Question

A closed organ pipe (closed at one end) is excited to support the third overtone. It is found that air in the pipe has

• A. three nodes and three antinodes
• B. three nodes and four antinodes
• C. four nodes and three antinodes
• D. four nodes and four antinodes

Answer 1

The Correct Option is D

To solve the problem, first, we need to understand the concept of standing waves in a closed organ pipe. A closed organ pipe is one that is closed at one end and open at the other. The closed end is a node (where air movement is minimal), and the open end is an antinode (where air movement is maximal).

In a closed pipe, the fundamental frequency or first harmonic is supported with a quarter wavelength inside the pipe:

L = \frac{\lambda_1}{4}

where L is the length of the pipe and \lambda_1 is the wavelength of the fundamental tone.

The overtone numbers in a closed pipe refer to the odd harmonics (1st, 3rd, 5th, ... etc.) because of the node-antinode pattern. For a closed pipe, the sequence of wavelengths for the harmonics is:

• 1st harmonic (fundamental) has 1 node and 1 antinode
• 1st overtone (3rd harmonic) has 2 nodes and 2 antinodes
• 2nd overtone (5th harmonic) has 3 nodes and 3 antinodes
• 3rd overtone (7th harmonic) has 4 nodes and 4 antinodes

Therefore, the third overtone (or 7th harmonic) in a closed pipe will consist of 4 nodes and 4 antinodes. The length of the pipe is equivalent to 7 quarter wavelengths of sound or:

L = \frac{7\lambda}{4}

Thus, the correct answer is: four nodes and four antinodes.