velocity of sound

Nicholas J. Pinto and Claudio Guerra-Vela. Department of Physics and Electronics. University of Puerto Rico at Humacao. Sponsored by the National Science Foundation (NSF) © All rights reserved (2006)

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Experiment 10

VELOCITY OF SOUND IN AIR – RESONANCE TUBE

Reference: Physics Laboratory Experiments – J. D. Wilson, DC Heath and Co.

Objective To measure the velocity of sound in air at room temperature

Theory Mechanical systems generally have one or more natural vibrating frequencies. When a system is driven at a natural frequency, there is a maximum energy transfer and the vibration amplitude increases to a maximum. In these conditions we say that the system is in resonance with the driving source and refer to the particular frequency at which this occurs as a resonance frequency. From the relationship between the frequency f, the wavelength ?, and the wave speed v, which is v = ?f, if the frequency and wavelength are known, the wave speed can be determined. Or, if the wavelength and speed are known, the frequency can be determined

Figure 10-1 A pipe of length L closed at its bottom end and opened at its upper end showing the fundamental or first

harmonic standing sound wave

Air columns in pipes or tubes of fixed lengths have particular resonant frequencies. The interference of the waves traveling down the tube and the reflected waves traveling up the tube produces longitudinal standing waves, which must have a node at the closed end of the tube and an anti-node at the open end of the tube. The resonance frequencies of a pipe or tube depend on its length L. As shown in Figure 10-1, and 10-2 a certain number of wavelengths or “loops” that can be “fitted” into the tube length with the node-anti-node requirements. Since each loop corresponds to one half-wave length, resonance occurs when the length of the tube is nearly equal to an odd 2 number of quarter wavelengths i.e. L = λ/4, 3λ/4, 5λ/4, etc, or in general,

L = n?/4, n = 1, 3, 5, etc 1


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Or

? = 4L/n 2

Incorporating the frequency, f and the speed, v through the relationship

?f = v, 3

Also called dispersion relation, f = v/?, and we have:

f n = nv/ 4λ, n = 1, 3, 5, etc 4

These fn frequencies are the resonance frequencies for all of the standing waves that can vibrate in the pipe

Figure 10-2 A pipe of length L closed at its bottom end and opened at its upper end showing the second harmonic

standing sound wave

Hence, an air column (tube) of length L has particular resonance frequencies and will be in resonance with the corresponding odd-harmonic driving frequencies. As can be seen from the above equation, the three experimental parameters involved in the resonance condition of an air column are f, v, and L. To study the resonance in this experiment, the length L of an air column will be varied for a given driving frequency. The length of the air column will be achieved by changing the position of the movable piston in the tube as seen in Figure 10-3. As the piston is removed, increasing the length of the air column, more wavelength segments will fit into the tube, consistent with the node-anti node requirements at the ends. The difference in the tube lengths when successive antinodes are at the open end of the tube and resonance occurs is equal to a half wavelength, for example:

? L = L2 - L1 = 3λ/4 - λ/4 = λ/2, 5


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as seen in Figure 10.3. When an anti node is at the open end of the tube, a loud resonance tone is heard. Hence, the tube lengths for antinodes to be at the open end of the tube can be determined by moving the piston away from the open end of the tube and “listening” for successive resonance frequencies. No end correction is needed for the anti node occurring slightly above the end of the tube in this case, since the differences in tube lengths for successive antinodes is equal to ?/2. Since the frequency of the driving source will be set up for us at the beginning of the experiment, and the difference in tube length between successive antinodes, ∆L is measured, the wavelength is determined from Equation 10-4, as λ = 2 ∆L, and the speed of the sound wave will be deduced from Equation 10-3

Figure 10-3 The first three standing sound waves in a pipe

Figure 10-4 Experiment set up

The speed of sound in air is temperature dependent and is given to a good approximation over the normal temperature range by:


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v = (331.5 + 0.6T) m/s 6

Where T is the air temperature in degrees Celsius. The equation above shows that the speed of sound in air at 0° C is 331.5 m/s and increases by 0.6 m/s for each degree of temperature increase. For example, at 20 C, the speed of sound is 343.5 m/s Examples 1. Weak background noises from a room set up the fundamental standing wave in a cardboard tube of

length L = 67.0 cm with two open ends. Assume that the speed of sound in the air within the tube is 343 m/s. What frequency do you hear from the tube if you jam your ear against one end of the tube?

Solution: With your ear closing one end, the fundamental frequency is given by f = v/4L = 343/(4)(0.67) = 128 Hz

2. Weak background noises from a room set up the fundamental standing wave in a cardboard tube of

length L = 67.0 cm with two open ends. Assume that the speed of sound in the air within the tube is 343 m/s. What frequency do you hear from the tube if you move your head away enough so that the tube has two open ends?

Solution: With both ends open, the fundamental frequency is given by f = v/2L = 343/(2)(0.67) = 256 Hz

3. The audible frequency range for normal hearing is about 20 Hz to 20 kHz. What are the wavelengths

of sound waves at these frequencies at 20 C?

Solution: ? = v / f for 20 Hz, ? = 343/20 = 17.15 m for 20 kHz, ? = 343/20000 = 1.715 cm

4. The shortest wavelength emitted by a bat is about 3.3 mm. What is the corresponding frequency?

Solution: f = v / ? = 343/0.0033 = 104 kHz

5. Diagnostic ultrasound of frequency 4.5 MHz is used to examine tumors in soft tissue. What is the

wavelength in air of such a sound wave?

Solution: ? = v / f = 343/(4.5x106) = 76.2 µm

6. Diagnostic ultrasound of frequency 4.5 MHz is used to examine tumors in soft tissue. If the speed of

sound in tissue is 1500 m/s, what is the wavelength of this wave in tissue?

Solution: ? = v / f = 1500/(4.5x106) = 333.3 µm

7. A conical loudspeaker has a diameter of 15.0 cm. At what frequency will the wavelength of the sound

it emits in air be equal to its diameter?

Solution: ? = 15.0 cm, and f = v / ? = 343/0.15 = 2.286 kHz


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Equipment and Materials Voltage Sensor Resonance Tube with accessories Microphone Computer System Data Studio™

Procedure

Overview 1. Use the ‘output’ feature of the interface to drive a speaker to vibrate the air inside the resonance tube at

a fixed frequency of 3,000 Hz

2. Use Data Studio™ to control the speaker’s output frequency with the signal generator

3. Use a microphone mounted in the resonance tube to measure the amplitude of sound

4. Use the voltage sensor to measure the signal from the microphone. A piston inside the tube is used to adjust the length of the column of air inside the tube

5. Use the software to display the output signal of the speaker, and the input signal from the microphone

6. Change the position of the piston to determine the distances between successive antinodes in the standing sound waves that occur inside the tube

7. Use the distance to determine the wavelength of the sound and then calculate the speed of sound

Part I

1. Connect the speaker to the voltage source and program the source to output a sine wave of 3.0 kHz and an amplitude of 1.0 V

2. Position the movable piston so that it is all the way inside the tube

3. Slowly move the piston out while listening to the tone inside the tube or watching the virtual oscilloscope screen in the computer’s monitor

4. As you move the piston outward you will hear distinct maxima and minima in the sound

5. Carefully locate the position of the maxima, starting with the piston all the way inside the tube

6. For each maxima that you locate, draw a diagram similar to Figure 1 to represent pictorially what you hear happening inside the tube

7. Calculate the wavelength of the standing wave inside the tube

8. Calculate the velocity of sound in air and compute the % difference

9. Repeat the same experiment for one more frequency Part II

1. Have your lab partner set a frequency that he/she does not disclose

2. Repeat the same steps as in Part I to locate the maxima inside the tube in order to find the wavelength

3. Use the velocity of sound that you found in Part I to determine the frequency of the sound wave

4. Calculate the % difference

5. Try one more frequency, and repeat the exercise


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Experiment 10. Laboratory Report Velocity of sound in air – Resonance Tube Section _______ Laboratory bench _______ Date: _______________________________________________________________ Student’s names: ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ Part I (Assume that the accepted value of the speed of sound at 20 °C and a pressure of 1.0 atm is 343 m/s)

1. Once you set up the frequency of the audio generator, position the movable piston to locate three consecutive maxima and write down their positions in Table I below

2. Make the necessary calculations to fill out the data in Table 1

Table I (f = 3.0 kHz) Position of first maximum

Position of second maximum

Position of third maximum

Average distance between adjacent maxima

Speed of sound in air (measured)

% Difference with accepted value

3. Repeat the exercise for any other frequency greater that 3.0 kHz and write down your results in Table II below

Table II Position of first maximum

Position of second maximum

Position of third maximum

Average distance between adjacent maxima

Speed of sound in air (measured)

% Difference with accepted value

Part II

1. Have your lab partner set a frequency that he/she does not disclose

2. Repeat the same steps as in Part I to locate the maxima inside the tube in order to find the wavelength

3. Use the best velocity of sound that you found in Part I to determine the frequency of the sound wave

4. Calculate the % difference

5. Try one more frequency


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Questions 1. For a resonance tube apparatus with a total tube length of 1.0 m, how many resonance positions would

be observed when the movable piston is slowly moved away from the speaker. The frequency of the speaker is 500 Hz. Show your calculations

2. Suppose that the laboratory temperature were 5°C higher than the temperature at which you performed

the experiment. Explain what effect(s) this would have on the experimental results


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Experiment 10. Questions Velocity of Sound in Air – Resonance Tube This questionnaire has some typical questions on experiment 10. All students who are taking the laboratory course of University Physics I must be able to correctly answer it before trying to make the experiment. 1. Weak background noises from a room set up the fundamental standing wave in a cardboard tube of

length L = 67.0 cm with two open ends. Assume that the speed of sound in the air within the tube is 343 m/s. What frequency do you hear from the tube if you jam your ear against one end of the tube? a. 128 Hz b. 256 Hz c. 512 Hz d. 230 Hz

2. Weak background noises from a room set up the fundamental standing wave in a cardboard tube of

length L = 67.0 cm with two open ends. Assume that the speed of sound in the air within the tube is 343 m/s. What frequency do you hear from the tube if you move your head away enough so that the tube has two open ends? a. 128 Hz b. 256 Hz c. 512 Hz d. 230 Hz


of sound waves at 20 Hz at 20 C? a. 6860 m b. 17 m c. 0.06 m d. 400 m


of sound waves at 20 kHz at 20 C? a. 1.7 m b. 6860 km c. 58 m d. 400 km

5. The shortest wavelength emitted by a bat is about 3.3 mm. What is the corresponding frequency?

a. 1132 Hz b. 104 Hz c. 104 kHz d. 104 MHz

6. Diagnostic ultrasound of frequency 4.5 MHz is used to examine tumors in soft tissue. What is the

wavelength in air of such a sound wave? a. 76 mm b. 76 µm c. 76 cm d. 76 m


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7. Diagnostic ultrasound of frequency 4.5 MHz is used to examine tumors in soft tissue. If the speed of sound in tissue is 1500 m/s, what is the wavelength of this wave in tissue? a. 333 mm b. 333m c. 333 km d. 333 µm

8. A conical loudspeaker has a diameter of 15.0 cm. At what frequency will the wavelength of the sound

it emits in air be equal to its diameter? a. 5145 Hz b. 2.3 kHz c. 51 Hz d. 23 Hz

velocity of sound

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