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The displacement of a periodically vibra...

The displacement of a periodically vibrating particle is ` y = 4 cos ^(2)((1)/(2)t)` sin (1000t) . Calculate the number of harmonic waves that are superposed .

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`y = 4 cos^(2) ((1)/(2)t) ` sin (1000t)
`2 * 2 cos^(2) ((1)/(2)t)* ` sin (1000 t)
= 2 (1 + cos t ) sin (1000 t )
= 2 sin (1000 t) + 2 sin (1000 t ) cos t
= 2 sin (1000 t ) + sin (1000 t + t ) + sin ( 1000 t - t)
= 2 sin (1000 t) + 1 sin (101 t ) + 1 sin (999 t )
` = y_(1) + y_(2) + y_(3)`
Here each of `y_(1),y_(2) and y_(3)` in the form of `a sin omegat` . Thus,each of them represents a harmonic wave .
Hence, the number of superposed harmonic waves = 3 .
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