Two sounding bodies are producing progressive waves given by `y_1 = 4 sin (400 pi t) `and `y_2 = 3 sin (404 pi t)` , where t is in second which superpose near the ears of a person. The person will hear

Options

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

Answer

To solve the problem, we will analyze the two progressive waves given by their equations and determine what the person will hear when these waves superpose.

### Step-by-Step Solution:

1. **Identify the wave equations**:
The two waves are given as:
\[ y_1 = 4 \sin(400 \pi t) \]
\[ y_2 = 3 \sin(404 \pi t) \]

2. **Determine the angular frequencies**:
From the wave equations, we can identify the angular frequencies (\(\omega\)):
- For \(y_1\): \(\omega_1 = 400 \pi\)
- For \(y_2\): \(\omega_2 = 404 \pi\)

3. **Calculate the frequencies**:
The relationship between angular frequency and frequency is given by:
\[ \omega = 2 \pi f \]
Thus, we can find the frequencies:
- For \(y_1\):
\[ f_1 = \frac{\omega_1}{2\pi} = \frac{400 \pi}{2\pi} = 200 \, \text{Hz} \]
- For \(y_2\):
\[ f_2 = \frac{\omega_2}{2\pi} = \frac{404 \pi}{2\pi} = 202 \, \text{Hz} \]

4. **Determine the beat frequency**:
The beat frequency (\(f_b\)) is given by the difference in frequencies:
\[ f_b = |f_2 - f_1| = |202 - 200| = 2 \, \text{Hz} \]
This means the person will hear 2 beats per second.

5. **Calculate the intensity ratio**:
The intensity of a wave is proportional to the square of its amplitude. The amplitudes are:
- \(A_1 = 4\)
- \(A_2 = 3\)

The maximum intensity (\(I_{max}\)) and minimum intensity (\(I_{min}\)) can be calculated as follows:
\[ I_{max} \propto (A_1 + A_2)^2 = (4 + 3)^2 = 7^2 = 49 \]
\[ I_{min} \propto (A_1 - A_2)^2 = (4 - 3)^2 = 1^2 = 1 \]

Therefore, the intensity ratio is:
\[ \frac{I_{max}}{I_{min}} = \frac{49}{1} = 49 \]

### Conclusion:
The person will hear 2 beats per second, and the intensity ratio of the maximum to minimum intensity is 49:1.