If two waves represented by `y_(1)=4sinomegat` and `y_(2)=3sin(omegat+(pi)/(3))` interfere at a point find out the amplitude of the resulting wave

Options

A. 7
B. 6
C. 5
D. `3.5`

Explanation

To find the amplitude of the resulting wave from the interference of two waves given by \( y_1 = 4 \sin(\omega t) \) and \( y_2 = 3 \sin\left(\omega t + \frac{\pi}{3}\right) \), we can follow these steps:

### Step 1: Identify the Amplitudes and Phase Difference
The amplitudes of the two waves are:
- \( A_1 = 4 \)
- \( A_2 = 3 \)

The phase difference \( \phi \) between the two waves is:
- \( \phi = \frac{\pi}{3} \)

### Step 2: Use the Resultant Amplitude Formula
The formula for the resultant amplitude \( A \) when two waves interfere is given by:
\[ A = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\phi)} \]

### Step 3: Substitute the Values into the Formula
Substituting the values we have:
\[ A = \sqrt{4^2 + 3^2 + 2 \cdot 4 \cdot 3 \cdot \cos\left(\frac{\pi}{3}\right)} \]

### Step 4: Calculate Each Term
Calculate \( A_1^2 \) and \( A_2^2 \):
- \( A_1^2 = 4^2 = 16 \)
- \( A_2^2 = 3^2 = 9 \)

Now, calculate \( 2 A_1 A_2 \):
- \( 2 \cdot 4 \cdot 3 = 24 \)

Next, calculate \( \cos\left(\frac{\pi}{3}\right) \):
- \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \)

Now substitute these values back into the equation:
\[ A = \sqrt{16 + 9 + 24 \cdot \frac{1}{2}} \]
\[ = \sqrt{16 + 9 + 12} \]
\[ = \sqrt{37} \]

### Step 5: Calculate the Result
The value of \( \sqrt{37} \) is approximately \( 6.08 \). Therefore, the amplitude of the resulting wave is:
\[ A \approx 6.08 \]

### Final Answer
Thus, the amplitude of the resulting wave is approximately \( 6 \).

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