Two standing bodies producing progressive waves are given by
` y_(1) = 4 sin 400 pi t and y_(2) = 3 sin 404 pi t`
One of these bodies situated very near to the ears of a person who will hear :

Options

A. ` 2 beats//s` with intensity ratio `4//3` between maxima and minima.
B. ` 2 beats//s` with intensity ratio `49//1` between maxima and minima.
C. ` 4 beats//s` with intensity ratio `7//2` between maxima and minima.
D. ` 4 beats//s` with intensity ratio `4//3` between maxima and minima.

Explanation

To solve the problem, we need to analyze the two progressive waves given by the equations:

1. \( y_1 = 4 \sin(400 \pi t) \)
2. \( y_2 = 3 \sin(404 \pi t) \)

### Step 1: Identify Angular Frequencies
From the equations, we can identify the angular frequencies (\( \omega \)) of the two waves:
- For wave 1: \( \omega_1 = 400 \pi \)
- For wave 2: \( \omega_2 = 404 \pi \)

### Step 2: Calculate Frequencies
Using the relationship \( \omega = 2 \pi f \), we can find the frequencies (\( f \)):
- For wave 1:
\[ f_1 = \frac{\omega_1}{2\pi} = \frac{400\pi}{2\pi} = 200 \text{ Hz} \]
- For wave 2:
\[ f_2 = \frac{\omega_2}{2\pi} = \frac{404\pi}{2\pi} = 202 \text{ Hz} \]

### Step 3: Calculate Beats Frequency
The beat frequency (\( n \)) is given by the difference in frequencies:
\[ n = f_2 - f_1 = 202 \text{ Hz} - 200 \text{ Hz} = 2 \text{ Hz} \]

### Step 4: Calculate Intensities
Intensity (\( I \)) is proportional to the square of the amplitude (\( A \)). The ratio of intensities can be calculated as follows:
\[ \frac{I_1}{I_2} = \frac{A_1^2}{A_2^2} \]
Where:
- \( A_1 = 4 \)
- \( A_2 = 3 \)

### Step 5: Calculate Maximum and Minimum Intensities
The maximum intensity occurs during constructive interference:
\[ I_{max} \propto (A_1 + A_2)^2 = (4 + 3)^2 = 7^2 = 49 \]
The minimum intensity occurs during destructive interference:
\[ I_{min} \propto (A_1 - A_2)^2 = (4 - 3)^2 = 1^2 = 1 \]

### Step 6: Calculate the Ratio of Intensities
The ratio of maximum to minimum intensity is:
\[ \frac{I_{max}}{I_{min}} = \frac{49}{1} = 49 \]

### Final Answers
- The beat frequency is \( 2 \text{ Hz} \).
- The ratio of maximum to minimum intensity is \( 49:1 \).