Two superimposig waves are represented by equation ` y_1=2 sin 2 pi ( 10t-0.4x) and y_2=4 sin2 pi ( 20 t- 0.8x)` . The ratio of `l_("max") " to " l_("min")` is

Options

A. `36:4`
B. `25:9`
C. `1:4`
D. `4:1`

Answer

To find the ratio of \( I_{\text{max}} \) to \( I_{\text{min}} \) for the given superimposing waves represented by the equations \( y_1 = 2 \sin(2\pi(10t - 0.4x)) \) and \( y_2 = 4 \sin(2\pi(20t - 0.8x)) \), we can follow these steps:

### Step 1: Identify the Amplitudes
From the equations, we can identify the amplitudes of the waves:
- For \( y_1 \): \( A_1 = 2 \)
- For \( y_2 \): \( A_2 = 4 \)

### Step 2: Calculate Intensities
The intensity \( I \) of a wave is proportional to the square of its amplitude:
\[ I \propto A^2 \]
Thus, we can write:
- \( I_1 = k A_1^2 = k (2^2) = 4k \)
- \( I_2 = k A_2^2 = k (4^2) = 16k \)

### Step 3: Find the Ratio of Intensities
Now, we can find the ratio of the intensities:
\[ \frac{I_1}{I_2} = \frac{4k}{16k} = \frac{4}{16} = \frac{1}{4} \]

### Step 4: Express \( I_{\text{max}} \) and \( I_{\text{min}} \)
For constructive interference (maximum intensity):
\[ I_{\text{max}} = (\sqrt{I_1} + \sqrt{I_2})^2 = (\sqrt{4k} + \sqrt{16k})^2 = (2\sqrt{k} + 4\sqrt{k})^2 = (6\sqrt{k})^2 = 36k \]

For destructive interference (minimum intensity):
\[ I_{\text{min}} = (\sqrt{I_1} - \sqrt{I_2})^2 = (\sqrt{4k} - \sqrt{16k})^2 = (2\sqrt{k} - 4\sqrt{k})^2 = (-2\sqrt{k})^2 = 4k \]

### Step 5: Calculate the Ratio \( \frac{I_{\text{max}}}{I_{\text{min}}} \)
Now we can find the ratio of maximum intensity to minimum intensity:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{36k}{4k} = \frac{36}{4} = 9 \]

### Final Answer
Thus, the ratio of \( I_{\text{max}} \) to \( I_{\text{min}} \) is:
\[ \boxed{9} \]