8.4 Resonance in Air Columns

Transcription

1 Resonance in Air Columns Sound waves from one source can cause an identical source to vibrate in resonance. Just as with mechanical resonance (section 6.7), a small force produces a large vibration. For example, look at Figure 1. When the first tuning fork is struck and then silenced, a sound of the same frequency comes from the second fork even though it was not struck. Resonance has occurred because the forks have the same natural frequency. Energy has been transferred from one fork to the other by sound waves. matched tuning forks resonant air column sound waves from nearby fork Figure 1 Two tuning forks with identical frequency are mounted on identical wooden boxes open at one end and placed about 1 m apart. But why is the tuning fork mounted on a wooden box? Why is the box open at one end and closed at the other, and why is the box designed to be of a specific length? You will look at these variables in the next investigation. Investigation Resonance in Closed Air Columns An air column that is closed at one end and open at the other is called a closed air column. When a vibrating tuning fork is held over the open end of such a column and the length of the column is increased, the loudness increases sharply at very specific lengths. If a different tuning fork is used, the same phenomenon is observed except the maxima occur at different lengths. In this investigation, you will examine the relationship between the frequency of the tuning fork and the resonant length of a closed air column. Questioning Hypothesizing Predicting Planning Conducting IQUIRY SKILLS Recording Analyzing Evaluating Communicating closed air column: column closed at one end and open at the other Question What lengths of a closed air column will resonate in response to a tuning fork of a known frequency? Materials 80 cm of plastic pipe large graduated cylinder at least two tuning forks (e.g., and 1024 Hz) metre stick thermometer Music, Musical Instruments, and Acoustics 287

2 plastic pipe water tuning fork Prediction (a) Use the diagrams in Figure 3 to help you predict two lengths of a closed air column that will resonate in response to a 512-Hz tuning fork and a 1024-Hz tuning fork. Procedure 1. Place the plastic pipe in the graduated cylinder, as shown in Figure 2. Fill the graduated cylinder with water as close to the top as possible. 2. Sound the tuning fork and hold it over the mouth of the plastic pipe. Have your partner move the pipe slowly out of the water and listen for the first resonant point. At points of resonance, the intensity of the sound originating from the tuning fork will increase dramatically. Ignore points of slightly increased intensity that are not of the same frequency as the tuning fork. 3. Use the metre stick to measure the length of the air column for the first resonant point. Record your measurements in a chart similar to Table 1. Table 1 Tuning fork 1 (f = ) Tuning fork 2 (f = 1024 Hz) Resonant point Length (cm) Length Length (cm) Length (wavelengths) (wavelengths) first second third adjustable resonance tube Figure 2 Setup for Investigation Continue to raise the pipe, finding and measuring other resonant points. 5. Repeat steps 2 through 4 with a tuning fork of a higher frequency. 6. Record the air temperature in the room. Analysis (b) What is the speed of sound at the air temperature you recorded? (c) What is the wavelength of the sound wave emitted by each tuning fork used in the investigation? (d) For each tuning fork, what is the relationship between the length of the closed air column for the first resonant point you encountered and the wavelength of the tuning fork? (e) For each tuning fork, what is the relationship between the length of the closed air column for the second resonant point you encountered and the wavelength of the tuning fork? (f) As a general rule, what are the resonant lengths, expressed in wavelengths, for a closed air column? Evaluation (g) How did your measured lengths compare with your predicted lengths? Determine a percent difference in each case. (h) Describe the sources of error in this investigation and suggest improvements. 288 Chapter 8

3 8.4 Resonance in Closed Air Columns As you saw in section 6.8, when a series of transverse waves was sent down a rope to a fixed end, the wave was reflected back and interfered with the incident waves. A node always formed at the fixed end where the reflection occurred. In a similar way, when longitudinal sound waves are emitted by a tuning fork, some of them travel down the closed air column. The end of the tube reflects the sound waves back in the same way that the waves in a rope are reflected from the fixed end. A node is formed at the bottom of the column (Figure 3). Resonance first occurs when the column is 1 4 l in length, since a single node is formed. The next possible lengths with a node at one end are 3 4 l, 5 4 l,etc.thus, the resonant lengths in a closed air column occur at 1 4 l, 3 4 l, 5 4 l, 7 4 l, and so on. The resonant length of a wooden box that is open at one end and attached to a tuning fork is 1 4 l. For a 256-Hz tuning fork, 1 4 l would be approximately 34 cm at room temperature (20 C). (a) (b) 256 Hz 256 Hz 1 4 Sample Problem 1 A vibrating tuning fork is held near the mouth of a column filled with water. The water level is lowered, and the first loud sound is heard when the air column is 9.0 cm long. Calculate the following: (a) the wavelength of the sound from the tuning fork (b) the length of the air column for the second resonance 3 4 Solution (a) 1 l = 9.0 cm 4 l = 36 cm (c) 256 Hz (b) The air column length is increased by 1 l, or 18 cm, to obtain the second 2 resonance. Thus, the total length is 9.0 cm + 18 cm = 27 cm,or 3 4 l. Sample Problem 2 The first resonant length of a closed air column occurs when the length is 16 cm. (a) What is the wavelength of the sound? (b) If the frequency of the source is, what is the speed of sound? 5 4 Solution (a) first resonant length = 1 4 l 1 l = 16 cm 4 l = 64 cm or 0.64 m (b) The wavelength is 0.64 m. v = f l = ()(0.64 m) v = 328 m/s The speed of sound is m/s. Figure 3 Resonant lengths of a closed air column with a sound of given frequency. The diagrams show the nodes and s for the longitudinal displacement sound wave. (a) First resonant length (b) Second resonant length (c) Third resonant length Music, Musical Instruments, and Acoustics 289

4 Practice Answers cm; 90 cm; 150 cm cm; 45 cm cm Hz cm; 52.4 cm; 87.5 cm m; 346 m/s 7. (a) 92.0 cm; cm; 152 cm (b) 371 Hz; 284 Hz; 224 Hz Understanding Concepts 1. The first resonant length of a closed air column occurs when the length is 30.0 cm. What will the second and third resonant lengths be? 2. The third resonant length of a closed air column is 75 cm. Determine the first and second resonant lengths. 3. What is the shortest air column, closed at one end, that will resonate at a frequency of Hz when the speed of sound is 352 m/s? 4. A signalling whistle measures 5.0 cm from its opening to its closed end. Find the wavelength of the sound emitted and the frequency of the whistle if the speed of sound is 344 m/s. 5. The note B 4 (f = 494 Hz) is played at the open end of an air column that is closed at the opposite end. The air temperature is 22 C. Calculate the length of the air column for the first three resonant sounds. 6. A tuning fork causes resonance in a closed pipe. The difference between the length of the closed tube for the first resonance and the length for the second resonance is 54.0 cm. If the frequency of the fork is 320 Hz, find the wavelength and speed of the sound waves. 7. An organ pipe resonates best when its length is 1 4 l. Three pipes have lengths of 23.0 cm, 30.0 cm, and 38.0 cm. (a) Find the wavelength of the sound emitted by each pipe. (b) Find the frequency of each pipe, if the speed of sound is 341 m/s. Questioning Hypothesizing Predicting Planning Conducting IQUIRY SKILLS Recording Analyzing Evaluating Communicating ever touch a glass cylinder with a vibrating tuning fork the cylinder might shatter. Investigation Speed of Sound in a Closed Air Column You now have knowledge of the resonant lengths and resonant frequencies for closed air columns. You can use a similar approach to that used in Investigation to design and carry out a method to determine the speed of sound in a closed air column. Question What is the speed of sound in a closed air column? Prediction (a) Using information from section 7.3, predict the speed of sound in a closed air column. Design Discuss with your group and teacher the method you will follow to perform your experiment. Decide on the materials and how you will use them to obtain data. Write out your experimental design, including your procedure, and prepare a table to record your observations. Draw a labelled diagram showing the proposed experimental setup. Submit your procedure to your teacher for approval before commencing with the investigation. Materials The materials will depend on your design. Materials that you may consider include plastic tube open at both ends, tall glass cylinder, metre stick, ther- 290 Chapter 8

5 8.4 mometer, tuning fork, and striking pad for the tuning fork. Prepare a list of materials and get your teacher s approval. open air column: column open at both ends Analysis (b) Show the relevant equation(s) and calculations used to determine the numerical solution to the problem. Evaluation (c) Answer the original question and describe how you determined the speed of sound in a closed air column. (d) Discuss the accuracy of your answer, including percentage error, and identify possible sources of error. (e) Calculate the percentage difference between your predicted value and the experimental value for the speed of sound in a closed air column. (f) Evaluate your design for this experiment. If you were to perform it again, what changes would you make to improve it? (g) Compare the value you obtained for the speed of sound with the value found in a standard source (such as a textbook) and determine the percent difference. (h) Why is your procedure an easier and more accurate way to determine the speed of sound in air than to measure it directly? 1 2 Synthesis (i) How would you use the same approach to measure the speed of sound in a gas, such as carbon dioxide or helium? Resonance in Open Air Columns Resonance may also be produced in an open air column, that is, a column that is open at both ends. If a standing wave interference pattern is created by reflection at a free end, an occurs at the free end. Since a pipe is open at both ends, s occur at both ends. The first length at which resonance occurs is 1 2 l. Succeeding resonant lengths will occur at l, 3 2 l,2l, and so on (see Figure 4). 3 2 Sample Problem 3 An organ pipe, 3.6 m long and open at both ends, produces a musical note at its fundamental frequency. (a) What is the wavelength of the note produced? (b) What is the frequency of the pipe if the speed of sound in air is 346 m/s? Solution (a) The fundamental frequency corresponds to the simplest resonance pattern, which has a resonant length of 1 2 l. 1 l = 3.6 m 2 l = 7.2 m The wavelength of the note is 7.2 m. Figure 4 Resonant lengths of an open air column with a sound of a given frequency 2 Music, Musical Instruments, and Acoustics 291

6 (b) v = f l v f = l = 3 46 m/s 7. 2 m f = 48 Hz The frequency of the pipe is 48 Hz. SUMMARY Resonance in Air Columns Resonance occurs in closed air columns at lengths of 1 4 l, 3 4 l, 5 4 l, and so on of the original sound wave. The resonant lengths of open air columns are 1 2 l, l, 3 2 l,2l, and so on of the original sound wave. Section 8.4 Questions Understanding Concepts 1. What is the length of an open air column that resonates at its first resonant length with a frequency of 560 Hz? (The speed of sound is 350 m/s.) 2. The second resonant length of an open air column is 48 cm. Determine the first and third resonant lengths. 3. An organ pipe, open at both ends, resonates at its first resonant length with a frequency of 128 Hz. What is the length of the pipe if the speed of sound is 346 m/s? 4. A Hz tuning fork is sounded and held near the mouth of an adjustable column of air open at both ends. If the air temperature is 20.0 C, calculate the following: (a) the speed of the sound in air (b) the wavelength of the sound (c) the minimum length of the air column that produces resonance 5. A closed air column is 60.0 cm long. Calculate the frequency of forks that will cause resonance at (a) the first resonant length (b) the third resonant length (The speed of sound is 344 m/s.) 6. In an air column, the distance from one resonance length to the next is 21.6 cm. What is the wavelength of the sound producing resonance if the column is (a) closed at one end? (b) open at both ends? 7. What vibrates to create sound in a column of air? 8. How would a higher air temperature affect the lengths of the resonating air columns in Investigation 8.4.1? Why? 9. When water is added to a bottle, what happens to the pitch of the sound as the water is added? Explain why. Applying Inquiry Skills 10. Explain how the external ear and ear canal magnify sounds entering the ear. 292 Chapter 8