Question

Two waves given as \( y_1 = 10 \sin t \, \text{cm} \) and \( y_2 = 10 \sin \left( \omega t + \frac{\pi}{3} \right) \, \text{cm} \) are superimposed. What is the amplitude of the resultant wave?

• A. \( 10\sqrt{2} \, \text{cm} \)
• B. \( 5\sqrt{3} \, \text{cm} \)
• C. \( 10\sqrt{3} \, \text{cm} \)
• D. \( 10 \, \text{cm} \)

Hint:

When two waves of the same frequency superimpose, their resultant amplitude depends on their phase difference. Use the formula \( A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos \phi} \) to calculate the resultant amplitude.

Answer 1

The Correct Option is C

Step 1: Understanding superposition of waves.
When two sinusoidal waves of the same frequency and different phases superimpose, the resultant amplitude \( A \) is given by: \[ A = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos \phi} \] where \( A_1 \) and \( A_2 \) are the amplitudes of the two waves and \( \phi \) is the phase difference. In this case, \( A_1 = A_2 = 10 \) cm and the phase difference \( \phi = \frac{\pi}{3} \).
Step 2: Calculating the amplitude.
Substitute the values into the equation: \[ A = \sqrt{10^2 + 10^2 + 2 \cdot 10 \cdot 10 \cdot \cos \frac{\pi}{3}} = \sqrt{100 + 100 + 100} = \sqrt{300} = 10\sqrt{3} \, \text{cm} \] Step 3: Conclusion.
Thus, the amplitude of the resultant wave is \( 10\sqrt{3} \, \text{cm} \), corresponding to option (C).