If `lambda_1, lambda_2 and lambda_3` are the wavelength of the waves giving resonance to the fundamental, first and second overtone modes respectively in a string fixed at both ends. The ratio of the wavelengths `lambda_1:lambda_2:lambda_3` is

Options

A. `1:2:3`
B. `1:3:5`
C. `1:1/2:1/3`
D. `1:1/3:1/5`

Explanation

To solve the problem of finding the ratio of the wavelengths \( \lambda_1 : \lambda_2 : \lambda_3 \) for the fundamental and overtone modes of a string fixed at both ends, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Modes**:
- The fundamental mode (first harmonic) corresponds to \( n = 1 \).
- The first overtone (second harmonic) corresponds to \( n = 2 \).
- The second overtone (third harmonic) corresponds to \( n = 3 \).

2. **Wavelengths and Frequencies**:
- The wavelength \( \lambda \) is related to the mode number \( n \) by the formula:
\[ \lambda_n = \frac{2L}{n} \]
where \( L \) is the length of the string.

3. **Calculating Wavelengths**:
- For the fundamental mode (\( n = 1 \)):
\[ \lambda_1 = \frac{2L}{1} = 2L \]
- For the first overtone (\( n = 2 \)):
\[ \lambda_2 = \frac{2L}{2} = L \]
- For the second overtone (\( n = 3 \)):
\[ \lambda_3 = \frac{2L}{3} = \frac{2L}{3} \]

4. **Finding the Ratio**:
- Now we have:
\[ \lambda_1 = 2L, \quad \lambda_2 = L, \quad \lambda_3 = \frac{2L}{3} \]
- To find the ratio \( \lambda_1 : \lambda_2 : \lambda_3 \), we can express them in terms of \( L \):
\[ \lambda_1 : \lambda_2 : \lambda_3 = 2L : L : \frac{2L}{3} \]
- To simplify, we can divide each term by \( L \):
\[ 2 : 1 : \frac{2}{3} \]
- To eliminate the fraction, multiply each term by 3:
\[ 2 \times 3 : 1 \times 3 : \frac{2}{3} \times 3 = 6 : 3 : 2 \]

5. **Final Ratio**:
- Thus, the ratio of the wavelengths is:
\[ \lambda_1 : \lambda_2 : \lambda_3 = 6 : 3 : 2 \]

### Conclusion:
The final answer is:
\[ \lambda_1 : \lambda_2 : \lambda_3 = 6 : 3 : 2 \]