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

Question #195544

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


1
Expert's answer
2021-05-31T15:37:53-0400

y1=0.2sin(3πx20t+ϕ)y_1=0.2\sin(3\pi x-20t +\phi)

y2=0.2sin(3πx20t)y_2=0.2\sin(3\pi x-20t)


y=y1+y2y=y_1+y_2

y=0.04sin(3πx20t+ϕ2)cosϕ2y=0.04\sin\bigg(3\pi x-20t+\dfrac{\phi}{2}\bigg)\cos\dfrac{\phi}{2}

Now,

k=2πλ=3πk=\dfrac{2\pi}{\lambda}=3\pi

λ=2π3π=23\lambda=\dfrac{2\pi}{3\pi}=\dfrac{2}{3}


Path difference for destructive interference,

Δx=(n+12)λ               n=0,1,2,3\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

Δx=λ2=13\Delta x=\dfrac{\lambda}{2}=\dfrac{1}{3}

Therefore path difference, Δx=13 m\Delta x=\dfrac{1}{3}\space m


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