Principle of superposition of waves Questions in English

(B) The resultant amplitude $R$ of two interfering waves with individual amplitudes $A_1$ and $A_2$ and phase difference $\phi$ is given by $R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos \phi}$.
Given $A_1 = A_2 = A$.
The path difference $\Delta x = \frac{\lambda}{4}$.
The phase difference $\phi$ is related to path difference by $\phi = \frac{2\pi}{\lambda} \Delta x$.
Substituting the values: $\phi = \frac{2\pi}{\lambda} \times \frac{\lambda}{4} = \frac{\pi}{2}$.
Now, calculate the resultant amplitude $R$:
$R = \sqrt{A^2 + A^2 + 2A^2 \cos(\frac{\pi}{2})}$
Since $\cos(\frac{\pi}{2}) = 0$, we get:
$R = \sqrt{A^2 + A^2 + 0} = \sqrt{2A^2} = \sqrt{2}A$.