The Fundamental Frequency of a Closed-End Organ Pipe

Introduction:
An organ pipe is a musical instrument that produces sound by blowing air through a tube. It can be classified as an open-end or closed-end pipe, depending on whether one or both ends of the pipe are open. In this question, we are considering a closed-end organ pipe.

Fundamental Frequency:
The fundamental frequency of an organ pipe is the lowest frequency at which the pipe can vibrate, producing a standing wave pattern. It is determined by the length and speed of sound in the pipe.

Effect of Changing Length:
When the length of a closed-end organ pipe is doubled, its fundamental frequency is halved. This can be explained by the relationship between the length of the pipe and the wavelength of the sound wave produced.

Explanation:
When the length of the pipe is doubled, the wavelength of the fundamental frequency also doubles. This is because the fundamental frequency corresponds to the first harmonic, which has one complete wavelength in the length of the pipe.

- If the original length of the pipe is L, then the wavelength of the fundamental frequency is 2L.
- When the length is doubled to 2L, the wavelength of the fundamental frequency becomes 4L.

Since the speed of sound remains constant, the frequency is inversely proportional to the wavelength. Therefore, when the wavelength is doubled, the frequency is halved.

Effect of Changing Radius:
The radius of the pipe does not affect the fundamental frequency of a closed-end organ pipe. This is because the fundamental frequency depends on the length of the pipe and the speed of sound, but not on the radius.

Conclusion:
When the length of a closed-end organ pipe is doubled and the radius is halved, the only factor affecting the fundamental frequency is the change in length. As a result, the frequency will be halved, approximately becoming n/2.