Two waves represented by the following equations are travelling in the same medium `y_(1)= 5sin 2 pi (75 t-0.25x),y_(2)=10 sin2 pi (150 t-0.50x)`
The intensity ratio `I_(1)//I_(2)` of the two waves is

Options

A. `1 : 2`
B. `1 : 4`
C. `1 : 8`
D. ` 1 : 16`

Answer

To find the intensity ratio \( I_1/I_2 \) of the two waves represented by the equations \( y_1 = 5 \sin(2\pi(75t - 0.25x)) \) and \( y_2 = 10 \sin(2\pi(150t - 0.50x)) \), we can follow these steps:

### Step 1: Identify the Amplitudes
The general form of a wave equation is given by:
\[ y = A \sin(\omega t - kx) \]
where \( A \) is the amplitude.

From the equations:
- For \( y_1 = 5 \sin(2\pi(75t - 0.25x)) \), the amplitude \( A_1 = 5 \).
- For \( y_2 = 10 \sin(2\pi(150t - 0.50x)) \), the amplitude \( A_2 = 10 \).

### Step 2: Write the Intensity Formula
The intensity \( I \) of a wave is proportional to the square of its amplitude:
\[ I \propto A^2 \]
Thus, we can express the intensities of the two waves as:
\[ I_1 \propto A_1^2 \quad \text{and} \quad I_2 \propto A_2^2 \]

### Step 3: Calculate the Intensities
Now we can calculate the intensities based on the amplitudes:
\[ I_1 = k A_1^2 = k (5^2) = k \cdot 25 \]
\[ I_2 = k A_2^2 = k (10^2) = k \cdot 100 \]
where \( k \) is a proportionality constant.

### Step 4: Find the Intensity Ratio
Now, we can find the ratio of the intensities \( I_1/I_2 \):
\[ \frac{I_1}{I_2} = \frac{k \cdot 25}{k \cdot 100} = \frac{25}{100} = \frac{1}{4} \]

### Step 5: Express the Ratio
Thus, we can express the intensity ratio as:
\[ I_1 : I_2 = 1 : 4 \]

### Conclusion
The intensity ratio \( I_1/I_2 \) of the two waves is \( 1 : 4 \).