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

The displacement of a periodically vibrating particle isy = 4 cos ^(2)

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