llllllllllilau '9 i9 .9 I6 A 19 III '9 '9 an I9 I. Q ID ID a P3. P4. P5. P6. Experiment 12: Wave Motion: Standing Waves and the Speed of Sound and Beats TUBE OPEN AT BOTH ENDS Remove the cap from the end of the tube and repeat steps P1 and P2, this time using two frequencies between 2000 Hz and 5000 Hz. The measurements taken in steps P1 to P3 will allow a determination of the speed of sound in air. Calculations from the data will give the wavelength which, when coupled with the measured frequency, permit the velocity to be found easily. These observations were taken at xed frequencies. But the frequency of the oscillator is variable, so it would seem that a natural extension of the experiment would be to x the length of the air column and vary the frequency, noting the frequencies which produce resonance. This turns out to be difcult, due to the nature of the psychological relationship between perceived loudness and actual sound intensity: perceived loudness is frequency dependent. For example, a signal directly from the speaker at 2000 Hz, with no resonance occurring, might appear just as loud as a signal of 150 Hz on resonance. For this reason it is difcult to judge the occurrences ofresonances if the air column is held xed in length and the frequency of the function generator is varied. Nevertheless, it is interesting to try. Only qualitative observations are required. Set the length of the air column to about 25 cm; since the speed of sound in the laboratory is approximately 344 m/s, a frequency of 344 Hz would produce a wavelength of one meter, and a column length of 25 cm would correspond to the rst resonance. Adjust the frequency of the oscillator to around 344 Hz until resonance is heard. Record the precise frequency from the multimeter. N ow, without changing the length of the tube, calculate the next two positions or resonances expected and tune the function generator (without looking at the dial) to ascertain if they actually occur at these positions. (You may need to vary the tube length by a small amount to test whether resonance is occurring.) Read the thermometer in the laboratory and record the temperature. Select a tuning fork and note its frequency, stamped at the base. This value is accurate to 0.5%. Connect the speaker to the multimeter by plugging the two leads from the speaker into the terminals on the multimeter marked VQHZ and COM. Then use another set of leads to connect these same two terminals of the meter to the blue and yellow terminals on the oscillator. Adjust the dial 0n the function generator to the range appropriate for your timing fork. Adjust the frequency dial on the function generator to match approximately the value of the fork; the match should not be exact, but rather adjust this frequency so that a beat note between the fork, when struck, and the speaker is obtained. You want to determine the frequency of this heat note, 50 try for a frequency which will produce 1520 beats in about 20 seconds, something that can be counted comfortably. Once a convenient beat frequency has been established, use a stopwatch to time 2030 beats; the heat period is then this time interval divided by the number of beats observed, and the beat frequency is the reciprocal of this period. Record also the frequency of the lnction generator as measured by the multimeter. Note any discrepancy between the setting of the function generator dial and the reading of the meter; the meter is accurate to :1 Hz. Repeat this process starting once with a frequency from the function generator that is higher than that of the tuning fork, and then one that is lower. 83 The Physics Lab Manual 1 82 PROCEDURE TUBE OPEN AT ONE END P 1 . Connect the speaker to the function generator, using P2. one set of the leads provided. Use the second set of leads to connect the VHz and COM terminals of the multimeter to the function generator and set the dial on kHz. (See Figure 12-4.) This will allow you to get an accurate measurement of the frequency of the function generator. Then place the adjustable tube (with the cap attached on one end) on the supports provided, and place the speaker about 2 cm from the open end of the tube. (See Figure 12-5.) Set the function generator to produce a frequency somewhere between 1000 Hz and 2000 Hz, and record the frequency as measured with the multimeter. (It's best if different groups do not use exactly the same frequency.) Use the amplitude control on the function generator to set the volume of the speakers so that you can just barely hear it. (This will make it easier to recognize the resonance positions, where the sound gets noticeably louder.) Starting with the inner tube pushed all the way into the outer tube, so that the air column inside is at its shortest length, gradually pull the inner tube out, thus lengthening the air column. Record the length of the air column for each resonance, i.e., for each position where the sound is noticeably louder than it is on either side of that position. You may nd it useful to download an app to your smart phone that measures the decibel level of a sound. (There are many free ones available.) You can then hold the phone next to the tube opening and note the tube lengths where the decibel level suddenly rises. The length of the air column can be read directly from the tape that is afxed to the inner tube (Figure 12 6). (The length of the air column is actually several centimeters longer than the reading on the tape but, since it is only differences in length that matter, the reading from the tape for each resonance position is sufcient.) Repeat this process several times as you move the inner tube back and fo Figure 12-4: Multimeter Set to lVIearnre Frequency Figure 12-5: Setnpfar Finding Resonant Length: Figure 12-6: A'Icasuring the Length aft/19 Air Column rth to make sure you have located all the resonance positions. (The interval between each pair of adjacent resonances should be approximately the same. If the g3 Repeat step Pl using ano between 1000 Hz and 3000 Hz. p between one pair seems larger than the rest, check to see ifyou missed one between them ) ther frequency, different from the one you used above, but somewhere llt$l$l$l\\\\\\\\\\l\\\\\\\\|'lIIIIIIIIIIJJIJIJJIGOI Experiment 12: Wave Motion: Standing Waves and the Speed of Sound and Beats The apparatus here allows for two different scenarios: the cap on one end of the tube can be left on so that one end of the tube is open and the other end is closed, or the cap can be removed so that both ends are open. If the cap is left on, there is an amplitude node at the closed end and an amplitude andnode at the open end. As a result, resonance occurs when the shortest air column is onequarter of a wavelength. (See Figure 12-2.) As the tube is lengthened, additional resonances occur whenever the length of the tube has increased a multiple of a half- wavelength beyond the position of the rst resonance. (Actually the situation is a bit more complicated, because the amplitude antinode at the open end of the tube occurs somewhat beyond the end, but that will not be important here since we will be using the difference in length from one resonance point to the next for our calculation.) If the cap is removed, both ends are open, and thus there is an amplitude antinode at each end. In this case, resonance occurs when the shortest air column is one-half of a wavelength. (See Figure 123.) But as before, as the tube is lengthened, each successive resonance occurs as the length of the tube is increased by one half-wavelength beyond that of the previous resonance. Once the wavelength which produces resonance for a given frequency is found, the analysis consists of determining the corresponding velocity of the sound wave and comparing with the value for the speed of sound predicted by theory. In addition to the determination of the velocity of the sound, this experiment provides an apparatus, which enables you to investigate beat phenomena, i.e., the uctuation in amplitude produced when two waves of slightly different frequencies interfere. A tuning fork of known frequency is sounded simultaneously with the signal from the function generator. The frequency of the function generator is measured with a multimeter, and the difference between the measured frequency and the frequency of the fork is compared with the observed beat frequency. Figure 12-2: Resonance Modesfbr an Air Column Closed at One End Figure 12-3: Resonance Modesfor an Air Column Open at Bath Ends 81 80 The Physics Lab Manual 1 INTRODUCTION In the vibrating string experiment, waves were generated in a string xed at both ends, and a relationship was determined empirically between the tension T and the wavelength A of standing waves at resonance. In that experiment the frequency of the source and the length and density of the string remained constant, and resonance was obtained by varying the wavelengths of the propagating waves. This was done by varying the tension in the string, which in turn varied the wave speed, which in turn varied the wavelength. When the wavelength was such that an integer number of half-waves exactly t between the xed ends of the suing, resonance was detected through observation of the large amplitude response ofthe string. Coupling the observations with the predictions of theory enabled a detcnninau'on of the frequency of the oscillator. In the experiment you are about to do, resonance is also to be observed. But in this case the waves in question are the longitudinal disturbances produced by the passage of sound through air. The moving medium consists of the air molecules, whose motion cannot be observed visually; instead, resonance is detected by listening for the large amplitude response. Also, in this experiment, no property of the medium can be varied, i.e., there is no way to vary any property ofthe air analogous to the tension of the suing in the vibrating string experiment. Sound here is produced by a function generator connected to a loudspeaker; the frequency of the function generator is variable, but for reasons discussed below you will take your measurements at a xed frequency. The question then becomes: if the frequency isn't to be varied, and no property of the medium can be altered to change the wavelength, how can the system be tuned to resonance? The answer lies in the way the air through which the waves pass is physically dened. In this case, the apparatus used is a pair of cylindrical tubes, one ofwhich can be slid back and forth inside the other in order to adjust the length of the air column inside. Hence it is the length of the medium which is varied to tune the system to resonance. When, for a given Frequency, the length of the air column is just right, the xed wavelength will be such that interference between waves traveling down the column and those traveling up (after reection from the other end) will produce a large amplitude standing wave, the presence of which is revealed by an increase in the intensity of the sound.4 In the vibrating string experiment, the resonance condition occurred when an integer number of half-wavelengths t between the xed ends of the string, but in this experiment the boundary conditions are different. While there is an amplitude node at the closed end of the tube, the situation at the open end of the tube is somewhat more complicated. Sound waves consist of regions of compression and rarefaction of air molecules (analogous to the crests and troughs of transverse waves like those in the sring). While at the closed end, a compression is reected as a compression and a rarefacu'on as a rarefaction, at the open end of the tube a compression is reected as a rarefaction and vice versa; this occurs because a compression, when reaching the open end of the tube, rushes out, leaving a rarefaction to be reected back into the tube. Note 4 It should be noted that the longitudinal sound waves can be described in terms of the variation in the amplitude of the vibrations (i.e., displacement of air molecules from their equilibrium positions), or in terms of the pressure variations in the air (or other medium). The pressure is at a maximum when the displacement is at a minimum, so that an amplitude node is a pressure antinode and an amplitude antinodc is a pressure node. 0. '---'..--'A \fExperiment 12: Wave Motion: Standing Waves and the Speed of Sound and Beats Here R is the ideal gas constant (8.314J/mol-K), Tthe temperature (in Kelvins), and M the molar mass of the air (0.0288 kg/mol). Use the temperature from step P5 and the constants given above to calculate the theoretically predicted value of 1) from Equation (5), and compare with the values obtained in steps CI to C3. 05. Using the time recorded for a given number of beats, as observed in step P6, determine the beat frequency for both cases (one with the function generator set to a frequency higher than that of the tuning fork and one for a Frequency lower than the tuning fork). Compare these values with the difference between the frequency of the tuning fork and the frequency read from the multimeter in each case. In addition, noting any discrepancy between the reading of the multimeter and the setting on the dial of the function generator, discuss the precision of the dial. 85 -' t vvv'Uv-UU'UU'UIIA'I'IU'Ui""-"'-'-""""" l 84 The Physics Lab Manual 1 CALCULATIONS C1 . Using the data from steps P1 and P2, nd the difference between each adjacent pair of resonance positions and record these distances as M2. Take the average of these values and multiply that average by 2 to determine the wavelength of the sound. Also calculate the uncertainty of this wavelength. (See the introductory pages of your lab manual or the tutorial on uncertainty if you don't remember how to do this.) C2. From the wavelength obtained in C1 and the measured frequency, determine the speed of sound in the air column, using the equation hf: v (1) C3. Repeat steps C1 and C2, but this time use the data from step P3. C4. In the vibrating string experiment a theoretical equation was used to relate the physical properties of the string to the speed of the propagating waves, viz T v: (2) sign A similar equation is obtained in the analysis of any mechanical wave: the wave speed is determined by two parameters, one characterizing the elasticity of the medium and one characterizing the inertia (T and 7t, respectively, in Equation (2)). The form of this dependence is always the same: elastic factor v = 7. (3) inertial factor In the case of a sound wave propagating in a gas, the explicit expression is v : (4) p where Y is a constant characteristic of the molecular nature of the gas, and for air is 1.40. P is the pressure of the gas, and p is the density. Contrary to intuition and the appearance of Equation (4), perhaps, the speed of sound is actually independent of pressure. This is because as the pressure increases, so does the density, if the temperature remains constant. This implies that the speed depends only on temperature, and the explicit temperature dependence can be found if a relation is known between pressure, temperature, and volume. To the extent that the air in the laboratory can be treated as an ideal gas, the ideal gas law, PV= MRI is just such a relation. Using it to rewrite Equation (4), {YRT '0 = (5) M --A---'(----'-'---- QUESTION 2 Suppose you set up the apparatus as shown in figure 12-5, on page 82 of your lab manual. You set the frequency on the function generator to 1040 Hz, and you obtain resonance for the tube lengths shown in table below. You measure the temperature of the room to be 21"C. Use the instructions in C1 on page 83 of your lab manual to determine the wavelength, in cm of these sound waves. Tube Lengths for Resonance in cm 9.3 26.5 42.4 58.9 QUESTION 3 If you set up the apparatus as shown in figure 12-5 on page 82, and you find the average difference between resonance lengths for the tube is 8.13 cm, when the frequency on the function generator is set at 2050 Hz, what is the speed of sound in the tube, in m/s? QUESTION 4 Suppose you set up the apparatus as shown in figure 12-5 on page 82 of your lab manual. With the cap on, so that the tube is closed at one end, you measure the average spacing between resonance positions to be 17.1 cm. If you now leave the frequency produced by the function generator unchanged, but remove the cap, so that the tube is now open at both ends, what would you expect the average spacing (in cm) between the new resonance positions to be