Two sounding bolies are producing progressive waves given by `y_(1) = 2 sin (400 pi t)` and `y_(2) = sin (404 pi t)` where `t` is in second, which superpose near the ears of a persion. The person will hear

Options

A. `2 beats//s` with intensity ratio `9//4` between maximum and minima
B. `2 beats//s` with intensity ratio `9` between maximum and minima
C. `4 beats//s` with intensity ratio `16` between maximum and minima
D. `4 beats//s` with intensity ratio `16//9` between maximum and minima

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the equations of the waves
The two progressive waves are given by:
- \( y_1 = 2 \sin(400 \pi t) \)
- \( y_2 = \sin(404 \pi t) \)

### Step 2: Determine the angular frequencies
The angular frequency (\( \omega \)) for each wave can be identified from the equations:
- For \( y_1 \): \( \omega_1 = 400 \pi \)
- For \( y_2 \): \( \omega_2 = 404 \pi \)

### Step 3: Calculate the frequencies
The frequency (\( f \)) can be calculated using the formula:
\[ f = \frac{\omega}{2\pi} \]
- For \( y_1 \):
\[ f_1 = \frac{400 \pi}{2 \pi} = 200 \, \text{Hz} \]
- For \( y_2 \):
\[ f_2 = \frac{404 \pi}{2 \pi} = 202 \, \text{Hz} \]

### Step 4: Calculate the beat frequency
The beat frequency (\( f_b \)) is given by the difference in frequencies:
\[ f_b = |f_2 - f_1| = |202 - 200| = 2 \, \text{Hz} \]

### Step 5: Determine the amplitudes
From the wave equations, we can identify the amplitudes:
- Amplitude of \( y_1 \) is \( A_1 = 2 \)
- Amplitude of \( y_2 \) is \( A_2 = 1 \)

### Step 6: Calculate the intensity ratio
The intensity ratio is calculated using the formula:
\[ \text{Intensity Ratio} = \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} \]
Substituting the values:
\[ \text{Intensity Ratio} = \frac{(2 + 1)^2}{(2 - 1)^2} = \frac{3^2}{1^2} = \frac{9}{1} = 9 \]

### Conclusion
The person will hear:
- **Beat frequency**: 2 beats per second
- **Intensity ratio**: 9 between maximum and minimum intensities.

### Final Answer
The person will hear 2 beats per second with an intensity ratio of 9 between maximum and minimum.

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