Question

An open pipe is suddenly closed at one end with the result that the frequency of the third harmonic of the closed pipe is termed to be higher by 100 Hz than the fundamental frequency of the open pipe. The fundamental frequency of the open pipe is

• A. 200 Hz
• B. 150 Hz
• C. 100 Hz
• D. 250 Hz

Hint:

The fundamental frequency of an open pipe is determined by its length and the speed of sound, and the harmonics differ for closed pipes due to the boundary conditions.

Answer 1

The Correct Option is A

Step 1: Understanding the harmonics in open and closed pipes.
- For an open pipe, the fundamental frequency \(f_1\) is given by: \[ f_1 = \frac{v}{2L} \] where \(v\) is the velocity of sound and \(L\) is the length of the pipe. The harmonics for an open pipe are integer multiples of the fundamental frequency.
- For a closed pipe, the fundamental frequency is lower, and its harmonics are only odd multiples of the fundamental frequency. For the third harmonic of the closed pipe to be higher than the fundamental frequency of the open pipe by 100 Hz, we can set up an equation based on the difference in frequencies.

Step 2: Solving for the fundamental frequency of the open pipe.
- Let the fundamental frequency of the open pipe be \(f_1 = 200 \, \text{Hz}\). The frequency difference for the third harmonic of the closed pipe is 100 Hz. So, the difference in frequencies is: \[ f_3 (\text{closed pipe}) - f_1 (\text{open pipe}) = 100 \, \text{Hz} \] This gives \(f_3 (\text{closed pipe}) = 300 \, \text{Hz}\), and by calculating the frequencies, we can deduce that the fundamental frequency of the open pipe is \(200 \, \text{Hz}\).

Step 3: Conclusion.
The fundamental frequency of the open pipe is 200 Hz. Hence, the correct answer is (1).