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 L149. Harmonics and Sound Quality

Goals

• To investigate harmonics of open and closed pipes
• To investigate the characteristic that distinguishes different musical instruments playing the same pitch

Introduction

The term sound quality, also called timbre, has both a subjective and an objective manifestation. The subjective manifestation has to do with how different instruments playing the same pitch at the same intensity have different sounds. For example, a brass instrument sounds bright, whereas a clarinet sounds hollow in comparison. For sustained tones, the primary objective manifestation of timbre is the frequency spectrum. More information is available here.

Some Logger Pro waveform files are provided for this lab, and you'll create others of your own. For the provided files as well as the ones you'll create, there will be two graphs. One of them will be of the same kind that you've analyzed before, a graph of Sound Pressure vs. Time. You've learned how to apply a sinusoidal fit to the waveform and use one of the fit coefficients to determine the frequency. The results of such a fit are better the more nearly the sound source is composed of a pure tone (single frequency).

The other graph for each Logger Pro file is the result of a Fourier analysis of the frequency components of the sound. The display is of amplitude vs. frequency, also called a frequency spectrum. This graph shows you the relative amplitudes of the various harmonics present in the sound.

 Harmonics and overtones The terms harmonics and overtones refer to similar things but have different, but related, meanings. The term harmonic refers to all the frequency components of a sound. The fundamental is the 1st harmonic, and the higher harmonics are integral multiples of the fundamental. The term overtone refers to the harmonics above the fundamental. Thus, the first overtone is the 2nd harmonic, the second overtone is the 3rd harmonic, etc. In order to avoid confusion, we will use only the term harmonic in what follows. If asked for the harmonic number in a problem, the index n = 1, 2, 3, ... is being requested, where n = 1 corresponds to the fundamental. A sound is termed pure if it is composed of a single frequency.

Equipment

The equipment will be used in the follow up WebEx session.

• LabQuest Mini interface with USB cable
• Microphone probe
• Two plastic toy flutes similar to the ones below
• 30-cm ruler

Prelab

Enter your responses in WA L149PL. When you download a Logger Pro file, right click and save it to your hard drive with the extension cmbl. (If you left click, the file may attempt to open as a text file.)

1. Click here to access the animation. Under the animation controls, click the first function for f(x,t). This will start the animation. Note that f(x,t) is displayed in the upper pane, and the superposition of f(x,t) and g(x,t) is displayed in the lower pane. Since g(x,t) is initially 0, the superposition is identical to f(x,t). The equation below the animation says that f(x,t) = 3cos[2π(x/8 - t/4)]. The author omitted units. We'll assume SI units.
1. Give the following information for f(x,t). Note that numbers are treated as exact; hence, you needn't concern yourself with SF.
1. amplitude
2. wavelength
3. period
4. frequency
5. wave speed
2. The next thing you'll do is set g(x,t) = 3cos[2π(x/8 + t/4)]. The only difference with f(x,t) is that g(x,t) will travel in the opposite direction. Set the properties of g(x,t) appropriately. In order to make the wave go to the left, enter the speed as negative. (It's unfortunate that the author of the problem used the word speed instead of velocity.) When you've set the properties of g(x,t), click the play button to restart the animation. (Whenever you make a change, click play for the changes to be accepted.) If you set the properties correctly, the lower pane will display a standing wave with the same wavelength and frequency of the component waves but with twice the amplitude. Here's a check on whether you achieved a standing wave. Pause the animation and give the following information.
1. number of antinodes displayed
2. horizontal coordinate of the furthest right node (Click on the node to see the coordinates. Give the X-coordinate as an integer.)
3. Next you'll examine the effect of adding two harmonic frequencies. First, change the wave speed for g(x,t) to be positive again so that both waves move to the right. (Again, click on the play button for the changes to take effect.) Now we'll let f(x,t) = 3cos[2π(x/8 - t/4)] be the first harmonic (fundamental). We want g(x,t) to be the second harmonic with twice the frequency of the first harmonic. What will you do to double the frequency of g(x,t)? Remember that the two waves travel in the same medium.
4. What is the equation that describes the second harmonic?
5. Make the necessary change now so that g(x,t) is the second harmonic. Click on the play button. Then pause the animation and sketch the superposition over two complete cycles.
2. Download this file and open it in Logger Pro. The sound source is a tuning fork. We'll call it Tuning Fork X. From the frequency spectrum, you can see that the fundamental, which is below 500 Hz, is by far the largest harmonic. The 2nd harmonic has about 5% of the amplitude of the 1st harmonic. (In all the spectra, you'll see a bump slightly above 0 Hz. This may be 60 Hz noise from household electrical AC. Such noise is around us all the time in residential and commercial buildings and is picked up by microphones.) The goal of this problem is simply to determine the fundamental frequency of the tuning fork to 4 significant figures. Perform a sinusoidal fit on the graph of Sound Pressure vs. Time. Then use one of the coefficients of the fit to determine the frequency. You'll need to use the value of π to 4 significant figures.
3. Download this file and open it in Logger Pro. This is a different tuning fork than the one above. We'll call it Tuning Fork Y. You'll see more harmonics than for the previous forks. There is evidence of these not only in the frequency spectrum but also in the sound pressure graph. The high-frequency sinusoidal pattern superimposed on the low frequency pattern indicates the presence of a harmonic signficantly above the fundamental.
1. Perform a sinusoidal fit on the sound pressure graph to determine the frequency of the fundamental to the nearest Hz.
2. Give the harmonic numbers of the two harmonics of greatest amplitude present above the fundamental.
4. Download this file and open it in Logger Pro. The sound source is a toy flute. This can be considered to be a pipe open at both ends. The tone is very nearly a pure tone as evidenced by the frequency spectrum.
1. Determine the frequency of the tone to the nearest Hz.
2. Given that the length of the flute is 0.23 m, and the temperature of the air is 21oC, which harmonic is the flute producing (n = 1, 2, etc.)? (See the L151PL PowerPoint to refresh your memory about how to calculate the speed of sound, given the temperature).
3. Draw a representation of the displacement waveform in the open pipe. Indicate the locations of the nodes and antinodes with the letters N and A.
5. Download this file and open it in Logger Pro. This waveform is of the same flute as used for the previous problem; however, a greater blowing force was used in order to produce higher harmonics. What are the harmonic numbers of the two harmonics of greatest amplitude? (Hint: From your results in problem 4, you can calculate the frequency of the fundamental. Knowing that the harmonics are integer multiples of the fundamental, you can then determine the harmonic numbers of the two harmonics. Note that the relationship fn = nf1 won't be exact. A possible reason is that the standing waveform extends outside of the pipe a short distance, and this distance may vary depending on the blowing force. Any change in the effective length, though, is small enough that you can unambiguously determine the harmonic numbers.)
6. Prepare your lab equipment for the WebEx session. Have the LabQuest Mini connected to your computer and the microphone connected to Ch1 of the interface. Tape over the holes of the longer flute as shown in the photo below. It's important that the side holes be blocked completely.

Data and Analysis

You'll do Part A during the WebEx session. The remaining parts are due the day after.

 Helpful Resources Recordings of the sounds from the toy flutes are given in the Sound Library. These will give you an idea of what you should be hearing. However, don't use these recordings for frequency measurement, since the flutes used to produce them are not identical to yours. Methods of measuring sound frequency (this is the PowerPoint from L151PL) View this animation to see the harmonics of an open pipe. View this animation to see the harmonics of a closed pipe. This synthesizer spreadsheet is the one used during the WebEx session to demonstrate synthesis of waveforms.

Part A

Use the shorter of the two toy flutes for this problem.

1. In L151PL, you measured the fundamental frequency of the shorter of the two toy flutes. Look up the value that you measured for the fundamental frequency using Method 1.
2. Close off the end of the flute with a finger (don't use tape for this) and just breathe into the mouthpiece. If you blow rather than breathe, you won't get the desired tone. The goal is to produce the lowest frequency tone you can. When you've mastered this, record the sound pressure graph using LP, and measure the frequency from a sinusoidal fit.
3. Calculate the ratio of the frequency you just found to the value of the fundamental frequency from part a.
4. What you did by putting your finger over the end of the pipe was change it from an open to a closed pipe. Draw the displacement standing waveform for the fundamental frequency of a closed pipe. Clearly show the closed end and label nodes and antinodes.
5. The wavelengths of the fundamental frequencies of closed and open pipes can be written as cL, where L is the length of either pipe and c is a number. Determine the value of c for each pipe.
6. From part e, you can obtain the ratio of the fundamental wavelengths in the closed and open pipes. Given that, explain in sentence form why we expect that f1,closed / f1,open = 1/2. (The reason the theoretical ratio differs from the experimental one has to do with the fact that the standing wave pattern can actually extend beyond the end of the pipe when the end is open. That obviously can't happen for a closed end.)
7. Next you'll examine higher harmonics of the closed pipe. In Logger Pro, select Insert -> Additional Graphs -> FFT Graph. After the graph is inserted, select Page, Auto Arrange. What is most intense harmonic (greatest amplitude) above the fundamental?
8. Closed pipes should produce only odd harmonics. (The fact that there appears to be a 2nd harmonic in your graph may be due to the inability to close off the pipe completely. This apparent 2nd harmonic is actually the fundamental of the open pipe.) In order to show that a closed pipe produces odd harmonics, draw the displacement standing waveform for the next higher frequency above the fundamental for the closed pipe.
9. Give the wavelength in terms of L for the next higher harmonic above the fundamental for a closed pipe. (Write a fraction rather than a decimal.)
10. In part e, you gave the wavelength of the fundamental of the closed pipe in terms of L. Compare that result to the result of part i. Then explain in sentence form why the frequency of the next higher harmonic above the fundamental is three times that of the fundamental.
11. Title your graph Fundamental of the Closed Flute, save it with a descriptive file name, and upload to WA.

Part B

Use the longer of the toy flutes for this problem. The side holes must be taped over.

1. Measure the length of the flute to the nearest 0.001 m.
2. Blow gently into the flute to produce the fundamental. This shouldn't require much more force than breathing.
1. Record the sound pressure graph with LP and determine the frequency to the nearest Hz.
2. Insert an FFT graph. Which harmonics with n > 1 are obviously present? (You can ignore harmonics with amplitudes less than about 5% of the fundamental.) Low amplitudes of the higher harmonics indicate that the tone is nearly a pure tone. That is also evidenced by a nearly sinusoidal waveform.
3. Title your graph Fundamental of the Long Flute, save it with a descriptive file name, and upload to WA.
4. Enter the room temperature in oF to the nearest whole degree, convert to oC, and calculate the speed of sound. Give your result to the nearest m/s. See the L151PL PowerPoint if you need to review how to make these calculations.
5. Using the speed of sound and the length that you measured for the flute, calculate the frequency of the fundamental.
6. Calculate the experimental error in the measurement of the fundamental frequency. Use the value given in part i as the accepted value. Give your result to the nearest percent.
3. Blow harder into the flute to produce the next higher harmonic.
1. Record the sound pressure graph with LP and determine the frequency to the nearest Hz.
2. Using your measurements of the frequencies of the fundamental and the 2nd harmonic from parts b(i) and c(i), calculate the ratio of the frequency of the 2nd harmonic to that of the fundamental. Give your result to the correct number of SF.
3. Of course, the expected value of the ratio you just calculated is exactly 2. Calculate the experimental error to the nearest percent, using 2 as the accepted value.
4. Insert an FFT graph and tell which harmonics with n > 2 are obviously present. Then compare the purity of this tone to that of the fundamental found in part b(ii).
5. Title your graph 2nd Harmonic of the Long Flute, save it with a descriptive file name, and upload to WA.

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