Fourier Analysis p. 1/5
FOURIER ANALYSIS
Topic: Superposition Principle and Fourier Analysis
Objectives:
•
•
To gain an intuitive understanding of Fourier Analysis
To understand how a wave packet is composed of a distribution of Fourier components
Overview:
In this lab, you will carry out several investigations. In the first investigation, you will use a
computer-based animation to learn how different functions can be created by superposition of
harmonic functions (sines and cosines). You will also create a Gaussian wave packet and explore
how the width of the wave packet relates to the width of the distribution of Fourier components.
This is related to the Uncertainty Principle in Quantum Mechanics but is a more general property of
wave solutions. In the second investigation, you will use a computer and signal analysis program to
analyze some real sounds.
There is no lab report for this lab. Your assignment is to learn something about Fourier analysis and
waves!
Investigation: Fourier Analysis
In this investigation you will use an interactive simulation developed by the Physics Education group
at University of Colorado (PhET). To get started, open the simulation “Fourier: Making Waves” by
double clicking on the “fourier_en.jar” shortcut on the desktop. Ask your TA if you can’t find it.
Play around with the controls to familiarize yourself with the program.
1. Superposition of Sine Waves
(a) Select the “Reset All” button near the bottom right to reset the program to its default
mode. The Function “sine/cosine” should be selected in the “Preset Functions” box at the
top right. You should see a single sine wave in the bottom two graphs. Click “Auto scale”
which is located to the bottom right of the bottom graph (see figure). This will cause the
graph scale to automatically adjust each time you click it. The scale can also be adjusted
manually.
(b) Adjust the amplitude of the A1 harmonic in the top panel by
dragging the red bar up and down. Make sure you understand what the
“Amplitude” adjustment does to a single sine wave before proceeding!
P15 Fall 2014
R. Millan
11/2012
Fourier Analysis p. 2/5
(c) Now add a sine wave of a different frequency/wavelength by adjusting the amplitude of
one of the other amplitude bars (for example A10). Try adding in different “harmonics” to
see what effect it has. The middle graph shows each sine wave individually and the bottom
graph shows the sum of all the harmonics (different frequency waves).
(d) Click the checkbox in the “Math Mode” box on the right. This will display the equations
that represent each graph. The equations can be expressed in terms of wavelength or wave
number. Play around with these and discuss with your partner and/or your TA.
(e) What happens if you include only odd harmonics (A1, A3, A5, A7, A9, A11)?
(f) Now select the “square” function in the “Preset Functions” box. Compare with what you
found in part (e). Click on the “Function with infinite number of harmonics” button in the
same box to see the result when an infinite number of harmonics are included (gray line in
the bottom graph). What is the effect of adding the higher frequency harmonics (an infinite
number of them!) and why?
2. A Narrow Wave Pulse
(a) Select the “Reset All” button near the bottom right to reset the program to its default
mode. The Function “sine/cosine” should be selected in the “Preset Functions” box at the
top right. Now select “cos” in the “Graph Controls” box to make a single cosine function.
(b) Start adding in harmonics each with amplitude A=1 (all the same amplitude). Click “Auto
Scale” to adjust the bottom graph scale as needed. What happens as you add in each higher
frequency harmonic? Describe the resulting function and explain why there is a central peak.
How does the width of the central peak depend on the number of harmonics included?
What do you think would happen if you were able to add in an infinite number of
harmonics? Discuss with your TA and partner.
P15 Fall 2014
R. Millan
11/2012
Fourier Analysis p. 3/5
3. A Gaussian Wave Packet
We saw in part (2) above that we can create a localized pulse by adding together many different
Fourier components (frequencies). In this part, we will create a different kind of localized wave.
(a) First start with a single sine wave A6 with amplitude of 1. What is the wavelength of this
harmonic in meters (roughly)? How does it relate to the wavelength of the A1 harmonic?
What is the wave number, k?
(b) Now add the A7 harmonic with an amplitude of 0.9. Using some trig identities, it is
possible to show that
Does your graph qualitatively agree with this?
(c) Now add in A5 with an amplitude of 0.90, and A4 and A6 each with amplitude 0.66. You
are creating a Gaussian distribution of the Fourier components. A Gaussian distribution
is also called a “normal distribution”. It’s the classic “bell curve”. Check out the
Wikipedia page on the Gaussian function! What’s the result of adding these components?
(d) Now select the “wave packet” function from the “Preset Functions” box. It will add in
the remaining components of your Gaussian distribution with the proper amplitudes.
The result is a spatially localized “wave packet” that has a Gaussian shape. Note that the
wave packet repeats so there is a wave packet at other locations in space (not just one).
This is a result of the fact that we are only including a discrete set of harmonic
frequencies in our sum. You may need to adjust the scale to see this.
(e) Finally, click on the “Discrete to Continuous” tab at the top of the page. The top panel
shows a distribution of amplitudes similar to your Gaussian distribution of part (d). Click
on “continuous waveform” to see the Gaussian envelope of the amplitudes. The bottom
graph shows the wave packets that result from adding together all of these Fourier
components. Expand the scale on the bottom graph so you can see several repeating
wave packets.
(f) Now reduce the spacing between the Fourier components. Change the spacing to pi/2 by
moving the slider to the left. This is equivalent to adding in frequencies between the
existing discrete frequencies. What happens? In particular, what happens to the repeating
wave packets in the bottom graph?
P15 Fall 2014
R. Millan
11/2012
Fourier Analysis p. 4/5
(g) Reduce the spacing all the way to zero. What happened and why? Discuss this with your
TA to make sure you understand the difference between a discrete Fourier series and a
continuous Fourier integral.
(h) You can also change the location and width of the wave packet by playing with the
controls in the “Wave packet center” and “Wave Packet Width” windows. You may want
to adjust the scale on the bottom graph for this part. What happens to the Gaussian
Amplitude distribution (top graph) when you make the wave packet narrower (more
localized in space)? How does this relate to the uncertainty principle in Quantum
Mechanics? Hint: think about how momentum relates to the wave number (or
wavelength) and relate this to the top and bottom graphs as you change the width.
Investigation: Fourier Analysis of Real Signals
In this investigation, you will apply what you learned in the previous investigation to analyze some
different sounds.
1. Fourier Analysis of a musical note
a. Tuning Fork:
i. Use the provided tuning fork to generate a musical note. You are going to record and
analyze the sound with the computer using the UltimaSound II program. To open the
program, click on the shortcut for it in the toolbar at the bottom of the desktop. After
launching the program, set the sample rate to 44,100 Hz, and change the FFT size as
desired. The higher the value the more precise the measurement. To measure the
"ultrasonic xylophone" frequencies, you'll want to use the lowest value. Ask your TA if
you need help with the settings.
ii. After clicking the start button (upper left), select logarithmic frequency scaling, uncheck
3D, then select "bottom to top". All of this is on the right hand side. You can now record
the sound of the tuning fork. You can pause the program if needed. Clicking in the blue
part of the window, whether the program is running or paused, displays the frequency at
the mouse point.
iii. What frequency or frequencies are present in the tuning fork sound?
b. Guitar String:
i. Use the provided string to produce the same musical note that your tuning fork produced.
First, you will need to calculate the tension required to produce this note and then use the
provided masses to create that value of tension.
P15 Fall 2014
R. Millan
11/2012
Fourier Analysis p. 5/5
ii. Again, record the sound with the computer and analyze the frequencies of the sound.
What frequency or frequencies are present? Which frequency is strongest? You may also
sketch the frequency spectrum of the sound here:
c. Other Instrument:
i. Find a third “instrument” to generate the same note. You can either bring in an
instrument from home if you play something, use an organ pipe, fill a bottle with water
and blow across the top, or else somebody in your lab group can whistle, sing or hum the
note. Record the sound and analyze it as you did for parts (a) and (b). Describe how this
sound signal is similar or different from the signals in parts (a) and (b).
2. Fourier Analysis of a Clap
a. Use the same setup as in Part 1 to analyze the frequency spectrum of a clap. Describe your
results here and discuss with your TA. What frequencies are present and why?
3. Ultrasonic Xylophone
a. Determine the fundamental frequency of each note in the xylophone. Which is the highest
frequency you can hear with your own ears?
P15 Fall 2014
R. Millan
11/2012