PHYS112 : Labs

2 Sound Waves

In addition to reading this assignment, you may also need to refer to Appendix A on uncertainties and Appendix B on linear regressions.

Introduction

The speed of waves can be determined from measurements of the wavelength and frequency of standing waves as we did with transverse waves on a string in Lab 1. Here, we will apply this strategy to sound waves traveling in air. Recall that the speed $v$ of a wave is related to its wavelength $\lambda$ and frequency $f$ as

$$v = \lambda f$$ (7)

As we observed in Lab 1, standing waves occur in a one-dimensional medium with both ends fixed when whole multiples of half wavelengths fit into the length of the medium. This along with Eq. 7 can be used to show that these standing-wave modes of vibration have frequencies

$$f_n = \frac{v}{2L} n \hspace{1cm} (n = 1,2,3,...)$$ (8)

where $L$ is the length of the medium.

Standing sound waves in a tube have displacement nodes at closed ends and antinodes at open ends. Hence, in a tube closed at both ends, we expect standing sound waves to have nodes at the end points, like standing waves on a string. Sketches of the displacement profiles of three standing-wave modes of a tube closed at both ends are shown in Figure 2. It is also possible to free both ends of the medium by opening both ends of the tube. This leads to an identical set of standing-wave frequencies. The pressure profiles of standing waves in a tube with both ends open are identical to the displacement profiles for the string shown in Figure 1 of Lab 1.

Figure 2: Displacement profiles of three standing waves with the same frequency and wavelength in a closed tube with variable length.

In a tube with one end closed and one end open (an organ pipe, e.g.), standing waves must have displacement nodes at the closed end and antinodes at the open end. In this case, odd multiples of quarter wavelengths fit into the tube, giving standing-wave modes with frequencies

$$f_n = \frac{v}{4L} n \hspace{1cm} (n = 1,3,5,...)$$ (9)

Sketches of the pressure profiles of the first three standing-wave modes of a tube closed at one end are shown in Figure 3.

Figure 3: Displacement profiles of the first three standing-wave modes of a tube closed at one end.

You will study standing sound waves in a tube driven by a speaker. The apparatus enables you to investigate both a tube of variable length closed at both ends, and a tube of fixed length open at one end. You will test empirically the above description of the standing-wave modes of these systems and consider the question of whether or not the speed of sound in air depends on its frequency.

The speed of waves traveling in a medium can also be determined just as one would determine the speed of a car or a baseball - by measuring the time it takes the waves to travel a known distance. However, this can be a difficult measurement to make for sound waves, since the waves are invisible and travel at speeds in excess of 300 m/s. You will try to measure the speed of sound pulses traveling in air with the help of a microphone and Logger Pro.

Experiments

Standing Waves in a Closed Tube

1. Your group will be assigned a frequency in the range 1000 Hz $\leq f \leq$ 2000 Hz. Set the frequency dial on the function generator to your assigned frequency.

2. Slowly adjust the length of the tube with the plunger, listening for resonances. Record the length of the tube at each resonance you can find. If you're not sure what to listen for, consult your instructor.

Standing Waves in a Tube Open at One End

1. Remove the plunger and its support from the far end of the tube from the speaker. Lay the support on its side and rest the end of the tube on it.

2. Use a meter stick to measure the length of the tube (from the speaker to the open end).

3. Slowly adjust the frequency of the function generator, listening for resonances.

4. Find all of the standing-wave frequencies of the tube in the range 200 Hz $\leq f \leq$ 2000 Hz.

Measuring the Speed of Sound Pulses

For this experiment, you'll need a long cardboard tube. We have four tubes to share among lab groups. Do this experiment when one becomes available to your group.

1. Lay the tube on its side on your lab table with its open end facing the far wall. (If your tube is open at both ends, block one end with a textbook.) Remove any obstructions from your lab table that might reflect sound into the tube.

2. Measure the length of the tube with a meter stick.

3. Place your microphone on the lab table, flush with the open end of the tube and aimed into the opening.

4. Assuming a speed of sound of about 340 m/s, use the measured length of the tube to estimate how long it takes sound to travel from the microphone to the closed end of the tube and back again. It will be difficult to distinguish your data from background sound. This estimate will help you recognize clean measurements.

5. Run Logger Pro.

6. Click on File -> Open ... and open the file Probes & Sensors -> Microphone -> Microphone.

7. A Sensor Confirmation window will open. Make sure that the Interface and Channel setting is CH1 on LabPro: 1 and the Sensor is set to Microphone. (These should be the default settings.) Then, click the Connect button.

8. Click on Experiment -> Data Collection ... (or the button). On the Triggering tab, check the checkbox next to Triggering. Start with the triggering level at 2.

9. Click the Collect button.

10. Snap your finger as loudly as possible at the opening of the tube.

11. This sound pulse should trigger Logger Pro to collect a Sound Level vs. Time graph. If Logger Pro collects its data immediately after you press Collect, the background noise in the lab is above the sound level threshold set to trigger data acquisition. Click on Experiment -> Data Collection ... (or ) and increase the trigger level on the Triggering tab.

12. If your snap triggers Logger Pro to acquire data, take a look at the Sound Level vs. Time graph. If you can identify two sound pulses on the graph that stand out above the background, and that have a separation in time of roughly the right size, save the graph for analysis.

13. Collect at least five good measurements.

Analysis

Standing Waves

1. Determine the $n$ values of your closed-tube measurements. They should be in order of increasing length, with the shortest length corresponding to $n=1$.

2. For your open-tube measurements, use Eq. 9 to predict the expected standing-wave frequencies of the tube in the range 200 Hz $\leq f \leq$ 2000 Hz. Then use these predictions as a guide for assigning $n$ values to your measured frequencies.

3. Your measurements from the two standing-wave experiments are a collection of (length, frequency) pairs. In a spreadsheet, set up ten columns with titles: Description, $L$ [m], $\sigma_L$ [m], $\lambda$ [m], $\sigma_\lambda$ [m], $n$, $f$ [Hz], $\sigma_f$ [Hz], $v$ [m/s], $\sigma_v$ [m/s]. Put your data into the Description, $L$, $\sigma L$, $f$, and $\sigma f$ columns. (The $\sigma$'s stand for uncertainties.) In the Description column, indicate whether the data are from the closed or open tube and include any other notes you find helpful.

4. Set up calculations to fill out the $\lambda$, $\sigma \lambda$, $v$, and $\sigma v$ columns. You may need to think a little about how to convert your measured lengths into wavelengths.

5. Calculate a best value for the speed of sound, along with its uncertainty, from your closed-tube measurements. Put your best value and the corresponding frequency on the board at the front of the lab.

6. Collect all of the closed-tube speed measurements from the board. Put them, along with the results of all of your open-tube measurements, in an $f [Hz]$, $v [m/s]$, $\sigma_v [m/s]$ table in your spreadsheet. Use the Excel chart wizard to make a graph of frequency vs. speed including error bars.

7. Devise an quantitative strategy for concluding, based on your data, whether or not the speed of sound waves in air depends on the frequency of the sound.

Measuring the Speed of Sound Pulses

1. Open each of your Sound Level vs. Time measurements in Logger Pro, and use Analyze -> Examine (or the button) to measure the times of the snap pulse and its echo.

2. In a spreadsheet, turn your snap/echo times into time intervals and combine these with the length of the tube to calculate the speeds of the pulses.

3. From these results, calculate a best value of the speed of sound and its uncertainty.

Before You Leave Lab

Discuss with your instructor preliminary answers to the questions below.

Hand In ...

... a printout of your spreadsheet(s) and answers to the following questions.

1. Based on your observations, does the speed of sound depend on frequency in the range 200 Hz $\leq f \leq$ 2000 Hz? Show your graph of $v$ vs. $f$, and give explicit reasoning for your conclusion.

2. Do the standing-wave frequencies you predicted for the open tube using Eq. 9 agree with the frequencies you actually measured within uncertainty? If not, what do you think might be the reason(s)?

3. Report your best value of the speed of sound pulses along with its uncertainty. Does this result agree with your standing-wave results within uncertainty?

Copyright © 2006-2009, L.A. Riley, T. J. Carroll, J.S. Scott

Updated Sun Apr 26 23:00:14 2009

This work is licensed under a Creative Commons License.