Two waves are represented by the following equations
`y_(1) = 5 sin 2pi (10t -0.1 x)`
`y_(2) = 10 sin 2pi (20t - 0.2 x)`
Ratio of intensites `I_(2)//I_(1)` will be

Options

A. 1
B. 2
C. 4
D. 16

Explanation

To find the ratio of intensities \( \frac{I_2}{I_1} \) for the two waves given by the equations:

1. \( y_1 = 5 \sin(2\pi(10t - 0.1x)) \)
2. \( y_2 = 10 \sin(2\pi(20t - 0.2x)) \)

we can follow these steps:

### Step 1: Identify the Amplitudes
From the equations, we can identify the amplitudes of the two waves:
- Amplitude of \( y_1 \) (A1) = 5
- Amplitude of \( y_2 \) (A2) = 10

### Step 2: Determine the Angular Frequencies
The angular frequency \( \omega \) can be extracted from the wave equations:
- For \( y_1 \):
\[ \omega_1 = 2\pi \times 10 = 20\pi \, \text{rad/s} \]
- For \( y_2 \):
\[ \omega_2 = 2\pi \times 20 = 40\pi \, \text{rad/s} \]

### Step 3: Determine the Wave Numbers
The wave number \( k \) can also be extracted:
- For \( y_1 \):
\[ k_1 = 0.1 \times 2\pi = 0.2\pi \, \text{rad/m} \]
- For \( y_2 \):
\[ k_2 = 0.2 \times 2\pi = 0.4\pi \, \text{rad/m} \]

### Step 4: Calculate the Intensities
The intensity \( I \) of a wave is proportional to the square of its amplitude:
\[ I \propto A^2 \]
Thus, we can express the intensities as:
- \( I_1 \propto A_1^2 = 5^2 = 25 \)
- \( I_2 \propto A_2^2 = 10^2 = 100 \)

### Step 5: Find the Ratio of Intensities
Now we can find the ratio of the intensities:
\[ \frac{I_2}{I_1} = \frac{100}{25} = 4 \]

### Step 6: Consider the Frequencies
Since the intensity also depends on the frequency, we need to include the frequency ratio:
\[ \frac{I_2}{I_1} = \frac{\omega_2^2}{\omega_1^2} \cdot \frac{A_2^2}{A_1^2} \]
Calculating the frequency ratio:
\[ \frac{\omega_2}{\omega_1} = \frac{40\pi}{20\pi} = 2 \]
Thus,
\[ \frac{I_2}{I_1} = \left(2\right)^2 \cdot \frac{100}{25} = 4 \cdot 4 = 16 \]

### Final Result
The ratio of intensities \( \frac{I_2}{I_1} \) is \( 16 \).

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