Answer to Question #195549 in Mechanics | Relativity for ghinwa

Question

Two waves travelling in the same direction are given by: y1 = 0.2 sin⁡(2πx-20t+φ) and y2 = 0.2 sin⁡(2πx-20t), where x and y are in meters and t is in seconds. If the two waves start at the same moment, then the path difference, ∆x, corresponding to a fully destructive interference is:

2/3 m

3/4 m

1/3 m

1/2 m

4/3 m

Answer 1

$y_1=0.2\sin(2\pi x-20t+\phi)$

$y_2=0.2\sin(2\pi x-20t)$

$y=y_1+y_2$

$y=\bigg(0.2\cos\dfrac{\phi}{2}\bigg)\sin\bigg(2\pi x-20t+\dfrac{\phi}{2}\bigg)$

Phase difference = $\phi$

For destructive interference,

Amplitude of resultant wave = 0

$\therefore\cos\dfrac{\phi}{2}=0$

$\phi=n\pi$

for Path difference, $\Delta=\dfrac{\lambda}{2\pi}\phi$

$\lambda=1$

$\Delta=\dfrac{1}{2\pi}n\pi$

for n=1

$\Delta=\dfrac{1}{2}\space m$