Ch16 Verified Test Bank Waves

Chapter: Chapter 16

Learning Objectives

LO 16.1.0 Solve problems related to transverse waves.

LO 16.1.1 Identify the three main types of waves.

LO 16.1.2 Distinguish between transverse waves and longitudinal waves.

LO 16.1.3 Given a displacement function for a traverse wave, determine amplitude y_m, angular wave number k, angular frequency ω, phase constant φ, and direction of travel, and calculate the phase kx + ωτ + φ and the displacement at any given time and position.

LO 16.1.4 Given a displacement function for a traverse wave, calculate the time between two given displacements.

LO 16.1.5 Sketch a graph of a transverse wave as a function of position, identifying amplitude y_m, wavelength λ, where the slope is greatest, where it is zero, and where the string elements have positive velocity, negative velocity, and zero velocity.

LO 16.1.6 Given a graph of displacement versus time for a transverse wave, determine amplitude y_m and period T.

LO 16.1.7 Describe the effect on a transverse wave of changing phase constant φ.

LO 16.1.8 Apply the relation between the wave speed v, the distance traveled by the wave, and the time required for that travel.

LO 16.1.9 Apply the relationships between wave speed v, angular frequency ω, angular wave number k, wavelength λ, period T, and frequency f.

LO 16.1.10 Describe the motion of a string element as a transverse wave moves through its location, and identify when its transverse speed is zero and when it is maximum.

LO 16.1.11 Calculate the transverse velocity u(t) of a string element as a transverse wave moves through its location.

LO 16.1.12 Calculate the transverse acceleration a(t) of a string element as a transverse wave moves through its location.

LO 16.1.13 Given a graph of displacement, transverse velocity, or transverse acceleration, determine the phase constant.

LO 16.2.0 Solve problems related to wave speed on a stretched string.

LO 16.2.1 Calculate the linear density μ of a uniform string in terms of the total mass and total length.

LO 16.2.2 Apply the relationship between wave speed v, tension τ, and linear density μ.

LO 16.3.0 Solve problems related to energy and power of a wave traveling along a string.

LO 16.3.1 Calculate the average rate at which energy is transported by a transverse wave.

LO 16.4.0 Solve problems related to the wave equation.

LO 16.4.1 For the equation giving a string-element displacement as a function of position x and time t, apply the relationship between the second derivative with respect to x and the second derivative with respect to t.

LO 16.5.0 Solve problems related to interference of waves.

LO 16.5.1 Apply the principle of superposition to show that two overlapping waves add algebraically to give a resultant (or net) wave.

LO 16.5.2 For two transverse waves with the same amplitude and wavelength and that travel together, find the displacement equation for the resultant wave and calculate the amplitude in terms of the individual wave amplitude and the phase difference.

LO 16.5.3 Describe how the phase difference between two transverse waves (with the same amplitude and wavelength) can result in fully constructive interference, fully destructive interference, and intermediate interference.

LO 16.5.4 With the phase difference between two interfering waves expressed in terms of wavelengths, quickly determine the type of interference the waves have.

LO 16.6.0 Solve problems related to phasors.

LO 16.6.1 Using sketches, explain how a phasor can represent the oscillations of a string element as a wave travels through its location.

LO 16.6.2 Sketch a phasor diagram for two overlapping waves traveling together on a string, indicating their amplitudes and phase difference on the sketch.

LO 16.6.3 By using phasors, find the resultant wave of two transverse waves traveling together along a string, calculating the amplitude and phase and writing out the displacement equation.

LO 16.7.0 Solve problems related to standing waves and resonance.

LO 16.7.1 For two overlapping waves (same amplitude and wavelength) that are traveling in opposite directions, sketch snapshots of the resultant wave, indicating nodes and antinodes.

LO 16.7.2 For two overlapping waves (same amplitude and wavelength) that are traveling in opposite directions, find the displacement equation for the resultant wave and calculate the amplitude in terms of the individual wave amplitude.

LO 16.7.3 Describe the SHM of a string element at an antinode of a standing wave.

LO 16.7.4 For a string element at an antinode of a standing wave, write equations for the displacement, transverse velocity, and transverse acceleration as functions of time.

LO 16.7.5 Distinguish between “hard” and “soft” reflections of string waves at a boundary.

LO 16.7.6 Describe resonance on a string tied taut between two supports, and sketch the first several standing-wave patterns, indicating nodes and antinodes.

LO 16.7.7 In terms of string length, determine the wavelengths required for the first several harmonics on a string under tension.

LO 16.7.8 For any given harmonic, apply the relationship between frequency, wave speed, and string length.

Multiple Choice

1. A traveling sinusoidal wave is shown below. At which point is the motion 180 out of phase with the motion at point P?

A) A

B) B

C) C

D) D

E) E

Difficulty: E

Section: 16-1

Learning Objective 16.1.0

2. What are the three main types of waves?

A) transverse, longitudinal, linear

B) plane, spherical, transverse

C) mechanical, electromagnetic, matter

D) transverse, linear, water

E) plane, longitudinal, mechanical

Difficulty: E

Section: 16-1

Learning Objective 16.1.1

3. What is the difference between transverse and longitudinal waves?

A) Mechanical waves are transverse waves while electromagnetic waves are longitudinal.

B) Plane waves are transverse waves while spherical waves are longitudinal.

C) Only longitudinal waves transmit matter.

D) Only transverse waves transmit energy.

E) In transverse waves the displacement is perpendicular to the direction of propagation of the wave, while in longitudinal waves the displacement is parallel to the direction of propagation.

Difficulty: E

Section: 16-1

Learning Objective 16.1.2

4. Three traveling sinusoidal waves are on identical strings, with the same tension. The mathematical forms of the waves are y₁(x,t) = y_msin(3x – 6t), y₂(x,t) = y_msin(4x – 8t), and y₃(x,t) = y_msin(6x – 12t), where x is in meters and t is in seconds. Match each mathematical form to the appropriate graph below.

A) y₁: i, y₂: ii, y₃: iii

B) y₁: iii, y₂: ii, y₃: i

C) y₁:i, y₂: iii, y₃: ii

D) y₁: ii, y₂: i, y₃: iii

E) y₁: iii, y₂: i, y₃: ii

Difficulty: E

Section: 16-1

Learning Objective 16.1.3

5. A wave is described by y(x,t) = 0.1 sin(3x + 10t), where x is in meters, y is in centimeters and t is in seconds. The angular wave number is:

A) 0.10 rad/m

B) 3 rad/m

C) 10 rad/m

D) 10 rad/m

E) 3.0 rad/m

Difficulty: E

Section: 16-1

Learning Objective 16.1.3

6. A wave is described by y(x,t) = 0.1 sin(3x – 10t), where x is in meters, y is in centimeters and t is in seconds. The angular frequency is:

A) 0.10 rad/s

B) 3.0 rad/s

C) 10 rad/s

D) 20 rad/s

E) 10 rad/s

Difficulty: E

Section: 16-1

Learning Objective 16.1.3

7. The displacement of a string carrying a traveling sinusoidal wave is given by

At time t = 0 the point at x = 0 has a displacement of 0 and is moving in the positive y direction. The phase constant is:

A) 0

B) 90

C) 135

D) 180

E) 270

Difficulty: M

Section: 16-1

Learning Objective 16.1.3

8. The displacement of a string carrying a traveling sinusoidal wave is given by

At time t = 0 the point at x = 0 has a velocity of 0 and a positive displacement. The phase constant is:

A) 45

B) 90

C) 135

D) 180

E) 270

Difficulty: M

Section: 16-1

Learning Objective 16.1.3

9. The displacement of a string carrying a traveling sinusoidal wave is given by

At time t = 0 the point at x = 0 has velocity v₀ and displacement y₀. The phase constant is given by tan=:

A) v₀/y₀

B) y₀/v₀

C) v₀/y₀

D) y₀/v₀

E) v₀y₀

Difficulty: M

Section: 16-1

Learning Objective 16.1.3

10. A wave is described by y(x,t) = 0.1 sin(3x – 10t), where x is in meters, y is in centimeters and t is in seconds. At time t = 0, the point at x = 0 has a vertical displacement y = 0.0 cm. When is its displacement equal to 0.1 cm?

A) 0.16 s

B) 0.47 s

C) 2.5 s

D) 7.5 s

E) 10 s

Difficulty: M

Section: 16-1

Learning Objective 16.1.4

11. A sinusoidal wave is traveling toward the right as shown. Which letter correctly labels the amplitude of the wave?

A) A

B) B

C) C

D) D

E) E

Difficulty: E

Section: 16-1

Learning Objective 16.1.5

12. A sinusoidal wave is traveling toward the right as shown. Which letter correctly labels the wavelength of the wave?

A) A

B) B

C) C

D) D

E) E

Difficulty: E

Section: 16-1

Learning Objective 16.1.5

13. In the diagram below, the interval PQ represents:

A) wavelength/2

B) wavelength

C) 2  amplitude

D) period/2

E) period

Difficulty: E

Section: 16-1

Learning Objective 16.1.5

14. This plot shows the displacement of a string as a function of time, as a sinusoidal wave travels along it. Which letter corresponds to the amplitude of the wave?

A) A

B) B

C) C

D) D

E) E

Difficulty: E

Section: 16-1

Learning Objective 16.1.6

15. This plot shows the displacement of a string as a function of time, as a sinusoidal wave travels along it. Which letter corresponds to the period of the wave?

A) A

B) B

C) C

D) D

E) E

Difficulty: E

Section: 16-1

Learning Objective 16.1.6

16. Two waves are traveling on two different strings. The displacement of one is given by y₁(x,t) = y_msin(kx + t) and of the other by y₂(x,t) = y_msin(kx + t + φ). What is the difference between these two waves?

A) The displacement of wave y₁ is always greater than the displacement of wave y₂.

B) Wave y₁ has a smaller amplitude than wave y₂.

C) Wave y₁ has a higher frequency than wave y₂.

D) Wave y₁ has a shorter wavelength than wave y₂.

E) The two waves are identical except for their displacement at time t = 0.

Difficulty: E

Section: 16-1

Learning Objective 16.1.7

17. A wave is described by y(x,t) = 0.1 sin(3x – 10t), where x is in meters, y is in centimeters and t is in seconds. How long does it take the wave to travel 2.0 m?

A) 0.6 s

B) 1.0 s

C) 3.0 s

D) 6.7 s

E) 10 s

Difficulty: M

Section: 16-1

Learning Objective 16.1.8

18. The displacement of a string is given by

The wavelength of the wave is:

A) 2k/

B) k/

C) k

D) 2/k

E) k/2

Difficulty: E

Section: 16-1

Learning Objective 16.1.9

19. For a transverse wave on a string the string displacement is described by y(x,t) = f(x–at) where f is a given function and a is a positive constant. Which of the following does NOT necessarily follow from this statement?

A) The shape of the string at time t = 0 is given by f(x).

B) The shape of the waveform does not change as it moves along the string.

C) The waveform moves in the positive x direction.

D) The speed of the waveform is a.

E) The speed of the waveform is x/t.

Difficulty: E

Section: 16-1

Learning Objective 16.1.9

20. The displacement of a string is given by

y(x,t) = y_msin(kx + t).

The speed of the wave is:

A) 2k/

B) /k

C) k

D) 2/k

E) k/2

Difficulty: E

Section: 16-1

Learning Objective 16.1.9

21. Water waves in the sea are observed to have a wavelength of 300 m and a frequency of 0.07 Hz. The speed of these waves is:

A) 0.00023 m/s

B) 2.1 m/s

C) 21 m/s

D) 4300 m/s

E) none of these

Difficulty: E

Section: 16-1

Learning Objective 16.1.9

22. Sinusoidal water waves are generated in a large ripple tank. The waves travel at 20 cm/s and their adjacent crests are 5.0 cm apart. The time required for each new whole cycle to be generated is:

A) 100 s

B) 4.0 s

C) 2.0 s

D) 0.5 s

E) 0.25 s

Difficulty: E

Section: 16-1

Learning Objective 16.1.9

23. For a given medium, the frequency of a wave is:

A) independent of wavelength

B) proportional to wavelength

C) inversely proportional to wavelength

D) proportional to the amplitude

E) inversely proportional to the amplitude

Difficulty: E

Section: 16-1

Learning Objective 16.1.9

24. Let f be the frequency, v the speed, and T the period of a sinusoidal traveling wave. The correct relationship is:

A) f = 1/T

B) f = v + T

C) f = vT

D) f = v/T

E) f = T/v

Difficulty: E

Section: 16-1

Learning Objective 16.1.9

25. Let f be the frequency, v the speed, and T the period of a sinusoidal traveling wave. The angular frequency is given by:

A) 1/T

B) 2/T

C) vT

D) f/T

E) T/f

Difficulty: E

Section: 16-1

Learning Objective 16.1.9

26. A source of frequency f sends waves of wavelength  traveling with speed v in some medium. If the frequency is changed from f to 2f, then the new wavelength and new speed are (respectively):

A) 2, v

B) /2, v

C) , 2v

D) , v/2

E) /2, 2v

Difficulty: E

Section: 16-1

Learning Objective 16.1.9

27. A long string is constructed by joining the ends of two shorter strings. The tension in the strings is the same but string I has 4 times the linear mass density of string II. When a sinusoidal wave passes from string I to string II:

A) the frequency decreases by a factor of 4

B) the frequency decreases by a factor of 2

C) the wavelength decreases by a factor of 4

D) the wavelength decreases by a factor of 2

E) the wavelength increases by a factor of 2

Difficulty: E

Section: 16-1

Learning Objective 16.1.9

28. A sinusoidal transverse wave is traveling on a string. Any point on the string:

A) moves in the same direction as the wave

B) moves in simple harmonic motion with a different frequency than that of the wave

C) moves in simple harmonic motion with the same angular frequency as the wave

D) moves in uniform circular motion with a different angular speed than the wave

E) moves in uniform circular motion with the same angular speed as the wave

Difficulty: E

Section: 16-1

Learning Objective 16.1.10

29. Any point on a string carrying a sinusoidal wave is moving with its maximum speed when:

A) the magnitude of its acceleration is a maximum

B) the magnitude of its displacement is a maximum

C) the magnitude of its displacement is a minimum

D) the magnitude of its displacement is half the amplitude

E) the magnitude of its displacement is one fourth the amplitude

Difficulty: E

Section: 16-1

Learning Objective 16.1.10

30. Here are equations for three waves traveling on separate strings. Rank them according to the maximum transverse speed, least to greatest.

A) 1, 2, 3

B) 1, 3, 2

C) 2, 1, 3

D) 2, 3, 1

E) 3, 1, 2

Difficulty: M

Section: 16-1

Learning Objective 16.1.11

31. The transverse wave shown is traveling from left to right in a medium. The direction of the instantaneous velocity of the medium at point P is:

A) 

B) 

C) 

D)

E) no direction since v = 0

Difficulty: E

Section: 16-1

Learning Objective 16.1.11

32. A wave traveling to the right on a stretched string is shown below. The direction of the instantaneous velocity of the point P on the string is:

A) 

B) 

C) 

D)

E) no direction since v = 0

Difficulty: E

Section: 16-1

Learning Objective 16.1.11

33. The mathematical forms for the three sinusoidal traveling waves are given by

where x is in meters and t is in seconds. Of these waves:

A) wave 1 has the greatest wave speed and the greatest maximum transverse string speed

B) wave 2 has the greatest wave speed and wave 1 has the greatest maximum transverse string speed

C) wave 3 has the greatest wave speed and the greatest maximum transverse string speed

D) wave 2 has the greatest wave speed and wave 3 has the greatest maximum transverse string speed

E) wave 3 has the greatest wave speed and wave 2 has the greatest maximum transverse string speed

Difficulty: M

Section: 16-1

Learning Objective 16.1.11

34. Suppose the maximum speed of a string carrying a sinusoidal wave is v_s. When the displacement of a point on the string is half its maximum, the speed of the point is:

A) v_s/2

B) 2v_s

C) v_s/4

D) 3v_s/4

E) v_s/2

Difficulty: M

Section: 16-1

Learning Objective 16.1.11

35. A string carries a sinusoidal wave with an amplitude of 2.0 cm and a frequency of 100 Hz. The maximum speed of any point on the string is:

A) 2.0 m/s

B) 4.0 m/s

C) 6.3 m/s

D) 13 m/s

E) unknown (not enough information is given)

Difficulty: M

Section: 16-1

Learning Objective 16.1.11

36. A transverse traveling sinusoidal wave on a string has a frequency of 100 Hz, a wavelength of 0.040 m and an amplitude of 2.0 mm. The maximum velocity of any point on the string is:

A) 0.20 m/s

B) 1.3 m/s

C) 4.0 m/s

D) 15 m/s

E) 25 m/s

Difficulty: M

Section: 16-1

Learning Objective 16.1.11

37. A transverse traveling sinusoidal wave on a string has a frequency of 100 Hz, a wavelength of 0.040 m and an amplitude of 2.0 mm. The maximum acceleration of any point on the string is:

A) 0 m/s²

B) 200 m/s²

C) 390 m/s²

D) 790 m/s²

E) 1600 m/s²

Difficulty: M

Section: 16-1

Learning Objective 16.1.12

38. In the figure, a wave is traveling from left to right. If the point marked “D” represents the origin at time t = 0, and the displacement of the wave is given by y(x,t) = y_msin(kx – t – ), what is the phase constant 

A) 0

B) π/6

C) π/4

D) π/2

E) 3π/2

Difficulty: E

Section: 16-1

Learning Objective 16.1.13

39. A transverse wave travels on a string of length 1.3 m and diameter 1.1 mm, whose mass is 10 g and which is under a tension of 16 N. What is the linear mass density of the string?

A) 7.7 x 10^-3 kg/m

B) 0.13 kg/m

C) 7.7 kg/m

D) 46 kg/m

E) 130 kg/m

Difficulty: E

Section: 16-2

Learning Objective 16.2.1

40. Sinusoidal waves travel on five identical strings. Four of the strings have the same tension, but the fifth has a different tension. Use the mathematical forms of the waves, gives below, to identify the string with the different tension. In the expressions given below x and y are in centimeters and t is in seconds.

A) y(x,t) = (2 cm) sin (2x – 4t)

B) y(x,t) = (2 cm) sin (4x – 10t)

C) y(x,t) = (2 cm) sin (6x – 12t)

D) y(x,t) = (2 cm) sin (8x – 16t)

E) y(x,t) = (2 cm) sin (10x – 20t)

Difficulty: M

Section: 16-2

Learning Objective 16.2.2

41. The speed of a sinusoidal wave on a string depends on:

A) the frequency of the wave

B) the wavelength of the wave

C) the length of the string

D) the tension in the string

E) the amplitude of the wave

Difficulty: E

Section: 16-2

Learning Objective 16.2.2

42. The time required for a small pulse to travel from A to B on a stretched cord shown is NOT altered by changing:

A) the linear mass density of the cord

B) the length between A and B

C) the shape of the pulse

D) the tension in the cord

E) none of the above (changes in all alter the time)

Difficulty: E

Section: 16-2

Learning Objective 16.2.2

43. The diagram shows three identical strings that have been put under tension by suspending masses of 5 kg each. For which is the wave speed the greatest?

A) 1

B) 2

C) 3

D) 1 and 3 tie

E) 2 and 3 tie

Difficulty: M

Section: 16-2

Learning Objective 16.2.2

44. The tension in a string with a linear density of 0.0010 kg/m is 0.40 N. A 100 Hz sinusoidal wave on this string has a wavelength of:

A) 0.20 cm

B) 2.0 cm

C) 5.0 cm

D) 20 cm

E) 400 cm

Difficulty: M

Section: 16-2

Learning Objective 16.2.2

45. When a 100-Hz oscillator is used to generate a sinusoidal wave on a certain string the wavelength is 10 cm. When the tension in the string is doubled the generator produces a wave with a frequency and wavelength of:

A) 200 Hz and 20 cm

B) 141 Hz and 10 cm

C) 100 Hz and 20 cm

D) 100 Hz and 14 cm

E) 50 Hz and 14 cm

Difficulty: M

Section: 16-2

Learning Objective 16.2.2

46. Three separate strings are made of the same material. String 1 has length L and tension , string 2 has length 2L and tension 2 and string 3 has length 3L and tension 3. A pulse is started at one end of each string. If the pulses start at the same time, the order in which they reach the other end is:

A) 1, 2, 3

B) 3, 2, 1

C) 2, 3, 1

D) 3, 1, 2

E) they all take the same time

Difficulty: M

Section: 16-2

Learning Objective 16.2.2

47. A long string is constructed by joining the ends of two shorter strings. The tension in the strings is the same but string I has 4 times the linear mass density of string II. When a sinusoidal wave passes from string I to string II:

A) the frequency decreases by a factor of 4

B) the frequency decreases by a factor of 2

C) the wave speed decreases by a factor of 4

D) the wave speed decreases by a factor of 2

E) the wave speed increases by a factor of 2

Difficulty: M

Section: 16-2

Learning Objective 16.2.2

48. A stretched string, clamped at its ends, vibrates at a particular frequency. To double that frequency, one can change the string tension by a factor of:

A) 2

B) 4

C)

D) 1/2

E)

Difficulty: M

Section: 16-2

Learning Objective 16.2.2

49. Two identical but separate strings, with the same tension, carry sinusoidal waves with the same amplitude. Wave A has a frequency that is twice that of wave B and transmits energy at a rate that is __________ that of wave B.

A) half

B) twice

C) one-fourth

D) four times

E) eight times

Difficulty: E

Section: 16-3

Learning Objective 16.3.1

50. Two identical but separate strings, with the same tension, carry sinusoidal waves with the same frequency. Wave A has an amplitude that is twice that of wave B and transmits energy at a rate that is __________ that of wave B.

A) half

B) twice

C) one-fourth

D) four times

E) eight times

Difficulty: E

Section: 16-3

Learning Objective 16.3.1

51. A sinusoidal wave is generated by moving the end of a string up and down periodically. The generator must supply the greatest power when the end of the string:

A) has its greatest acceleration

B) has its greatest displacement

C) has half its greatest displacement

D) has one fourth its greatest displacement

E) has its least displacement

Difficulty: M

Section: 16-3

Learning Objective 16.3.1

52. A sinusoidal wave is generated by moving the end of a string up and down periodically. The generator does not supply any power when the end of the string

A) has its least acceleration

B) has its greatest displacement

C) has half its greatest displacement

D) has one fourth its greatest displacement

E) has its least displacement

Difficulty: M

Section: 16-3

Learning Objective 16.3.1

53. The displacement of an element of a string is given by y(x,t) = 4.3sin(1.2x – t – π), with x in meters and t in seconds. Given that , what is v?

A) 0.065 m/s

B) 0.25 m/s

C) 2.0 m/s

D) 3.9 m/s

E) 15 m/s

Difficulty: M

Section: 16.4

Learning Objective 16.4.1

54. The sum of two sinusoidal traveling waves is a sinusoidal traveling wave only if:

A) their amplitudes are the same and they travel in the same direction

B) their amplitudes are the same and they travel in opposite directions

C) their frequencies are the same and they travel in the same direction

D) their frequencies are the same and they travel in opposite directions

E) their frequencies are the same and their amplitudes are the same

Difficulty: E

Section: 16-5

Learning Objective 16.5.1

55. Two traveling sinusoidal waves interfere to produce a wave with the mathematical form

If the value of is appropriately chosen, the two waves might be:

A) y₁(x,t) = (y_m/3) sin (kx + t) and y₂(x,t) = (y_m/3) sin (kx + t + )

B) y₁(x,t) = 0.7y_m sin (kx – t) and y₂(x,t) = 0.7y_m sin (kx – t + )

C) y₁(x,t) = 0.7y_m sin (kx – t) and y₂(x,t) = 0.7y_m sin (kx + t + )

D) y₁(x,t) = 0.7y_m sin [(kx/2) – (t/2)] and y₂(x,t) = 0.7y_m sin [(kx/2) – (t/2) + ]

E) y₁(x,t) = 0.7y_m sin (kx + t) and y₂(x,t) = 0.7y_m sin (kx + t + )

Difficulty: E

Section: 16-5

Learning Objective 16.5.2

56. Two sinusoidal waves have the same angular frequency, the same amplitude y_m, and travel in the same direction in the same medium. If they differ in phase by 50, the amplitude of the resultant wave is given by

A) 0.64 y_m

B) 1.3 y_m

C) 0.91 y_m

D) 1.8 y_m

E) 0.35 y_m

Difficulty: M

Section: 16-5

Learning Objective 16.5.2

57. Fully constructive interference between two sinusoidal waves of the same frequency occurs only if they:

A) travel in the same direction and are 270 out of phase

B) travel in the same direction and are 45 out of phase

C) travel in the same direction and are in phase

D) travel in the same direction and are 180 out of phase

E) travel in the same direction and are 90 out of phase

Difficulty: E

Section: 16-5

Learning Objective 16.5.3

58. Fully destructive interference between two sinusoidal waves of the same frequency and amplitude occurs only if they:

A) travel in the same direction and are 270 out of phase

B) travel in the same direction and are 45 out of phase

C) travel in the same direction and are in phase

D) travel in the same direction and are 180 out of phase

E) travel in the same direction and are 90 out of phase

Difficulty: E

Section: 16-5

Learning Objective 16.5.3

59. Two sinusoidal waves travel in the same direction and have the same frequency. Their amplitudes are y_1m and y_2m. The smallest possible amplitude of the resultant wave is:

A) y_1m + y_2m and occurs when they are 180 out of phase

B) y_1m – y_2m and occurs when they are 180 out of phase

C) y_1m + y_2m and occurs when they are in phase

D) y_1m – y_2m and occurs when they are in phase

E) y_1m – y_2m and occurs when they are 90 out of phase

Difficulty: E

Section: 16-5

Learning Objective 16.5.3

60. Two separated sources emit sinusoidal traveling waves that have the same wavelength  and are in phase at their respective sources. One travels a distance ℓ₁ to get to the observation point while the other travels a distance ℓ₂. The amplitude is a minimum at the observation point if ℓ₁ − ℓ₂ is:

A) an odd multiple of /2

B) an odd multiple of /4

C) a multiple of 

D) an odd multiple of /2

E) a multiple of 

Difficulty: E

Section: 16-5

Learning Objective 16.5.4

61. Two separated sources emit sinusoidal traveling waves that have the same wavelength  and are in phase at their respective sources. One travels a distance ℓ₁ to get to the observation point while the other travels a distance ℓ₂ The amplitude is a maximum at the observation point if ℓ₁ − ℓ₂ is:

A) an odd multiple of /2

B) an odd multiple of /4

C) a multiple of 

D) an odd multiple of /2

E) a multiple of 

Difficulty: E

Section: 16-5

Learning Objective 16.5.4

62. Two sources, S₁ and S₂, each emit waves of wavelength  in the same medium. The phase difference between the two waves, at the point P shown, is . The quantity  is:

A) the distance S₁S₂

B) the angle S₁PS₂

C) /2

D) the phase difference between the two sources

E) zero for transverse waves,  for longitudinal waves

Difficulty: E

Section: 16-5

Learning Objective 16.5.4

63. Two sinusoidal waves travel along the same string. They have the same wavelength and frequency. Their amplitudes are y_m1 = 2.5 mm and y_m2 = 4.5 mm, and their phases are π/4 rad and π/2 rad, respectively. What are the amplitude and phase of the resultant wave?

A) cannot solve without knowing the wavelength

B) 5.1 mm, 0.51 rad

C) 5.1 mm, 0.79 rad

D) 6.5 mm, 1.3 rad

E) 7.0 mm, 1.3 rad

Difficulty: M

Section: 16-6

Learning Objective 16.6.3

64. The sinusoidal wave

y(x,t) = y_msin(kx – t)

is incident on the fixed end of a string at x = L. The reflected wave is given by:

A) y_msin(kx + t)

B) –y_msin(kx + t)

C) y_msin(kx + t – kL)

D) y_msin(kx + t – 2kL)

E) –y_msin(kx + t + 2kL)

Difficulty: H

Section: 16-7

Learning Objective 16.7.0

65. A standing wave:

A) can be constructed from two similar waves traveling in opposite directions

B) must be transverse

C) must be longitudinal

D) has motionless points that are closer than half a wavelength

E) has a wave velocity that differs by a factor of two from what it would be for a traveling wave

Difficulty: E

Section: 16-7

Learning Objective 16.7.0

66. When a certain string is clamped at both ends, the lowest four resonant frequencies are 50, 100, 150, and 200 Hz. When the string is also clamped at its midpoint, the lowest four resonant frequencies are:

A) 50, 100, 150, and 200 Hz

B) 50, 150, 250, and 300 Hz

C) 100, 200, 300, and 400 Hz

D) 25, 50 75, and 100 Hz

E) 75, 150, 225, and 300 Hz

Difficulty: M

Section: 16-7

Learning Objective 16.7.0

67. When a certain string is clamped at both ends, the lowest four resonant frequencies are measured to be 100, 150, 200, and 250 Hz. One of the resonant frequencies (below 200 Hz) is missing. What is it?

A) 25 Hz

B) 50 Hz

C) 75 Hz

D) 125 Hz

E) 225 Hz

Difficulty: E

Section: 16-7

Learning Objective 16.7.0

68. When a string is vibrating in a standing wave pattern the power transmitted across an antinode, compared to the power transmitted across a node, is:

A) more

B) less

C) the same (zero)

D) the same (non-zero)

E) sometimes more, sometimes less, and sometimes the same

Difficulty: E

Section: 16-7

Learning Objective 16.7.0

69. Which of the following represents a standing wave?

A) y = (6.0 mm)sin[(3.0 m^–1)x + (2.0 s^–1)t] – (6.0 mm)cos[(3.0 m^–1)x + 2.0]

B) y = (6.0 mm)cos[(3.0 m^–1)x – (2.0 s^–1)t] + (6.0 mm)cos[(2.0 s^–1)t + (3.0 m^–1)x]

C) y = (6.0 mm)cos[(3.0 m^–1)x – (2.0 s^–1)t] – (6.0 mm)sin[(2.0 s^–1)t – 3.0]

D) y = (6.0 mm)sin[(3.0 m^–1)x – (2.0 s^–1)t] – (6.0 mm)cos[(2.0 s^–1)t + (3.0 m^–1)x]

E) y = (6.0 mm)sin[(3.0 m^–1)x] + (6.0 mm)cos[(2.0 s^–1)t]

Difficulty: E

Section: 16-7

Learning Objective 16.7.2

70. Which of the following represents the motion of a string element at an antinode of a standing wave?

A) y = (6.0 mm)sin[(3.0 m^–1)x + (2.0 s^–1)t]

B) y = (6.0 mm)cos[(3.0 m^–1)x – (2.0 s^–1)t]

C) y = (6.0 mm)cos[(3.0 m^–1)x + (2.0 s^–1)t]

D) y = (6.0 mm)sin[(3.0 m^–1)x

E) y = (6.0 mm)cos[(2.0 s^–1)t

Difficulty: E

Section: 16-7

Learning Objective 16.7.3

71. A wave on a stretched string is reflected from a fixed end P of the string. The phase difference, at P, between the incident and reflected waves is:

A) 0 rad

B)  rad

C) /2 rad

D) depends on the velocity of the wave

E) depends on the frequency of the wave

Difficulty: E

Section: 16-7

Learning Objective 16.7.5

72. A wave on a string is reflected from a fixed end. The reflected wave:

A) is in phase with the original wave at the end

B) is 180 out of phase with the original wave at the end

C) has a larger amplitude than the original wave

D) has a larger speed than the original wave

E) cannot be transverse

Difficulty: E

Section: 16-7

Learning Objective 16.7.5

73. Two traveling waves, y₁ = A sin[k(x – vt)] and y₂ = A sin[k(x + vt)], are superposed on the same string. The distance between the adjacent nodes is:

A) vt/

B) vt/2

C) /2k

D) /k

E) 2/k

Difficulty: M

Section: 16-7

Learning Objective 16.7.6

74. If  is the wavelength of the each of the component sinusoidal traveling waves that form a standing wave, the distance between adjacent nodes in the standing wave is:

A) /4

B) /2

C) 3/4

D) 

E) 2

Difficulty: E

Section: 16-7

Learning Objective 16.7.6

75. A standing wave pattern is established in a string as shown. The wavelength of one of the component traveling waves is:

A) 0.25 m

B) 0.5 m

C) 1 m

D) 2 m

E) 4 m

Difficulty: E

Section: 16-7

Learning Objective 16.7.6

76. Standing waves are produced by the interference of two traveling sinusoidal waves, each of frequency 100 Hz. The distance from the 2nd node to the 5th node is 60 cm. The wavelength of each of the two original waves is:

A) 50 cm

B) 40 cm

C) 30 cm

D) 20 cm

E) 15 cm

Difficulty: M

Section: 16-7

Learning Objective 16.7.6

77. A string of length 100 cm is held fixed at both ends and vibrates in a standing wave pattern. The wavelengths of the constituent traveling waves CANNOT be:

A) 400 cm

B) 200 cm

C) 100 cm

D) 67 cm

E) 50 cm

Difficulty: E

Section: 16-7

Learning Objective 16.7.6

78. A string of length L is clamped at each end and vibrates in a standing wave pattern. The wavelengths of the constituent traveling waves CANNOT be:

A) L

B) 2L

C) L/2

D) 2L/3

E) 4L

Difficulty: E

Section: 16-7

Learning Objective 16.7.6

79. Two sinusoidal waves, each of wavelength 5 m and amplitude 10 cm, travel in opposite directions on a 20-m stretched string which is clamped at each end. Excluding the nodes at the ends of the string, how many nodes appear in the resulting standing wave?

A) 3

B) 4

C) 5

D) 7

E) 8

Difficulty: E

Section: 16-7

Learning Objective 16.7.6

80. A 40-cm long string, with one end clamped and the other free to move transversely, is vibrating in its fundamental standing wave mode. The wavelength of the constituent traveling waves is:

A) 10 cm

B) 20 cm

C) 40 cm

D) 80 cm

E) 160 cm

Difficulty: E

Section: 16-7

Learning Objective 16.7.7

81. A 30-cm long string, with one end clamped and the other free to move transversely, is vibrating in its second harmonic. The wavelength of the constituent traveling waves is:

A) 10 cm

B) 30 cm

C) 40 cm

D) 60 cm

E) 120 cm

Difficulty: E

Section: 16-7

Learning Objective 16.7.7

82. A string, clamped at its ends, vibrates in three segments. The string is 100 cm long. The wavelength is:

A) 33 cm

B) 67 cm

C) 150 cm

D) 300 cm

E) need to know the frequency

Difficulty: E

Section: 16-7

Learning Objective 16.7.8

83. A 40-cm long string, with one end clamped and the other free to move transversely, is vibrating in its fundamental standing wave mode. If the wave speed is 320 cm/s the frequency is:

A) 32 Hz

B) 16 Hz

C) 8 Hz

D) 4 Hz

E) 2 Hz

Difficulty: M

Section: 16-7

Learning Objective 16.7.8