounds
are just vibrations which travel through the air. Your eardrum detects the changing air pressure caused by these
vibrations, and interprets the result as Mozart, the Beatles, a dog barking,
the wind in the trees, the sizzle of meat on a grill, or (if you're
particularly unlucky) Britney Spears.
In this project we will describe these vibrations mathematically. Like most vibrations, sounds (at least pure
tones) are periodic, so it is natural to model them using trigonometric
functions. A sound is generally
described by its frequency, measured in Hertz (1 Hz = 1 sec-1), and
by its loudness or volume.
1.
What
do frequency and volume correspond to in our usual descriptions of
trigonometric functions (period, amplitude, etc.)?
Data
We
measure the air vibrations of a sound with a microphone. You will be provided with three sets of
data. The first was obtained by ringing
a 256 Hertz tuning fork, the second from a 288 Hertz tuning fork, and the third
by ringing both simultaneously. In the
first two sets of data, the x values
are time, from 0 to 20 milliseconds, measured every 0.1 millisecond. In the third set, the x values range from 0 to 160 milliseconds, measured every 0.4
milliseconds. In each case, the y-values in volts (measuring the
electricity passing through the microphone, which is proportional to the
intensity of the sound waves). The
lists are arranged as follows:
L1: x-values
for 256 Hz tuning fork (time in seconds, from 0 to 0.02)
L2: y-values
for 256 Hz tuning fork (volts)
L3: x-values
for 288 Hz tuning fork (time in seconds, from 0 to 0.02)
L4: y-values
for 288 Hz tuning fork (volts)
L5: x-values
for both tuning forks together (time in seconds, from 0 to 0.16)
L6: y-values
for both tuning forks together (volts)
Problems
2.
Plot
the first set of data. Describe the
graph. What are the period and
amplitude of the graph? Use this
information to model the data by a trigonometric function. Adjust your model until you obtain the best
visual fit you can. Using linear
regression, describe how good the fit is between your model and the data.
3.
Do
the same for the second data set.
4.
The
third data set is the result of hitting both tuning forks simultaneously. Given your results in the previous parts,
what function do you think should model this data?
5.
Now
plot the third data set. Describe what
you see.
The
behavior of the third data set has a practical application for musicians when
they tune their instruments. If an
instrument is almost in tune, it is
very difficult to hear the difference.
But if a note is played at the same time as another instrument (which is in tune), or a tuning fork, the
musician will hear the beats that you
can see in your graph.
6.
Most
likely, the function you proposed in part 4 would not lead you to expect the
behavior you saw in part 5. Does this
mean the function was wrong? Graph it
and compare it to the plot. How do the
graphs compare (visually)?
7.
Now
we will try to find another model which is more natural. The graph in part 5 seems to involve two
separate trigonometric functions, one of which is modulating the amplitude of
the other. The "outer"
function is called the envelope -
determine its period and amplitude, and find a function which models it. Then find the period of the "inner"
function, and model it. How might you
combine these to get a model of the whole?
[Hint: think of the outer
function as the amplitude of the inner function.] How well does your resulting model fit the data (visually, and
using linear regression)?
8.
It
seems (or should) that the two models have very similar graphs, but appear to
be very different algebraically. To
resolve this, use trigonometric identities to prove that:
[Hint: 
]
9.
Using
this relation, compare your two models.
How close are they?
Graded out of 35 points