Is a resonant frequency any of the harmonics?

In summary, the problem asks for the length of a pipe needed to achieve a resonant frequency of 280.0 Hz when a drum skin is stretched over one end, creating a resonant air column with one open end and one fixed end. Using the equation V = fλ, the wavelength is calculated to be 1.225m. The first harmonic for a fixed and free end system is 1/4 λ, resulting in a pipe length of ~0.30m. However, other harmonics can also be used, such as 3/4 λ for a pipe length of ~0.92m and 5/4 λ for a pipe length of ~1.5m. It could also be interpreted as
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1. The problem statement,λ all variables and given/known data
A drum skin is stretched over one end of a pipe, creating a resonant air column with one open end and one fixed end. How long must the pipe be to achieve a resonant frequency of 280.0 Hz? (Use 343 m/s for the speed of sound.)

Homework Equations


V = fλ

The Attempt at a Solution


My assumption is that a resonant air column means that any of the harmonics for a fixed and open end could be used in order to solve for pipe length.

1. Solve for wavelength:
V = fλ
343/280 = λ
1.225m = λ

2. Solve for "pipe length"
Knowing that the first harmonic is a 1/4 λ for a fixed and free end system, I divided 1.225 by 4 to get ~0.30m, which the book states is correct. But if that works for a resonant air column, the rest of the harmonics should as well. So I multiplied 1.225 by 3/4 to get ~0.92m. The book also states this is correct, but these are the only two lengths the book shows

However, could I not keep going to other harmonics to get greater and greater lengths? For example, the 3rd harmonic, I multiply 1.225 by 5/4 to get ~1.5m, which the book does not list. But is this not also correct?

Thanks for your help
 
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  • #2
AAAA said:
1. The problem statement,λ all variables and given/known data
A drum skin is stretched over one end of a pipe, creating a resonant air column with one open end and one fixed end. How long must the pipe be to achieve a resonant frequency of 280.0 Hz? (Use 343 m/s for the speed of sound.)

Homework Equations


V = fλ

The Attempt at a Solution


My assumption is that a resonant air column means that any of the harmonics for a fixed and open end could be used in order to solve for pipe length.

1. Solve for wavelength:
V = fλ
343/280 = λ
1.225m = λ

2. Solve for "pipe length"
Knowing that the first harmonic is a 1/4 λ for a fixed and free end system, I divided 1.225 by 4 to get ~0.30m, which the book states is correct. But if that works for a resonant air column, the rest of the harmonics should as well. So I multiplied 1.225 by 3/4 to get ~0.92m. The book also states this is correct, but these are the only two lengths the book shows

However, could I not keep going to other harmonics to get greater and greater lengths? For example, the 3rd harmonic, I multiply 1.225 by 5/4 to get ~1.5m, which the book does not list. But is this not also correct?

Thanks for your help
I agree with you. Or you could read the question as "how long does the pipe need to be", suggesting a minimum length.
 
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  • #3
Thanks for your help! :)
 

Related to Is a resonant frequency any of the harmonics?

1. What is a resonant frequency?

A resonant frequency is the natural frequency at which an object vibrates when it is subjected to a periodic force. This frequency is determined by the object's physical properties, such as its mass, length, and elasticity.

2. How is a resonant frequency related to harmonics?

A resonant frequency can also be referred to as a harmonic frequency because it is one of the possible frequencies at which an object can vibrate in a system with multiple natural frequencies. Harmonics are integer multiples of the resonant frequency.

3. Can a resonant frequency be any of the harmonics?

Yes, a resonant frequency can be any of the harmonics, as long as it is within the object's natural frequency range. However, the first harmonic (also known as the fundamental frequency) is typically the strongest and most commonly observed resonant frequency.

4. What happens when an object is subjected to its resonant frequency?

When an object is subjected to its resonant frequency, it absorbs the energy of the periodic force and begins to vibrate with a large amplitude. This phenomenon is known as resonance and can cause the object to vibrate or even break if the force is strong enough.

5. How is knowledge of resonant frequencies useful?

Understanding resonant frequencies is important in various fields, such as music, engineering, and science. It allows us to analyze the properties of objects and predict how they will respond to different forces or vibrations. It also helps in designing structures and devices to avoid potential damage from resonance.

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