Answer to Question #195544 in Mechanics | Relativity for Mohammad Balhas

Question

Two waves travelling in the same direction are given by: y1 = 0.2 sin⁡(3πx-20t+φ) and y2 = 0.2 sin⁡(3πx-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:

1/2 m

1/3 m

3/4 m

2/3 m

4/3 m

Answer 1

$y_1=0.2\sin(3\pi x-20t +\phi)$

$y_2=0.2\sin(3\pi x-20t)$

$y=y_1+y_2$

$y=0.04\sin\bigg(3\pi x-20t+\dfrac{\phi}{2}\bigg)\cos\dfrac{\phi}{2}$

Now,

$k=\dfrac{2\pi}{\lambda}=3\pi$

$\lambda=\dfrac{2\pi}{3\pi}=\dfrac{2}{3}$

Path difference for destructive interference,

$\Delta x=\bigg(n+\dfrac{1}{2}\bigg)\lambda\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space n=0,1,2,3$

for n = 0

$\Delta x=\dfrac{\lambda}{2}=\dfrac{1}{3}$

Therefore path difference, $\Delta x=\dfrac{1}{3}\space m$