Two waves are represented by: `y_(1)=4sin404 pit` and `y_(2)=3sin400 pit`. Then :

Options

A. beat frequency is 4 Hz and the ratio of maximum to minimum intensity is 49 : 1
B. beat frequency is 2 Hz and the ratio of maximum to minimum intensity is 49 : 1
C. beat frequency is 2 Hz and the ratio of maximum to minimum intensity is 1 : 49
D. beat frequency is 4 Hz and the ratio of maximum to minimum intensity is 1 : 49

Answer

To solve the problem, we need to find the beat frequency and the ratio of maximum to minimum intensity for the two waves represented by:

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

### Step 1: Determine the Frequencies of the Waves

The general form of a wave is given by \( y = A \sin(\omega t) \), where \( \omega \) is the angular frequency. The relationship between angular frequency \( \omega \) and frequency \( f \) is given by:

\[ \omega = 2 \pi f \]

For wave \( y_1 \):
- \( \omega_1 = 404 \pi \)
- Therefore, the frequency \( f_1 \) is:

\[ f_1 = \frac{\omega_1}{2 \pi} = \frac{404 \pi}{2 \pi} = 202 \text{ Hz} \]

For wave \( y_2 \):
- \( \omega_2 = 400 \pi \)
- Therefore, the frequency \( f_2 \) is:

\[ f_2 = \frac{\omega_2}{2 \pi} = \frac{400 \pi}{2 \pi} = 200 \text{ Hz} \]

### Step 2: Calculate the Beat Frequency

The beat frequency \( f_b \) is given by the absolute difference of the two frequencies:

\[ f_b = |f_1 - f_2| = |202 - 200| = 2 \text{ Hz} \]

### Step 3: Calculate the Maximum and Minimum Intensities

Intensity \( I \) is proportional to the square of the amplitude \( A \). For two waves, the maximum and minimum intensities can be calculated using the principle of superposition.

- The maximum intensity \( I_{\text{max}} \) is given by:

\[ I_{\text{max}} = (A_1 + A_2)^2 = (4 + 3)^2 = 7^2 = 49 \]

- The minimum intensity \( I_{\text{min}} \) is given by:

\[ I_{\text{min}} = (A_1 - A_2)^2 = (4 - 3)^2 = 1^2 = 1 \]

### Step 4: Calculate the Ratio of Maximum to Minimum Intensity

The ratio of maximum to minimum intensity is:

\[ \text{Ratio} = \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{49}{1} = 49 \]

### Final Answers

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