In case of superposition of waves (at x = 0). `y_1 = 4 sin(1026 pi t)` and `y_2 = 2 sin(1014 pi t)`

Options

A. the frequency of resulting wave is 510 Hz
B. the amplitude of resulting wave varies of frequency 3 Hz
C. the frequency of beats is 6 Hz
D. the ratio of maximum to minimum intensity is 9

Answer

To solve the problem of superposition of waves given by the equations \( y_1 = 4 \sin(1026 \pi t) \) and \( y_2 = 2 \sin(1014 \pi t) \), we will follow these steps:

### Step 1: Identify the wave parameters
We have two waves:
- Wave 1: \( y_1 = 4 \sin(1026 \pi t) \)
- Wave 2: \( y_2 = 2 \sin(1014 \pi t) \)

From these equations, we can identify:
- Amplitude of wave 1, \( A_1 = 4 \)
- Amplitude of wave 2, \( A_2 = 2 \)
- Angular frequency of wave 1, \( \omega_1 = 1026 \pi \)
- Angular frequency of wave 2, \( \omega_2 = 1014 \pi \)

### Step 2: Determine the resultant wave
The resultant wave \( y \) due to the superposition of \( y_1 \) and \( y_2 \) can be expressed as:
\[ y = y_1 + y_2 = 4 \sin(1026 \pi t) + 2 \sin(1014 \pi t) \]

### Step 3: Use the principle of superposition
The principle of superposition states that when two or more waves overlap, the resultant displacement at any point is the algebraic sum of the displacements due to the individual waves. Therefore, we can write:
\[ y = 4 \sin(1026 \pi t) + 2 \sin(1014 \pi t) \]

### Step 4: Analyze the resultant wave
The resultant wave is a combination of two sine waves with different frequencies. Since the frequencies are different, the resultant wave will exhibit a phenomenon known as "beats," where the amplitude of the resultant wave will vary over time.

### Step 5: Calculate the beat frequency
The beat frequency \( f_b \) can be calculated using the formula:
\[ f_b = |f_1 - f_2| \]
where \( f_1 \) and \( f_2 \) are the frequencies of the two waves. The frequencies can be derived from the angular frequencies:
\[ f_1 = \frac{\omega_1}{2\pi} = \frac{1026 \pi}{2\pi} = 513 \text{ Hz} \]
\[ f_2 = \frac{\omega_2}{2\pi} = \frac{1014 \pi}{2\pi} = 507 \text{ Hz} \]
Now, calculating the beat frequency:
\[ f_b = |513 - 507| = 6 \text{ Hz} \]

### Final Result
The resultant wave can be expressed as:
\[ y = 4 \sin(1026 \pi t) + 2 \sin(1014 \pi t) \]
with a beat frequency of \( 6 \text{ Hz} \).

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