When beats are produced by two progressive waves of same amplitude and of nearly same frequencies then the maximum loudness of the resulting sound is `n` times the loudness of each of the component wave trains. The value of `n` is

Options

A. `1`
B. `2`
C. `4`
D. `8`

Explanation

To solve the problem of determining the value of \( n \) when beats are produced by two progressive waves of the same amplitude and nearly the same frequencies, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have two progressive waves that have the same amplitude and nearly the same frequencies. When these two waves interfere, they produce beats. The maximum loudness of the resulting sound is \( n \) times the loudness of each of the component wave trains.

2. **Intensity and Loudness Relation**: The loudness of a sound is proportional to the intensity of the sound. If the intensity of each wave is \( I_0 \), then the loudness of each wave can be represented as \( L_0 \).

3. **Resultant Intensity Calculation**: When two waves interfere, the resultant intensity \( I_{max} \) can be calculated using the principle of superposition. If both waves have the same amplitude \( A \), the resultant amplitude when they are in phase is:
\[ A_{resultant} = A + A = 2A \]
The intensity \( I \) is proportional to the square of the amplitude:
\[ I_{max} \propto (2A)^2 = 4A^2 \]

4. **Intensity of Each Wave**: The intensity of each individual wave is given by:
\[ I_0 \propto A^2 \]

5. **Relating Intensities**: Since the intensity of each wave is \( I_0 \), we can express the resultant intensity in terms of \( I_0 \):
\[ I_{max} = 4I_0 \]

6. **Finding the Value of \( n \)**: The maximum loudness \( L_{max} \) of the resultant wave is proportional to the maximum intensity:
\[ L_{max} \propto I_{max} = 4I_0 \]
Therefore, the ratio of the maximum loudness to the loudness of each individual wave is:
\[ n = \frac{L_{max}}{L_0} = \frac{4I_0}{I_0} = 4 \]

### Conclusion:
The value of \( n \) is \( 4 \).