Calculating Resonant Lengths of Air Columns

In summary, the question is asking for the first and second resonant lengths of a closed air column, with the given information that the third resonant length is 75cm. A closed air column resonates in the same way as a stringed instrument, with the fundamental frequency having a half-sine wave shape and each subsequent harmonic having an additional half-period. The first and second harmonics can be determined by adding in half-periods to the fundamental frequency.
  • #1
soulja101
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Homework Statement



The 3rd resonant length of a closed air colum is 75cm. determine the 1st and second length.

Homework Equations


the rensonant patterns


The Attempt at a Solution



i just don't get this resonant things.
 
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  • #2
How many sinusoidal periods are in a resonant length? I'm not familiar with air column resonances, but for stringed instruments like a guitar, the fundamental is when there is a half-sine wave shape on the string as it vibrates. That is, the ends are fixed, and the middle is what moves back and forth to make the sound. So the 2nd harmonic has a full sine wave shape on the string as it vibrates (the middle doesn't move, and each half of the string vibrates back and forth. The 3rd harmonic has 3 half-sines on the string, etc.

Now, you need to first figure out what the fundamental consists of in an air column (it's a pressure wave for air, obviously), and then think in terms of adding in half-periods as you move up through the harmonics...
 
  • #3
A closed air column behaves in the same way as a string, in as far as resonant harmonics are concerned...
 

Related to Calculating Resonant Lengths of Air Columns

1. What is the purpose of calculating the resonant lengths of air columns?

The purpose of calculating the resonant lengths of air columns is to determine the frequencies at which a given air column will vibrate most strongly, producing a sound with a particular pitch. This is useful in the design of musical instruments and in understanding the physics of sound.

2. How do you calculate the resonant length of an air column?

The resonant length of an air column can be calculated using the formula L = (n * λ) / 4, where L is the resonant length, n is the harmonic number, and λ is the wavelength of the sound wave.

3. What factors influence the resonant length of an air column?

The resonant length of an air column is influenced by factors such as the speed of sound in the medium, the temperature of the air, the density of the medium, and the shape and size of the air column.

4. How does the length of the air column affect the pitch of the sound produced?

The length of the air column directly affects the pitch of the sound produced. As the length of the air column increases, the pitch decreases, and vice versa. This is because longer air columns have longer wavelengths, resulting in lower frequencies and lower pitches.

5. Can the resonant length of an air column be changed?

Yes, the resonant length of an air column can be changed by altering the length or shape of the air column, or by changing the properties of the medium (such as temperature or density). This is how instruments such as trumpets and clarinets are able to produce different pitches by adjusting the length of the air column with valves or keys.

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