The vibration of a stretched string consists of

Options

A. only the fundamental frequency (n)
B. frequency `n, 2n, 3n, 4n,....`
C. frequency `n, 3n, 5n,....`
D. frequency of `n,2n,4n,6n...`

Answer

To solve the question regarding the vibration of a stretched string, we can break it down into a series of steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a stretched string fixed at both ends. The length of the string is denoted as \( L \).

2. **Identifying Nodes and Antinodes**:
- In a vibrating string, the points where the string does not move are called nodes, and the points where the string moves with maximum amplitude are called antinodes.
- For the fundamental frequency (first harmonic), there are two nodes at the ends and one antinode in the middle.

3. **Wavelength Calculation**:
- The wavelength \( \lambda \) for the fundamental frequency can be determined. For a string of length \( L \) with fixed ends, the relationship is:
\[ L = \frac{\lambda}{2} \]
- Therefore, the wavelength \( \lambda \) is:
\[ \lambda = 2L \]

4. **Frequency Calculation**:
- The frequency \( f \) of the wave can be calculated using the wave speed \( v \) and the wavelength \( \lambda \):
\[ f = \frac{v}{\lambda} \]
- Substituting the value of \( \lambda \):
\[ f = \frac{v}{2L} \]
- This frequency is known as the fundamental frequency \( f_1 \).

5. **Higher Harmonics**:
- The string can vibrate in higher harmonics (overtones). The frequencies of the higher harmonics are integer multiples of the fundamental frequency:
- First harmonic (fundamental frequency): \( f_1 = \frac{v}{2L} \)
- Second harmonic: \( f_2 = 2f_1 = \frac{v}{L} \)
- Third harmonic: \( f_3 = 3f_1 = \frac{3v}{2L} \)
- And so on, for the \( n \)-th harmonic:
\[ f_n = n \cdot f_1 = n \cdot \frac{v}{2L} \]

6. **Conclusion**:
- The vibration of a stretched string consists of multiple frequencies, specifically the fundamental frequency and its harmonics. The frequencies can be expressed as:
\[ f_n = n \cdot \frac{v}{2L} \]
- Thus, the vibration of a string can be described as including the fundamental frequency \( f_1 \) and its harmonics \( 2f_1, 3f_1, \ldots \).