If the ratio of amplitude of wave is `2 : 1`, then the ratio of maximum and minimum intensity is

Options

A. ` 9 : 1`
B. `1 : 9`
C. `4 : 1`
D. ` 1 :4`

Answer

To solve the problem of finding the ratio of maximum and minimum intensity given the ratio of amplitudes of two waves as \(2:1\), we can follow these steps:

### Step-by-Step Solution

1. **Understand the Relationship Between Intensity and Amplitude**:
The intensity \(I\) of a wave is directly proportional to the square of its amplitude \(A\). Mathematically, this can be expressed as:
\[ I \propto A^2 \]

2. **Assign Amplitudes**:
Let the amplitudes of the two waves be:
\[ A_1 = 2A \quad \text{and} \quad A_2 = A \]
where \(A\) is a common unit of amplitude.

3. **Calculate Maximum Intensity**:
The maximum intensity occurs when the two waves are in phase (constructive interference). The resultant amplitude \(A_{max}\) is given by:
\[ A_{max} = A_1 + A_2 = 2A + A = 3A \]
The maximum intensity \(I_{max}\) can be calculated as:
\[ I_{max} = k(A_{max})^2 = k(3A)^2 = 9kA^2 \]
where \(k\) is the proportionality constant.

4. **Calculate Minimum Intensity**:
The minimum intensity occurs when the two waves are out of phase (destructive interference). The resultant amplitude \(A_{min}\) is given by:
\[ A_{min} = A_1 - A_2 = 2A - A = A \]
The minimum intensity \(I_{min}\) can be calculated as:
\[ I_{min} = k(A_{min})^2 = k(A)^2 = kA^2 \]

5. **Find the Ratio of Maximum to Minimum Intensity**:
Now, we can find the ratio of maximum intensity to minimum intensity:
\[ \frac{I_{max}}{I_{min}} = \frac{9kA^2}{kA^2} = 9 \]

### Final Answer:
The ratio of maximum to minimum intensity is:
\[ \boxed{9:1} \]