1: INTRODUCTION TO SOUND SIGNALS INTRODUCTION

Transcription

1 1: INTRODUCTION TO SOUND SIGNALS INTRODUCTION Acoustics is the interdisciplinary science that studies sound, or mechanical pressure waves in gases, liquids, and solids. The acoustic labs will use various tools to visualize and measure properties of sounds. The labs will study the production, transmission, resonance, absorption, digitization, filtering, and amplification of sounds. For example, we will record sound with computers that store the recording as a list of numbers (similar to the way a CD represents sound). Today, you will talk into a microphone, which converts the sound pressure into an electrical signal, and the computer will measure the strength of that electrical signal about 22,000 times each second. The result is that each second of recorded sound is represented by a time-series of 22,000 numbers that we will call a "sound signal". Once the sound data is stored on the computer, there are many things that the computer allows us to do to analyze or modify the sound recording. One thing the computer can do is to make a graph of this data. Here is how the computer graphs a part of the recording of the word hello. The vertical axis is the strength of the electrical signal that is proportional to sound pressure in Pascal [Pa=N/m 2 ], and the horizontal axis is time. The computer connects the points on the plot to help us to see the patterns. RECORDING AND REPLAYING SOUND Time (s) A. Praat program: Turn on the computer if it is not already on. After a short time, the computer monitor will light up and you will see the desktop. Use the mouse, double-click at the Praat Icon. This will start up the program called PRAAT. PRAAT is a computer application that phoneticians and other researchers use to analyze, synthesize, and manipulate speech and other sounds as well a providing high quality figures to document their work. [see Wait a minute for the program to load itself. Two windows should appear: a Praat objects window and a Praat picture window. B. To record from the microphone: perform the following steps: 1. Choose Record mono Sound from the New menu in the Praat objects window. A SoundRecorder window will appear on your screen. 2. Press the button on the computer s microphone to activate the microphone. Use the Record and Stop buttons to record a few seconds of your speech. Experiment with how loud to speak and how close to the microphone. The level meter will indicate the Acoustic Lab#1: Introduction to Sound

2 appropriate combination, i.e., you want to maintain the level to show green with the rare spike into the yellow or red levels. 3. Use the Play button to hear what you have recorded. Repeat the two steps above until you are satisfied with what you have recorded. To make sure that the sound file does not get too large, don t record more than a few seconds of your voice for now. 4. Type in a name for your sound file in the text box below the Save to list: button. Hit the Save to list: button and the text string Sound name should appear in the Praat objects window that indicates file where your sound is recorded. 5. The right part of the Praat objects window shows what you can do with the sound. Try the Play, Edit, and Draw buttons. C. Understand the Displays: We will be examining waveforms that consist of sound pressure on the vertical axis and time on the horizontal axis. 1. Edit window Again, you get the Edit window by clicking the Edit button in the Praat objects window when your sound file is selected. It will show a waveform of your sound displayed in the top half of a window that has your sound file as its name. The bottom half of the display will be a spectrogram of your sound that we will learn about later in the course. An example is shown in the following graph: Top: waveform of "test, test, test" spoken in to the microphone, with sound pressure along the vertical axis, and time along the horizontal axis. Bottom: corresponding spectrogram that will be discussed later in the course. With this window up, hit the long horizontal Window button below the displays. A vertical line will sweep across the display, indicating what portion of the sound that is being played at a given time, and your sound will be played back. Click and drag (i.e., point the cursor to some part of the sound signal, click and hold the mouse button down, and drag to another part of the sound signal) in the display to change the size of a red rectangle in the display. This selects a portion of the waveform that can subsequently be played back by clicking on the button that is lined up directly below the red selection rectangle. Play back different parts of your recording. Try selecting less than an entire word and playing it back. Can you still understand what is being said? Go through the whole chain several times, i.e., saying (or singing) different things into the microphone, recording the results, storing it in to a sound file, and then using the Edit button to examine the different sound waveforms. We will develop the tools to analyze the differences systematically during the semester. Introduction to Sound

3 2. Praat picture window: You can also draw the waveform of your sound by clicking and dragging a red window in the Praat picture window, and then hitting the Draw button in the Praat objects window when your sound file is selected. The picture will always fill the drawn red rectangle. Drawing always overlays a new picture on top of an existing picture, so you will have to choose Erase all from the Edit menu of the Praat picture window to get rid of the previous drawing. When you hit the Draw button, a choice of range of times (from time t start to t end ) to display is given. If the second, t end, time entered is 0.0, the entire waveform will be displayed by default. Try hitting the Draw button a few times with different time ranges (remembering to Erase all from the Edit menu each time) to zoom in on interesting portions of the waveform. 3. Print your recording: Once you have a good recording in the Praat picture window, try printing the display. To do so, choose Print from the File menu of the Praat picture window. On your printout, label the loudest and softest parts of your recording, and if possible, the words or letters corresponding to each section of the waveform. WHAT DO WE MEAN BY SOUND? When people talk about sound, they may mean one of several things. Usually, they mean either the sensation people have when vibrations of the air hit their ears, or they mean the actual vibration of the air, whether or not anyone is there to hear it. The relationship between vibrations and sensual perceptions is complicated, largely because perception itself is so complicated. How loud a particular sound you perceived may depend on your mood. Nevertheless, there are three most basic qualities that are often used to describe perceived sound loudness, pitch, and timbre. They correspond in a fairly straightforward way to physical properties of vibration that can be measured. Loudness: Vibrations can be large or small. Large vibrations are said to have a large amplitude. In the world of sound, amplitude corresponds to loudness the greater the amplitude, the louder the sound. PITCH: Vibrations can happen quickly or slowly. This property of vibration is called frequency. Frequency tells how often something happens in a given unit of time. For example, the sound you just heard caused your eardrum to move back and forth about 1000 times in each second, so the frequency was about 1000 per second. The unit per second has an abbreviation, Hz, named after Heinrich Hertz and pronounced hurts. So a vibration that happens 1000 times each second is said to have a frequency of 1000 Hz. The physical property frequency corresponds to the sensation of pitch. Try this out for yourself: put on the headphones, turn up the amplitude to a comfortable volume, and then turn the knob on the function generator that is labeled FREQUENCY. When you move the knob clockwise, both the frequency and the pitch go (up/down)? (please choose one) Acoustic Lab#1: Introduction to Sound

4 A. Set the frequency The function generator can produce any frequency from less than 1 Hz up to more than 5,000,000 Hz. This wide range is obtainable by using the bank of grey buttons labeled RANGE. The 5K button is currently pushed in. The k is an abbreviation for kilo meaning 1000 (as in kilometer or kilogram). So the generator is now set to generate frequencies near 5000 Hz. Frequency adjustments within this range are provided by the coarse and fine frequency knobs. The current frequency setting is displayed on the screen. Note the lights which indicate whether the units are Hz or khz. To get higher or lower pitches you push a different RANGE button. Push in the 50k button. This puts you in the frequency range from about 4000 to 50,000 Hz, which contains the top of the human hearing range. Be sure you understand the relationship between the dial and button settings on the generator and the frequency. The dial is not very precise. Later in the semester we will learn how to measure the frequency more exactly. 1. With the RANGE set to 50k, what range of frequencies can you obtain? Lowest frequency: Hz; Highest frequency: Hz 2. Use the generator to determine the highest frequency you can hear: Hz 3. Set the RANGE button at 5k Hz. Without changing the amplitude knob, turn the coarse frequency knob from a low setting to a high setting. Does the loudness you perceive change? At a fixed amplitude knob setting, what is the frequency when the sound seems loudest? Hz. 4. Set the RANGE to 500 Hz. What range of frequencies can you get? Lowest frequency: Hz; Highest frequency: Hz 5. What is the lowest frequency you can hear? Note that to measure this, you may need to change to an even lower range as well as turn up the amplitude. Lowest frequency: Hz; Highest frequency: Hz (Nore: If you only hear clicking, you are not hearing a frequency because the headphones are small, they do not produce low sounds very efficiently, so you will need to turn the amplitude all the way up on both the generator and on the volume control on the headphone box. You will probably want to turn it down again later, or it the sounds will be annoyingly loud.) TIMBRE (pronounced tamber ) You have seen (and heard) that sounds can be loud or soft, depending on amplitude, and they can be high or low, depending on frequency. But that is not all there is to sound. You can distinguish sounds that have the same pitch and the same volume. For example, you can sing the vowel sound a and then the vowel e, and they sound different, even though they have the same frequency and amplitude. The sounds a and e have different tone qualities or timbres. Timbre much more complicated than loudness or pitch, and you will spend much of the semester studying it. Introduction to Sound

5 B. Setting the Function Generator To begin your investigation of the timbre, listen to the effect of changing the shape of the vibration. The function generator can produce three different types of oscillation, called sine, triangle, and square. The smooth sine function is the simplest, purest, cleanest, least complex sound there is. Every other sound is, at least to some degree, brighter, sharper, or harsher than the sine sound. You can choose one type of oscillation or another by pushing one of three MODE buttons, labeled for sine, for triangle, and for square. Connect the function generator output to the speaker-box, and use the earphone to listen the output. 1. First, set the frequency to around 250 Hz. Then push the triangle button and listen to the sound. Typically, the triangle function is said to be brighter or sharper than the sine function. You should avoid calling it higher, since that implies a higher pitch. Do you agree that the sound is brighter? 2. Now try the square function. Is it much brighter, possibly even buzzy? Does it sound louder to you? If so, you may want to turn down the volume to make a good comparison. 3. Switch back and forth, listening to the three timbres. Try listening to the different shapes at widely differing frequencies. Explain your findings. 4. Is it easier to distinguish the sounds at low frequency, of 100 Hz or at high frequency of 8 khz? 5. Is there any frequency at which you cannot tell the three waveforms apart? In summary, the three qualities of sound with a description of what we could hear, see (using PRAAT), or control (using knobs and buttons on the function generator). Quality Description of sound Appearance of sound Control Pitch Sounds high or low Many or few oscillations Frequency (horizontally) Loudness Sounds loud or soft Big or small oscillations Amplitude (vertically) Timbre Sounds pure or complex Shape of sound signal Function We discovered how to associate changes in the "Controls" with changes in the sound and the appearance of the sound signal. We found that: higher frequency sounded higher and produced more oscillations in the same time period; higher amplitude sounded louder and made the oscillations larger; different wave form changed the timbre of the sound without changing the pitch (but some people also noticed a change in the loudness). Now, we will use the program PRAAT to measure quantities that are related to Pitch, Loudness, and Timbre. These "quantities" are related to the "Controls" in the above table: Frequency, Amplitude, and Function. Acoustic Lab#1: Introduction to Sound

6 QUANTIFYING SOUND Let's define each of the quantities we will use today to describe periodic sound signals (a periodic signal is made up of a small unit called a cycle that repeats over and over again): Frequency: Count the number of cycles during some time interval and divide by the length of the time interval. [Standard definition] This definition is not quite complete, what's a cycle? Cycle: The smallest repeating unit in a periodic signal. [Standard definition] Period: Measure the length of time for a known number of cycles, and then divide the time by the number of cycles. [Standard definition] Is it obvious what the relationship between frequency and period are? Let's call the number of cycles N and the length of time t: f = Frequency = N t, T = Period = t N, therefore, f = 1 T ; or T= 1 f or literally: Frequency = 1 Cycle Period or Period = (Time for one Cycle). Amplitude (Peak-to-peak): Difference (either sound pressure in Pascals or the electrical signal in volts) between the highest point of a cycle and the lowest point of a cycle. We will use this definition in this lab. The usual definition of amplitude without peak-to-peak qualifier is the coefficient of the sine or cosine function describing the wave, and usually equal to one-half the peak-to-peak amplitude. Function: The shape of one cycle [Standard definition]. A. Setting up 1. Start up PRAAT by double-clicking on the icon marked "Praat" on the computer desktop. While it is starting up, turn on your function generator and set the Range to 500Hz. Adjust the Coarse frequency knob until the screen reads 200 Hz. As you get close to 200 Hz, you may want to use the Fine frequency knob. Set the Output Level knob (Amplitude) to the minimum. Set the function to the sine wave. For this purpose, you can connect the output of the function generator to speaker-box inputs, and connect headphone to a speaker-box output. Put on the headphones and have a listen. You may need to increase the amplitude a little if the sound is not loud enough. Connect the other speaker-box output to MIC input of the computer. 2. When you started the program, two windows should have appeared: a Praat objects window and a Praat picture window. Recall how you record a sound with PRAAT: choose Record mono Sound... from the New menu in the Praat objects window. A SoundRecorder window will appear on your screen. 3. Use the Record and Stop buttons to record a few seconds of your sine wave signal. Type in a name for your sound file (e.g., "sine") in the text box below the Save to list: button, hit the Save to list: button and the text string "sound sine" should appear in the Praat objects window that indicates the file where your sound is recorded. Introduction to Sound

7 4. With the name of the file highlighted in the Praat objects window, hit the Edit button to see the waveform that you just recorded in all its glory. 5. To simplify the display (if it is not already done), go across the menu of the Edit window (that should be titled "Sound sine"), Spectrogram menu, pull down, Show spectrogram deselect (i.e., if a check mark is shown, select it. If no check mark, exit without doing anything). Pitch menu, pull down, deselect Show pitch Intensity menu, pull down, deselect Show intensity Formants menu, pull down, deselect Show formants Pulses menu, pull down, deselect Show pulses B. Function measurement 1. The first thing to do is to identify the shape of cycles (this is how we quantify timbre). To zoom in to show individual cycles, go to the Edit window, View menu, pull down to Zoom.. Select a start time of 0.0 seconds and a stop time of 0.1 seconds. Draw the waveform and its shape: 2. Does your picture match the function setting of the generator? C. Frequency measurement 1. If you followed the directions above, you should be displaying 0.1 second of the signal from the function generator. a. Count the number of cycles that the computer recorded. To calculate the frequency, use our definition: Number of cycles Frequency = Length of time = 0.1 second = Hz b. Is the answer close to what the frequency generator said? D. Measuring Period The most accurate way to measure the frequency is to measure the length of one cycle (the period). We can do this very accurately using "Markers", which give the horizontal and vertical coordinates for the sound signal. There are two markers, indicated by the dashed vertical red lines in the display. 1. Zoom in horizontally so that roughly two cycles are visible in the display as shown below. Do this by using the View menu, Zoom, and entering say 0.0 as a start time and 0.01 sec as an end time (or any relevant combination of Zoom in, Zoom out, Zoom to Acoustic Lab#1: Introduction to Sound

8 selection.) Click to place the "start" marker, then drag to place the "stop" marker and let go. The region of signal selected is shown highlighted in pink. You want to measure the time period between adjacent maximum points, or minimum points. Don't worry, you can do it over and over until you get it just right. 2. The time where the start and stop markers have been placed is shown as red numbers at the top of the display. More conveniently, the difference between these two times, and hence the period, is shown in the bar between the two red numbers. The value of 1/T, i.e., the frequency is also shown. For example the period of the plot is s, and the frequency is Hz. 3. Measure the period of your signal using the markers: One period = seconds 4. Since the period is the time for one cycle, the frequency is just: Frequency = 1 Cycle Period = 1 Cycle = Hz [cycles per second] second E. Amplitude measurement We can use PRAAT DISPLAY to measure amplitude as well. Leave the display zoomed in on approximately two cycles. 1. You can click anywhere on the waveform (don t drag, just click) and a dashed red vertical marker will appear. The red number at the top indicates the time of the marker location. The blue number to the left of the display gives the amplitude of the waveform at that time as a sound pressure in units of Pascals. Click at several points to try it out. 2. Now try to click at a waveform maximum. You can keep trying, checking that you are getting the largest positive amplitude that you can, e.g., the sound picture shows the maximum amplitude is Pascals at a time of seconds. Record your maximum amplitude: Pascals. 3. Do the same clicking on an adjacent waveform minimum. It should be a negative number. Record your minimum amplitude: Pascals 4. The difference in these two numbers is simply the peak-to-peak amplitude. Record your peak-to-peak amplitude = Max. amp. Min. amp. = Pascals. 5. Change the function to the triangle wave. Record this new sound waveform. To do so, you have to close the SoundRecord window, go back to the Praat Object window, use the New menu, and choose Record mono Sound... Record for a few seconds, save the file with file name Introduction to Sound

9 "triangle", and Save to list... With this file name highlighted, hit the Edit button to display the waveform, zoom in on it, and make measurements as before. peak-to-peak amplitude = Pascals. 6. Draw the shape of one cycle. Measure the period and amplitude. Are period and amplitude independent of the wave shape? F. Frequency and period of a spoken sound 1. Plug in your microphone, and record a vowel sound. Softly singing "eeeeee", "aaaaah", or "ooooooo" work best. See below for an example of the waveform of a sustained vowel "eeeee". 2. Measure the period, frequency (in Hz), amplitude (in Pascals), and function (show pictorially) of your own sustained vowel. 3. Put on the headphones again. Set the function generator to a frequency whose pitch you can approximate with your voice. Record a vowel sound at that pitch. Use PRAAT to measure the frequency of the voice signal. Write down the frequency of the function generator and the frequency of your voice. Are they close? 4. Make a different vowel sound at the same pitch. Measure the period and frequency. Are they close to what you measured in the previous part? Acoustic Lab#1: Introduction to Sound

10 Introduction to Sound

11 2: PRELAB: FREQUENCY AND PERIOD INTRODUCTION The frequency of a sound vibration is associated with its perceived pitch, and since pitch is one of the basic properties of sound, frequency is very important. This hand out will help you understand the relationship between frequency and period, a related quantity. First, let us have a quick review of frequency FREQUENCY Frequency tells you how many events happen in a certain unit of time. As an example, try taking your pulse. That is, count the number of times your heart beats in one minutes. Heart beat rate = per minute This is a frequency, because it tells how many beats there are in one unit of time. The unit of time you used was the minute, which is customary for pulse measurements. In science, however, the second is the standard unit of time. Since there are 60 seconds in a minutes, you will have t divide the number above by 60 to get your pulse in standard units. Heat beat rate = per second Since a typical pulse is between 60 to 85 beats per minute, your pulse in beats per second, or Hertz (Hz), will probably be between 1.0 and 1.5. PERIOD Anything that happens over and over again at a regular interval is said to be periodic. This event is characterized by a certain period of time, namely the period of time it takes for the repeating event to happen once. This characteristic time is called, guess what, the period. As an example, try tapping your finger on the table once every second (you can come close to one per second by counting one thousand one, one-thousand two, one-thousand three, ). You are now making a periodic tapping sound with a period of 1 second there is one second between taps. Now tap twice as often, every half second. This makes the period 0.5 s. Period and Frequency The period (T) is the time interval of repetitive events. The frequency (f) is the number of repetitive events in a time interval. Thus f=1/t. 2: Simple harmonic motion prelab 011

12 You know that your heart beats in a regular way, so it must be periodic. Since it is periodic, it must have a period. You also know that it has a frequency, because you measured it earlier. Pulse has both period and frequency. As another example, you have already seen that each pitch you can hear has a frequency somewhere above 15 Hz and below Hz. Since you know that there are steady, periodic vibrations, you know that each pitch must also have a period. In fact, everything that has a period must also have a frequency. Frequency and period are really just two different ways of saying the same thing. So what is exactly the relationship between the period and frequency? It will be easier for you to remember the relationship between two quantities if you figure it out yourself. To see the general relationship, look at a few specific examples. As you write down both the frequency and period of several periodic events, a rather simple pattern will emerge, which you should summarize in a sentence. (Hint: you will see the pattern more quickly if you express the number less than one as fraction rather than decimals.) Remember, frequency tells you how many events happen in a certain unit of time. Period tells you how long it takes for one of those events to happen. 1. Tap your finger once every second a. What is the period? second b. What is the frequency? Hz 2. Tap twice as often, i.e. every ½ second. a. What is the period? second b. What is the frequency? Hz 3. Tap as fast as you can. This will be about 8 times per second. a. What is the period? second b. What is the frequency? Hz 4. Now tap slower at once every two seconds. a. What is the period? second b. What is the frequency? Hz 5. Now tap slower at once every 4 seconds. a. What is the period? second b. What is the frequency? Hz Do you see the pattern? Write a sentence that describes the general relationship between period and frequency. 2: Frequency and Period Prelab 012

13 2: SIMPLE HARMONIC MOTION Motion of a mass hanging from a spring If you hang a mass from a spring, stretch it slightly, and let go, the mass will go up and down over and over again. That is, you will get Periodic Motion. Using Logger Pro you can see the repetition in position and velocity graphs. The length of time to go through one cycle of the motion is called the period. If you remember that the period refers to a period of time, you will remember the correct units for the period, namely seconds. (a) Learning about Logger Pro In this activity you will examine the graphical display of the motion of a mass hanging from a spring. 1. Log on to the computer. Launch LoggerPro by double clicking on the LoggerPro 3.3 icon. Open a New file. File Open New. 2. Play with the motion detector to learn about what it does. Aim the detector horizontally off the bench. Click on Collect (or hit the F11 key) and the motion sensor will begin to emit clicks, which means that it is measuring. Start from 4-5 feet (sensor is sensitive up to 2m), move towards the sensor, stop for a second or two when you are 2-3 feet away, then move away from the sensor. 3. Try to watch the top part of the display as you move or stop: which way does the line move when you are moving toward the sensor? Away from the sensor? Not moving? 4. Now look at the bottom part of the display. It plots your velocity at the same time as it measured your distance. Note that velocity can be positive or negative: which was it as you were moving toward the sensor? away from the detector? at rest? Click Data - Clear All Data (b) Measuring periodic motion 1. Prepare a mass on a spring. If it hasn t already been done for you, hang a spring vertically from the support which is mounted to the table and attach the weight holder to the spring. The mass of the holder itself is 20g. In addition, put one slotted mass of 20g on the holder. Place the motion detector face-up directly below the spring. 2. Measure the mass at rest. Try to make the mass stay as still as possible, then click on the Collect button (or use the F11 key). After the program is done measuring, save this data by clicking Experiment (in the menu bar) Store latest run Data New Data Set. Now click Experiment (in the menu bar) Data Collection and set the length for 5 seconds. 3. Make distance and velocity graphs of the motion of the mass. Push the mass straight upward 5-10 cm, and let it go. Warning: To avoid making the amplitude too big, push the mass up to start it moving instead of pulling down. Click Collect to begin graphing. The program will measure the motion for 5 seconds. Note: Be sure the motion detector sees the mass over its full motion and that there are no flat portions of your graph where the mass came too close or too far from the detector. Acoustic Lab#2: Simple Harmonic Motion

14 4. Adjust the distance and velocity scales on the screen so that your graphs fill the axes better, and are easier to read. To do this, click the icon on the menu bar that is an A inside a box (to the left of the magnifying glass icons). This is the autoscale button. 5. Print your graph. Select File (in the menu bar), Print Graph, Ok. Click Ok and then Ok again to print your graph. 6. On your print out, label both graphs with all letters: B at the Beginning of one cycle. E at the End of the same complete cycle. A on each spot where the mass is moving Away from the detector fastest. T on each spot where the mass is moving Toward the detector fastest. Z on each spot where the mass has Zero velocity. F on each spot where the mass is Farthest from the motion detector. C on each spot where the mass is Closest to the motion detector. Comment: Note that when an object returns to the same position, it does not necessarily mean that a cycle is ending. It must return to the same position, and the velocity must also return to the same value in both magnitude and direction for this to be the start of a new cycle. Question: Compare the distance and velocity graphs. Do they appear to have the same period? Do their peaks occur at the same times? If not, how are the peaks related in time? Select Analyze (in the menu bar), Examine to help you read the graph accurately. (c) Properties of Periodic Motion Period and Amplitude 1. Measure the period of the motion represented by the graphs above. Click Analyze (in the menu bar), Examine. Then point the cursor at the beginning of the first full cycle (for example, your first peak) on your distance graph. (You may of course start with any part of the cycle, but be consistent once you pick a starting point.) Note the Time reading (the bottom one of two) in the examine box and write down the number in the table below as the Starting Time. Now find the end of the last full cycle (continuing the example, your last peak), point the cursor at that point and record that as the Ending Time. To calculate the period of the motion, you need to know how many full cycles there are between the two times you just marked. Enter your data and calculation in the first five cells of the top row of the table below. Leave the other cells blank for now. Period and Amplitude Data Amplitude Starting Time [s] Ending Time [s] Time Elapsed [s] Number of Cycles Period [s] Equilibrium Position [m] Maximum Position [m] Amplitude [m] Large Small The midpoint of the motion is also the point where the mass will hang at rest. This position is called the equilibrium position. The amplitude of the motion is the maximum displacement 2: Simple Harmonic Motion

15 (change in distance) from the equilibrium position. (For the motion of a mass and spring it should be the same in both directions.) See the diagram below. Highest Position Amplitude Equilibrium Position Amplitude Lowest Position Detector Zero Position 2. Find the equilibrium position and the amplitude. a) Now point your cursor to find the position of the mass when it is at rest. Write down that position in the table above. (This position will be the top one listed in the box.) b) Also read the maximum distance of the mass when it was oscillating and record it in the table. Finally calculate the amplitude by taking the difference between the maximum and equilibrium positions, and record it in the table. The motion of a mass hanging from a spring that you looked at in the previous investigation is a close approximation to a kind of periodic motion called simple harmonic motion, often abbreviated SHM. It is relatively easy to show mathematically that if the restoring force (whatever produces the motion) is proportional to the distance away from the equilibrium position, the result is SHM. (d) The Dependence of the Period on the Amplitude 1. Repeat the previous investigation with a smaller amplitude. Using the same SHM setup as in the previous investigation (i.e., one spring with the weight holder only plus the 20g mass), make distance and velocity graphs with a smaller amplitude than before. Make the amplitude about half as large as before by only lifting the hanger and masses about half as high as you did the first time. When you have a good graph, find the period and the amplitude, and record all your data in the second row of the Period and Amplitude data table on the previous page. 2. Compare the periods. Take ratio of the periods and the ratio amplitudes for the two cases. Ratio of Amplitudes: Ratio of Periods: Question: Is there evidence that the period depends on amplitude? (Did the change in amplitude result in a comparable change in period?) Acoustic Lab#2: Simple Harmonic Motion

16 (e) The Dependence of the Period on the Mass In this activity you will find out what happens to the period of oscillation when you change the hanging mass. Repeat the period measuring procedure three times, each time adding a 20g slotted mass to the holder. Fill in the appropriate rows of the table. Then, carefully remove the weight holder and repeat the period measuring procedure with a 10g mass and a 5g mass. These can be attached by slipping the bottom of the spring through a hole in the mass. Fill in the appropriate rows of the table. (Note: Because the masses are so small, the oscillations will also be very small. It is convenient to double click on the velocity graph and use the Auto Scale feature on the velocity axis. (You can also do this with the distance graph.) When you have just started the oscillation, there will probably be other small vibrations that make the oscillation of the mass hard to see, but these extraneous vibrations will eventually die away enough for you to get good graphs. Keep trying until your graphs are smooth enough for you to measure the period. Then fill in the last two rows of the table.) Period versus Mass Hanging Mass Mass [g] Starting Time [s] Ending Time [s] Time Elapsed [s] Number of Cycles Period Holder + 3 masses 80 Holder +2 masses 60 Holder +1 mass 40 Holder alone 20 No Holder + 10g mass 10 No Holder + 5g mass 5 Question: Does the period depend on the mass. Does it regularly increase or decrease as mass is increased? 2: Simple Harmonic Motion

17 (f) Graphing the period versus mass Plot the period and mass data. Launch the Graphical Analysis program. a.) Click Data New Manual Column Insert a column with name Mass; enter your experimental mass data in this column Insert a column with name Period; enter your experimental period data in this column b.) Click Insert graph; a new graph will appear; change the axes to mass and period. c.) Click Analyze curve fit; choose proper curve fit to your data, you may consider power fit AM^B. Print this fit page and explain what you find in the graph. A= B= Explanation: Comment: It should be clear from your graph that while the period does increase as mass increases, it is not directly proportional to mass since the fitted power B is not 1. You may find B is near 0.5. In the next part (g), you will raise mass to a fixed power and try linear fit to your data, and find out the intercept. (g) Determining the mathematical relationship between period and mass. Now you want to use linear fit to find out how the period depends on the mass. You should plot period T vs mass to a power, m z, where z varies from 0.2, 0.25, 0.333, 0.5,.. to 3 as shown in the Table in the next page. Since you now have enough data to find the quantitative relationship between the period and the hanging mass. The next several steps will allow you to test several hypotheses concerning the period and mass relation. For this purpose you will plot the observed periods versus various powers of mass, e.g. period versus mass to power: 1/5, 1/4, 1/3, 1/2, 1, 2 and 3. Try to fit a straight line for each case. From the parameters of the fit (intercept, coefficient of regression (C.O.R.)) you will be able to determine which hypothesis fits the data best. Follow the steps described below. 1. Make a column which is mass raised to a power. Label the x data column mass by double clicking on the x heading. Type mass and hit enter. Next, create column to show mass raised to a power by selecting Data (from the menu bar) New Column Calculated. For new column name, enter Mass ^ and then the power to which you want to raise the mass. From the Columns pull down menu, select Mass. Then type ^ and the power you wish to use. Click Ok. Do this for the 1/5, 1/4, 1/3, 1/2, 1, 2, and 3 powers. 2. Look at your graphs for each of your masses raised to a power. Click on the label on the horizontal axis of the graph to move between your graphs. Click Analyze Linear fit to analyze your data. Then, fill the table in next page and decide which is the most linear. 3. Find a power of mass which has a COR closest to 1. COR is a measure of how close the data is to a straight line. That is, try to find a power which gives the straightest line which goes through the origin. Use a Regression Line to find the best fit. Write down the values of the slope, intercept and C.O.R. for each tried power of mass. Fill the Table on the next page. Acoustic Lab#2: Simple Harmonic Motion

18 Comment: You will find it impossible to get the line to go exactly through the origin. This is because the spring has some mass, which contributes to the effective size of the hanging mass. 4. When you have the best fit you can get, show it to the instructor, print copies for yourself and your lab partner and attach to your report. Comments: Although, your points probably fit fairly well to a line. It could be that the period increases linearly with mass, but offset by a constant. Or it could be that the period depends on mass in some other way, such as being proportional to a power of mass. The only way to find out which of these is correct is to collect more data. Power of Mass Slope Intercept C.O.R Comments/Notes Question: Which of the tried hypotheses fits the data best? In other words, what is the mathematical relationship between the period and mass (to which power of m is period proportional)? Mathematical relationship between period and mass: f 1 2 k m T 2 m k 2: Simple Harmonic Motion

19 PRELAB 3: ELECTRIC CIRCUITRY A. ELECTRIC CURRENT AND CIRCUIT Electric charge Q is a fundamental property of matter. Charges are measured in Coulombs [C], which is a large number. An electron carries a unit of negative charge e, a proton carries a unit of positive charge +e, and neutrons carry no charge. Each unit of charge is e= C. An atom is made of electrons orbiting outside a nucleus, which is composed of protons and neutrons, called nucleons. The oxygen-16 nucleus is composed of 8 protons, and 8 neutrons. Thus the charge of an oxygen nucleus is +8e. The radius of nucleus is of the order of to m, depending on the number of nucleons. Atoms are normally neutral, and thus the number of electrons is equal to the number of protons. The neutral atom of oxygen-16 is made of an oxygen nucleus and 8 electrons orbiting outside the nucleus. Matters can be classified into conductors and insulators. Some materials are called semiconductors. A conductor has free electrons that can move in the presence of electric potential, an applied voltage, resembling a ball falling from a height by the gravitational potential. When electrons move in a conductor, we call electric current flows in the conductor. The electric current is I= Q/ t, the flow rate of charge flow per unit time. Since the charge of electrons is negative, the current is flowing opposite to the direction of electrons. As electric potential is applied to an electric circuit, electric current flows from the high voltage point to a lower voltage point, resembling water flows from higher ground to lower position. The electric potential is measured in Volts, and the electric current is measured in Amperes, related by the Ohm s law: V=IR. Here V is the electric potential in Volts, I is the electric current in Amperes, and R is the resistance (impedance) measured in Ohms (Ω). Major electric circuit elements are resistors, capacitors, and inductors. Resistors are made of materials impede motion of electrons, capacitors are made of parallel plates that can hold charges, and inductors are made of coils that can hold magnetic energy. For example, a car battery is typically 12 V. Connect the battery to a 1000Ω resistor, the current in the circuit is I=12V/1000Ω=12 ma=0.012 A. a. The home electric outlet is typically 120 V. When connected to a light bulb that has filament with 240 Ω resistor, what is the current flowing through the light bulb? b. Charge of an electron is I = A. c. Charge of a carbon nucleus (made of 6 protons and 6 neutrons) is 1 019

20 A capacitor (also known as condenser) is normally made of two conductor plates. If positive charges Q are placed on one plate, negative charges Q will be induced on the other plate. An electric potential V and electric field will be established between these two plates. Thus a capacitor can store electric energy! The positive charge plate has a higher potential. The capacitance C is given by C=Q/V. The capacitance is measured in Farads [F]=[C/V]. Typical capacitance of capacitors is measured in picofarad, i.e Farad, nanofarad [nf]=1000[pf], or microfarad [μf]=1000,000[pf]. An inductor is made of conducting coils. As the electric current flows, magnetic field is induced. When conducting wires are closely wound around a cylinder, electric field is enhanced at the inner tube of the cylinder, and canceled each other outside the cylinder. Thus the inductor can store magnetic energy. The voltage between two ends of an inductor is equal to V=L( I/ t), where L is the inductance. The amount of energy stored in an inductor is equal to (½)LI 2, where I is the current in Ampere, and L is the inductance measured in Henry [H], and energy stored in the inductor is measured in Joules [J]. Typical inductance of inductors are measured in milli-henry [mh]=10 3 H and Henry [H]. B. PROPERTIES OF A RESONANCE: Often a system has natural frequencies of vibration. In a column of air in the PVC tube, the natural frequencies are called the harmonic frequencies, f 1, f 2,. A vibrating source, such as the speaker, can drive the column of air. As the frequency, f, of the driver (speaker) is slowly varied, the amplitude of the driven system (air column) gets larger and larger until it reaches a peak at one of its own resonant frequencies, f 1, f 2,. When we sweep the speaker frequency past one of the harmonic frequencies of the air column, we hear the increase in amplitude of the vibrating air column as an increase in loudness of the radiated sound from the tube. At resonance the driving system passes energy very efficiently on to the driven system and the amplitude of the driven system reaches a maximum called A max. (See Rossing, the Science of Sound, Ch. 4.) A graph of the amplitude as a function of frequency is shown below

21 The resonant amplitude peaks at frequency f 0 and that the peak has a width, f called the linewidth. The linewidth is usually measured at amplitude of 71% of V max. For a system which loses energy rapidly through damping or friction, the maximum amplitude, V max, is small and the linewidth large, and the resonance is said to be broad. Similarly, for a resonating system which loses energy very slowly, the maximum amplitude is very large and the linewidth is very small. The resonance is said to be sharp. The quality, Q, characterizes the sharpness of one of the resonances. For example, for the fundamental frequency the Q value is: f 0 Q. (1.0) f A high Q system is one with a sharp resonance. Once set oscillating it loses energy very slowly. A tuning fork is an example of an object with a very high Q. To drive an object with high Q, the driving frequency must be very close to the resonant frequency, f 0. C. ELECTRIC RESONANCE OF LC CIRCUIT A system of electric circuit can also have a resonance. Since capacitors can store electric energy and inductors can store magnetic energy, a circuit made of a capacitor, a resistor, and an inductor in series can become a resonant circuit. Resembling simple harmonic oscillators that can transform gravitational potential energy to kinetic energy and vice versa, LC circuit can transform electric energy to magnetic energy and vice versa. Such a circuit forms a resonance circuit. Resistors are dissipative, resembling friction. The RLC electric circuit is shown below. 1 The resonance frequency of this circuit is f 0 2 LC Here the capacitor C is in Farad [F], the inductance L is in Henry [H], and the resulting resonance frequency f 0 is in Hertz or [Hz] or [1/s]. Typical number of L is mh (10 3 H) to H, and C is 100 pf to pf, where 1 pf = F. As the frequency of the sine wave generator is varied at constant input voltage Vin, the output voltage Vout shows a maximum at the resonance frequency f 0, which is analog to the resonance frequency of a simple harmonic oscillator. a) L= 100 mh, C=1000 pf, what is the resonance frequency? 3 021

22 b) L= 1 H, C=1000 pf, what is the resonance frequency? Data shown below is a typical experimental data for an RLC circuit. Plot Voltage vs frequency and find the resonance frequency f 0 and find the Q-value. (You can make a better by using the whole scale of horizontal axis from 200 to 800 Hz, and the vertical scale from 0.5 V to 3.5 V.) Frequency (Hz) Output voltage (V) The resonance frequency f 0 = The Q value is Q= 4 022

23 3A: THE OSCILLOSCOPE Introduction The oscilloscope is a universal measuring instrument with applications in physics, biology, chemistry, medicine, and many other scientific and technological areas. It is used to give a visual representation of electrical voltages. Thus, any quantity which can be converted to a voltage can be displayed on an oscilloscope. Although the oscilloscope looks very complicated, once you familiarize yourself with its controls and functions, it is surprisingly easy to use. The purpose of this experiment is to develop familiarity with the oscilloscope and with the types of measurements that can be made with it. How the Oscilloscope Works The most important component of the oscilloscope is the cathode ray tube (CRT), a vacuum tube in which a filament is heated to boil off electrons which are then focused into a beam and shot toward the screen with an electron gun. In the photograph above, screen is the rectangular, gridded area on the left of the oscilloscope. The screen is coated with fluorescent material which glows when it is hit by the electron beam. On its way to the screen, the beam passes between two sets of deflection plates (horizontal and vertical) and a voltage applied to these plates will cause the beam to curve. The sketch illustrates the CRT components with a negative voltage applied only to the vertical plates (V y ), causing the beam to bend downward. The amount of deflection d shown on the screen is proportional to the voltage applied to the plates, so you can measure a voltage by seeing where the beam hits the screen. Glass Tube V y V x Fluorescent Screen d Filament Electron Gun The oscilloscope you are using has many electronic refinements associated with the CRT to allow electronic signals to be conveniently displayed and measured. The following sections describe these controls and tell you how to get the oscilloscope ready for the experiment. Acoustic Lab#3A Oscilloscope

24 Getting Started Look at all those knobs! Looks complicated, doesn't it? Relax. Prepare to enjoy this. We tell you step by step how to work the scope and the function generator. It really isn't as bad as it looks. As you work through this experiment, feel free to twiddle the knobs as much as you want. As long as you don't wrench them off their shafts, there are NO control settings that will harm the equipment or endanger you. On the next page is a drawing of the scope with the controls labeled using circled numbers. As much as possible, controls have been paired. For example, numbers 5A, 6A, 7A etc. control the same functions for channel 1 (CH1) as numbers 5B, 6B, 7B etc. control for channel 2 (CH2). Note that the scope is divided into four major areas, the SCREEN area on the left, the VERTICAL section, in the middle, the Horizontal section and the TRIGGER section on the right. We will look what the controls for each of these sections do, one at a time. Don t worry if you can t remember everything. The oscilloscope is surprisingly intuitive in its operation. Once you have used the scope a few times its operation becomes almost automatic. By this time you should be practically bored to death so let s do something! First, let s turn on the scope and observe some electrical signals. Push in the POWER button (2) and push in the X-Y switch located in the HORIZONTAL section. Turn all three POSITION controls (11A, 11B and 19), to 12 o clock (mark up, like so: ). Turn the INTENSITY control up until a dot appears on the screen. If no dot appears, ask your instructor for help. Play with the FOCUS(3) and INTENSITY (4) controls. Use the focus knob to get a sharp round dot. Use only as much intensity as you need; excessive brightness for prolonged periods can damage the screen. On the screen is a grid to help you measure the position of the dot, as shown here. 1 Division Hatch Marks 1 Division Some of the knobs on the scope refer to divisions (DIV). Each division is marked off by a solid line. Within the divisions are small hatch marks (or subdivisions). These marks are not divisions. Since there are five marks per division, each mark is 0.2 divisions. Acoustic Lab#3A Oscilloscope

25 Acoustic Lab#3A Oscilloscope

26 Control Function SCREEN SECTION 1 DISPLAY SCREEN Electrical Signal is displayed here. 2 POWER Push on, push off. 3 FOCUS Adjusts the focus of the trace. 4 INTEN Adjusts the intensity (brightness) of the trace. VERTICAL SECTION 5A, 5B VOLTS/DIV Selects the vertical voltage scale for channel 1 (A), channel 2 (B). 6 A, 6B AC/DC Selects whether the channel measures all voltages (DC) or AC voltages only (AC). 7 A, 7B GND Grounds the channel. 8 A, 8B CH1, CH2 Input Jack Channel Input Jack 9 A, 9B VAR. Reduces channel gain. Should normally be all the way clockwise (CW) 10 CHOP Forces display of traces to chopped mode regardless of frequency when pushed in. Normally out. 11 A, 11B POSITION Controls the vertical position of the trace. 12 MODE Determines whether channel one,, channel two or whether both channels are displayed. Also allows the display of the sum of the channels added. 13 CH2 INV Inverts the display of channel 2. HORIZONTAL SECTION 14 TIME/DIV Selects the horizontal sweep rate. 15 SWP. UNCAL. Turns on the SWP VAR control which reduces the sweep rate. Should normally be off (out). 16 GROUND JACK This is the ground terminal. Used for both channels one and two. 17 SWP. VAR. When SWP UNCAL is out the SWP VAR control reduces the sweep rate. 18 X10 MAG Increases the sweep rate by a factor of 10 when pushed in. Should 19 POSITION normally be out Horizontal Position. Moves the trace right and left. It affecfts both channels one and two together. 20 X-Y Disables the time sweep function of the scope. Channel one is applied to the vertical amplifier, channel two to the horizontal amplifier. TRIGGER SECTION 21 SLOPE Selects whether the scope is triggered on a rising (+) voltage when out, or a falling voltage (-) when pushed in. 22 EXT External input jack for trigger signal. 23 TRIG ALT Forces the scope to alternately trigger on channel one, then on channel COUPLING Selects how the trigger signal is processed. Normally set to DC. 25 SOURCE Selects the source of the trigger signal. 26 NORM/AUTO Selects whether the trigger free runs if no signal is applied (AUTO) or whether trigger is held until an actual trigger signal is present. 27 LOCK Allows the scope to automatically determine the trigger voltage. 28 LEVEL Determines voltage level at which the scope triggers. 29 HOLDOFF Assists in triggering complex waveforms. Not normally used. Acoustic Lab#3A Oscilloscope

27 Acoustic Lab#3A Oscilloscope

28 Item Name Description 1 POWER Switch Turns power ON and OFF 2 RANGE Switch Selects frequency range. Switch indicates maximum frequency of the range. 3 FUNCTION Switch Selects SINE, SQUARE or TRIANGLE waveform at OUTPUT jack. 4 OUTPUT LEVEL Control Controls amplitude of the signal at the OUTPUT jack. 5 DC OFFSET Control Enabled by the DC OFFSET Switch. 6 OUTPUT Jack Output waveform selected by the FUNCTION Switch appears here. May be offset by the DC OFFSET Control if the DC OFFSET Switch is ON. 7 TTL/CMOS Jack TTL or CMOS square wave depending on the setting of the CMOS LEVEL SWITCH. This output is independent of the OUTPUT LEVEL and DC OFFSET controls. 8 CMOS LEVEL Control Adjusts the amplitude of the cmos square wave at the TTL/CMOS jack. 9 VCG Jack Voltage Controlled Generator Input. Not Used 10 DUTY CYCLE Control Enabled by the DUTY CYCLE Switch. Adjusts the duty cycle of the signal at the OUTPUT jack DB Switch When engaged the signal at the OUTPUT jack is attenuated by 20 DB. 12 DC OFFSET Switch When engaged enables operation of the DC OFFSET control. 13 CMOS LEVEL Switch When engaged changes the TTL signal to a CMOS signal at the TTL/CMOS jack. 14 DUTY CYCLE Switch When engaged enables operation of the DUTY CYCLE control. 15 FINE FREQUENCY Control 16 COARSE FREQUENCY Control Vernier adjustment of frequency. Coarse adjustment of frequency. Varies frequency from.1 to 1 times the selected range. 17 COUNTER Display Displays frequency of the output waveform. 18 GATE LED Indicates when the frequency counter is updating. Updating can take as long as 10 seconds on the lowest frequency range. 19 HZ and KHZ LEDs Indicates whether the counter is displaying Hertz (Hz) or kilohertz (KHZ) Acoustic Lab#3A Oscilloscope

29 Measuring DC Voltages The oscilloscope is fundamentally, a voltmeter. You are provided with a box which produces eight different voltages to measure. The box has already been connected to a power supply, so you can simply turn on the power supply and you are ready to go. Defining Zero In the HORIZONTAL section, place the X-Y switch (20) in the IN position. This turns off the time base so the dot will not move horizontally across the screen. In the VERTICAL, CH2 section, place the MODE switch (12) in the CH2 position. Then push the CH2 AC/DC coupling switch (6B) IN to DC couple the signal to the scope. Set the CH2 VOLTS/DIV switch (5B) to the 5V/DIV position. Using one of the leads provided, connect the ground terminal of the scope (16) to the black ground terminal of the voltage box, both of which are marked with the ground symbol. All voltages are measured relative to this ground. Now use another lead to connect the CH2 input (8B) to the ground. Since ground is defined to be zero, you are now measuring zero volts. Use the CH2 position control (11B) to move the dot on the screen up or down to some convenient place. Since all of the voltages you will be measuring in this section will be positive, the best place is the bottom line of the screen grid. This allows you to use the entire screen for measurement. Moving the dot to this position when the CH2 input is grounded defines the bottom line of the grid as zero volts. In the left-right direction, you will want to have the dot in the middle of the screen, so that you can take advantage of the hatch marks to help you measure the position of the dot accurately. To move the dot left-and-right, use the horizontal POSITION control (19). The easiest way to ground CH2 is to push the CH2 GND switch (7B) in. The dot on the screen will drift around some over time, particularly as the scope warms up, so it is important to check it occasionally as you make these measurements. You check it by grounding the input and then looking at the dot to make sure it s in the correct position. If necessary, use the position controls to move the dot. Then put the GND switch in its OUT position to make measurements. While you are flipping switches, rotate the variable controls (9A and 9B) to their full clockwise positions. This is the calibrated position. If these controls are in any other position than full clockwise, the voltages will not be displayed correctly. Any time you get voltages lower than expected these are the first controls to check! Practicing with Known Voltages Connect the CH2 input (8B) to the largest Known Voltage on your box, about 17 volts. You should see the dot jump up about three and a half divisions. To measure the voltage, carefully count the number of major and minor divisions above the zero line and multiply by the number on the VOLTS/DIV knob. 3.4 div 5 volts/div = 17 V Note that on the 5 V/DIV scale, each minor division corresponds to 1 volt. Also notice that with careful measurement, you can estimate the third significant digit, but only if the scope is well zeroed. Check the zero now to see if it has drifted. Now connect CH2 to the 3.7 V terminal. The line should be less than one division above zero. You can get a rough measurement now: 0.8 div 5 volts/div = 4 V However, you can do much better than this with the scope. Slowly turn the VOLTS/DIV knob (5B) clockwise, to more sensitive settings. The dot will get higher and higher on the screen and will Acoustic Lab#3A Oscilloscope

30 eventually go off the screen. Turn the knob back to the position which puts the line highest on the screen but not off the screen. (This should be 0.5 V/div.) This is the setting which will allow you to make the most accurate measurement of the voltage. 7.4 div 0.5 volts/div = 3.7 V Note now, that each minor division has a value of 0.1 volts. Each box also has a known voltage between 80 mv and 125 mv. Check your measuring technique by measuring it as accurately as you can and comparing your measurement to the value written on the box. For this particular voltage, you should use the 20 mv/div setting. Remember that this is 20 millivolts per division, not 20 volts. For this setting, the minor divsions represent 4 millivolts each. Mystery Voltages Now you are ready for the mystery voltages, labeled A E. You will measure each one and record it on the data sheet which will be handed out in lab. Then have your instructor check your work. Start by writing the box number on the data sheet. Your instructor will be looking to see not only that you have measured the correct voltage, but also that you have measured it as accurately as possible by using the optimum VOLTS/DIV setting. To be sure you have the right setting, use the following procedure: Turn the VOLTS/DIV knob (5B) to the least sensitive setting, 5 volts/div. Connect the CH2 input (8B) to the mystery voltage. Slowly turn the VOLTS/DIV knob (5B) and watch the dot, which should rise up on the screen. Keep turning the knob until the dot goes entirely off the screen. Then turn the knob back one notch. This is the best setting. Check the zero by pushing the GND button (7B) in.. Then carefully measure the number of divisions that the dot moves when you switch the GND button back to its OUT position. Record the number of divisions and the VOLTS/DIV setting (including units) in the table. Multiply to get the voltage. Measuring AC Voltages Setting Up the Function Generator Push in the red POWER button. Push in the black RANGE button labeled "5". Three gray buttons in the upper right shape the wave. Push in the sine wave. Make sure all 4 gray buttons beneath the POWER switch are OFF (OUT). Turn the OUTPUT LEVEL knob all the way up (CW clockwise). Turn the coarse FREQUENCY dial all the way down (CCW counter clockwise). There are two outputs on the function generator, each of which has an adapter on it with one red and one black jack (connector). To look at the output of the function generator on the scope, hook up the right-hand output (which is labeled OUTPUT and 50 ) to the scope input as follows. Connect the black jack to the ground terminal (16) of the scope. Connect the red jack to the CH2 input (8B) of the scope. On the scope, set the CH2 VOLTS/DIV (5B) knob to 5 volts/div. You should see the dot moving slowly up and down. Notice that it goes below the zero line. This indicates that the function generator alternates between positive and negative voltages. To deal with this, you will need to redefine zero volts to be in the center of the screen. Ground the CH2 input and Acoustic Lab#3A Oscilloscope

31 use the position controls to move the dot to the center of the screen. Then look at the dot moving up and down. The function generator has a digital frequency display. It should read about 0.3 Hz. Slowly turn the coarse frequency dial all the way up (CW). As you do so you will notice that the dot moves up and down more rapidly. (At these low frequencies the display takes about 10 sec. to "catch up" to your change it should eventually read around 5.8 Hz.) Now switch the RANGE to 50. The dot is now going up and down about 50 times per second. This is too fast to follow with your eye, and you see a solid line on the screen. (You may need to turn up the intensity.) To see this kind of rapidly oscillating voltage, you need to make use of another feature of the scope called the sweep. Sweep Rate You will now spend some time with the HORIZONTAL controls (14, 17, 18, ). But first a word about the controls 15, 17, 18 and 26. In general, you will want to check these controls for correct settings, then leave them alone. The SWP UNCAL switch (15) should be out. If this control is not in this position, the sweep rate is not calibrated and will not be exactly what is listed on the TIME/DIV knob. The SWP VAR (sweep variation) control (17) works only if the SWP UNCAL switch is in. The SWP VAR control is used to adjust the sweep rate to some value other than that shown on the TIME/DIV control. The X10 MAG button (18) should always be left out. The NORM/AUTO switch (26), found in the TRIGGER section should be set to AUTO (pushed in) unless you are specifically told set it otherwise. To make sure things got smoothly, also make the following settings: Set the trigger SOURCE switch (25) to CH2. Set the trigger LEVEL knob (28) to 12 o clock. Set the COUPLING knob (24) to DC. Set the TRIG ALT switch (23) to CH2. Now you are ready to vary the sweep rate using the TIME/DIV knob (14). Because the X-Y switch is pushed in, the sweep is currently disabled and the signal on the screen does not move horizontally. To get the signal to sweep across the screen, put the X-Y switch (20) in the OUT position and set the TIME/DIV control (14) to the 0.2 sec/div setting. You should now see the vertical line from before sweeping slowly across the screen. How long does it take to get across the screen grid? Since there are 10 divisions in the horizontal direction, it should take 10 div 0.2 sec/div = 2.0 seconds. Try timing the sweep to see if this is correct. (Use a wrist stopwatch, or just count one-1000, two-1000,... ) Now increase sweep rate by setting the TIME/DIV setting to 50 ms/div. How long should it take the line to get across the screen grid? Does this seem to be correct? While you watch the screen, slowly increase the sweep rate to 2 ms/div. This sweep rate is fast enough for you to see the oscillations spread out across the screen. (You may need to turn the intensity up again at this point.) The picture which you now have on your screen is basically a graph of the voltage produced by the Acoustic Lab#3A Oscilloscope

32 generator (plotted vertically) versus time (plotted on the horizontal axis). The scale of each axis is determined by the settings of the TIME/DIV and VOLTS/DIV controls. With the scope set up like this you can see in detail the way the voltage changes in time. For example, you can see why this type of oscillation is labeled. Try changing to the and shapes. Then go back to the shape. In addition, you can measure the amplitude of the oscillation, as described below. Measuring Amplitude The amplitude of a voltage oscillation is measured in exactly the same way as any other voltage: Measure the number of divisions vertically and multiply by the VOLTS/DIV setting. It is traditional to measure the peak-to-peak amplitude with the scope, which is the voltage difference from the top of the signal to the bottom. Be sure to use the most sensitive setting you can to get the most accurate measurement. Use the position controls to make the measurement easier. For example, you can line up the lowest part of the voltage trace with one of the division lines. For practice, measure some amplitudes and record them on the data sheet. Measure the amplitude for the following settings: The output level knob is turned fully clockwise, to MAX. The output level knob points to 12 o clock. The output level knob is set to 9 o'clock. Record these amplitudes on the data sheet. The Display as a Voltage versus time Graph As you use the scope here and in future experiments, it is important that you understand what you are looking at: a graph of voltage versus time. Go to the data sheet and follow the graphing instructions. When you return, look at some different frequencies. Set the generator to the 50K range and adjust the sweep rate to display the wave. Then go to the generator's highest frequency and display that. Play around with different settings until you understand how to display any frequency. This is a common adjustment you will need to make whenever you use the scope, so make sure you are comfortable doing it. Triggering Have you wondered why the waveform display is stationary on the screen? Why when you rock the coarse FREQUENCY control back and forth the waveform expands and compresses on the screen but it always starts at the same point on the left-hand side, no matter where it ends on the right? This happens because the scope is "triggered" after the beam sweeps left to right it jumps back to the left and waits in this "cocked" position until the incoming waveform reaches a predetermined voltage level, at which point the trigger releases the beam, it sweeps, and it resets again. As you have it set from the last part, the beam triggers about halfway up the positive (+) slope of the wave. There are four scope controls that have to do with triggering: trigger LEVEL (28); trigger SOURCE (25); trigger SLOPE and trigger COUPLING (24). For this course you only need to be familiar with the trigger LEVEL, trigger SLOPE and trigger SOURCE. The COUPLING control should always be set to DC. The trigger SOURCE control selects what source causes the trace to trigger. This will always be either Acoustic Lab#3A Oscilloscope

33 CH1 or CH2 unless told otherwise. Triggering is one of the most important features of a scope because it makes the pattern stand still so you can examine it. The trigger LEVEL control is one you will use quite often, play with it until you understand how it works. It sets the voltage level at which sweep begins, either on positive slope (push in) or negative slope (pull out) of the triggering signal. Notice how synchronization is lost if you set the trigger level too high or too low. Whenever you see that rambling pattern of multiple waves, reach for the LEVEL control. The way this scope works, you lose synchronization whenever the voltage of the displayed signal is less than the trigger level. If it falls below the trigger level for any reason, including change of the volts/div control, you will have to re-adjust the level to get it back. For example, set the amplitude on the function generator to 9 o clock. Set the VOLT/DIV to make the display as large as is can be and still fit completely on the screen. Set the level so it triggers near the top of the trace. Now flip the volts/div switch to a less sensitive setting for a smaller picture. How do you get the display back in synch? Dual Trace Feature This scope has two separate inputs, CH1 (8A) and CH2 (8B), and can display both signals at the same time. It s easy to do the only tricky part is triggering. Just place the MODE switch (12) in the DUAL position. We ll now connect the second output of the function generator to CH1 of the scope and observe both signals from the function generator. The function generator has two outputs. The one on the right is currently displayed on CH2. The other output, on the left, labeled TTL/CMOS, always puts out a square voltage which jumps from 0 V to about 20 V. Use a banana lead to connect the TTL/CMOS output (red post) to the CH1 input (8A) of the scope. (You could hook the black post to the scope ground too, but it's not necessary because that ground connection between the two instruments already exists.) Flip the vertical section MODE switch (12) to CH1 to see this signal. You may need to adjust the CH1 VOLTS/DIV knob (5A), and you will need to set the CH1 mode select (6A) to DC. You should now see both both channels at the same time. Use the vertical position controls to put CH1 above CH2 on the screen so they are easily distinguished. Adjust the gain of each vertical channel using the VOLTS/DIV switches so the signals don t overlap. The scope can also ADD the two signals and display the sum, often with interesting results. To add the signals select ADD on the MODE switch (12). Put the MODE switch back to DUAL so you can look at triggering in dual trace operation. So far you have triggered off of CH2, which you can see by looking at the trigger section SOURCE (25) switch. Now pull the test lead out of the CH2 input. Not only does the CH2 display become a straight line at zero volts (since there is no signal coming into that channel), but the CH1 display is not stationary. That s because the scope is trying to trigger off channel 2, but there is no signal there anymore. To synchronize it with the sweep. Move the trigger SOURCE switch (25) to CH1 so it triggers on that signal. If it still isn t stationary, adjust the trigger LEVEL control (28). Conclusion By now you should understand the basic operation of the scope. Congratulations! As you use the scope in other experiments, it will become easier to use each time. Have your instructor look over your Data Sheet, and you're all set. Please turn off the scope and the function generator, and check that the power supply is off. Acoustic Lab#3A Oscilloscope

34 Oscilloscope Setup Sheet This sheet is provided to help you set up the oscilloscope. CONTROL GENERAL POWER Switch INTENSITY Control FOCUS Control VERTICAL SECTION: CHANNEL 1 POSITION Control CHOP Switch VOLTS/DIV Switch AC/DC Switch GND Switch VAR Control VERTICAL SECTION: MODE MODE Switch VERTICAL SECTION: CHANNEL 2 POSITION Control CH2 INV Switch VOLTS/DIV Switch AC/DC Switch GND Switch VAR Control HORIZONTAL POSITION Control X10 MAG Switch X-Y Switch TIME/DIV Switch SWP UNCAL Switch SWP VAR Control TRIGGER HOLDOFF LEVEL Control NORM/AUTO Switch LOCK Switch COUPLING Switch SOURCE POSITION IN (ON) Full Clockwise Straight Up Straight Up IN (Chopped) Full Counter Clockwise (5 V/Div) IN (DC) OUT Full Clockwise (Cal) Dual Straight Up OUT Full Counter Clockwise (5 V/DIV) IN (DC) OUT Full Clockwise (Cal) Straight Up OUT OUT 5 ms/s OUT Full Clockwise SLOPE Switch OUT (+) TRIG ALT Switch OUT Full Counter Clockwise Vertical (Mid Position) OUT (Normal) OUT DC CH1 or CH2 (should match the signal input channel) Acoustic Lab#3A Oscilloscope

35 DATA SHEET --- LAB 3A REPORT Measuring DC Voltages Put your data for the mystery voltages here: Box # Unknown number of divisions volts/div Voltage A B C D E When you are done, have your instructor check your work. Then turn off the power supply and disconnect the leads from the scope. The dot on the oscilloscope should still be at the bottom of the screen. Measuring AC Voltages Measuring Amplitudes Setting number of divisions volts/div Amplitude (p-p) Maximum 12 o clock 9 o clock Graphing Voltage versus time On the function generator: Set the OUTPUT LEVEL to maximum Set the frequency to 50 Hz (use the fine frequency control) On the oscilloscope: Adjust the display so that it is a sine wave: starting at the left-hand edge of the grid at zero; rising to maximum voltage; falling through zero to the maximum negative voltage; and finally rising back to the zero line. [The sweep rate should still be set at 2 ms/div. Temporarily ground the CH2 input to set the zero volt line across the middle o the screen. Fine-tune the trigger LEVEL control ƒ to get the wave to start at zero.] On the grid here, sketch the sine wave from the screen. Then label the axes of this graph with the appropriate voltage and time scales. Acoustic Lab#3A Oscilloscope

36 Sketch your measurement of the peak-to-peak voltage on the graph. Measure the period T of the sine wave. Then calculate the frequency f 1 T. How does this agree with the frequency indicated on the function generator's display? Measuring Period and Frequency Set the frequency multiplier to 1000 and study the accuracy of the frequency dial at the low end (250 Hz) in the middle (1000 Hz) and at the high end of the dial (1900 Hz). Function Generator Setting 250 Hz 1000 Hz 1900 Hz number of divisions for one period time/div Period Frequency How accurate is the dial on the function generator? Is it consistently high or low? For a little more practice, try measuring some very high and very low frequencies: Function number of time/div Period Frequency Generator Setting divisions for one period 15 khz 400 khz 19 Hz Acoustic Lab#3A Oscilloscope

37 3B: ELECTRIC RESONANCES A. ELECTRIC RESONANCE CIRCUIT Electrical circuits made of inductors and capacitors can produce electric resonance. An example is shown in the following diagram, where a capacitor C, an inductance L, and a resistor R are in series with the function generator. The resonance frequency of this circuit is f LC 2 Here the unit of L is in Henry [H], C is in Farad [F], the resulting resonance frequency f 0 is in [Hz], and ω 0 is the angular frequency in radian/s. The angular resonance frequency ω 0 is measured in rad/s. Typical number of L is in milihenry (mh) to H, and typical number of C is in 100 pf to pf, where 1 pf = F. As the frequency of the sine wave generator is varied at constant input voltage V in, the output voltage V out shows a maximum at the resonance frequency f 0, which is analog to maximum amplitude we saw before for the resonating air column. The maximum output voltage and the sharpness of the resonance, Q, are determined by the total resistance and the inductance in the circuit, of the order of Q=2πf 0 L/R. Note that the inductor has some internal resistance. 2. Measurement of the Quality Factor Q a) Use the provided RLC circuit with values of L and C specified on the relevant circuit elements (each circuit box may have different L [H] and C [F]). Find the L and C of your circuit box and calculate the resonance frequency of your box. L= H, C= F, and f 0 = Hz b) Connect the Function Generator to the input of the circuit. Also connect the scope to show V in and V out. Determine the resonance without recording any data and determine roughly the range of frequencies required to go completely across the resonance. Compare your result with the theoretical calculated number in part a) f 0expeeriment = Hz c) Now decide upon a set of 15 to 20 frequencies over this range for your measurements. Set V in to a suitable value (e.g. 0.4 V peak-to-peak) so that you can measure V out with the oscilloscope, and measure V out at each chosen frequency. When you change frequency, the voltage from the generator may also be changed. Be sure to set V in to the same amplitude each time, if necessary, and verify that it remains sinusoidal. To achieve a more precise measurement of resonance and Q-value, choose the starting and ending frequencies to be about 0.75f 0 to 1.25f 0. 3B: Resonance in electric LC circuit

38 frequency in Hz V in in volts V out in volts V out /V in c. Plot this new data in the same manor as above. Calculate the width of this response curve at 71% of the maximum value of the V out /V in ratio, f, and the Q for this resonance. i. A max : ii. 0.71A max : iii. f: iv. f: v. Q f f : vi. Your comments: Note that the observed fact that the output voltage can be much greater than the input voltage shows why an LC resonance circuit is so useful in radio communication circuits (transmitters, receivers). In this experimental setup, the resistor was added to make the resonance broader. Otherwise the quality factor Q would have been too high to make the measurements readily. In communication resonance circuits the resistance is kept to a minimum to provide a very sharp resonance and thus a good selectivity of stations on nearby frequencies. 3B: Resonance in electric LC circuit

39 PRELAB4: METRIC UNIT PREFIX AND ERROR ANALYSIS INTRODUCTION In this course it is often necessary to calculate frequencies by taking reciprocals of periods. Sometimes the period will be is units of seconds. With sound signals, however, the period will usually be in the millisecond range, and occasionally even the microsecond range. Note that our Mean Reaction Time (RT) is approximately 180~200 msec to detect visual stimulus, and approximately milliseconds to detect an auditory stimulus. Many students have difficulty taking reciprocal of period which is in milliseconds. For example, they will write 1/(4.2 ms) = 0.24 Hz or even 0.24 mhz. This is not correct. Remember that 4.2 ms is shorthand for 4.2 x s = s: x s = s = 238 Hz This homework will give you some practice take reciprocals of numbers which have prefixes like milli- and micro- on their units. You will need a calculator to do this homework. Since physical quantities can vary many factors of 10s, they are expressed in standard prefix as follows: T = terra = m = milli = 10 3 = G = giga = 10 9 μ (or u) = micro = 10 6 = M = mega = 10 6 = 1,000,000 n = nano = 10 9 k = kilo = 10 3 = 1,000 p = pico = These prefix will be needed in future labs. Please remember the prefix notation! CALCULATION PROCEDURE Method 1 The most direct way to calculate a reciprocal is simply to write the number without a prefix before you take its reciprocal. This is what was done in the introduction to take the reciprocal of 4.2 ms. Just write out 4.2 ms as s, put into your calculator and push the 1/x button (x -1 on some calculators), or you can divide 1 by The only problem with this approach is that you may make a mistake when translating the prefix, for example, writing 4.2 ms = s, which is not correct. But if you are careful, you can use this method perfectly well. Method 2 Since you will use your calculator to take the reciprocal, it is often more reliable to make use of the scientific notation feature of your calculator to deal with the metric unit prefix. A prefix like m (milli-) means multiply by That is, 4.2 ms = s. On most calculators today, you can enter a number directly into the calculator using scientific notation. This is done using a button which is marked either EE or EXP, depending on what brand of calculator you have. To get s into your calculator, you take the following steps: 4: Metric Unit Prefix Prelab 1 039

40 a) 4.2 b) EE Button for scientific notation (called EXP on some calculators). c) +/- Button to get a minus sign (called CHS on some calculators). d) 3 The steps c) and d) are reversed in some calculators. At this point, your calculator display should be something like: or 4.2E-3. Notice that this means , not Then, to take the reciprocal (to find the frequency) just hit the 1/x button. Your calculator display may be , which means = Hz. (Note: some calculators will display directly on their screen after hitting the 1/x button.) The advantage to this method is that you put the number part (4.2) directly into the calculator, and then put the unit prefix (milli-) directly into the calculator as -03. You never make a mistake counting zeros while moving the decimal point. Calculations of period from frequencies are done in the same way. Say you have a frequency of 2 khz and you want to find the period: T = 1 1 f = Hz = 5 x 10-4 s = 0.5 ms To do this on your calculator, you would push: 2 EE 3 to get a display of Then your push 1/x, and the calculator displays 5-04, which means = This is the same as , so the period is 0.5 ms. Metric Unit Prefix Worksheet If your answer is less than 0.01 or greater than 1000, then express your answer using a prefix. For example, if f = 44 khz, then write your answer as: 1 T = 44 x 1000 Hz = s =23 µs. f = 16 Hz T = f = 16 khz T = f = 750 Hz T = T = µs f = T = ms f = 4: Metric Unit Prefix Prelab 2 040

41 Uncertainties and Error Propagation The accuracy (correctness) and precision (number of significant figures) of a measurement are always limited by the apparatus used, by the skill of the observer, and by the basic physics in the experiment. In doing experiments we are trying to establish the best values for certain quantities, or trying to validate a theory. We must also give a range of possible true values based on our limited number of measurements. The variation in measured data is expressed in synonymous terms uncertainty, error, or deviation. The Systematic error is the result of a mis-calibrated device, or a measuring technique which always makes the measured value larger (or smaller) than the true value. Careful design of an experiment will allow us to minimize or to correct for systematic errors. The random errors can be dealt with in a statistical manner. In your oscilloscope experiment, you can change the knob of voltage per division to measure more precisely a small voltage value. For example, if you use 0.1 V/div then the LEAST COUNT of the scope will be 0.02 V. Your measurement error is a fraction of 0.02 V, say ±0.01V, called the instrument limited error (ILE). If I measure a voltage 5 times, and each time measured is 0.36 V. My measurement is (0.36 ±0.01) V. Besides ILE, the measured values may vary. For example, you measure the voltage of a terminal of a mystery box, you may find 0.32 V, 0.34 V, 0.37 V, 0.37 V, 0.40V. The average of these 5 measurements is 0.36 V, and the standard deviation is 0.03V. You should choose the larger of the ILE and standard deviation. For example, since the ILE of ±0.005V is smaller than the standard deviation, your answer is (0.36 ±0.03) V. measurements Data (V) deviation (V) (ΔV) 2 (Volt 2 ) average The average of the last column is obtained by σ 2 = ( V) 2 /(N-1), where the N is the number of measurements, and σ is the standard deviation. Fill the table measured below using a stop-watch and find the standard deviation. Note that the σ 2 defined above is the mean-square error for the ordinary least square, whereas the maximum-likelyhood estimation (MLE) is =(N-1)/N σ 2. The two estimates are quite similar in large samples; σ 2 is unbiased, while is biased but minimizes the mean squared error of the estimator. In practice σ 2 is used more often, since it is more convenient for the hypothesis testing. The square root of σ 2 is called the standard error of the regression (SER), or called standard deviation. When you see a number reported as (0.36 ± 0.03) V, your might thought that all the readings lie between 0.33V (= ) and 0.39V (= ). A quick look at the data shows that 3 of the 5 readings are in this range. Statistically we expect 68% of the values to lie in the range of <V> ± σ V, but that 95% lie within of <V> ± 2σ V. In the first example all the data lie between 0.30 (= *0.3) and 0.42 (= *0.3) V. As a rule of thumb for this course we usually expect the actual value of a measurement to 4: Metric Unit Prefix Prelab 3 041

42 lie within two deviations of the mean. In the language of statistics, it is said to have 95% confidence levels. Now try to analyze the following data measurements Time t (sec.) deviation ( t) (s) (Δt) 2 (s 2 ) average The SER or standard deviation is σ= Significant figures In Table 1, you find that the voltage between two terminals is 0.36 V. Your measurement has two digits of significant figures. You will NOT write your answer as V or 0.3 V. The basic rules are 1. Zeros at the beginning of a number are not significant. For example, m has 3 significant figures. 2. Zeros after the decimal point are significant. For example and has respectively 3 and 4 significant figures. 3. Zeros following a whole number may or may not be significant. For example, 500 kg may have 1 or 2 or 3 significant figures. To be precise, one writes , or , or for 1 or 2, or 3 significant figures. 4. Arithmetic operations such as multiplication, division, addition and subtraction are round off to the smaller number of significant figures. For example a) 5.3 m/(1.67 m/s) (=3.1736s) = 3.2 s 2 significant figures b) 3.4 m 3.65 m (= 8.76 m 2 ) = 8.8 m 2 2 significant figures c) 725 m/0.125s (=5800 m/s) = m/s 3 significant figures d) In rare situation, the number of significant figures may increase by 1 in addition. For example: = The answer has 4 significant figures. Answer the following questions: m/s significant figures: miles significant figures: 3. ( m/s) * 0.2 s significant figures: miles/2.50hr significant figures: significant figures: significant figures: significant figures: 4: Metric Unit Prefix Prelab 4 042

43 4: PROPERTIES OF WAVES Definition of Wave: A wave is a disturbance traveling in a medium. A. SMALL GROUP ACTIVITIES WITH SLINKIES Several basic properties of wave behavior can be demonstrated with long springs and slinkies. Quite a bit of work will be done in this class with springs and strings because you can see the waves on a spring, which you can t do with sound waves. Your study of sound will be easier for you if you can learn to make analogies to other kinds of waves which you can visualize. 1. Wave Propagation and Reflection Have one person hold one end of the spring and another person hold the other end. Stretch the spring until it is 15 ft long. One person should send pulses down the spring by jerking the spring to the right and back to center. Do this very fast, to make the pulse as short as possible, but try not to overshoot when you bring your hand back to the center. The pulse should be only on one side: This Not This All waves have the property that they can be reflected from boundaries. Waves are reflected differently from different types of boundaries. Please answer the following questions: a. What is the medium in this case? b. What is the disturbance? c. Is the wave transverse or longitudinal? d. What happens to the pulses when they are reflected from the fixed end? 2. Speed of a Wave Speed of sound wave in air is an important property of sound, and you need to know what speed is and how to calculate it. Speed is the distance something moves in one unit of time. The easiest way to measure speed is to measure the time it takes it to travel a known distance and then divide the distance by the time. Speed = distance/time. Have a third person use a stopwatch to measure the amount of time it takes for a pulse to go down and back once. It may take several tries to get a good value for the time. Calculate the speed of the wave. Remember, distance = 30 ft! (15 down, 15 back.) How far does this wave travel in one second? How far would it travel in 15 seconds? Acoustic Lab 4: Properties of Waves; page 1 043

44 3. Wave Speed Depends on the Medium Stretch the spring out to 20 ft., and once again send pulses down the spring. From casual observation, does the pulse travel faster or slower than before? Measure the speed with the stopwatch. 4. Is there Reflection from a Free End? Tie a long string (10 ft. or more) to the end of the spring. Tie here Hold here Before you send pulses down the spring, predict what will happen to the pulses when they hit the string. Will they be reflected, or disappear, or what? (The spring/string junction is called a free end because the string allows the spring back and forth freely. Since this is a transverse wave, only the back-and-forth direction matters as far as the wave is concerned.) Now try it. What happens? Draw sketches showing how this is the same or different from when the wave hit the fixed end. 5. Longitudinal Waves on a Slinky Coil the long spring neatly in the box and get out the slinky. Most demonstrations work better with the slinky if you only use half of its length, so have the person who holds the fixed end keep half of the slinky all bunched up in his or her hands. Stretch the remaining half of the slinky until it is about 10 ft long. The slinky is useful because you can use it to make both transverse and longitudinal waves. First, make a couple of transverse waves in the same way you did with the long spring. Now try making longitudinal waves (waves in which the disturbance is in the same direction as the direction of propagation): thrust your hand suddenly toward the person holding the fixed end and then back to its starting point. Do this as quickly as possible, but try not to get any side to side motion. Watch the compression pulse move down the slinky. This is analogous to a region of high pressure moving as a sound pulse in the air; sound waves are longitudinal. Longitudinal wave are also called compression-rarefaction waves. Places where the medium is all bunched together are called regions of compression, whereas places where the medium is a spread out are called regions of rarefaction. In air this corresponds to regions of high and low pressure. If you are careful, you can see a pure rarefaction pulse moving along the slinky. To do this, let out the entire slinky and stretch it to that it covers only 8 ft. Now suddenly jerk your hand away from the person holding the fixed end but do not return it to its original position until the wave has died out. If you push the slinky back and forth you have alternating regions of high and low pressure, like a continuous sound wave in the air. 4: Properties of Waves; page 2 044

45 B. STANDING WAVES ON THE LONG SPRING Vibrating strings are important sound sources, particularly for musical instruments. Violins, guitars, pianos, and a whole host of instruments make sounds which begin with vibrating strings. The vocal cords are similar to vibrating strings. Furthermore, strings provide an exact visual analogy to air columns, which are used in many other instruments and which allow humans to control the timbre of the sounds that they make in speech. The best way to get a feel for standing waves is to make some yourself on a long rope or spring. 1. Standing Waves on a Spring with Fixed Ends With the spring still at 15 ft, oscillate your arm back and forth, sending regularly spaced pulses down the spring. Vary the frequency of oscillation until you set up a standing wave that looks like this: Each hump is one half of a wavelength, so this spring now has one and a half wavelengths on it. Since the spring is 15 ft long, the wavelength of the standing wave is 10 ft. Wavelength is the length of two humps, and is symbolized by the lower case Greek letter (lambda). a. With 3 humps on the spring, measure the amount of time required to make 20 complete oscillations. This is twenty periods. What is one period of oscillation of this standing wave? What is the frequency of oscillation? b. Create a standing wave with one wavelength on the spring (two humps). What is? (It is not 2. If you think the wavelength is 2, ask for help.) Measure the amount of time required to make 20 complete oscillations. This is 20T. What is the period of oscillation of this standing wave? Put this and the previous data into the appropriate spaces in the table below. Then complete the table by setting up standing waves with two complete wavelengths and with one half of a wavelength, measuring 20T for each one. Include units. Leave the last column empty. Data from standing waves on a spring fixed at both ends looks like # humps 20T T f 1 f 1 = ft 4 c. The frequencies should form a harmonic series. To find out if they do, divide each frequency by the fundamental frequency f 1. Label the last column f /f 1 and fill it in. Are the other frequencies two, three, and four times the frequency of the fundamental? 4: Properties of Waves; page 3 045

46 Do not be discouraged if your results are not perfect. It is difficult to time the periods of the long spring accurately. Unfortunately, the fundamental is usually the least accurate of the measurements. If you find, for example, that the harmonics are 1.6, 2.5, and 3.4 times the fundamental, the problem is probably that the f 1 you calculated is too large. You might try assuming that the upper harmonics are correct and try to determine the expected fundamental. d. Compare the periods to the travel time you measured previously: one of them should be close. This is no accident: the relationship between the speed, v, wavelength,, and period, T, of a wave is: v = T The relationship between the speed, the distance traveled, D, and the time traveled, t, for a wave is: v = D t Therefore, the time it takes a wave to travel down the spring and back (D = 2L) will be the same as the period of a wave that has a wavelength 2L. C. Standing Waves on a Spring with a Free End (optional) Different kinds of standing waves can be created on a spring on which one end is free to move back and forth. To study these types of standing waves, have one person hold the end of the string which is attached to the end of the spring. The spring itself should still be stretched to 15 ft; the string just serves as a way to hold the spring without restricting its side-to-side movement. If the spring is stretched to the same length as before, then the speed of waves on the spring will also be the same as before. Set up a standing wave that looks like this: This is one and a quarter wavelengths. Each bulge in the spring is (1/2), so the part left over at the end, half a bulge, is (1/4). Since the spring is 15 ft long, the wavelength is (15 ft)/1.25 = 12 ft. You should be able to check this by counting the number of floor tiles that are taken up by the full wavelength. a. Vary the frequency to generate different patterns. Notice that every pattern you make has a quarter of a wavelength (half of a hump) at the end held by the string. The lowest frequency pattern should have only half a hump (one-quarter wavelength). b. Measure the period of the four lowest frequency standing waves you can make, filling in the table below in the same way as you did for the fixed-end spring. Remember that the string is not part of the standing wave pattern, and should not be shown in the picture you put in the looks like column. 4: Properties of Waves; page 4 046

47 It is more difficult to determine for the free-end standing waves. The easiest way is to divide the length of the spring (15 ft) by the number of wavelengths on the spring. For this reason, an extra column for number of wavelengths has been added to the table. Data from standing waves on a spring with a free end looks like # humps # wavelengths 20T T f f/f 1 f 1 = 2 1 /2 1 1 /4 12 ft c. How does the fundamental frequency compare to the fundamental for the fixed-end spring? Why? d. Do the frequencies form a harmonic series? C. THE STANDING WAVE APPARATUS It is difficult to get accurate results with the spring and stopwatch. In contrast, very accurate results can be achieved using the apparatus provided in the lab room. A string is driven by a speaker which is controlled by the function generator. Measurements are made on the frequency counter. 1. Turn on the counter and function generator. Set the frequency to 90 Hz and turn the amplitude all the way up. Hang 500 g (50 g weight holder grams) from the end of the string. Move the clear plastic bridge until you get a fundamental standing wave. Play with it until you get the largest possible amplitude. If the hanging mass which provides the tension is swinging around, it will make it hard to adjust, so you may want to stop the mass if it is swinging. Once you get the best standing wave, do not move the bridge until you see you should do so in the manual. Record the length of the string. L = It is important to know the frequency of the fundamental as accurately as possible. 2. Turn the dial on the function generator to increase the frequency until you get the next standing wave, which should have two humps, called antinodes. Record the frequency in the table below. 3. Continue in this way until you get to the eighth harmonic. Then turn down the amplitude while you do the calculations in the next part. # humps f (Hz) f/f 1 Harmonic number : Properties of Waves; page 5 047

48 Calculate f/f 1. You should find that these values are very close to being whole numbers. Round these values to whole numbers and put the result under Harmonic Number. It should be obvious that the number of humps is the number of the harmonic for a string fixed at both ends. Use this knowledge to predict the frequency which will be necessary to create the 11th harmonic. Then try it. D. SPEED AND HARMONICS The speed of a wave on a string depends on the stiffness, density, and tension in the string. It does not depend on the wavelength of the wave sent down the string. In the same way, the speed of sound in air does not depend on the wavelength of the sound. When standing waves are created on a string, the wavelength, frequency, and speed are related according to the formula v = f. In first part of the lab, you set up standing waves on the long spring. Copy your data from lab for the long spring into the table below. Then calculate the speed v for each standing wave. # humps f v Is v roughly constant? 2. Are these v s similar to the v which you measured directly by timing a pulse going down the spring (recall that this was the first thing you did with the spring)? The data you took with the standing-wave apparatus should be much more accurate. Determine the velocity of waves on the string for the four harmonics indicated in the table. # humps f v Is the speed constant? E. Length and Frequency You have seen that a vibrating string can vibrate at a lowest frequency, called the fundamental, and also in an entire harmonic series of frequencies above the fundamental. The frequency of the fundamental can be increased either by making the string shorter or by making the speed higher. The speed can be increased either by increasing the tension in the string or by making the string lighter (for a given length). Vocal chords are not really chords, but are more like flaps. Nonetheless, the factors of length, weight, and tension affect vocal chords and other vibrating objects in the same way that they affect strings. Thus, women s voices are generally higher in pitch than men s because they are 4: Properties of Waves; page 6 048

49 both shorter and lighter. People can change the pitches of their voices by increasing the tension in the chords with muscles. Speed, frequency, and wavelength are related by the equation v = f. For a string of fixed length, only certain wavelengths will fit onto the string, resulting in a harmonic series. The least number of wavelengths which will fit onto a fixed string is one-half wavelength, so for this situation = 2L. Therefore, for a string fixed at both ends, v = 2Lf 1, where f 1 is the fundamental frequency. Since you have already studied the harmonic series in some detail, the standing waves referred to in this lab will always be fundamental standing waves. 1. Hang 500 g (50 g weight holder grams) from the end of the string. Put the plastic bridge at 90.0 cm (0.900 m). 2. Set the amplitude of the function generator to maximum. Find the frequency that gives the best fundamental standing wave. Best, in this case, means largest amplitude. Measure the frequency with the green frequency counter. Record your data in the first row of the data table below. 3. Repeat for L = 45.0 cm, 30.0 cm, and 22.5 cm. Look at your data. Use the space here to describe the pattern which you see. Frequency versus Length for a string with a fixed tension L (m) f (Hz) Prediction f(hz) v = 2Lf 1 (m/s) f = f = Use your data to predict what frequency will resonate for a string 50.0 cm and 72.0 cm long. Show your reasoning and calculations in the space below. Then determine the frequency experimentally and put it in the table above. Also put your prediction in the third column. 5. Use you data to determine what length of string that will vibrate at 280 Hz, showing your work below. Then set the function generator a close as you can to 280 Hz, which will take a little patience. Then move the bridge until you get the best standing wave. Fill in the table. 6. How well do your predictions compare with the actual values? 7. In the last column of the table put the heading v = 2Lf (m/s). Then calculate the speed of the wave for each row in the table. Is the speed constant? 8. You may have answered questions 3, 4, and 5 above using ratios. This is a good approach. However, if you have to answer several questions about the same spring, the table suggests a faster approach. Since the speed of the wave on the string is constant, the best approach is to use the first set of data to calculate the speed of the wave using v = 2Lf. Then use that speed 4: Properties of Waves; page 7 049

50 to calculate L or f for each new situation. Try it. Use the known speed of the string to calculate the resonant frequency for a length of 43 cm. Then test your result. F. TENSION AND FREQUENCY For a string of fixed length it is possible to raise the fundamental frequency by increasing the tension in the string. This is because increasing the tension increases the wave speed. On your apparatus, tension is provided by the weight of a hanging mass; by changing the mass you can change the tension. The tension in the string is the mass multiplied by the earth gravitational constant g = 9.8 m/s 2. In your data table, however, it will be easier to see what is going on if you record the total hanging mass m. You must always remember that the mass hanger already has a mass of 50 g. 1. Put the bridge at 90 cm. Hang 125 g on the string (50 g + 75 g). Find the best fundamental standing wave, which will be quite low for such a small tension. Record the frequency in the table below. Frequency versus Tensionfor a string with a fixed length m (g) f (Hz) v T (m/s) v f (m/s)=λf Double the mass. That is, put on a mass of 250 g (50 g g). Find the new frequency. Record your data in the table. What is the ratio of these two frequencies? How is the ratio compared with the ratio of two masses? 3. Measure the frequency for masses of 500 g and 1000 g; put the data in the table. Theoretically, the speed of a wave on a string is given by v T = T µ 1/2 where T =mg, g= 9.8 m/s 2 is the tension and is the mass density of the string (measured in g/m). Use the density of the string 0.38 grams per meter, measured with an accurate balance and calculate the speed in meters per second (m/s). Record your result in the above table. 4. Now use v f =λf= 2Lf to calculate the wave speed, where the subscript f stands for speed determined using frequency measurement, and record your result in the above table. Are the speeds the same? (Of course, experimental uncertainties in the measurement mean that they will never be exactly the same. However, they should be within a few percent of each other.) 5. Predict the frequency which will resonate for a mass of 1125 g. That is, calculate the speed for a mass of 1125 g using the known density. Then use the speed and the length of the string to calculate the resonant frequency. After you have made your prediction, try it to see if you were right. Explain your results below. 4: Properties of Waves; page 8 050

51 INTRODUCTION PRELAB5: SOUND WAVES IN TUBES Sound is longitudinal mechanical vibrations that are transmitted as pressure wave through mediums such as gases, liquids, solids, and plasmas. Sound cannot travel in vacuum. Experimental measurements of the speed of sound in air were carried out successfully between 1630 and 1680 by a number of investigators, prominently Mersenne. Newton ( ) derived the relationship for wave velocity in solids, a cornerstone of physical acoustics (Principia, 1687). Sound travels through air at a velocity of approximately 343 m/s (1050ft/s) at 1 atm and 20 C, or v = m/s + (0.6 m/s) T (1) where temperature T is in degrees Celsius. Because its energy is spread over larger and larger area, the amplitude of a sound wave decreases as it travels away from a source such as a vibrating speaker or tuning fork. These waves also reflect from surfaces and in reflection lose amplitude through absorption. The sound wave can be a pulse, or continuous perturbation. The wave can be decomposed into sinusoidal components in time. The eighteenth century saw major advances in acoustics at the hands of the great mathematicians of that era, who applied the new techniques of the calculus to the elaboration of wave propagation theory. In the nineteenth century the giants of acoustics were Helmholtz in Germany, who consolidated the field of physiological acoustics, and Lord Rayleigh in England, who combined the previous knowledge with his own copious contributions to the field in his monumental work The Theory of Sound. Also in the 19th century, Wheatstone, Ohm, and Henry developed the analog between electricity and acoustics. The twentieth century saw a burgeoning of technological applications of the large body of scientific knowledge that was by then in place. Wallace Clement Sabine ( ) produced a groundbreaking work on architectural acoustics, and many others followed. Underwater acoustics was used for detecting submarines during World Wars. Sound recording and the telephone globally transformed our society. Sound measurement and analysis reached new levels of accuracy and sophistication through the use of electronics and computers. Electronic instruments were introduced to musicians. The ultrasonic frequency range enabled wholly new kinds of application in medicine and industry. New kinds of transducers (generators and receivers of acoustic energy) were discovered and invented. Sound Reflection Sound wave will be reflected by a surface. The reflection of sound obeys the law of reflection, i.e. Angle of incidence equals angle of reflection. r i PRELAB5: Sound waves in Tubes page 1 051

52 Wave reflected from a fixed end will be 180 out of phase with respect to the incident wave, i.e. an incident wave with positive sign will produce a reflected wave with a negative sign. On the other hand, the reflected wave from an open end will be in-phase with respect to the incident wave, i.e. the reflected wave has the same sign as that of the incident wave. In this lab, you will determine the sign of the reflected sound wave from open tube and closed tubes respectively. Standing wave The reflected waves can interfere with incident waves, producing patterns of constructive and destructive interference. This can lead to resonances called standing waves in rooms. It also means that the sound intensity near a hard surface is enhanced because the reflected wave adds to the incident wave, giving pressure amplitude that is twice as great in a thin pressure zone near the surface. This is used in pressure zone microphones to increase sensitivity. The doubling of pressure gives a 6 decibel increase in the signal picked up by the microphone. Reflection of waves in strings and air columns are essential to the production of resonant standing waves in those systems. For example, the graph below shows a standing wave of an open-closed tube. The schematic standing wave (pressure) of open-open, closed-closed, or open-closed systems are shown in the graph below. Acoustic Lab 5: Sound waves in Tubes page 2 052

53 1. Open-open or closed-closed: the fundamental frequency is, and the higher harmonics are f 2 =2f 1, f 3 =3f 1,. 2. Open-closed: the fundamental frequency is and higher harmonics are f 2=3f 1, f 3 =5f 1,. In human, standing wave plays an important role in speech of vowel sounds: The vocal tract can be considered a single tube extending from the vocal folds to the lips, with a side branch leading to the nasal cavity. The length of the vocal tract is typically about 17 centimeters, though this can be varied slightly by lowering or raising the larynx and by shaping the lips. The pharynx connects the larynx (as well as the esophagus) with the oral cavity. The oral cavity is the most important component of the vocal tract because its size and shape can be varied by adjusting the relative positions of the palate, the tongue, the lips, and the teeth. The characteristic resonance produced by our vocal track is called Formant. A musical instrument may have several formant regions dictated by the shape and resonance properties of the instrument. The human voice also has formant regions determined by the size and shape of the nasal, oral and pharyngeal cavities (i.e. the vocal tract), which permit the production of different vowels and voiced consonants. Formant regions are not directly related to the pitch of the fundamental frequency and may remain more or less constant as the fundamental changes. If the fundamental is well below or low in the formant range, the quality of the sound is rich, but if the fundamental is above the formant regions the sound is thin and in the case of vowels may make them impossible to produce accurately - the reason singers often seem to have poor diction on the high notes. Acoustic Lab 5: Sound waves in Tubes page 3 053

54 Prelab Report The result of open-open system (e.g. flute and other wind instruments) or closed-closed tube of length L has standing waves where the wave length of the nth harmonic is shown in the Table below. The corresponding frequency is listed in the third column, where v is the speed of sound in the tube. harmonics wavelength frequency 1 2L f 1 =v/2l 2 L 2 f 1 3 2L/3 3 f 1 n 2L/n n f 1 Question: An open-open PVC pipe of length 60 cm. What is the fundamental resonance frequency in the air at 1 atm and 20 C? What is the frequency of the next higher harmonic? 2. On the other hand, an open-closed tube of length L has standing waves with only odd harmonics shown in the Table below: harmonics wavelength frequency 1 4L f 1 =v/4l 3 4L/3 3 f 1 5 4L/5 5 f 1 2n+1 4L/(2n+1) (2n+1) f 1 Question: An open-closed PVC pipe of length 60 cm. What is the fundamental resonance frequency in the air at 1 atm and 20 C? What is the frequency of the next higher harmonic? Acoustic Lab 5: Sound waves in Tubes page 4 054

55 INTRODUCTION 5A: SOUND WAVES IN TUBES So far we have studied oscillations and waves on springs and strings. We have done this because it is comparatively easy to observe wave behavior directly in these media. Sound is also a wave. In this lab we will study the behavior of sound waves in tubes. It is important that you compare the things you observe in this lab to your observations of the more easily visible waves in previous labs. Wave behavior is universal. A. TRAVELING WAVES IN TUBES 1. Getting Ready Measure the length of your tube to the nearest millimeter. Then glue the glass plate to the one end of the tube with rubber cement. 2. Hooking up the Pulse Generator L = You can make pulses of sound with a pulse generator. You are provided with a pulse generator which is similar to the function generator you have been using, except that it only generates square pulses of various widths and spacings. a. Hook the VARiable output of the 4001 Pulse Generator to the speaker. b. There are two time adjustments on the pulse generator: Pulse Spacing and Pulse Width. Pulse Width i. Each one has a control knob with a coarse setting (which clicks into place) and a fine setting (which is continuously variable). Set the Pulse Spacing to 100 ms and turn the fine adjustment fully clockwise. Set the Pulse Width to 10 μs and turn the fine adjustment to 10 o clock. ii. Pulse Spacing Turn on the pulse generator and turn the Amplitude knob to 10 o clock. Connect the VAR out to a speaker. You should hear a soft clicking sound coming from the speaker. Sound Waves in Tubes Page 1 055

56 3. Looking at the Pulse on the Scope a. Connect a BNC/BNC cable from the TTL output of the generator to the TRIGger input on the scope and set the Trigger Select to EXTernal. Set the sweep rate to 0.1 ms/div and the Trace MODE to NORMal. Connect the VARout to CH1; Set CH1 to 1 Volt/div and the input selector switch to AC, and set the pulse generator to RUN mode.look at channel 1. You should now be able to see the voltage pulse created by the pulse generator if you turn the intensity all the way up. Adjust the fine control of the Pulse Width until the pulse is 0.1 ms long and adjust the Amplitude until the height of the pulse is 1.0 V. The pulse will look like this: 1 V 0.1 ms b. Connect VAR out with a BNC T-connector to the speaker and to the CH1 of the oscilloscope. c. Connect the BNC terminal from the microphone to CH2. Set CH2 to 20 or 50 mv/div and the input selector to AC. Switch the scope to look at dual mode. Turn on the mike and then hold the mike near the speaker. You should be able to see the pulse as detected by the microphone. This sort of very abrupt, square pulse is quite difficult for a speaker to reproduce, so the pulse you see detected by the microphone will have a more complicated shape, going sharply up but then overshooting and going slightly negative. Consider only the very first part of the pulse. A typical shape for the detected pulse is shown below. If the first part of the pulse goes negative, switch the leads Positive Negative going into the speaker to make it positive. 4. Seeing Reflections from the Closed End of the Tube Change the sweep rate to 1 ms/div. Put the microphone at the open end of the tube. Now place the speaker near the open end. You should now be able to see two pulses detected by the mike the direct pulse and the same pulse after it has gone down the tube, bounced of the glass and returned. To make it easier to see, turn the fine spacing adjust fully clockwise, so that the pulses come more often. You may even be able to put the pulse spacing on 10 ms/div, depending on the length of your tube. However, if you send pulses too often, each individual pulse will not be able to get down the tube and back before a new one is sent out. a. Slide the mike farther into the tube. What happens to the two pulses? Why? b. Is the pulse inverted when it is reflected? Is this similar to a fixed or a free end on the spring? Sound Waves in Tubes Page 2 056

57 5. Measuring the Speed of Sound Place the mike just at the mouth of the tube. Now very carefully measure the time between the direct and reflected pulses. You may need to adjust the sweep rate or the scope settings to get the most accurate measurement for your particular length of tube. Calculate the speed of the sound pulses in air. Compare your result to the accepted value of 340 m/s at room temperature. 6. Opening the Closed End What do you think will happen to the reflected pulse when you open the closed end? (Write down your prediction before you open it.) Now slowly remove the glass. What does actually happen to the reflected pulse? Compare this to your experiences with pulses on springs. 7. End Correction When you removed the glass plate you may have noticed that the reflected pulse shifts to a slightly later time implying that the open tube is effectively longer than the closed one. This is an interesting effect called the end correction. The end correction depends somewhat on the frequency of the sound but is approximately 0.61r for a tube of radius r. Try to measure the small shift in time between the open and closed case to estimate the end correction ( v - denotes speed of sound). Time difference between open and closed tube: t = ms Effective increase in tube length ( v t / 2 ) = Expected end correction ( 0.61 r ) = B. STANDING WAVES IN THE OPEN TUBE. 1. Introduction For the reminder of this laboratory you will concentrate on periodic waves where the wavelength, λ, the frequency, f, and the velocity, v are related by v = λf As with mechanical waves on strings, a reflected wave from the end of a tube will be superimposed on the original wave and a standing wave will result for certain frequencies and tube lengths. Let's study standing waves in a tube created by a periodic signal from a speaker near a tube which is open at both ends. If a speaker produces a positive pressure pulse, it will Sound Waves in Tubes Page 3 057

58 travel down the tube, reflect from the open end as a negative pulse because of the 180 o phase shift. The negative pulse travels back, reflecting from the starting end, now as a positive pressure pulse because of the 180 o shift, again. If, just at that moment the next positive pulse from the speaker starts down the tube reinforcing the pulse already in the tube, a standing wave will result. This happens, for example, when the time between speaker s pulses (the period, T) is exactly the time for the pulse to travel down the tube and back (2L). This period is T = 2L / v, where L is the length of the tube and v velocity of sound. The corresponding frequency is f = v / 2L. This is the lowest frequency which will resonate in the tube and is called the fundamental frequency, f1. Resonances (i.e. standing waves) will also occur at multiples of the fundamental frequency, f2 = 2 f1, f3 = 3 f1, f4 = 4 f1,... This is the same harmonic series we found for a string fixed at each end. For both fixed ends of strings and open ends of tubes, the wave is inverted on reflection. The open ends of the tube correspond to pressure nodes (the fixed ends of strings were displacement nodes). 2. Predictions for the fundamental frequency Calculate the expected value of the fundamental frequency using the measured length of your tube and velocity of sound v = 340 m/s. Do your calculations twice, first without the end correction: f1 = v / 2L =... then, taking into account the end correction, EC, determined on the revious page with the end correction: f1 = v / 2(L+2 EC) =... 3 Switching to the 4011A Function Generator. When you made waves on springs, you could either watch individual pulses, or you could send continuous signals to create standing waves. Recall that it was only possible to create standing waves for certain frequencies. You can also do this with sound waves in tubes. To do so you need to switch to the function generator. a. Turn off your 4001 pulse generator and disconnect it. Disconnect the cable from the trigger input of the scope and switch the trigger select to INTernal and the mode to AUTO. Disconnect the speaker from the pulse generator and the scope and then reconnect the speaker to the output of the 4011A Function Generator. Hook the TTL output of the function generator into the high level input of the frequency counter. Take the mike out of the tube, turn it off, and set it aside. b. Set the speaker at the mouth of the tube (which is open on both ends) so that there is only about 1/4 inch between the speaker and the tube. Set the amplitude of the function generator to 12 o clock, the frequency multiplier to 1k Hz, and the frequency knob fully clockwise, all the way down past 0.2. Slowly turn up the frequency until you hear a resonance. The resonance should sound comparatively loud and very pure and hollow. If you re not sure what a resonance sounds like, ask your instructor for help. The lowest resonance is the hardest to find, so it may be helpful to listen to higher resonances for practice. Sound Waves in Tubes Page 4 058

59 c. The lowest frequency is called the fundamental or first harmonic and is denoted by f 1. Record it and the next six or seven resonances above it in the first column of a data table that looks like this: Resonant Frequencies for a Tube Open at Both Ends f (Hz) f /f 1 Harmonic # antinodes Comments Since the fundamental is the hardest frequency to measure, go back now and check it (without looking at the frequency counter). d. Fill in the f/f 1 column by dividing each resonant frequency by the fundamental. Then fill in the Harmonic column by rounding this number to the nearest whole number. What pattern do you see? Are any harmonics missing? How does this compare to your experience with resonance in a string (Last week s experiment)? e. Compare the measured fundamental frequency with your predictions. Do you need the end corrections to predict the measured value? f. Now tape the microphone on to the meter stick as shown: 0 cm 100 cm Masking Tape Now set the frequency so that the tube is resonating in its second harmonic mode. Use the meter-stick to slide the mike slowly into the tube. Watch the pressure variations on the scope, and observe the number of nodes (no oscillations) and antinodes (maximum oscillations) that occur in the tube. Record the number of antinodes in the data table. Repeat for the first, third, and forth harmonics. Does this agree with your expectations? C. STANDING WAVES IN A TUBE CLOSED AT ONE END 1. Introduction We use the term closed tube when discussing a tube with one closed end and one open end. A periodic sound wave, generated by a small speaker near the open end, travels down the tube, reflects from the closed end of the tupe and is superimposed on the Sound Waves in Tubes Page 5 059

60 original wave. One would expect a standing wave or resonance to occur if the next wave to start down the tube reinforces the waves already bouncing up and down the tube. When a positive pressure pulse hits a closed tube end, it is reflected as a positive, not a negative, pulse (no phase shift). If a positive pulse is started at an open end, it travels down the tube, reflects back as a positive pulse, reflects at an open end as a negative pulse, reflects at the close end as a negative pulse and finally, after two round trips, (4L), relects down the tube again as a positive pulse, reinforced by the next pulse from the speaker. The fundamental frequency, f1, is thus: f1 = v / 4L Now, consider what happens if the frequency is doubled to produce the second harmonic. One pulse goes down, comes back and turns into a negative pulse at the same time that the next positive pulse is applied. Instead of adding amplitudes to make a stronger wave, the old and the new wave cancel each other so that the resonance does not occur for the second harmonics. Resonances will occur for odd harmonics only. 2. Predict the expected fundamental frequency for your tube Calculate the expected fundamental frequency using the measured length of your tube. Use v = 340 m/s. Tube length:... Fundamental frequency (no end correction, f1 = v / 4L ): Fundamental frequency (after an end correction,.f1 = v / 4(L+EC) ): 3. Measure resonance frequencies a. Close up again one end of the tube: First completely cover the end with masking tape. Then glue the glass plate over the tape. It is critical to get an air-tight seal. If the seal is not complete, you will essentially have a mixture of a closed and an open tube, and it will be very difficult to interpret your results. b Measure the frequencies at which the closed tube resonates (using the hints listed below). You should be able to measure at least five resonances above the fundamental. Record your results in the data table below. Hints: In the noisy room it is much easier to find resonances with the aid of the microphone. Put the microphone just inside the open end of the tube. It is important to watch the microphone signal on the scope and listen to the sound. The resonances will be louder if you hold the speaker very close to the tube. However, this end of the tube must remain open, so don t press the speaker hard against the tube. A good way is to hold the speaker slightly off to the side, which also gives room to put the microphone in. The lowest frequency is hardest to find. Look for several higher ones first and look Sound Waves in Tubes Page 6 060

61 for a pattern which might help you find the lowest one. Leave space in your data table for this. Most of the speakers have a resonance in the range from 250 to 300 Hz, which can give you very confusing results. Be suspicious of all resonances in this range. Resonant Frequencies for a Tube Closed at One End f (Hz) f /f 1 Harmonic Comments D. Comparison of Standing Waves in the Tube Closed at One End and Open at Both ends Please, discuss the following questions: a. How does the new series of frequencies compare to the harmonic series generated in the open tube? b. How does the fundamental frequency of the closed tube compare to the fundamental when the tube is open? c. How does the period of the fundamental for each tube compare to the time it takes for a single pulse to go down the tube and back? Try to explain the observed relation. [Here is a hint: Recall, that a standing wave is a result of a constructive interference between incident pulses and pulses reflected from the ends of the tube. Calculate for each case (open-open, closed-open tube) the number of reflections a pulse has to undergo to reach the same polarity as the incoming pulse. Hopefully, you will find that the period of the fundamental is equal to the time the reflected pulse needs to travel within the tube before its polarity matches that of the incident pulse.] d. What are the similarities between wave behavior in tubes and on strings? Sound Waves in Tubes Page 7 061

62 A. INTRODUCTION: 5B: ACOUSTIC RESONANCES Often a system has natural frequencies of vibration. In the case of the column of air in the PVC tube we are studying, these natural frequencies are called the harmonic frequencies, f 1, f 2,. A vibrating source, such as the speaker, is said to drive the column of air. As the frequency, f, of the driver (speaker) is slowly varied, the amplitude of the driven system (air column) gets larger and larger until it reaches a peak at one of its own resonant frequencies, f 1, f 2,. Therefore, as we sweep the speaker frequency past one of the harmonic frequencies of the air column, we hear the increase in amplitude of the vibrating air column as an increase in loudness of the radiated sound from the tube. At resonance the driving system passes energy very efficiently on to the driven system and the amplitude of the driven system reaches a maximum called A max. (See Rossing, the Science of Sound, Ch. 4.) A graph of the amplitude as a function of frequency is shown here. Note that the resonant amplitude peaks at frequency f 0 and that the peak has a width, f called the linewidth. The linewidth is usually measured at amplitude of 71% of A max. For a system which loses energy rapidly through damping or friction, the maximum amplitude, A max, is small and the linewidth large, and the resonance is said to be broad. Similarly, for a resonating system which loses energy very slowly, the maximum amplitude is very large and the linewidth is very small. The resonance is said to be sharp. The quality, Q, characterizes the sharpness of one of the resonances. For example, for the fundamental frequency the Q value f 0 is: Q. f B. Measurement of the Q of a Resonance: 1. High Q System: A high Q system is one with a sharp resonance. Once set oscillating it loses energy very slowly. A tuning fork is an example of an object with a very high Q. To drive an object with high Q, the driving frequency must be very close to the resonant frequency, f 1. It is convenient to measure the Q of one of the harmonics of your long PVC tube with one end Sound Waves in Tubes Page 8 062

63 closed. If your tube does not already have one end closed, close it at described in the first part of the lab. a. Set up the scope with CH1 looking at the signal from the function generator 4011A and with CH2 looking at the signal from the microphone. Set the scope to trigger on CH1. With a small sine wave signal (about 1 volt) from the signal generator, attach the small speaker to the generator. b. Find a convenient harmonic of your open tube with a frequency above 350 Hz. (The small speakers we are using have a resonant frequency of their own at about 260 Hz. It s nice to avoid this frequency region for this reason.) Choose sensitivity (VOLTS/DIV) for the microphone input into CH2 so that the full signal can be seen on the scope face at resonance. c. Now, starting below the resonance frequency and scanning over the resonance in 10 or 12 steps, record the amplitude of the microphone signal in millivolts. As a first try you might start about 50 Hz below resonance and step over the resonance in 10 Hz steps. You may want to fill in a few extra points where the amplitude changes rapidly. Driving Frequency Amplitude (mv) Comments d. On a separate sheet of graph paper make a plot of amplitude (on the vertical scale) against frequency (on the horizontal scale). You may also plot your data points with the Graphical Analysis program and print the graph. A max 0.71A max f f Q f / f Sound Waves in Tubes Page 9 063

64 C. COMPARISON OF ACOUSTICAL AND ELECTRICAL RESONATING SYSTEMS You have now measured the resonance behavior of an air column and an electrical RLC circuit. In both cases you used a Function Generator to excite the system at various frequencies. You displayed input and output voltages on an oscilloscope and measured their peak-to-peak amplitudes. The traces on the scope looked the same, except that the air column and the electric circuit resonated at different frequencies. If you now put the air column or the electric circuit into a black box with two sockets, one for the input to drive the system and one for the output to measure the amplitude of the resonating system, you would not be able to distinguish between the two systems. Both systems show the same resonant behavior. However, there are differences between a resonating air column and an electric RLC circuit. What is the major difference? Does a resonating RLC circuit have higher harmonics like the air column? Sound Waves in Tubes Page

65 PRELAB6: SOUND PULSES Sound waves propagate (travel) through air at a velocity of approximately 343 m/s (1125 ft/s). As a sound wave travels away from a point (small) source of sound such as a vibrating speaker or tuning fork, its amplitude decreases as its energy is spread over larger and larger area. These waves also reflect from surfaces and in reflection lose amplitude through absorption. A. 1 R2 Decrease in Sound Intensity: Since the energy of a sound wave from a source must spread over a larger and larger sphere as it moves away from the source and since the area of this sphere is 4πR 2, one expects the energy of any wave to decrease as 1/R 2 as the distance to the source, R, is increased. Similarly, since the energy (and the corresponding intensity or energy per second per square meter) is proportional to the square of the amplitude of the wave, the amplitude should decrease with distance as 1/R. a. Energy in physics is measured in Joules [J]. The power in physics is measured in amount of energy emitted per unit time, e.g. Joules per second, called Watts [W], i.e. 1 W = 1 J/s. The intensity is measured in power per unit area, i.e. I=P/A, and this the unit of intensity is Watts/m 2. What is the light intensity at 1 m away from a 60 W light bulb? b. The solar constant is the light intensity from sun reaching on Earth. The solar constant is I = 1.37 kw / m². The distance between the Sun and the Earth is 149,597,892 km. What is the power of our Sun? B. Early Sound Early sound refers to the arrival of reflected sound from surfaces in a room (or concert hall). Reverberation is a result of multiple REFLECTIONs. A SOUND WAVE in an enclosed or semi-enclosed environment will be broken up as it is bounced back and forth among the reflecting surfaces. Reverberation is, in effect, a multiplicity of Echoes whose speed of repetition is too quick for them to be perceived as separate from one another. W.C. Sabine established the official period of reverberation as the time required by a sound in a space to decrease to onemillionth of its original strength (i.e. for its intensity level to change by 60 db), i.e. Reverberation time = RT 60 = time to drop 60 db below the original level Architectural acousticians stress the importance of early reflections (arriving within the first 80 ms) which reinforce the direct sound as long as the angle of reflection is not too wide. Reflections arriving after 80 ms add reverberant energy which is often described as giving the PRELAB: Sound pulses Page 1 065

66 sound spaciousness, warmth and envelopment. The acoustic design of such spaces usually involves creating a balance between clarity and definition on the one hand, and spaciousness on the other. Listeners often have different preferences as to this balance. Sabine equation of RT 60 in [seconds] is given by The Sa product is called the Sabine number. For a room composed of different materials, the effective Sabine number is given by Sa= S i a i. Note that the surface area includes ceiling, floor and 4 walls. a. Estimate the reverberation time of SW166, where the room is about 15 m 15 m with ceiling at 3 m height. We assume that the absorption coefficient of the walls is 0.1, and no furniture inside the room. b. Estimate the reverberation time of your living room where you put your audioentertainment system. C. Sound Absorptions by Materials Absorption refers to the absorption of sound waves by a material. As the sound wave hits a material surface, the acoustic energy can be absorbed and converted into heat energy. The absorption depends on frequency-dependent, size, shape, location, and environment. A good sound absorber is normally a porous material. Mineral wool, glass wool, and micro perforated plates work as sound absorbers. Absorption is not a single mechanism of sound attenuation. Propagation through a heterogeneous system is affected by scattering as well. Acoustical absorption is represented with the symbol A. Acoustical absorption in the ocean is an important part of the analysis of sonar. The primary substance in seawater that is responsible for absorption is magnesium sulfate. The secondary substance is boric acid. The most common sea salt, sodium chloride has virtually zero effect on sound absorption. The fraction of the incident sound energy absorbed is called the absorption coefficient, a. An ideal absorbing surface would have a = 1, while a perfect reflecting surface would have a = 0. An example of an ideal absorbing surface might be an open window in a room. It reflects none of the sound hitting it, effectively absorbing it all. A wooden floor (or hard table) typically has an absorption coefficient, a, of less than 10% (that is 0.1) for most frequencies of sound. To measure the absorption coefficient of material 1, we compare the intensities reflected from the material 1 with that from the material 2, which has known absorption coefficient a 2. The absorption coefficient a 1 can be obtained from the following equation: 1 1 PRELAB: Sound pulses Page 2 066

67 6: EXPERIMENTS WITH SOUND PULSES Sound waves propagate (travel) through air at a velocity of approximately 340 m/s (1115 ft/s). As a sound wave travels away from a small source of sound such as a vibrating speaker or tuning fork, its amplitude decreases as its energy is spread over larger and larger area. These waves also reflect from surfaces and in reflection lose amplitude through absorption. The purpose of these experiments is to explore these ideas by using short pulses of sound. We will generate these short pulses of sound by sending short voltage pulses to a speaker. Physics Concepts Involved: The velocity of sound in air depends on the temperature of the air. The formula: v = m/s + (0.6 m/s) T gives the velocity in m/s (meters per second) where temperature T is in degrees Celsius. Sound reflects from a flat surface much as light reflects from a mirror. In analogy to light one can draw rays of sound where the incoming and reflected angles are equal. From this view you can calculate the time it takes for a sound wave from the speaker to bounce from a flat surface and back to the mike. Assume the room temperature to be 20 degrees Celsius. A. Velocity of Sound: A convenient arrangement for these experiments is one where the directional speaker is mounted on a ring stand about 0.5 m above the lab table. The computer microphone must be mounted on a ring stand in fixed position for careful measurements. See the drawing. 1. Simple velocity measurement: Look at the VARiable OUTput of the 4001 Pulse Generator on channel 1 of your oscilloscope. Convenient settings of the pulse generator are: power and run buttons pushed in, pulse spacing = 10ms, pulse width = 100μs, spacing vernier = 1, and width vernier = 1. Adjust the amplitude of the pulse to approximately 0.5 V. When you are happy with your pulse, attach your speaker in parallel with the oscilloscope. You should hear a (gentle) buzzing. Loud buzzing is unnecessary. To measure the velocity of sound, you need only measure the distance the sound travels and the time it takes to get there. 2. To measure the time, you can see a close pair of pulses (the first pulse represents the sound signal from the speaker passing the microphone, the second is the reflected sound). Point the cursor at the first pulse and click the mouse, then point it at the corresponding part of the reflected pulse and click again. The time (in seconds) and signal level (in volts) of each Acoustic Lab 6: Sound pulses Page 1 067

68 point you clicked are recorded in the bottom window. To find the time between the pulses, subtract the two times. The distance (in meters) between the speaker and microphone is also needed. Calculate the velocity and compare it with the value you expect from equation (1). Be sure to measure both the time and distance as carefully as the instruments allow and record your labeled measurements and calculations: pulse travel time = distance traveled: expected velocity from Equation (1): velocity from your measurements (v = d/t): comments: B. 1 R2 Decrease in Sound Intensity: Since the energy of a sound wave from a source must spread over a larger and larger sphere as it moves away from the source and since the area of this sphere is 4πR 2, one expects the energy of any wave to decrease as 1/R 2 as the distance to the source, R, is increased. Similarly, since the energy (and the corresponding intensity or energy per second per square meter) is proportional to the square of the amplitude of the wave, the amplitude should decrease with distance as 1/R. You can make a measurement of this effect by simply observing the amplitude of the mike pulse as you space the mike farther and farther from the speaker. To measure the amplitude, turn the microphone towards the speaker, place it at the distance you want, and record the sound signal. Then zoom in on several pulses and use the mouse to click the most positive and negative peaks of each pulse. The difference between the signal levels (which are recorded in the lower window) represent the peak-to-peak amplitude of each pulse. Measure the peak-to-peak amplitudes of several pulses in each recording and average them. Make a table of amplitude as a function of distance of the mike from the speaker. Start your measurements with the mike close to the speaker where the amplitude is the largest (10 centimeters away) and then measure every 10 centimeters. Be careful to avoid saturation effects. The largest peak to peak amplitude should never exceed V. distance (cm) = R 1 amplitude (volts) = A 1/amplitude (1/volts) = 1/A Acoustic Lab 6: Sound pulses Page 2 068

69 On a separate piece of graph paper plot your amplitudes as a function of the distance from the speaker. Does the amplitude decrease with distance as 1/R? Do you see any strange effects as you get very close to the speaker? These are called proximity effects. (plot A vertical axis vs R1 as horizontal axis) When measuring the distance to the speaker it s difficult to decide precisely where to take the position of the speaker. The speaker really isn t a point source, so it s hard to select the center of the sphere. We will try to re-plot your data in such a way that the position of the center of the speaker can be extracted. The arguments follow. The expected amplitude-distance relationship is A = k/r, where R is the distance from the speaker to the mike and k is a constant determined by the amplifier gain, speaker type, etc. Since the distance, R 1, is from the front edge of the speaker to the mike, the distance from the center to the mike, R, can be expressed as R = R 1 + R 0, where R 0 represents the distance from the edge of the speaker to its center. In order to determine R 0 we can write: A = k/r = k / (R 1 + R 0 ). This equation can be rearranged to read: R 1 = k/a R 0. Plotting distance R 1 on the vertical axis against one over amplitude, 1/A, on the horizontal axis should give a straight line with slope k and intercept, R 0, Make this plot. Use a ruler to draw a straight line through your data points. From the plot, read the intercept, which is equal to the vertical coordinate where the line crosses the vertical axis (horizontal coordinate 1/A=0 ). Calculate the slope k of the straight line by drawing a line through the most points: Acoustic Lab 6: Sound pulses Page 3 069

70 slope = (change in the vertical coordinates) / (change in horizontal the coordinates) The unit of the intercept is cm, and the unit for the slope is cm*v. Comment on whether you believe your data fits this model. What values do you find for R 0 and k? R 0 = k = (plot R1 vs 1/A; read off R 0 and the slope k see above) C. Early Sound Early sound refers to the arrival of reflected sound from surfaces in a room (or concert hall). As mentioned in the physics section above, sound reflects from a flat surface much as light reflects from a mirror with equal incident and reflected angles. You can calculate the time it takes for a sound wave from the speaker to bounce from a flat surface and back to the mike. With the speaker and mike mounted on separated ring stands 50 cm or so above the lab table, you can observe the arrival of both the direct sound and the early sound reflected from the table. With the computer, we can measure the difference between the arrival times. Place the mike and the speaker about 1 meter apart, facing each other. Record the sound signal and zoom in on a pulse (this is the direct pulse). About 1 ms (millisecond) later, you should see a much smaller pulse, which represents the early sound. Use the mouse to measure the time interval between the pulses. To make sure you have identified the early sound pulse, put a piece of acoustic foam on the table and record again. Check to make sure your reflected pulse disappears from its previous position (if the feature you measured did not disappear, you probably did not identify it correctly, contact your instructor). Early Time - Direct Time = Acoustic Lab 6: Sound pulses Page 4 070

71 The difference between the direct time and the early time should reflect the difference in distances traveled by the sound in each case. Without moving your apparatus, use a meter stick to measure the direct sound distance and the reflected sound distance. In measuring the reflected sound distance how do you decide the path to measure? Make a sketch, approximately to scale, where you have labeled the two paths and the lengths of each. On your sketch be sure to show the two angles that are known to be equal. Direct Distance = Early Distance = Early Distance - Direct Distance Using your best velocity determination from part A and your measured time difference calculate the expected path difference for the early and direct sounds. Does it agree with your estimate using a meter stick? D. Sound Absorptions by Materials When sound bounces off different surfaces, the amount of reflected sound is always less than the incident sound. The sound energy which is not reflected is absorbed by the surface. The fraction of the incident sound energy absorbed is called the absorption coefficient, a. An ideal absorbing surface would have a = 1. while a perfect reflecting surface would have a = 0. An example of an ideal absorbing surface might be an open window in a room. It reflects none of the sound hitting it, effectively absorbing it all. A wooden floor (or hard table) typically has an absorption coefficient, a of less than 10% (that is 0.1) for most frequencies of sound. With your apparatus placed either as in part A of this lab manual or as in part B (your choice) record a series of pulses. You can ignore the direct sound pulse for this experiment. Measure the amplitude, A of the reflected pulse (the same way as in part B: measure the peak-to-peak amplitude of several pulses and average them). You are now in a position to be able to compare the absorption of various surfaces with the board. The technique is simply to place different surfaces at the same position and compare the new reflected amplitude, Ac, with the board (table) amplitude, Ab. Since the energy in the reflected wave is proportional to the amplitude squared of the wave, you need to square the ratios of the amplitudes you obtain. For example, suppose the reflected amplitude with a layer of carpet, Ac, is only half of that for the board, A b. That indicates that the reflected energy ratio of carpet to board, (Ac/Ab ) 2 = ( ½ ) 2 = 1/4. Since the reflected energy is proportional to (1-a), we can relate the absorption coefficient for the board, a b, to the absorption coefficient for the carpet, a c, by the equation: 1 ac 1 ab = A c 2 Ab 2 = I c (3) I b We can rearrange equation (3) to find the absorption coefficient of the carpet as: Acoustic Lab 6: Sound pulses Page 5 071

72 ac = 1 (1 ab) A c 2 Ab 2 (4) Measure the reflected amplitude for the board and measure it again after the insertion of : (i) a layer of carpet, (ii) a piece of acoustic tile, (iii) a piece of acoustic foam, and (iv) regular foam. Assume an absorption coefficient, ab = 0.1, for the board, and calculate the coefficient of absorption a c for various materials. A b (board amplitude) = Fill the rest of the Table below: Are the relative values of your measured absorption coefficients consistent with your expectations? Compare them with the values from the attached table of absorption coefficients for various materials taken from the textbook by Rossing et al. Material Amplitude (volts) = A Absorption coefficient: a use Equation (4) Compare and comment on the absorption coefficients Board (table) 0.1 assumption Carpet Acoustic tile Acoustic foam Regular foam Acoustic Lab 6: Sound pulses Page 6 072

73 A. INTRODUCTION PRELAB 7: SYNTHESIS In previous labs you have looked at and listened to a number of oscillations called sine oscillations. The sine oscillation is the most basic of all oscillations, and is often called simple harmonic oscillation. It turns out that any kind of oscillation can be produced by an appropriate combination of sine oscillations of different frequencies and amplitudes. Indeed, the study of oscillations often begins by analyzing the oscillation into its sine components. This is particularly true in the area of sound, where the number and type of sine components determine the timbre or tone quality of the sound. B. PHASE To construct a given complex oscillation from sine components, it is important to know the relative amplitudes and frequencies of the components, and also to know their PHASES relative to each other. Phase has to do with where the oscillation begins. Below are three oscillations which are otherwise identical but differ in phase: 0 t The function in diagram (1) is zero when t = 0 and is called the sine function. The function in (2) is identical except that is starts 1/4 cycle later. This particular phase relationship has a special name: cosine. The cosine is sometimes said to be 90 degrees outof-phase with the sine function. The curve in (3) is a sine function starting 1/2 cycle late and is said to be 180 degrees out of phase with (1) or exactly out-of-phase. If you look carefully at these graphs you will realize that (3) can be thought of either as (1) out of phase or as (1) with a negative amplitude. Later in this lab 180 degrees out of phase oscillations will be referred to as having negative amplitude. C. Harmonic series ANY oscillation which is periodic (repeats exactly over and over again) can be produced by combining a very particular set of sine oscillations known as the harmonic Series. The harmonic series starting on 100 Hz is:. The lowest frequency in the series (the first harmonic) is called the fundamental. Each possible fundamental frequency has its own harmonic series, which is obtained by multiplying the fundamental by 1, then by 2, then by 3, etc. In music, the frequency of the fundamental determines the perceived pitch of the note, whereas the relative amplitudes of PRELAB Synthesis

74 the other harmonics (the "overtones") determine the timbre. The harmonic series starting on 123 Hz is: The analysis of any periodic oscillation that happens 123 times every second will reveal only these frequencies. Ever 150Hz, for example, will never be present. This extremely powerful fact was first studied by Euler and by Bernoulli, who first published the mathematical equations of the series in Extensive later work by Fourier resulted in the name "Fourier Analysis" for this kind of study of periodic functions. It is important to realize that while Fourier Analysis is exact for periodic oscillations, real life oscillations are, at their least complex, only approximately periodic. Musical sounds, in particular, are usually much more complex. Treating them as approximately periodic is, nonetheless, often an excellent way to begin an analysis, which can later be made more subtle. D. ADDING OSCILLATIONS (SUPERPOSITION) In this lab you will be studying oscillating voltages. An oscillating voltage is a voltage that changes in time. You can add two oscillating voltages simply by adding the two voltages at each point in time. Consider the two oscillating voltage signals, the two thin lines "l" and "2," shown below. You can find their sum by using four-step processes. 1: Find the times where Signal 2 crosses through V=0. Circle these points. For each of these these times, place a dot on top of Signal 1. 2: Find the times when Signal 2 has a maximum and mark these by up-arrows. For each of these times, place a dot the same distance above Signal 1 as the length of these arrows. 3: Find the times when Signal 2 has a minimum and mark these by down arrows. For each of these times, place a dot the same distance below Signal 1 as the length of these arrows. 4: Connect the dots with a smooth curve, representing Signal 1 + Signal~2 (in this case, the thick line). PRELAB Synthesis

75 E. Beat, Fusion Frequency, and Critical Band: A typical experimental setup for psychoacoustic measurement is shown in the drawing below (see books by Rosederer, Rossing, et al.). Careful study of psychoacoustic study shows that when tones of frequencies f 1 and f 2 are superimposed, we may perceive these two tone in beat. Experiments show disappearance of beats occurs around frequency difference of about Δf= f 1 f 2 =15 Hz regardless of the values of f 1 and f 2. Demonstration of beats with electrical signal generators will be carried out in our experimental set up as shown above (see also Fig. 8.8 of Rossing) When two pure tones are so close in frequency that there is considerable overlap in their amplitude envelopes on the basilar membrane, they are said to lie within the same critical band, introduced by Harvey Fletcher in the 1940s. The critical band is associated with the overlap of the basilar membrane in the inner ear. Based on the placement theory of hearing, vibrations set up a travelling wave along the membrane giving it a different maximum displacement for different frequencies causing a shearing motion for the hair cells of the organ of Corti, and generating electrical potentials that stimulate the auditory nerve. The critical bandwidth (c.b.) depends on the frequency. (see Georg von Békésy s work1960). Central frequency (Hz) Critical bandwidth (Hz) Similarly, Critical fusion frequency (CFF) or marked as F below is defined as the rate at which stimuli can be presented and still be perceived as a separate stimuli. Stimuli presented at a higher rate than CFF are perceived as continuous stimuli. The fusion frequency F is schematically shown in the figure below (see Roederer, J.G. Introduction to the Physics and Psychophysics of Music, 2 nd Ed (Springer, Verlag, New York, 1975).) Finally the just noticeable difference (jnd) in frequency determined by modulating the frequency of a tone of about 4 Hz. Note that the jnd at each frequency is nearly a constant percentage of the critical bandwidth as show in the graph above (see also Zwicker, Flottorp and Stevens, critical bandwidth in loudness Loudness summation, J. Acoustic. Soc 29, 548 (1957).) PRELAB Synthesis

76 F. Equal Loudness Curve The loudness of a sound is defined in unit of decibel, which is a number (with no units) by comparing two numbers. A decibel is always a comparison to some reference value, and is meaningless unless you know what that reference value is. A typical reference value for sound pressures is pressure of the softest sound the average person can hear. To get the decibel value, you divide your value by the reference value, take the logarithm, and multiply by 20. db = 20 log (p/p 0 ) For example, if a sound pressure is 2000 times the reference pressure, it has a value of db = 20 log (p/p 0 ) = 20 log (2000) = 20 x 3.3= 66 db, which is about the level of sound near a busy street when compared to the threshold of hearing. With voltages, it is common to take the largest typical voltage as the reference, which means that the other voltages will be a negative number of decibels in comparison. For example, if the reference signal has amplitude of 3 volts, then a signal with amplitude of 1 V is db= 20 log (V/V 0 ) = 20 log (1/3) = 20 x ( 0.477)= 9.5 db Audio Equal-loudness contours are often referred to as Fletcher-Munson curves, after the earliest experimenters, but today, the curves are defined in the international standard ISO 226:2003 (most recent revision) which are based on a review of several modern determinations made in various countries. See G. Masking: Auditory masking occurs when the perception of one sound is affected by the presence of another sound. Following the textbook by Thomas D. Rossing, The Science of Sound, (Addison-Wesley Pub. Co, ISBN ): The following figure summarizes a simplified response of the basilar membrane for two pure tones A and B. a) The excitations barely overlap; little masking occurs. b) There is an appreciable overlap; tone B masks tone A and somewhat more than the reverse. c) The more intense tone B almost completely masks the higher frequency tone A. d) The more intense tone A does not completely mask the lower frequency tone B. PRELAB Synthesis

77 H. PRELAB 7 (TURN IN THE FOLLOWING TWO PAGES) ANSWER THE FOLLOWING QUESTIONS 1. Determine the frequencies of the first six harmonics of 342 Hz. 2. If 227 Hz is the fundamental, which harmonic is 1589 Hz? 3. Sometimes, the amplitude of some harmonics will be zero. These harmonics are missing. It is even possible for the fundamental to be missing. What is the highest frequency that could be the fundamental for a harmonic series containing the following frequencies: 500 Hz, 750 Hz, 875 Hz, and 1000 Hz. (Of course, 1 Hz could be the fundamental for a series containing these harmonics, but that is not really what we are after. Please find the highest frequency which could be the fundamental.) 4. Which harmonics (including fundamental) in the above series in item 3 are missing? 5. If the reference voltage is 2.0 V at 0 db, what is the db value for the voltage at 0.5 V? 6. If the reference voltage is 2.0 V at 0 db, what is the voltage at 20 db? 7. When a sound of two pure sinusoidal waves at 343 Hz and 345 Hz are heard, what is the beat frequency? 8. From past psychoacoustic experiments, would a higher frequency sound masks more on the lower frequency sound or the vice versa? 9. Find the values of the following sine functions. Note that the sine function sin(ωt)=sin(2πft), where ω is the angular frequency measured in radian/s, and f is the frequency measured in Hertz (Hz) or 1/s. In your calculator, you should use radian mode. If you use degree, you should use transformation that π radian is equal to 180 degrees. a. Frequency f=100 Hz, t= s, sin(2πft)= b. In the second example, we us frequency f=100 Hz, t= s, and 180 degrees out of phase from the above example, i.e sin(2πft+π)= PRELAB Synthesis

78 10. Add the voltages shown below (use color other than black). The period of 1 cycle of the superposition is s. PRELAB Synthesis

79 A. INTRODUCTION 7A: SYNTHESIS Of the three most basic properties of a sound, loudness, pitch, and timbre, surely the most interesting is timbre. Timbre is what makes a cello sound like a cello, a howling wolf like a howling wolf, Roger Rabbit like Roger Rabbit. All of spoken communication is based on differences in timbre. As you have heard, different types of vibration sound differently, and when they are displayed on the oscilloscope they have different shapes. But it turns out that studying signal shape is not the best way to study timbre. It was discovered in the 19th century that the most basic type of oscillation is the sine oscillation, which is available from your function generator. All other vibrations can be produced simply by adding together sine vibrations. Since all sounds are just combinations of sine sounds, all sounds other that sine sounds are called complex. When you make a complex sound by adding together a bunch of sine signals, you are synthesizing the complex sound. Synthesizing simply means putting together. In this lab, you will synthesize several complex sounds using a synthesizer which has a harmonic series of nine oscillators each of which has separate amplitude and phase controls. The synthesizer has a summing amplifier which allows you to hear and see the result of adding all these signals together. You may be surprised by how much you can do with just nine sine sounds. B. PHASE To construct a given complex oscillation from sine components, it is important to know the relative amplitudes and frequencies of the components, and also to know their PHASES relative to each other. Phase has to do with where the oscillation begins. Below are three oscillations which are otherwise identical but differ in phase: 0 t The function in diagram (1) is zero when t = 0 and is called the sine function. The function in (2) is identical except that is starts 1/4 cycle later. This particular phase relationship has a special name: cosine. The cosine is sometimes said to be 90 degrees outof-phase with the sine function. The curve in (3) is a sine function starting 1/2 cycle late and is said to be 180 degrees out of phase with (1) or exactly out-of-phase. If you look carefully at these graphs you will realize that (3) can be thought of either as (1) out of phase or as (1) with a negative amplitude. Later in this lab 180 degrees out of phase oscillations will be referred to as having negative amplitude. Lab 7: Synthesis

80 C. LEARNING TO USE THE SYNTHESIZER The synthesizer consists of 10 oscillators. The first two are both labeled 1 and are the same frequency. The reason that there are two signals of the same frequency is so you can study the result of adding two signals that have the same frequency but different phases. The remaining eight oscillators form a harmonic series above the lowest frequency oscillators (1- Left and 1-Right). Using the descriptions listed below, try out each of the controls on oscillator 1-Left until you understand them; the controls on the rest of the oscillators are the same. Using the synthesizer controls 1. At the very bottom is the in/out switch. Switch it in. 2. Above that is an output jack for this particular oscillator. You won t use it. 3. Above that is the amplitude knob, labeled Try it. Listen to the sound with the headphones. The sound is louder with larger wave amplitude. Note: The best way to control the loudness of the sound you hear is to turn the GAIN knob of the summing amplifier all the way up and then use the volume control on the headphone box if you want a quieter sound. 4. The next knob up (marked something like this: + 0 ) is the fine phase control. Try it. Turning the knob changes the phase of the signal, but you can t get any old phase you want with this knob alone. For big changes in phase you need to use the switches above this knob. How does the sound change when you change the phase? 5. These two switches change the phase by either 180 or 90. Try them. You would like to think that if you put both of these switches to 0, and then turned the fine phase control knob to 0 that you would have 0 phase, but no such luck. The only way to set the phase of these oscillators is with the scope. Use whatever settings are necessary to get the phase you want on the scope. How does the sound change when you change the phase? 6. The two lowest oscillators have the ability to make triangle and square functions as well as the sine. The next-to-top switch can be set to sine (left) or other (right). The top-most switch determines whether the other option is triangle or square. Try it. D. Adding Signals of the same frequency In this section you will use the two oscillators labeled 1. Because they are both called 1, in this section they will be referred to as the left and right oscillators. Now that you have tried the controls, use them to create a specific signal: First set the scope to 0.5 ms/div and 0.2 V/div. Then, using oscillator 1 left (and all others switched out), make a sine function with exactly 0 degrees of phase shift and a peak-to-peak amplitude of 0.4 V. Draw this signal (include units on axes) and record the period of this oscillation below. (Hint: count the number of divisions in one cycle then multiplies by 0.5 ms/div)? Lab 7: Synthesis

81 1. Identical Phase a. Switch the left oscillator out. b. Switch in the right oscillator and adjust it until it is identical to the left one phase of 0 and the same amplitude. c. To add them together all you have to do is switch them both in at the same time. Do this, and measure the resultant amplitude. Amplitude of the sum: Period of the sum: Is this what you expected? Why? 2. Opposite phase You might think that when you add together two signals with the same frequency and amplitude that you will obviously get a resultant signal that has twice the amplitude, but this is not always the case. To see for yourself, try adding two signals of opposite phase, using the instructions below. a. Switch both oscillators out. b. Switch in the left one to check to be sure it still has the correct amplitude and phase. Then switch it out. c. Switch in the right one. Flip the switch, which should shift the phase of this signal by 180. Look at the scope, draw the signal: (Include units on the axis.) Be sure you understand why this is called a 180 phase shift, and also how this is equivalent to a signal with a negative amplitude. d. Add the signals by switching them both in. What is the resultant amplitude? Lab 7: Synthesis

82 e. Try turning the fine phase adjustment knob. By now you should see that when you add two signals, the amplitude of the result depends on phase. Does the resulting period depend on the phase of the right oscillator? f. Does the loudness depend on the phase of the right oscillator? You are now done with the right oscillator 1. Switch it out. As you go through the rest of the lab it is important that you not confuse this right oscillator 1 with oscillator 2. You may want to put a piece of masking tape on the in/out switch of right oscillator 1 so you don t forget this. E. Adding waves with Different Frequencies of the Harmonic Series 1. Next, we want to find out what happens when we oscillations at different harmonics of the fundamental frequency. Switch the left oscillator 1 out and switch oscillator 2 in. Set the amplitude of oscillator 2 to 0.4 Volts. Draw the signal: Measure the period the oscillation: What is the frequency of the oscillation? How do these compare to the period and frequency of oscillator 1? How does the sound compare to the sound of oscillator 1? 2. Next, set the phase of oscillator 2 to 0. Switch both oscillator 1-Left and oscillator 2 in. How does the sound change? How does the sound signal change on the scope (amplitude, period, shape)? 3. With both oscillators 1-Left and 2 in change the phase of oscillator 2. Which of the following change (and how): The qualities of the sound (loudness, pitch, and timber): The quantities that describe a periodic sound signal (period, amplitude, shape): 4. Alternately switch oscillators 1-Left and 2 out and in momentarily. Does the pitch of the sum sound more like the pitch of oscillator 1 or oscillator 2? Lab 7: Synthesis

83 5. Now switch in any number of the other oscillators. How does the sound change? After you have turned on 4 or 5 more oscillators the signal on the scope is probably pretty complicated, but there is something very interesting: what is the period of the sound signal? Change a few amplitudes and phases of the different oscillators. What do you conclude about the period of the signal made up of adding together members of a harmonic series? 6. Leave the oscillators from step 5 in, except now switch oscillator 1 out. What is the period of the sound signal now? Are you surprised at the answer? 7. Now switch out all the odd-numbered oscillators (including oscillator 1), and measure the period of the signal. Does your result make sense? 8. Try this psychoacoustic experiment: Switch in oscillators 2, 3, and 4, set the amplitudes so that they are approximately the same, and listen. Now switch oscillator 3 out. How does the sound change? Explain using what you learned in steps 5-7. F. Synthesis of Periodic Sounds As we have seen, pure sine sound waves can be added together to produce more complex sounds. To simplify matters further, any steady state sound (any sound which is not changing over time) can be made from a special set of sine oscillations called the harmonic series. A harmonic series is a series of frequencies which are whole number multiples of the lowest frequency. We have discussed the harmonic series several times, so you should already be familiar with it. Here is another example: Say you want to make a complex sound which has a period of 1.0 ms and which looks like this on the scope: 1.0 ms The frequency of this sound is f = 1/T = 1000 Hz. In order to synthesize it, you need a harmonic series starting on 1000 Hz. That is, you would have to add together sine oscillations with frequencies 1000 Hz, 2000 Hz, 3000 Hz, 4000 Hz, 5000 Hz, etc. To get exactly the shape shown above, you would need an infinite number of oscillators. But since you can t possibly hear any of the oscillators above 20,000 Hz, it wouldn t sound any different if you just had twenty oscillators, and with twenty oscillators it would look pretty good, too. To get the right shape, you need to use the right amplitude and phase for each of the different oscillators. Lab 7: Synthesis

84 G. SYNTHESIS WITH THE HARMONIC SERIES Oscillators 2 9 work the same way as number one except the each has a higher frequency. Oscillator 2 has twice the frequency of 1, oscillator 3 has three times the frequency of 1, and so on in a harmonic series. 1. Setting the phases For the first exercise, you will need to set all of the phases to 0. It is easiest to set the phases accurately when the amplitude is large. Follow the procedure below to set the phase of each oscillator. Start with oscillator 1. a. Switch the oscillator in. b. Turn the amplitude all the way up. Set the volts/div on the scope so that you have the largest trace that will fit on the screen. If necessary, momentarily switch CH1 on the scope to GND and reposition the trace so that it starts at the exact left edge of the grid along the center line. You need to check this from time to time since the scope may drift. c. Adjust the phase controls until the phase is 0 at the left edge of the grid d. Switch the oscillator out. Then move on to the next oscillator and repeat. You will probably only need to make the trace adjustments from part (b) every 5 minutes or so. 2. Setting the amplitudes Signal shape depends on the relative amplitudes of the harmonics. In this first example, you will set oscillator 2 to have 1/2 the amplitude of 1, oscillator 3 to have 1/3 the amplitude of 1, oscillator 4 to have 1/4 the amplitude of 1, and so on up to oscillator 9, which will have 1/9 the amplitude of oscillator 1. Make sure that you do not change any of the phases set above during this process. You can make this task considerably easier by setting the amplitude of 1 (which is called the fundamental) to some easy to remember voltage. All of the synthesizers have fundamental oscillators which are capable of producing amplitudes of at least 1.0 V. By far the easiest thing to do is just set it to be exactly 1.0 V, which makes all of the other amplitudes easy to set. The 7th harmonic, for example, is to have 1/7 the amplitude of the fundamental it should be set at 1/7 V = 0.14 V Switch in each oscillator, one at a time, and set it to its appropriate amplitude. Use whichever volts/div setting allows you to set the amplitude most accurately. 3. Synthesizing a Complex Signal a. With all of the other oscillators out, switch in the fundamental, look at it on the screen, and listen to it one the headphones. Draw the fundamental (include units on axes): Lab 7: Synthesis

85 b. Add the second harmonic. Notice that immediately after switching in this higher signal you can hear it as a separate tone, but that after a few seconds it tends to fade into the overall sound of the fundamental. Observe the effect of this addition on the shape of the signal on the scope (which part of the signal is higher than the fundamental, which is lower?) Draw the sum on top of the fundamental above. c. Now add the third harmonic, again noticing how the sound stands out at first but then tends to blend into a single tone with the lower harmonics. Observe the effect on the shape (which part of the signal is higher than the fundamental, which is lower?). d. Continue this process, slowly adding new harmonics, listening to the sound and looking at the shape. Notice that by the time you get up to the 9th harmonic, the individual character of the first four or five harmonics is completely lost; they are no longer heard as separate tones but only contribute to the timbre of the fundamental pitch, which is now bright, almost buzzy. e. You should also see how this arrangement of nine harmonics is well on its way to producing a characteristic saw tooth shape. If you continued adding harmonics in this way, you would eventually get to a shape like this: f. Take the headphones off for a moment and then put them back on. You should hear a single bright tone rather than a series of tones. Because of the unnaturally sudden cutoff of harmonics at nine, the top two or three harmonics may not totally blend in. For your convenience, do not change any of the amplitude or phase settings until part 6! Show your sawtooth wave to your instructor for grading. 4. Synthesizing a Square Voltage It is possible for the amplitude of a harmonic to be zero. If it is, the harmonic is said to be missing. A square voltage, which you have heard before, is the same as the sawtooth except that it contains only odd harmonics all the even harmonics are missing. a. Switch out all of the harmonics except the fundamental. Listen with the headphones. Draw the fundamental: b. Now add back the third harmonic, Draw the sum on top of the fundamental in the above graph paper. Notice also what has happened to the overall amplitude. You have added a new signal, making the sound louder, and yet the peak-to-peak amplitude is actually a bit smaller. c. Continue adding the odd harmonics one at a time. Watch and listen as the signal approaches a square voltage. Lab 7: Synthesis

86 d. Show your instructor your square wave. e. Compare the sound of your synthesized square voltage to the square voltage from one of the fundamental oscillators (You need to switch out all of the harmonics except the fundamental and then use the function switch at the top of the panel) 5. Synthesizing a new wave. Switch out all of the odd harmonics (including number 1) and switch in all of the even harmonics. What is the result? Why? (Hint: Note that oscillator 2 is now the fundamental; make a table showing the harmonic number, the amplitude of the harmonic, and An/A1.) 6. Synthesizing a Triangle Voltage A triangle voltage is similar to the square voltage in that it has all odd harmonics, only at much lower levels relative to the fundamental. Also, some of the harmonics have negative amplitude, which is different from both the saw tooth and square voltages. Synthesize your own triangle voltage. The relative amplitudes of the first nine harmonics are given below. All of the even harmonics are missing. Harmonic Number Amplitude Relative to Fundamental /9 5 +1/ / /81 Show your triangle wave to your instructor. Compare the sound of your synthesized triangle voltage to the triangle wave from one of the fundamental oscillators. While listening to your synthesized voltage, try changing the phases of some of the harmonics. Notice that the phases have a very dramatic impact on the shape of the signal, but very little effect on the sound. This is why phase is usually ignored in describing a particular sound. This is also why shape is not the most useful way of describing sounds. Far and away the most important factors in determining tone quality are the relative amplitudes of the harmonics. 7. Relative harmonic amplitudes for some complex waves Wave Contributing Harmonics Relative Amplitude of the n-th harmonic: A n / A 1 Sawtooth All ±1/n Square Odd 1/n Triangle Odd 1/n 2 Wave types triangular sawtooth rectangular harmonics Relative amplitude Lab 7: Synthesis

87 2 0-1/ /9 1/3 1/ / /25 1/5 1/ / /49 1/7 1/7 7B: ADDING SOUND SIGNALS INTRODUCTION Both loudness and pitch are complicated subjective phenomena. The purpose of this lab is to give you some direct personal experience of some of the hearing phenomena. This lab should also give you more practice using the oscilloscope as a measurement tool. NOTES ON THE EQUIPMENT At your station you will find some of the equipment which has already been hooked up for you: a dual-function generator, a headphone box, an oscilloscope, and a frequency counter. The function generators can be connected to a mixing amplifier, which combines (adds) two signals so you can hear them at the same time. The combined signals are displayed on channel 1 of the oscilloscope (and audible through the headphones). You can control the frequency and amplitude of each independently, or eliminate them from the combined signal by selecting "OUT." Another thing you need to know is that the mixing amplifier can only put out so much power. If you ask for more power, the amplifier will simply clip off the top and bottom of the output signal, which you will hear as a nasty buzzing sound. You can solve this problem by turning down the amplitudes of the generators. DECIBELS As you should know from the previous lab, a decibel is a number (with no units) which compares two numbers. A decibel is always a comparison to some reference value, and is meaningless unless you know what that reference value is. A typical reference value for sound pressures is pressure of the softest sound the average person can hear. To get the decibel value, you divide your value by the reference value, take the logarithm, and multiply by 20. db = 20 log (p/p 0 ) For example, if a sound pressure is 2000 times the reference pressure, it has a value of db = 20 log (p/p 0 ) = 20 log (2000) = 20 x 3.3 = 66 db, which is about the level of sound near a busy street when compared to the threshold of hearing. Lab 7: Synthesis

88 With voltages, it is common to take the largest typical voltage as the reference, which means that the other voltages will be a negative number of decibels in comparison. For example, if the reference signal has amplitude of 3 volts, then a signal with amplitude of 1 V is db= 20 log (V/V 0 ) = 20 log (1/3) = 20 ( 0.477) = 9.5 db compared to the fundamental. When you put a number less than 1 into your calculator and push log, it will automatically give you a negative number, but it is important that you know what positive and negative decibel mean. PITCH AND SOUND LEVEL In a rough way, we can say that perceived pitch corresponds to frequency. However, a number of other factors besides frequency can affect pitch perception. In The Science of Sound, Rossing discusses several of these factors, such as the duration of the sound, whether or not other sounds are present, etc. Many observers report changes in pitch if the loudness of the sound changes dramatically. However, this effect differs from person to person. Use one generator to see if you notice such pitch changes. 1. Use the scope to set the frequency to exactly 1000 Hz: figure out the period of a 1000 Hz signal and then adjust the frequency knob of Generator 1 until you have that period on the scope. Listen to the sound on the headphones, listening particularly to the pitch. Then change the amplitude of Generator 1. Do you notice any change in pitch? If so, is the softer signal higher or lower in pitch than the loud one? 2. Repeat this procedure for 200 Hz and 10k Hz. Each time, set the frequency exactly using the scope. 3. Rossing says that most people hear a rise in the pitch of high frequency sounds as they get louder, but a drop in pitch of low frequency sounds as the get louder, with little noticeable change at middle frequencies. Do you follow this pattern? MASKING The presence of a loud sound can make it impossible to hear another softer sound. This is called masking. It is easier for a sound to mask a sound of similar frequency than a sound of very different frequency. Also signals at lower frequency can better mask a weaker signal at higher frequency than vice versa. Follow the steps outlined below: 1. Use the scope to set the frequency of the Generator 1 to 500 Hz. This will be your masking signal. Then set it "OUT". Set the other generator (Generator 2) to produce a 500 Hz sine signal which is down 30 db from the other generator. It means the amplitude of this signal is 10 (30/20) = = 32 times smaller than that of the masking signal. Can you hear the softer tone? Lab 7: Synthesis

89 2. Turn the frequency dial of the Generator 2 all the way up, which will allow you to hear the soft, high tone. 3. Turn the Generator 1 back IN. Then reduce the frequency of the second generator until you can no longer hear the soft tone. If you do not hear both signals when they are widely separated in frequency, repeat steps 1 and 2 with the two signals differing in power by 20 db rather than 30 db. 4. What is the frequency difference between the two generators? Please, indicate what the difference in power of the two signals is. BEATS When you are presented with two different pitches at once, what you hear depends on how different the two frequencies are. If the frequencies are very different, you will hear the two frequencies as separate tones. However, if the two frequencies are only different by a few Hz, you will hear only one pitch which gets alternately louder and softer, called beating." Actually, the beat frequency (the number of times per second that the sound gets loud) has a simple relationship to the frequencies of the two pitches which are being added. If you have two tones, f 1 and f 2, which have almost the same frequency, and you add them together, you hear a single tone half way in between which is beating at a rate f b = f 2 f 1. HEARING BEATS 1. To hear beats the most clearly, you need the two tones to have the same amplitude. Set you oscillators so that each one produces a 1000 Hz tone (set as accurately as you can with the scope) with amplitude of 5 V peak-to-peak. Then turn them both on at the same time (turn down the headphone volume to eliminate distortion if you need to). You should clearly hear the beating, and you should see it on the scope, too. If you don t, ask your instructor. 2. Vary the frequency of one of the oscillators. Make it so that the tones are very far apart and then slowly bring them together. When you get into the range where you hear beating, notice that the closer the frequencies are together, the slower the beats. 3. Determine the beat frequency: Adjust the oscillators so that you hear beats which are fast but which are still slow enough for you to count. Count the number of beats that occur in 30 seconds, and divide by 30 to get the number of beats per second. Is this the same as the difference in the two signal frequencies? 4. To measure frequencies accurately, you can use either (a) the frequency counter, or (2) oscilloscope. Turn the frequency counter on. This measurement has to be very accurate, so you need to find the button on the counter labeled GATE (sec.) and switch it to the setting called This tells the counter to count the signal for 10 seconds, which allows it to tell you the frequency to the nearest 0.1 Hz. Measure and record frequencies for both generators to that precision. To measure the frequency using the oscilloscope, carefully measure the period T in the oscilloscope, and the frequency is 1/T. Lab 7: Synthesis

90 5. Now subtract the two frequencies. Is the difference in the two original frequencies the same as the beat frequency? 6. Since you are done using the 10 second gate, switch the frequency counter s gate switch back to the 1 second position. Fused, Rough, and Separate Tones Any two frequencies are characterized by the difference between the two frequencies. In science, differences are often notated using the capital Greek letter Δ (delta). So the difference between two frequencies f 1 and f 2 is written Δf = f 2 f 1. If the two frequencies are sufficiently close together, you hear a single tone which is beating at a rate equal to Δf. This is only true, however, if the frequencies are very close together. If the frequencies are very different, you hear them as two completely separate tones, and there is no beating. Somewhere in between very close together and very far apart is a region of transition. In this region, the tones are fused into a single tone which has a rough quality, and it is not possible to hear distinguishable beats. You can use your lab set-up to determine just what it means for two sounds to be close together or far apart. 1. Set both generators to within a few Hz of 1000 Hz. You can do this quickly using your WAVETEX frequency counter. 2. Listen to the tones. As you listen, turn up the frequency on one generator to about 2000 Hz. Notice that you can clearly hear two distinct tones. Now slowly bring the upper frequency back toward 1000 Hz. As you do this, keep your attention fixed on the lower tone. Eventually you will reach a point where you can no longer hear the lower frequency as a separate tone. Instead, you will hear a single sound with a rough texture. When you get there, stop and record the frequencies of the two tones. Calculate Δf. This difference is called the fusion frequency. (Record your data in the table below) 3. Listen again, and keep lowering the upper frequency until you can hear beats. This means you can hear a fluttering in the tone at a definite rate, though it will be too fast for you to count. Once again, record the frequencies and calculate Δf. This is the maximum beat frequency. (Record your data in the table below) 4. You have now determined the fusion frequency and the maximum beat frequency for a center frequency of about 1000 Hz. (The center frequency is the frequency half way in between f 1 and f 2.) Repeat this experiment at 200 Hz and at 5000 Hz. That is, determine the fusion frequency and the maximum beat frequency when the center frequency is around 200 Hz and around 5000 Hz. 5. Is the maximum beat frequency about the same at each center frequency? Are the fusion frequencies the same at each center frequency? Read the pages in PRELAB from The Science of Sound by Rossing. Does your experiment agree with the text? Center frequency (Hz) Fusion frequency (Hz) Beat Frequency (Hz) Lab 7: Synthesis

91 PRELAB8. ANALYSIS OF COMPLEX SOUNDS Amplitude, loudness, and decibels Last week we found that we could synthesize complex sounds with a particular frequency, f, by adding sine waves from the harmonic series at (f, 2f, 3f, 4f, ). We can reverse the process: a complex sound with particular frequency can be analyzed and quantified by the amplitude spectrum: the relative amplitudes of the harmonics. For the last two weeks we have used the amplitudes to represent the different waves. We can represent these in a table (the choice for the amplitude of the fundamental wave to be 2V is completely arbitrary): Another way to represent the relative amplitudes, which is used by PRAAT, is to graph the power for each harmonic. Since the sound power is proportional to the square of the amplitude for each harmonic, these numbers often get quite small for the higher harmonics. For this reason, the power is often expressed as a decibel. The power for each harmonic in decibels (db) is: Relative Power (db) = P n = 10 log 10 [ (A n /A 1 ) 2 ] = 20 log 10 [ (A n /A 1 )] Since the logarithm of zero is negative infinity, the Relative Power (db) for the even harmonics is "Error". We can display the harmonic power spectrum as a bar graph below. Note that the bars for the even harmonics are missing since these harmonic are "missing" from a square wave. Acoustic PRELAB 8: page 1 091

92 Analysis of a Triangle Wave Predictions based on known relative amplitudes Try to make a table similar to Table 1 for the triangle wave you synthesized several weeks ago. Assume the amplitude of the fundamental wave to be 2V. Remember that the even harmonics had zero amplitude and the amplitudes for the odd harmonics were given by: A n = A 1 /n 2. Fill up the following table and mark on the graph their relative sound intensity db level. Now graph your results for the relative power in db for each harmonic. Note that the fundamental is 0 db by definition: Acoustic PRELAB 8: page 2 092

93 8. ANALYSIS OF COMPLEX SOUNDS AND SPEECH ANALYSIS Amplitude, loudness, and decibels A complex sound with particular frequency can be analyzed and quantified by the amplitude spectrum: the relative amplitudes of the harmonics. Our goal today is to understand how the pitch, loudness, and timbre of a sound are represented in a spectrum. However, rather than use an analog Band Filter, we will use the PRAAT program, which has powerful analysis software built in to calculate and graph the spectrum of a recorded sound. A. Warm up : Analysis of a measured sine wave Before using PRAAT to analyze a more complex wave, first analyze a simple sine wave to better understand frequency spectra. Take the output from the frequency synthesizer ground (black socket) to the headphone box ground input (also black socket). Use another banana plug cable from the top red 8 output socket of the synthesizer to the other input on the headphone box. Use the synthesizer to generate a sine wave (use oscillator 1-Left, change the function using the switches at the top left corner of the synthesizer box; make sure to switch all other oscillators out). 1. Send the signal from the headphone box to the microphone input of the computer using the cable supplied. When you start the program, two windows should have appeared: a Praat objects window and a Praat picture window. Recall how you record a sound with PRAAT: choose Record mono Sound... from the New menu in the Praat objects window. A SoundRecorder window will appear on your screen. 2. Use the Record and Stop buttons to record a few seconds of your sine wave signal. You can repeat this several times until you are satisfied with your levels. Adjust the gain of the synthesizer output so that the level meter of the SoundRecorder window registers and is "green", or occasionally "yellow". Type in a name for your sound file (e.g., "sine") in the text box below the Save to list: button. hit the Save to list: button and the text string "sound sine" should appear in the Praat objects window that indicates the file where your sound is recorded. 3. With the name of the file highlighted in the Praat objects window, hit the Edit button to see the waveform that you just recorded. To simplify the display (if it is not already done), go across the menu of the Edit window (that should be titled "Sound sine"), Spectrum menu, pull down, Show spectrogram deselect (i.e., if a check mark is shown, select it. If no check mark, exit without doing anything). Pitch menu, pull down, deselect Show pitch Intensity menu, pull down, deselect Show intensity Formants menu, pull down, deselect Show formants Pulses menu, pull down, deselect Show pulses. 4. Set up by going to the Spectrum menu, going to Advanced Spectrum settings..., and in Pre-Emphasis (db/out) type "0.0" if it is "6.0". This latter option will be used when we analyze speech in the next lab. Acoustic Lab 8: Synthesis and Speech analysis page 1 093

94 5. Check that you have a clean sine wave by using the View menu and combinations of Zoom in (<CTRL>-i), Zoom out (<CTRL>-o), or selecting and Zoom to selection. 6. Now generate a frequency spectrum of this input waveform. From the Spectrum menu, choose View spectral slice. A Spectrum Slice window should appear as shown in the graph below. Frequency spectrum of a sine wave. This frequency spectrum shows frequency in Hz along the horizontal axis and power of that frequency component on the vertical axis. Use the cursor to click at the tip of the first frequency peak (the fundamental). The frequency of the peak is given as a red number at the top of the display (in this case, 442 Hz, very close to the 440 Hz sine wave expected from the synthesizer). Its power level is given as a number at the left hand side. Note that the frequency spectrum of a pure sine wave is dominated by the power of the frequency of the sine wave. The other harmonics simply show that this is not a perfect sine wave; an ideal one would have only the first peak. B. Analysis of a measured triangle wave Now use the synthesizer to generate a triangle wave (use oscillator 1-Left, change the function using the switches at the top left corner of the synthesizer box; switch other oscillators out). a) Follow the same procedure as with the sine wave to record the wave with PRAAT. In the window obtained from using Edit, use the Spectrogram menu, View spectral slice, to generate a power spectrum of the triangle wave. It should appear similar to the example below. Frequency Spectrum of a triangle wave. b) Notice that the frequency power spectrum is made up of a series of peaks. Each peak represents one of the harmonics. You can use the mouse to click at the tip of each peak. The number above the display shows the frequency and the number to the left the power (in decibels) for that frequency. In the example above, the frequency peak of 1329 Hz has a power of 31.5 db. Record the frequency and power for each peak in the table below. To obtain the last line of the table, you need to subtract the power level of the fundamental (P 1 ) from each harmonic power level (P n ). Acoustic Lab 8: Synthesis and Speech analysis page 2 094

95 c) How do your relative power numbers compare to what you calculated in the Table 2 of this PRELAB? d) Return to the initial Sound object window obtained from the Edit button. Zoom in to a few cycles of the triangle wave. Use the mouse click on a peak and an adjacent valley in order to measure the peak-to-peak amplitude of the triangle wave. Pascals. e) Now reduce the amplitude of the oscillator; record the sound again using PRAAT. Measure the peak-to-peak amplitude of the sound signal: Pascals. Record the power levels of all of the harmonics in the table below. f) Compare your results to those from Table 3: Do all of the power levels change? By how much do they change? Acoustic Lab 8: Synthesis and Speech analysis page 3 095

96 Do they change by different amounts or the same amount? Do the relative power levels change? g) The change in the power levels for each harmonics present should be the same as the change in the overall power level, which you can calculate from your measurements of the new and old amplitudes. Change in Power (db) = 10 log 10 [ (A new /A old ) 2 ] Are your measurements consistent with this observation? C. Analysis of a superposition of a triangle wave and another harmonic, not present in the triangle wave Now switch in oscillator 4 (summing it to the triangle wave). Turn its amplitude halfway up of the oscillator 1 level (make this adjustment by recording signal for oscillator 4 first, before summing the two signals. You will probably have to go through several cycles of recording, adjusting amplitude, recording, adjusting amplitude, etc.). Note that this harmonic was not present in the triangle wave signal a) Record the sound signal. How do the sound signal and the power spectrum change? b) Change the phase of oscillator 4 by 180 and record the signal again. Does the sound change? Qualitatively, how do the sound signal and power spectrum change? D. Analysis of a superposition of a triangle wave and another harmonic, already present in the triangle wave Now switch out oscillator 4 and switch in oscillator 3. Adjust its amplitude (by recording the signal separately first) to be approximately half of that from oscillator 1-Left. Note that this harmonic was present in the original triangle wave. a) Record the sound signal. Qualitatively, how do the sound signal and the power spectrum change? Acoustic Lab 8: Synthesis and Speech analysis page 4 096

97 b) Change the phase of oscillator 3. Does the sound change? Record the signal again. Qualitatively, how do the sound signal and power spectrum change? E. Discussion of expected effects. If the Power Spectrum of the sound changes, you can probably hear the difference. If the Power Spectrum does not change, you can probably not hear the difference. For example, you should have found that changing the phase of the fourth harmonic did not change the power spectrum, but changing the phase of the third harmonic did matter. Why? The third harmonic was present in the original sound. Phase may matter when you add sounds that are close together (or the same) in frequency. Switch Oscillator 1-Left back to a sine wave. a. Now switch in or out additional oscillators, one at a time, listen to the differences and record the signal after each switch. The sound should change every time, and new peaks should appear or peaks should disappear in the Power Spectrum. If you see no change, the amplitude for that oscillator is probably close to zero. b. Changing the phases of any of the oscillators should not change the sound or the power spectrum. The lesson is that the shape of the power spectrum determines the timbre. We can now give very precise ways to define how to measure sounds made up of a harmonic series using the Power Spectrum: Pitch: the frequency of the fundamental of the harmonic series (f 1 ). Timbre: the relative heights of peaks in the power spectrum (compared to the fundamental) Loudness: the overall heights of the peaks in the power spectrum SPEECH ANALYSIS In the previous section of this lab we found that we could analyze a complex sound with particular frequency using the power spectrum. We gave very precise ways to define how to measure sounds made up of a harmonic series using the power spectrum: In this part of the lab we will try to better understand the acoustical features of speech. Speech sounds, called phonemes are classified either as vowels or consonants. Frequency analysis of a vowel sound always reveals a clear harmonic spectrum, with a pitch (fundamental frequency) varying from individual to individual, with the pitch of female speakers being on the average twice as high as that of male speakers. The pitch is determined by the vibration of vocal cords. The features that distinguish specific vowels are called formants. Formants are vocal tract resonances, the frequency of which depends on such effects as the tongue height or tongue advancement. The first two formants are particularly important in speech recognition. The frequency of the first formant increases as we open our mouth wider and lower the tongue. Acoustic Lab 8: Synthesis and Speech analysis page 5 097

98 The frequency of the second formant increases as we advance our tongue (see positions of formants for selected vowels in the attached figures pp19-21.). Frequencies of formants change only within 15% between female and male speakers. Most consonants do not have harmonic frequency spectra. The features that distinguish consonants are periods of silence, voice bars, noise, and the consonant s effects on the frequency spectra of adjacent vowels. Consonants are classified by: (i) manner of articulation, (ii) place of articulation, and (iii) as voiced or unvoiced. The consonant types, classified by manner of articulation, include: 1. plosive or stop (p, b, t, d, k, g) produced by blocking the flow of air somewhere in the vocal tract 2. fricative (f, s, sh, h, v, th, z) produced by constricting the air flow to produce a turbulence 3. nasal (m, n, ng) produced by lowering the soft palate 4. liquid (l, r) generated by raising the tip of the tongue 5. semi-vowel (w, y), always followed by a vowel. A. Looking at a power spectrum of a spoken sound The power spectrum is quite complicated; nevertheless, we can spot some of the same features as those found in the spectrum of a simpler sound (such as a triangle wave): a. Identify on the graph the position of the peak corresponding to the fundamental (hint: it is the biggest peak on the graph); label it as "1." Note that the horizontal position of the peak shows the frequency corresponding to the peak. The fundamental peak is about halfway between 0 and 440 Hz; therefore, a good estimate for the frequency of the fundamental would be 220 Hz. b. Now label the other peaks (important: label only the peaks that correspond to harmonics of the fundamental frequency - ignore the small peaks before the first big peak or between the other big ones). You should be able to find 11 distinct harmonic peaks. What are the prominent harmonic numbers for this sound? (with peak heights above the 0 db line). Acoustic Lab 8: Synthesis and Speech analysis page 6 098

99 c. Generally, the lowest harmonics have higher power. Do you notice that, some higher harmonics have greater power than some lower harmonics? Give some examples: d. We will now use PRAAT to record and analyze your own vowel sound: Use the microphone as input to the computer. Following the same instructions as in the previous lab, record a few seconds of someone in your group singing the continuous vowel "aw" (the sound of "a" in "call"). Use the Zoom controls as before in zoom in to see a number of consistent cycles with repeating shape. Drag and select an approximate 0.1 seconds of the sound. In the Spectrum menu, choose View spectral slice. In the Spectrum slice window, click and start dragging at the left and select the approximate region of 0 to approximately 3000 Hz. From the View menu, use Zoom to selection to magnify the interesting frequency region. Below is shown an example frequency spectrum of someone saying the vowel "aw": Frequency spectrum of the vowel "aw". What is similar about your "aw" power spectrum and mine? What is different? B. Quantifying the power spectrum of a spoken sound Formants. We want to understand what makes each sound identifiable. We know that different periodic sound signals have different harmonic power spectra. Let's look at one way to quantify the power spectrum. The sound power measured by the microphone is the output of the vocal cords as filtered through the rest of the vocal tract and as transmitted out of the mouth. The net output of the vocal cords (called "Source" in the table) represents the combined effects of the vocal cords and the transmitted efficiency of the mouth: This Source spectrum resembles triangular wave form, which diminish in amplitude 1/n 2, i.e. 12dB/octave. The mouth radiates more efficiently at high frequencies. This increases the transmission power by 6dB/octave. The resulting change is 6 db per octave {net} = 12 db per octave {vocal cords} + 6 db per octave {mouth}. Acoustic Lab 8: Synthesis and Speech analysis page 7 099

100 Therefore (other than telling us the pitch), the Source spectrum does not tell us much about what distinguishes the different sounds of speech. What changes when we talk is the filter function. This is a function, representing relative power changes as a function of frequency. It describes how we modify the sound produced by the vocal cords. The filter function changes as we shape our mouth, adjust tongue position etc to pronounce a given sound. In particular, a sound corresponding to a specific vowel is characterized by a specific filter function that changes little from one individual to another. A filter function for a given vowel, plotted as a function of frequency, exhibits characteristic maxima, called formants. The filter function for your sound can be approximated by subtracting the "Source" from "Power" for each harmonic: Filter function (db) = Observed Power (db) Source Power (db) The PRAAT program can be set up to do this automatically for you. Set it up: in the Spectrum menu, go to Advanced spectrogram settings..., and under Pre-emphasis (db/oct.), type in "6.0" if it is not already at this value. Re-analyze your "aw" vowel with this setting. Table 1a: Here is a sample table for the example spoken "aw" sound: Harmonic No Frequency [Hz] = f n Power [db] = P n Relative Filter Magnitude [db] = P n P 1 Approx. Formant Position (peak) F 1 F 2 Now Use the mouse and cursor to measure the power of each harmonic in your aw sound. a) First determine the fundamental frequency for your sound by moving the cursor by clicking the mouse at the tip of the first large peak in the spectrum. The red numbers that appear show the frequency and power (in db) for that peak. Record these numbers in the first column of the table following. b) Now move the cursor so that it is aligned with the next big peak in the spectrum. Make sure that the peak really corresponds to a harmonic of the fundamental (the frequency should be very close to a whole-number multiple of the fundamental frequency that you recorded from part a.) Record the frequency and power of each peak in the table following: Table 1b: Measured power spectrum for your "aw" sound Harmonic No Frequency [Hz] = f n Power [db] = P n Relative Filter Magnitude [db] = P n P 1 Approx. Formant Position (peak) Acoustic Lab 8: Synthesis and Speech analysis page 8 100

101 To get the relative filter magnitude, subtract the filter magnitude of the first harmonic from the filter magnitude of the other harmonics. Scan across the row of "Relative Filter Magnitude", and "peaks" or areas of enhancement should be evident. The points of greatest enhancements can be identified as the approximate positions of formants (and in the example of Table 1a, are observed roughly at harmonic no. 7 and 10). This method of obtaining the filter function is only an approximation because we are only obtaining the function at a small set of frequency values (the harmonics). The true filter function is a continuous function of frequency; we need to draw a smooth line between the points in our approximation. As mentioned earlier, the peaks in the filter function are called formants. Two are apparent in Table 1a (at harmonic numbers 7 and 10). We can compare our numbers to those in Rossing's The Science of Sound [shown in brackets]. Table 1c: Formants obtained from the Power Spectrum of spoken "aw" sound: Formant number Frequency Relative Magnitude (db) F [730] -1.2 [-3] F [1090] -3.5 [-5] The PRAAT program will also calculate the frequency locations of these formants by performing a fit to the filter function across all frequencies. Click anywhere in your waveform window (instead of selecting), and under the Formant menu, choose Formant listing. The first five formants should be given. Table 2a gives the formants calculated by PRAAT for the "aw" sound analyzed in Table 1a. These differ slightly from the numbers in Table 1a, but they were much easier to obtain! For bandwidth (the "width" of each formant peak; some are wider than others), use the Formant menu, Get bandwidth..., and fill in for which formant you want it calculated. Do the same for your own recording of your "aw" sound in Table 2b. Table 2a: Formants for spoken "aw" sound, corresponding to the power spectrum recorded in Table 1a Formant number F 1 F 2 F 3 Frequency [Hz] Bandwidth [Hz] Relative Magnitude [db] Table 2b: measured formants of your "aw" sound, corresponding to your power spectrum recorded in Table 1b Formant number Frequency [Hz] F 1 F 2 F 3 Bandwidth [Hz] Acoustic Lab 8: Synthesis and Speech analysis page 9 101

102 How do these peaks compare with your numbers from the power spectrum (Relative Filter Magnitude line) recorded in Table 1b? 3) Try a few more vowel sounds (recommend "ee" as in "heat", "u", as in "blue", "i" as in "bit", etc). Record your data in the tables below. Record formant frequencies only let the program do the calculations for you. Compare your results to the "averages" found in the attached Table 15.3 from Rossing s textbook. Write your comments (does it agree/disagree with the averages ) next to the relevant Table. Table 3: measured formants of your /i/ ( ee ), as in heat, sound Frequency [Hz] Bandwidth [Hz] Formant number F 1 F 2 F 3 Table 4: measured formants of your /u/, as in blue, sound Formant number Frequency [Hz] Bandwidth [Hz] F 1 F 2 F 3 Table 5: measured formants of your.. sound Formant number Frequency [Hz] Bandwidth [Hz] F 1 F 2 F 3 Table 6: measured formants of your.. sound Frequency [Hz] Bandwidth [Hz Formant number F 1 F 2 F 3 Acoustic Lab 8: Synthesis and Speech analysis page

103 C. Vowel/Consonant Recognition using Spectrograms (optional) We can also the PRAAT program to plot spectrograms of our waveforms, that is, frequency plotted versus time with the intensity or level of grey scale representing the relative power of contributing frequencies. Start with recording several vowels, e.g. "ooo","ee","aw","u". In the window obtained with the Edit button, under the Spectrum menu, select Show Spectrogram. A frequency spectrogram should show up below the waveform as shown below. You can zoom in and out as before. Spectrogram of the vowel "aw". Regular dark, horizontal bands, representing contributing harmonics should be clearly visible. In the example above, the cursor in centered at approximately the third formant at 2560 Hz. The power spectrum (spectrogram structure) should look different for different vowels. To identify different parts of the spectrogram click and drag mouse on the waveform display to select a portion of the recording, and then play back the selected part. In addition, for the recorded vowels, you can also display their formants. To do this, under the Formant menu, select Show formants and they will appear as red dots overlaid on the frequency spectrum. In some cases, they will change slightly with time, but generally should be relatively constant for vowels. The figure at right shows frequency spectrogram of spoken "aw" vowel, with formants overlaid. Record the spectrograms and formants of several vowels and then print them out. Unfortunately for these spectrogram menus, printing is not directly supported. The way around it: Open Microsoft Word. Go back to the window showing the spectrogram you would like to capture and make sure it is "active" by clicking on it. Do a "screen shot" of that window by <ALT>-Print Screen. Go back to Microsoft Word, and under the Edit menu, select Paste, and the spectrogram should now be inserted in to the Microsoft Word document. Acoustic Lab 8: Synthesis and Speech analysis page

104 Collect the spectrograms of a number of vowels in the same Word document, print it out, and identify the recorded vowels on each plot. Consonants also have distinctive features that define them on a spectrogram. These do not include formants like for the vowels, unless a consonant is combined with a vowel and one observes formant transition. Formant transition is a formant frequency change associated with phonetic transcription moving from a vowel to a consonant. Consonants are classified by manner of their articulation. This manifests itself by such features of a spectrogram as: periods of silence, burst, noise, and presence of voice bars. Consider stop type consonants as an example. The spectrogram should reveal a period of silence, followed, right after, by a powerful burst. Finally, a detection of voice bars would indicate whether the stop sound is a voiced consonant or not. Fricative consonants are characterized by noise, recognized on a spectrogram by its aperiodic and irregular pattern (i.e., "ssssh" or "sss" should fill all frequencies). Affricates (a combination of stops and fricatives) are described as having a brief period of silence, followed by a weak burst, and some noise. Record and make spectrograms of different consonants pronounced between two a vowels. Zoom into the relevant part of the waveform and spectra in each case. Suggested combinations are listed in the Table below. Print two or three of the relevant graphs (using the procedure described earlier) above as an illustration. Identify and mark on the spectrograms features like periods of silence, bursts, voice bars and list them in the Table below (see an example for /aba/, with an example spectrogram shown below). Consonant type Voiced Voiceless Stops /aba/ = "boy" silence, voice bar, burst /apa/ = "pot" Fricatives /aza/ ="measure" /asa/ = "sheet" Affricates /aja/ = "judge" /aca/ = "church" Frequency spectrogram with formants overlaid of the vowel-consonant combination "/aba/".the first "a" vowel is clearly seen, then a period of silence with a faint voice bar, a sharp burst, and then the second "a" vowel. Acoustic Lab 8: Synthesis and Speech analysis page

105 I. INTRODUCTION PRELAB 9: TRANSDUCERS Transduction is a process that changes energy (or information) from one form to another. Microphones transduce acoustical energy into electrical energy (voltage); loudspeakers transduce voltage back into sound. Tape deck record and playback heads transduce electrical signals into magnetized regions on magnetic tape, and vice versa. CD players transduce the information recorded on a CD back into electrical energy. Thus, transducers are a crucial part of the study of acoustics. A microphone is a transducer that converts acoustic energy to electrical energy. There are five key types of microphone you may use Moving coil (dynamic microphone) Ribbon Condenser Electret Crystal All employ different mechanisms to convert sound energy to electrical energy. Hence all have different advantages and disadvantages. Parameters of microphone include sensitivity, frequency response and directional response. Since the microphone operates into HIGH ELECTRICAL IMPEDANCE such that there is (approximately) zero current and the VOLTAGE is the output variable, it is not expected to deliver electrical power. Consequently, it is conventional to talk of the "Open Circuit" response. The microphone may be sensitive to any combination of acoustic variables, but the simplest case is a microphone that is responsive to pressure input. II Microphone sensitivity A microphone s sensitivity (pressure sensitivity) is defined as the voltage generated in response choosing a microphone to a certain pressure input. The common unit is: M 0 (Volts/Pascal). For example: A microphone is exposed to 94dB SPL generates 50mV output, its sensitivity is 50mV/Pa because a SPL 94dB corresponds to an RMS acoustic pressure of 1Pa. An alternative unit for a microphone s sensitivity is expressed in logarithmic form, relative to a reference, i.e. db re 1 V/Pa. To converting to this scale, we have 20log(M 0 ) db re 1 V/Pa. The above microphone sensitivity is 20log(0.05)= 26 db re V/Pa or 26 dbv/pa. Question: A microphone is rated 52 db re 1 V/Pa. If an acoustic pressure of Pa is incident on the microphone, what is the open-circuit output voltage of the microphone? PRELAB 09: Transducer; Page 1 105

106 III FARADAY S LAW OF ELECTROMAGNETIC INDUCTION The magnetic field H, is measured in amperes per meter. However, the commonly called Magnetic field B is known as magnetic flux density that has the SI units TESLA. One Tesla is equal to or 10 4 Gauss. The magnetic field lines point away from N-pole (or North Pole) and point toward S-pole (or the South Pole). Typically, the Earth magnetic field is about 0.5 Gauss pointing toward Earth North Pole. Thus the Earth North Pole is a magnetic South Pole. Typical magnetic field strength of magnets of speakers and microphones is 0.1 T to 1 T. Pace makers should avoid magnetic field strength larger than 5 Gauss, and typical refrigerator magnets are about 50 G. The operation of most transducers is based on Faraday s Law, which says that if you move a wire through a magnetic field (or vary the magnetic field strength through a coil of wire), a voltage will be produced in the wire. The induced electromotive force (EMF) is equal to the rate change of the magnetic flux, i.e. E= Φ/Δt, where the negative sign arises from the Lenz s law, E is the induced EMF, Φ is the magnetic flux in [Tesla-m 2 ], given by (magnetic field B) (area of an electric circuit loop), and t is the time in [seconds]. The unit of EMF is [V]=[T m 2 ]/[s]. Example: A bar magnet is moved rapidly toward a 40-turn, circular coil of wire. As the magnet moves, the average value of magnetic flux density B over the area of the coil increases from T to T in 0.25 s. If the radius of the coil is 3.05 cm, the magnitude of the induced emf can be found as follows: Answer: The initial and final magnetic fluxes through the coil are i Bi A (0.0125T) (0.0305m) Tm B A (0.45T) (0.0305m) Tm f f The change of magnetic flux is ΔΦ=Φ f Φ i. The induced voltage over the 40-turn coil is N t 40 ( In a dynamic microphone, the induced voltage is small, and the induced voltage is of the order of mv (10 3 V) that needs amplification for an audible sound. Ribbon microphone uses Faraday s law for transduction. IV Condenser microphone f f t i i Two conductor sheets can hold charges between two plates. The amount of charge depends on the distance between these two sheets. When the sound wave exerts pressure on the condenser sheet, the distance change and the charge on each plate varies. This sets up electric current. This process converts sound energy to electric energy. What is the difference between a speaker and a microphone? 3 2 Tm ) / 0.25s 0.205V Acoustic Lab 09: Transducer; Page 2 106

107 I. INTRODUCTION 9: TRANSDUCERS Transduction is a process that changes energy (or information) from one form to another. A transducer is a device that converts energy from one form to another. Common examples include microphones, loudspeakers, thermometers, position and pressure sensors, antenna, etc. Transducers may include generators, turbine, photocells, LEDs (light-emitting diodes), and even common light bulbs. Microphones transduce acoustical energy into electrical energy (voltage); loudspeakers transduce voltage back into sound. Tape deck record and playback heads transduce electrical signals into magnetized regions on magnetic tape, and vice versa. CD players transduce the information recorded on a CD back into electrical energy. Thus, transducers are a crucial part of the study of acoustics. In this lab, you will measure the voltage generated by speaking to various kind of mike construction. The photo below shows a simple mike based on magnetic induction, a mike based on capacitive plates, a pc mike, a pc mike connected to an amplifier, and a speaker that can be used as a mike. Use an oscilloscope to measure the output voltage when you speak to the mike. II. FARADAY S LAW OF ELECTROMAGNETIC INDUCTION The operation of most transducers is based on Faraday s Law, which says that if you move a wire through a magnetic field (or vary the magnetic field strength through a coil of wire), a voltage will be produced in the wire. This physical fact is very useful in acoustics since sound is a mechanical vibration which can easily cause a small ribbon or membrane to move. Acoustic Lab 09: Transducer; Page 1 107

108 A. A SINGLE WIRE MOVING IN A MAGNETIC FIELD 1. Hook one of the long blue banana-banana wires directly from one of the scope inputs to scope ground. Set the scope to a sensitive setting, say 5 mv/div, and then move the wire back and forth between the jaws of your large horseshoe magnet. Try to keep the magnet as far away from the oscilloscope as possible, since its large magnetic field badly distorts the scope display. Notice that to generate a larger signal you need to move the wire faster. Also try moving the wire in different directions, for example back-and-forth between the jaws as opposed to parallel to the jaws. 2. Build a ribbon microphone. a. You will need: A strong horseshoe magnet A plastic frame Two clean 1/8 in. brass rods A strip of 1/2 mil aluminized (or copperized) mylar Masking tape b. Clean the brass rods with Emery-paper for better contact. Tape the rods onto the plastic frame so that they stick out from the frame on one side, which will allow for easy attachment of alligator clips. Stretch the mylar so that it runs through the central notch, over the rods (with the metalized side touching the brass rods), and tapes onto the back. The mylar should be as loose as possible, BUT IT MUST MAKE A GOOD CONNECTION WITH BOTH RODS. That s it. Simply connect the microphone to the scope using a shielded cable with alligator clips at one end. It may be necessary to pull out the 5 knob on the front of the Volts/Div knob to see the signal well. c. Hold the frame so that the ribbon is close to the magnet, talk into your microphone and look at the output on the scope. How big is the signal if you make a loud (but not shouting) sounds an inch from the microphone? Notice that the orientation of the ribbon with respect to the magnet is very important. Which orientation gives a highest output voltage (parallel or perpendicular to the ribbon)? d. Record yourself using your ribbon microphone: using an adaptor plug the end of the cable into the black cable that leads to the microphone input of the computer. Use the praat program to record. Play back the recording: How does it sound? e. Analyze the frequency spectrum of your recording (select a vowel) using the praat program: what do you notice about the frequency response of your microphone? Acoustic Lab 09: Transducer; Page 2 108

109 B. A COIL OF WIRE MOVING IN A MAGNETIC FIELD 1. Repeat the first experiment, moving a wire through the magnetic field of your large horseshoe magnet. Now wrap one of your long banana wires into a loop and moving ONE SIDE of the loop between the jaws of the magnet. The more turns of wire in the coil, the bigger the signal. How much bigger is the signal from a coil compared to a single wire? For this reason, many transducers use coils of wire having many, many turns. The most common example is the cone loudspeaker. In a microphone, the movement of the wire or coil causes a current to flow in the wire. In a speaker, a current is sent through the coil, which makes it move in the magnetic field. 2. In fact, a cone speaker is itself an excellent microphone. Hook your small speaker directly to your scope and talk into it. Notice the impressive signal generated with a relatively small magnet. How big is it? This is possible because of the large number of turns in the coil. Your speaker is acting as a moving-coil dynamic microphone, and this is also the principle used in moving-coil phono cartridges. Of course, it is also possible to hold the coil fixed and move the magnet. Such devices are called moving-magnet microphones and cartridges. III. Transduction using Capacitance Magnets are bulky and heavy, and stray magnetic fields can cause problems in surrounding electrical equipment. A number of transduction devices use other techniques. Often these transducers are simple and durable but low in quality, such as the carbon microphone in a telephone or piezo-electric phono cartridges. An exception is the Condenser Microphone which is rather difficult to make but can have an excellent response. Condenser is an old-fashioned word for a capacitor, a pair of conducting plates separated by an insulator. The plates are charged. When they are then moved back and forth, charge is forced off and on the plates, creating a current. Build a condenser microphone and see if you can get it to work! Your instructor will provide help. See the sample microphone at the demonstration station for hints. 1. Building a condensor microphone You will need: Small aluminum plate; 1 6-inch square 1 mil mylar sheets; a 3 to 4 inch square of 1/2 mil aluminized mylar; and masking tape Instructions: Tape the plain mylar to aluminum plate. Put a layer of masking tape all around the edge. Place the aluminized mylar square on top of this and tape it down metal side up (it does not need to be taped all the way around, and should be only Acoustic Lab 09: Transducer; Page 3 109

110 somewhat tight). Apply a piece of copper tape to the alumized mylar to make your electrical connection.. Connect the clips of your cable onto (a) the plate and (b) the copper tape attached to the diaphram. Connect the inputs to the scope. Try out your microphone. These should give signals large enough to distinguish between different vowel sounds on the oscilloscope. Do not try to record this signal with the computer the impendances of the two devices do not much. How big is the signal you can produce? Show your signal on the scope to your instructor. In general, the frequency response is better for a condenser mike than a ribbon mike. How would you explain this fact? In real condenser microphones, a battery is used to maintain a charge on the plates (here, a small amount of static charge is present), so it is important to TURN OFF real condenser microphones when you are done with them, or you run down the batteries. IV. Electret Microphone It is possible to use a plate which has a permanent charge on one plate. This is called an Electret Microphone, invented at Bell laboratories in 1962 by Gerhard Sessler and Jim West. A large condenser microphone run backwards is an Electrostatic Speaker. The externally-applied charge in condenser microphones is replaced by a permanent charge in an electret material. An electret is a ferroelectric material that has been permanently electrically charged or polarized. The name comes from electrostatic and magnet. An electret is made by alignment of the static charges in the material, much the way a magnet is made by aligning the magnetic domains in a piece of iron. Most microphones made today are electret microphones, estimated annual production at over one billion units. Nearly all cell-phone, computer, PDA and headset microphones are electret types. Use the alligator clip to connect an electret microphone to an oscilloscope. Talk, shout or whistle to it. What is the response? Please give comments to your observation. V. Use speaker as microphone The speaker on your lab bench uses magnetic induction to convert electric energy into mechanical energy and create pressure wave to produce sound. Speaker can also be used as a microphone. Connect the speaker input ports to the oscilloscope and speak to it. Can you describe what you see, What is the output voltage produced? Speak a constant ooo.. sound and adjust the time/second division to see the wave form. Change your tone to eee and describe what you have observed. Acoustic Lab 09: Transducer; Page 4 110

111 5/7/2012 PRELAB FOR FIRST OP-AMP LAB Electrical circuits are CIRCUITS 10: PRELAB AMPLIFIERS Every element must have at least two wires and the wiring must eventually come back to the power supply. Circuit diagrams are symbolic pictures of circuits. typical circuit diagram might look like this: S G In general, all branches of the circuit must eventually come back to the negative side of the power source. The negative side of the power supply is often called ground ("G"). Common symbols for circuit diagrams include: Resistor Battery (or power supply) Capacitor Inductor Ground S G Oscillating voltage source Operational Amplifier In complex circuits, putting in all the ground wires only makes the diagram harder to read. Using these symbols, the circuit diagram above could be redrawn as the left plot below. In general, it is assumed that all the grounds are hooked together. If you want to change the input supply, you can leave it out. In fact, it is usually just assumed that the input voltage is measured with respect to ground, so the simplified diagram is shown in the right plot: Input Some circuit elements have several inputs and require their own power supply (all of our op-amp circuits, for example). An operational amplifier is an amplifier circuit with very high open loop gain and differential inputs which employs external feedback for control of its transfer function or gain. A common op amp circuit is: Acoustic Lab 10: Amplifiers; Page 1 111

112 2 - S to scope G +15 V -15 V scope gr ound (pins 1, 5, and 8 on the op-amp are not used by us). All op amps require power, so there is really no need to show it. Using the ground symbol and leaving out the power supplies to the op amp will simplify things considerably. In addition, if we then assume that the input and output are both measured with respect to ground, we get a very simple, easy to read diagram: i nput - + out put Compare this to the first diagram of this circuit. The advantages of the simplified diagram are obvious. However, YOU are responsible for knowing that the op amp requires power and that the input and output devices must be hooked to ground. In fact, the input device, the output device, the power supplies to the op amp, and the side of the op amp must ALL BE HOOKED TOGETHER, TO A COMMON GROUND. In practice, to build a circuit, it is most important to understand how to un-simplify a schematic diagram. Try it: un-simplify this diagram. i nput - out put + Acoustic Lab 10: PRELAB Amplifiers; Page 2 112

113 Electric Current, Resistor and Ohm s Law When a voltage source (electric potential) is applied to a circuit, electrons move in the circuit. Electrons collide with atoms in the circuit elements and move in a zigzag path, creating a resistance to electron s motion. The resulting effect is that the current flow in a wire is proportional to the voltage applied to the wire. This is called the Ohm s law discovered by Georg Simon Ohm ( ): V IR where the voltage V is measured in [Volts] or [V], the current I is measured in [Amperes], or [A], and the resistance R is measured in [Ohms] or [Ω]. When two resistors are connected in parallel, the effective resistance becomes R 1 R 2 we find: R R R Thus if R 1 =R 2, we find R=R 1 /2. When two resistors are in series, the effective resistance becomes R R 1 R 2 we find: R R 1 R2 Thus if R 1 =R 2, we find R=2R 1. One can use this method to obtain the resistance that we need in a circuit. RESISTOR COLOR CODES Many times we use resistors with particular values to tune the properties of our circuits. The resistor values are often not shown as numbers but instead using a color code consisting of four colored bands. Color Code for resistors R 1st significant figure 2nd significant figure Significant figures Color Multiplier (Ohm) Tolerance 0 black 1 silver ±10% 1 brown 10 2 red 100 gold ±5% 3 orange 1,000 4 yellow 10,000 5 green 100,000 6 blue 1,000,000 7 violet 8 gray 9 white gold 0.1 silver multiplier tolerance Acoustic Lab 10: PRELAB Amplifiers; Page 3 113

114 Prelab Report 10 Name: For example, the color code of a 20 kohm resistor with 5% tolerance is red-black-orange-gold (20 x 1000 Ohm ±5%). Answer following questions: 1. What is the color code for the following resistors (all 5% tolerance)? a. a: 10 Ohm b. 100 Ohm c. 1 kohm d. 10 kohm e. 220 kohm f. 450 Ohm 2. What is the value of a resistor that has bands colored orange-red-green-silver? 3. What is the effective resistance for two 10 kω resistors connected in parallel? 4. What is the effective resistance for two two 10 kω resistors connected in series? Acoustic Lab 10: PRELAB Amplifiers; Page 4 114

115 Circuit Connections in the Laboratory 10: AMPLIFIERS From now on you will construct electrical circuits and test them. The usual way of constructing circuits would be to solder each electrical connection to assure good electrical contact. Since each circuit has to be changed quite often during the lab period and also be dismantled at the end of the lab, soldering is impractical. Thus you will be provided a compact bread-board box which allows quick and good electrical connections without soldering. A copy of this breadboard box is shown on the next page. This box is made of aluminum and contains a variety of plug-in sockets with which circuit connections can be made. On the periphery of the box are banana-plug and BNC-type sockets, which have wire sockets connected to them. The central region of the box consists of two floating socket boards each of which contains a matrix of small sockets which are connected to each other in various ways. Whit this useful box, you will be able to construct electrical circuits simply by inserting the jumper wires of the components into the appropriate sockets. By the way, this is the manner in which many circuit designers mock-up their circuits before connecting them permanently. Op-Amp An operational amplifier, often called an op-amp, is a DC-coupled high-gain electronic voltage amplifier with differential inputs and, usually, a single output. It can handle a DC potential or AC signals. The output of the op-amp is controlled either by negative feedback (Sec. III and IV), which largely determines the magnitude of its output voltage gain, or by positive feedback (Sec. V), which facilitates regenerative gain and oscillation. An ideal op-amp has high input impedance (ideally infinite) and low output impedance (ideally zero). Op-amps are among the most widely used electronic devices today in a vast array of consumer, industrial, and scientific devices. I. Introduction Amplifiers play an important part at all stages of sound recording and reproduction and in PA systems. It probably will never be necessary for any of you to design and build your own amplifier; however, it is helpful to understand some of the limitations of amplifiers, and how high quality in one area can be traded for high quality in another. In the next few labs you will start with a working amplifier. You will then modify the amplifier in various ways which will change its capabilities. The major purpose of this first lab is to introduce you to an integrated-circuit (IC) amplifier and the equipment needed to get it to work and to study it; bread-boards, power supplies, resistors, capacitors, and, of course, the oscilloscope. You will also learn some of the characteristics of amplifiers and how they are measured. Acoustic Lab 10: Amplifiers; Page 1 115

116 II. The 741 Op-Amp You will be using an operational amplifier, or op-amp, which comes in an 8-lead IC package. Release of the μa741 in 1968 was seen as a nearly ubiquitous chip. The opamp diagram is: 1: not used 2: inverting input 3: noninverting input 8: not used 7: +V supply 6: output 4: -V supply 5: not used +V and V are the places where the power supply is connected to the op-amp, and need to be at +15V and 15V respectively. Pins 1, 5, and 8 are not used. Use the oscilloscope to check the voltages coming out of the power supply on the bread board. V+ = V = Your instructor will show you how to mount your chip on the bread-board and how to power it. You can also look at the sample set-up for help. The first circuit we want to try is: 10kΩ input (from generator) 1kΩ output (to scope) Use the Function Generator on the bread-board for at 1Volt amplitude and 1000 Hz sine wave as the inverting input to the op-amp. Use the dual trace feature of the oscilloscope to display both the unamplified and the amplified signals on the screen. What is the gain (the ratio between the output voltage and the input voltage)? Acoustic Lab 10: Amplifiers; Page 2 116

117 Increase the amplitude of the input to find what is the maximum peak-to-peak voltage output the op-amp can produce without clipping (clipping is when the peaks and/or valleys of the signal are clipped-off because they have surpassed the maximum voltage which can be produced by the amplifier)? What is the phase of the output compared to the input? (Omit this item, if you are using the breadboard function generator)!: Observe the effect of changing the DC offset on the function generator (this is produced by pulling out the DC OFFSET knob and turning it). What is the largest positive voltage the opamp can produce? The most negative voltage? Congratulations! You have made your first amplifier. This amplifier employs "negative feedback" (notice that the output is connected back to the input), which reduces the gain of the amplifier, but greatly reduces distortion and gives a flat response from dc to well above 1 khz. To see why negative feedback is a good idea, we next measure the amplifier response without it. III. Frequency Response with Negative Feedback Your op-amp, as you have noticed, has an extremely large gain (which falls off rapidly with increasing frequency). You can trade away some of the low frequency gain for a flat frequency response by using negative feedback. Recall that the original signal was fed into the inverting input of the op-amp. You can feed some of the inverted output back into the input. When the output is very large, the (negative) feedback is large. This kind of feedback decreases the gain the most at exactly those frequencies where the gain of the op-amp is largest. The result is to make the gain constant over a wide range of frequencies. A graph of the gain as a function of frequency is a straight line, and so is called a flat response. Of course, the more negative feedback you use, the less gain you get. The amount of feedback (and gain) is controlled by the ratio of the sizes of the input and feedback resistors. In part two, the feedback resistor was 10 kω, which gave a gain of 10. Here, we will use a 100 kω feedback resistor, which will have more gain: gain feedback resistor input resistor 100,000 Ohm 1000Ohm 100 Build this amplifier (just put a 100 kω resistor back connecting pins 2 and 6 of the op amp), measure its gain at several frequencies. Remember that it is not important to take a lot of data in the region where the gain is constant (that is, where the response if flat). Acoustic Lab 10: Amplifiers; Page 3 117

118 Table: 741 Op-amp gain versus frequency Frequency(Hz) input amplitude (V) output amplitude (V) Gain k 20k 50k (Be sure to determine the frequency by measurement of wave period, and measure input voltage at each frequency.) Plot your results on the 3 cycle by 3 cycle log-log graph paper. For what range of frequencies do you get a gain close to 100? Where does the gain "roll off" (drop to 71% of its low frequency value)? By setting the scope mode to ALTernate and the INTernal TRIGger to CH1 you can observe the phase relationship between the input and the output. Why is this opamp/resistor configuration called an inverting amplifier? Acoustic Lab 10: Amplifiers; Page 4 118

119 IV. Frequency response for high gain with negative feedback Put a 1 MΩ resistor as the feedback resistor (replacing the 100 kω resistor). Measure the amplitude of the small signal (point A) directly with the scope from 100 Hz to 50 khz. Use a signal of 20 mv pp. You may need to play around with the DC offset knob on the generator to get the output to stay between the +15 and 15 V limits of the op-amp. Table: 741 Op-amp high gain Frequency (Hz) input amplitude (milivolts) output amplitude (volts) gain 100 Hz k 20k 50k Plot your results on the same 3 cycle by 3 cycle log-log graph paper which you used before. What is the low frequency gain for the 1 MΩ feedback resistor? Where does the gain roll off? Compare the frequency response for low gain and high gain. Acoustic Lab 10: Amplifiers; Page 5 119

120 V. Pure, Unadulterated Op-amp Op amps without negative feedback (operating "open loop") have a very high gain for low frequency signals. To see this, we need an input signal of no more than 10 mv. In order to produce a clean voltage signal this small from the Function Generator you need to make a "voltage divider" (on the bread board, to one side of your amplifier): input=v (from generator) R2 R1 output=v1 (to op-amp input) The amplitude of the voltage difference across the small resistor, R1, is V1 = V*R1/(R1+R2), where V is the voltage amplitude from the function generator. Since R1=10 Ohm here and R2 is 10 kohm, the amplitude V1 is about 1000 times smaller than V (which is less than 20 Volts peak-to-peak). The complete circuit will now be the same as your first circuit, but with the "feedback" resistor (10 kω) removed and the input coming from the "output" of your voltage divider instead of directly from the function generator. a) What is your observation? b) Is the output signal saturated (either clipping or pegged at its maximum output) for almost any frequency, even with very small or zero input voltage? That is, without feedback there is so much gain that the amplifier saturates. c) How is the output signal at higher frequencies (in 100 khz range)? Acoustic Lab 10: Amplifiers; Page 6 120

121 VI. Positive Feedback In the last circuit, some of the output was run into the inverting input (negative feedback) to reduce the gain of the amplifier. It is also possible to send some of the output to the non-inverting input, resulting in positive feedback. This feedback is often sent through circuit elements which are more resistive to some frequencies than others. In this case a combination of a capacitor and an inductor will be used. A variable resistor will allow you to control the amount of positive feedback; the less resistance, the more feedback. Turn off the function generator and bread board and construct the following circuit: 10 kω 10 kω input (from generator) kω resistor output (to scope) 0.1 µf 10 mh 10 kω Use the bread board 10 kω variable resistor for feedback. Use the function generator in the breadbox for input source. Set the frequency multiplier to 10k. Now turn on the function generator. Use the frequency dial to sweep through various frequencies while you watch the output on the scope. Is the response still completely flat? Turn the 10 kω knob on the variable resistor to about 1 kω so that the feedback is strong and the signal is small. Now the output should show a definite frequency preference. It may be necessary to reduce the input source amplitude or to lower the power supply (±15 V) to keep the output from clipping near the resonant frequency. If you lower the positive feedback resistor (now 1 kω), you may set the amplifier oscillating on its own, independent of the output! The amplifier should oscillate on its own. You have built a fixed frequency oscillator. By carefully measuring the period on the scope, you can determine the frequency of this oscillation, which is the natural resonant frequency of the inductor-capacitor (LC) combination we are 1 using: f 0. 2 LC Acoustic Lab 10: Amplifiers; Page 7 121

122 Acoustic Lab 10: Amplifiers; Page 8 122

123 PRELAB11: AUDIO AMPLIFIER I. CAPACITORS A capacitor is a device that can store electric energy, typically made of two parallel conductors. When an electric potential is applied, positive charge will accumulate on one plate and an equal amount of negative charge on the other plate. The charge Q is related to the voltage V by the relation: Q=CV, where C is the capacitance of the capacitor. The dimension of the capacitance is Farad, which is equal to 1 Coulomb per volt. Farad is a very large unit. Typical capacitance ranges from pico-farad (pf, or F), nano-farad (nf, or 10 9 F), micro-farad (μf, or 10 6 F). A capacitor with capacitance larger than 1μF is typically an electrolytic capacitor. An electrolytic capacitor is a type of capacitor that uses an ionic conducting liquid as one of its plates with a larger capacitance per unit volume than other types. They are used in relatively high-current and low-frequency electrical circuits, as coupling capacitors in circuits where AC should be conducted but DC should not. Electrolytic capacitors can have a very high capacitance, allowing filters made with them to have very low cutoff frequencies. Most electrolytic capacitors are polarized and may catastrophically fail if voltage is incorrectly applied. If a reverse-bias voltage above 1 to 1.5 V will destroy the center layer of dielectric material via electrochemical reduction (called redox reactions). Following the loss of the dielectric material, the capacitor will short circuit, and with sufficient short circuit current, the electrolyte will rapidly heat up and either leak or cause the capacitor to burst. When the aluminum foil, with a layer of aluminum oxide on it, is made positive the oxide ion may convert into oxygen gas generating a high pressure and hence may burst up the capacitor. Capacitors can be connected in parallel or in series, shown respectively as follows: When capacitors are in parallel, the effective capacitance is C eq =C 1 +C 2 +C 3. For example two 10 pf capacitors in series give an effective capacitance of 20 pf. When capacitors are in series, the effective capacitance is 1/C eq =1/C 1 + 1/C 2 + 1/C 3, For example two 10 pf capacitors in series give an effective capacitance of 5 pf. Acoustic Lab 11: page 1 123

124 The capacitance of a capacitor is color coded (similar code as that of the resistor) or indicated by three numbers printed on the body. The third digit in this case works like the multiplier band on a resistor; it tells the number of zeros to tack onto the end of the two significant digits. For example, if you see a capacitor marked 151, it is an ordinary capacitor with a capacitance of 150 pf. In this nomenclature, all values are given in pf. A capacitor with voltage rating is also indicated on the body. Question: 1. When a capacitor indicated with positive and negative end, what can happens if you connect the capacitor to wrong polarity? 2. If two capacitors with capacitance 10 pf and 20 pf are connected in parallel, what is the effective capacitance of the circuit? 3. If two capacitors with capacitance 10 pf and 20 pf are connected in series, what is the effective capacitance of the circuit? II. AMPLIFIER In the experiment last week, we study the circuit with a popular 741 op-amp. We discussed the advantage of a negative feedback, R 2. The circuit is shown at right. Answer the following question. Question: If R 1 is 2 kω, and R 2 is 22 kω, what is the voltage amplification of the circuit? Acoustic Lab 11: page 2 124

125 11: AUDIO AMPLIFIER I. INTRODUCTION The properties of an amplifying circuit using an op-amp depend primarily on the characteristics of the feedback network rather than on those of the op-amp itself. A typical feedback network consists of resistors and capacitors. Because these components are available in conditions of high precision and low drift, the accuracy and stability of circuits using them is very high. Audio receiver systems consist of a number of amplifying stages since the gain of each stage has to be kept low due to the frequency range required for audio signals. Because of thermal drift, component tolerance and variation, a DC level is produced. In order to prevent the amplification of such DC levels, coupling capacitors must be used between the stages. The coupling capacitor not only blocks the DC voltage but also sets a lowfrequency cutoff limit; it acts as a filter. Such an amplifier with a coupling capacitor is called an AC amplifier. The following figure shows a simple amplifier using an op-amp as a small-signal AC amplifier. The input signal comes from the voltage output of a microphone. Usually microphones have very high internal electrical impedance and therefore must be connected to a highimpedance circuit in order to maximize to power transfer to this circuit, and the high input impedance of the op-amp (several Meg-Ohms) is ideal for such an application. The op-amp output could drive the audio speaker directly so that the signal from the microphone, amplified by the op-amp, would be audible. However, the speaker has very Acoustic Lab 11: page 1 125

126 low impedance (8 Ohms) in the audio frequency range (like most HI-FI speakers), while the op-amp output impedance is approximately ten times this value. Therefore, the direct connection of the speaker to the op-amp output constitutes an excessive load, and the opamp output signal will be heavily attenuated. Thus, a push-pull (connected in a unity-gain emitter-follower configuration) is interposed between the op-amp and the speaker. The op-amp sees the several-kilo-ohm input impedance of the driver circuit, while the speaker is placed directly in the very-low-impedance output of the driver transistor. Wire up the above circuit and make sure it is working. Make a few gain measurements on the amplifier at some mid-range frequency (say f = 1 khz) for the following settings of the variable resistor: maximum, minimum and one intermediate setting. 1. The variable 10 kω feedback from the pre-amplifier can also come from the preamplifier output port #6. Try it, and write down your comment here: 2. From the result of your lab#10, what are the minimum and maximum theoretical gains of this circuit with feedback resistance 10kΩ? 3. What is the range of the measured gain of the amplifier? Does the measured gain agree with the theoretical gain calculated above? 4. Observe the maximum undistorted output voltage delivered to the speaker, or observed on the oscilloscope by increasing the input voltage. What is it? Acoustic Lab 11: page 2 126

127 BANDWIDTH The bandwidth of an amplifier is defined as the range (band) of frequencies for which the gain remains essentially constant. To fix the frequency boundaries, (V out /V in ) mid was chosen to be the gain cutoff level. The corresponding frequencies f 1 and f 2 are generally called the cutoff, band, or half-power frequencies. The bandwidth is then f 2 f 1. The multiplier was chosen because at this level the output power is half the mid-band power output, that is, at mid-frequency. Measure the frequency band width of the amplifier for a gain setting of 10kΩ. Remember that it is not necessary to take a lot of data in the region where the response is flat. Gain versus frequency for 741 plus transistor Frequency (Hz) V in V out db = 20 log V out /V in Plot your data on graph paper and find the bandwidth. Note that if you plot (Vout/Vin) vs frequency, you should use log-log graph paper, and if you plot db vs frequency, you should use the semi-log graph paper! Does this circuit satisfy the minimum requirements (20 hz 20 khz) of a hi-fidelity audio amplifier? Acoustic Lab 11: page 3 127

128 MONOLITHIC AUDIO AMPLIFIER In the previous circuit of an audio amplifier a general purpose op-amp is connected in series to a power booster (transistor connected in an emitter-follower configuration). Power amplifiers are also commercially available in monolithic form. Although power amplifiers differ from general-purpose op-amps in delivering various amounts of power, they are nearly as compact. In the following we will use the ECG704 or LM380 power audio amplifier. The 14 pins on the chip are hooked as follows: open +15 V 1 14 Inputs Output 7 3,4,5,10,11,12 Ground The following figure shows the simplest and most basic application of the ECG704 or LM380 as an audio power amplifier. This amplifier requires very few external components (and absolutely no 10 kω resistors!). Although the gain of the LM380 is internally fixed at 50, it can be changed with the use of external components. In the present configuration, variable gains up to 50 are obtained with the use of the potentiometer across the two input terminals. Note the polarities of the electrolytic capacitors having values greater than 1 µf. Most electrolytic capacitors are polarized and may catastrophically fail if voltage is incorrectly applied. This is because a reverse-bias voltage above 1 to 1.5 V will destroy the center layer of dielectric material via electrochemical reduction (see redox reactions). Following the loss of the dielectric material, the capacitor will short circuit, and with sufficient short Acoustic Lab 11: page 4 128

129 circuit current, the electrolyte will rapidly heat up and either leak or cause the capacitor to burst. Measure the frequency band width of the amplifier for a gain setting of 10 and 40. Record your results in the Table below. If necessary, reduce input voltage to avoid distortions. Gain versus frequency for ECG704 or LM380 audio amplifier Frequency (Hz) V in V out db = 20 log V out /V in Plot your data on graph paper and find the bandwidth. Note that if you plot (Vout/Vin) vs frequency, you should use log-log graph paper, and if you plot db vs frequency, you should use the semi-log graph paper! Compare it with the previous results. How does the monolithic stack up against your homebrewed amp? Observe the maximum undistorted output voltage delivered to the speaker by increasing the input voltage. What is it? Acoustic Lab 11: page 5 129

130 (Plot db vs frequency, where frequency is in log scale, and db in linear scale) Acoustic Lab 11: page 6 130

131 A. ABOUT FILTERS 12: PRELAB ACTIVE FILTERS An electric filter is a frequency-selecting circuit designed to pass a specified band of frequencies while attenuating signals of frequencies outside this band. Filters may be either active or passive depending on the type of elements used in their circuitry. Passive filters contain only resistors, capacitors, and inductors. Active filters employ transistors or op-amps in addition to resistors and capacitors. Active filters offer several advantages over passive filters. Since the op-amp is capable of providing a gain, the input signal is not attenuated as it is in a passive filter. Because of the high input and low output resistance of the op-amp, the active filter does not cause loading of the source or load. There are four types of filters: low-pass, high-pass, band-pass, and band-reject filters. The figure below shows the frequency response characteristics for a low- and high-pass filter. A low-pass filter has a constant gain (=Vout/Vin) from 0 Hz to a high cut off frequency f H. This cut off frequency is defined as the frequency where the voltage gain is reduced to 0.707, that is at f H the gain is down by 3 db; after that (f > f H ) it decreases as f increases. The frequencies between 0 Hz and f H are called pass band frequencies, whereas the frequencies beyond f H are the so-called stop band frequencies. A common use of a low-pass filter is to remove noise or other unwanted high-frequency components in a signal for which you are only interested in the dc or low frequency components. Low-pass filters are also used to avoid aliasing in analog-digital conversion (which we will encounter in a few weeks). A high-pass filter has a stop band for 0 < f < f L and where f L is the low cut off frequency. A common use for a high-pass filter is to remove the dc component of a signal for which you are only interested in the ac components (such as an audio signal). A bandpass filter has a pass band between two cut off frequencies f H and f L, (f H > f L ), and two stop bands 0 < f < f L and f > f H. The bandwidth of a bandpass filter is equal to f H f L. Recall that we used a tunable bandpass filter to do harmonic spectrum analysis several weeks ago. Acoustic PRELAB 12: Active Filters Page 1 131

132 As shown in the above figure, the actual response curves of the filters in the stop band either steadily decrease or increase with increase of frequency. The roll-off rate, measured at [db/decade] or [db/octave] is defined as rate change of power at 10 times (decade) or 2 times (octave) change of frequency in the stop band. The First-order filters attenuate voltages in the stop band 20 db/decade (for example, a first-order lowpass filter would attenuate a signal at a frequency 100 times (2 decades) higher than f H by 40 db. The second-order filters attenuate by about 40 db/decade. A simple low pass filter circuit employing resistors and capacitors is shown at the right: The cutoff frequency is f H =1/(2πRC). The power output is proportional to 1/(1+(f/f H ) 2 ). In decibel unit, the rolloff rate for the first order filter is typically 20log(10)=20 db/decade or 6dB/octave as shown in the semi-log plot on the right of db vs f. Similarly, a simple high pass filter is a) If a capacitor is C=0.01 μf, and R=4 kω, what is the cutoff frequency? b) If a capacitor is C=0.01 μf, and R=1 kω, what is the cutoff frequency? c) What is the typical roll-off rate in db/decade for the first order filter? Acoustic PRELAB 12: Active Filters Page 2 132

133 A. INTRODUCTION 12: ACTIVE FILTERS After last week s encounter with op-amps we will use them to build active filters. B. ABOUT FILTERS An electric filter is a frequency-selecting circuit designed to pass a specified band of frequencies while attenuating signals of frequencies outside this band. Filters may be either active or passive depending on the type of elements used in their circuitry. Passive filters contain only resistors, capacitors, and inductors. Active filters employ transistors or op-amps in addition to resistors and capacitors. Active filters offer several advantages over passive filters. Since the op-amp is capable of providing a gain, the input signal is not attenuated as it is in a passive filter. Because of the high input and low output resistance of the op-amp, the active filter does not cause loading of the source or load. There are four types of filters: low-pass, high-pass, band-pass, and band-reject filters. A low-pass filter has a constant gain (=Vout/Vin) from 0 Hz to a high cut off frequency f H. This cut off frequency is defined as the frequency where the voltage gain is reduced to 0.707, that is at f H the gain is down by 3 db; after that (f > f H ) it decreases as f increases. The frequencies between 0 Hz and f H are called pass band frequencies, whereas the frequencies beyond f H are the so-called stop band frequencies. A common use of a low-pass filter is to remove noise or other unwanted high-frequency components in a signal for which you are only interested in the dc or low frequency components. Low-pass filters are also used to avoid aliasing in analog-digital conversion (which we will encounter in a few weeks). Correspondingly, a high-pass filter has a stop band for 0 < f < f L and where f L is the low cut off frequency. A common use for a high-pass filter is to remove the dc component of a signal for which you are only interested in the ac components (such as an audio signal). A bandpass filter has a pass band between two cut off frequencies f H and f L, (f H > f L ), and two stop bands 0 < f < f L and f > f H. The bandwidth of a bandpass filter is equal to f H f L. Recall that we used a tunable bandpass filter to do harmonic spectrum analysis several weeks ago. The actual response curves of the filters in the stop band either steadily decrease or increase with increase of frequency. The roll-off rate, measured at [db/decade] or [db/octave] is defined as rate change of power at 10 times (decade) or 2 times (octave) change of frequency in the stop band. The First-order filters attenuate voltages in the stop band 20 db/decade (for example, a first-order lowpass filter would attenuate a signal at a frequency 100 times (2 decades) higher than f H by 40 db. The second-order filters attenuate by about 40 db/decade. Acoustic Lab 12: Active Filters Page 1 133

134 C. LOW PASS FILTER After these introductory notes we now want to build an active low-pass filter that uses an RC network and test it. The complete circuit is as follows. To make the cut-off frequency convenient to determine, use R 3.9 kω. Note that the op-amp is used in its non-inverting mode here (the input is connected to pin 3). The resistor-capacitor configuration between the input and the op-amp s non-inverting input does the filtering (it is a passive low-pass filter). For a better measurement, make Vin less than 2V peak-to-peak. Measure the frequency response and plot the resulting gain (Vout/Vin) as a function of the operating frequency, put your data in the table and graph it. (Use semi-log graph paper if you plot power in db vs frequency)! Table: Gain in db =20log(Vout/Vin) Frequency(Hz) input amplitude (V) output amplitude (V) Gain in db k 20k 50k 100k a. Determine the cut off frequency from your graph and compare it to the theoretical value: Acoustic Lab 12: Active Filters Page 2 134

135 b. What is the pass band gain? Compare it to the theoretical gain, 20 log(1 + RF/R1) c. What is the roll-off rate (in db/decade) in the stop band? D. HIGH-PASS FILTER Now build an active high-pass filter. The high-pass filter is formed by interchanging the resistor and capacitor in the low-pass filter that you made -- the rest of the circuit is the same! Measure the frequency response, i.e. measure V out, and convert your data into db as a function of the operating frequency. Fill in the table below and plot it on a semi-log graph paper in db vs frequency. Table: Gain in db =20log(Vout/Vin) Frequency(Hz) input amplitude (V) output amplitude (V) Gain in db k 20k 50k 100k a. Determine the cut off frequency and compare it to the theoretical value,. b. Compare the experimental and theoretical 20 log(1 + RF/R1) pass band gains. c. What is the roll-off rate (in db/decade) in the stop band? Acoustic Lab 12: Active Filters Page 3 135

136 E. HIGHER-ORDER FILTERS For these first-order low-pass and high-pass filters, the gain rolls off at the rate of about 20dB/decade in the stop band. In critical applications (such as digitization, which needs the flattest response possible in the pass band and most sharply-defined stop band) a higher-order filter is a necessity. The following diagram shows a second-order low-pass filter (it s second order because it contains two low-pass filters). Put it together and measure its gain versus frequency. Use R2=R3 3.9 kω. Fill the table below and graph your results. Table: Gain in db =20log(Vout/Vin) Frequency(Hz) input amplitude (V) output amplitude (V) Gain in db k 20k 50k 100k How does the roll-off (db/decade) compare to the first-order filter we measured in part C? Acoustic Lab 12: Active Filters Page 4 136

137 F. BAND-PASS FILTERS Band pass filters can be formed by simply "cascading" high-pass and low-pass sections (that is, put the output of one into the input of the next). Each section resembles the filter in the previous section (the high-pass versions have their capacitors and resistors interchanged from their positions in the low pass filter). We can also make a band-pass filter in one section: The RC combination R4 and C2 is a high-pass filter, which will determine the lower cutoff frequency. Try R4=1 kω, and C2 = C3 = 0.01 μf, R2=R3 3.9 kω, R1=Rf =RL=10 kω. Carry out measurements and fill the following Table and graph your result! Table: band pass filter gain in db =20log(Vout/Vin) Frequency(Hz) input amplitude (V) output amplitude (V) Gain in db k 20k 50k 100k Acoustic Lab 12: Active Filters Page 5 137

138 Acoustic Lab 12: Active Filters Page 6 138

139 13: PRELAB OF DIGITIZATION Binary (base 2) numbers All digital technologies use bits as their basic unit of information. A bit is a single piece of information which is in one of two states. These states are called on and off (lights), yes and no (logic), high and low (voltage), or 1 and 0 (base 2 numbers). Base 2 numbers are just like ordinary numbers except that there are only two symbols, 0 and 1. Here are base 2 counting from 0 to 111 (7 in base 10): 000, 001, 010, 011, 100, 101, 110, 111. For example: a number 101 means , which is equivalent to 5 in base 10, i.e a. Use four digits to count in base 2 from 1000 to b. What numbers would these be in base 10? Try adding some binary numbers, then check your answer in base 10. Remember that when you add a 1 and a 1 you get a 0 AND YOU MUST CARRY A 1 TO THE NEXT PLACE. Example: 0101 Base 10 equivalent: Acoustic PRELAB Lab 13 Page 1 139

140 Digital to Analog Converter A Digital to Analog Converter is a circuit which turns a binary number into a voltage, such as Notice that this circuit is like the inverting amplifier you have studied before, except that this one has four inputs. Since the inputs have different resistance, the signals coming in are amplified by different amounts. Each input corresponds to one binary digit. The input signals are either 0 (a low or zero voltage) or 1 (a high voltage). How does it work? An inverting amplifier with several inputs acts as a summing amplifier. With the non-inverting input grounded, the inverting input becomes a "virtual ground." When you apply a positive voltage, Vin, to one or more of the inputs, current will flow through the corresponding input resistor(s) towards the inverting input of the op-amp (for example, if Vin is applied to input a, current Vin/Ra will flow through input resistor a. Note that for a given Vin the most current will flow through the smallest input resistor, the least will flow through the largest. The total current, I, is the sum of the currents through each input resistor, i.e. I=Ia+Ib+Ic+Id, where Ia=Va/5kΩ, Ib=Vb/10kΩ, Ic=Vc/20kΩ, Id=Vd/40kΩ. All this current cannot flow into the op-amp's input, instead, almost all of it flows through the feedback resistor, RF. The result is that the other side of the feedback resistor, which is also the output of the opamp (pin 6), will be at a voltage I*RF with respect to virtual ground. Therefore, the output voltage is proportional to the total current flowing through the feedback resistors. QUESTIONS: 1. In a number, the digit on the left is called the Most Significant Digit, because it s size affects the number more than any other digit. Which input of this circuit (a, b, c, or d) corresponds to the most significant digit? (i.e., which will affect the voltage output the most?) 2. Let input a be on, b be off, c on, and d on. a. What binary number does this DAC input represent? b. If on stands for 5 Volt and off stands for 0 Volt. What voltage comes out of the above DAC? Acoustic PRELAB Lab 13 Page 2 140

141 A. DIGITAL TO ANALOG CONVERTER 13: DIGITIZATION A Digital to Analog Converter is a circuit which turns a binary number into a voltage, such as Notice that this circuit is like the inverting amplifier you have studied before, except that this one has four inputs. Since the inputs have different resistance, the signals coming in are amplified by different amounts. Each input corresponds to one binary digit. The input signals are either 0 (a low or zero voltage) or 1 (a high voltage). How it works: an inverting amplifier with several inputs acts as a summing amplifier. With the non-inverting input grounded, the inverting input becomes a "virtual ground." When you apply a positive voltage, Vin, to one or more of the inputs, current will flow through the corresponding input resistor(s) towards the inverting input of the op-amp (for example, if Vin is applied to input a, current Vin/Ra will flow through input resistor a. Note that for a given Vin the most current will flow through the smallest input resistor, the least will flow through the largest. The total current, I, is the sum of the currents through each input resistor. All this current cannot flow into the op-amp's input, instead, almost all of it flows through the feedback resistor, RF. The result is that the other side of the feedback resistor, which is also the output of the op-amp (pin 6), will be at a voltage I*RF with respect to virtual ground. Therefore, the output voltage is proportional to the total current flowing through the feedback resistors. B. DIGITIZATION The next two labs are a very basic introduction to digital recording and playback. There are three parts to digital recording: 1. Converting a voltage (derived from a sound source) into a series of numbers. 2. Storing the numbers in a computer or on a CD or digital tape (possibly with compression). 3. Converting those numbers back into a voltage (which drives a speaker to make sound). You will do the first and third of these. Acoustic Lab 13: Digitization Page 1 141

142 1. Analog to Digital Converter (ADC) The first step is to convert a voltage into a number. This is done by an analog to digital Converter, also called an A/D converter or just an ADC. An audio ADC is basically a very fast digital voltmeter (it records the voltage signal from a transducer many times per second). You will use an 8-bit ADC chip such as ECG2053 or ADC0804LCN, of which you will use only the four most significant bits. Build your ADC on the bottom half of the board of your breadboard. Keep the top half free for the Digital to Analog Converter, which you will build later. Hook up this circuit, following the instructions given after the diagram: First ground pins 1, 2, 7, 8, and 10. With a small wire connect pins 3 and 5. Connect pin 4 to pin 19 with a 10 k ohm resistor and connect a 1000 ohm resistor from the input pin (6) to some free place on the board. Run all inputs through this resistor. Hook pins 11, 12, 13, and 14 to the four indicator lights on your breadboard. Pin 11 is the most significant digit, so to get an output that looks like a normal binary number it should go to the light which is farthest to the left. Now connect pin 20 to the +5 volt power supply on your breadboard box and turn on the power. Since no voltage is hooked to the input, the lights should remain off. As an input we will use a 1000 ohm potentiometer hooked to the +5 V power supply. A potentiometer is usually called a pot and is really just a variable resistor with a center tap. It looks like this: Hook the a wire to ground, the b wire to the input of the ADC (through the 1000 ohm resistor) and the c wire to +5 V. Turn the knob back and forth and watch the lights count up and down in binary. Use one of the BNC in/out terminals to look at the input voltage on the scope. What is the smallest voltage that light each of the different combinations of lights? Acoustic Lab 13: Digitization Page 2 142

143 Light Combination Minimum Input Voltage [V] Light Combination Minimum Input Voltage [V] a. Does your measured voltage for the light combination 1011 agree with the result of your prelab? Explain b. Now remove the DC 5V input across the 1KΩ pot, and connect the 1KΩ pot to the output of the 4011A function generator to provide a constantly changing sine voltage. You will need to adjust the DC offset on the generator so that the signal is always positive. Set the frequency to be so slow that you can easily watch the lights count up and down. What frequency is just high enough to make the lights a blur? Turn off the function generator, and leave the circuit intact to be used in part #3. 2. Digital to Analog Converter Rather than store the numbers from your ADC, in this lab your will convert them directly back into voltage using a DAC. First, however, it will be necessary to build the DAC. Do this now on the top half of your board. The input circuit is the one from the prelab (see page 2). Don't forget to bring in +15 V and -15 V to power the op amp. (+15 V to pin 7 and -15 V to pin 4.) Because resistors are not available in factors of two, it will be necessary to use identical resistors in series and parallel to get the same effect. Use all 10 k-ohm resistors hooked up as shown: Acoustic Lab 13: Digitization Page 3 143

144 Connect four of the +5 V logic switches (above the breadboard) to the DAC inputs and display the output on the scope. Use Channel 2 because it allows you to invert the signal by pulling out the positional knob. This will correct for the fact that it is an inverting amplifier. Turn on the breadboard power supply. Using the logic switches as binary digits, count up from zero to fifteen and watch the output voltage on the scope increase by steps and record them. If it doesn t, you have the switches hooked up incorrectly to the inputs. Light Combination Minimum Input Voltage [V] Light Combination Compare the data of your DAC and ADC tables. Are they the same? Explain Minimum Input Voltage [V] 3. Connect the ADC output to the DAC input Now it is time to hook the ADC and DAC together. Move the wires that connect the DAC inputs from the logic switches over to the outputs of the ADC. Be sure to get them hooked up in the right order, i.e. MSB of ADC should be connected to MSB of the DAC, and so on. Use the function generator as an input to the ADC, making a 100 Hz sine voltage. Look at the DAC out put on the scope (CH2 inverted). With the scope in DC mode, note the effects of adjusting the DC offset and amplitude on the function generator. Look at the ADC input signal on CH1. What is the largest peak-to-peak amplitude that can be used without clipping? What is the smallest input signal which will assure you of getting an oscillating signal out? (Be sure to try varying the DC offset before answering this.) LEAVE YOUR CONVERTERS SET UP FOR NEXT LAB!!! Acoustic Lab 13: Digitization Page 4 144

145 PRELAB 14: ALIASING I. PRELAB FOR ALIASING LAB In data sampling, aliasing refers to effects that cause different signals to become indistinguishable (or aliases of one another). It also refers to the distortion or artifact due to the signal reconstruction from samples. Temporal aliasing is a major concern in the sampling of video and audio signals. For example, the sampled data (circles) below may come from sinusoidal waves of many different frequencies (see the graph below).. For example, to record a frequency of 4000 Hz (period=0.25 ms), you might think that you can sample it at a rate of 4000 Hz, or a sampling time of 0.25 ms! It turns out, however, that you actually have to sample at least TWICE that rate (the Nyquist theorem)! You must sample at least 8000 Hz, or a sampling time of ms to record a frequency of 4000 Hz! This is because you need to record a peak and a valley from each cycle. If you sampled at the same rate that the signal was oscillating, you would get a peak every time you sampled, and the output would be a constant voltage. When you sample at a rate that is less than twice the signal frequency, the output frequency is lower than it should be. This phenomenon, called aliasing or foldover, is best studied by trying it. Fill in the following worksheet pages. A colored pen or pencil works best. The procedure is: 1. Start at the left edge of the paper; at this point the signal is at its maximum value. 2. Draw a straight line to the right for the number of squares indicated in the left hand margin. 3. Drop straight down until your pen is BELOW OR EQUAL TO the signal. 4. Go right again for the appropriate number of squares. Each time you get to sample, move your pen to the highest corner which is below or equal to the signal. 5. Continue. Always stay on the grid. Never cut across squares. Trace the example to get a feel for it. Notice how the quality of reproduction decreases as the sampling rate decreases. In the last examples on the first worksheet (page 2) the sampling frequency is more than half the signal frequency. Observe the consequences. This is aliasing. Acoustic Lab 14 PRELAB: Page 1 145

146 The period of the wave is 8 boxes. After you complete the worksheet, answer following questions: 1. What is the period of the top ADC signal if you sample every square? 2. What is the period of the second ADC signal if you sample every 2 squares? 3. What is the period of the third ADC signal where you sample every 4 squares? 4. What is the period of the fourth ADC signal where you sample every 7 squares? 5. What happens if you sample every 8 squares? Acoustic Lab 14 PRELAB: Page 2 146