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Question

The equation of a stationary and a travelling waves are y1=asinkxcosωt and y2=asin(ωtkx) 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 D 67
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(ωtkx)
Wave number,
k=2πλ
Hence, two points on the travelling wave are,
x1=π3k=λ6, and
x2=3π2k=3λ4
Δx=x2x1=3λ4λ6=7λ12
Phase difference,
ϕ2=2πλ×Δx=2πλ×7λ12
=7π6 .............(2)
Now,
ϕ1ϕ2=π7π6 [from (1) and (2)]
ϕ1ϕ2=67

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