Question

The number of nodes and antinodes in a guitar string vibrating in the third harmonic is ________.

• A. 5 nodes, 4 antinodes
• B. 4 nodes, 3 antinodes
• C. 3 nodes, 2 antinodes
• D. 2 nodes, 3 antinodes
• E. 1 node, 2 antinodes

Hint:

Nodes are points of zero displacement; Antinodes are points of maximum displacement.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
This question is about standing waves on a string fixed at both ends, like a guitar string. A harmonic refers to a mode of vibration. The 'n-th harmonic' (or (n-1)-th overtone) is the standing wave pattern that has 'n' loops or segments. Nodes are points of zero displacement, and antinodes are points of maximum displacement.
Step 2: Key Formula or Approach
For a string of length L fixed at both ends, vibrating in its n-th harmonic:
- The number of antinodes (loops) is equal to \(n\).
- The number of nodes is equal to \(n+1\). The two fixed ends are always nodes.
We need to apply this for the third harmonic, which means \(n=3\).
Step 3: Detailed Explanation
1. Identify the harmonic number.
The string is vibrating in the third harmonic, so \(n=3\).
2. Visualize the standing wave pattern.
The third harmonic on a string fixed at both ends consists of three "loops" or segments oscillating between the fixed ends.
- The ends of the string must be nodes.
- Between each pair of nodes, there is an antinode.
The pattern will look like: Node - Antinode - Node - Antinode - Node - Antinode - Node.
3. Count the nodes and antinodes.
- Antinodes: The number of antinodes is the number of loops, which is equal to the harmonic number. Number of antinodes = \(n = 3\).
- Nodes: There are nodes at both ends, and there are nodes separating the loops. For 3 loops, there will be 2 nodes in between. Total number of nodes = (nodes at ends) + (nodes in between) = 2 + (n-1) = 2 + (3-1) = 4.
Alternatively, using the formula: Number of nodes = \(n+1 = 3+1 = 4\).
So, for the third harmonic, there are 4 nodes and 3 antinodes.
Step 4: Final Answer
There are 4 nodes and 3 antinodes.