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Question

Two waves have equations x1=asin(kxωt+ϕ1) and x2=asin(kxωt+ϕ2). If in the resultant wave the frequency and amplitude remain equal to amplitude and frequency of each superimposing waves, the phase difference between them is

A
π6
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B
2π3
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C
π4
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D
π3
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Solution

The correct option is B 2π3
Given,
x1=asin(kxωt+ϕ1)
x2=asin(kxωt+ϕ2)

Resultant,
x=x1+x2

x=a[sin(kxωt+ϕ1)+sin(kxωt+ϕ2)]

Using , sinC+sinD=2sin(C+D2)cos(CD2)

x=2 a sin(kxωt+ϕ1+ϕ22)cos(ϕ1ϕ22)

From the data given in the question, we can say that,

2acos(ϕ1ϕ22)=a
cos(ϕ1ϕ22)=12ϕ1ϕ22=π3
ϕ1ϕ2=2π3

Thus, option (b) is the correct answer.

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