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Two waves passing through a region are r...

Two waves passing through a region are respresented by
`y_(1) = 5 mm sin [(2pi cm^(-1))x - (50 pis^(-1))t]`
and `y_(2) = 10 mm sin [(pi cm^(-1))x - (100 pis^(-1))t]`
Find the displacement of the particle at `x = 1 cm` at time `t = 5.0 ms`.

Text Solution

Verified by Experts

Accoding to the principle of superposition, each wave produce its disturbance independent of the other and the resultant disturbance is equal to the vector sum of the individual disturbance. The displacements of the particle at `x = 1cm` at time `t = 5.0 ms` due to the two waves are.
`y_(1) = 2 mm [(2pi cm^(-1)) xx - (50 pi s^(-1))t]`
`y_(1) = 5 mm sin[(2pi cm^(-1)) xx 1 cm - (50 pi s^(-1))5 xx 10^(-3) sec]`
`= 5 mm sin [2pi - (p)/(4)] = -5 mm`
and `y_(2) = 10 mm sin [(pi cm^(-1)) xx - (100 pi s^(-1))t]`,
`y_(2) = 10 mm sin [(pi cm^(-1)) xx 1 cm - (100 pi s^(-1))5 xx 10^(-3) sec]`
`= 10 mm sin [pi - (pi)/(2)] = 10 mm`
The net displacement is : `y = y_(1) + y_(2) = 10 mm - 5 mm = 5 mm`
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