Question 1

The superposition of two waves,  and  , results in a wave with a wavelength of
A)  m.
B)2 m.
C)π m.
D)4 m.
E)4π m.

Accepted Answer

The answer of The superposition of two waves,  and...

Question 2

Which of the following frequencies could NOT be present as a standing wave in a 2-m-long organ pipe open at both ends? The fundamental frequency is 85 Hz.
A)85 Hz.
B)170 Hz.
C)255 Hz.
D)340 Hz.
E)382 Hz.

Accepted Answer

In a pipe open at both ends, the harmonics are integer multiples of the fundamental frequency. Given the fundamental frequency is 85 Hz, the harmonics would be 85 Hz, 170 Hz (2 x 85), 255 Hz (3 x 85), 340 Hz (4 x 85), etc. The frequency 382 Hz does not fit this pattern, making it the correct choice.

Question 3

Two waves are described by ​
Y1 = 6 cos(180t​) and y2 = 6 cos(186t) (both in meters).
​
What effective frequency does the resultant vibration have at a point?
A)92 Hz
B)183 Hz
C)6 Hz
D)3 Hz
E)366 Hz

Accepted Answer

To find the effective frequency, we need to find the frequency of the resultant wave. 
Using the formula:

y = Y1 + Y2 
y = 6 cos(180t) + 6 cos(186t)

Using the identity:

cos(A) + cos(B) = 2 cos[(A + B)/2] cos[(A - B)/2]

We can simplify the equation:

y = 12 cos(183t) cos(3t)

The frequency of this wave is the frequency of the cosine term, which is 3 Hz.

Therefore, the answer is choice B) 183 Hz, since that is the frequency of the wave that causes the 3 Hz vibration at a point.

Question 4

Two identical strings have the same length and same mass per unit length. String B is stretched with four times as great a tension as that applied to string A. Which statement is correct for all n harmonics on the two strings, n = 1, 2, 3...?
A) 
B) 
C) 
D)fn,B = 2fn,A
E)fn,B = 4fn,A

Accepted Answer

The answer of Two identical strings have the same length...

Question 5

Two point sources emit sound waves of 1.0-m wavelength. The sources, 2.0 m apart, as shown below, emit waves which are in phase with each other at the instant of emission. Where, along the line between the sources, are the waves out of phase with each other by π radians? 
A)x = 0, 1.0 m, 2.0 m
B)x = 0.50 m, 1.5 m
C)x = 0.50 m, 1.0 m, 1.5 m
D)x = 0.75 m, 1.25 m
E)x = 0.25 m, 0.75 m, 1.25 m, 1.75 m

Accepted Answer

The answer of Two point sources emit sound waves of...

Question 6

Two waves are described by ​
Y1 = 6cos(180t) and y2 = 6cos(186t) (both in meters).
​
With what angular frequency does the maximum amplitude of the resultant wave vary with time?
A)366 rad/s
B)6 rad/s
C)3 rad/s
D)92 rad/s
E)180 rad/s

Accepted Answer

The resultant wave is given by y= y1 + y2 = 6cos(180t) + 6cos(186t). Using the identity cos(A) + cos(B) = 2cos((A+B)/2)cos((A-B)/2), we can simplify this to y = 12cos(183t)cos(3t). The amplitude of this wave varies as 12cos(3t), so the angular frequency with which the maximum amplitude varies with time is 3 rad/s.

Question 7

As shown below, a garden room has three walls, a floor, and a roof, but is open to the garden on one side. The wall widths are L and w. The roof height is h. When traveling sound waves enter the room, standing sound waves can be present in the room if the wavelength of the standing waves is ​  ​
A)  , where n is a positive integer.
B)  , where n is an odd integer.
C)  , where n is an even integer.
D)in all cases listed above.
E)given by (a) or (b) above, but not by (c).

Accepted Answer

The answer of As shown below, a garden room has...

Question 8

An observer stands 3 m from speaker A and 4 m from speaker B. Both speakers, oscillating in phase, produce 170-Hz waves. The speed of sound in air is 340 m/s. What is the phase difference (in radians) between the waves from A and B at the observer's location, point P? ​  ​
A)0
B) 
C)π
D)2π
E)4π

Accepted Answer

The answer of An observer stands 3 m from speaker...

Question 9

Transverse waves y1 = A1 sin(k1x − ω1t) and y2 = A2 sin(k2x − ω2t), with A2 > A1, start at opposite ends of a long rope when t = 0. The magnitude of the maximum displacement, y, of the rope at any point is
A)A1 − A2.
B)A2 − A1.
C)A1 + A2.
D)  .
E)  .

Accepted Answer

The answer of Transverse waves y1 = A1 sin(k1x −...

Question 10

Which of the following wavelengths could NOT be present as a standing wave in a 2-m-long organ pipe open at both ends?
A)4 m
B)2 m
C)1 m
D)0.89 m
E)0.5 m

Accepted Answer

For a pipe open at both ends, the allowed wavelengths ($\lambda$) for standing waves are given by $\lambda = \frac{2L}{n}$, where $L$ is the length of the pipe and $n$ is a positive integer (1, 2, 3, ...). For a 2 m long pipe, the wavelengths can be 4 m, 2 m, 1 m, etc., but 0.89 m does not fit this pattern, making it the wavelength that cannot be present as a standing wave in the pipe.

Question 11

The figure below shows wave crests after a stone is thrown into a pond.  Use this exhibit to answer the following question(s).
- The phase difference in radians between points A and B is
A)0.
B)  .
C)  .
D)π.
E)  .

Accepted Answer

The answer of 
The figure below shows wave crests after...

Question 12

In a standing wave, not necessarily at the fundamental frequency, on a string of length L, the distance between nodes is
A)λ/4.
B)λ/2.
C)λ.
D)L/4.
E)L/2.

Accepted Answer

In a standing wave on a string, nodes are points where the amplitude of vibration is zero. The distance between nodes is always half of a wavelength, because nodes occur at the points where the string is not vibrating at all. Therefore, the distance between nodes in a standing wave on a string of length L is λ/2.

Question 13

The superposition of two waves ​  and  ​
At the location x = 0 in space results in
A)beats at a beat frequency of 3 Hz.
B)a pure tone at a frequency of 153 Hz.
C)a pure tone at a frequency of 156 Hz.
D)beats at a beat frequency of 6 Hz.
E)beats at a frequency of 156 Hz.

Accepted Answer

The answer of The superposition of two waves ​ ...

Question 14

Two strings are, respectively, 1.00 m and 2.00 m long. Which of the following wavelengths, in meters, could represent harmonics present on both strings?
A)0.800, 0.670, 0.500
B)1.33, 1.00, 0.500
C)2.00, 1.00, 0.500
D)2.00, 1.33, 1.00
E)4.00, 2.00, 1.00

Accepted Answer

The wavelengths of the harmonics present on both strings must be the same. The fundamental mode for the 1.00 m string has a wavelength of 2.00 m, and the fundamental mode for the 2.00 m string has a wavelength of 4.00 m. Therefore, any common harmonics must have wavelengths that are integer divisors of both 2.00 m and 4.00 m. The only common divisors are 2.00 m and 1.00 m, so the harmonics must have wavelengths of 2.00 m, 1.00 m, or 0.500 m. The only choice that includes all three of these options is C.

Question 15

Which of the following wavelengths could NOT be present as a harmonic on a 2-m-long string?
A)4 m
B)2 m
C)1 m
D)0.89 m
E)0.5 m

Accepted Answer

The wavelength of a harmonic on a string must satisfy the condition that it is equal to some multiple of half the length of the string. Thus, the allowed wavelengths are given by λ = 2L/n, where n is an integer. Plugging in L = 2 m, we find that the allowed wavelengths are λ = 4 m, 2 m, 4/3 m, 1 m, 4/5 m, 4/7 m, etc. The only wavelength that cannot be expressed in this form is 0.89 m, so the answer is D.

Question 16

Two speakers in an automobile emit sound waves that are in phase at the speakers. One speaker is 40 cm ahead of and 30 cm to the left of the driver's left ear. The other speaker is 50 cm ahead of and 120 cm to the right of the driver's right ear. Which of the following wavelengths is(are) in phase at the left ear for the speaker on the left and the right ear for the speaker on the right?
A)10 cm
B)20 cm
C)650 cm
D)All of the wavelengths listed above.
E)Only the wavelengths listed in (a) and (b).

Accepted Answer

For the sound waves from the left speaker to be in phase at the driver's left ear, the path difference between the two waves must be an integer multiple of the wavelength. Let's call the wavelength λ.

The path difference between the two waves is the difference in distance traveled by the waves from each speaker to the driver's left ear. This can be calculated using the Pythagorean theorem: 
√[(40 cm)² + (30 cm)²] - √[(50 cm)² + (120 cm)²] = λn, where n is an integer.

Simplifying this equation gives:
√(2500) - √(14500) = λn
50√(2) - 2√(3625) = λn
50√(2) - 5√(145) = λn

To find the possible wavelengths that satisfy this equation, we need to find the greatest common factor of 50 and 5√(145). 
50 = 2 x 5² and 5√(145) = 5√(5² x 29) = 5 x 5√(29), so their GCF is 10.

Therefore, the wavelength can be any multiple of 10 cm. Choices (a) and (b) are both multiples of 10 cm, so they are in phase at the left ear for the speaker on the left.

Similarly, for the sound waves from the right speaker to be in phase at the driver's right ear, the path difference between the two waves must be an integer multiple of the wavelength. Let's call the wavelength λ.

The path difference between the two waves is the difference in distance traveled by the waves from each speaker to the driver's right ear. This can be calculated using the Pythagorean theorem: 
√[(50 cm)² + (120 cm)²] - √[(40 cm)² + (30 cm)²] = λn, where n is an integer.

Simplifying this equation gives:
√(14500) - √(2500) = λn
5√(580) - 50√(2) = λn

To find the possible wavelengths that satisfy this equation, we need to find the greatest common factor of 5√(580) and 50√(2). 
5√(580) = 5√(4 x 145) = 10√(145) and 50√(2) = 10√(2² x 5²) = 100√(2), so their GCF is 10√(2).

Therefore, the wavelength must be a multiple of 10√(2) cm. Choice (c) is not a multiple of 10√(2) cm, so it is not in phase at the right ear for the speaker on the right.

Therefore, the only wavelengths that are in phase at the left ear for the speaker on the left and the right ear for the speaker on the right are choices (a) and (b). The answer is (E).

Question 17

The superposition of two waves,  and  , results in a wave with a frequency of
A)85 Hz.
B)170 Hz.
C)85π Hz.
D)340 Hz.
E)170π Hz.

Accepted Answer

The answer of The superposition of two waves,  and...

Question 18

The superposition of two waves,  and  , results in a wave with a phase angle of
A)0 rad.
B)0.5 rad.
C)  rad.
D)  rad.
E)π rad.

Accepted Answer

The answer of The superposition of two waves,  and...

Question 19

A very long string is tied to a rigid wall at one end while the other end is attached to a simple harmonic oscillator. Which of the following can be changed by changing the frequency of the oscillator?
A)The speed of the waves traveling along the string.
B)The tension in the string.
C)The wavelength of the waves on the string.
D)All of the above.
E)None of the above.

Accepted Answer

The frequency of the oscillator is directly proportional to the frequency of the waves on the string, which determines the wavelength according to the equation λ = v/f, where λ is the wavelength, v is the speed of the waves, and f is the frequency. Therefore, changing the frequency of the oscillator will change the wavelength of the waves on the string, but it will not change the speed of the waves or the tension in the string.

Question 20

An organ pipe open at both ends is 1.5 m long. A second organ pipe that is closed at one end and open at the other is 0.75 m long. The speed of sound in the room is 330 m/s. Which of the following sets of frequencies consists of frequencies which can be produced by both pipes?
A)110 Hz, 220 Hz, 330 Hz
B)220 Hz, 440 Hz, 660 Hz
C)110 Hz, 330 Hz, 550 Hz
D)330 Hz, 440 Hz, 550 Hz
E)220 Hz, 660 Hz, 1 100 Hz

Accepted Answer

For an open pipe at both ends, the wavelengths of the standing waves are given by $\lambda_n = \frac{2L}{n}$ where $L$ is the length of the pipe and $n$ is an integer. 
For a closed pipe at one end and open at the other, the wavelengths are given by $\lambda_n = \frac{4L}{2n+1}$.
The frequencies of the standing waves are given by $f_n = \frac{nv}{2L}\cdot$ 
For the open pipe, we have $\lambda_1 = 3L$, $\lambda_2 = L$, $\lambda_3 = \frac{2}{3}L$, $\lambda_4 = \frac{3}{4}L$,... Thus, the frequencies are $f_1 = \frac{v}{3L}, f_2 = \frac{2v}{L}, f_3 = \frac{3v}{2L}, f_4 = \frac{4v}{3L}, ...$
For the closed-open pipe, we have $\lambda_1 = 4L/3$, $\lambda_2 = 4L/5$, $\lambda_3 = 4L/7$, $\lambda_4 = 4L/9$,... Thus, the frequencies are $f_1 = \frac{3v}{4L}, f_2 = \frac{5v}{4L}, f_3 = \frac{7v}{4L}, f_4 = \frac{9v}{4L}, ...$
To find the frequencies that can be produced by both pipes, we need to look for any frequencies $f_n$ for which there exist integers $m$ and $k$ such that $f_n = mf_1 = kf_2$. 
Since $f_1$ and $f_2$ are both multiples of $v/L$, the only way $f_n$ can be expressed as a multiple of both $f_1$ and $f_2$ is if $f_n$ is a multiple of $v/L$ as well. 
Checking the answer choices, we see that only choice C contains frequencies that are all multiples of $v/L$: $110 = (1/3)(330/L)$, $330 = (1)(330/L)$, and $550 = (5/3)(330/L)$. Therefore, the answer is $\boxed{	ext{C}}$.

Quiz 18: Superposition and Standing Waves

The superposition of two waves, and , results in a wave with a wavelength of

Which of the following frequencies could NOT be present as a standing wave in a 2-m-long organ pipe open at both ends? The fundamental frequency is 85 Hz.

Two waves are described by Y1 = 6 cos(180t) and y2 = 6 cos(186t) (both in meters) . What effective frequency does the resultant vibration have at a point?

Two identical strings have the same length and same mass per unit length. String B is stretched with four times as great a tension as that applied to string A. Which statement is correct for all n harmonics on the two strings, n = 1, 2, 3...?

Two point sources emit sound waves of 1.0-m wavelength. The sources, 2.0 m apart, as shown below, emit waves which are in phase with each other at the instant of emission. Where, along the line between the sources, are the waves out of phase with each other by π radians?

Two waves are described by Y1 = 6cos(180t) and y2 = 6cos(186t) (both in meters) . With what angular frequency does the maximum amplitude of the resultant wave vary with time?

As shown below, a garden room has three walls, a floor, and a roof, but is open to the garden on one side. The wall widths are L and w. The roof height is h. When traveling sound waves enter the room, standing sound waves can be present in the room if the wavelength of the standing waves is

An observer stands 3 m from speaker A and 4 m from speaker B. Both speakers, oscillating in phase, produce 170-Hz waves. The speed of sound in air is 340 m/s. What is the phase difference (in radians) between the waves from A and B at the observer's location, point P?

Transverse waves y1 = A1 sin(k1x − ω1t) and y2 = A2 sin(k2x − ω2t) , with A2 > A1, start at opposite ends of a long rope when t = 0. The magnitude of the maximum displacement, y, of the rope at any point is

Which of the following wavelengths could NOT be present as a standing wave in a 2-m-long organ pipe open at both ends?

The figure below shows wave crests after a stone is thrown into a pond. Use this exhibit to answer the following question(s) . - The phase difference in radians between points A and B is

In a standing wave, not necessarily at the fundamental frequency, on a string of length L, the distance between nodes is

The superposition of two waves and At the location x = 0 in space results in

Two strings are, respectively, 1.00 m and 2.00 m long. Which of the following wavelengths, in meters, could represent harmonics present on both strings?

Which of the following wavelengths could NOT be present as a harmonic on a 2-m-long string?

The superposition of two waves, and , results in a wave with a frequency of

The superposition of two waves, and , results in a wave with a phase angle of

A very long string is tied to a rigid wall at one end while the other end is attached to a simple harmonic oscillator. Which of the following can be changed by changing the frequency of the oscillator?

An organ pipe open at both ends is 1.5 m long. A second organ pipe that is closed at one end and open at the other is 0.75 m long. The speed of sound in the room is 330 m/s. Which of the following sets of frequencies consists of frequencies which can be produced by both pipes?

Quiz 18: Superposition and Standing Waves

The superposition of two waves, and , results in a wave with a wavelength of

Which of the following frequencies could NOT be present as a standing wave in a 2-m-long organ pipe open at both ends? The fundamental frequency is 85 Hz.

Two waves are described by ​ Y1 = 6 cos(180t​) and y2 = 6 cos(186t) (both in meters) . ​ What effective frequency does the resultant vibration have at a point?

Two identical strings have the same length and same mass per unit length. String B is stretched with four times as great a tension as that applied to string A. Which statement is correct for all n harmonics on the two strings, n = 1, 2, 3...?

Two point sources emit sound waves of 1.0-m wavelength. The sources, 2.0 m apart, as shown below, emit waves which are in phase with each other at the instant of emission. Where, along the line between the sources, are the waves out of phase with each other by π radians?

Two waves are described by ​ Y1 = 6cos(180t) and y2 = 6cos(186t) (both in meters) . ​ With what angular frequency does the maximum amplitude of the resultant wave vary with time?

As shown below, a garden room has three walls, a floor, and a roof, but is open to the garden on one side. The wall widths are L and w. The roof height is h. When traveling sound waves enter the room, standing sound waves can be present in the room if the wavelength of the standing waves is ​ ​

An observer stands 3 m from speaker A and 4 m from speaker B. Both speakers, oscillating in phase, produce 170-Hz waves. The speed of sound in air is 340 m/s. What is the phase difference (in radians) between the waves from A and B at the observer's location, point P? ​ ​

Transverse waves y1 = A1 sin(k1x − ω1t) and y2 = A2 sin(k2x − ω2t) , with A2 > A1, start at opposite ends of a long rope when t = 0. The magnitude of the maximum displacement, y, of the rope at any point is

Which of the following wavelengths could NOT be present as a standing wave in a 2-m-long organ pipe open at both ends?

The figure below shows wave crests after a stone is thrown into a pond. Use this exhibit to answer the following question(s) . - The phase difference in radians between points A and B is

In a standing wave, not necessarily at the fundamental frequency, on a string of length L, the distance between nodes is

The superposition of two waves ​ and ​ At the location x = 0 in space results in

Two strings are, respectively, 1.00 m and 2.00 m long. Which of the following wavelengths, in meters, could represent harmonics present on both strings?

Which of the following wavelengths could NOT be present as a harmonic on a 2-m-long string?

The superposition of two waves, and , results in a wave with a frequency of

The superposition of two waves, and , results in a wave with a phase angle of

A very long string is tied to a rigid wall at one end while the other end is attached to a simple harmonic oscillator. Which of the following can be changed by changing the frequency of the oscillator?

An organ pipe open at both ends is 1.5 m long. A second organ pipe that is closed at one end and open at the other is 0.75 m long. The speed of sound in the room is 330 m/s. Which of the following sets of frequencies consists of frequencies which can be produced by both pipes?

Two waves are described by Y1 = 6 cos(180t) and y2 = 6 cos(186t) (both in meters) . What effective frequency does the resultant vibration have at a point?

Two waves are described by Y1 = 6cos(180t) and y2 = 6cos(186t) (both in meters) . With what angular frequency does the maximum amplitude of the resultant wave vary with time?

As shown below, a garden room has three walls, a floor, and a roof, but is open to the garden on one side. The wall widths are L and w. The roof height is h. When traveling sound waves enter the room, standing sound waves can be present in the room if the wavelength of the standing waves is

An observer stands 3 m from speaker A and 4 m from speaker B. Both speakers, oscillating in phase, produce 170-Hz waves. The speed of sound in air is 340 m/s. What is the phase difference (in radians) between the waves from A and B at the observer's location, point P?

The superposition of two waves and At the location x = 0 in space results in