Two sound waves of wavelengths `lambda_(1)` and `lambda_(2) (lambda _(2) gt lambda_(1))` produces `n beats//s` , the speed of sound is

Options

A. `(n lambda_(1) lambda_(2))/(lambda_(2) - lambda_(1))`
B. `n((1)/(lambda_(1))- (1)/(lambda_(2)))`
C. `n(lambda_(2)- lambda_(1))`
D. `n(lambda_(2) + lambda_(1))`

Explanation

To find the speed of sound given two sound waves of wavelengths \( \lambda_1 \) and \( \lambda_2 \) (where \( \lambda_2 > \lambda_1 \)) that produce \( n \) beats per second, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Beat Frequency**:
The beat frequency \( n \) is the difference between the frequencies of the two sound waves. Thus, we can write:
\[ n = |f_1 - f_2| \]
where \( f_1 \) and \( f_2 \) are the frequencies corresponding to wavelengths \( \lambda_1 \) and \( \lambda_2 \).

2. **Relate Frequency to Wavelength**:
The frequency of a wave is related to its wavelength and the speed of sound \( v \) by the formula:
\[ f = \frac{v}{\lambda} \]
Therefore, we can express \( f_1 \) and \( f_2 \) as:
\[ f_1 = \frac{v}{\lambda_1} \quad \text{and} \quad f_2 = \frac{v}{\lambda_2} \]

3. **Substitute Frequencies into Beat Frequency Equation**:
Substituting \( f_1 \) and \( f_2 \) into the beat frequency equation gives:
\[ n = \left|\frac{v}{\lambda_1} - \frac{v}{\lambda_2}\right| \]

4. **Factor Out the Speed of Sound**:
Factoring out \( v \) from the right side, we have:
\[ n = v \left|\frac{1}{\lambda_1} - \frac{1}{\lambda_2}\right| \]
This can be rewritten as:
\[ n = v \left(\frac{\lambda_2 - \lambda_1}{\lambda_1 \lambda_2}\right) \]

5. **Solve for Speed of Sound \( v \)**:
Rearranging the equation to solve for \( v \) gives:
\[ v = n \cdot \frac{\lambda_1 \lambda_2}{\lambda_2 - \lambda_1} \]

### Final Answer:
The speed of sound is given by:
\[ v = n \cdot \frac{\lambda_1 \lambda_2}{\lambda_2 - \lambda_1} \]