Chapter 16

Wave Motion

Multiple Choice

1. The wavelength of light visible to the human eye is on the order of 5 × 10–7 m. If the speed of light in air is 3 × 108 m/s, find the frequency of the lightwave.

a. 3 × 107 Hz b. 4 × 109 Hz c. 5 × 1011 Hz d. 6 × 1014 Hz e. 4 × 1015 Hz

2. The speed of a 10-kHz sound wave in seawater is approximately 1500 m/s. What is its wavelength in sea water?

a. 5.0 cm b. 10 cm c. 15 cm d. 20 cm e. 29 cm

3. Bats can detect small objects such as insects that are of a size on the order of a wavelength. If bats emit a chirp at a frequency of 60 kHz and the speed of soundwaves in air is 330 m/s, what is the smallest size insect they can detect?

a. 1.5 mm b. 3.5 mm c. 5.5 mm d. 7.5 mm e. 9.8 mm

4. Ocean waves with a wavelength of 120 m are coming in at a rate of 8 per minute. What is their speed?

a. 8.0 m/s b. 16 m/s c. 24 m/s d. 30 m/s e. 4.0 m/s

291

292 CHAPTER 16

5. An earthquake emits both S-waves and P-waves which travel at different speeds through the Earth. A P-wave travels at 9000 m/s and an S-wave travels at 5000 m/s. If P-waves are received at a seismic station 1.00 minute before an S-wave arrives, how far away is the earthquake center?

a. 88.9 km b. 1200 km c. 675 km d. 240 km e. 480 km

6. A piano string of density 0.0050 kg/m is under a tension of 1350 N. Find the velocity with which a wave travels on the string.

a. 260 m/s b. 520 m/s c. 1040 m/s d. 2080 m/s e. 4160 m/s

7. A 100-m long transmission cable is suspended between two towers. If the mass density is 2.01 kg/m and the tension in the cable is 3.00 × 104 N, what is the speed of transverse waves on the cable?

a. 60 m/s b. 122 m/s c. 244 m/s d. 310 m/s e. 1500 m/s

8. Transverse waves are traveling on a 1.00-m long piano string at 500 m/s. If the points of zero vibration occur at one-half wavelength, (where the string is fastened at both ends), find the frequency of vibration.

a. 250 Hz b. 500 Hz c. 1000 Hz d. 2000 Hz e. 2500 Hz

9. The lowest A on a piano has a frequency of 27.5 Hz. If the tension in the 2.00-m string is 308 N, and one-half wavelength occupies the string, what is the mass of the wire?

a. 0.025 kg b. 0.049 kg c. 0.051 kg d. 0.081 kg e. 0.037 kg

Wave Motion 293

10. If y = 0.02 sin (30x – 400t) (SI units), the frequency of the wave is

a. 30 Hz b. 15/π Hz c. 200/π Hz d. 400 Hz e. 800π Hz

11. If y = 0.02 sin (30x – 400t) (SI units), the wavelength of the wave is

a. π/15 m b. 15/π m c. 60π m d. 4.2 m e. 30 m

12. If y = 0.02 sin (30x – 400t) (SI units), the velocity of the wave is

a. 3/40 m/s b. 40/3 m/s c. 60π/400 m/s d. 400/60π m/s e. 400 m/s

13. If y = 0.02 sin (30x – 400t) (SI units), the angular frequency of the wave is

a. 30 rad/s b. 30/2π rad/s c. 400/2π rad/s d. 400 rad/s e. 40/3 rad/s

14. If y = 0.02 sin (30x – 400t) (SI units), the wave number is

a. 30 m–1

b. 30/2π m–1

c. 400/2π m–1

d. 400 m–1

e. 60π m–1

15. If y = 0.02 sin (30x - 400t) (SI units) and if the mass density of the string on which the wave propagates is .005 kg/m, then the transmitted power is

a. 1.03 W b. 2.13 W c. 4.84 W d. 5.54 W e. 106 W

294 CHAPTER 16

16. Write the equation of a wave, traveling along the +x axis with an amplitude of 0.02 m, a frequency of 440 Hz, and a speed of 330 m/sec.

a. y = 0.02 sin [880π (x/330 – t)] b. y = 0.02 cos [880π x/330 – 440t] c. y = 0.02 sin [880π(x/330 + t)] d. y = 0.02 sin [2π(x/330 + 440t)] e. y = 0.02 cos [2π(x/330 + 440t)]

17. For the wave described by y = 0.15 sin units) (SI )642(16 ⎥⎦

⎤⎢⎣⎡ − tx

π, determine the

first positive x-coordinate where y is a maximum when t = 0.

a. 16 m b. 8 m c. 4 m d. 2 m e. 13 m

18. For the wave described by y = 0.15 sin units) (SI )642(16 ⎥⎦

⎤⎢⎣⎡ − tx

π, determine

x coordinate of the second maximum when t = 0.

a. 20 m b. 18 m c. 24 m d. 28 m e. 16 m

19. For the wave described by y = 0.02 sin (kx) at t = 0 s, the first maximum at a positive x coordinate occurs where x = 4 m. Where on the positive x axis does the second maximum occur?

a. 20 m b. 18 m c. 24 m d. 28 m e. 16 m

20. For the transverse wave described by y = 0.15 sin units) (SI )642(16 ⎥⎦

⎤⎢⎣⎡ − tx

π,

determine the maximum transverse speed of the particles of the medium.

a. 0.192 m/s b. 0.6π m/s c. 9.6 m/s d. 4 m/s e. 2 m/s

Wave Motion 295

21. Which of the following is a solution to the wave equation, 2

2

22

2 1ty

vxy

∂∂

=∂∂

?

a. x

e x−

sin x

b. (cos kx) (sin t) c. e–x sin ωt d. e–x sin (kx – ωt) e. e–x cos t

22. Find the period of a wave of 100-m wavelength in deep water where πλ 2/gv = .

a. 5.0 s b. 8.0 s c. 12.5 s d. 15 s e. 0.125 s

23. A piano wire of length 1.5 m vibrates so that one-half wavelength is contained on the string. If the frequency of vibration is 65 Hz, the amplitude of vibration is 3.0 mm, and the density is 15 g/m, how much energy is transmitted per second down the wire?

a. 21 W b. 11 W c. 5.4 W d. 2.2 W e. 1.1 W

24. A student attaches a length of nylon fishing line to a fence post. She stretches it out and shakes the end of the rope in her hand back and forth to produce waves on the line. The most efficient way for her to increase the wavelength is to

a. increase the tension on the hose and shake the end more times per second. b. decrease the tension on the hose and shake the end more times per second. c. increase the tension on the hose and shake the end fewer times per second. d. decrease the tension on the hose and shake the end fewer times per second. e. keep the tension and frequency the same but increase the length of the hose.

296 CHAPTER 16

25. The figure below shows a sine wave at one point of a string as a function of time.

t

y

Which of the graphs below shows a wave where the amplitude and the frequency are doubled?

y

t

y

t

y

t

(a) (b)

(c) (d)

(e)

y

t

y

t

Wave Motion 297

26. The figure below shows a sine wave at one point of a string as a function of time.

y

t

Which of the graphs below shows a wave where the amplitude and frequency are each reduced in half?

y

t

y

t

y

t

(a) (b)

(c) (d)

(e)

y

t

y

t

298 CHAPTER 16

27. The figure below shows a sine wave on a string at one instant of time.

y

x

Which of the graphs below shows a wave where the frequency and wave velocity are both doubled?

y

x

y

x

y

x

(a) (b)

(c) (d)

(e)

y

x

y

x

Wave Motion 299

28. The figure below shows a sine wave on a string at one instant of time.

y

x

Which of the graphs below shows a wave where the wavelength is twice as large?

y

x

y

x

y

x

(a) (b)

(c) (d)

(e)

y

x

y

x

29. Superposition of waves can occur

a. in transverse waves. b. in longitudinal waves. c. in sinusoidal waves. d. in all ofthe above. e. only in (a) and (c) above.

300 CHAPTER 16

30. Two pulses are traveling towards each other at 10 cm/s on a long string at t = 0 s, as shown below.

1 cm Which diagram below correctly shows the shape of the string at 0.5 s?

1 cm

(a)

1 cm

(b)

1 cm

(c)

1 cm

(d)

1 cm

(e)

31. Suppose that you were selected for a “Survivor”-type TV show. To help keep your group connected, you suggest that long vines can be tied together and used to transmit signals in cases of emergency. To get the signals to travel faster, you should

a. select lighter vines. b. increase the tension on the vines. c. hang weights from the vines at evenly spaced intervals. d. do all of the above. e. do (a) and (b) above only.

Wave Motion 301

32. Two ropes are spliced together as shown.

A short time after the incident pulse shown in the diagram reaches the splice, the ropes appearance will be that in

(a)

(b)

(c)

(d)

(e)

33. Two ropes are spliced together as shown.

A short time after the incident pulse shown in the diagram reaches the splice, the ropes appearance will be that in

(a)

(b)

(c)

(d)

(e)

34. The fundamental frequency of a above middle C on the piano is 440 Hz. This is the tenor high A, but a convenient note in the mid-range of women’s voices. When we calculate the wavelength, we find that it is

a. much shorter than the length of either a man’s or woman’s lips. b. shorter than the length of a man’s lips, but about the length of a woman’s

lips. c. longer than a woman’s lips, but about the length of a man’s lips. d. much longer than the length of either a man’s or a woman’s lips. e. about the same length as either a man’s or woman’s lips.

302 CHAPTER 16

35. Ariel claims that a pulse is described by the equation

y(x,t) =2

x 2 − 6.0xt + 9t 2 + 9

where x and y are measured in cm and t in s. Miranda says that it is not possible to represent a pulse with this function because a wave must be a function of x + vt or x − vt . Which one, if either, is correct, and why?

a. Ariel, because x 2 − 6.0xt + 9t 2 = (x − 3.0t)2. b. Ariel, because a pulse is not an infinite wave. c. Miranda, because (x − 3.0t)2 is the same as (3.0t − x)2 . d. Miranda, because a pulse is not an infinite wave.

e. Miranda, because x 2 − 6.0xt + 9t 2 = x 2 1−6.00t

x+

9t 2

x 2

⎛

⎝ ⎜

⎞

⎠ ⎟ is infinite when x=0.

36. How does the wave function ′ y = sin(k 2x − ω 2t) differ from the wave function y = sin(kx −ωt)? (In ′ y , k 2 has units cm-1 and ω 2 units s . In -1 y , k has units cm-1 and ω units .)s-1

a. ′ λ =λ

2π.

b. ′ f = 2πf . c. . ′ v = v 2

d. All of the above are correct. e. The only difference is that ′ v = 2v .

37. The figure below represents a string which has a heavy section and a light section. The mass per unit length of the heavy section is 16 times as large as the mass per unit length of the light section. When the string is under tension, the speed of a pulse traveling in the heavy section is ____ times the speed of that same pulse traveling in the light section.

a. 1

16

b. 1

4

c. 1

2

d. 2

e. 4

Wave Motion 303

38. To transmit four times as much energy per unit time along a string, you must

a. double the frequency. b. double the amplitude. c. increase the tension by a factor of 16. d. do any one of the above. e. do only (a) or (b) above.

39. The wave equation is written down in an exam as

d 2ydx 2 = v 2 d 2y

dt 2 .

From dimensional considerations we see that

a. should be replaced by v 2 Tµ

.

b. should be replaced by v 2 Tµ

.

c. should be replaced by . v 2 v

d. should be replaced by v 2 1v

.

e. should be replaced by v 2 1v 2 .

40. Four wave functions are given below. Rank them in order of the magnitude of the wave speeds, from least to greatest.

I. y(x,t) = 5sin(4x − 20t + 4)

II. y(x,t) = 5sin(3x −12t + 5)

III. y(x, t) = 5cos(4 x + 24 t + 6)

IV. y(x, t) = 14 cos(2x − 8t + 3)

a. IV, II, I, III b. IV = II, I, III c. III, I, II, IV d. IV, I, II, III e. III, IV, II, I

304 CHAPTER 16

41. Four wave functions are given below. Rank them in order of the magnitude of the frequencies of the waves, from least to greatest.

I. y(x,t) = 5sin(4x − 20t + 4)

II. y(x,t) = 5sin(3x −12t + 5)

III. y(x, t) = 5cos(4 x + 24 t + 6)

IV. y(x, t) = 14 cos(2x − 8t + 3)

a. IV, II, I, III b. IV = II, I, III c. III, I, II, IV d. IV, I, II, III e. III, IV, II, I

42. Four wave functions are given below. Rank them in order of the magnitude of the wavelengths, from least to greatest.

I. y(x,t) = 5sin(4x − 20t + 4)

II. y(x,t) = 5sin(3x −12t + 5)

III. y(x, t) = 5cos(4 x + 24 t + 6)

IV. y(x, t) = 14 cos(2x − 8t + 3)

a. IV, II, I, III b. IV, I, II, III c. I, II, III, IV d. IV, II, III = I e. I = III, II, IV

43. You are holding on to one end of a long string that is fastened to a rigid steel light pole. After producing a wave pulse that was 5 mm high and 4 cm wide, you want to produce a pulse that is 4 cm wide but 7 mm high. You must move your hand up and down once,

a. the same distance up as before, but take a shorter time. b. the same distance up as before, but take a longer time. c. a smaller distance up, but take a shorter time. d. a greater distance up, but take a longer time. e. a greater distance up, but take the same time.

44. You are holding on to one end of a long string that is fastened to a rigid steel light pole. After producing a wave pulse that was 5 mm high and 4 cm wide, you want to produce a pulse that is 6 cm wide but still 5 mm high. You must move your hand up and down once,

a. the same distance up as before, but take a shorter time. b. the same distance up as before, but take a longer time. c. a smaller distance up, but take a shorter time. d. a greater distance up, but take a longer time. e. a greater distance up, but take the same time.

Wave Motion 305

45. You are holding on to one end of a long string that is fastened to a rigid steel light pole. After producing a wave pulse that was 5 mm high and 4 cm wide, you want to produce a pulse that is 6 cm wide and 7 mm high. You must move your hand up and down once,

a. the same distance up as before, but take a shorter time. b. the same distance up as before, but take a longer time. c. a smaller distance up, but take a shorter time. d. a greater distance up, but take a longer time. e. a greater distance up, but take the same time.

Open-Ended Problems 46. If the breakers at a beach are separated by 5.0 m and hit shore with a frequency

of 0.20 Hz, with what speed are they traveling?

47. Bats can detect small objects such as insects that are of a size approximately that of one wavelength. If bats emit a chirp at a frequency of 60 kHz, and the speed of sound in air is 340 m/s, what is the smallest size insect they can detect?

48. A circus performer stretches a tightrope between two towers. He strikes one end of the rope and sends a wave along it toward the other tower. He notes that it takes 0.8 s for the wave to travel the 20 m to the opposite tower. If one meter of the rope has a mass of 0.35 kg, find the tension in the tightrope.

306 CHAPTER 16

Wave Motion 307

Chapter 16

Wave Motion

1. d

2. c

3. c

4. b

5. c

6. b

7. b

8. a

9. c

10. c

11. a

12. b

13. d

14. a

15. b

16. a

17. c

18. a

19 a

20. b

21. b

22. b

23. d

24. c

25. d

26. e

27. a

28. d

29. d

30. b

31. e

32. a

33. c

34. d

35. a

36. d

37. b

38. e

39. e

40. b

41. a

42. e

43. e

44. b

45. d

46. 1.0 m/s

47. 5.7 mm

48. 219 N

308 CHAPTER 16