Stationary Waves & Vibrating Strings question from MHT CET 2024 (15 May, Shift 1)

Answer: (c)

1. $\sin(A - B) + \sin(A + B) = 2\sin A\cos B$, so $Y = 2\cos\left(\dfrac{2\pi x}{4}\right)\sin\left(\dfrac{2\pi t}{0.4}\right)$.
2. Amplitude at $x$: $A(x) = 2\cos\left(\dfrac{\pi x}{2}\right)$.
3. At $x = 0.5\text{ m}$: $A = 2\cos 45^\circ = 2 \times \dfrac{1}{\sqrt{2}} = \sqrt{2}\text{ m}$.
4. So the answer is (c) $\sqrt{2}\text{ m}$.