Question 1

A string is stretched and fixed at both ends, 200 cm apart. If the density of the string is 0.015 g/cm, and its tension is 600 N, what is the wavelength (in cm) of the first harmonic?

Accepted Answer

B

Question 2

The superposition of two waves  and $y_{2}=(0.006 \mathrm{~cm}) \cos \left[2 \pi\left(\frac{150}{\mathrm{~s}} t\right)\right]$ at the location x = 0 in space results in:

Accepted Answer

D

Question 3

Two tuning forks with frequencies 264 and 262 Hz produce 'beats'. What is the beat frequency (in Hz)?

Accepted Answer

The beat frequency is the absolute difference between the two frequencies, which is |264 Hz - 262 Hz| = 2 Hz.

Question 4

A clarinet behaves like a tube closed at one end. If its length is 1.0 m, and the velocity of sound is 344 m/s, what is its fundamental frequency (in Hz)?

Accepted Answer

A tube closed at one end can only produce odd harmonics, which means the fundamental frequency is the third harmonic. The formula for the frequency of a closed tube is: 
f = (2n - 1) v/4L
where n is the harmonic number, v is the velocity of sound, and L is the length of the tube. 
Plugging in the values given: 
f = (2(3) - 1) (344 m/s)/(4(1.0 m))
f = 86 Hz

Question 5

The path difference between two waves is 5 m. If the wavelength of the waves emitted by the two sources is 4 m, what is the phase difference (in degrees)?

Accepted Answer

The phase difference is calculated using the formula $\Delta \phi = \frac{2\pi}{\lambda} \Delta x$, where $\Delta x$ is the path difference and $\lambda$ is the wavelength. Substituting the given values, $\Delta \phi = \frac{2\pi}{4} \times 5 = \frac{5\pi}{2}$ radians. Converting to degrees, $\frac{5\pi}{2} \times \frac{180}{\pi} = 450$ degrees. Since phase differences are typically expressed in the range of 0 to 360 degrees, we subtract 360 degrees to get 90 degrees.

Question 6

A string is stretched and fixed at both ends, 200 cm apart. If the density of the string is 0.015 g/cm, and its tension is 600 N, what is the fundamental frequency?

Accepted Answer

The speed of the wave on the string can be calculated as follows:

v = sqrt(T/μ)

where T is the tension and μ is the linear density (mass per unit length) of the string.

μ = 0.015 g/cm = 0.00015 kg/m

T = 600 N

v = sqrt(600 N / 0.00015 kg/m) = 4900 m/s

The wavelength of the fundamental mode is twice the length of the string:

λ = 2L = 400 cm = 4 m

The fundamental frequency can be calculated as:

f = v / λ = 4900 m/s / 4 m = 1225 Hz

However, this is the frequency of the first harmonic, which has one antinode in the center of the string. The fundamental frequency corresponds to the second harmonic, which has two nodes at both ends of the string. Therefore, the fundamental frequency is half the frequency of the first harmonic:

f1 = 1225 Hz

f2 = f1/2 = 612.5 Hz

Rounding to the nearest integer gives:

Answer: C (158 Hz)

Question 7

Two harmonic waves are described by: $y_{1}=(3 \mathrm{~m}) \sin \left(\frac{4}{\mathrm{~m}} x-\frac{700}{\mathrm{~s}} t\right)$ $y_{2}=(3 \mathrm{~m}) \sin \left(\frac{4}{\mathrm{~m}} x-\frac{700}{\mathrm{~s}} t-2\right)$ What is the amplitude of the resultant wave?

Accepted Answer

The answer of Two harmonic waves are described by: \(y_{1}=(3...

Question 8

Two harmonic waves travelling in opposite directions interfere to produce a standing wave described by y = 2 sin (4x) cos (3t) where x is in m and t is in s. What is the speed (in m/s) of the interfering waves?

Accepted Answer

The speed of the interfering waves is given by v = ω/k, where ω is the angular frequency and k is the wave number. The angular frequency is given by ω = 2πf, where f is the frequency. The wave number is given by k = 2π/λ, where λ is the wavelength. The frequency is given by f = 3 Hz and the wavelength is given by λ = π/2 m. Therefore, the speed of the interfering waves is v = ω/k = (2πf)/(2π/λ) = fλ = 3 Hz * π/2 m = 1.5π m/s ≈ 4.71 m/s.

Question 9

Two waves are described by: y1 = 6 cos 180t and y2 = 6 cos 186t, (both in metres).With what angular frequency does the maximum amplitude of the resultant wave vary with time?

Accepted Answer

The equation for the resultant wave is given by y = y1 + y2 = 6(cos 180t + cos 186t). Using the identity cos(a) + cos(b) = 2cos((a+b)/2)cos((a-b)/2), we can rewrite the equation as y = 12(cos(183t)cos(3t)). The amplitude of the wave is 12|cos(183t)|, which varies with time. The angular frequency at which the amplitude of the wave varies with time is 183 rad/s - 3 rad/s = 180 rad/s. Therefore, the answer is C.

Question 10

A vertical tube one metre long is open at the top. It is filled with 75 cm of water. If the velocity of sound is 344 m/s, what will the fundamental resonant frequency be (in Hz)?

Accepted Answer

The answer of A vertical tube one metre long is...

Question 11

Two harmonic waves travelling in opposite directions interfere to produce a standing wave described by y = 3 sin (2x) cos 5t where x is in m and t is in s. What is the wavelength of the interfering waves?

Accepted Answer

The wavelength of a standing wave is given by lambda = 2*pi/k, where k is the wave number. The wave number is given by k = 2*pi/lambda. In this case, the wave number is k = 2*pi/2 = pi. Therefore, the wavelength is lambda = 2*pi/pi = 2 m.

Question 12

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?

Accepted Answer

The fundamental frequency of the open-open pipe is given by $f=\frac{nv}{2L}$, where $n$ is an odd integer, $v$ is the speed of sound, and $L$ is the length of the pipe. Thus, the possible frequencies are $f_1=\frac{v}{2L}$, $f_3=\frac{3v}{2L}$, $f_5=\frac{5v}{2L}$, etc.

The fundamental frequency of the closed-open pipe is given by $f=\frac{nv}{4L}$, where $n$ is an odd integer, $v$ is the speed of sound, and $L$ is the length of the open end of the pipe. Thus, the possible frequencies are $f_1=\frac{3v}{4L}$, $f_3=\frac{9v}{4L}$, $f_5=\frac{15v}{4L}$, etc.

To find the frequencies that can be produced by both pipes, we need to find the multiples of the fundamental frequency of the open-open pipe that are also multiples of the fundamental frequency of the closed-open pipe.

The fundamental frequency of the open-open pipe is $f_1=\frac{330	ext{ m/s}}{2(1.5	ext{ m})}=110	ext{ Hz}$. The multiples of this frequency are 220 Hz, 330 Hz, 440 Hz, 550 Hz, etc.

The fundamental frequency of the closed-open pipe is $f_1=\frac{3(330	ext{ m/s})}{4(0.75	ext{ m})}=110	ext{ Hz}$. The multiples of this frequency are 220 Hz, 330 Hz, 440 Hz, 550 Hz, etc.

Thus, the frequencies that can be produced by both pipes are 110 Hz, 330 Hz, and 550 Hz, which are only in choice C.

Question 13

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...?

Accepted Answer

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

Question 14

An organ pipe open at both ends has a radius of 4.0 cm and a length of 6.0 m. What is the frequency (in Hz) of the third harmonic? (Assume the velocity of sound is 344 m/s.)

Accepted Answer

The answer of An organ pipe open at both ends...

Question 15

Two harmonic waves are described by: $y_{1}=(7 m) \sin \left(\frac{5}{m} x-\frac{100}{s} t\right)$ $y_{2}=(7 \mathrm{~m}) \sin \left(\frac{5}{\mathrm{~m}} x-\frac{100}{\mathrm{~s}} t+2\right)$ What is the phase (in rad) of the resultant wave when x = t = 0?

Accepted Answer

The answer of Two harmonic waves are described by: \(y_{1}=(7...

Question 16

Two harmonic waves are described by: $y_{1}=(3 \mathrm{~cm}) \sin \left(\frac{8.0}{\mathrm{~m}} x+\frac{2.0}{\mathrm{~s}} t\right)$ $y_{2}=(3 \mathrm{~cm}) \sin \left(\frac{8.0}{\mathrm{~m}} x-\frac{2.0}{\mathrm{~s}} t\right)$ What is the magnitude of the speed (in m/s) of the two travelling waves?

Accepted Answer

The answer of Two harmonic waves are described by: \(y_{1}=(3...

Question 17

Two waves are described by: y1 = 6 cos 180t and y2 = 6 cos 186t, (both in metres).What effective frequency does the resultant vibration have at a point?

Accepted Answer

The effective frequency of the resultant vibration is the average of the frequencies of the two waves. 
Frequency of y1 = 180 Hz 
Frequency of y2 = 186 Hz 
Average frequency = (180 Hz + 186 Hz) / 2 = 183 Hz. 
Therefore, the answer is B) 183 Hz.

Question 18

Two harmonic waves travelling in opposite directions interfere to produce a standing wave described by y = 2 sin ( $\pi$x) cos (3 $\pi$t) where x is in m and t is in s. What is the distance (in m) between the first two antinodes?

Accepted Answer

The distance between two consecutive antinodes in a standing wave is half the wavelength. The wave number k is $\pi$, so the wavelength $\lambda$ is $2$ m. Therefore, the distance between the first two antinodes is $\lambda/2 = 1$ m.

Question 19

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  $\pi$ radians?

Accepted Answer

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

Question 20

Two harmonic waves are described by:   From the choices given, determine the smallest positive value of x (in cm) corresponding to a node of the resultant standing wave.

Accepted Answer

To find the node of the resultant standing wave, we need to find the points where the two waves interfere destructively, i.e., the amplitudes subtract from each other. This occurs at half-wavelength intervals of the standing wave.

Let's first find the wavelength of each wave:
Wave 1: wavelength = 22/5 cm = 4.4 cm
Wave 2: wavelength = 22/4 cm = 5.5 cm

The wavelengths are not the same, so we need to find the least common multiple (LCM) to determine the distance between consecutive nodes. 
The LCM of 4.4 and 5.5 is 22 cm.

So, the distance between consecutive nodes is 22 cm / 2 = 11 cm.

Therefore, the smallest positive value of x corresponding to a node of the resultant standing wave is 0 cm (at x = 11 cm), which is the midpoint between two nodes. Thus, the answer is choice C.

Quiz 18: Superposition and Interference

A string is stretched and fixed at both ends, 200 cm apart. If the density of the string is 0.015 g/cm, and its tension is 600 N, what is the wavelength (in cm) of the first harmonic?

Two tuning forks with frequencies 264 and 262 Hz produce 'beats'. What is the beat frequency (in Hz) ?

A clarinet behaves like a tube closed at one end. If its length is 1.0 m, and the velocity of sound is 344 m/s, what is its fundamental frequency (in Hz) ?

The path difference between two waves is 5 m. If the wavelength of the waves emitted by the two sources is 4 m, what is the phase difference (in degrees) ?

A string is stretched and fixed at both ends, 200 cm apart. If the density of the string is 0.015 g/cm, and its tension is 600 N, what is the fundamental frequency?

Two harmonic waves are described by: $y_{1}=(3 \mathrm{~m}) \sin \left(\frac{4}{\mathrm{~m}} x-\frac{700}{\mathrm{~s}} t\right)$ $y_{2}=(3 \mathrm{~m}) \sin \left(\frac{4}{\mathrm{~m}} x-\frac{700}{\mathrm{~s}} t-2\right)$ What is the amplitude of the resultant wave?

Two harmonic waves travelling in opposite directions interfere to produce a standing wave described by y = 2 sin (4x) cos (3t) where x is in m and t is in s. What is the speed (in m/s) of the interfering waves?

Two waves are described by: y1 = 6 cos 180t and y2 = 6 cos 186t, (both in metres) .With what angular frequency does the maximum amplitude of the resultant wave vary with time?

A vertical tube one metre long is open at the top. It is filled with 75 cm of water. If the velocity of sound is 344 m/s, what will the fundamental resonant frequency be (in Hz) ?

Two harmonic waves travelling in opposite directions interfere to produce a standing wave described by y = 3 sin (2x) cos 5t where x is in m and t is in s. What is the wavelength of the interfering waves?

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 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...?

An organ pipe open at both ends has a radius of 4.0 cm and a length of 6.0 m. What is the frequency (in Hz) of the third harmonic? (Assume the velocity of sound is 344 m/s.)

Two harmonic waves are described by: $y_{1}=(7 m) \sin \left(\frac{5}{m} x-\frac{100}{s} t\right)$ $y_{2}=(7 \mathrm{~m}) \sin \left(\frac{5}{\mathrm{~m}} x-\frac{100}{\mathrm{~s}} t+2\right)$ What is the phase (in rad) of the resultant wave when x = t = 0?

Two harmonic waves are described by: $y_{1}=(3 \mathrm{~cm}) \sin \left(\frac{8.0}{\mathrm{~m}} x+\frac{2.0}{\mathrm{~s}} t\right)$ $y_{2}=(3 \mathrm{~cm}) \sin \left(\frac{8.0}{\mathrm{~m}} x-\frac{2.0}{\mathrm{~s}} t\right)$ What is the magnitude of the speed (in m/s) of the two travelling waves?

Two waves are described by: y1 = 6 cos 180t and y2 = 6 cos 186t, (both in metres) .What effective frequency does the resultant vibration have at a point?

Two harmonic waves travelling in opposite directions interfere to produce a standing wave described by y = 2 sin ( $\pi$ x) cos (3 $\pi$ t) where x is in m and t is in s. What is the distance (in m) between the first two antinodes?

Two harmonic waves are described by: From the choices given, determine the smallest positive value of x (in cm) corresponding to a node of the resultant standing wave.

Quiz 18: Superposition and Interference

A string is stretched and fixed at both ends, 200 cm apart. If the density of the string is 0.015 g/cm, and its tension is 600 N, what is the wavelength (in cm) of the first harmonic?

The superposition of two waves and y2=(0.006 cm) cos⁡[2π(150 st) ]y_{2}=(0.006 \mathrm{~cm}) \cos \left[2 \pi\left(\frac{150}{\mathrm{~s}} t\right) \right]y2​=(0.006 cm) cos[2π( s150​t) ] at the location x = 0 in space results in:

Two tuning forks with frequencies 264 and 262 Hz produce 'beats'. What is the beat frequency (in Hz) ?

A clarinet behaves like a tube closed at one end. If its length is 1.0 m, and the velocity of sound is 344 m/s, what is its fundamental frequency (in Hz) ?

The path difference between two waves is 5 m. If the wavelength of the waves emitted by the two sources is 4 m, what is the phase difference (in degrees) ?

A string is stretched and fixed at both ends, 200 cm apart. If the density of the string is 0.015 g/cm, and its tension is 600 N, what is the fundamental frequency?

Two harmonic waves travelling in opposite directions interfere to produce a standing wave described by y = 2 sin (4x) cos (3t) where x is in m and t is in s. What is the speed (in m/s) of the interfering waves?

Two waves are described by: y1 = 6 cos 180t and y2 = 6 cos 186t, (both in metres) .With what angular frequency does the maximum amplitude of the resultant wave vary with time?

A vertical tube one metre long is open at the top. It is filled with 75 cm of water. If the velocity of sound is 344 m/s, what will the fundamental resonant frequency be (in Hz) ?

Two harmonic waves travelling in opposite directions interfere to produce a standing wave described by y = 3 sin (2x) cos 5t where x is in m and t is in s. What is the wavelength of the interfering waves?

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 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...?

An organ pipe open at both ends has a radius of 4.0 cm and a length of 6.0 m. What is the frequency (in Hz) of the third harmonic? (Assume the velocity of sound is 344 m/s.)

Two waves are described by: y1 = 6 cos 180t and y2 = 6 cos 186t, (both in metres) .What effective frequency does the resultant vibration have at a point?

Two harmonic waves travelling in opposite directions interfere to produce a standing wave described by y = 2 sin ( π\piπ x) cos (3 π\piπ t) where x is in m and t is in s. What is the distance (in m) between the first two antinodes?

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\piπ radians?

Two harmonic waves are described by: From the choices given, determine the smallest positive value of x (in cm) corresponding to a node of the resultant standing wave.

Two harmonic waves travelling in opposite directions interfere to produce a standing wave described by y = 2 sin ( $\pi$ x) cos (3 $\pi$ t) where x is in m and t is in s. What is the distance (in m) between the first two antinodes?