Two waves `y_1 = 3 sin omegat cm and Y = 4 cos (omegat+pi/2)` cm product the interference pattern. Then the amplitude of the resultant wave will be

Options

A. 1 cm
B. 3 cm
C. 4 cm
D. 5 cm

Answer

To find the amplitude of the resultant wave produced by the interference of the two given waves, we can follow these steps:

### Step 1: Write down the equations of the waves
The two waves are given as:
- Wave 1: \( y_1 = 3 \sin(\omega t) \) cm
- Wave 2: \( y_2 = 4 \cos(\omega t + \frac{\pi}{2}) \) cm

### Step 2: Convert the second wave into a sine form
Using the trigonometric identity \( \cos(\theta + \frac{\pi}{2}) = -\sin(\theta) \), we can rewrite the second wave:
\[ y_2 = 4 \cos(\omega t + \frac{\pi}{2}) = 4 \cdot (-\sin(\omega t)) = -4 \sin(\omega t) \]

### Step 3: Combine the two waves
Now, we can express the two waves as:
- \( y_1 = 3 \sin(\omega t) \)
- \( y_2 = -4 \sin(\omega t) \)

The resultant wave \( y \) can be expressed as:
\[ y = y_1 + y_2 = 3 \sin(\omega t) - 4 \sin(\omega t) = -1 \sin(\omega t) \]

### Step 4: Determine the amplitude of the resultant wave
The amplitude of the resultant wave is the coefficient of \( \sin(\omega t) \):
\[ \text{Amplitude} = |-1| = 1 \text{ cm} \]

### Conclusion
Thus, the amplitude of the resultant wave is **1 cm**.

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