The ratio of intensities between two coherent sound sources is `4:1` the difference of loudness in decibels between maximum and minimum intensities, when they interfere in space, is

Options

A. 10 log (2)
B. 20 log (3)
C. 10 log (3)
D. 20 log (2)

Answer

To solve the problem, we need to find the difference in loudness between the maximum and minimum intensities when two coherent sound sources interfere. The given ratio of their intensities is \(4:1\). Let's break down the solution step by step.

### Step 1: Understand the relationship between intensity and amplitude
We know that intensity (\(I\)) is directly proportional to the square of the amplitude (\(A\)):
\[ I \propto A^2 \]
Given the intensity ratio \(I_1:I_2 = 4:1\), we can express this relationship in terms of amplitudes.

### Step 2: Set up the ratio of amplitudes
From the intensity ratio:
\[ \frac{I_1}{I_2} = \frac{4}{1} \]
We can write this in terms of amplitudes:
\[ \frac{A_1^2}{A_2^2} = \frac{4}{1} \]
Taking the square root of both sides gives us:
\[ \frac{A_1}{A_2} = \frac{2}{1} \]
This means that \(A_1 = 2A_2\).

### Step 3: Calculate maximum and minimum intensities
The maximum intensity (\(I_{max}\)) occurs when the amplitudes add up:
\[ I_{max} = (A_1 + A_2)^2 \]
The minimum intensity (\(I_{min}\)) occurs when the amplitudes subtract:
\[ I_{min} = (A_1 - A_2)^2 \]
Substituting \(A_1 = 2A_2\) into these equations:
\[ I_{max} = (2A_2 + A_2)^2 = (3A_2)^2 = 9A_2^2 \]
\[ I_{min} = (2A_2 - A_2)^2 = (A_2)^2 = A_2^2 \]

### Step 4: Find the ratio of maximum to minimum intensities
Now, we can find the ratio of maximum to minimum intensities:
\[ \frac{I_{max}}{I_{min}} = \frac{9A_2^2}{A_2^2} = 9 \]

### Step 5: Calculate the difference in loudness in decibels
The difference in loudness (\(L\)) in decibels is given by:
\[ L = 10 \log \left(\frac{I_{max}}{I_{min}}\right) \]
Substituting the ratio we found:
\[ L = 10 \log(9) \]
Using the logarithmic property \(9 = 3^2\):
\[ L = 10 \cdot 2 \log(3) = 20 \log(3) \]

### Final Answer
The difference of loudness in decibels between maximum and minimum intensities is:
\[ \boxed{20 \log(3)} \]