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Sound Resonance: How to Calculate Speed of Sound

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Author: Erin Bjornsson

Sounds, whether from talking, music, an earthquake, a fire or a sonic boom, travel in waves. Wave travel, called propagation, has a different velocity through each material, like water or air.

Frequency is the number of times an event occurs per unit of time. For sound, the frequency is measured in Hertz, abbreviated Hz, which means is period cycle, from the top of one wave to another, per second. Humans can normally hear anywhere from 20Hz to 20000Hz, while dogs can hear up to 60,000 Hz, and some bats can hear sound as low as 1 Hz. Sound is perceived in pitch; higher frequencies sound like higher pitches (like a whistle) and lower frequencies have lower pitches (like a tuba).

A tuning fork is an object that resonates, or vibrates, at a specific frequency and pitch. The sound of a tuning fork depends on the length of the prongs and can be used to tune musical instruments.

Problem:

Calculate the speed of sound in air.

Materials:

  • Tuning fork with known frequency
  • Large, tall cylindrical glass container
  • Water
  • Rubber mallet
  • Ruler

Procedure                                    

  1. Fill the glass container halfway up with water.
  2. Strike the tuning fork with the rubber mallet. Why should you use a mallet to strike the tuning for rather than a hard surface like a table?
  3. Hold the tuning fork over the opening of the glass container as pictured. Record your observations. How does holding the fork over the container change the sound you hear? Why does this happen?

Sound Resonance Diagram

  1. While keeping the tuning fork over the opening of the container, slowly pour more water into the cylinder. If you cannot do it yourself, ask a friend or family member to help you. Listen for the point when there is a change in pitch (frequency).
  2. Remove some water and experiment with adding it again until you can identify the height of the water at which the frequency jumps. Why does the frequency jump?
  3. Calculate the velocity of sound at this height using the following equation:
v = 4f (H + 0.4D)

v is the velocity of sound

f is the frequency of the tuning fork in Hz

is the height distance between the top of the water and the lip of the cylinder

D is the diameter of the water container in meters

Results

The speed of sound in air is 343 m/s (nearly a mile in 5 seconds!) at 20°C. Compare your experimental value to the true value. If it is warmer, the speed will be faster.

Why?

The change in sound in the water column will occur when the graduated cylinder produces the same resonance as the tuning fork. Simply put, this means that one period of the wave goes to the bottom of the cylinder and back in the same amount of time it takes for the tuning fork to vibrate one time.

The fundamental frequency is the frequency of one period of a wavelength. The second harmonic, also called the 1st overtone, is an integer (whole number) multiple of the fundamental frequency. Many people will not hear the difference between harmonics, but they are commonly used in music, and examples of harmonics can even be found in the human voice.

Here are a few examples of frequencies applied to musical notes:

Approximate Note

Frequency (Hz)

Wavelength (m)

Fundamental height (m)

2nd Harmonic height (m)

C

256

1.34

0.33

0.17

C#

278

1.23

0.31

0.16

D

294

1.17

0.29

0.15

Eb

311

1.10

028

0.14

E

330

1.04

0.26

0.13

F

349

0.98

0.24

0.12

F#

370

0.93

0.23

0.12

G

392

0.88

0.22

0.11

Ab

415

0.83

0.20

0.11

A

440

0.78

0.19

0.10

B

494

0.69

0.18

0.09

 

Wavelength is the length in meters of one period of the wave, from peak to peak. It is often denoted by the Greek symbol lambda λ and can be calculated by the equation

v = λ f

 v is the velocity in m/s

 λ is wavelength in meters

 f is frequency in Hz, or s-1​

 
Sound in water is 1484 m/s, more than 4 times the speed of air. This is why whales and dolphins can communicate quickly over large distances.
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