Two progressive waves `y_(1) = 4 sin 400 pi t and y_(2) = 3 Sin 404 pi t` moving in the same direction superpose on each other producing beats. Then the number of beats per second and the ratio of maximum to minimum intensity of the resultant waves are respectively

Options

A. `2 and (5)/(1)`
B. `4and (49)/(1)`
C. `4 and (16)/(9)`
D. `2 and (49)/(1)`

Answer

To solve the problem, we need to find the number of beats per second produced by the superposition of two progressive waves and the ratio of maximum to minimum intensity of the resultant wave.

### Step-by-Step Solution:

1. **Identify the given wave equations**:
- The first wave is given by \( y_1 = 4 \sin(400 \pi t) \)
- The second wave is given by \( y_2 = 3 \sin(404 \pi t) \)

2. **Extract the angular frequencies**:
- From \( y_1 \), we identify \( \omega_1 = 400 \pi \)
- From \( y_2 \), we identify \( \omega_2 = 404 \pi \)

3. **Convert angular frequencies to linear frequencies**:
- The relationship between angular frequency (\( \omega \)) and linear frequency (\( f \)) is given by:
\[ \omega = 2 \pi f \]
- For the first wave:
\[ 400 \pi = 2 \pi f_1 \implies f_1 = \frac{400 \pi}{2 \pi} = 200 \, \text{Hz} \]
- For the second wave:
\[ 404 \pi = 2 \pi f_2 \implies f_2 = \frac{404 \pi}{2 \pi} = 202 \, \text{Hz} \]

4. **Calculate the beat frequency**:
- The beat frequency (\( f_b \)) is given by the absolute difference between the two frequencies:
\[ f_b = |f_1 - f_2| = |200 - 202| = 2 \, \text{Hz} \]

5. **Calculate the ratio of maximum to minimum intensity**:
- The intensity \( I \) of a wave is proportional to the square of its amplitude. The maximum and minimum intensities can be calculated using the formula:
\[ \text{Ratio} = \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} \]
- Here, \( A_1 = 4 \) and \( A_2 = 3 \):
\[ \text{Maximum Intensity} = (4 + 3)^2 = 7^2 = 49 \]
\[ \text{Minimum Intensity} = (4 - 3)^2 = 1^2 = 1 \]
- Thus, the ratio of maximum to minimum intensity is:
\[ \text{Ratio} = \frac{49}{1} = 49 \]

6. **Final Answer**:
- The number of beats per second is \( 2 \, \text{Hz} \) and the ratio of maximum to minimum intensity is \( 49 \).

### Summary:
- **Number of beats per second**: \( 2 \, \text{Hz} \)
- **Ratio of maximum to minimum intensity**: \( 49 \)