5: SOUND WAVES IN TUBES AND RESONANCES INTRODUCTION

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1 5: SOUND WAVES IN TUBES AND RESONANCES INTRODUCTION 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 tape the glass plate to the one end of the tube with the masking tape. L = 2. Hooking up the Pulse Generator 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 Pulse Spacing 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. 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

2 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

3 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 remainder 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

4 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. Use the TTL signal for the oscilloscope trigger input. Disconnect the speaker from the pulse generator and the scope and then reconnect the speaker to the output of the 4011A Function Generator. 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 range suitable for finding resonances. 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. c. The lowest frequency is called the fundamental or first harmonic and is denoted by Sound Waves in Tubes Page 4

5 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 The fundamental is the hardest frequency to measure, go back and check it. 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 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. Sound Waves in Tubes Page 5

6 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), reelects 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 for a pattern which might help you find the lowest one. Leave space in your data table for this. Sound Waves in Tubes Page 6

7 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

8 E. ACOUSTIC RESONANCES I. INTRODUCTION: 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 f 0 the Q value is: Q. f II. Measurement of the Q of a Resonance: 1. High Q System: (Choose Open-Closed configuration) 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 closed. Sound Waves in Tubes Page 8

9 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. Choose a convenient harmonic of your open-closed tube in the Table in p.7 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

10 III. 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 10

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