The equation of a stationary and a travelling waves are y1=asinkxcosωt and y2=asin(ωt−kx) respectively. The phase difference between two points, x1=π3k and x2=3π2k is ϕ1 in the standing wave y1 and is ϕ2 in travelling wave y2 then the ratio ϕ1ϕ2 is
A
13
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B
56
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C
34
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D
67
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Solution
The correct option is D67 For the stationary wave, y1=asinkxcosωt
Wave number, k=2πλ
Hence, two points on standing wave are, x1=π3k=λ6, and x2=3π2k=3λ4
It is clear from the above figure that points x1 and x2 lie in the adjacent loops of standing wave.
So, the phase difference between them will be π. ⇒ϕ1=π ..............(1)
For the travelling wave, y2=asin(ωt−kx)
Wave number, k=2πλ
Hence, two points on the travelling wave are, x1=π3k=λ6, and x2=3π2k=3λ4 Δx=x2−x1=3λ4−λ6=7λ12
Phase difference, ϕ2=2πλ×Δx=2πλ×7λ12 =7π6 .............(2)
Now, ϕ1ϕ2=π7π6 [from (1) and (2)] ⇒ϕ1ϕ2=67