Q: A guitar string is tuned to 440 Hz. If the tension in the string is doubled, what will be the new frequency produced by the string?

$f_{new} = 440 \times \sqrt{2} \approx 622 \ \mathrm{Hz}$

Q: A bottle is filled with water so that the air column is 15 cm long. If the speed of sound is $343 \ \mathrm{m/s}$, what is the fundamental frequency produced?

$f_1 = \dfrac{343}{4 \times 0.15} = 572 \ \mathrm{Hz}$

Question 1

A student blows across the top of a partially filled bottle and produces a sound of frequency 440 Hz. If the speed of sound in air is $v = 343 \ \mathrm{m/s}$, what is the effective length of the air column in the bottle, assuming it acts as a closed pipe (one end closed, one end open)?

Accepted Answer

$0.195 \ \mathrm{m}$

Question 2

If the frequency of the sound produced by blowing across a bottle increases, what happens to the height of the water inside the bottle?

Accepted Answer

The height of the water increases.

Question 3

A bottle acts as a closed pipe. If the fundamental frequency is $f_1 = \dfrac{v}{4L}$, what is the next higher resonance frequency (first overtone) for this system?

Accepted Answer

$f_3 = 3f_1$

Question 4

A guitar string is tuned to 440 Hz. If the tension in the string is doubled, what will be the new frequency produced by the string?

Accepted Answer

$f_{new} = 440 	imes \sqrt{2} \approx 622 \ \mathrm{Hz}$

Question 5

Which of the following best explains why tightening a guitar string increases its pitch?

Accepted Answer

Increasing the tension increases the wave speed, which increases the frequency.

Question 6

A bottle is filled with water to a height $h$. If the speed of sound in air is $v$, what is the formula for the fundamental frequency produced when blowing across the top?

Accepted Answer

$f_1 = \dfrac{v}{4(L_{eff})}$

Question 7

If a guitar string is shortened while keeping the tension constant, what happens to the frequency of the sound produced?

Accepted Answer

The frequency increases.

Question 8

A bottle resonates at 440 Hz when the air column is 19.5 cm long. What would be the new resonance frequency if the air column is shortened to 10 cm? (Assume speed of sound is constant.)

Accepted Answer

$f_{new} = 858 \ \mathrm{Hz}$

Question 9

Which equation correctly relates the frequency $f$ of a vibrating string to its tension $T$, length $L$, and mass per unit length $\mu$?

Accepted Answer

$f = \dfrac{1}{2L} \sqrt{\dfrac{T}{\mu}}$

Question 10

A musician wants to lower the pitch of a guitar string without changing its tension. Which of the following actions will achieve this?

Accepted Answer

Increasing the length of the vibrating part of the string.

Question 11

If the speed of sound in air increases due to a rise in temperature, what happens to the resonance frequency of a bottle with a fixed air column length?

Accepted Answer

The resonance frequency increases.

Question 12

A bottle is filled with water so that the air column is 15 cm long. If the speed of sound is $343 \ \mathrm{m/s}$, what is the fundamental frequency produced?

Accepted Answer

$f_1 = \dfrac{343}{4 	imes 0.15} = 572 \ \mathrm{Hz}$

Question 13

If a guitar string is replaced with a thicker string of the same material and length, and the tension is kept the same, what happens to the frequency?

Accepted Answer

The frequency decreases.

Question 14

A bottle produces a note of 440 Hz when the air column is 19.5 cm. If the bottle is filled so the air column is now 9.75 cm, what is the new frequency?

Accepted Answer

$880 \ \mathrm{Hz}$

Waves & Sound: Resonance in Air Columns and Strings

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