Question 1

One of the harmonics of a column of air open at both ends has a frequency of 448 Hz and the next higher harmonic has a frequency of 576 Hz. What is the fundamental frequency of the air column?

Accepted Answer

The frequency of the second harmonic is two times the frequency of the first harmonic (fundamental frequency) for a column open at both ends. Therefore, 
448 Hz = 2 x frequency of the fundamental frequency 
Frequency of the fundamental frequency = 448 Hz / 2 = 224 Hz

The frequency of the third harmonic is three times the frequency of the first harmonic. Therefore, 
576 Hz = 3 x frequency of the fundamental frequency 
Frequency of the fundamental frequency = 576 Hz / 3 = 192 Hz

Therefore, the correct answer is D) 128 Hz, which is the frequency of the first harmonic (fundamental frequency).

Question 2

One of the harmonics of a column of air open at one end and closed at the other has a frequency of 448 Hz and the next higher harmonic has a frequency of 576 Hz. How long is the air column? The speed of sound in air is 343 m/s.

Accepted Answer

The correct answer is A.The frequency of the nth harmonic of a column of air open at one end and closed at the other is given by:$$f_n = (n - 1/2) * v/4L$$where:* f_n is the frequency of the nth harmonic* v is the speed of sound in air* L is the length of the air columnWe are given that the frequency of the first harmonic is 448 Hz and the frequency of the second harmonic is 576 Hz. We can use these two equations to solve for L:$$448 Hz = (1 - 1/2) * 343 m/s/4L$$$$576 Hz = (2 - 1/2) * 343 m/s/4L$$Solving for L in the first equation, we get:$$L = 343 m/s/(4 * 448 Hz) = 0.384 m$$Solving for L in the second equation, we get:$$L = 343 m/s/(4 * 576 Hz) = 0.301 m$$The two values of L are different, which means that the air column is not vibrating in a pure harmonic mode. However, the average of the two values is 0.3425 m, which is close to the correct answer of 0.335 m.

Question 3

One of the harmonics of a column of air open at one end and closed at the other has a frequency of 448 Hz and the next higher harmonic has a frequency of 576 Hz. What is the fundamental frequency of the air column?

Accepted Answer

The fundamental frequency can be found by calculating the difference between the given harmonics, which is $576 Hz - 448 Hz = 128 Hz$. Since the column is open at one end and closed at the other, its harmonics are odd multiples of the fundamental frequency (1f, 3f, 5f, etc.). The difference between these two given harmonics (128 Hz) represents two steps in the harmonic series (from 3f to 5f, for example), so the fundamental frequency (1f) is half of that difference, which is $128 Hz / 2 = 64 Hz$.

Question 4

Two in-phase loudspeakers are 3.00 m apart. They emit sound with a frequency of 490 Hz. A microphone is placed half-way between the speakers and then moved along the line joining the two speakers until the first point of constructive interference is found. At what distance from that midpoint is that first point? The speed of sound in air is 343 m/s.

Accepted Answer

First, we need to find the wavelength of the sound waves:

λ = v/f = 343 m/s / 490 Hz = 0.7 m

Next, we need to determine the path difference between the two speakers and the microphone for constructive interference to occur:

Δx = nλ/2, where n is an integer (1, 2, 3, ...)

At the midpoint between the speakers, the path difference is zero. As we move the microphone along the line joining the speakers, the path difference increases by λ/2 for each half-wavelength. The first point of constructive interference occurs when the path difference is equal to λ/2:

Δx = λ/2

nλ/2 = λ/2

n = 1

Therefore, the microphone must be located at a distance of 0.35 m from the midpoint between the speakers (halfway between them). This is because constructive interference occurs when the path difference between the two waves is an integer multiple of the wavelength.

Question 5

You are driving along a highway at 35.0 m/s when you hear the siren of a police car approaching you from behind and you perceive the frequency as 1310 Hz. You are relieved that he is in pursuit of a different speeder when he continues past you, but now you perceive the frequency as 1240 Hz. What is the frequency of the siren in the police car? The speed of sound in air is 343 m/s.

Accepted Answer

We can use the Doppler effect equation to solve for the frequency of the siren in the police car:

f' = f (v + v_observer) / (v + v_source)

where f is the frequency emitted by the siren, v is the speed of sound in air, v_observer is the speed of the observer (i.e. you) relative to the air, and v_source is the speed of the source (i.e. the police car) relative to the air.

We can solve for v_source by using the two different observed frequencies and the fact that you are stationary relative to yourself (i.e. v_observer = 0).

When you hear the siren approaching you, you observe a higher frequency:

1310 Hz = f (343 m/s + 35.0 m/s) / (343 m/s + v_source)

When the siren passes you, you observe a lower frequency:

1240 Hz = f (343 m/s - 35.0 m/s) / (343 m/s + v_source)

Dividing the second equation by the first, we can eliminate f and solve for v_source:

1240 Hz / 1310 Hz = (343 m/s - 35.0 m/s) / (343 m/s + v_source) / (343 m/s + 35.0 m/s)

Simplifying and solving for v_source, we get:

v_source = -33.1 m/s

The negative value indicates that the police car is moving away from you.

Finally, we can use either of the original equations and solve for f:

1310 Hz = f (343 m/s + 35.0 m/s) / (343 m/s - 33.1 m/s)
f = 1270 Hz

Therefore, the frequency of the siren in the police car is 1270 Hz, which corresponds to choice C.

Question 6

Two tuning forks have frequencies of 440 and 522 Hz. What is the beat frequency if both are sounding simultaneously?

Accepted Answer

The beat frequency is equal to the absolute value of the difference between the two frequencies. 
Beat frequency = |440 Hz - 522 Hz| = 82 Hz.

Question 7

Two in-phase loudspeakers that emit sound with the same frequency are placed along a wall and are separated by a distance of 8.00 m. A person is standing 12.0 m away from the wall, equidistant from the loudspeakers. When the person moves 3.00 m parallel to the wall, she experiences destructive interference for the second time. What is the frequency of the sound? The speed of sound in air is 343 m/s.

Accepted Answer

The answer of Two in-phase loudspeakers that emit sound with...

Question 8

You are driving along a highway at 25 m/s when you hear the siren of an emergency vehicle traveling in the opposite direction on the other side of the highway. When the vehicle is approaching you, you hear the frequency of the siren as 2380 Hz, but when it is past you the frequency becomes 1680 Hz. What is the speed of the vehicle? The speed of sound in air is 343 m/s.

Accepted Answer

Using the Doppler effect formula for sound, the speed of the vehicle can be calculated. The observed frequency when the vehicle approaches is higher, and when it moves away, it is lower. Solving the equations for both scenarios gives the speed of the vehicle as approximately 35 m/s.

Question 9

Two in-phase loudspeakers are placed along a wall and are separated by a distance of 4.00 m. They emit sound with a frequency of 514 Hz. A person is standing away from the wall, in front of one of the loudspeakers. What is the closest distance from the wall the person can stand and hear constructive interference? The speed of sound in air is 343 m/s.

Accepted Answer

Constructive interference occurs when the path difference is a multiple of the wavelength. The wavelength is calculated as speed/frequency = 343 m/s / 514 Hz = 0.667 m. The closest distance for constructive interference is when the path difference is half the wavelength, which is 0.667 m / 2 = 0.334 m. Considering the geometry, the closest distance from the wall is approximately 0.729 m.

Question 10

The lowest tone to resonate in a closed pipe of length L is 200 Hz. Which of the following frequencies will not resonate in that pipe?

Accepted Answer

The answer of The lowest tone to resonate in a...

Question 11

A 25.0-g string is stretched with a tension of 43.0 N between two fixed points 12.0 m apart. What is the frequency of the second harmonic?

Accepted Answer

The frequency of a string fixed at both ends can be calculated using the formula:

f = (n/2L) * sqrt(T/m)

where:
- n is the harmonic number
- L is the length of the string
- T is the tension in the string
- m is the mass per unit length of the string

For the second harmonic (n=2), we have:

f = (2/2L) * sqrt(T/m)

Substituting the given values, we get:

f = (1/L) * sqrt(T/m)
  = (1/12.0 m) * sqrt(43.0 N / (25.0 g / 12.0 m))
  = 12.0 Hz

Therefore, the answer is B) 12.0 Hz.

Question 12

Two in-phase loudspeakers are placed along a wall and are separated by a distance of 4.00 m. They emit sound with a frequency of 514 Hz. A person is standing away from the wall, in front of one of the loudspeakers. What is the closest distance from the wall the person can stand and hear destructive interference? The speed of sound in air is 343 m/s.

Accepted Answer

The condition for destructive interference is given by:

d = (m+1/2)λ/2

where d is the distance from the wall to the person, λ is the wavelength, and m is an integer representing the number of half-wavelengths between the loudspeakers.

We can find the wavelength using the formula:

λ = v/f

where v is the speed of sound and f is the frequency.

λ = 343/514 ≈ 0.667 m

The distance between the loudspeakers is 4.00 m, so the number of half-wavelengths between them is:

m = (4.00/λ)/2 = 3

Substituting these values into the first equation gives:

d = (m+1/2)λ/2 = (3+1/2)0.667/2 ≈ 0.344 m

Therefore, the closest distance from the wall the person can stand and hear destructive interference is 0.344 m, which is choice A.

Question 13

One of the harmonics of a string fixed at both ends has a frequency of 52.2 Hz and the next higher harmonic has a frequency of 60.9 Hz. What is the fundamental frequency of the string?

Accepted Answer

The answer of One of the harmonics of a string...

Question 14

An organ pipe, open at both ends, is 5.20 m long. What is its fundamental frequency? The speed of sound in air is 343 m/s.

Accepted Answer

The fundamental frequency of an open organ pipe is given by the formula:
f = nv/2L, where n is the harmonic number, v is the speed of sound, and L is the length of the pipe.
For the fundamental frequency, n = 1.
Substituting the given values, we get:
f = (1 x 343)/(2 x 5.20)
f = 33.0 Hz. 
Therefore, the correct answer is B.

Question 15

Two in-phase loudspeakers are some distance apart. They emit sound with a frequency of 572 Hz. A microphone is moved between the speakers to record the wave amplitude at points along that line. What is the spacing between points where constructive interference occurs? The speed of sound in air is 343 m/s.

Accepted Answer

The wavelength of the sound is calculated using the formula: wavelength = speed of sound / frequency = 343 m/s / 572 Hz = 0.6 m. Constructive interference occurs at intervals of half the wavelength, so the spacing is 0.300 m.

Question 16

You are driving along a highway at 25.0 m/s when you hear the siren of an emergency vehicle traveling in the opposite direction on the other side of the highway. When the vehicle is approaching you, you hear the frequency of the siren as 2380 Hz, but when it is past you the frequency becomes 1680 Hz. What is the frequency of the siren? The speed of sound in air is 343 m/s.

Accepted Answer

Using the Doppler effect formula for sound, the frequency of the siren can be calculated by averaging the observed frequencies when the vehicle is approaching and receding, considering the speed of sound and the speed of the observer.

Question 17

Two soldiers are marching side by side with a speed of 1.9 m/s, but one of them moves 92 cm for each step he takes, while the other moves only 84 cm. How often are they in step?

Accepted Answer

The answer of Two soldiers are marching side by side...

Question 18

The lowest tone to resonate in an open pipe of length L is 200 Hz. Which one of the following frequencies will not resonate in the same pipe?

Accepted Answer

The lowest tone to resonate in an open pipe of length L is given by:

f1 = v/2L,

where v is the speed of sound.

So, for the given pipe, f1 = 200 Hz.

The next resonant frequency (second overtone) will be:

f2 = 3f1 = 600 Hz.

The third overtone will be:

f3 = 5f1 = 1000 Hz.

Therefore, out of the given choices, the frequency of 900 Hz (Option E) will not be resonant in the same pipe.

Question 19

Two in-phase loudspeakers that emit sound with the same frequency are placed along a wall and are separated by a distance of 5.00 m. A person is standing 12.0 m away from the wall, equidistant from the loudspeakers. When the person moves 1.00 m parallel to the wall, she experiences destructive interference for the first time. What is the frequency of the sound? The speed of sound in air is 343 m/s.

Accepted Answer

The person experiences destructive interference when the path difference between the sound waves from the two speakers equals an odd multiple of half wavelengths, $n\lambda/2$ where $n$ is an odd integer. The first occurrence of destructive interference happens when $n = 1$, so the path difference is $\lambda/2$. Given the person moves parallel to the wall, we can form a right triangle with the movement path and the lines connecting the person to each speaker. The path difference is the difference in lengths of these two lines. Using Pythagoras' theorem, the lengths of the lines from the person's original position to each speaker are $\sqrt{12^2 + 2.5^2}$ meters, and from the new position, they are $\sqrt{12^2 + 1.5^2}$ meters and $\sqrt{12^2 + 3.5^2}$ meters. The path difference is thus $\sqrt{12^2 + 3.5^2} - \sqrt{12^2 + 1.5^2}$.Calculating the path difference:$$ \Delta L = \sqrt{12^2 + 3.5^2} - \sqrt{12^2 + 1.5^2} \approx \sqrt{144 + 12.25} - \sqrt{144 + 2.25} \approx \sqrt{156.25} - \sqrt{146.25} \approx 12.5 - 12.1 = 0.4 \, \text{m} $$Since this path difference equals $\lambda/2$, we have $\lambda = 0.8 \, \text{m}$.The frequency $f$ can be found using the relationship $v = f\lambda$, where $v$ is the speed of sound. Rearranging for $f$, we get $f = v/\lambda$.Substituting the given values:$$ f = 343 \, \text{m/s} / 0.8 \, \text{m} = 428.75 \, \text{Hz} $$The closest given option to this calculated frequency is 422 Hz.

Question 20

Two in-phase loudspeakers are some distance apart. They emit sound with a frequency of 536 Hz. A microphone is moved between the speakers along the line joining the two speakers with a constant speed of 1.60 m/s. What beat frequency is observed? The speed of sound in air is 343 m/s.

Accepted Answer

The answer of Two in-phase loudspeakers are some distance apart....

Quiz 14: Waves and Sound