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Two waves passing through a region are r...

Two waves passing through a region are represented by
`y = ( 1.0 m) sin [( pi cm^(-1)) x - ( 50 pi s^(-1))t]`
and `y = ( 1.5 cm) sin [( pi //2 cm^(-1)) x - ( 100 pi s^(-1)) t]`.
Find the displacement of the particle at ` x = 4.5 cm` at time ` t = 5.0 ms`.

Text Solution

Verified by Experts

According to the principle of superposition , each wave produces its disturbance independent of the other and the resultant disturbnce is equal to the vector sum of the individual time ` t= 5.0 ms` due to the waves are,
` y_(1) = ( 1.0 cm) sin [( pi cm^(-1)) ( 4.5 cm) - ( 50 pi s^(-1)) ( 5.0 xx 10^(-3) s)]`
` = ( 1.0 cm) sin [ 4.5 pi - (pi)/( 4)]`
`= ( 1.0 cm) sin [ 4 pi + (pi)/( 4)] = ( 1.0 cm)/( sqrt( 2))`
and
`y_(2) = ( 1.5 cm) sin[( pi//2 cm^(-1)) ( 4.5 cm) - ( 100 pi s^(-1) ( 5.0 xx 10^(-3) s)]`
` = ( 1.5 cm) sin [ 2.25 pi - (pi)/(2))]`
`= ( 1.5 cm) sin [ 2 pi - (pi)/(4)] = - ( 1.5 cm) sin ( pi)/(4) = - ( 1.5 cm)/( sqrt (2))`
The net displacement is `y = y_(1) + y_(2) = - 0.5 // sqrt(2) cm`
` = - 0.35 cm`
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