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Question

A harmonic wave propagates in a medium. Find the average energy density of the wave, if at any point, the energy density becomes equal to W0 at an instant t=t0+T/6, where t0 is the instant when amplitude is maximum at this location and T is the time period of oscillation.

A
32W0
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B
W02
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C
2W0
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D
2W03
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Solution

The correct option is D 2W03
Let us consider the wave y=Acos(ωtkx)
Then, the energy density (U) of the wave is given by
U=ρA2ω2sin2(ωtkx) [ρ=density of medium]
Let us consider x=0, where amplitude is maximum at t0=0.
Then, at t=t0+T6,
y=Acos[ω(t0+T6)kx]
y=Acos(ωT6)[t0=0,x=0]
and the energy density is
U=ρA2ω2sin2(ωT6)
=ρA2ω2sin2(2πT×T6)[ω=2πT]
=ρA2ω2sin2(π/3)
=ρA2ω2×34
From the question,
U=W0
ρA2ω2=43W0
Hence, average energy density of given wave is
U=12ρA2ω2=12×43W0=2W03

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