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NAME____________________________________ DATE________________ Project 2: Fourier Analysis of Waves Physics of Sound and Music: PHYS 152 The French mathematician Fourier showed that even complex-looking wave forms can be synthesized by summing simple sine and cosine waves of different frequencies, amplitudes and phases. For example, a sawtooth wave, synthesized using the Fourier Series Applet found at: http://www.falstad.com/fourier/ is shown at the top (in red, slightly wiggly lines) of the following figure: Figure A, Fourier Series Applet For this project, we will be using the Fourier Series Applet to better understand Fourier synthesis and Fourier analysis (‘measurement’ of the content (sines/cosines) within a given wave. The other applet we will use is Bar Waves Applet: http://www.falstad.com/barwaves/ 1 Activity: Analysis of a Sawtooth Wave Load the Fourier Series Applet on your web browser. You should see something like Figure A, above. If you don’t, then your computer doesn’t have java properly loaded. My advice is to go to a computer in one of UO’s computer lab to finish this project. Once the applet is running, you can play with it by clicking different buttons, dragging the white Sine and Cosine balls up and down, turning on Sound, adjusting the Playing Frequency, increasing or decreasing the Number of Terms, etc. 1. Push the Sawtooth button of the applet. Describe the Fourier components (sines and cosines) that comprise the sawtooth wave. You can get information from each component by holding your mouse cursor over a white ball on the line below Sines. What are the frequencies, amplitudes and phases of each component? What is the relationship between higher harmonics and the fundamental (lowest frequency sine wave) in terms of frequency? 2. With Sound checked, listen to the sawtooth as you change the phase of the entire waveform by click on Phase Shift. Can you hear any difference for different start phases? Explain. 3. Make a table of the Fourier components (sines) of the Sawtooth wave. List the harmonic, its frequency, the frequencies algebraic relationship to the fundamental, amplitude and phase. Write a row for the Nth component, predicting the frequency in terms of the fundamental frequency. Try to write an expression for its amplitude relative to the fundamental in terms of N. 2 4. Do you think the applet has a bug regarding the sine wave it shows for the 2nd harmonic? Explain. 5. Decrease the Number of Terms. Explain what this is doing in terms of the Fourier synthesis of the Sawtooth wave (the red summed wave at top). What is the effect on the sound of the sawtooth wave? Activity: Analysis of Noise Click on the Noise button. Be sure to keep Sound checked so you can hear this. Set the Number of Terms to be relatively low (~20 sines) and set the Playing Frequency to a relatively low value. 6. What are the characteristics of the Fourier components of noise? 7. Increase the Number of Terms. Describe what happens to the (red) waveform when you do this. Describe also how the sound changes. Relative to the range of human hearing (approx. 40- 20,000 Hz), what is the range of frequencies of the Noise like when you have a limited Number of Terms. Is this noise white, blue or red? What is the effect of increasing the Number of Terms on the ‘color’ of the noise? 8. Make some ‘blue’ noise by adjusting the individual sine components. Draw what your Sine components (white balls) look like on the next page, and describe the sound of the noise. 3 Activity: Don’t be a square, man Click on the Square button. Be sure to keep Sound checked so you can hear this. Use the default Number of Terms and set the Playing Frequency to a relatively low value. 9. Describe some aspects of the Fourier series that makes up the Square wave. How is it different than that of the Sawtooth wave? What type of musical resonator (clamped string, open/open pipe, etc.) does this wave resemble. 10. Drag the white ball representing the third harmonic up and down and describe what happens to the wave form (also, draw it below). Also, describe what happens the sound. 11. Drag the even harmonics up from the original settings of zero amplitude. How does this affect the sound? After dragging up several of them, describe what the new waveform looks like. 4 Activity: Bar Waves Close the Fourier Series Applet and choose “More Applets” from the web page. Choose the Bar Waves Applet for the next series of activities. The Bar Waves Applet allows one to make musical sounds like those from a marimba or xylophone. Before we get going, click on the Sound and Stopped buttons (should be checked), pull down and choose the Mouse=Applied Static Force selection, and choose Display Modes. This will let you pull down the bar with the mouse and release it, making a sound, showing the harmonics you excited, and freezing the Fourier components (white balls) where you first excited them. 5 12. Choosing Setup: bar, free, try pulling down the bar in the center and releasing it. What harmonics are most emphasized by this applied force? [note, this applet doesn’t report out amplitudes for the individual components, but they can be crudely estimated by comparing the background grid (faint, horizontal lines).] 13. What is the relationship between the frequency of the second harmonic (first overtone) vs. the fundamental. Is it an integer number times the fundamental or something else. In terms of musical intervals (3rds, 5ths, etc.) what does the second harmonic sound closest to? 14. How would you hit the bar to emphasize higher harmonics? Try it and see (an “applied static force” is meant to be the equivalent of hitting the bar with a mallet at the spot you grab with the mouse). Devise a rule (and state below) where do you need to ‘hit’ the bar to excite higher order harmonics [hint, think antinode]. Choose Setup: bar, hinged and make some sounds. 15. How does this change the relationship between the second harmonic and the first (fundamental)? Can you explain why? 16. Changing from “free” to “hinged” affects the overtone content of the bar. Is it easier to excite overtones for a “free” bar or a “hinged” bar? Explain. 6 Activity: Bar Waves—You design the experiment Using the Bar Waves Applet, design a simple experiment using the applet. Do this by comparing two different setups, by ‘striking’ the bar differently and giving data as to what happens, or by changing one of the sliders (e.g., Damping, Base Frequency) to measure the impact. Fill out the blanks below to describe your inquiry. Design: I will be testing the affect of …. on ….. by changing the …… Prediction: When I change the …., I expect …. to happen because I anticipate the relationship between…. and … to be ….. Data: Here is my data table and/or graphs Analysis: The data in graph 1 show that….. so {my prediction and hypothesis were correct; or my prediction and hypothesis weren’t correct). The data show that …….. 7 Conclusion: My experiment showed that, by {clamping, unclamping, hinging, freeing, changing damping, changing where I strike the bar, etc.} the {frequency, overtone content, relationship between harmonics and fundamentals, etc.} was changed in this particular way……… Definitions: Name Symbol Definition Formula wavelength λ (m) Distance between successive peaks or troughs of a wave at a given instant in time. period T (s) Time between successive peaks or troughs of a wave at a given position. frequency f (Hz) Number of waves passing a given position each f=1/T second. wave speed v (m/s) Speed at which waves travel. v=f·λ Transverse wave A wave that looks like the wave in Figure 2. The displacement of the media (e.g., slinky) is perpendicular to the direction the wave travels Longitudinal A wave that looks like Figure 5. The short-range wave motion of the media (e.g., compression of air) is in the direction the wave is traveling. Restoring force FR (Newtons) The force that acts to restore the transport media to its original state. Example: Tension on a stretched slinky mass density µ (kg/m), ρ For strings, slinkys, etc., the mass per unit length. (kg/m3) For air, the density (mass per unit volume)..In general, that which opposes displacement when waves are passing through. 8 9