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Question

A standing wave is maintained in a homogenous string of cross-sectional area s and density ρ. It is formed by the superposition of two waves travelling in opposite directions given by the equations y1=asin(ωtkx) and y2=asin(ωt+kx). The total mechanical energy confined between the sections corresponding to adjacent nodes is

A
92ρω2a2πsk
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B
72ρω2a2πsk
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C
ρω2a2πsk
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D
32ρω2a2πsk
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Solution

The correct option is C ρω2a2πsk
Let u1 and u2 be the energy densities due to individual waves.
So,
u1=12ρω2a2, and
u2=12ρω2(a)2
Now, volume of string in a loop of standing wave,
V=A×l=s×λ2=πsk


Total mechanical energy confined in a loop,
E=V(u1+u2)
E=πsk(ρω2a2)=ρω2a2πsk

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