
Concept explainers
A pipe that is open at both ends will form standing waves, if properly excited, with antinodes near both ends of the pipe. Suppose we have an open pipe 60 cm in length.
a. Sketch the standing-wave pattern for the fundamental standing wave for this pipe. (There will be a node in the middle, and antinodes at either end.)
b. What is the wavelength of the sound waves that interfere to form the fundamental wave?
c. If the speed of sound in air is 340 m/s, what is the frequency of this sound wave?
d. If the air temperature increases so that the speed of sound is now 358 m/s. by how much does the frequency change?
e. Sketch the standing-wave pattern and find the wavelength and frequency for the next harmonic in this pipe.
(a)

Sketch the sound-wave pattern for this pipe.
Answer to Problem 3SP
The fundamental wave pattern in the pipe is drawn.
Explanation of Solution
Conclusion:
Therefore, the fundamental sound-wave pattern is drawn.
(b)

The wavelength of the sound wave.
Answer to Problem 3SP
The wavelength is
Explanation of Solution
Given Info: The length of the pipe is
Write the formula to calculate the fundamental wavelength inside the pipe.
Here,
L is the length of the pipe
Substitute
Conclusion:
Therefore, the wavelength is
(c)

The frequency of the sound wave.
Answer to Problem 3SP
The frequency of the sound wave is
Explanation of Solution
Given Info: The wavelength of the sound wave is
Write the expression to calculate the speed of the sound wave.
Here,
f is the frequency of the sound wave
Substitute
Conclusion:
Therefore, the frequency of the sound wave is
(d)

The change in frequency of the sound wave.
Answer to Problem 3SP
The change in frequency of the sound wave is
Explanation of Solution
Given Info: The wavelength of the sound wave is The longest possible wavelength is
Write the expression to calculate the speed of the sound wave.
Here,
Substitute
Write the formula to calculate the frequency change for the sound wave.
Here,
Substitute
Conclusion:
Therefore, the change in frequency of the sound wave is
(e)

Sketch the standing-wave pattern for the next harmonic and find wavelength and frequency.
Answer to Problem 3SP
The frequency of the second harmonic wave is
Explanation of Solution
For the second harmonic, the frequency is twice as that of the fundamental standing-wave and wavelength is half of that value of the fundamental standing wave. Therefore the frequency of the second harmonic wave is
The sketch of the second harmonic wave is shown below.
Conclusion:
Therefore, the frequency of the second harmonic wave is
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Chapter 15 Solutions
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