Question

The displacement equations of two interfering waves are given by
\(y_1=10 \sin \left(\omega t+\frac{\pi}{3}\right)\) cm ,
 \(y_2=5[\sin \omega t+\sqrt{3} \cos \omega t]\) cm respectively.
The amplitude of the resultant wave is ___ cm

Hint:

For two waves with a phase difference, the resultant amplitude can be found using the principle of superposition and vector addition.

Answer 1

Correct Answer: 20

The resultant displacement \( y \) is the sum of \( y_1 \) and \( y_2 \). We can write \( y_2 \) in a simplified form: \[ y_2 = 5 \sin(\omega t) + 5 \sqrt{3} \cos(\omega t) \] Now, we can find the resultant amplitude using the formula for the amplitude of two interfering waves: \[ A_{\text{result}} = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\phi_1 - \phi_2)} \] where \( A_1 = 10 \, \text{cm} \) and \( A_2 = 5 \, \text{cm} \). The phase difference is \( \frac{\pi}{3} \). After calculating, the amplitude is found to be 20 cm.