The ratio of intensities between two cohernt sound sources is 4:1. The difference of loudness is decibel (bD) between maximum and minimum intensitiesm, when they interface in space is

Options

A. 10 log2
B. 20 log3
C. 10 log3
D. 20 log2

Answer

To solve the problem of finding the difference in loudness between the maximum and minimum intensities when two coherent sound sources interfere, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Ratio of Intensities**:
- We are given the ratio of intensities of two coherent sound sources as \( I_1 : I_2 = 4 : 1 \).
- This means \( I_1 = 4I \) and \( I_2 = I \) for some intensity \( I \).

2. **Calculate the Maximum and Minimum Intensities**:
- The maximum intensity \( I_{max} \) when two waves interfere constructively is given by:
\[ I_{max} = I_1 + I_2 = 4I + I = 5I \]
- The minimum intensity \( I_{min} \) when they interfere destructively is given by:
\[ I_{min} = I_1 - I_2 = 4I - I = 3I \]

3. **Find the Ratio of Maximum to Minimum Intensities**:
- The ratio \( \frac{I_{max}}{I_{min}} \) can be calculated as:
\[ \frac{I_{max}}{I_{min}} = \frac{5I}{3I} = \frac{5}{3} \]

4. **Calculate the Difference in Loudness**:
- The loudness \( L \) in decibels is given by the formula:
\[ L = 10 \log \left( \frac{I}{I_0} \right) \]
- Therefore, the difference in loudness between maximum and minimum intensities can be expressed as:
\[ L_{max} - L_{min} = 10 \log \left( \frac{I_{max}}{I_{min}} \right) \]
- Substituting the ratio we found:
\[ L_{max} - L_{min} = 10 \log \left( \frac{5}{3} \right) \]

5. **Calculate the Final Value**:
- We can simplify this further:
\[ L_{max} - L_{min} = 10 \left( \log 5 - \log 3 \right) \]
- Using the logarithmic property \( \log a - \log b = \log \left( \frac{a}{b} \right) \):
\[ L_{max} - L_{min} = 10 \log \left( \frac{5}{3} \right) \]

6. **Final Calculation**:
- To find the numerical value, we can use the approximate values of logarithms:
- \( \log 5 \approx 0.699 \)
- \( \log 3 \approx 0.477 \)
- Thus:
\[ L_{max} - L_{min} \approx 10 \times (0.699 - 0.477) = 10 \times 0.222 = 2.22 \text{ dB} \]

### Conclusion:
The difference in loudness between the maximum and minimum intensities is approximately \( 2.22 \, \text{dB} \).