Question

If an open pipe suddenly closed, the frequency of the third harmonic of the closed pipe is 50Hz more than the fundamental frequency of the open pipe. Find the fundamental frequency.

Hint:

For open and closed pipes, the frequencies of the harmonics are calculated differently. Remember the formula for the closed pipe involves only odd harmonics.

Answer 1

Step 1: Use the relationship between the harmonics of the open and closed pipes.
For an open pipe, the frequencies of the harmonics are given by: \[ f_n^{\text{open}} = n \times f_1 \] where \( f_1 \) is the fundamental frequency and \( n \) is the harmonic number. For a closed pipe, the frequencies are given by: \[ f_n^{\text{closed}} = (2n - 1) \times f_1 \]
Step 2: Use the given frequency difference.
We are told that the frequency of the third harmonic of the closed pipe is 50Hz more than the fundamental frequency of the open pipe. This gives the equation: \[ f_3^{\text{closed}} = f_1^{\text{open}} + 50 \] The third harmonic for the closed pipe is given by: \[ f_3^{\text{closed}} = 5 f_1 \] So, we have: \[ 5 f_1 = f_1 + 50 \]
Step 3: Solve for the fundamental frequency.
Solving for \( f_1 \): \[ 5 f_1 - f_1 = 50 \quad \Rightarrow \quad 4 f_1 = 50 \quad \Rightarrow \quad f_1 = \frac{50}{4} = 12.5 \, \text{Hz} \] Thus, the fundamental frequency is: \[ \boxed{12.5 \, \text{Hz}} \]