PHYS 164 LAB #21 The Speed of Sound in Air DATA f (Hz) 1000 1500 2000 2500 3000 3500 4000 3rd 39.5 25.3 18.4 15.2 12.5 10.1 8.8 dL (cm) 0.1 0.1 0.1 0.1 0.1 0.1 0.1 l/2 2nd-1st 17.2 11.1 9.1 6.5 5.8 4.8 4.3 l/2 3rd-2nd 17.5 11.5 8.6 7 5.9 4.9 4.3 d (l/2) (cm) 0.2 0.2 0.2 0.2 0.2 0.2 0.2 Avg l (m) 0.347 0.226 0.177 0.135 0.117 0.097 0.086 dl (m) 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 v = fl (m/s) 347 339 354 337.5 351 339.5 344 dv (m/s) 0.547 0.526 0.577 0.635 0.717 0.797 0.886 vaverage d(vaverage) vexpected d(vexpected) (m/s) 344.571429 (m/s) 0.66928571 (m/s) 345 (m/s) 1 % Diff 0.1242236 Agree? YEP 1st 4.8 2.7 0.7 1.7 0.8 0.4 0.2 L (cm) 2nd 22 13.8 9.8 8.2 6.6 5.2 4.5 Avg l/2 (cm) 17.35 11.3 8.85 6.75 5.85 4.85 4.3 ANALYSIS f (Hz) 1000 1500 2000 2500 3000 3500 4000 Dev from Average 2.42857143 5.57142857 9.42857143 7.07142857 6.42857143 5.07142857 0.57142857 46 #21 The Speed of Sound in Air Objectives 1. To determine whether the speed of sound in air is a function of the frequency of the sound. 2. To extend our study of standing waves and resonance. Introduction and Theory In order to determine whether the speed of sound in air is a function of the frequency of the sound, we will measure the speed of sound for a range of frequencies. The relationship between frequency f, wavelength 2, and the speed v for all waves is v=fa.doc (21.1) As a result, v can be calculated if we know f and 2. We will use our understanding of standing waves to measure the wavelength à for a given input frequency f. An audio oscillator attached to a speaker will be used to produce tones of different fre- quencies. The speaker in turn will be used to produce standing sound waves in a resonance tube, and the wavelength will be calculated from measurements of the nodes in the standing wave patterns. The nodes are detected using a small microphone positioned at the end of the tube which is connected to an oscilloscope in order to detect the intensity of the sound wave in the tube. A picture of the apparatus is shown in Fig. 21.1. The resonance apparatus consists of a large diameter tube with an adjustable plung- er/piston that can be used to adjust the effective length of the air column in the tube. The tube is marked with a scale so that the length of the air column can be measured. The small loudspeaker mounted at the left end of the tube is driven by the frequency generator to produce a tone of one frequency. This sound wave excites the air column and will cause the column to resonate if the length of the air column is just the right length to produce a standing wave pattern having a node at the end of the plunger and an antinode at the speaker end of the tube. Figure 21.2 illustrates several of the possible standing wave patterns. Note that the distance between nodes (locations of resonance) is always 1/2. Thus, if we measure the locations for successive resonance patterns, 2/2 and thus a can easily be determined. The speed of sound can then be found using Eq. 21.1 above. 02 47 ++TER 0 1594 SEDANCE ES Fig. 21.1. Resonance apparatus used to measure standing waves in an air column. The length of the air column is changed by adjusting the position of the plunger. 09 A N First resonance _ Stov Α Ν N A N. Second Resonance obroton above A N A N A N Third resonance H N A 22 - Α Ν Α N AN Fourth resonance E E A= antinode N=node Fig. 21.2. Several possible standing wave patterns in air columns open at one end. 48 Apparatus Frequency generator (audio oscillator) - Resonance tube apparatus with speaker and microphone Oscilloscope Procedure 1. Start with the plunder all of the way in. Before recording data it will pay to practice detecting the resonances. It is important that the volume of the tone from the loud- speakers be as low as possible so that it will not interfere with other students in the room. It should be set so that you can observe a clear sinusoidal wave on the oscillo- scope when the tube is in resonance. Practice using a frequency of 2000 Hz. Move the plunger to the right and observe the locations of resonance. As the length of the air column increases the amplitude of the wave on the oscilloscope will increase as a node is approached and will be a maximum at resonance. Once you are confident that you can detect the resonances and under- stand how they are spaced, you are ready to begin taking data. You should find a defi- nite regularity to the spacing of the nodes (resonances). 2. Set the frequency as close to 4,000 Hz as you can and record the actual frequency. Starting with the plunder all of the way in, move it slowly to the right to detect the first resonance. Continue moving it to the right in order to find at least four resonances. The plunger can be moved to the left and right in order to “fine tune” the position of the resonance. Measurements can be repeated several times if you have any doubt about the results. Once you are sure of a position, record it along with its uncertainty in your spreadsheet. 3. Repeat step 2 until data is collected for all frequencies from 1000 Hz to 4000 Hz, in 500 Hz steps. 4. Recall that the difference between any two consecutive resonance levels corresponds to one-half a wavelength. In your spreadsheet calculate the differences between reso- nances 2 and 1, between resonances 3 and 2, and between resonances 4 and 3 for each frequency. If, for a given frequency, these values differ from one another by a consid- erable amount, you have inadvertently skipped a resonance point in taking your data. If this has occurred, repeat your measurements for that frequency. 49 Analysis of Data 1. For each frequency calculate the average value of 1/2 (in cm), along with its uncertain- ty. From this, calculate the wavelength 2 (in m), along with its uncertainty. 2. Using these average wavelength values, calculate the speed of the sound wave for each frequency along with its uncertainty from Eq. 21.1. 3. Plot a graph of speed vs. frequency for your data, making sure to include the error bars. In plotting the graph be sure to adjust the v-axis limits so as to spread the data fully along the y-direction. You may have to fiddle with this several times in order that the limits of the error bars can be seen. 4. Is it possible to draw a horizontal line through your data so that it passes within all of the error bars? If so draw this line using the drawing tools in Excel. If not, draw the best fit straight line through your data points. Conclusion Include the following in your discussion: 1. Discuss the dependence of the speed of sound on frequency. 2. Discuss the relationship between fand 2 in Eq. 21.1. 3. How could the experiment be modified to extend the range of validity of your conclu- sion, and what problems would be encountered in doing this? 4. Compare the average value obtained for v with the expected value at room temperature (71°F) of 345 + 1 m/s. oroda
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