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Updated on : 2022-09-05

Solution

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$y_{1}=asinωt$

$y_{2}=asin(ωt+ϕ)$

Resultant displacement is: $y=y_{1}+y_{2}$

$y=asinωt+asin(ωt+ϕ)=asinωt(1+cosϕ)+cosωt(asinϕ)$

$Rcosθ=a(1+cosϕ)$

$Rsinθ=asinϕ$

$y=Rsin(ωt+θ)$

Where, $R$ is resultant amplitude at $P$, $I$ is intensity, squaring the equations we get,

$I=R_{2}=a_{2}(1+cosϕ)_{2}+a_{2}(sinϕ)_{2}=2a_{2}(1+cosϕ)=4a_{2}cos_{2}2ϕ $

Maximum intensity:

$cos_{2}2ϕ =1$

$ϕ=2nπ$ where $n=0,1,2,3,$....

Therefore, $I_{max}=4a_{2}$

Minmum intensity:

$cos_{2}2ϕ =0$

$ϕ=(2n+1)π$ where $n=0,1,2,3,$....

Minmum intensity:

$cos_{2}2ϕ =0$

$ϕ=(2n+1)π$ where $n=0,1,2,3,$....

Therefore, $I_{max}=0$

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