Deck 16: Waves I

Question 1

The displacement of a string is given by y(x,t) = y_msin(kx + $\omega$ t).
The speed of the wave is:

A) 2 $\pi$ k/ $\omega$
B) $\omega$ /k
C) $\omega$ k
D) 2 $\pi$ /k
E) k/2 $\pi$

Answer 1

$\omega$ /k

Question 2

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

Answer 2

inversely proportional to wavelength

Question 3

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 $\phi$ is given by tan $\phi$ =:

A) v₀/ $\omega$ y₀
B) $\omega$ y₀/v₀
C) $\omega$ v₀/y₀
D) y₀/ $\omega$ v₀
E) $\omega$ v₀y₀

Answer 3

$\omega$ y₀/v₀

Question 4

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

A) A
B) B
C) C
D) D
E) E

Question 5

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

Question 6

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

Question 7

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.

Question 8

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 $\pi$ rad/m
C) 10) rad/m
D) (10 $\pi$ ) rad/m
E) 3.0 rad/cm

Question 9

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

Question 10

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) 0.00021 m/s
B) 2.1 m/s
C) 21 m/s
D) 210 m/s
E) none of these

Question 11

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 $\phi$ is:

A) 45 $\degree$
B) 90 $\degree$
C) 135 $\degree$
D) 180 $\degree$
E) 270 $\degree$

Question 12

In the diagram below, the interval PQ represents:

A) wavelength/2
B) wavelength
C) 2 * amplitude
D) period/2
E) period

Question 13

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

Question 14

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 $\pi$ rad/s
D) 20 $\pi$ rad/s
E) (10/ rad/s

Question 15

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

Question 16

The displacement of a string is given by The wavelength of the wave is:

A) 2 $\pi$ k/ $\omega$
B) k/ $\omega$
C) $\omega$ k
D) 2 $\pi$ /k
E) k/2 $\pi$

Question 17

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 $\pi$ /T
C) vT
D) f/T
E) T/f

Question 18

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)

Question 19

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 $\phi$ is:

A) 45 $\degree$
B) 90 $\degree$
C) 135 $\degree$
D) 180 $\degree$
E) 270 $\degree$

Question 20

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

Question 21

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) $\uparrow$
B) $\downarrow$
C) $\rightarrow$
D)
E) no direction since v = 0

Question 22

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

Question 23

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

Question 24

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) $\uparrow$
B) $\downarrow$
C) $\rightarrow$
D)
E) no direction since v = 0

Question 25

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) 0.05 cm
B) 2.0 cm
C) 5.0 cm
D) 20 cm
E) 100 cm

Question 26

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

Question 27

A long string is constructed by joining the ends of 2 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

Question 28

A source of frequency f sends waves of wavelength $\lambda$ 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 $\lambda$ , v
B) $\lambda$ /2, v
C) $\lambda$ , 2v
D) $\lambda$ , v/2
E) $\lambda$ /2, 2v

Question 29

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

Question 30

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)

Question 31

The mathematical forms for the three sinusoidal traveling waves are gives 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 maxmium 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

Question 32

Here are the equations for the three waves traveling on separate strings. Rank them according to the maxium 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

Question 33

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)

Question 34

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

Question 35

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

Question 36

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

Question 37

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

Question 38

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

Question 39

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)

Question 40

Three separate strings are made of the same material. String 1 has length L and tension $\tau$ , string 2 has length 2L and tension 2 $\tau$ and string 3 has length 3L and tension 3 $\tau$ . 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

Question 41

If $\lambda$ 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) $\lambda$ /4
B) $\lambda$ /2
C) 3 $\lambda$ /4
D) $\lambda$
E) 2 $\lambda$

Question 42

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

Question 43

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 $\degree$ out of phase
B) $\mid$ y_1m - y_2m $\mid$ and occurs when they are 180 $\degree$ out of phase
C) y_1m + y_2m and occurs when they are in phase
D) $\mid$ y_1m - y_2m $\mid$ and occurs when they are in phase
E) $\mid$ y_1m - y_2m $\mid$ and occurs when they are 90 $\degree$ out of phase

Question 44

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

A) the distance S₁S₂
B) the angle S₁PS₂
C) $\pi$ /2
D) the phase difference between the two sources
E) zero for transverse waves, $\pi$ for longitudinal waves

Question 45

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) zero
B) $\pi$ rad
C) $\pi$ /2 rad
D) depends on the velocity of the wave
E) depends on the frequency of the wave

Question 46

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/ $\pi$
B) vt/2 $\pi$
C) $\pi$ /2k
D) $\pi$ /k
E) 2 $\pi$ /k

Question 47

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

A) an odd multiple of $\lambda$ /2
B) an odd multiple of $\lambda$ /4
C) a multiple of $\lambda$
D) an odd multiple of $\pi$ /2
E) a multiple of $\pi$

Question 48

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]

Question 49

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

Question 50

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 $\degree$ 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

Question 51

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

A) travel in opposite directions and are in phase
B) travel in opposite directions and are 180 $\degree$ out of phase
C) travel in the same direction and are in phase
D) travel in the same direction and are 180 $\degree$ out of phase
E) travel in the same direction and are 90 $\degree$ out of phase

Question 52

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

Question 53

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 in m/s of any point on the string is:

A) 0.2
B) 1.3
C) 4
D) 15
E) 25

Question 54

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 $\degree$ , 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) 1.8y_m
E) 0.35 y_m

Question 55

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 in m/s² of any point on the string is:

A) 0
B) 130
C) 395
D) 790
E) 1600

Question 56

Two traveling sinusoidal waves interfere to produce a wave with the mathematical form If the value of $\phi$ is appropriately chosen, the two waves might be:

A) y₁(x,t) = (y_m/3) sin (kx + $\omega$ t) and y₂(x,t) = (y_m/3) sin (kx + $\omega$ t + $\phi$ )
B) y₁(x,t) = 0.7y_m sin (kx - $\omega$ t) and y₂(x,t) = 0.7y_m sin (kx - $\omega$ t + $\phi$ )
C) y₁(x,t) = 0.7y_m sin (kx - $\omega$ t) and y₂(x,t) = 0.7y_m sin (kx + $\omega$ t + $\phi$ )
D) y₁(x,t) = 0.7y_m sin [(kx/2) - ( $\omega$ t/2)] and y₂(x,t) = 0.7y_m sin [(kx/2) - ( $\omega$ t/2) + $\phi$ ]
E) y₁(x,t) = 0.7y_m sin (kx + $\omega$ t) and y₂(x,t) = 0.7y_m sin (kx + $\omega$ t + $\phi$ )

Question 57

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

A) travel in opposite directions and are in phase
B) travel in opposite directions and are 180 $\degree$ out of phase
C) travel in the same direction and are in phase
D) travel in the same direction and are 180 $\degree$ out of phase
E) travel in the same direction and are 90 $\degree$ out of phase

Question 58

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

A) an odd multiple of $\lambda$ /2
B) an odd multiple of $\lambda$ /4
C) a multiple of $\lambda$
D) an odd multiple of $\pi$ /2
E) a multiple of $\pi$

Question 59

The sinusoidal wave y(x,t) = y_msin(kx - $\omega$ t)
Is incident on the fixed end of a string at x = L. The reflected wave is given by:

A) y_msin(kx + $\omega$ t)
B) -y_msin(kx + $\omega$ t)
C) y_msin(kx + $\omega$ t - kL)
D) y_msin(kx + $\omega$ t - 2kL)
E) -y_msin(kx + $\omega$ t + 2kL)

Question 60

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

Question 61

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

Question 62

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

A) 33.3 cm
B) 66.7 cm
C) 150 cm
D) 300 cm
E) need to know the frequency

Question 63

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

A) 2
B) 4
C)
D) 1/2
E)

Question 64

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

Question 65

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

Question 66

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

Question 67

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) 66.7 cm
E) 50 cm

Question 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

Question 69

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

Question 70

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

Question 71

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