If you set up the seventh harmonic on a string fixed at both ends, how many nodes and antinodes are set up in it

Answer

To solve the problem of determining the number of nodes and antinodes when setting up the seventh harmonic on a string fixed at both ends, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Harmonics**:
- The nth harmonic of a string fixed at both ends corresponds to the formation of n loops (or segments) in the standing wave pattern.

2. **Identifying the Seventh Harmonic**:
- For the seventh harmonic (n = 7), there will be 7 loops formed along the length of the string.

3. **Counting Nodes**:
- In a standing wave, nodes are points where there is no displacement (minimum amplitude).
- The number of nodes in a standing wave is always one more than the number of loops. Therefore, for the seventh harmonic:
\[ \text{Number of nodes} = n + 1 = 7 + 1 = 8 \]

4. **Counting Antinodes**:
- Antinodes are points of maximum displacement (maximum amplitude).
- The number of antinodes is equal to the number of loops. Thus, for the seventh harmonic:
\[ \text{Number of antinodes} = n = 7 \]

5. **Final Answer**:
- Therefore, when the seventh harmonic is set up on the string, there are 8 nodes and 7 antinodes.

### Summary:
- **Number of Nodes**: 8
- **Number of Antinodes**: 7