The displacement at a point due to two waves are `y_1=4 sin (500pit)` and `y_2=2sin(506 pi t)`. The result due to their superposition will be

Options

A. 3 beats per second with intensity relation between maxima and minima equal to 2
B. 3 beats per second with intensity relation between maxima and minima equal to 9
C. 6 beats per second with intensity relation between maxima and minima equal to 2
D. 6 beats per second with intensity relation between maxima and minima equal to 9

Answer

To find the resultant displacement due to the superposition of the two waves given by \( y_1 = 4 \sin(500 \pi t) \) and \( y_2 = 2 \sin(506 \pi t) \), we can follow these steps:

### Step 1: Identify the Frequencies
The angular frequencies (\( \omega \)) for the two waves are given as:
- For \( y_1 \): \( \omega_1 = 500 \pi \)
- For \( y_2 \): \( \omega_2 = 506 \pi \)

From the angular frequency, we can find the frequencies (\( f \)) using the formula:
\[ f = \frac{\omega}{2\pi} \]
Thus, we calculate:
- \( f_1 = \frac{500 \pi}{2\pi} = 250 \, \text{Hz} \)
- \( f_2 = \frac{506 \pi}{2\pi} = 253 \, \text{Hz} \)

### Step 2: Calculate the Beat Frequency
The beat frequency (\( f_b \)) is the difference between the two frequencies:
\[ f_b = |f_2 - f_1| = |253 - 250| = 3 \, \text{Hz} \]

### Step 3: Determine the Amplitudes
From the equations of the waves, we can identify the amplitudes:
- Amplitude of \( y_1 \) (\( A_1 \)) = 4
- Amplitude of \( y_2 \) (\( A_2 \)) = 2

### Step 4: Calculate Maximum and Minimum Amplitudes
The maximum amplitude (\( A_{\text{max}} \)) and minimum amplitude (\( A_{\text{min}} \)) due to superposition can be calculated as:
\[ A_{\text{max}} = A_1 + A_2 = 4 + 2 = 6 \]
\[ A_{\text{min}} = |A_1 - A_2| = |4 - 2| = 2 \]

### Step 5: Calculate Intensities
Intensity (\( I \)) is proportional to the square of the amplitude. Thus:
\[ I_{\text{max}} \propto A_{\text{max}}^2 = 6^2 = 36 \]
\[ I_{\text{min}} \propto A_{\text{min}}^2 = 2^2 = 4 \]

### Step 6: Find the Ratio of Intensities
The ratio of maximum intensity to minimum intensity is:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{36}{4} = 9 \]

### Final Result
The resultant displacement due to the superposition of the two waves is characterized by:
- Beat frequency: 3 Hz
- Ratio of maximum to minimum intensity: 9