Question

The fundamental frequency of an air column in pipe 'A' closed at one end coincides with the second overtone of pipe 'B' open at both ends. The ratio of the length of pipe 'A' to that of pipe 'B' is

• A. 3 : 8
• B. 3 : 4
• C. 1 : 6
• D. 2 : 3

Hint:

Always be careful with terminology: for an open pipe, the $k^{\text{th}}$ overtone is always equal to the $(k+1)^{\text{th}}$ harmonic, so the second overtone is the third harmonic ($3 \times \text{fundamental}$). For a closed pipe, overtones skip directly across odd numbers, but here we only need the basic fundamental frequency ($1 \times \text{fundamental}$). Setting $\frac{1}{4L_A} = \frac{3}{2L_B}$ quickly yields the answer.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem compares two acoustic organ pipes: Pipe A is closed at one end, and Pipe B is open at both ends. We are given that the fundamental frequency of Pipe A equals the second overtone frequency of Pipe B, and we need to find the ratio of their physical lengths ($\frac{L_A}{L_B}$).

Step 2: Key Formula or Approach:
1. For an organ pipe closed at one end (length $L_A$), the fundamental frequency is given by:
$$n_A = \frac{v}{4L_A}$$ 2. For an organ pipe open at both ends (length $L_B$), the fundamental frequency is $n_B = \frac{v}{2L_B}$. The open pipe produces all integer harmonics, so its second overtone corresponds to the third harmonic ($3n_B$):
$$n_B' = 3 \left(\frac{v}{2L_B}\right) = \frac{3v}{2L_B}$$

Step 3: Detailed Explanation:
According to the problem, the fundamental frequency of Pipe A is equal to the second overtone of Pipe B:
$$n_A = n_B'$$ Substitute our length equations into this frequency equality:
$$\frac{v}{4L_A} = \frac{3v}{2L_B}$$ Since the speed of sound $v$ is identical in both pipes, we can cancel it from both numerators:
$$\frac{1}{4L_A} = \frac{3}{2L_B}$$ Now, rearrange the terms to isolate the length ratio $\frac{L_A}{L_B}$ by cross-multiplying:
$$\frac{L_A}{L_B} = \frac{2}{4 \times 3} = \frac{2}{12} = \frac{1}{6}$$ This simplifies to a length ratio of exactly 1 : 6.

Step 4: Final Answer:
The ratio of the length of pipe 'A' to that of pipe 'B' is 1 : 6, which corresponds to option (C).