Physics of Music Laboratory Manual (fall 1999)



[pic] [pic] [pic]

Physics of Music

Laboratory Manual (Fall 2005)

Note: Labs may not be done in this order!

Guide to writing a lab report ……………………….……………..p2

Lab 1 Measurements of Frequency and Computer Aided Wave Analysis .…….p3

Lab 2 Complex Waveform Analysis………....p9

Lab 3 Construction of a Flute………….…….p13

Lab 4 Variations on Standing Waves…….…p18

Lab 5 Resonant Tones in a Column of Air.…p21

Lab 6 Room Acoustics…………………….….p24

Appendices: A. Scientific Notation and It’s Prefixes………p28

B. Notes on the Piano……………………….…p29

C. Equal Tempered Music Scale ……………..p30

D. The 12 Pitch Classes………………………..p31

E. The speed of sound in air ………………….p32

Guide to Writing a Lab Report

In General

• You do not need a data notebook, however you should keep notes on your experimental set up and observations so you can describe them later.

• Your report should be neat.

• Use plain white paper. Use only one side.

Sections of the Report

1. Heading (Top of first page) [1 point]

• Name of you, and your partner(s) and TA/TI

• Name of lab experiment, lab section and date.

2. Abstract (Middle of first page) [2 points]

• Brief and clear summary of the experiment and your results

• About 4 sentences

3. Theory/Introduction (Begin on page 2) [5 points]

• Brief description of the ideas underlying the experiment.

• Explain equations and basic physical principles of experiment.

• Include formulas when appropriate.

• Use your own words. DO NOT COPY THE MANUAL.

4. Data Analysis [10 points]

• This is the most important part of the lab report. This is what YOU have done.

• Include tables, graphs, schematic drawings, error analysis, etc.

• Remember to define axis of plots and define variables used.

• In the text describe what is seen in the plots and figures, let the reader know why and what you have plotted or drawn.

• Data sheets go at the end—not here!

5. Conclusions [5 points]

• Summarize in a paragraph or two what you have learned.

• Compare with what was expected. Comment on differences.

• Answer any questions posed in the lab manual.

6. Remarks (NOT optional) [1 point]

• You could discuss the accuracy or limitations of your measurements.

• You could discuss future types of measurements or experiments

• You could describe your ideas on how to improve the lab.

• Any other comments here.

7. Data [1 point]

• Attach data sheets and materials to back of the report.

• Do not make any changes in them. All changes or modification go into the data analysis section.

Physics of Music

Lab 1

Measurements of Frequency And

Computer Aided Wave Analysis

Last revised: March 2003 by Kelly Gresham

EQUIPMENT

Part A:

• Tenma oscilloscope (new black oscilloscopes; our lab does not have enough, so get some from Thang)

• BK Precision function generator (get from Thang)

• Pasco counter/timer

• Connectors

Part B:

• Pasco Signal Interface with appropriate PC and software

• 1 pair Headphones (with 8-pin DIN adapter)

• 1 sound sensor microphone with cable

• optional musical instruments or sound-making devices

INTRODUCTION

In part A of this lab, we will be measuring the frequency of a signal using three different methods. The first method involves the use of the function generator. A function generator is an instrument that can produce sine, square, and triangular waves at a given frequency. The second method makes use of the oscilloscope. An oscilloscope is an instrument principally used to display signals as a function of time. The final method for measuring the frequency uses the counter/timer. A counter/timer is an instrument that can give a very accurate measurement of the frequency of a signal.

The frequency and period of a wave are related in the following way:

[pic] (Equation 1.1)

The electrical signals created by a function generator can become “sound waves” when passed through a speaker. In part B of the lab, you will finally bridge the gap between electronics and music by doing just this: generating and observing signals while listening to them. In addition, you will have the chance to create some “noise” of your own, and observe it in the same way. In the process, you will begin to see the power of technology in bringing together Physics and Music through the Fast Fourier Transformation (FFT). FFT is an algorithm that can, through the mathematical process of Fourier Analysis, break down a signal into a number of basic sine waves of specific frequencies. If this seems confusing, don’t worry: you’ll have many more opportunities to hear about this process throughout the semester.

PURPOSE

The purpose of this lab is to gain a working knowledge of a function generator, an oscilloscope, and a counter/timer and to familiarize you with the Apple Macintosh and the Pasco Signal Interface (with its software). In addition, this lab will introduce you to four different types of waveforms, and to two different ways of scientifically analyzing sound (with FFT and with the function generator, oscilloscope, and counter/timer). The Pasco Signal Interface contains its own function generator, (along with the Macintosh computer’s internal processor) oscilloscope, and Fourier transformation capabilities. Thus for part B, all of your data acquisition will take place using the computer, although you’ll naturally use your ears to listen to the sound.

Part A: Measurements of Frequency

PROCEDURE

Part A.I – The Function Generator

1. Turn on the function generator.

2. Press in the button on the front panel with the sine wave picture.

3. Where it is labeled “Range,” press the “500” button.

4. Use the “Course” and “Fine” tuning dials (where it is labeled “Frequency” on the front panel) until the display reads 300 Hz.

5. In your lab notebook, write “300 Hz” for the value you obtained for your frequency using the function generator on your first trial.

6. Turn the amplitude knob on the function generator so the output is half of the maximum.

Part A.II – The Oscilloscope

1. Turn on the oscilloscope by pushing in the power button in the “Display” section. It will take a few seconds for a trace to appear on the screen. If one does not appear in a few seconds, try increasing the trace intensity by following step 2. If this does not work, ask your TA for assistance.

2. There is an INTENSITY knob in the “Display” section. Turn this knob clockwise.

3. There is a FOCUS knob also in the “Display” section. Adjust this knob to focus the line in the monitor.

4. There is a MODE switch in the “Vertical” section. Move this switch to “Ch 1.” This allows you to look at channel 1. There are two channels available so that it is possible to compare two traces at once. To show both traces at once, select the DUAL mode. The X-Y mode allows you to look at an x versus y display. The horizontal (x) signal is connected through the CH1 X input connector and the vertical (y) signal is connected through the Ch2 Y input connector. The ADD mode allows two traces to be added together.

5. There is a SOURCE switch in the “Trigger” section. Make sure this switch is on “Ch1.”

6. Using the CH1 Y shift control (the vertical arrows) in the “Vertical” section, make sure that the trace is in the middle of the screen vertically.

7. Using the X shift control (the horizontal arrows) in the “Horizontal” section, make sure that the trace is in the middle of the screen horizontally.

8. Turn the VARIABLE knob, in the “Horizontal” section, clockwise to the CAL position.

9. In the “Vertical” section, there is a GND button. Make sure that this button is sticking out (i.e. not pushed in). This will make CH1 not grounded.

10. If the waveform on your screen is not stationary, adjust the LEVEL knob in the “Trigger” section. When the waveform is not stationary it is said to be “freerunning”.

11. The TIME/CM knob in the “Horizontal” section specifies how long it takes for the trace to sweep through a centimeter on the oscilloscope screen. Adjust the TIME/CM knob so you can see one full cycle on the monitor.

12. The VOLTS/CM knob in the “Vertical” section determines how tall the signal will be on the screen. Adjust the VOLTS/CM knob for channel 1 to obtain a waveform that fills up almost all of the screen.

13. Adjust the INTENSITY and FOCUS knobs to get a clear trace.

14. At this point you should have a stationary waveform positioned nicely on your screen. If this is not the case, get help from the TA.

15. Count the number of squares in one period of the wave. Remember that each square is 1 cm by 1 cm.

16. Multiply this number by the value that the TIME/CM knob is set to. Make sure to convert the value that the TIME/CM knob is set at into seconds per centimeter. For example, .2 ms/cm = .0002 s/cm. A conversion table is provided at the end of the lab manual in appendix A. The number that you have just calculated is the period of the signal in seconds.

17. Determine the frequency of the signal using formula (1.1).

18. In your lab notebook, write the value you obtained where it asks for the value of your frequency using the oscilloscope on your first trial.

Part A.III – The Counter/Timer

1. Ask the TA to connect the counter/timer to the oscilloscope and function generator setup.

2. Turn on the counter/timer.

3. Flip the mode switch to measure Hz.

4. In your lab notebook, write the value you obtain for the frequency where it asks for this value using the counter/timer on your first trial.

Part A.IV – Additional Measurements

1. For your second trial, measure the frequency of a 600 Hz signal using the three methods you just learned.

2. For your third trial, measure the frequency of a 900 Hz signal using the three methods you just learned.

DATA ANALYSIS (Part A)

1. Explain how you obtained the frequency for all three methods.

2. Plot the frequencies you obtained vs. methods used [function generator, oscilloscope, and counter/timer] for the three trials as shown below.

QUESTIONS (Part A)

1. Did all three methods that you used to determine a given frequency provide the same result? Why or why not?

2. Which method did you find most accurate? How could you tell it was the most accurate?

Part B: Computer Aided Wave Analysis

PROCEDURE

Part B.I – Setup

1. Make sure that your Pasco Signal Interface (the black box near your computer) is turned “on.” The green light on the front panel should be illuminated. The power switch is on the back of the interface.

2. Connect your headphones to the small (approx. 1 in. long) black adapter supplied by your TA.

3. Plug this adapter (and your partner’s adapter) into Analog Channels A (and B) on the front of the Interface.

4. Plug your Pasco Sound Sensor into Analog Channel C using the black cable provided.

5. Log into the computer terminal as 103user with password: music.

6. Click on the Pasco Science Workshop icon, and allow this software (which will allow you to control and monitor the Interface’s actions) to load.

Part B.II - Listening

1. Click on the button that reads Sample. This will bring to view the controls for the Interface’s built-in function generator.

2. Set the output amplitude to 1.00 V. When using speakers or headphones, amplitude-of-signal is directly related to volume-of-sound. By changing this value, you are directly adjusting the maximum and minimum voltage passed to your headphones in one “wave”.

3. Set your frequency to 440.00 Hz, which is indicative of the note A4.

4. Don your headphones and click the on button. You should hear a tone. If this volume is uncomfortable, adjust the volume control on your headphones. If you don’t hear a tone, contact a TA. The drawings of waves (on the panel currently being displayed) are indicative of how the electric signal coming into your headphones changes with time. The sine wave (to which you are currently listening) changes smoothly with time, and sounds somewhat “pure,” although terms like this are subjective and can be somewhat confusing.

5. The square wave (the next button to the right) allows only two intensities in its “wave.” Click this button and listen to the “buzzing” nature of the tone, as the speakers in your headphones abruptly switch in and out, 440 times per second.

6. The triangle wave (the third button on your panel) varies linearly with time. Click this button and listen to this third, “electronic” type of wave.

7. All of the above adjectives are rather subjective terms. Listen again to the above wave forms, at frequencies of 440.00 Hz (A4) as before, and at 587.33 Hz (D5). (Note the accuracy to which the Pasco Interface can generate functions.) In the space provided on the data sheet, list one or two adjectives describing how the three waveforms sound to you.

8. Return the frequency setting to 440.00 Hz.

Part B.III - Looking

You are now going to observe (on a virtual oscilloscope) the previous three types of waveforms.

1. Click on the window in the background labeled untitled. Drag your mouse (by holding down the button) from the scope icon to the circular “OUT” icon (above the sample button). This should bring an oscilloscope display to your screen, and a smile to your face.

2. Now, click again on the untitled window, and press the mon button (with the triangle). The scope that you just “created” is now monitoring the tones being emitted by the function generator.

3. Find the window labeled scope and select it again. As in the previous lab, you must adjust the horizontal and vertical sweep of the oscilloscope to obtain a “clean trace.” Use the buttons below the ms/div label to adjust the horizontal sweep. As you can see, this sweep is measured in milliseconds per division.

4. Now, use the buttons to the right of the V/div label to adjust the vertical sweep. As you can guess, this sweep is measured in Volts per division. When you obtain a clean trace for your 440hz sine wave, record the scale used for your horizontal and vertical sweep on your data sheet.

5. You are now able to compare this sine wave to the square and triangle waves as before. Switch waveforms and frequencies as in the previous section. Observe how the wavelength (the distance covered on the screen before the trace repeats) changes or does not change for the three different types of waves.

6. Use the ms/div reading to determine the periods of a 440.00 Hz triangle wave and an 880.00 Hz square wave, as you did in the previous experiment. Record these values on your data sheet. For your own vindication, apply these periods in equation 2.1, and compare these frequencies to the frequency set on your signal generator. You can use the built-in calculator, which is activated by clicking the third button under the data column in the untitled window. If these frequencies are not approximately equal to those that you entered into the signal generator, reassess your method of measuring periods of non-sine waves.

7. Feel free to experiment with different frequencies and waveforms. When finished, press the stop button in the untitled window.

Part B.IV - Transforming

You are now going to begin using a tool that will be very useful during the remainder of this course: the Fourier Transformation. This tool will give you a graphical representation on your screen of the fundamental and harmonic frequencies that go into making the tones that you have been observing throughout this lab.

1. Drag the FFT icon to the out jack, as you did before for the scope.

2. Reset the signal generator to a 440.00 Hz sine wave.

3. Now, press the mon button in the untitled window, and observe the frequency spectrum window. The horizontal sweep of this window determines the total range of frequencies being observed at any time. For example, if you had a 2000hz wave, you would have to set your sweep higher that 2000hz to begin observing the wave’s fundamental frequency and its (possible) harmonics. Adjust the horizontal sweep until you can clearly see a peak at or near 440.00 Hz. Record the scale used to obtain a clean sweep on your data sheet.

4. Adjust the signal generator to a 587.33 Hz wave. Note that the position of the fundamental frequency changes on the frequency spectrum. At any time, you can still examine the scope to confirm the period of the wave.

5. Set the signal generator to a 587.33 Hz triangle wave. Now, return to the frequency spectrum window and note the overtones (additional frequencies) which have appeared. You may need to adjust the sweep until at least two overtones are seen. You can use the smart cursor to estimate their frequencies. Click on the x-y button in the bottom corner of your window, and move the crosshairs to the first peak (the “fundamental”). Record this frequency (measured in Hz below the display), and the frequencies of the next two overtones, in your data sheet.

6. Switch to a square wave, and record the frequencies of the first three peaks as above. Note the number of overtones necessary to create a square wave. This will be discussed in an upcoming lecture. For now, simply observe how any wave (except for a sine wave) can be broken into sine waves of specific frequencies.

8. Experiment with another frequency of your choosing. When finished, click the stop button, and then the off button in the signal generator window. Close all windows except for the untitled window.

Part B.V - Creating

For the remainder of this lab, you will create your own sounds, and observe them in the same manner as you have previously observed sounds from a function generator. You will use the sound sensor microphone supplied by your TA to turn sound waves into electrical signals, which you can analyze with the help of the Pasco Interface.

1. Drag the analog input icon (at the bottom-right corner of the untitled window) onto Analog Channel C. When prompted, select the Sound Sensor (which looks like an ear). The Pasco software is now ready to analyze input from the microphone.

2. Drag the Scope and FFT icons (separately) onto Analog Channel C. Press the mon button to begin monitoring your sound signal.

3. To begin, sing a note into the microphone. (Don’t be shy. Your “sensitivity” far exceeds that of the microphone.) Observe how the wave appears in the scope. You may wish to adjust the scope between 2 and 100 ms while singing, to observe different ways of analyzing your voice.

4. While continuing to sing, observe the Frequency Spectrum. Increase the horizontal sweep until you see a peak that moves as you change the pitch of your voice. (The equipment may need a second to “catch up” with your changing pitch. Be patient, please.) Speak a few words slowly into the microphone, and observe how many frequencies go into making your voice unique. Remember: the quantity and value of overtones, with which your voice is laden, determine timbre.

5. Experiment with different sounds, including whistling (across the microphone, not into it), humming and clapping. If a musical instrument or tuning fork is available, play it near the microphone and observe its data. Pick your favorite-looking sound, and sketch its scope reading and frequency spectrum on your data sheet.

6. When finished, close all windows and log out of the computer. In a later lab, you will revisit the Pasco software to complete a more detailed analysis of sound waves.

DATA ANALYSIS (Part B)

1. Explain in your own words the difference between a sine wave, a square wave, and a triangle wave. No diagrams, please.

QUESTIONS (Part B)

1. Why don’t sine, triangle, and square waves with the same frequencies and amplitudes sound the same?

2. How did the waveform and frequency spectrum of your voice differ from those generated by the Pasco Interface internally?

3. How accurate is the Pasco software? Do you expect it to always remain this accurate (or inaccurate)? Remembering that this is a solid state device (basically, full of microchips), what factors might determine its accuracy during a given experiment?

*Remember to attatch your data sheets to every lab write-up [this is worth one point!].

**For your lab write-up, consider labeling subsection “Part A” and “Part B” under Heading, Abstract, Theory, ect. This will make the write-up easier to read.

Physics of Music

Lab 2

Complex Waveform Analysis

(Last revised: March 2003 by Kelly Gresham)

EQUIPMENT

• Pasco Signal Interface with appropriate PC and software

• 1 sound sensor microphone with cable

• 1 pair of headphones (with 8-pin DIN adapter)

• Train whistle

• Ruler

INTRODUCTION

In the previous lab, you learned how to use the Pasco Sound Interface to break down both simple waves (like sine, triangle, and square waves) and complex waves (like those of your voice or a whistle). In this lab, you will utilize those skills to do a deeper analysis of two complex (and beautiful) instruments: the train whistle and the human voice.

The train whistle appears to be a simple instrument: a tube, open at one end, which resonates at many frequencies. Yet the train whistle is actually a complex instrument, whose tones consist of such advanced musical qualities as pressure nodes and forced overtones. Many of these issues will be addressed in a later lab on standing waves in a resonance tube. For the moment, we will only explore the simplest quality of the train whistle: the relationship between the position of the stopper and the tone produced. You probably are already familiar with this relationship, as you have seen a trombone player extend the length of his resonance tube by moving his slide, in order to produce a lower tone. The relationship between length (L) and frequency (f) is given by the following formula:

[pic] (Equation 2.1)

Adjusting the length of a resonance cavity is something that you do every day when you speak. Speech is actually facilitated through changing the length and shape of the resonance cavity of your mouth. Many sounds which we classify as single letters (like the vowel u) are actually combinations of two different tones (an “ee” which slowly slides into an “oo.”) Through a Fast Fourier Transformation, you can observe the overtones that your mouth (like a train whistle) produces above a fundamental frequency. In addition, you can observe and, as is done in some speech therapy programs, imitate the visualized “waveform” of a given sound.

PURPOSE

The purpose of this lab is to use the Pasco Signal Interface to analyze the waveforms and frequency spectra produced from a train whistle and your own voice. You will adjust the stopper of a train whistle and establish a relationship between the length of the pipe and the frequency produced. In addition, you will study vowel sounds and attempt to match a given waveform to a particular vowel sound.

PROCEDURE

Part I – Setup

1. Make sure that your Pasco Signal Interface (the black box near your computer) is turned “on.” The green light on the front panel should be illuminated. The power switch is on the back of the interface.

2. Plug your Pasco Sound Sensor into Analog Channel C using the black cable provided.

3. Log into the computer terminal as PHY 103 (password phy103).

4. Click on the Pasco Science Workshop icon, and allow this software (which will allow you to control and monitor the interface’s actions) to load.

5. Drag the analog input icon (at the bottom-right corner of the untitled window) onto Analog Channel C. When prompted, select the Sound Sensor (which looks like an ear). The Pasco software is now ready to analyze input from the microphone.

6. Drag the Scope and FFT icons (separately) onto Analog Channel C. To the extent that your machine can keep up with the data entered, you can resize these windows to your taste. If too large, your waveform will become “jagged”. Press the mon button to begin monitoring your sound signal.

7. Test your microphone by speaking into the sound sensor and observing the screen.

Part II – Train Whistle

1. Place the stopper as far as it will go into the train whistle and measure the distance from the end of the stopper to where it fits into the train whistle. Record this as L0 on your data sheet.

2. Now move the stopper to the first note on the scale. The entire box that contains the note should be showing on the stopper. Measure the distance from the end of the stopper to where it fits into the train whistle and record this on your data sheet in the appropriate spot. This will be used to determine the length of the air column.

3. Blow a steady stream of air into the train whistle and observe the frequency spectrum. You should adjust the horizontal sweep of your spectrum until you can clearly see the fundamental and at least two overtones. Use the scope to help you establish a steady tone. Be careful how forcefully you blow into the train whistle. Too strong a stream of air will cause the whistle to resonate more in the first harmonic than in the fundamental. If you observe the second peak on your frequency spectrum to be higher than the first, you are hearing more of a harmonic than of the actual tone.

4. While one of you continues to blow this whistle tone, the other person can record on your data sheet the frequency of the fundamental and its first two overtones in the space provided. You can use the “smart cursor” as you did in the previous lab to pinpoint this frequency. So long as you hold down your mouse button over the window, the display will freeze. Feel free to breathe.

5. Now continue to record the length of the stopper that is exposed, the frequency of the fundamental, and the frequency of the harmonics for the rest of the notes on the scale.

6. When finished, mark on your data sheet how much deviation your frequency measurements had from the frequencies around them. That is, using the smart cursor, discover how many Hz “lie between” any two pixels on the screen. This will be used to estimate your error graphically.

Part III – Voice

1. You will now use the same setup with which you analyzed the train whistle to analyze your own voice. In the process, you will discover the difference between vowels that you may have thought to have the same frequency. That is, you believe to have spoken them at the same pitch. You may even discover that vowels that you originally believed to sound “the same” have very different waveforms.

2. Begin by speaking (actually, singing) the vowel sound “ee”. You may have to adjust the scale of the scope to register a good trace of one full waveform of your voice. Examine how that waveform changes as you change the volume of your voice. Note, however, that changing the volume is different from “forcing” a note. Try to “force” the “ee” sound and observe the change in the waveform as well. Remember that you may need to change the volume of your voice in the upcoming section of the lab in order to match the given waveform.

3. Experiment with the following vowel sounds. Use your lab notebook to sketch the waveforms and spectra for each vowel. A brief sketch is fine. This is for your benefit in writing up the lab.

Ah (as in paw)

Eh (as in sketch)

A (as in play)

E (as in tree)

I (as in fly)

O (as in know)

Oo (as in true)

U (as in use)

Uh (as in smut)

4. On the next page are two waveforms and spectra for vowel sounds sung (spoken) at 90 Hz. Examine these on your data sheet and in the appropriate space, designate which vowel sounds you feel they represent.

Voice wave #1:

[pic]

Voice wave #2:

[pic]

DATA ANALYSIS

1. Using graph paper, plot a graph of the frequency produced by your train whistle versus the length of the pipe. Physicists often use error bars to account for uncertainties in measurement. Using the difference in Hz between two pixels on the screen of the computer, add error bars to each of your points. Plot a line of best fit on your graph as well.

(

2. Add to the plot, in a recognizably different line, points and error bars for each of the two harmonics you observed for each note.

QUESTIONS

1. What kind of relationship did you find between frequency and length of the pipe? How might this discovery relate to the equation at the beginning of the lab? Justify any observed deviations from the given equation.

2. What relationship could you determine existed between each fundamental and its two harmonics? Was there a ratio between each tone and its first and second visible harmonics?

3. When you spoke vowels at the same apparent pitch did the fundamental frequency shift at all? What does this tell you about the subjectivity of your sense of pitch?

4. Of the nine vowels that you sketched, rank them in order of complexity of waveform.

5. What is the definition of a “diphthong”? What did you notice when trying to draw the waveforms of “diphthongs” like “I” and “U”?

6. Use what you learned about the train whistle to explain how the mouth changes to resonate at higher and lower pitches.

Physics of Music

Lab 3

Construction of a Flute

(Last revised: March 2003 by Kelly Gresham)

EQUIPMENT

• PVC pipe

• Cork or dowel

• Ruler

• Various tools (power hand-drill, hacksaws, clamps)

INTRODUCTION

Now you will get the chance to build an instrument for yourself – a PVC pipe flute. After the flute has been constructed, you will get to play your flute for an audience! You have your choice of which type of flute you want to make and what scale you want to make your flute in.

When constructing your flute, you will need to figure out where your holes should go. If you want to change the length by (L, the change in frequency (f can be found using equation 5.1:

[pic] (Equation 3.1)

If you wish to make a flute in G, we can provide the hole placements for you. However, be sure to show that they still follow from equation (5.1).

PURPOSE

The purpose of this lab is to learn how to use your knowledge of physics and music to construct an actual musical instrument.

PROCEDURE

Sideblown Flute

(There is a helpful insert on page 16)

1. Cut your PVC tubing to 16 and 7/16 inches.

2. Mold 1” of cork so it fits tightly in the end of the tube. This can be done by using a knife or by using a sharp pipe of the same diameter as your tube to shape your cork.

3. Tap the cork into one end, to a depth of ½ an inch.

4. To form the blowhole, drill a 3/8 inch hole in the side of the tube one inch on center from the corked end.

5. Ream the hole out slightly so that you end up with an oval shape, about ½ an inch in the long dimension, with the long dimension along the line of the tube.

6. Smooth and de-burr the edge.

7. Decide on a scale for your flute and create a chart showing the value of the note and the corresponding frequency of that note. If you wish to make a G flute, follow the insert on p. 16. If you wish to be different, use formula (3.1) in conjunction with the chart of scales in appendix D and the list of frequencies in appendix C.

8. Drill the fingerholes, starting with the one farthest down the tube. You should end up with six holes in all. They will be 3/8-inch holes, spaced according to the scale you plan on creating. The holes need not be in a straight line down the tube; it makes fingering easier if some are offset to one side or the other to accommodate the natural fall of the fingers.

9. After you have drilled a hole, you can fine-tune a hole if necessary by enlarging a given hole to raise its pitch or reducing it (by back-filling with autobody filler, epoxy glue, or bee’s wax) to lower its pitch.

10. The stopper position of the cork can also be used to fine-tune if you still need to do so after you have drilled all of your holes.

11. Write down (in your data notebook) the actual frequencies of the notes in your flute’s scale. To do this, note the number of cents that each note is off. Then use “The 12 Tone Music Scale” table on the next page and the list of frequencies in appendix C to determine the actual frequencies of the flute’s notes. NOTE: if you are trying to figure out the actual frequency of your C note and you are 15 cents off, do the following:

You know by appendix C that the C note is supposed to be 261.63 Hz.

You know that you are off by +15 cents. By looking at the formula in “The 12 Tone Music Scale” table, you can calculate 15 cents = 3986 Log [f/261.63] where Log is base 10.

So, [pic]

So, the actual frequency of the C note is 263.9Hz. [pic]

12. Practice playing your flute and be sure that you can get a clear tone from each one of your holes.

Fipple Flute

(There is a helpful insert on page 17)

1. Flatten one side of your cork by cutting and/or sanding, to create a sloped flat surface along one side.

2. Cut your PVC tubing to 16 and 9/16 inches.

3. Tap the cork into one end of your tube so that the end is flush. This will leave a narrow windway between the tube and the flattened part of the cork.

4. Cut the beveled opening in the tube, to form the aperture and edge. Make the angle cut as shown in the figure and end it with the vertical cut at one inch from the tube end.

5. Carefully de-burr the edge as need be.

6. Blow through the windway to check for tone. If you don’t get a clear tone, try adjusting the cork to shift the air stream angle over the edge. If after some effort you still can’t get a clear tone, try making another cork with a slightly different bevel.

7. Decide on a scale for your flute and create a chart showing the value of the note and the corresponding frequency of that note.

8. Drill the fingerholes, starting with the one farthest down the tube. You should end up with six holes in all. They will be 3/8-inch holes, spaced according to the scale you plan on creating. Use formula (3.1) to determine the placement of your holes. The holes need not be in a straight line down the tube; it makes fingering easier if some are offset to one side or the other to accommodate the natural fall of the fingers.

9. After you have drilled a hole, you can fine-tune a hole if necessary by enlarging a given hole to raise its pitch or reducing it (by back-filling with autobody filler, epoxy glue, or bee’s wax) to lower its pitch.

10. The stopper position of the cork can also be used to fine-tune if you still need to do so after you have drilled all of your holes.

11. Write down (in your data notebook) the actual frequencies of the notes in your flute’s scale.

12. Practice playing your flute and be sure that you can get a clear tone from each one of your holes.

[pic]

DATA ANALYSIS

1. If you made a G flute: Use equation (3.1) to determine the theoretical hole placement for your flute. Make a table with the theoretical hole placement (found with the equation) in one column and the actual hole placement (where the holes actually are on your flute) in another column. Be sure to demonstrate how to use equation (3.1) by showing your work for at least one calculation. If you made a flute with a different scale: demonstrate how you decided where you drilled your holes by applying formula (3.1). Make a table of these values. Be sure to show your work for at least one calculation.

2. Make a table listing the theoretical (accepted) frequencies of the scale of your flute (i.e. the frequencies you can look up in appendix C), the actual frequencies you obtained for the notes in your scale, and the number of cents you were off for each note.

QUESTIONS

1. Is your flute accurate? What would you have done differently to increase the accuracy if you were to make another flute?

2. What was the most difficult part of constructing the flute? Can you think of ways that this part could have been done easier?

3. Have you found playing your flute to be difficult? What techniques have you used to generate a tone from your flute?

4. [pic]

[pic]

Physics of Music

Lab 4

Variations on Standing Waves

(Last revised: March 2003 by Kelly Gresham)

EQUIPMENT

• Pasco sonometer

• BK Precision function generator

• Pasco counter/timer

• Tenma oscilloscope

• Tuner

• Chladni plates

INTRODUCTION

Many musical instruments, such as guitars, pianos, and violins, operate by creating standing waves in strings. This process has much in common with the creation of standing waves in tubes, as you will examine in the next lab. The formula for the frequency is given by the following formula:

[pic] (Equation 4.1)

This formula explains the relationship between the length L of the string and the frequencies f at which the string will resonate. Two features of the equation should be noted. First, the string is fixed at both ends, which is mathematically equivalent to a tube that is open or closed at both ends. Second, the velocity of the sound wave is determined by qualities of the string. The velocity of sound is given by the following equation:

[pic] (Equation 4.2)

In this equation, T is the tension in the wire and ( is the linear density of the string. The linear density is the mass of a specific length of the string.

If you know the linear density of and tension on a string, you can calculate the frequency at which the string will vibrate for a given length (like the length of a guitar neck or of a piano backboard). In our case, we will mount the string on a Pasco sonometer and this will allow us to vary the tension on the string and see how it changes the frequency at which the string will resonate.

The sonometer also allows us to “drive” the string into vibration (without plucking the string) through the use of a magnetic field. This permits us to test which harmonics can themselves be made to ring within a certain length and type of string. The sonometer will also be used to analyze those parts of the string that are “louder” than others. In this portion of the experiment, the following equation will be used:

[pic] (Equation 4.3)

The formula for the tension applied to the string is:

[pic] (Equation 4.4)

where m is the mass on the end of the string. Notch number one is the notch closest to the sonometer string, while notch five is that farthest from the sonometer string. Therefore:

In the second part of the experiment, the Chladni plates will demonstrate wave patterns on a stretched membrane. The plates represent percussion instruments such as the drum.

PURPOSE

The purpose of this lab is to analyze the relationship between mass and frequency in a vibrating string and to analyze the vibrations of the circular Chladni plates.

PROCEDURE

Part I – Sonometer

1. Use the scale provided to determine the mass of the string and a yardstick to measure its length from bridge to bridge (in meters). The linear mass density is the mass divided by the length of the string. If the linear mass density of the string is already known, then the TA will provide this information and you may skip this step.

2. The sonometer will be set up for you by your TA. Make sure the oscilloscope is on dual mode. The trigger should be for CH 2, DC. If you are not getting a signal, check the DC offset on the function generator. Make sure the driver is plugged into the GND and OUT on the function generator. You will hang your three different trial masses from the end of the sonometer’s tension rod. In the fifth notch, the tension applied to the string will be given by formula (4.4). Position the driver 5 cm from the bridge and the detector near the center. Attach 1 kg of mass to the end of the sonometer and record this value on your data sheet. In addition, record the length of the portion of the string that is resonating on your data sheet.

3. Use the instrument tuner to estimate the starting frequency of resonance (The frequency of resonance occurs when the string is plucked.) Set your function generator approximately 50 Hz below the frequency of resonance. The function generator will attempt to magnetically “drive” the string at the rate you have set. Slowly increase frequency until you find the first point at which the string resonates. You will know that the string is resonating because you will hear the string vibrating and you will observe a change in the waveform displayed by the pickup on your oscilloscope. Remember, the pickup and “driver” both create magnetic fields, so if you see a perfect sine wave coming from the pickup, you know that you are seeing traces of the driving signal. The real resonance will be a more complex waveform. Read this frequency from the counter/timer and record it on your data sheet.

4. Continue to increase the frequency until you find several more resonating frequencies and record these on your data sheet. Do not go past 1000 Hz. These resonating points will correspond to the string vibrating with two, three, and four nodes respectively. Move slowly here and try not to miss a resonant frequency between two others. It is easy to skip over a crucial point!

5. Repeat the above procedure using the third and first notch.

Part II – Chladni Plates

1. You will be observing both the round and square chladni plates.

2. Using the setup provided, sprinkle some sand on top of one of the plates. Make sure the plate is level.

3. Increase the frequency slowly starting at around 10 Hz. When you find a pattern start to form with the sand on the plate, note the frequency and draw the pattern on your data sheet.

4. Continue until you have found several patterns.

5. Repeat process for the second plate.

DATA ANALYSIS

1. Make two plots. First, plot the tension of the mass attached to the end of the string against the frequencies at which the string vibrated. Make three unconnected points – one for the fundamental, and one for each harmonic.) With three trials, you should be able to see a non-linear relationship between the tension and the frequencies of vibration. Next, plot the square root of the tension against the frequencies at which this tension allowed the string to resonate. In this plot, estimate a line of best fit.

2. Using the length of the string, the string tension, and the linear mass density, calculate the fundamental (n = 1) frequency of vibration for each of the three trials. How does this theoretical model compare to the resonant frequencies observed? Show your work.

3. From your experiments with the Chladni plates, make a chart showing the pattern observes and the frequencies at which you observed these patterns.

QUESTIONS

1. Does your second plot confirm that the frequency is linearly proportional to the square root of tension and thus the square root of mass?

2. When you varied your pickup location, did you see any difference in output amplitude? Explain.

3. Knowing what factors go into changing the resonant frequencies of a string, how can you make that low string on your booty-shaking bass go even lower? Give at least three examples.

4. When you were using the Chladni plates, were you able to create all of the patterns that you should have been able to? Were some patterns harder to create than others?

Physics of Music

Lab 5

Resonant Tones in a Column of Air

(Last revised: August 2005 by Alice Quillen)

EQUIPMENT

• Kundt’s tube apparatus

• Pasco Signal Interface with appropriate PC and software

• 1 pair Headphones (with 8-pin DIN adapter)

• 1 sound sensor microphone with cable

• Tuning forks

• Meter stick

• Thermometer

• Masking tape

INTRODUCTION

All wind instruments operate on the same basic principle. They all have a method of creating vibrations. For example, a French horn player buzzes her lips and a clarinetist makes a wooden reed vibrate. All wind instruments have a characteristic length that determines what notes or vibrations can be produced by the instrument. Finally, they all have some type of device to project the sound that they produce. It is the second item in this list that we are concerned with here – the characteristic length. Only certain resonant frequencies can be produced in a tube of a particular length. Understanding the boundary conditions of a column of air will determine these frequencies.

[pic]

Above are shown the fundamental modes of a column of air with two open ends (left) and one open, open closed end (right). We can calculate the allowable frequencies of a column of air by using the formulas:

[pic] (open-open) or [pic] (closed-open) (Equations 5.1)

In the above formulas, f is the frequency, v is the speed of sound in air, and L is the tube length. The allowed frequencies are known as harmonics of the tube. The velocity depends on the temperature in the room, as you have already demonstrated in your homework. Be sure to use the correct form of the equation for the boundary conditions you are considering!

In the first part of the lab, we will use a tuning fork to create longitudinal vibrations that are transmitted through the air into a plastic tube with an adjustable piston at one end. By subtly adjusting the length of the air column via moving the piston, standing waves are produced in the tube. The lengths of the air columns that produce resonance determine the location of the nodes of the standing waves. Knowing the distance between the nodes, the frequency can be calculated using equations (5.1).

In the second part of the lab, we will use a setup similar to that of part one. Instead of a tuning fork, we will use a pair of headphones driven by the frequency generator built into your Pasco box. We will detect resonance in two ways: with our ears and with the oscilloscope. Instead of adjusting the length of the tube with the piston, we will vary the driving frequency of the speaker and use a tube that is open at both ends.

PURPOSE

The purpose of this lab is to examine the resonant frequencies of a closed and open tube.

PROCEDURE

Part I – The Open/Closed Tube

1. Set up the apparatus as demonstrated by your TA.

2. The TA will divide up the tuning forks among the groups. After you have finished using your tuning forks, you will switch with another group so that you will eventually use all of the tuning forks. Record the frequencies of the tuning forks you are given on your data sheet.

3. Firmly strike the tuning fork with a rubber mallet or the bottom of your sneaker (should you be wearing one) and hold it about a centimeter away from the open end of the tube. Do not strike the tuning forks on the table! Be sure not to let it touch the tube.

4. Vary the effective length of the tube by moving the piston through its complete range and determine the positions of the nodes by listening for resonance points. In general, the larger the tuning fork you are working with, the harder it will be to locate the resonance points. Listen closely!

5. Record the distance of each node from the open end of the tube.

6. Note the temperature in the lab.

7. Repeat the above procedure until you have worked with every tuning fork.

Part II – The Open/Open Tube

1. Prop your sound sensor next to the end of your tube and hold your headphones next to the support at the other end of your tube. Be careful not to “close off” either end of your tube (or else you may alter your data).

2. Connect your sound sensor to channel C and your headphones to channel A.

3. Start your Pasco software and open the signal generator by clicking on the sample button. Drag an analog plug to channel C and select the “sound sensor” icon.

4. Drag a “scope” to channel C.

5. Set the signal generator to create a frequency lower than 150 Hz. Click the ON button and then click the MON button to begin viewing the sound at the end of your tube on the cope.

6. Increase the frequency slowly until you find resonance. You will both hear the intensity of the sound increase and see the trace on the scope increase in amplitude. You may need to adjust the vertical sweep of the scope as the sound coming from the headphones is not that loud. This isn’t all that easy, so be patient. Once you find resonance, record the frequency from the counter/timer.

7. Continue increasing the frequency to find 4 or 5 more resonance points.

8. Measure the length of the tube and record it on your data sheet.

DATA ANALYSIS

1. For part one, calculate the frequency of each tuning fork using equations (5.1). Do this for each set of data you took. For example, first calculate the frequency using the first resonance point and then using the second, and so on making sure to make the appropriate adjustments to the formula. Be sure to use the correct form of the equation! One is for an open tube, and the other is for a closed tube!

2. Calculate the theoretical harmonic spectrum of the open tube.

3. For part one, make a table with one column the calculated spectrum and another column the measured spectrum.

4. For part two, make a table with the calculated spectrum in one column and the measured spectrum in the other column.

QUESTIONS

1. When you calculated the frequency of each tuning fork in part one, did each set of data for one tuning fork give the same frequency? Why or why not? How do these calculated frequencies compare to the actual (labeled) frequencies of the tuning forks?

2. When you constructed the two tables above, did you find any discrepancies between the calculated spectrum and the measured spectrum? How do you account for this?

3. A stopped organ pipe has a stopper in the top of it, forcing a node to be there. Why do think organ makers’ use stopped pipes when they are building organs, especially for smaller buildings?

4. In this lab, we studied the resonant properties of a cylindrical tube. What do you think would change if we had a conical “tube”?

5. What effect would an increase in air temperature have on your results in this experiment? Explain.

6. The tunnel below the Eastman Quad at the UR is approximately 50 meters long. What frequencies would resonate in this tube? Are there any frequencies that would resonate in the tunnel below your range of hearing?

Physics of Music

Lab 6 (optional)

Room Acoustics

(Last revised: August 2002 by Kelly Gresham)

EQUIPMENT

• Ruler

• Meter stick

• Tape measure

• Noise level meter

• Noise making device

INTRODUCTION

One important application of the study of sound is in the area of acoustics. The complete acoustical analysis of a room is both difficult and time-consuming. The benefits of such an analysis, however, are tremendous for lecture halls, auditoriums, and libraries where the sound level is an important feature of the room. What we plan to do in this lab is give you an introduction to room acoustics by analyzing two rooms: the anechoic chamber and an individual room assigned to your group by your TA.

An anechoic chamber is a room built specifically for the purpose of absorbing sound. Such a room should be considerably more quiet than a normal room. By using the anechoic chamber as a reference point, your individual room can be analyzed.

The reverberation time T, as described in lecture, is the time it takes for a short, loud sound to decay by a certain amount. The reverberation time is given by the following equation:

[pic] (Equation 6.1)

The internal building resonances can also be calculated. By using different combinations of integers, many resonant frequencies can be found. The formula for these frequencies is:

[pic] (Equation 6.2)

Some important acoustical characteristics to consider are:

• Liveness – This just refers to the length of the reverberation time. The longer the reverberation time, the more “live” the room is.

• Intimacy – This refers to how close the performing group sounds to the listener.

• Fullness – This refers to the amount of reflected sound intensity relative to the intensity of the direct sound. The more reflected sound, the more fullness the room will have.

• Clarity – This is the opposite of fullness. In general, greater clarity implies a shorter reverberation time.

• Warmth – This is obtained when the reverberation time for low-frequency sounds is somewhat greater than the reverberation time for high frequencies.

• Brilliance – This is the opposite of warmth.

• Texture – This refers to the time structure of the pattern in which the reflections reach the listener. The first reflection should quickly follow the direct sound.

• Blend – This refers to how well the mixing of sound occurs between all of the instruments playing.

• Ensemble – This refers to the ability of the members of the performing group to hear each other during the performance.

Some problems in acoustical design are:

• Focusing of sound – This refers to the undesirable effect that occurs when sound is much louder at one point in the room than surrounding points in the room.

• Echoes – To obtain good texture, it is desirable to avoid any particularly large single echoes.

• Shadows – These are quiet areas that can be produced when there are large overhanging balconies or other structure jutting into the room.

• Resonances – It is necessary to separate the resonances of a speaker and the box enclosure of a tuned-port system to provide the best sound.

• External noise – This refers to noise coming from outside of the room.

• Double-valued reverberation time – If a recording of a musical event is played back in a room that it was not originally recorded in, this phenomenon can result.

Once you have completed your acoustical analysis, you can use your knowledge of room acoustics in your every day life. It’s useful when deciding where to place speakers in your room or when you want to set up a room for a party so that the music is not too soft, but not too loud, and everyone can still hear whoever they are talking to.

PURPOSE

The purpose of this lab is to understand the basics of the acoustical analysis of a room. The anechoic chamber is used as a standard to measure the individual room by.

PROCEDURE

Part I – Anechoic Chamber

1. Using the various measuring devices, measure the volume of the room. Record this on your data sheet.

2. Measure the surface area of all absorbing materials in the room. Remember to include yourself if you happen to be in the room! Record these values on your data sheet.

3. Have one member of your group go into the anechoic chamber and determine the ambient noise level using the sound level meter. Ambient noise is the noise that a room naturally has in it due to such things as air conditioners, lights, and sound from nearby rooms and hallways. Record this value on your data sheet.

Part II – Individual Room

1. Go to the room your group has been assigned. Be sure to find a room that you can work in without disturbing anyone else and where there isn’t anyone else to disturb you. Once there, make a map of your room. Make your map as specific as possible by indicating what objects are in the room and where they are found.

2. Measure the volume of your room. If your room is not rectangular, be sure to make enough measurements so that you are able to calculate the volume.

3. Measure the surface area of all absorbing materials in the room. If you have a large number of similar objects, it suffices to measure the surface area of one of them and multiply. For example, if you are measuring the surface area of seats in an auditorium, measure one seat and multiply that value by the number of seats in the auditorium. There is no need to measure the surface area of everything in the room, but all of the larger objects should be measured. Don’t forget to measure doorways, windows, and yourselves.

4. Use the sound level meter to determine the ambient noise level. Take measurements from several different spots in your room. On your map, show where you took measurements of the ambient noise level and what the measurements were.

5. Now, using your noise making device, have one person stay in one spot and continually produce a sound while the other person walks around the room and records the values of the sound level meter. Again, mark on your map where you took a measurement and what the result of the measurement was. If applicable, have the person making the noise stand where a performer or lecturer may typically stand.

6. If you haven’t already done so, take a minute to look over the important acoustical characteristics. After you have done so, go through each one and apply it to your room. You may need to test a few out with your noise making device and the sound level meter. Record your thoughts about each characteristic on the data sheet.

7. Now, take a look at the problems in acoustical design. Go through each one and determine how it applies to your room. You may need to test some things out again. Record your results on the data sheet.

DATA ANALYSIS

1. Calculate the reverberation time from formula (6.1) for both your room and the anechoic chamber at 250 Hz, 100 Hz, 200 Hz, and 5000 Hz. Use the tables given for absorption coefficients. If a material in you room is not found on the chart, make a guess at what the absorption coefficient is based on what material it is most like on the list.

2. Graph the reverberation time versus frequency for both your room and the anechoic chamber. Draw a line of best fit.

3. Calculate the internal building resonances using formula (6.2). Calculate at least six different values by using various integers. If your room was not perfectly rectangular, be sure to account for this in the formula.

QUESTIONS

1. Were you able to find any “dead spots” in your room? These are places where, during the part of the lab where a noise was being made from a fixed point in the room, the sound level meter recorded significantly lower values than at surrounding points. If you did find any, can you see any reason why they are there? Indicate them on your map of your room.

2. What differences did you find between the anechoic chamber and the room you analyzed?

3. Go through the important acoustical characteristics and describe how they applied to your room. Was your room lacking any important acoustical characteristic that it should have had? How would you improve the room to help it achieve what it is missing?

4. Go through the problems in acoustical design and describe how they applied to your room. If your room had any major problems, discuss how you might get rid of them.

5. Suppose a new sound system was to be installed in your room. Where would you place the speakers (two, four, or six based on the size of your room) so as to create the best possible room acoustics? Why would you place them where you would?

Acoustics Insert with Helpful Information:

[pic]

Appendix A: Scientific Notation and It’s Prefixes

|SI PREFIXES |

|Factor |Prefix |Symbol |

|1024 |yotta |Y |

|1021 |zetta |Z |

|1018 |exa |E |

|1015 |peta |P |

|1012 |tera |T |

|109 |giga |G |

|106 |mega |M |

|103 |kilo |k |

|102 |hecto |h |

|101 |deka |da |

|10-1 |deci |d |

|10-2 |centi |c |

|10-3 |milli |m |

|10-6 |micro |( |

|10-9 |nano |n |

|10-12 |pico |p |

|10-15 |femto |f |

|10-18 |atto |a |

|10-21 |zepto |z |

|10-24 |yocto |y |

Examples of use:

89 ns = 89 ( 10-9 s = 8.9 ( 10-8 s

230 km = 230 ( 103 m = 2.3 ( 105 m

Appendix B: Notes on the Piano

[pic]

Appendix C: Equal Tempered Music Scale

[pic]

Appendix D: The 12 Pitch Classes

[pic]

\

Appendix E: The speed of sound in air

V = 331.4 + 0.6 Tc where V is the speed of sound in m/s and Tc is the air temperature in Celcius.

This is a good approximation at sea level.

-----------------------

Frequency

Oscilloscope

Function Generator

Counter/Timer

To the left is an example of a data point with error bars. The line going through the data point is twice as long as the error with which you obtained the point (i.e. the difference in Hz between two pixels). Notice that the line is centered on the data point and the ends of the line are marked on both the top and bottom, so the error bars look like a capital “I.”

Freq.

Length

1

5

3

sonometer

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download