Deck 16: Superposition and Standing Waves

Question 1

The air columns in two identical pipes vibrate at frequencies of 150 Hz.The percentage of change needed in the length of one of the pipes to produce 3 beats per second is

A)1%
B)2%
C)3%
D)4%
E)5%

Answer 1

2%

Question 2

When a piano tuner strikes both the A above middle C on the piano and a 440 Hz tuning fork,he hears 4 beats each second.The frequency of the piano's A is

A)440 Hz
B)444 Hz
C)880 Hz
D)436 Hz
E)either 436 Hz or 444 Hz

Answer 2

either 436 Hz or 444 Hz

Question 3

If two identical waves with a phase difference of 6 $\pi$ are added,the result is

A)a wave with the same frequency but twice the amplitude.
B)a wave with the same amplitude but twice the frequency.
C)a wave with zero amplitude.
D)a wave with zero frequency.
E)This problem cannot be solved without knowing the wavelengths of the two waves.

Answer 3

a wave with the same frequency but twice the amplitude.

Question 4

Two waves with the same frequency and wavelength but with different amplitudes are added.If A₁ = 2A₂ and the waves are 180º out of phase,then the amplitude of the resultant wave is

A)zero.
B)the same as A₁.
C)the same as A₂.
D)equal to A₁ + A₂.
E)coherent.

Question 5

Tuning fork A has a frequency of 440 Hz.When A and a second tuning fork B are struck simultaneously,four beats per second are heard.When a small mass is added to one of the tines of B,the two forks struck simultaneously produce two beats per second.The original frequency of tuning fork B was

A)448 Hz
B)444 Hz
C)438 Hz
D)436 Hz
E)432 Hz

Question 6

Two speakers face each other at a distance of 1 m and are driven by a common audio oscillator.A first minimum in sound intensity is found 16.1 cm from the midpoint.If the velocity of sound is 330 m/s,find the frequency of the oscillator.

A)256 Hz
B)1024 Hz
C)512 Hz
D)341 Hz
E)683 Hz

Question 7

The interference of waves refers to the

A)slowing down of one wave in the presence of another.
B)resultant disturbance of two or more waves at every point in the medium.
C)change in wavelength that occurs when two waves cross one another.
D)phase change of 180º that occurs on reflection of a wave at a fixed end.
E)ability of waves to go around corners.

Question 8

Two sound waves,one wave is given by y₁ = p_o sin (kx - $\omega$ t)and the other by y₂ = p_o sin (kx - $\omega$ t + $\pi$ /4).The phase constant resulting from the interference of the two waves is

A) $\pi$ /8
B) $\pi$ /4
C) $\pi$ /2
D) $\pi$
E)0

Question 9

A wave on a string has a frequency of 100 Hz and travels at a speed of 24 m/s.The minimum distance between two points with a phase difference of 60º is

A)0.040 m
B)0.12 m
C)0.14 m
D)0.24 m
E)25 m

Question 10

Two wave trains of the same frequency are traveling in opposite directions down a string.When they meet,these wave trains will not

A)be described by the principle of superposition.
B)reflect from each other.
C)pass through one another.
D)continue to carry energy.
E)remain transverse.

Question 11

Two tones of equal amplitude but slightly different frequencies are emitted by a sound source.This gives rise to

A)standing waves.
B)destructive interference.
C)constructive interference.
D)beats.
E)amplification.

Question 12

Two wave trains travel on a string under a constant tension T.Which of the following statements is NOT correct?

A)The two waves can have different speed.
B)The two waves can have different frequency.
C)The two waves can have different wavelength.
D)The superposition principle applies for the two waves.
E)At any point on the string,the resultant amplitude is the algebraic sum of the amplitudes of the two waves.

Question 13

What is the phase difference at any given instant between two points on a wave which are 1.52 m apart if the wavelength of the wave is 2.13 m?

A)0.430 rad
B)2.70 rad
C)4.48 rad
D)44.0 rad
E)119 rad

Question 14

Middle C on a piano has a frequency of 262 Hz.Sometimes it is said that middle C is actually 2⁸ = 256 Hz,and tuning forks are made with this frequency.How many beats per second would be heard if such a tuning fork were sounded simultaneously with the middle C of a (well-tuned)piano?

A)3
B)6
C)12
D)4
E)8

Question 15

A violinist is tuning the A string on her violin by listening for beats when this note is played simultaneously with a tuning fork of frequency 440 Hz.She hears a beat frequency of 4 Hz.She notices that,when she increases the tension in the string slightly,the beat frequency decreases.What was the frequency of the mistuned A string?

A)448 Hz
B)444 Hz
C)436 Hz
D)432 Hz
E)438 Hz

Question 16

If two identical waves with the same phase are added,the result is

A)a wave with the same frequency but twice the amplitude.
B)a wave with the same amplitude but twice the frequency.
C)a wave with zero amplitude.
D)a wave with zero frequency.
E)This problem cannot be solved without knowing the wavelengths of the two waves.

Question 17

Two sound waves,one wave is given by y₁ = p_o sin (kx - $\omega$ t)and the other by y₂ = p_o sin (kx - $\omega$ t + $\pi$ /2).The amplitude resulting from the interference of the two waves is

A)2p_o
B)
C)1.25p_o
D)
E)0

Question 18

Two trumpet players are both playing a pitch with a frequency of 440 Hz,corresponding to the musical pitch A above middle C.However,one of the trumpet players is marching away from you so that you hear a beat frequency of 4 Hz from the two trumpets.With what speed is the departing trumpet player moving away from you? (The speed of sound in air is 340 m/s)

A)3.12 m/s
B)3.09 m/s
C)3.06 m/s
D)3.00 m/s
E)2.95 m/s

Question 19

If two identical waves with a phase difference of 3 $\pi$ are added,the result is

A)a wave with the same frequency but twice the amplitude.
B)a wave with the same amplitude but twice the frequency.
C)a wave with zero amplitude.
D)a wave with an intensity equal to the sum of the intensities of the two waves.
E)This problem cannot be solved without knowing the wavelengths of the two waves.

Question 20

Two whistles produce sounds with wavelengths 3.40 m and 3.30 m.What is the beat frequency produced? (the speed of sound is 340 m/s)

A)0.1 Hz
B)1.0 Hz
C)2.0 Hz
D)3.0 Hz
E)4.0 Hz

Question 21

The fundamental frequency of a vibrating string is f₁.If the tension in the string is decreased by 50% while the linear density is held constant,the fundamental frequency becomes

A)0.5f₁
B)0.7f₁
C)0.9f₁
D)f₁
E)None of these is correct.

Question 22

The fundamental frequency of a vibrating string is f₁.If the tension in the string is quadrupled while the linear density is held constant,the fundamental frequency becomes

A)f₁
B)1.2f₁
C)1.5f₁
D)1.7f₁
E)2f₁

Question 23

If both the tension and the length of a vibrating string are doubled while the linear density remains constant,the fundamental frequency of the string is multiplied by

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

Question 24

The fundamental frequency of a vibrating string is f₁.If the tension in the string is doubled,the fundamental frequency becomes

A)f₁/2
B)
C)f₁
D)
E)2f₁

Question 25

Two sound waves,one wave is given by y₁ = p_o sin ( $\omega$ ₁t),and the other by y₂ = p_o sin ( $\omega$ ₂t),where $\omega$ ₁ differs from $\omega$ ₂ by a rad/s.The maximum sound intensity of the beat frequency is

A)p_o²
B)2p_o²
C)4p_o²
D)8p_o²
E) 0

Question 26

Two sources are said to be coherent if

A)they are of the same frequency and has a phase of zero.
B)they are of the same frequency and maintain a constant non-zero phase.
C)they are of the same intensity but different frequency and has a phase of zero.
D)they are of the same intensity but different frequency and maintain a constant non-zero phase.
E)(A)and (B)

Question 27

If the amplitude of a standing wave is doubled,the energy in the wave increases by a factor of

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

Question 28

Which of the following equations represents a standing wave? (The symbols have their usual meaning.)

A)
B)
C)
D)
E)(B)and (D)

Question 29

A string 2.0 m long has a mass of 2.4 $\times$ 10^-2 kg.When fixed at both ends,it vibrates with a fundamental frequency of 150 Hz.The speed of a transverse wave in the string is

A)3.6 m/s
B)75 m/s
C)0.30 km/s
D)0.60 km/s
E)0.63 km/s

Question 30

A standing wave is shown in the figure on the right.If the period of the wave is T,the shortest time it takes for the wave to go from the solid curve to the dashed curve is

A)T/4
B)T/3
C)T/2
D)3T/4
E)None of these is correct.

Question 31

A string fixed at both ends is driven by a tuning fork to produce standing waves.If the tension in the string is increased,

A)the frequency increases.
B)the frequency decreases and the wave velocity remains constant.
C)the wavelength decreases.
D)the wave velocity increases.
E)the wave velocity decreases.

Question 32

A microphone is placed at the node of a standing sound wave.What does the microphone pick up?

A)A constant and very high intensity sound.
B)A constant and very low intensity sound.
C)A varying high intensity sound.
D)A varying low intensity sound.
E)Unable to tell.

Question 33

In a vibrating-string experiment,three loops are observed between points A and B when the mass on one end of the string is 100 g.The number of loops between A and B can be changed to two by replacing the 100-g mass with a mass of

A)150 g
B)225 g
C)44.4 g
D)66.7 g
E)300 g

Question 34

A standing wave is created by oscillating a taut string at a frequency that corresponds to one of the resonant frequencies.The amplitude of the antinodes is very much larger than the amplitude of the oscillator.Does this violate the conservation of energy principle? Explain why.

A)Yes,since E is proportional to amplitude squared.
B)Yes,since there is large kinetic energy of the string,and this is much bigger than the energy from the oscillator.
C)No,energy from waves does not obey the conservation of energy principle in the first place.
D)No,the energy at the antinodes builds up after the first few cycles,after which the dissipation due to friction equals the energy supplied by the oscillator.
E)Whether it obeys the conservation of energy principle depends on the tension in the string.

Question 35

A string of linear density $\mu$ and length L is under a constant tension T = mg.One end of the string is attached to a tunable harmonic oscillator.A resonant standing wave is observed

A)at any frequency.
B)when the frequency where n = 1,2,3,...
C)when the frequency where n = 1,2,3,...
D)when the frequency where n = 1,2,3,...and v_s is the speed of sound.
E)unable to tell

Question 36

A string 2.0 m long has a mass of 2.4 $\times$ 10^-2 kg.When fixed at both ends,it vibrates with a fundamental frequency of 150 Hz.The frequency of the third harmonic of this fundamental is

A)50 Hz
B)75 Hz
C)0.15 kHz
D)0.45 kHz
E)1.1 kHz

Question 37

The figure shows a standing wave in a pipe that is closed at one end.The frequency associated with this wave pattern is called the

A)first harmonic.
B)second harmonic.
C)third harmonic.
D)fourth harmonic.
E)fifth harmonic.

Question 38

The fundamental frequency of a vibrating string is f₁.If the tension in the string is increased by 50% while the linear density is held constant,the fundamental frequency becomes

A)f₁
B)1.2f₁
C)1.5f₁
D)1.7f₁
E)2f₁

Question 39

Four pendulums are hung from a light rod that is free to rotate about its long axis.The pendulums have lengths L,2L,L/2 and L,and masses m,m/2,2m and 4m respectively.Pendulum 1 is set to swing at its natural frequency.Which of the other three will,over time,also oscillate at the same frequency?

A)(2)
B)(3)
C)(4)
D)(2)and (3)
E)all three

Question 40

A string whose length is 1 m is fixed at both ends and vibrates according to the equation where the units are SI.The total number of nodes exhibited by the string is

A)1
B)2
C)3
D)4
E)5

Question 41

The fundamental frequency of a pipe that has one end closed is 256 Hz.When both ends of the same pipe are opened,the fundamental frequency is

A)64.0 Hz
B)128 Hz
C)256 Hz
D)512 Hz
E)1.02 kHz

Question 42

The third harmonic of a tube closed at one end is 735 Hz.If the speed of sound in air is 335 m/s,the length of the tube must be

A)11.6 cm
B)22.9 cm
C)34.1 cm
D)45.7 cm
E)57.3 cm

Question 43

A clarinet,which is essentially a tube that is open at one end,is properly tuned to concert A (440 Hz)indoors,where the temperature is 20ºC and the speed of sound is 340 m/s.The musician then takes the instrument to play an outdoor concert,where the temperature is 0ºC and the speed of sound is 331 m/s.What is the frequency of the A played on the cold clarinet? (Ignore any thermal changes in the body of the clarinet itself.)

A)417 Hz
B)428 Hz
C)434 Hz
D)445 Hz
E)451 Hz

Question 44

Standing waves exist in a string of length L that is fixed at one end and free at the other.The speed of the waves on the string is v.The three lowest frequencies of vibration are

A)v/4L,v/2L,and 3v/4L
B)v/2L,v/L,and 3v/2L
C) $\lambda$ /4, $\lambda$ /2,and 3 $\lambda$ /4
D)v/4L,3v/4L,and 5v/4L
E) $\lambda$ /3,2 $\lambda$ /3,and 3 $\lambda$ /3

Question 45

For a tube of length 57.0 cm that is open at both ends,what is the frequency of the fundamental mode? (the speed of sound in air is 340 m/s)

A)149 Hz
B)447 Hz
C)596 Hz
D)298 Hz
E)746 Hz

Question 46

A 1.00 m string fixed at both ends vibrates in its fundamental mode at 440 Hz.What is the speed of the waves on this string?

A)220 m/s
B)440 m/s
C)660 m/s
D)880 m/s
E)1.10 km/s

Question 47

The standing waves in air in a pipe of length L that is open at one end and closed at the other have a speed v.The frequencies of the three lowest harmonics are

A)v/4L,v/2L,and 3v/4L
B)v/2L,v/L,and 3v/2L
C) $\lambda$ /4, $\lambda$ /2,and 3 $\lambda$ /4
D)v/4L,3v/4L,and 5v/4L
E) $\lambda$ /3,2 $\lambda$ /3,and 3 $\lambda$ /3

Question 48

When an organ pipe,which is closed at one end only,vibrates with a frequency that is three times its fundamental (first harmonic)frequency,

A)the sound produced travels at three times its former speed.
B)the sound produced is its fifth harmonic.
C)beats are produced.
D)the sound produced has one-third its former wavelength.
E)the closed end is a displacement antinode.

Question 49

A string fixed at both ends is vibrating in a standing wave.There are three nodes between the ends of the string,not including those on the ends.The string is vibrating at a frequency that is its

A)fundamental.
B)second harmonic.
C)third harmonic.
D)fourth harmonic.
E)fifth harmonic.

Question 50

A string fixed at both ends is 50.0 cm long and has a tension that causes the frequency of its fundamental to be 262 Hz.If the tension is increased by 4%,what does the fundamental frequency become?

A)252 Hz
B)257 Hz
C)264 Hz
D)267 Hz
E)272 Hz

Question 51

On a standing-wave pattern,the distance between two consecutive nodes is d.The wavelength is

A)d/2
B)d
C)3d/2
D)2d
E)4d

Question 52

The standing waves in air in a pipe of length L that is open at both ends have a speed v.The frequencies of the three lowest harmonics are

A)v/L,2v/L,and 3v/L
B)v/2L,v/L,and 3v/2L
C) $\lambda$ /2, $\lambda$ ,and 3 $\lambda$ /2
D)L/v,2L/v,and 3L/v
E) $\lambda$ /3,2 $\lambda$ /3,and 3 $\lambda$ /3

Question 53

A vibrating tuning fork of frequency 1080 Hz is held above a tube filled with water.Assume the speed of sound to be 330 m/s.As the water level is lowered,consecutive maxima in intensity are observed at intervals of about

A)7.65 cm
B)15.3 cm
C)23.0 cm
D)30.6 cm
E)53.6 cm

Question 54

Sound has a velocity of 335 m/s in air.For an air column that is closed at both ends to resonate to a frequency of 528 Hz,the length of the air column could be

A)79.2 cm
B)55.5 cm
C)47.5 cm
D)31.7 cm
E)15.8 cm

Question 55

The standing waves on a string of length L that is fixed at both ends have a speed v.The three lowest frequencies of vibration are

A)v/L,2v/L,and 3v/L
B)v/2L,v/L,and 3v/2L
C) $\lambda$ /2, $\lambda$ ,and 3 $\lambda$ /2
D)L/v,2L/v,and 3L/v
E) $\lambda$ /3,2 $\lambda$ /3,and 3 $\lambda$ /3

Question 56

The human vocal tract can be thought of as a tube that is open at one end.If the length of this tube is 17 cm (about average for an adult male),what are the lowest two harmonics?

A)500 Hz,1500 Hz
B)500 Hz,1000 Hz
C)1000 Hz,2000 Hz
D)1000 Hz,3000 Hz
E)1500 Hz,2500 Hz

Question 57

The ratio of the fundamental frequency (first harmonic)of an open pipe to that of a closed pipe of the same length is

A)2:1
B)7:8
C)4:5
D)3:2
E)1:2

Question 58

In a pipe that is open at one end and closed at the other and that has a fundamental frequency of 256 Hz,which of the following frequencies cannot be produced?

A)768 Hz
B)1.28 kHz
C)5.12 kHz
D)19.7 kHz
E)all of these can be produced

Question 59

A stretched string of length L,fixed at both ends,is vibrating in its third harmonic.How far from the end of the string can the blade of a screwdriver be placed against the string without disturbing the amplitude of the vibration?

A)L/6
B)L/4
C)L/5
D)L/2
E)None of these is correct.

Question 60

A vibrating tuning fork of frequency 640 Hz is held above a tube filled with water.Assume the speed of sound to be 330 m/s.As the water level is lowered,consecutive maxima in intensity are observed at intervals of about

A)12.9 cm
B)19.4 cm
C)25.8 cm
D)51.7 cm
E)194 cm

Question 61

The reason we can tell the difference between a trumpet and a clarinet when they both play the same pitch is that they have

A)the same overtones.
B)the same harmonics.
C)different fundamental frequencies.
D)different waveforms.
E)harmonic syntheses.

Question 62

Wire A is the same mass per unit length as wire B.However wire A is twice as long as wire B and has three times as much tension on it.Calculate the fundamental frequency of wire A divided by wire B.

A)0.87
B)0.66
C)0.43
D)0.75
E)1.50

Question 63

The wave function y(x,t)for a standing wave on a string fixed at both ends is given by y(x,t)= 0.080 sin 6.0x cos 600t where the units are SI.The amplitudes of the traveling wave that result in this standing wave are

A)0.04 m
B)0.08 m
C)0.02 m
D)0.16 m
E)impossible to tell given this information about the standing wave.

Question 64

The wave function y(x,t)for a standing wave on a string fixed at both ends is given by y(x,t)= 0.080 sin 6.0x cos 600t where the units are SI.The speed of the traveling waves that result in this standing wave is

A)6.00 m
B)1.05 m
C)600 m
D)0.010 m
E)impossible to tell given this information about the standing wave.

Question 65

A string with mass density equal to 0.0025 kg/m is fixed at both ends and at a tension of 290 N.Resonant frequencies are found at 558 Hz and the next one at 744 Hz.What is the length of the wire?

A)0.8 m
B)1.6 m
C)3.2 m
D)1.2 m
E)0.4 m

Question 66

Two pipes closed at one end of length L₁ and L₂ are excited at their resonant frequencies.If the beat period is B_f Hz,then the velocity of sound is given by:

A)B_f $\times$ L₁ $\times$ L₂ / (4L₁ $\times$ 4L₂)
B)4 $\times$ B_f $\times$ L₁ $\times$ L₂ / (4L₁ - 4L₂)
C)16 $\times$ B_f $\times$ L₁ $\times$ L₂ / (4L₁ + 4L₂)
D)4 $\times$ B_f $\times$ L₁ $\times$ L₂ / (L₁ -L₂)
E)4 $\times$ B_f $\times$ L₁ $\times$ L₂ / (L₁ + L₂)

Question 67

A pipe produces successive harmonics at 300 Hz and 350 Hz.Calculate the length of the pipe and state whether it is closed at one end or not.Assume the speed of sound to be 340 m/s.

A)
B)
C)
D)
E)

Question 68

A vibrating tuning fork of 725 Hz is held above a tube filled with water.Successive resonances are heard when the water level is lowered by 11.5 cm and 34.5 cm from the top of the tube.Calculate a value for the speed of sound.(Hint: remember the small end correction $\Delta$ L at the top of the tube.)

A)333 m/s
B)343 m/s
C)325 m/s
D)315 m/s
E)338 m/s

Question 69

The wave function y(x,t)for a standing wave on a string fixed at both ends is given by y(x,t)= 0.080 sin 6.0x cos 600t where the units are SI.The wavelength of this wave is

A)6.00 m
B)1.05 m
C)600 m
D)0.010 m
E)impossible to tell given this information about the standing wave.

Question 70

The wave function y(x,t)for a standing wave on a string fixed at both ends is given by y(x,t)= 0.080 sin 6.0x cos 600t where the units are SI.The distance between successive nodes on the string is

A)0.24 m
B)0.08 m
C)0.02 m
D)0.52 m
E)impossible to tell given this information about the standing wave.

Question 71

A string with mass density equal to 0.0025 kg/m is fixed at both ends and at a tension of 290 N.Resonant frequencies are found at 558 Hz and the next one at 744 Hz.To what harmonic does the 558 Hz resonance correspond?

A)1
B)2
C)3
D)4
E)5

Question 72

A piano tuner hears a beat every 0.33 seconds when he hits a note and compares it to his reference tone at 163 Hz.What is the lowest possible frequency of the piano note?

A)44.9 Hz
B)166.0 Hz
C)162.7 Hz
D)163.3 Hz
E)160.0 Hz

Question 73

A wire of mass 1.1 g is under a tension of 100 N.If its third overtone is at a frequency of 750 Hz,calculate the length of the wire.

A)72 cm
B)101 cm
C)36 cm
D)65 cm
E)None of the above

Question 74

A string with mass density equal to 0.0025 kg/m is fixed at both ends and at a tension of 290 N.Resonant frequencies are found at 558 Hz and the next one at 744 Hz.What is the fundamental frequency of the string?

A)558 Hz
B)372 Hz
C)93 Hz
D)186 Hz
E)none of the above

Question 75

A vibrating tuning fork of 850 Hz is held above a tube filled with water.The first and third resonances occur when the water level is lowered by 8.8 cm and 47.6 cm from the top of the tube.If there is a small end correction that adds a small extra length $\Delta$ L to the effective length of the air column,calculate $\Delta$ L.Assume the speed of sound to be 330 m/s.

A)0.2 cm
B)0.9 cm
C)0.4 cm
D)0.6 cm
E)1.1 cm

Question 76

Two identical loudspeakers are driven in phase by the same amplifier.The speakers are positioned a distance of 3.2 m apart.A person stands 4.1 m away from one speaker and 4.8 m away from the other.Calculate the second lowest frequency that results in destructive interference at the point where the person is standing.Assume the speed of sound to be 340 m/s.

A)245 Hz
B)735 Hz
C)1225 Hz
D)490 Hz
E)1470 Hz

Question 77

A vibrating tuning fork is held above a tube filled with water.The first two resonances occur when the water level is lowered by 14.2 cm and 44.2 cm from the top of the tube.If there is a small end correction that adds a small extra length $\Delta$ L? to the effective length of the air column,calculate the frequency of the tuning fork.Assume the speed of sound to be 330 m/s.

A)560 Hz
B)581 Hz
C)550 Hz
D)1100 Hz
E)1120 Hz

Question 78

What is the third harmonic of an open-both-ends organ pipe of length 1.5 m? Assume the speed of sound to be 340 m/s.

A)229 Hz
B)340 Hz
C)457 Hz
D)686 Hz
E)none of the above

Question 79

A vibrating tuning fork of 300 Hz is held above a tube filled with water.The first resonance is heard when the water level is lowered by 26.1 cm.A second tuning fork of 400 Hz is held above the tube,and its first resonance occurs when the water level is lowered by 19.3 cm from the top.Calculate a value for the speed of sound.(Hint: remember the small end correction $\Delta$ L at the top of the tube.)

A)333 m/s
B)343 m/s
C)325 m/s
D)315 m/s
E)338 m/s

Question 80

A guitar string of length 105 cm is in resonance with a tuning fork of frequency f.Using the fret board the length of the string is shortened by 1.5 cm while keeping the tension in the string constant.Now a beat frequency of 10 Hz is heard between the string and the tuning fork.What is the frequency of the tuning fork?

A)230 Hz
B)1380 Hz
C)345 Hz
D)690 Hz
E)none of the above