Q: A guitar string has a length of 0.86 m and a mass per unit length of 0.015 kg/m. If the string vibrates at its fundamental frequency of 41 Hz, what is the tension in the string?

$$F_T = (2Lf_1)^2$ \mu = [2 \times 0.86 \times 41]^2 \times 0.015 = 76\ \mathrm{N}$$

Q: A tube closed at one end and open at the other resonates at a frequency of 25 kHz. If the distance from the closed end to the nearest displacement node is 0.026 m, what is the speed of sound in the tube?

$$v = 2fL = 2 \times 25,000 \times 0.026 = 1,300\ \mathrm{m/s}$$

Q: A commuter train moves at 40.0 m/s and sounds its horn at 320 Hz. If the speed of sound is 340 m/s, what frequency does a stationary observer hear as the train approaches?

$$f_L = \frac{v}{v - v_S} f_S = \frac{340}{340 - 40} \times 320 = 363\ \mathrm{Hz}$$

Q: If two sound waves of frequencies 256 Hz and 260 Hz are played together, what is the beat frequency heard?

$$f_{\text{beat}} = |260 - 256| = 4\ \mathrm{Hz}$$

Question 1

Which of the following statements best distinguishes a mechanical wave from an electromagnetic wave?

Accepted Answer

A mechanical wave requires a medium to propagate, while an electromagnetic wave does not.

Question 2

A sound wave travels through air at a speed of 344 m/s. What is the wavelength of a sound wave with a frequency of 440 Hz (A above middle C)?

Accepted Answer

$$\lambda = \frac{v}{f} = \frac{344}{440} = 0.782\ \mathrm{m}$$

Question 3

Which of the following is an example of a transverse wave?

Accepted Answer

A wave on a stretched rope

Question 4

A guitar string has a length of 0.86 m and a mass per unit length of 0.015 kg/m. If the string vibrates at its fundamental frequency of 41 Hz, what is the tension in the string?

Accepted Answer

$$F_T = (2Lf_1)^2$ \mu = [2 	imes 0.86 	imes 41]^2 	imes 0.015 = 76\ \mathrm{N}$$

Question 5

For a pipe open at both ends, which of the following equations gives the allowed frequencies of standing waves?

Accepted Answer

$$f_n = n \frac{v}{2L}$$, $$n = 1, 2, 3, \ldots$$

Question 6

A tube closed at one end and open at the other resonates at a frequency of 25 kHz. If the distance from the closed end to the nearest displacement node is 0.026 m, what is the speed of sound in the tube?

Accepted Answer

$$v = 2fL = 2 	imes 25,000 	imes 0.026 = 1,300\ \mathrm{m/s}$$

Question 7

Which of the following best describes the condition for maximum constructive interference between two waves?

Accepted Answer

$$|d_1$ - d_2$| = n\lambda$$, $$n = 0, \pm1, \pm2, \ldots$$

Question 8

If the intensity of a sound is doubled, by how many decibels does the sound level increase?

Accepted Answer

$$\Delta \beta = 10 \log_{10} 2 = 3.01\ \mathrm{dB}$$

Question 9

A standing wave on a string fixed at both ends has a length $$L$$ and supports a fundamental frequency $$f_1. $$What is the wavelength of the second harmonic?

Accepted Answer

$$\$$lambda_2$$ = L$$

Question 10

A commuter train moves at 40.0 m/s and sounds its horn at 320 Hz. If the speed of sound is 340 m/s, what frequency does a stationary observer hear as the train approaches?

Accepted Answer

$$f_L = \frac{v}{v - v_S} f_S = \frac{340}{340 - 40} 	imes 320 = 363\ \mathrm{Hz}$$

Question 11

Which equation correctly relates the speed of a wave on a string to the tension $$F_T$$ and the linear mass density $$\mu$$?

Accepted Answer

$$v = \sqrt{\frac{F_T}{\mu}}$$

Question 12

If two sound waves of frequencies 256 Hz and 260 Hz are played together, what is the beat frequency heard?

Accepted Answer

$$f_{	ext{beat}} = |260 - 256| = 4\ \mathrm{Hz}$$

Question 13

A pipe closed at one end and open at the other has a length $$L. $$Which harmonics are present in its standing wave pattern?

Accepted Answer

Only odd harmonics: $$n = 1, 3, 5, \ldots$$

Question 14

The intensity of a sound wave is defined as:

Accepted Answer

$$I = \frac{P}{A}$$, where $$P is $$power and $$A is $$area

Question 15

A 2000-seat auditorium is designed to produce a sound intensity of $$1.0\ \mathrm{W/m^2$}$$ over the surface of a hemisphere of radius 20 m. What total acoustic power is needed from the speakers?

Accepted Answer

$$P = I 	imes A = 1.0 	imes 2\pi (20)^2$ = 2,513\ \mathrm{W}$$

Question 16

Which of the following best describes a standing wave?

Accepted Answer

A wave formed by the superposition of two waves traveling in opposite directions with the same frequency and amplitude.

Question 17

If the path difference between two sound waves from two speakers is $$\frac{3}{2}\lambda$$, what type of interference occurs at that point?

Accepted Answer

Destructive interference

Question 18

The decibel level $$\beta of a $$sound with intensity $$I is $$given by:

Accepted Answer

$$\beta = (10\ \mathrm{dB}) \log_{10} \left(\frac{I}{I_0$}\right)$$

Chapter 12: Mechanical Waves & Sound – Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards