Q: A flute (open at both ends) and a clarinet (open at one end, closed at the other) are both approximately 66 cm long. If the speed of sound in air is $$350\ \mathrm{m/s}$$, what is the fundamental frequency of the clarinet?

$$f_1$$ = \dfrac{v}{4L} = \dfrac{350}{4 \times 0.66} \approx 133\ \mathrm{Hz}$$

Q: Two sound waves of frequencies $$f_1 = 440$$\ \mathrm{Hz}$$ and f_2$ = 444\ \mathrm{Hz}$$ are played together. What is the beat frequency heard?

$$f_{\text{beat}} = |f_1$ - f_2$| = 4\ \mathrm{Hz}$$

Q: A standing wave on a string has a node spacing of $$20\ \mathrm{cm}. $$What is the wavelength of the wave?

$$\lambda = 2 \times 20\ \mathrm{cm} = 40\ \mathrm{cm}$$

Q: A thin film of magnesium fluoride ($$n = 1.39) is $$used as an antireflection coating for light of wavelength $$510\ \mathrm{nm}. $$What is the minimum thickness $$d of $$the film for destructive interference?

$$d = \dfrac{\lambda}{4n} = \dfrac{510\ \mathrm{nm}}{4 \times 1.39} \approx 92\ \mathrm{nm}$$

Question 1

Which of the following best describes the principle of superposition for waves?

Accepted Answer

The displacement of the medium at a point is the sum of the displacements due to each individual wave present at that point.

Question 2

A string of length $$L is $$fixed at both ends. Which of the following equations gives the allowed wavelengths for standing waves on this string?

Accepted Answer

$$\lambda_m = \dfrac{2L}{m}$$, where $$m = 1, 2, 3, \ldots$$

Question 3

A flute (open at both ends) and a clarinet (open at one end, closed at the other) are both approximately 66 cm long. If the speed of sound in air is $$350\ \mathrm{m/s}$$, what is the fundamental frequency of the clarinet?

Accepted Answer

$$f_1$$ = \dfrac{v}{4L} = \dfrac{350}{4 	imes 0.66} \approx 133\ \mathrm{Hz}$$

Question 4

Which of the following statements about nodes and antinodes in a standing wave is correct?

Accepted Answer

Nodes are points where the displacement is always zero, and antinodes are points where the displacement is maximum.

Question 5

Two sound waves of frequencies $$f_1 = 440$$\ \mathrm{Hz}$$ and f_2$ = 444\ \mathrm{Hz}$$ are played together. What is the beat frequency heard?

Accepted Answer

$$f_{	ext{beat}} = |f_1$ - f_2$| = 4\ \mathrm{Hz}$$

Question 6

A standing wave on a string has a node spacing of $$20\ \mathrm{cm}. $$What is the wavelength of the wave?

Accepted Answer

$$\lambda = 2 	imes 20\ \mathrm{cm} = 40\ \mathrm{cm}$$

Question 7

Which of the following best describes constructive interference?

Accepted Answer

When two waves are in phase and their amplitudes add to produce a larger amplitude.

Question 8

A thin film of magnesium fluoride ($$n = 1.39) is $$used as an antireflection coating for light of wavelength $$510\ \mathrm{nm}. $$What is the minimum thickness $$d of $$the film for destructive interference?

Accepted Answer

$$d = \dfrac{\lambda}{4n} = \dfrac{510\ \mathrm{nm}}{4 	imes 1.39} \approx 92\ \mathrm{nm}$$

Question 9

A standing sound wave in a closed-closed tube of length $$L$$ has which of the following allowed frequencies?

Accepted Answer

$$f_m = \dfrac{m v}{2L}$$, where $$m = 1, 2, 3, \ldots$$

Question 10

Two loudspeakers emit sound waves of the same frequency and are in phase. At a certain point, the path-length difference between the waves from the two speakers is $$\lambda/2. $$What type of interference occurs at this point?

Accepted Answer

Destructive interference

Question 11

A string with linear density $$\mu = 5.0\ \mathrm{g/m} is $$stretched with a tension $$T = 8.0\ \mathrm{N}. $$What is the speed of waves on the string?

Accepted Answer

$$v = \sqrt{\dfrac{T}{\mu}} = \sqrt{\dfrac{8.0}{0.0050}} = 40\ \mathrm{m/s}$$

Question 12

Which of the following is a real-world example where standing waves play a critical role?

Accepted Answer

The collapse of the Tacoma Narrows Bridge due to resonance with a standing wave mode.

Question 13

A laser cavity of length $$L = 0.310372\ \mathrm{m}$$ supports a standing wave of wavelength $$\lambda = 632.9924\ \mathrm{nm}. $$What is the mode number $$m of $$this standing wave?

Accepted Answer

$$m = \dfrac{2L}{\lambda} = \dfrac{2 	imes 0.310372}{6.329924 	imes 10$$^{-7}$$} \approx 980,650$$

Question 14

Which of the following equations gives the beat frequency when two waves of frequencies $$f_1$$ and $$f_2$$ are superimposed?

Accepted Answer

$$f_{	ext{beat}} = |f_1$ - f_2$|$$

Question 15

A tube open at one end and closed at the other has length $$L. $$Which of the following is true about the allowed standing wave frequencies?

Accepted Answer

$$f_m = \dfrac{m v}{4L}$$, where $$m = 1, 3, 5, 7, \ldots$$

Question 16

If two waves of equal amplitude $$a$$ and phase difference $$\Delta \phi$$ interfere, what is the amplitude $$A of $$the resulting wave?

Accepted Answer

$$A = 2a \cos\left(\dfrac{\Delta \phi}{2}\right)$$

Question 17

A shower stall is 2.45 m tall and closed at both ends. What is the fundamental frequency of standing sound waves in the stall if the speed of sound is $$343\ \mathrm{m/s}$$?

Accepted Answer

$$f_1$$ = \dfrac{v}{2L} = \dfrac{343}{2 	imes 2.45} \approx 70\ \mathrm{Hz}$$

Chapter 17: Superposition and Interference of Waves

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