Which of the following statements about the 3rd overtone of a vibrating air column of an open pipe is correct? It has

Analyze the question

We need the number of nodes and antinodes in the 3rd overtone of an open pipe.

Concept

An open pipe has an antinode at both ends. The harmonics present are n = 1, 2, 3, 4, … (all integers), where:

• Fundamental = 1st harmonic (n = 1)
• 1st overtone = 2nd harmonic
• 2nd overtone = 3rd harmonic
• 3rd overtone = 4th harmonic (n = 4)

For the nth harmonic in an open pipe:

• Number of antinodes = n + 1
• Number of nodes = n

Solution

For the 3rd overtone, n = 4:

• Nodes = 4
• Antinodes = 5

Check by picture: the fundamental has A–N–A (1 node, 2 antinodes). Each higher harmonic adds one node and one antinode. So the 4th harmonic has 4 nodes and 5 antinodes: A N A N A N A N A.

Checking the options

• A. 4 nodes ✔ correct
• B. 5 nodes ✘ (5 is the number of antinodes)
• C. 3 antinodes ✘
• D. 4 antinodes ✘ (there are 5)

Tip: In an open pipe, the nth overtone is the (n+1)th harmonic. Then nodes = harmonic number and antinodes = harmonic number + 1. (For a closed pipe, it's different: only odd harmonics occur, with a node at the closed end and an antinode at the open end.)