Calculating the Length of a Pipe Closed at One End

In physics, understanding how sound waves behave within different types of vibrating systems is crucial. This article focuses on calculating the length of a pipe that is closed at one end, given its lowest frequency and the speed of sound. The key concepts involve the relationship between the frequency, length of the pipe, and the speed of sound. We'll explore the derivation and application of these principles to ensure accurate and efficient calculation methods.

The Physics Behind the Calculation

A pipe closed at one end behaves differently from an open pipe or a pipe open at both ends. When a pipe is closed at one end, it resonates at a frequency corresponding to a quarter wavelength. This means that the length of the pipe is related to the wavelength of the sound wave, which is crucial for determining the pipe's dimensions.

Relationship Between Wavelength and Frequency

The relationship between the wavelength (λ) and frequency (f) of a sound wave can be described by the equation:

λ v / f

Where:

λ is the wavelength, v is the speed of sound, and f is the frequency.

Given that the speed of sound is v 344 m/s and the lowest frequency is f 540 Hz, we can calculate the wavelength as follows:

λ 344 m/s ÷ 540 Hz 0.637 meters

Calculating the Length of the Pipe

The length of the pipe (L) is equal to one quarter of the wavelength (since it is resonating at the fundamental frequency for a closed pipe at one end):

L λ / 4 0.637 meters / 4 0.159 meters or 15.9 cm or 6.27 inches.

This calculation can also be represented using algebra:

L v / 4f 344 m/s / (4 × 540 Hz) 0.159 meters

Verification of Calculation

Another approach to verify the calculation is by using the general relationship between the length of the pipe, the wavelength, and the frequency. Given:

Speed of sound, (v 344 , m/s) Frequency, (f 450 , Hz)

The length of the pipe is given by:

L frac{v}{4f} frac{344 , m/s}{4 times 450 , Hz} 0.160 , meters

This length also matches the quarter wavelength calculated earlier, confirming the correctness of the derived formula.

Conclusion

Determining the length of a closed pipe based on its fundamental frequency and the speed of sound in the medium is a fundamental concept in acoustics. By understanding the relationship between the frequency, the speed of sound, and the geometry of the pipe, we can accurately calculate the length required for specific resonant frequencies.

The key takeaway is using the formula (L frac{v}{4f}) to find the length of the pipe when the fundamental frequency and speed of sound are known. This method provides a reliable and efficient means of solving similar problems in the field of acoustics.