    Question

# Three travelling waves in same direction are superimposed. The equation of wave are y1=A0sin(kx−ωt),y2=3√2A0sin(kx−ωt+ϕ) and y3=4A0cos(kx−ωt). If 0≤ϕ≤π/2 and the phase difference between resultant wave and first wave is π/4, then ϕ is

A
π6
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B
π3
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C
π12
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D
none of these
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Solution

## The correct option is C π12Given, y1=A0sin(kx−ωt)y2=3√2A0cos(kx−ωt+ϕ)and y3=4A0cos(kx−ωt),These waves can be represented by phase diagram as shown.From phase diagram,tan (π4)=BCAC=A0(4+3√2sinϕ)A0(1+3√2cosϕ)⟹4+3√2sinϕ=1+3√2cosϕ⟹cosϕ−sinϕ=dfrac1√2squaring both sides, we getcos2ϕ+sin2ϕ−2sinϕcosϕ=121−2sinϕcosϕ=12 or 2sinϕcosϕ=12or 2 sin2ϕ=12=sinπ6 or 2ϕ=π6⟹ϕ=π12   Suggest Corrections  0      Similar questions  Related Videos   Area under the Curve
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