When two coherent waves interfere, the minimum and maximum intensities are in the ratio 16 : 25. Then
a) the maximum and minimum amplitudes will be in the ratio 5 : 4
b) the amplitudes of the individual waves will be in the ratio 9 : 1
c) the intensities of the individual waves will be in the ratio 41 : 9
d) the intensities of the individual waves will be in the ratio 81 : 1.

Options

A. a, b and c are true
B. a, b and d are true
C. a and b are true
D. b and c are true

Answer

To solve the problem, we need to analyze the relationship between the intensities and amplitudes of two coherent waves that interfere. Given the ratio of minimum and maximum intensities, we can derive the required ratios step by step.

### Step-by-Step Solution:

1. **Understanding Intensity Ratio**:
The problem states that the minimum and maximum intensities are in the ratio of \( I_{min} : I_{max} = 16 : 25 \). This can be expressed mathematically as:
\[ \frac{I_{min}}{I_{max}} = \frac{16}{25} \]

2. **Relating Intensity to Amplitude**:
We know that intensity \( I \) is proportional to the square of the amplitude \( A \):
\[ I \propto A^2 \]
Therefore, we can express the intensities in terms of amplitudes:
\[ I_{min} = k A_{min}^2 \quad \text{and} \quad I_{max} = k A_{max}^2 \]
where \( k \) is a proportionality constant.

3. **Setting Up the Ratio**:
Using the relationship between intensity and amplitude, we can write:
\[ \frac{A_{min}^2}{A_{max}^2} = \frac{16}{25} \]
Taking the square root of both sides gives us the ratio of the amplitudes:
\[ \frac{A_{min}}{A_{max}} = \frac{4}{5} \]

4. **Expressing Amplitudes of Individual Waves**:
Let the amplitudes of the two individual waves be \( A_1 \) and \( A_2 \). The maximum amplitude \( A_{max} \) is given by \( A_1 + A_2 \) and the minimum amplitude \( A_{min} \) is given by \( |A_1 - A_2| \). Thus:
\[ \frac{|A_1 - A_2|}{A_1 + A_2} = \frac{4}{5} \]

5. **Cross-Multiplying**:
Cross-multiplying gives:
\[ 5|A_1 - A_2| = 4(A_1 + A_2) \]
This leads to two cases, but we will assume \( A_1 \geq A_2 \) for simplicity:
\[ 5(A_1 - A_2) = 4(A_1 + A_2) \]

6. **Solving for Amplitudes**:
Rearranging the equation:
\[ 5A_1 - 5A_2 = 4A_1 + 4A_2 \]
Combining like terms:
\[ 5A_1 - 4A_1 = 5A_2 + 4A_2 \implies A_1 = 9A_2 \]

7. **Finding the Ratio of Amplitudes**:
Therefore, the ratio of the amplitudes of the individual waves is:
\[ \frac{A_1}{A_2} = \frac{9}{1} \]

8. **Finding the Ratio of Intensities**:
Since intensity is proportional to the square of the amplitude:
\[ \frac{I_1}{I_2} = \left(\frac{A_1}{A_2}\right)^2 = \left(\frac{9}{1}\right)^2 = \frac{81}{1} \]

### Conclusion:
- The ratio of the amplitudes of the individual waves is \( 9 : 1 \).
- The ratio of the intensities of the individual waves is \( 81 : 1 \).